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आचार्य जी: मन की शांति
मन क्यों उदास हो जाता है, मन की इच्छाएं क्यों असीमित हो जाती हैं, मन में लालसाएं व जिज्ञासाएं हर पल क्यों जन्म लेती हैं, क्यों मन की शांति के लिये भटकना पडता है, मन की शांति का उपाय क्या है !!!मन चंचल है, भूखा-प्यासा है, लालची है, व्याकुल है, अस्थिर है!!!मन पर नियंत्रण बेहद सरल कार्य है, जैसे ही मन में कोई इच्छा जागृत हो उसे तत्काल शांत कर दीजिये, मन को संदेश दो अभी रुक जाओ एक घंटे के बाद प्रयास करते हैं, चाय पीने की इच्छा जागृत हुई चाय पिओ पर एक घंटे के बाद, धीरे धीरे इस अवधि को बढाते जाईये ... आपका मन स्वमेव शांत होने लगेगा!!!!!
इस अशांति के माहौल में आपने मन को शांति प्रदान की है........आचार्य वर !
वाह .. आपका पहला वचन तो प्रयोग करने लायक है ... कहीं ऐसा तो नही होगा .. इच्छा को टालते टालते वो फट पड़े और तीव्र उत्कंठा से जागृत हो जाए ....
@दिगम्बर नासवा वत्स इच्छा को शनै शनै नियंत्रित करना है, यह प्रयोग धीरे धीरे अमल में लाना है, समय अवधि शनै शनै बढते जायेगी और एक समय आयेगा मन को पूर्णरुपेण शांति प्राप्त होने लगेगी।आचार्य जी
जनाब हर रोज पाकसमर्थक मुसलिम आतंकवादी और चीन समर्थक माओवादी आतंकवादी कहर वर्पा रहें हैं ऐसे में हमारे जैसे लोगों का मन देश को इन गद्दारों से मुक्त किए विना शांत हो असम्वभव नहीं तो मुसकिल हैं।जिस दिन हमें चैन महसूस होता है उसी दिन ये लोग आम जनता का कत्ल कर हमें फिर असांत कर देते हैं और हम निकल पड़ते हैं देश को इन गद्दारों से मुक्त करने के उपाए तलासने।आपको भी इन सेकुलर गद्दारों का कोई समाधान दिखे तो जरूर बताना जी।
सही मार्ग बताया आपने आचार्य जी।
हम निकल पड़ते हैं देश को इन गद्दारों से मुक्त करने के उपाए तलासने।
आत्मा तृप्त भई! | hindi |
\begin{document}
\title{The edge spectrum of $K_4^-$-saturated graphs
\thanks{The work was supported by NNSF of China (No. 11671376) and NSF of Anhui Province (No. 1708085MA18).}
}
\author{Jun Gao$^a$, \quad Xinmin Hou$^b$,\quad Yue Ma$^c$\\
\small $^{a,b,c}$ Key Laboratory of Wu Wen-Tsun Mathematics\\
\small School of Mathematical Sciences\\
\small University of Science and Technology of China\\
\small Hefei, Anhui 230026, China.
}
\date{}
\maketitle
\begin{abstract}
Given graphs $G$ and $H$, $G$ is $H$-saturated if $G$ does not contain a copy of $H$ but the addition of any edge $e\notin E(G)$ creates at least one copy of $H$ within $G$. The edge spectrum of $H$ is the set of all possible sizes of an $H$-saturated graph on $n$ vertices. Let $K_4^-$ be a graph obtained from $K_4$ by deleting an edge. In this note, we show that (a) if $G$ is a $K_4^-$-saturated graph with $|V(G)|=n$ and $|E(G)|>\lfloor \frac{n-1}{2} \rfloor \lceil \frac{n-1}{2} \rceil +2$, then $G$ must be a bipartite graph;
(b) there exists a $K_4^-$-saturated non-bipartite graph on $n\ge 10$ vertices with size being in the interval $\left[3n-11, \left\lfloor \frac{n-1}{2} \right\rfloor \left\lceil \frac{n-1}{2} \right\rceil +2\right]$.
Together with a result of Fuller and Gould in [{\it On ($\hbox{K}_t-e$)-Saturated Graphs. Graphs Combin., 2018}], we determine the edge spectrum of $K_4^-$ completely, and a conjecture proposed by Fuller and Gould in the same paper also has been resolved.
\end{abstract}
\section{Introduction}
Given a graph $H$, we say a graph $G$ is {\it $H$-saturated} if $G$ does not contain a copy of $H$ but the addition of any edge $e\notin E(G)$ creates at least one copy of $H$ within $G$. The minimum (resp. maximum) number of edges of an $H$-saturated graph on $n$ vertices is known as the {\it saturation }(resp. {\it Tur\'an) number}, and denoted by $sat(n,H)$ (resp. $ex(n, H)$). A natural question is to determine all possible values $m$ between $sat(n, H)$ and $ex(n,H)$ such that there is an $H$-saturated graph whose size equals $m$. We call the set of all possible sizes of an $H$-saturated graph on $n$ vertices the {\it edge spectrum} of $H$ and denoted by $ES(n, H)$.
Clearly, for $m\in ES(n, H)$, we have $sat(n, H)\le m\le ex(n, H)$.
Write $K_n$ for a complete graph on $n$ vertices and $K_n^-$ for a graph obtained from $K_n$ by deleting one edge. Write $[m, n]$ for the set $\{m, m+1, \ldots, n\}$.
For complete graphs, Barefoot et al.~\cite{K_3} determined $ES(n, K_3)$;
Amin, Faudree, and Gould~\cite{K_4} evaluated $ES(n, K_4)$, and, more generally, $ES(n, K_p)$ for $p\ge 3$ was studied and given by Amin et al. in~\cite{K_p}. { Continuing the work, Gould et al. \cite{small-path} found the edge spectrum of small paths.} Recently, Faudree
et al. completely determined the edge spectrum of stars and partially gave the edge spectrum of paths in~\cite{Paths}.
For $K_t^-$, there is no $ES(n, K_t^-)$ for $t\ge 4$ has been completely determined so far.
It is well known that $ex(n, K_4^-)=\lceil\frac n2\rceil\lfloor\frac n2\rfloor$ and the upper bound can be realized by the complete bipartite graph $K_{\lceil\frac n2\rceil,\lfloor\frac n2\rfloor}$. Chen, Faudree, and Gould~\cite{Chen-Faudree-Gould-08} determined that $sat(n, K_4^-)=\left\lfloor\frac{3(n-1)}2\right\rfloor$ and the lower bound can be realized by the graph obtained from $K_{1,n-1}$ by adding $\lfloor\frac{n-1}2\rfloor$ independent edges.
In \cite{K_t-e}, Fuller and Gould proved that
\begin{equation}\label{EQN: e1}
\left\{\left\lfloor\frac{3(n-1)}2\right\rfloor\right\}\cup \left[2n-4, \left\lfloor\frac n2\right\rfloor\left\lceil\frac n2\right\rceil-n+6\right]\subseteq ES(n, K_4^-)
\end{equation}
and proposed the following conjecture.
\begin{conj}[Fuller, Gould \cite{K_t-e}]\label{CONJ: c1}
The $K_4^-$-saturated graphs with sizes in the interval $\left[ \lfloor \frac{n}{2} \rfloor \lceil \frac{n}{2} \rceil -n+7,\lfloor \frac{n}{2} \rfloor \lceil \frac{n}{2} \rceil\right] $ are of two types: complete bipartite graphs with partite sets of nearly equal size, and 3-partite graphs with two partite sets of nearly equal size and one partite set of order one.
\end{conj}
In this paper, we first show that Conjecture~\ref{CONJ: c1} is true when the $K_4^-$-saturated graphs with sizes in the interval $\left[ \lfloor \frac{n-1}{2} \rfloor \lceil \frac{n-1}{2} \rceil +3, \lfloor \frac{n}{2} \rfloor \lceil \frac{n}{2} \rceil\right]$ and we also give $K_4^-$-saturated non-bipartite graphs with sizes in the interval $\left[3n-11, \lfloor \frac{n-1}{2} \rfloor \lceil \frac{n-1}{2} \rceil +2\right]$. Combining with (\ref{EQN: e1}) we completely determine the edge spectrum of $K_4^-$.
Specifically, we show the following theorem.
\begin{thm}\label{THM: Main}
(a) If $G$ is a $K_4^-$-saturated graph with $|V(G)|=n$ and $|E(G)|>\lfloor \frac{n-1}{2} \rfloor \lceil \frac{n-1}{2} \rceil +2$, then $G$ must be a bipartite graph.
(b) There exists a $K_4^-$-saturated non-bipartite graph on $n\ge 10$ vertices and $m$ edges where $3n-11\le m\le \lfloor \frac{n-1}{2} \rfloor \lceil \frac{n-1}{2} \rceil +2$.
(c) For $n\ge 10$,
$$
ES(n, K_4^-)=\left\{\left\lfloor \frac{3(n-1)}{2} \right\rfloor \right\} \cup \left[2n-4 , \left\lfloor \frac{n-1}{2} \right\rfloor \left\lceil \frac{n-1}{2} \right\rceil +2\right] \cup \left\{ i(n-i) : i\in [1, n-1]\right\}.$$
\end{thm}
Clearly, (c) is a direct corollary from (\ref{EQN: e1}), (a) and (b). We give the proof of (a) and (b) in Section 2.
\section{Proof of Theorem~\ref{THM: Main}}
We first prove (a) of Theorem~\ref{THM: Main}.
\noindent{\bf Proof of Theorem~\ref{THM: Main} (a):}
Suppose to the contrary that $G$ is non-bipartite. Let $C$ be a shortest odd cycle in $G$ and $G'=G-V(C)$. Assume $|V(C)|=2t+1$ for some integer $t$.
Since $C$ is a shortest odd cycle and $G$ is $K_4^-$-free, we have $t\ge 1$, and for any $v \in V(G')$, $e(v, V(C))\le t$.
So
\begin{eqnarray*}e(G) &=&e(C)+e(G')+e(V(G'),V(C))\\
&\le& 2t+1+\left\lfloor \frac{n-2t-1}{2} \right\rfloor \left\lceil \frac{n-2t-1}{2} \right\rceil+(n-2t-1)t\\
&=&\left\lfloor \frac{n-1}{2} \right\rfloor \left\lceil \frac{n-1}{2} \right\rceil +2-(t-1)^2\\
&\le& \left\lfloor \frac{n-1}{2} \right\rfloor \left\lceil \frac{n-1}{2} \right\rceil +2,
\end{eqnarray*}
a contradiction.
\rule{1mm}{2mm}
To show (b) of Theorem~\ref{THM: Main}, we first construct a family of $K_4^-$-saturated non-bipartite graphs. We write $K(X, Y)$ for the complete bipartite graph with partite sets $X$ and $Y$.
\noindent{\bf Construction A:} Given nonnegative integers $n, a, b$ with $n\ge a+b+5$ and sets $I, A_1, A_2, B_1,$ $B_2, C$ with $|I|=1, |A_1|=|B_1|=2, |A_2|=a, |B_2|=b$ and $|C|=n-a-b-5$, let $M$ be two independent edges connecting $A_1$ and $B_1$. Define $F_n(a,b)$ be the graph with vertex set
$$V(F_n(a,b))=I\cup A_1 \cup A_2 \cup B_1 \cup B_2\cup C$$
and edge set
$$E(F_n(a,b))=E(K( A_1\cup A_2\cup C, B_2))\cup E(K(A_2, B_1))\cup E(K(I, A_1\cup B_1\cup C))\cup M.$$
We can count the number of edges of $F_n(a,b)$ directly from the construction.
\begin{lem}\label{LEM: l0}
$e(F_n(a, b))=b(n-b-3)+n+a-b+1$.
\end{lem}
\begin{lem}\label{LEM: l1}
If $b\ge 2$, then $F_n(a,b)$ is $K_4^-$-saturated for any $a\ge 0$.
\end{lem}
\begin{proof}
Denote $I=\{x\}$, $A_1=\{u_1, u_2\}$, and $B_1=\{v_1, v_2\}$. Assume $u_iv_i \in E(F_n(a,b))$ for $i=1,2$.
Clearly, $F_n(a,b)$ is $K_4^-$-free since $F_n(a,b)$ has only two triangles $xu_1v_1$ and $xu_2v_2$ sharing a common vertex $x$.
In the following, we show that $F_n(a,b)$ is $K_4^-$-saturated. Choose any edge $e=t_1t_2\notin E(F_n(a,b))$, we show that $F_n(a,b)+e$ contains a copy of $K_4^-$.
(i) $t_1, t_2\in A_1\cup A_2 \cup C$. Choose $b_1, b_2 \in B_2$ (this can be done since $|B_2|=b\ge 2$). Then $\{t_1,t_2,b_1,b_2\}$ induces a copy of $K_4^-$ in $F_n(a,b)+e$.
(ii) $t_1, t_2\in B_2$. Then $\{t_1,t_2, u_1, u_2\}$ induces a $K_4^-$ in $F_n(a,b)+e$.
(iii) $t_1, t_2\in B_1$. Then $\{t_1,t_2, u_1, x\}$ induces a $K_4^-$ in $F_n(a,b)+e$.
(iv) $t_1\in B_1$ and $t_2 \in B_2$. Assume $t_1=v_1$. Then $\{t_1,t_2,x,u_1\}$ induces a $K_4^-$ in $F_n(a,b)+e$.
(v) $t_1=x, t_2\in A_2$. Then $\{x,t_2, v_1, v_2\}$ induces a $K_4^-$ in $F_n(a,b)+e$. Similar argument for the case $t_1=x, t_2 \in B_2$.
(vi) $t_1=u_i, t_2=v_{3-i}$ for $i=1,2$. Then $\{x, u_i, v_1, v_2\}$ induces a $K_4^-$ in $F_n(a,b)+e$.
(vii) $t_1\in C, t_2\in B_1$. Assume $t_2=v_1$. Then $\{x, t_1, v_1, u_1\}$ induces a $K_4^-$ in $F_n(a,b)+e$.
This completes the proof.
\end{proof}
Now, define $f_n(a,b)=|E(F_n(a,b))|$ and $$\mathcal{F}_n(a,b)=\{f_n(a,b) : b\ge 2, a\ge 0\}.$$ By Lemma~\ref{LEM: l1}, $\mathcal{F}_n(a,b)\subseteq ES(n,K_4^-)$.
\begin{lem}\label{LEM: l2}
$$\left[3n-11, \left\lfloor \frac{n-1}{2} \right\rfloor \left\lceil \frac{n-1}{2} \right\rceil +2\right]\subseteq \mathcal{F}_n(a,b).$$
\end{lem}
\begin{proof}
By Lemma~\ref{LEM: l0}, $$f_n(a, b)=b(n-b-3)+n+a-b+1.$$
Clearly, for fixed $a$, $f_n(a, b)$ is a concave function of $b$ with maximum $f_n(a, \lfloor (n-5)/2 \rfloor)$, and for fixed $b$, $f_n(a,b)$ is an increasing function of $a$. Note that $n-5-b \ge a\ge 0$. We have
$$\mathcal{F}_n(a,b)=\bigcup_{b=2}^{\lfloor (n-5)/2 \rfloor}\left[f_n(0,b), f_n(n-b-5,b)\right].$$
We claim that $\mathcal{F}_n(a,b)$ is an interval. To show the claim, it is sufficient to check that two consecutive intervals are overlap.
In fact,
\begin{equation*}
\begin{split}
f_n(0,b+1)&=(b+1)(n-b-1-3)+n-(b+1)+1\\
&=b(n-3-b)+2n-3b-4\\
&\le f_n(n-b-5, b).
\end{split}
\end{equation*}
Note that $f_n(0,2)=3n-11$ and $f_n(n-\lfloor (n-5)/2 \rfloor-5, \lfloor (n-5)/2 \rfloor)=\lfloor \frac{n-1}{2} \rfloor \lceil \frac{n-1}{2} \rceil +2$.
We have
$$\left[3n-11, \left\lfloor \frac{n-1}{2} \right\rfloor \left\lceil \frac{n-1}{2} \right\rceil +2\right]\subseteq \mathcal{F}_n(a,b).$$
\end{proof}
(b) of Theorem~\ref{THM: Main} follows directly from Lemmas~\ref{LEM: l1} and~\ref{LEM: l2}.
\end{document} | math |
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package org.apache.isis.applib.util;
import com.google.common.base.Function;
import com.google.common.base.Joiner;
import com.google.common.base.Splitter;
import com.google.common.collect.Iterables;
public final class Enums {
private Enums(){}
public static String getFriendlyNameOf(Enum<?> anEnum) {
return getFriendlyNameOf(anEnum.name());
}
public static String getFriendlyNameOf(String anEnumName) {
return Joiner.on(" ").join(Iterables.transform(Splitter.on("_").split(anEnumName), LOWER_CASE_THEN_CAPITALIZE));
}
public static String getEnumNameFromFriendly(String anEnumFriendlyName) {
return Joiner.on("_").join(Iterables.transform(Splitter.on(" ").split(anEnumFriendlyName), UPPER_CASE));
}
public static String enumToHttpHeader(final Enum<?> anEnum) {
return enumNameToHttpHeader(anEnum.name());
}
public static String enumNameToHttpHeader(final String name) {
final StringBuilder builder = new StringBuilder();
boolean nextUpper = true;
for (final char c : name.toCharArray()) {
if (c == '_') {
nextUpper = true;
builder.append("-");
} else {
builder.append(nextUpper ? c : Character.toLowerCase(c));
nextUpper = false;
}
}
return builder.toString();
}
public static String enumToCamelCase(final Enum<?> anEnum) {
return enumNameToCamelCase(anEnum.name());
}
private static String enumNameToCamelCase(final String name) {
final StringBuilder builder = new StringBuilder();
boolean nextUpper = false;
for (final char c : name.toCharArray()) {
if (c == '_') {
nextUpper = true;
} else {
builder.append(nextUpper ? c : Character.toLowerCase(c));
nextUpper = false;
}
}
return builder.toString();
}
private static Function<String, String> LOWER_CASE_THEN_CAPITALIZE = new Function<String, String>() {
@Override
public String apply(String input) {
return capitalize(input.toLowerCase());
}
};
private static Function<String, String> UPPER_CASE = new Function<String, String>() {
@Override
public String apply(String input) {
return input.toUpperCase();
}
};
private static String capitalize(final String str) {
if (str == null || str.length() == 0) {
return str;
}
if (str.length() == 1) {
return str.toUpperCase();
}
return Character.toUpperCase(str.charAt(0)) + str.substring(1);
}
}
| code |
'गली बॉय' की प्रशंसा, पर कभी रणवीर को लगता था नहीं मिलेगा मुकाम - लाइफबेरीस.कॉम हिंदी
'गली बॉय' की प्रशंसा, पर कभी रणवीर को लगता था नहीं मिलेगा मुकाम
बर्लिन फिल्म समारोह (बर्लिन फिल्म फेस्टिवल) में रणवीर सिंह (रणवीर सिंह) आलिया भट्ट (एलिया भट्ट) की फिल्म गली बॉय (गुल्ली बॉय) का प्रीमियर किया गया जहाँ पर इस फिल्म को समीक्षकों और दर्शकों ने बहुत सराहा है। भारतीय समीक्षकों ने जमकर इसकी तारीफ की है। हालाकि अभी तक किसी भी विदेशी फिल्म समीक्षक की तारीफ सामने नहीं आई है। उनकी यह फिल्म आगामी १४ फरवरी को प्रदर्शित होने जा रही है। इस फिल्म के प्रमोशन में व्यस्त रणवीर सिंह (रणवीर सिंह) ने हाल ही में दिए एक साक्षात्कार में कहा है कि उन्हें ऐसा लगता था कि बॉलीवुड में वो कोई मुकाम हासिल नहीं कर पाएंगे।
रणवीर सिंह (रणवीर सिंह) की पिछली फिल्म सिम्बा ने वैश्विक स्तर पर कुल मिलाकर ३५० करोड़ का कारोबार किया है। इस बात की जानकारी भी स्वयं रणवीर सिंह (रणवीर सिंह) ने ही दी है। गत शनिवार ९ फरवरी को सोनी टीवी पर प्रसारित कार्यक्रम कपिल शर्मा शो में रणवीर सिंह आलिया भट्ट (एलिया भट्ट) के साथ अपनी फिल्म गली बॉय (गुल्ली बॉय) का प्रमोशन करने आए थे जहाँ उन्होंने कहा कि मेरे पास सिम्बा की कमाई का एक रुपया नहीं आया है। पूरा का पूरा ३५० करोड़ रोहित शेट्टी (रोहित शेट्टी) के पास गया है। | hindi |
Described as ‘body-worn video for vehicles’, the ASPECT 360 vehicle-mounted panoramic camera from Observant Innovations allows you to see in all directions, all the time. Built to withstand the harshness of mounting on mobile platforms, the camera is heavily used in the security sector.
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Improve performance, professionalism and front-line staff protection with ASPECT 360 – the powerful and robust 360° panoramic camera sensor for autonomous vehicles.
This product comes with the Review software tool and a unit-specific restricted basic license for the Aperture Imaging Workbench, which is locked to the unit. Please note that this product does not have a recording facility.
View the Aspect panoramic camera data sheet from Observant Innovations below. | english |
package gov.nasa.gsfc.seadas.processing.l2gen.userInterface;
import gov.nasa.gsfc.seadas.processing.core.L2genData;
import gov.nasa.gsfc.seadas.processing.l2gen.productData.L2genWavelengthInfo;
import gov.nasa.gsfc.seadas.processing.common.GridBagConstraintsCustom;
import javax.swing.*;
import java.awt.*;
import java.awt.event.ActionEvent;
import java.awt.event.ActionListener;
import java.awt.event.ItemEvent;
import java.awt.event.ItemListener;
import java.beans.PropertyChangeEvent;
import java.beans.PropertyChangeListener;
import java.util.ArrayList;
/**
* Created by IntelliJ IDEA.
* User: knowles
* Date: 5/11/12
* Time: 3:42 PM
* To change this template use File | Settings | File Templates.
*/
public class L2genWavelengthLimiterPanel extends JPanel {
private L2genData l2genData;
private JPanel waveLimiterJPanel;
private ArrayList<JCheckBox> wavelengthsJCheckboxArrayList = new ArrayList<JCheckBox>();
private boolean waveLimiterControlHandlersEnabled = false;
private JButton
infraredButton,
visibleButton,
nearInfraredButton;
private final static String
SELECT_ALL_INFRARED = "Select All Infrared",
DESELECT_ALL_INFRARED = "Deselect All Infrared",
SELECT_ALL_NEAR_INFRARED = "Select All Near-Infrared",
DESELECT_ALL_NEAR_INFRARED = "Deselect All Near-Infrared",
SELECT_ALL_VISIBLE = "Select All Visible",
DESELECT_ALL_VISIBLE = "Deselect All Visible";
// private InfraredButton infraredButton = new InfraredButton();
L2genWavelengthLimiterPanel(L2genData l2genData) {
this.l2genData = l2genData;
initComponents();
addComponents();
}
public void initComponents() {
waveLimiterJPanel = createWaveLimiterJPanel();
infraredButton = createInfraredButton();
nearInfraredButton = createNearInfraredButton();
visibleButton = createVisibleButton();
}
public void addComponents() {
setLayout(new GridBagLayout());
setBorder(BorderFactory.createTitledBorder("Wavelength Limiter"));
setToolTipText("The wavelengths selected here are applied when you check a wavelength dependent product. Not that any subsequent change ...");
JPanel innerPanel = new JPanel(new GridBagLayout());
innerPanel.add(visibleButton,
new GridBagConstraintsCustom(0, 0, 1, 0, GridBagConstraints.NORTHWEST, GridBagConstraints.NONE));
innerPanel.add(nearInfraredButton,
new GridBagConstraintsCustom(0, 1, 1, 0, GridBagConstraints.NORTHWEST, GridBagConstraints.NONE));
innerPanel.add(infraredButton,
new GridBagConstraintsCustom(0, 2, 1, 0, GridBagConstraints.NORTHWEST, GridBagConstraints.NONE));
innerPanel.add(waveLimiterJPanel,
new GridBagConstraintsCustom(0, 3, 1, 0, GridBagConstraints.NORTHWEST, GridBagConstraints.NONE));
innerPanel.add(new JPanel(),
new GridBagConstraintsCustom(0, 4, 1, 1, GridBagConstraints.NORTH, GridBagConstraints.BOTH));
JScrollPane innerScroll = new JScrollPane(innerPanel);
innerScroll.setBorder(null);
add(innerScroll,
new GridBagConstraintsCustom(0, 0, 1, 1, GridBagConstraints.NORTHWEST, GridBagConstraints.BOTH));
}
private JButton createInfraredButton() {
final JButton jButton = new JButton(SELECT_ALL_INFRARED);
jButton.addActionListener(new ActionListener() {
@Override
public void actionPerformed(ActionEvent e) {
if (jButton.getText().equals(SELECT_ALL_INFRARED)) {
l2genData.setSelectedAllWaveLimiter(L2genWavelengthInfo.WaveType.INFRARED, true);
} else if (jButton.getText().equals(DESELECT_ALL_INFRARED)) {
l2genData.setSelectedAllWaveLimiter(L2genWavelengthInfo.WaveType.INFRARED, false);
}
}
});
l2genData.addPropertyChangeListener(L2genData.WAVE_LIMITER, new PropertyChangeListener() {
@Override
public void propertyChange(PropertyChangeEvent evt) {
updateInfraredButton();
}
});
l2genData.addPropertyChangeListener(L2genData.IFILE, new PropertyChangeListener() {
@Override
public void propertyChange(PropertyChangeEvent evt) {
updateInfraredButton();
}
});
return jButton;
}
private void updateInfraredButton() {
// Set INFRARED 'Select All' toggle to appropriate text and enabled
if (l2genData.hasWaveType(L2genWavelengthInfo.WaveType.INFRARED)) {
nearInfraredButton.setEnabled(true);
if (l2genData.isSelectedAllWaveLimiter(L2genWavelengthInfo.WaveType.INFRARED)) {
if (!infraredButton.getText().equals(DESELECT_ALL_INFRARED)) {
infraredButton.setText(DESELECT_ALL_INFRARED);
}
} else {
if (!infraredButton.getText().equals(SELECT_ALL_INFRARED)) {
infraredButton.setText(SELECT_ALL_INFRARED);
}
}
} else {
nearInfraredButton.setEnabled(false);
}
}
private class InfraredButton {
private static final String selectAll = SELECT_ALL_INFRARED;
private static final String deselectAll = DESELECT_ALL_INFRARED;
InfraredButton() {
}
private JButton createInfraredButton() {
final JButton jButton = new JButton(selectAll);
jButton.addActionListener(new ActionListener() {
@Override
public void actionPerformed(ActionEvent e) {
if (jButton.getText().equals(selectAll)) {
l2genData.setSelectedAllWaveLimiter(L2genWavelengthInfo.WaveType.INFRARED, true);
} else if (jButton.getText().equals(deselectAll)) {
l2genData.setSelectedAllWaveLimiter(L2genWavelengthInfo.WaveType.INFRARED, false);
}
}
});
l2genData.addPropertyChangeListener(L2genData.WAVE_LIMITER, new PropertyChangeListener() {
@Override
public void propertyChange(PropertyChangeEvent evt) {
updateInfraredButton();
}
});
l2genData.addPropertyChangeListener(L2genData.IFILE, new PropertyChangeListener() {
@Override
public void propertyChange(PropertyChangeEvent evt) {
updateInfraredButton();
}
});
return jButton;
}
private void updateInfraredButton() {
// Set INFRARED 'Select All' toggle to appropriate text and enabled
if (l2genData.hasWaveType(L2genWavelengthInfo.WaveType.INFRARED)) {
nearInfraredButton.setEnabled(true);
if (l2genData.isSelectedAllWaveLimiter(L2genWavelengthInfo.WaveType.INFRARED)) {
if (!infraredButton.getText().equals(deselectAll)) {
infraredButton.setText(deselectAll);
}
} else {
if (!infraredButton.getText().equals(selectAll)) {
infraredButton.setText(selectAll);
}
}
} else {
nearInfraredButton.setEnabled(false);
}
}
}
private JButton createNearInfraredButton() {
final JButton jButton = new JButton(SELECT_ALL_NEAR_INFRARED);
jButton.addActionListener(new ActionListener() {
@Override
public void actionPerformed(ActionEvent e) {
if (jButton.getText().equals(SELECT_ALL_NEAR_INFRARED)) {
l2genData.setSelectedAllWaveLimiter(L2genWavelengthInfo.WaveType.NEAR_INFRARED, true);
} else if (jButton.getText().equals(DESELECT_ALL_NEAR_INFRARED)) {
l2genData.setSelectedAllWaveLimiter(L2genWavelengthInfo.WaveType.NEAR_INFRARED, false);
}
}
});
l2genData.addPropertyChangeListener(L2genData.WAVE_LIMITER, new PropertyChangeListener() {
@Override
public void propertyChange(PropertyChangeEvent evt) {
updateNearInfraredButton();
}
});
l2genData.addPropertyChangeListener(L2genData.IFILE, new PropertyChangeListener() {
@Override
public void propertyChange(PropertyChangeEvent evt) {
updateNearInfraredButton();
}
});
return jButton;
}
private void updateNearInfraredButton() {
// Set NEAR_INFRARED 'Select All' toggle to appropriate text and enabled
if (l2genData.hasWaveType(L2genWavelengthInfo.WaveType.NEAR_INFRARED)) {
nearInfraredButton.setEnabled(true);
if (l2genData.isSelectedAllWaveLimiter(L2genWavelengthInfo.WaveType.NEAR_INFRARED)) {
if (!nearInfraredButton.getText().equals(DESELECT_ALL_NEAR_INFRARED)) {
nearInfraredButton.setText(DESELECT_ALL_NEAR_INFRARED);
}
} else {
if (!nearInfraredButton.getText().equals(SELECT_ALL_NEAR_INFRARED)) {
nearInfraredButton.setText(SELECT_ALL_NEAR_INFRARED);
}
}
} else {
nearInfraredButton.setEnabled(true);
}
}
private JButton createVisibleButton() {
final JButton jButton = new JButton(SELECT_ALL_VISIBLE);
jButton.addActionListener(new ActionListener() {
@Override
public void actionPerformed(ActionEvent e) {
if (jButton.getText().equals(SELECT_ALL_VISIBLE)) {
l2genData.setSelectedAllWaveLimiter(L2genWavelengthInfo.WaveType.VISIBLE, true);
} else if (jButton.getText().equals(DESELECT_ALL_VISIBLE)) {
l2genData.setSelectedAllWaveLimiter(L2genWavelengthInfo.WaveType.VISIBLE, false);
}
}
});
l2genData.addPropertyChangeListener(L2genData.WAVE_LIMITER, new PropertyChangeListener() {
@Override
public void propertyChange(PropertyChangeEvent evt) {
updateVisibleButton();
}
});
l2genData.addPropertyChangeListener(L2genData.IFILE, new PropertyChangeListener() {
@Override
public void propertyChange(PropertyChangeEvent evt) {
updateVisibleButton();
}
});
return jButton;
}
private void updateVisibleButton() {
// Set VISIBLE 'Select All' toggle to appropriate text and enabled
if (l2genData.hasWaveType(L2genWavelengthInfo.WaveType.VISIBLE)) {
visibleButton.setEnabled(true);
if (l2genData.isSelectedAllWaveLimiter(L2genWavelengthInfo.WaveType.VISIBLE)) {
if (!visibleButton.getText().equals(DESELECT_ALL_VISIBLE)) {
visibleButton.setText(DESELECT_ALL_VISIBLE);
}
} else {
if (!visibleButton.getText().equals(SELECT_ALL_VISIBLE)) {
visibleButton.setText(SELECT_ALL_VISIBLE);
}
}
} else {
visibleButton.setEnabled(false);
}
}
private JPanel createWaveLimiterJPanel() {
final JPanel jPanel = new JPanel(new GridBagLayout());
l2genData.addPropertyChangeListener(L2genData.WAVE_LIMITER, new PropertyChangeListener() {
@Override
public void propertyChange(PropertyChangeEvent evt) {
updateWaveLimiterSelectionStates();
}
});
l2genData.addPropertyChangeListener(L2genData.IFILE, new PropertyChangeListener() {
@Override
public void propertyChange(PropertyChangeEvent evt) {
updateWavelengthLimiterPanel();
updateWaveLimiterSelectionStates();
}
});
return jPanel;
}
private void updateWavelengthLimiterPanel() {
waveLimiterJPanel.removeAll();
// clear this because we dynamically rebuild it when input file selection is made or changed
wavelengthsJCheckboxArrayList.clear();
ArrayList<JCheckBox> wavelengthGroupCheckboxes = new ArrayList<JCheckBox>();
for (L2genWavelengthInfo waveLimiterInfo : l2genData.getWaveLimiterInfos()) {
final String currWavelength = waveLimiterInfo.getWavelengthString();
final JCheckBox currJCheckBox = new JCheckBox(currWavelength);
currJCheckBox.setName(currWavelength);
// add current JCheckBox to the externally accessible arrayList
wavelengthsJCheckboxArrayList.add(currJCheckBox);
// add listener for current checkbox
currJCheckBox.addItemListener(new ItemListener() {
@Override
public void itemStateChanged(ItemEvent e) {
if (waveLimiterControlHandlersEnabled) {
l2genData.setSelectedWaveLimiter(currWavelength, currJCheckBox.isSelected());
}
}
});
wavelengthGroupCheckboxes.add(currJCheckBox);
}
if (l2genData.hasWaveType(L2genWavelengthInfo.WaveType.INFRARED)) {
l2genData.setSelectedAllWaveLimiter(L2genWavelengthInfo.WaveType.INFRARED, true);
}
if (l2genData.hasWaveType(L2genWavelengthInfo.WaveType.VISIBLE)) {
l2genData.setSelectedAllWaveLimiter(L2genWavelengthInfo.WaveType.VISIBLE, true);
}
if (l2genData.hasWaveType(L2genWavelengthInfo.WaveType.NEAR_INFRARED)) {
l2genData.setSelectedAllWaveLimiter(L2genWavelengthInfo.WaveType.NEAR_INFRARED, true);
}
// some GridBagLayout formatting variables
int gridyCnt = 0;
int gridxCnt = 0;
int NUMBER_OF_COLUMNS = 2;
for (JCheckBox wavelengthGroupCheckbox : wavelengthGroupCheckboxes) {
// add current JCheckBox to the panel
waveLimiterJPanel.add(wavelengthGroupCheckbox,
new GridBagConstraintsCustom(gridxCnt, gridyCnt, 1, 1, GridBagConstraints.NORTHWEST, GridBagConstraints.NONE));
// increment GridBag coordinates
if (gridxCnt < (NUMBER_OF_COLUMNS - 1)) {
gridxCnt++;
} else {
gridxCnt = 0;
gridyCnt++;
}
}
// just in case
l2genData.fireEvent(l2genData.WAVE_LIMITER);
// updateWaveLimiterSelectionStates();
}
/**
* Set all waveLimiterInfos controls to agree with l2genData
*/
private void updateWaveLimiterSelectionStates() {
// Turn off control handlers until all controls are set
waveLimiterControlHandlersEnabled = false;
// Set all checkboxes to agree with l2genData
for (L2genWavelengthInfo waveLimiterInfo : l2genData.getWaveLimiterInfos()) {
for (JCheckBox currJCheckbox : wavelengthsJCheckboxArrayList) {
if (waveLimiterInfo.getWavelengthString().equals(currJCheckbox.getName())) {
if (waveLimiterInfo.isSelected() != currJCheckbox.isSelected()) {
currJCheckbox.setSelected(waveLimiterInfo.isSelected());
}
}
}
}
// Turn on control handlers now that all controls are set
waveLimiterControlHandlersEnabled = true;
}
}
| code |
Allows you to train behind a door.
Door Anchor gives you the opportunity to train suspension training travel, office or wherever you want it.
Close the door as Door Anchor squeezed between the door frame and door mount your Liana system in Door Anchor and you are ready to train. | english |
<?php
namespace App\Http\Controllers;
use Validator;
use App\Http\Controllers\Controller;
use App\Repositories\FeedbackRepository;
use Input;
use Response;
use Auth;
use App\Http\Requests;
use Illuminate\Http\Request;
class FeedbackController extends Controller
{
/**
* The user repository instance.
*
* @var FeedbackRepository
*/
protected $feedback;
/**
* Create a new controller instance.
*
* @param FeedbackRepository $feedback
* @return void
*/
public function __construct(FeedbackRepository $feedback)
{
$this->feedback = $feedback;
}
protected function feedbackValidator($data)
{
return Validator::make($data, [
'rating' => 'required',
'comment' => 'required',
'name' => 'required|max:255',
'email' => 'required|email|max:255|',
],
[
'required' => 'The :attribute is required.',
]);
}
public function jsModalfeedback()
{
return view('feedback.feedback');
}
public function saveFeedback(Request $request) {
$validator = $this->feedbackValidator($request->all());
if ($validator->fails()) {
$this->throwValidationException(
$request, $validator
);
}
$status = $this->feedback->saveFeedback($request);
$request->session()->flash($status['type'], $status['msg']);
$response = array(
'type' => $status['type'],
'msg' => $status['msg']
);
return Response::json($response);
exit();
}
} | code |
انگٕر چھ ہمیشہٕ لۄکٹ آسان | kashmiri |
\begin{document}
\author{Yuri A. Rylov}
\title{Classification of Finite Subspaces of Metric Space Instead of Constraints on
Metric}
\date{Institute for Problems in Mechanics, Russian Academy of Sciences, 101-1,
Vernadskii Ave., Moscow, 117526, Russia\\
e-mail: rylov@ipmnet.ru}
\maketitle
\begin{abstract}
A new method of metric space investigation, based on classification of its
finite subspaces, is suggested. It admits to derive information on metric
space properties which is encoded in metric. The method describes geometry
in terms of only metric. It admits to remove constraints imposed usually on
metric (the triangle axiom and nonnegativity of the squared metric), and to
use the metric space for description of the space-time and other geometries
with indefinite metric. Describing space-time and using this method, one can
explain quantum effects as geometric effects, i.e. as space-time properties.
\end{abstract}
\section{Introduction}
Let $M=\{\rho ,\Omega\}$ be a metric space, where $\Omega$ is a set of
points, and $\rho $ be the metric, i.e.
\begin{equation}
\rho :\quad \Omega \times \Omega \rightarrow [0,\infty )\subset {\Bbb R}
\label{a1.1}
\end{equation}
\begin{equation}
\rho (P,P)=0,\qquad \rho (P,Q)=\rho (Q,P),\qquad \forall P,Q\in \Omega
\label{a1.2}
\end{equation}
\begin{equation}
\rho (P,Q)\geq 0,\qquad \rho (P,Q)=0,\quad {\rm \; \; \; \;} P=Q,\qquad
\forall P,Q\in \Omega \label{a1.3}
\end{equation}
\begin{equation}
\rho (P,Q)+\rho (Q,R)\geq \rho (P,R),\qquad \forall P,Q,R\in \Omega
\label{a1.4}
\end{equation}
\begin{opred}
\label{d1} Any subset $\Omega ^{\prime }\subset \Omega $ of points of the
metric space $M=\{\rho ,\Omega \}$, equipped with the metric $\rho ^{\prime }
$ which is a contraction $\rho |_{\Omega ^{\prime }\times \Omega ^{\prime }}$
of the mapping (\ref{a1.1}). on the set $\Omega ^{\prime }\times \Omega
^{\prime }$ is called the metric subspace $M^{\prime }=\{\rho ^{\prime
},\Omega ^{\prime }\}$ of the metric space $M=\{\rho ,\Omega \}$.
\end{opred}
It is easy to see that the metric subspace $M^{\prime }=\{\rho ^{\prime
},\Omega ^{\prime }\}$ is a metric space.
\begin{opred}
\label{d5} The metric space $M_n({\cal P}^n)=\{\rho ,{\cal P}^n\}$ is called
a finite one, if it consists of a finite number of points ${\cal P}^n\equiv
\{P_i\}$, $i=0,1,\ldots n$.
\end{opred}
\begin{opred}
\label{d6} Finite metric space $M_n({\cal P}^n)=\{\rho ,{\cal P}^n\}$ is
called oriented one $\overrightarrow{M_n({\cal P}^n)}$, if the order of its
points ${\cal P}^n=\{P_0,P_1,\ldots P_n\}$ is given.
\end{opred}
\begin{opred}
\label{d7} An oriented finite subspace $\overrightarrow{M_n({\cal P}^n)}$ of
the metric space $M=\{\rho ,\Omega \}$ is called a multivector. It is
designed by means of $\overrightarrow{M_n({\cal P}^n)}\equiv \overrightarrow{
{\cal P}^n}\equiv \overrightarrow{\{P_0,P_1,\ldots P_n\}}$
\end{opred}
\begin{opred}
\label{d2} A description is called $\sigma $-immanent one, if it does not
contain any references to objects or concepts other than subspaces of the
metric space or its metric.
\end{opred}
$\sigma$-immanency of a description means that it is immanent to the metric
space and it is carried out in terms of its metric and subspaces. The prefix
"$\sigma$" associates with the world function $\sigma$, which is connected
with the metric by means of the relation $\sigma =\rho ^2/2$. The name
"world function" was suggested by Singe \cite{S60}, who introduced it for
the Riemannian space description and used it for description of the
space-time geometry. A use of the world function $\sigma$ instead of the
metric $\rho$ appears to be more convenient.
The shortest, connecting two arbitrary points $P,Q\in\Omega$, is the basic
geometrical object which is constructed usually in the metric space $\{\rho
,\Omega\}$ \cite{ABN86}. One can construct an angle, triangle, different
polygons from segments of the shortest. Construction of two-dimensional and
three-dimensional planes in the metric space is rather problematic. At any
rate it is unclear how one could construct these planes, using the shortest
as the main geometrical object. It gives rise to think that the metric
geometry, i.e. geometry generated by the metric space, is less pithy, than
the geometry of the Euclidean space, where such geometric objects as
two-dimensional and three-dimensional planes can be build without any
problems. In fact it is not so. In the scope of the metric geometry one can
construct almost all geometric objects which can be constructed in the
Euclidean geometry, including $n$-dimensional planes. It is necessary to use
more effective method of the metric space description, than that based on
the use of the shortest.
Let ${\cal P}^n=\{P_0,P_1,\ldots ,P_n\}\subset\Omega$ be the set of $n+1$
points $P_i\in\Omega $, $(i=1,2,\ldots n)$ in $D$-dimensional Euclidean
space $\Omega ={\Bbb R}^D$, $(D>n)$. Let $\rho$ be the Euclidean metric in $
{\Bbb R}^D$. Let us consider $(n+1)$-edr with vertices at the points ${\cal P
}^n$. Its volume $S_n({\cal P}^n)$ may be presented in $\sigma$-immanent
form.
\begin{equation}
S_n({\cal P}^n)=\frac1{n!}\sqrt{F_n\left( {\cal P}^n\right)}, \label{a1.0}
\end{equation}
\begin{equation}
F_n\left( {\cal P}^n\right) =\det ||\left( {\bf P}_0{\bf P}_i.{\bf P}_0{\bf P
}_k\right) ||,\qquad P_0,P_i,P_k\in \Omega ,\qquad i,k=1,2,...n \label{a1.6}
\end{equation}
\begin{equation}
\left( {\bf P}_0{\bf P}_i.{\bf P}_0{\bf P}_k\right) \equiv \Gamma \left(
P_0,P_i,P_k\right) \equiv \sigma \left( P_0,P_i\right) +\sigma \left(
P_0,P_k\right) -\sigma \left( P_i,P_k\right) ,\qquad i,k=1,2,...n.
\label{a1.7}
\end{equation}
\begin{equation}
\sigma (P,Q)\equiv \frac 12\rho ^2(P,Q),\qquad \forall P,Q\in \Omega ,
\label{a1.8}
\end{equation}
where ${\bf P}_0{\bf P}_i$, $i=1,2,\ldots n$ are $n$ vectors of the
Euclidean space. If these $n$ vectors are linear independent, the volume $
S_n({\cal P}^n)$ of $(n+1)$-edr does not vanish. If they are linear
dependent, $S_n({\cal P}^n)=0$. In virtue of (\ref{a1.0})-(\ref{a1.8}) the
condition $F_n\left( {\cal P}^n\right)=0$ is a $\sigma$-immanent criterion
of linear dependence of vectors ${\bf P}_0{\bf P}_i$, $i=1,2,\ldots n$.
The value of the $\sigma$-immanent function $F_n\left( {\cal P}^n\right)$ of
points ${\cal P}^n$ may serve as a criterion of "linear independence of
vectors" even in the case, when a linear vector space cannot be introduced
and the concept of linear independence cannot be defined via it. This shows
that the concept of linear independence is in reality something more
fundamental, than an attribute of a linear vector space. Essentially the
quantity $F_n\left( {\cal P}^n\right)$ is a characteristic of $(n+1)$-point
metric space $M_n=\{\rho ,{\cal P}^n\}$. This quantity appears, when one
identifies the quantity $\sqrt{F_n\left( {\cal P}^n\right)}$ with the length
$|M_n|\equiv |{\cal P}^n|$ of the finite metric space $M_n=\{\rho ,{\cal P}
^n\}$.
Let $n$ vectors ${\bf P}_0{\bf P}_i$, $i=1,2,\ldots n$ are linear
independent. Then $F_n\left( {\cal P}^n\right)\ne 0$. Let us construct the
linear span of vectors ${\bf P}_0{\bf P}_i$, $i=1,2,\ldots n$, consisting of
vectors ${\bf P}_0{\bf R}$. Then $n+2$ vectors ${\bf P}_0{\bf P}_i$, $
(i=1,2,\ldots n)$, ${\bf P}_0{\bf R}$ are linear dependent, and the point $R$
satisfies the $\sigma$-immanent relation $F_{n+1}\left( {\cal P}^n,
R\right)= 0$. The set ${\cal L}({\cal P}^n)=\{R|F_{n+1}\left( {\cal P}^n,
R\right)= 0\}$ of points $R$ is a $n$-dimensional plane, passing through $
n+1 $ points ${\cal P}^n$. As far as this relation is $\sigma$-immanent, it
determines some set ${\cal T}({\cal P}^n)=\{R|F_{n+1}\left( {\cal P}^n,
R\right)= 0\} \subset\Omega$ of points in any metric space $M=\{\rho
,\Omega\}$. This set ${\cal T}({\cal P}^n)$, called the $n$th order tube, is
an analog of $n$-dimensional plane of the Euclidean space. $n+1$ points $
{\cal P}^n$, determining the tube, will be referred to as basic points of
the tube, or its $(n+1)$-point $\sigma$-basis. The $n$th order tube may be
considered to be a natural geometric object (NGO) of the metric space,
determined by $(n+1)$-point metric space $\{\rho ,{\cal P}^n\}$. The $n$th
order tube is a $\sigma$-immanent geometric object.
Thus, always there exists a subspace of the metric space which is an analog
of $n$-dimensional Euclidean plane, but linear operations on vectors cannot
be defined always. This takes place, because the tubes ${\cal T}({\cal P}^n)$
have another structure than corresponding planes ${\cal L}({\cal P}^n)$ of
the Euclidean space. Let ${\cal Q}^n\subset {\cal L}({\cal P}^n)$ be other $
(n+1)$-point $\sigma$-basis ($F_n\left( {\cal Q}^n\right)\ne 0$), belonging
to the plane ${\cal L}({\cal P}^n)$. Then ${\cal L}({\cal Q}^n)={\cal L}(
{\cal P}^n)$. In the arbitrary metric space it is not so, and, in general. $
{\cal T}({\cal Q}^n)\ne {\cal T}({\cal P}^n)$. In particular, in the
Euclidean space any two different points of a straight determine this
straight. In many cases for a metric space two basic points determine the
tube which coincides with the shortest (For instance, it is so for a
Riemannian space, considered as a metric space). Then any two points of the
shortest are the basic points of this shortest and determine it. But there
are cases, when it is not so. Then two different points other than basic
points determine a tube, but it is another tube. In the case of the second
order tubes (analog of two-dimensional Euclidean plane) the inequality $
{\cal T}({\cal Q}^2)\ne {\cal T}({\cal P}^2)$ is more likely to be a rule
than an exception from the rule. For instance, it is true for the Riemannian
space considered as a metric one.
A possibility of the metric space description in terms of only the shortest
is restricted. Although exhibiting ingenuity, such a description may be
constructed. For instance, A.D.~Alexandrov showed that internal geometry of
two-dimensional boundaries of convex three-dimensional bodies may be
represented in the $\sigma$-immanent form \cite{A48}. Apparently, without
introducing tubes of the order higher than unity, the solution of similar
problem for three-dimensional boundaries of four-dimensional bodies is very
difficult.
The $\sigma$-immanent conception of the metric space description can be
formulated as a classification of all finite metric subspaces $M_n({\cal P}
^n)=\{\rho ,{\cal P}^n\}$, $(n=1,2,\ldots )$, where ${\cal P}^n\subset\Omega$
, is a set of $n+1$ points $P_i\in\Omega $, $(i=0,1,\ldots n)$. Any $M_n(
{\cal P}^n)$ associates with the number $|M_n({\cal P}^n)|=|{\cal P}
^n|=n!S_n({\cal P}^n)=\sqrt{F_n({\cal P}^n)}$, called the length.
From mathematical viewpoint the classification of metric subspaces reduces
to equipping the metric space $\{\rho ,\Omega\}$ with a series of $\sigma$
-immanent mappings
\begin{equation}
F_n:\quad \Omega ^{n+1}\rightarrow {\Bbb R},\qquad \Omega
^{n+1}=\bigotimes\limits_{k=1}^{n+1}\Omega ,\qquad n=1,2,\ldots \label{a1.5}
\end{equation}
where $F_n({\cal P}^n)$ is defined by the relations (\ref{a1.6})-(\ref{a1.8}
). As one can see from (\ref{a1.6}), (\ref{a1.7}), in the case $n=1$ $
F_1(P,Q)=2\sigma (P,Q)=\rho ^2(P,Q)$.
Further it will be shown that the classification of finite metric subspaces,
carried out by means of the series of mappings (\ref{a1.5}), admits to
derive information on the metric space properties contained in its metric.
The metric geometry (i.e. the geometry generated by the metric space)
appears to be not less pithy, than the Euclidean geometry. It means,
particularly, that the Euclidean geometry may be formulated in the $\sigma$
-immanent form. Furthermore the geometry of any subset of points of the
proper Euclidean space may be formulated in the $\sigma$-immanent form. In
other words, the metric geometry, constructed on the basis of the metric, is
insensitive to continuity or discreteness of the space.
The situation of constructing the metric geometry may be presented
conveniently as follows. In the $D$-dimensional Euclidean space $E_D=\{\rho
,\Omega \}$, $\Omega ={\Bbb R}^D$ the $n$-dimensional plane ${\cal L}({\cal P
}^{n})$, $n=1,2,\ldots D$, which passes through points ${\cal P}^n$, forming
$(n+1)$-point $\sigma$-basis $(F_n({\cal P}^n)\ne 0)$, is described as a set
of points $R$, satisfying $\sigma$-immanent equation $F_{n+1}({\cal P}^n,R)=
0$. $n$-dimensional plane ${\cal L}({\cal P}^{n})$, $(n=1,2,\ldots D)$ is
determined by only metric. It is NGO for the Euclidean space $E_D$.
The metric space can be conceived as a result of a deformation of $D$
-dimensional Euclidean space $E_D$ with rather large $D$. The deformation
means a variation of distances between points of $E_D$, accompanied by
removing some set $U$ of points belonging to $E_D$. Under such a deformation
NGOs are deformed, turning to sets of points of more complicated
configuration, but they continue to be attributes of the metric space,
because they are $\sigma$-immanent and determined only by metric.
Restrictions (\ref{a1.3}), (\ref{a1.4}) on the metric $\rho$ are used by no
means. They are needed for constructing the shortest. They may be removed,
if geometrical objects are constructed on the basis of a classification of
finite metric subspaces $\{\rho ,{\cal P}^n\}$.
In this case, replacing the metric by the world function $\sigma =\frac
12\rho ^2$, one obtains the more general metric space $V=\{\sigma ,\Omega\}$
instead of the usual metric space $M=\{\rho ,\Omega\}$. This metric space
will be referred to as $\sigma $-space. The geometry, generated by the $
\sigma $-space will be referred to as T-geometry. The T-geometry is a
generalization of the metric geometry on the case of indefinite metric.
T-geometry may be used for a description of the space-time geometry.
Under above described deformation of the Euclidean space the Euclidean
straights turn to hallow tubes. (In general, it is possible such a case,
when the straights turn to curves, remaining to be lines, but it ia a very
special case of deformation). The hallow tubes appear in the general case,
because one equation, determining the tube, describes generally a surface.
This fact explains the name of the geometry: tubular geometry, or
T-geometry. From viewpoint of the more general T-geometry the conventional
metric geometry is a degenerated geometry, where the tubes degenerate to
lines (the shortests). The T-geometry is a natural geometry (which is
nondegenerated, in general). It is the most general geometry. A strong
argument in favour of T-geometry is the circumstance that on the basis of
T-geometry one can construct such a space-time model, where quantum effects
are explained as simple T-geometric effects, and the quantum constant is an
attribute of the space-time \cite{R91}.
In the second section definitions of main objects of $\sigma $-space are
given. The third section is devoted to the formulation and proof of the
theorem, stating that the Euclidean geometry can be described in terms of
only metric. In the fourth section the role of the triangle axiom is
discussed.
\section{$\sigma $-space and its properties.}
\begin{opred}
\label{d3.1.1} $\sigma $-space $V=\{\sigma ,\Omega \}$ is nonempty set $
\Omega $ of points $P$ with given on $\Omega \times \Omega $ real function $
\sigma $
\begin{equation}
\sigma :\quad \Omega \times \Omega \to {\Bbb R},\qquad \sigma (P,P)=0,\qquad
\sigma (P,Q)=\sigma (Q,P)\qquad \forall P,Q\in \Omega . \label{a2.1}
\end{equation}
\end{opred}
The function $\sigma $ is called world function, or $\sigma $-function.
\begin{opred}
\label{d3.1.1 }. Nonempty subset $\Omega ^{\prime }\subset \Omega $ of
points of the $\sigma $-space $V=\{\sigma ,\Omega \}$ with the world
function $\sigma ^{\prime }=\sigma |_{\Omega ^{\prime }\times \Omega
^{\prime }}$, which is a contraction $\sigma $ on $\Omega ^{\prime }\times
\Omega ^{\prime }$ is called $\sigma $-subspace $V^{\prime }=\{\sigma
^{\prime },\Omega ^{\prime }\}$ of $\sigma $-space $V=\{\sigma ,\Omega \}$.
\end{opred}
Further the world function $\sigma ^\prime = \sigma |_{\Omega ^{\prime
}\times\Omega ^\prime }$, which is a contraction of $\sigma $ will be
designed by means of $\sigma $. Any $\sigma$-subspace of $\sigma$-space is a
$\sigma$-space.
\begin{opred}
\label{d3.1.1ba}. $\sigma $-space $V^{\prime }=\{\sigma ^{\prime },\Omega
^{\prime }\}$ is called isometrically embedded in $\sigma $-space $
V=\{\sigma ,\Omega \}$, if there exists such a monomorphism $f:\Omega
^{\prime }\rightarrow \Omega $, that $\sigma ^{\prime }(P,Q)=\sigma
(f(P),f(Q))$,\quad $\forall P,\forall Q\in \Omega ^{\prime },\quad
f(P),f(Q)\in \Omega $,
\end{opred}
Any $\sigma$-subspace $V^{\prime}$ of $\sigma$-space $V=\{\sigma ,\Omega \}$
is isometrically embedded in it.
\begin{opred}
\label{d3.1.1b}. Two $\sigma $-spaces $V=\{\sigma ,\Omega \}$ and $V^{\prime
}=\{\sigma ^{\prime },\Omega ^{\prime }\}$ are called to be isometric
(equivalent), if $V$ is isometrically embedded in $V^{\prime }$, and $
V^{\prime }$ is isometrically embedded in $V$.
\end{opred}
\begin{opred}
\label{d3.1.1bc}. $\sigma $-space $M_n({\cal P}^n)=\{\sigma ,{\cal P}^n\}$,
consisting of $n+1$ points ${\cal P}^n$ is called the finite $\sigma $-space
of $n$th order.
\end{opred}
\begin{opred}
\label{d3.1.1bd}. The number $\sqrt{F_n({\cal P}^n)}$, where $F_n({\cal P}
^n) $ is defined by relations (\ref{a1.6})-(\ref{a1.8}), is called the
length (volume) of the finite $n$th order $\sigma $-space $M_n({\cal P}^n)$.
\end{opred}
If the set of points ${\cal P}^n$ of a finite $\sigma$-space $M_n({\cal P}
^n) $ is ordered, such a finite $\sigma$-space $M_n({\cal P}^n)$ is called
multivecor. Practically only multivectors which are $\sigma$-subspaces of
the same $\sigma$-space and described by the same world function will be
considered. In this case one may not mention on the world function in the
definition of the multivector and define it as follows.
\begin{opred}
\label{d3.1.2b}. The ordered set $\{P_l\},\quad l=0,1,\ldots n$ of $n+1$
points $P_0,P_1,...,P_n$, belonging to the $\sigma $-space $V$ is called the
$n$th order multivector $\overrightarrow{P_0P_1...P_n}$. The point $P_0$ is
the origin of the multivector $\overrightarrow{P_0P_1...P_n}$
\end{opred}
Let us use the following designation for the multivector $\overrightarrow{
P_0P_1...P_n}$. $\overrightarrow{P_0P_1...P_n}\equiv {\bf P}_0{\bf P}_1...
{\bf P}_n \equiv\overrightarrow{{\cal P}^n}$.
\begin{opred}
\label{d3.1.2}. The vector ${\bf PQ}$ in the $\sigma $-space $V$ is the
first order multivector, or the ordered set $\{P,Q\}$ of two points $P,Q$.
The point $P$ is the origin, nd $Q$ is the end of the vector.
\end{opred}
\begin{opred}
\label{d3.1.3}. The scalar $\sigma $-product $({\bf P}_0{\bf P}_1.{\bf P}_0
{\bf P}_2)$ of two vectors ${\bf P}_0{\bf P}_1$ and ${\bf P}_0{\bf P}_2$,
having a common origin, is called a real number
\begin{equation}
({\bf P}_0{\bf P}_1.{\bf P}_0{\bf P}_2)\equiv \Gamma (P_0,P_1,P_2)\equiv
\sigma (P_0,P_1)+\sigma (P_0,P_2)-\sigma (P_1,P_2), \label{a2.3}
\end{equation}
\[
P_0,P_1,P_2\in \Omega
\]
\end{opred}
In the case, when it does not lead to a misunderstanding, the term "scalar
product" will be used instead of the term "scalar $\sigma$-product".
\begin{opred}
\label{d3.1.4}. According to the definition \ref{d3.1.1bd}, the length $\mid
{\bf PQ}\mid $ of the vector ${\bf PQ}$ is the number
\begin{equation}
\mid {\bf PQ}\mid =\sqrt{2\sigma (P,Q)}=\left\{
\begin{array}{c}
\mid \sqrt{({\bf PQ.PQ})}\mid ,\quad ({\bf PQ.PQ})\ge 0 \\
i\mid \sqrt{({\bf PQ.PQ})}\mid ,\quad ({\bf PQ.PQ})<0
\end{array}
\right. \qquad P,Q\in \Omega \label{a2.4}
\end{equation}
\end{opred}
\begin{opred}
\label{d3.1.5}. Vectors ${\bf P}_0{\bf P}_1$, ${\bf P}_0{\bf P}_2$ are
parallel or antiprallel, if the following relations are fulfilled
respectively
\begin{equation}
{\bf P}_0{\bf P}_1\uparrow \uparrow {\bf P}_0{\bf P}_2:\qquad ({\bf P}_0{\bf
P}_1.{\bf P}_0{\bf P}_2)=\mid {\bf P}_0{\bf P}_1\mid \cdot \mid {\bf P}_0
{\bf P}_2\mid \label{a2.5}
\end{equation}
\begin{equation}
{\bf P}_0{\bf P}_1\uparrow \downarrow {\bf P}_0{\bf P}_2:\qquad ({\bf P}_0
{\bf P}_1.{\bf P}_0{\bf P}_2)=-\mid {\bf P}_0{\bf P}_1\mid \cdot \mid {\bf P}
_0{\bf P}_2\mid \label{a2.6}
\end{equation}
\end{opred}
\begin{opred}
\label{d3.1.6}. Vectors ${\bf P}_0{\bf P}_1$, ${\bf P}_0{\bf P}_2$ are
collinear, if they are parallel, or antiparallel.
\begin{equation}
{\bf P}_0{\bf P}_1\parallel {\bf P}_0{\bf P}_2:\qquad ({\bf P}_0{\bf P}_1.
{\bf P}_0{\bf P}_2)^2=\mid {\bf P}_0{\bf P}_1\mid ^2\cdot \mid {\bf P}_0{\bf
P}_2\mid ^2 \label{a2.7}
\end{equation}
\end{opred}
\begin{opred}
\label{d3.1.6b}. The scalar $\sigma $-product $(\overrightarrow{{\cal P}^n}.
\overrightarrow{{\cal Q}^n})$ of $n$th order multivectors $\overrightarrow{
{\cal P}^n}$ and $\overrightarrow{{\cal Q}^n}$, having the common origin $
P_0=Q_0$ is the real number
\begin{equation}
(\overrightarrow{{\cal P}^n}.\overrightarrow{{\cal Q}^n})=\det \Vert \Gamma
(P_0,P_i,Q_k)\Vert ,\,\,\,\,\,\,\,\,\,\,\,i,k=1,2,...n \label{a2.8}
\end{equation}
\end{opred}
\begin{opred}
\label{d3.1.6c}. In accordance with the definition \ref{d3.1.1bd} the length
$|\overrightarrow{{\cal P}^n}|$ of the multivector $\overrightarrow{{\cal P}
^n}$ is the number
\begin{equation}
|\overrightarrow{{\cal P}^n}|=\left\{
\begin{array}{c}
\mid \sqrt{(\overrightarrow{{\cal P}^n}.\overrightarrow{{\cal P}^n})}\mid =|
\sqrt{F_n({\cal P}^n)}|,\quad (\overrightarrow{{\cal P}^n}.\overrightarrow{
{\cal P}^n})\ge 0 \\
i\mid \sqrt{(\overrightarrow{{\cal P}^n}.\overrightarrow{{\cal P}^n})}\mid
=i|\sqrt{F_n({\cal P}^n)}|,\quad (\overrightarrow{{\cal P}^n}.
\overrightarrow{{\cal P}^n})<0
\end{array}
\right. \qquad \overrightarrow{{\cal P}^n}\subset \Omega \label{a2.9}
\end{equation}
where the quantity $F_n({\cal P}^n)$ is defined by the relatons (\ref{a1.6}
)-(\ref{a1.8})
\end{opred}
\begin{opred}
\label{d3.1.5c}. Two $n$th order multivectors $\overrightarrow{{\cal P}^n}$ $
\overrightarrow{{\cal Q}^n}$, having the common origin, are collinear $
\overrightarrow{{\cal P}^n}\parallel \overrightarrow{{\cal Q}^n}$, if
\begin{equation}
(\overrightarrow{{\cal P}^n}.\overrightarrow{{\cal Q}^n})^2=|\overrightarrow{
{\cal P}^n}|^2\cdot |\overrightarrow{{\cal Q}^n}|^2 \label{a2.10}
\end{equation}
\end{opred}
\begin{opred}
\label{d3.1.5e}. Two collinear $n$th order multivectors $\overrightarrow{
{\cal P}^n}$ and $\overrightarrow{{\cal Q}^n}$ are similar oriented $
\overrightarrow{{\cal P}^n}\uparrow \uparrow \overrightarrow{{\cal Q}^n}$
(parallel), if
\begin{equation}
(\overrightarrow{{\cal P}^n}.\overrightarrow{{\cal Q}^n})=|\overrightarrow{
{\cal P}^n}|\cdot |\overrightarrow{{\cal Q}^n}| \label{a2.11}
\end{equation}
They have opposite orientation $\overrightarrow{{\cal P}^n}\uparrow
\downarrow \overrightarrow{{\cal Q}^n}$ (antiparallel), if
\begin{equation}
(\overrightarrow{{\cal P}^n}.\overrightarrow{{\cal Q}^n})=-|\overrightarrow{
{\cal P}^n}|\cdot |\overrightarrow{{\cal Q}^n}| \label{a2.12}
\end{equation}
\end{opred}
{\it Example 2.1.} Let us consider $D$-dimensional point proper Euclidean
space. It may be considered as a metric space $E_D=\{\rho ,{\Bbb R}^D\}$ or
as a $\sigma $-space $E_D=\{\sigma ,{\Bbb R}^D\}$, the world function $
\sigma =\frac 12\rho ^2$ being given by the relations
\begin{equation}
\sigma (P,Q)=\sigma (x,y)={\frac{1}{2}}\sum^{D}_{i,k=1}g_{ik}(x^{i}- y^{i})
(x^{k}- y^{k}), \qquad x,y\in {\Bbb R}^n, \label{a2.13}
\end{equation}
\noindent where $x=\{x^{i}\}$ and $y=\{y^{i}\}$, $(i=1,2,\ldots D)$ are
contravariant coordinates of points $P$ and $Q$ respectively in some
rectilinear coordinate system $K$. Here $g_{ik}$ = const, $(i,k= 1,2,\ldots
D)$ is the metric tensor, $\det ||g_{ik}||\ne 0$. Eigenvalues of the matrix $
g_{ik}$, $i,k= 1,2,\ldots D$ of the metric tensor are positive, and $
\sum^{D}_{i,k=1}g_{ik}x^{i}x^{k}=0$, if and only if $x=0$. The above made
definitions of the vector, its length, scalar product of two vectors and
relations of collinearity agree with the use of these concepts for the
Euclidean space. Indeed, the length of the vector ${\bf PQ}$ in the
Euclidean space is
\begin{equation}
\mid {\bf PQ}\mid = \sqrt{g_{ik}(x^i-y^i)(x^k-y^k)}= \sqrt{2\sigma (P,Q)}
\label{a2.14}
\end{equation}
\noindent that agrees with (\ref{a2.4}).
In the proper Euclidean space according to the cosine theorem for two
vectors ${\bf P}_{0}{\bf P}_{1}$ and ${\bf P}_{0}{\bf P}_{2}$
\begin{equation}
\mid {\bf P}_{1}{\bf P}_{2}\mid ^{2}=\mid {\bf P}_{0}{\bf P}_{2}-{\bf P}_{0}
{\bf P}_{1}\mid ^{2}=\mid {\bf P}_{0}{\bf P}_{2}\mid ^{2}+\mid {\bf P}_{0}
{\bf Q}_{1}\mid ^{2}-2({\bf P}_{0}{\bf P}_{1}.{\bf P}_{0}{\bf P}_{2})
\label{a2.15}
\end{equation}
It follows from this relation
\begin{equation}
({\bf P}_{0}{\bf P}_{1}.{\bf P}_{0}{\bf P}_{2})={\frac{1}{2}}\{\mid {\bf P}
_{0}{\bf P}_{2}\mid ^{2}+\mid {\bf P}_{0}{\bf P}_{1}\mid ^{2}-\mid {\bf P}
_{1}{\bf P}_{2}\mid ^{2}\} \label{a2.16}
\end{equation}
\noindent that agrees with (\ref{a2.3}), if one takes into account (\ref
{a2.4}).
In the proper Euclidean space the vectors ${\bf P}_{0}{\bf P}_{1}$ and ${\bf
P}_{0}{\bf P}_{2}$ are parallel or antiparallel, if cosine of the angle $
\vartheta $ between them is equal respectively to $1$ or $-1$. As far as
\begin{equation}
\cos \vartheta =({\bf P}_{0}{\bf P}_{1}.{\bf P}_{0}{\bf P}_{2}) \mid {\bf P}
_{0}{\bf P}_{2}\mid ^{-1}\cdot\mid {\bf P}_{0}{\bf P}_{1} \mid ^{-1},
\label{a2.17}
\end{equation}
one obtains an accord with the definitions (\ref{a2.5}), (\ref{a2.6}).
In the proper Euclidean space the $n$th order multivector ${\bf m}$ is
defined as an external (skew) product of vectors
\begin{equation}
{\bf m=e}_1\wedge {\bf e}_2\wedge ...\wedge {\bf e}_n\qquad {\bf e}_i={\bf P}
_0{\bf P}_i,\qquad i=1,2,...n \label{a2.18}
\end{equation}
The scalar product of two $n$th order multivectors ${\bf m}$ and ${\bf q}$
\begin{equation}
{\bf q}=\bigwedge\limits_{i=1}^{i=n}{\bf k}_i,\qquad {\bf k}_i={\bf P}_0{\bf
Q}_i{\bf \qquad }i=1,2,...n \label{c.1.12b}
\end{equation}
is defined by means of the relation
\begin{equation}
\left( {\bf m.q}\right) =\det \left\| \left( {\bf e}_i.{\bf k}_l\right)
\right\| =\det \left\| \left( {\bf P}_0{\bf P}_i.{\bf P}_0{\bf Q}_l\right)
\right\| =\det \left\| \Gamma \left( P_0,P_i,Q_l\right) \right\| ,\qquad
i,l=1,2,...n, \label{a2.19}
\end{equation}
that agrees with the relation (\ref{a2.8}).
The difference between the definition \ref{d3.1.2b} of the multivector and
its conventional definition (\ref{a2.18}) consists in that that the first
definition does not use the summation operation and that of multiplication
of vectors by a number which are not defined in $\sigma $-space, whereas the
conventional definition (\ref{a2.18}) refers to the concept of manifold and
linear space, where these operations are defined.
In principle, summation of vectors and multiplication of them by a number
may be defined in $\sigma$-space $V=\{\sigma ,\Omega \}$ as follows. The
vector ${\bf P}_0{\bf R}$ is a sum of vectors ${\bf P}_0{\bf P}_1$ and ${\bf
P}_0{\bf P}_2$, if $\exists R\in\Omega $ such, that
\begin{equation}
\left( {\bf P}_0{\bf R}.{\bf P}_0{\bf Q}\right) =\left( {\bf P}_0{\bf P}_1.
{\bf P}_0{\bf Q}\right) +\left( {\bf P}_0{\bf P}_2.{\bf P}_0{\bf Q}\right)
,\qquad \forall Q\in \Omega . \label{a2.20}
\end{equation}
The vector ${\bf P}_0{\bf R}$ is a result of multiplication of ${\bf P}_0
{\bf P}$ by the real number $a$:\quad ${\bf P}_0{\bf R}=a{\bf P}_0{\bf P}$,
if $\exists R\in \Omega $ such, that
\begin{equation}
\left( {\bf P}_0{\bf R}.{\bf P}_0{\bf Q}\right) =a\left( {\bf P}_0{\bf P}.
{\bf P}_0{\bf Q}\right) ,\qquad \forall Q\in \Omega \label{a2.21}
\end{equation}
But such definitions are not effective, because in general case there is no
point $R$, satisfying the relations (\ref{a2.20}), (\ref{a2.21}). In the
case of the proper Euclidean space, when the points $R$, satisfying (\ref
{a2.20}), (\ref{a2.21}), exist, the operation of summation of vectors (\ref
{a2.20}) and that of multiplication of the vector by a number (\ref{a2.21})
coincide with the conventional definition of these operations in the
Euclidean space.
Operation of permutation of the multivector points can be effctively defined
in the $\sigma $-space. Let us consider two $n$th order multivectors $
\overrightarrow{{\cal P}^n}=\overrightarrow{P_0P_1P_2...P_n}$ and $
\overrightarrow{{\cal P}_{(1\leftrightarrow 2)}^n}= \overrightarrow{
P_0P_2P_1P_3P_4...P_n}$, $(n\ge 2)$, which differ by the order of points $
P_1 $ ¨ $P_2$.
\begin{equation}
( \overrightarrow{{\cal P}^n}.\overrightarrow{{\cal Q}^n} ) =\det \parallel
\Gamma ( P_0,P_{i},Q_k)\parallel , \qquad i,k=1,2,...n, \qquad \forall {\cal
Q}^n\subset\Omega , \label{a2.22}
\end{equation}
\begin{equation}
( \overrightarrow{{\cal P}_{(1\leftrightarrow 2)}^n}.\overrightarrow{{\cal Q}
^n}) =\det \parallel \Gamma ( P_0,P_{i}^{\prime },Q_k) \parallel ,\qquad
i,k=1,2,...n, \qquad n\ge 2, \qquad \forall {\cal Q}^n\subset\Omega .
\label{a2.23}
\end{equation}
Here the ordered set of points $\{P_i^{\prime }\},\,\,\,\,\,\,\,
(i=1,2,...n) $ is obtained from the ordered set of points $
\{P_i\},\,\,\,\,\,\,\,(i=1,2,...n)$ by permutation of points $P_1$ ¨ $P_2$.
This means that the determinant (\ref{a2.23}) is obtained from the
determinant (\ref{a2.22}) by permutation of the first and second rows. Then
one obtains
\begin{equation}
( \overrightarrow{{\cal P}^n}.\overrightarrow{{\cal Q}^n}) =-(
\overrightarrow{{\cal P}_{(1\leftrightarrow 2)}^n}.\overrightarrow{{\cal Q}^n
}), \qquad n\ge 2,\qquad \forall {\cal Q}^n\subset\Omega . \label{a2.24}
\end{equation}
As far as $\overrightarrow{{\cal Q}^n}$ is an arbitrary multivector, the
relation (\ref{a2.24}) may be written in the form
\begin{equation}
\overrightarrow{{\cal P}^n} =-\overrightarrow{{\cal P}_{(1\leftrightarrow
2)}^n}, \qquad n\ge 2. \label{a2.25}
\end{equation}
It may be interpreted in the sense, that permutation of any two points $P_i$
and $P_k\,\,\,\,\,\,i,k=1,2,...n$, $(n \ge 2)$ (except for the origin $P_0$
) at the multivector $\overrightarrow{{\cal P}^n}$ leads to a change of its
sign. The negative sign of the multivector means by definition that
\begin{equation}
(- \overrightarrow{{\cal P}^n}.\overrightarrow{{\cal Q}^n}) =-(
\overrightarrow{{\cal P}^n}.\overrightarrow{{\cal Q}^n}) , \qquad \forall
{\cal Q}^n\subset\Omega . \label{a2.26}
\end{equation}
Permutating the points $P_0$ and $P_1\,\,\,\,\,$ at the multivector $
\overrightarrow{{\cal P}^n}$, $(n\ge 2)$, one turns it in multivector $
\overrightarrow{{\cal P}_{(0\leftrightarrow 1)}^n}$, having the origin at
the point $P_1$. Strictly, one cannot compare multivectors $\overrightarrow{
{\cal P}^n}$ and $\overrightarrow{{\cal P}_{(0\leftrightarrow 1)}^n}$ at the
point $P_0$. But they have other common points $P_2,P_3,...P_n$, and one may
compare them at these points, forming scalar product with the multivector $
\overrightarrow{{\cal Q}^n}$, having the origin, for instance, at the point $
P_2$ $(Q_2=P_2)$
\begin{equation}
( \overrightarrow{{\cal P}^n}.\overrightarrow{{\cal Q}^n}) _{P_2}=\det
\left\| \Gamma \left( P_2,P_i,Q_k\right) \right\|
,\,\,\,\,\,\,\,\,\,\,\,i,k=0,1,3,4,...n,\qquad n\ge 2, \label{a2.27}
\end{equation}
\begin{equation}
( \overrightarrow{{\cal P}_{(0\leftrightarrow 1)}^n}.\overrightarrow{{\cal Q}
^n}) _{P_2}=\det\parallel\Gamma ( P_2,P_{i}^{\prime }, Q_k)\parallel,\qquad
i,k=0,1,3,4,...n,\qquad n\ge 2, \label{a2.28}
\end{equation}
where $P_0^{\prime }=P_1$,\,\, $P_1^{\prime }=P_0,$ $P_i^{\prime}=P_i,\,\,\,
\,\,\,i=3,4,...n$, and index $P_2$ shows, that the point $P_2$ is considered
as the origin of the multivector $\overrightarrow{{\cal Q}^n}$. Comparison
of rhs of (\ref{a2.27}) and (\ref{a2.28}) shows that
\[
( \overrightarrow{{\cal P}^n}.\overrightarrow{{\cal Q}^n}) _{P_2}=-(
\overrightarrow{{\cal P}_{(0\leftrightarrow 1)}^n}.\overrightarrow{{\cal Q}^n
}) _{P_2},\qquad n\ge 2.
\]
The same result is obained, choosing any point of $P_i\,\,\,\,\,\,\,\,
\,i=3,4,...n$ as an origin. It means that the relation (\ref{a2.25}) is
valid for permutation of any two points of the multivector $\overrightarrow{
{\cal P}^n}$, and one may write
\begin{equation}
\overrightarrow{{\cal P}_{(i\leftrightarrow k)}^n}=-\overrightarrow{{\cal P}
^n},\qquad i,k=0,1,...n,\qquad i\neq k,\qquad n\ge 2. \label{a2.29}
\end{equation}
Thus, a change of the $n$th order multivector sign $(n\ge 2)$
(multiplication by the number $a=-1$) may be always defined as an odd
permutation of points.
For the vector (the first order multivector) the multiplication (\ref{a2.21}
) by the number $a=-1$ is defined directly as a permutation of the origin
and the end of the vector by means of the relatons
\[
-{\bf P}_0{\bf P}_1={\bf P}_1{\bf P}_0,
\]
It means by definition that
\[
\left( -{\bf P}_0{\bf P}_1.{\bf P}_0{\bf Q}\right) =-\left( {\bf P}_0{\bf P}
_1.{\bf P}_0{\bf Q}\right)= -\sigma \left( P_0,P_1\right) -\sigma \left(
P_0,Q\right) +\sigma \left( P_1,Q\right) ,\qquad \forall Q\in \Omega ,
\]
\[
\left( -{\bf P}_0{\bf P}_1.{\bf P}_1{\bf Q}\right) =\left( {\bf P}_1{\bf P}
_0.{\bf P}_1{\bf Q}\right)= \sigma \left( P_1,P_0\right) +\sigma \left(
P_1,Q\right) -\sigma \left( P_0,Q\right) ,\qquad \forall Q\in \Omega .
\]
Thus multiplication of any multivector by the number $a=\pm 1$ may be always
defined in $\sigma$-space as a result of permutation of points, forming the
multivector.
In the properEuclidean space, where the multivector is defined in the form
\begin{equation}
\overrightarrow{{\cal P}^n}=\bigwedge_{i=1}^{i=n}{\bf P}_0{\bf P}_i ,
\label{a2.30}
\end{equation}
it is antisymmetric with respect to permutation of any two indices $
i,k=0,1,...n$,\quad $i\neq k$. For indices $i,k=1,2,...n,$ it follows from
the external product properties.
For permutation of points $P_0\leftrightarrow P_1$ one has
\[
\overrightarrow{{\cal P}_{(0\leftrightarrow 1)}^n}={\bf P}_1{\bf P}
_0\bigwedge_{i=2}^{i=n}{\bf P}_1{\bf P}_i=-{\bf P}_0{\bf P}
_1\bigwedge_{i=2}^{i=n}\left( {\bf P}_0{\bf P}_i-{\bf P}_0{\bf P}_1\right) =
\]
\begin{equation}
-{\bf P}_0{\bf P}_1\bigwedge_{i=2}^{i=n}{\bf P}_0{\bf P}_i=-\overrightarrow{
{\cal P}^n} \label{a2.31}
\end{equation}
A similar result is obtained for permutation of points $P_0\leftrightarrow
P_i,\,\,\,\,\,\,\,i=1,2,...n$. Thus, the multivector in $\sigma $-space is
the geometrical object antisymmetric with respect to permutation of any two
points.
\begin{opred}
\label{d3.1.7}. $n+1$ points ${\cal P}^n$ , $P_i\in \Omega \quad (i=0,1,..n)$
form $(n+1)$-point $\sigma $-basis of the tube in $\sigma $-space, if the
multivector $\overrightarrow{{\cal P}^n}$ has nonvanishing length
\begin{equation}
|\overrightarrow{{\cal P}^n}|^2\equiv F_n({\cal P}^n)\ne 0. \label{a2.32}
\end{equation}
\end{opred}
Let us illustrate this definition of the tube $\sigma $-basis in the example
of the $D$-dimensional proper Euclidean space. Let $n$ vectors ${\bf e}_{i}=
{\bf P}_{0}{\bf P}_{i}$, $i=1,2,\ldots n$ be given in $D$-dimensional proper
Euclidean space $(n\le D)$. In this case (\ref{a1.6}) is the Gram's
determinant
\begin{equation}
F_{n}({\cal P}^{n})=\det \parallel ({\bf e}_{i}.{\bf e}_{k})\parallel
=(n!S_{n}({\cal P}^{n}))^{2},\qquad i,k=1,2,\ldots n \label{a2.33}
\end{equation}
\noindent and $S_{n}$ is the volume of $(n+1)$-edr with vertices at points $
{\cal P}^{n}$. Vanishing of this determinant is the necessary and sufficient
condition of linear independence of vectors ${\bf e}_{i},\quad (i=1,2,\ldots
n)$ in the proper Euclidean space.
If the condition (\ref{a2.32}) is fulfilled, $n$ vectors ${\bf e}_{i}$ are
linear independent and may serve as a basis in the $n$-dimensional plane $
{\cal L}({\cal P}^{n})$, passing through points ${\cal P}^{n}$. In
particular, if one uses the expression (\ref{a2.13}) for calculation of the
scalar product of the vectors ${\bf e}_{i}={\bf P}_{0}{\bf P}_{i}$,\quad $
(i=1,2,\ldots D)$, considering the $(D+1)$-point tube $\sigma$-basis ${\cal P
}^{D}$, as the system of coordinate vectors, one obtains by means of (\ref
{a2.3}) ¨ (\ref{a2.13}) that
\begin{equation}
({\bf e}_{i}.{\bf e}_{k})=({\bf P}_{0}{\bf P}_{i}.{\bf P}_{0}{\bf P}_{k})
=g_{ik}({\cal P}^{D})=\Gamma (P_{0},P_{i},P_{k}),\qquad i,k=1,2,\ldots D
\label{a2.34}
\end{equation}
\begin{opred}
\label{d3.1.8}. The $n$th order tube ${\cal T}({\cal P}^n)$, (n=0,1,\ldots
), formed by $(n+1)$-point tube $\sigma $-basis ${\cal P}^n\subset \Omega $
(or by the $n$th order multivector $\overrightarrow{{\cal P}^n}\subset
\Omega $), is the set of points $P\in \Omega $
\begin{equation}
{\cal T}({\cal P}^n)\equiv {\cal T}_{{\cal P}^n}=\{P\mid F_{n+1}(P,{\cal P}
^n)=0\},\qquad F_n({\cal P}^n)\neq 0. \label{a2.35}
\end{equation}
\end{opred}
The relation (\ref{a2.35}) may be written also in terms of multivector $
\overrightarrow{{\cal P}^n}$
\begin{equation}
{\cal T}({\cal P}^{n}) = \left\{P_{n+1}\left| |\overrightarrow{{\cal P}^{n+1}
} |=0\right.\right\}, \qquad | \overrightarrow{{\cal P}^{n}} |\neq 0.
\label{a2.36}
\end{equation}
The tube ${\cal T}({\cal P}^{n})$ is the $n$th order natural geometrical
object (NGO), i.e the set of points, determined by geometry and parameters: $
n+1$ points ${\cal P}^{n}$. The set of all possible NGOs is a set of $\sigma$
-immanent geometric objects on the set $\Omega$. Each NGO contains at least
basic points ${\cal P}^{n}$.
\begin{opred}
\label{d3.1.9}. Section ${\cal S}_{n;P}$ of the tube ${\cal T}({\cal P}^n)$
at the point $P\in {\cal T}({\cal P}^n)$ is the set ${\cal S}_{n;P}({\cal T}(
{\cal P}^n))$ of points, belonging to the tube ${\cal T}({\cal P}^n)$
\begin{equation}
{\cal S}_{n;P}({\cal T}({\cal P}^n))=\{P^{\prime }\mid
\bigwedge_{l=0}^{l=n}\sigma (P_l,P^{\prime })=\sigma (P_l,P)\},\qquad
P,P^{\prime }\in {\cal T}({\cal P}^n). \label{a2.38}
\end{equation}
\end{opred}
In the proper Euclidean space the $n$th order tube is the $n$-dimensional
plane, containing points ${\cal P}^{n}$ , and its sectiom ${\cal S}_{n;P}(
{\cal T}({\cal P}^{n}))$ at the point $P$ consists of one point $P$.
The zeroth and first order tubes are the most interesting and important. For
$F_{1}$ one obtains from (\ref{a1.6}) and (\ref{a2.3})
\[
F_{1}(P_{0},P_{1}) = 2\sigma (P_{0},P_{1})
\]
Then
\begin{equation}
{\cal T}(P_{0})\equiv {\cal T}_{P_{0}}=\{P\mid\sigma (P_{0},P)=0\},
\label{a2.39}
\end{equation}
In the properEuclidean space the zeroth order tube ${\cal T}_{P_{0}}
=\{P_{0}\}$ consists of one point $P_{0}$, and its section ${\cal S}
_{0;P_{0}}({\cal T}_{p_{0}}) =\{P_{0}\}$ consists of one point $P_{0}$ also.
But in the pseudo-Euclidean space (for instance, in the space-time of the
special relativity) ${\cal T}_{P_{0}}$ is the light cone with the vertex at
the point $P_{0}$, and its section
\[
{\cal S}_{0;P}({\cal T}(P_{0}))=\{P^{\prime}\mid \sigma
(P_{0},P^{\prime})=0\wedge \sigma (P_{0},P^{\prime})=\sigma (P_{0},P)\}=
{\cal T}_{P_{0}}
\]
\noindent coincides with the light cone.
Describing the first order tubes, it is convenient to use the circumstance
that the function $F_{2}({\cal P}^{2})$ can be presented in the form of a
product
\begin{equation}
F_{2}(P_{0},P_{1},P_{2})=S_{+}(P_{0},P_{1},P_{2})S_{2}(P_{0},P_{1},P_{2})S_{2}(P_{1},P_{2},P_{0})S_{2}(P_{2},P_{0},P_{1})
\label{a2.40}
\end{equation}
\noindent where
\begin{equation}
S_{+}(P_{0},P_{1},P_{2})\equiv S(P_{0},P_{1})+S(P_{1},P_{2})+S(P_{0},P_{2})
\label{c.1.22}
\end{equation}
\begin{equation}
S_{2}(P_{0},P_{1},P_{2})\equiv S(P_{0},P_{1})+S(P_{1},P_{2})-S(P_{0},P_{2})
\label{c.1.23}
\end{equation}
Here $S=\sqrt{2\sigma }$. $S_{+}$ vanishes, if and only if any term of the
sum (\ref{c.1.22}) vanishes. Then no two points form $\sigma$-basis, and the
tube is not defined. The tube ${\cal T}({\cal P}^{2})$ may be presented as
consisting of parts, and any multiplier in (\ref{a2.40}) (except for $S_{+}$
) is responsible for one of these parts.
Let us set
\begin{equation}
{\cal T}_{[P_{0}P_{1}]}={\cal T}_{[P_{1}P_{0}]}=\{P\mid
S_{2}(P_{0},P,P_{1})=0\} \label{a2.41}
\end{equation}
\begin{equation}
{\cal T}_{P_{0}[P_{1}}={\cal T}_{P_{1}]P_{0}}=\{P\mid
S_{2}(P_{0},P_{1},P)=0\} \label{c.1.25}
\end{equation}
Let us refer to ${\cal T}_{[P_{0}P_{1}]}$ as the tube segment between the
points $P_{0}$, $P_{1}$, and to ${\cal T}_{P_{0}[P_{1}}$ as the tube ray
outgoing from $P_{1}$ towards th point $P_{0}$.
It is evident from (\ref{a2.40}), (\ref{a2.41}), (\ref{c.1.25}) that
\begin{equation}
{\cal T}_{P_{0}P_{1}}={\cal T}_{P_{0}]P_{1}}\bigcup {\cal T}_{[P_{0}P_{1}]}
\bigcup {\cal T}_{P_{0}[P_{1}} \label{a2.42}
\end{equation}
As far as the relation (\ref{a2.7}) is equivalent to the equation $F_{2}(
{\cal P}^{2})=F_{2}(P_{2},{\cal P}^{1})=0$, the first order tube ${\cal T}
_{P_{0}P_{1}}$ may be defined also as a set of such points $P$ that ${\bf P}
_{0}{\bf P}\parallel {\bf P}_{0}{\bf P}_{1}$.
\begin{equation}
{\cal T}_{P_{0}P_{1}}=\{P\mid {\bf P}_{0}{\bf P}_{1}\parallel {\bf P}_{0}
{\bf P\}} \label{a2.43}
\end{equation}
For the tube rays one can use the definitions
\begin{equation}
{\cal T}_{P_{0}[P_{1}}=\{P\mid {\bf P}_{1}{\bf P}\uparrow\downarrow {\bf P}
_{1}{\bf P}_{0}\} \label{a2.44}
\end{equation}
\begin{equation}
{\cal T}_{[P_{0}P_{1}}=\{P\mid {\bf P}_{0}{\bf P}\uparrow\uparrow {\bf P}_{0}
{\bf P}_{1}\} \label{a2.45}
\end{equation}
\begin{opred}
\label{d3.1.8 }. The oriented segment $\overrightarrow{{\cal T}_{[P_0P_1]}}$
of the first order tube, formed by by the vector $\overrightarrow{P_0P_1}
\subset \Omega $ of unvinishing length is a totality $\{\overrightarrow{
P_0P_1},{\cal T}_{[P_0P_1]}\}$ of the vector $\overrightarrow{P_0P_1}$ and
segment ${\cal T}_{[P_0P_1]}$, formed by this vector. The length of the
oriented segment $\overrightarrow{{\cal T}_{[P_0P_1]}}$ is the quantity
\begin{equation}
|\overrightarrow{{\cal T}_{[P_0P_1]}}|=|\overrightarrow{P_0P_1}|=\sqrt{
2\sigma (P_0,P_1)}. \label{a2.46}
\end{equation}
The scalar $\sigma $-product $(\overrightarrow{{\cal T}_{[P_0P_1]}}.
\overrightarrow{P_0Q})$ of the oriented segment $\overrightarrow{{\cal T}
_{[P_0P_1]}}$ and vector $\overrightarrow{P_0Q}$ is the number
\begin{equation}
(\overrightarrow{{\cal T}_{[P_0P_1]}}.\overrightarrow{P_0Q})=(
\overrightarrow{P_0P_1}.\overrightarrow{P_0Q})=\sigma (P_0,P_1)+\sigma
(P_0,Q)-\sigma (P_1,Q),\qquad P_0,P_1,P,Q\in \Omega . \label{a2.47}
\end{equation}
The scalar $\sigma $-product $(\overrightarrow{{\cal T}_{[P_0P_1]}}.
\overrightarrow{P_1Q})$ of the oriented segment $\overrightarrow{{\cal T}
_{[P_0P_1]}}$ and vector $\overrightarrow{P_1Q}$ is the number
\begin{equation}
(\overrightarrow{{\cal T}_{[P_0P_1]}}.\overrightarrow{P_1Q})=-(
\overrightarrow{P_1P_0}.\overrightarrow{P_1Q})=-\sigma (P_0,P_1)-\sigma
(P_1,Q)+\sigma (P_0,Q),\qquad P_0,P_1,P,Q\in \Omega . \label{a2.48}
\end{equation}
\end{opred}
In other words, $\overrightarrow{{\cal T}_{[P_0P_1]}}=-\overrightarrow{{\cal
T}_{[P_1P_0]}}$.
Describing in terms of differential geometry, the geodesic in $D$
-dimensional Riemannian space is considered as {\sl special kind of a curve,
having the following properties}.
\noindent (i) {\sl Extremality}. The distance $(2\sigma )^{1/2}$, measured
along the geodesic between two points is the shortest (extremal) as compared
with the distance measured along other curves.
\noindent (ii) {\sl Definiteness}. Any two different points of the geodesic
determine uniquelly the geodesic, passing through these points.
\noindent (iii) {\sl Minimality of the section} (one-dimensionality). Any
section of the geodesic consists of one point.
At the conventional approach the property (ii) is a corollary of the
property (i) (for rather small regions of the space), but the property (iii)
is the property of any curve (but not only of geodesic).
In T-geometry the geodesic is considered as a special kind of the tube,
degenerating into a line. Then the properties (ii) and (iii) are supposed to
be fulfilled. The property (i) is not defined, because the concept of line
is not defined.
Let us try to determine the geodesic as the tube, having the properties of
definiteness and of the section minimality at the same time.
\begin{opred}
\label{d3.1.10}. The tube ${\cal T}({\cal P}^n)$ has the definiteness
property, if for any $(n+1)$-point tube $\sigma $-basis ${\cal Q}^n\subset
{\cal T}({\cal P}^n)$ (or for any multivector $\overrightarrow{{\cal Q}^n}
\subset {\cal T}({\cal P}^n)$ of unvanishing length) the following condition
is fulfilled
\begin{equation}
{\cal T}({\cal Q}^n)={\cal T}({\cal P}^n) \label{a2.49}
\end{equation}
\end{opred}
\begin{opred}
\label{d3.1.11}. The tube ${\cal T}({\cal P}^n)$ has the minimality section
property, if $\forall P\in {\cal T}({\cal P}^n)$
\begin{equation}
{\cal S}_{n;P}({\cal T}({\cal P}^n))=\{P\},\qquad \forall P\in {\cal P}^n
\label{a2.50}
\end{equation}
\end{opred}
\begin{opred}
\label{d3.1.12}. $\sigma $-space is extremal on the tube ${\cal T}({\cal P}
^n)$, if for ${\cal T}({\cal P}^n)$ the conditions of definiteness and
section minimality are fulfilled.
\end{opred}
\begin{opred}
\label{d3.1.13}. $\sigma $-space is extremal on the set ${\cal T}$ of tubes $
{\cal T}({\cal P}^n)$, if it is extremal on any tube of the set ${\cal T}$.
\end{opred}
\begin{opred}
\label{d3.1.14}. $\sigma $-space is extremal in the $n$th order, if it is
extremal on all $n$th order tubes ${\cal T}({\cal P}^n)$.
\end{opred}
\begin{opred}
\label{d3.1.15}. The tube ${\cal T}({\cal P}^n)$ is the geodesic tube ${\cal
L}({\cal P}^n)$, if the $\sigma $-space is extremal on the tube ${\cal T}(
{\cal P}^n)$.
\end{opred}
\section{Euclidean space as a special case of $\sigma $-space.}
\begin{opred}
\label{d3.2.1}. $n$-dimensional Euclidean space $E_n$ is a set ${\Bbb R}^n$
of all ordered sets $x=\{x_1,x_2,\ldots x_n\}$ of $n$ real numbers on which
for $\forall x\in {\Bbb R}^n$, $\forall y\in {\Bbb R}^n$ is given the real
function $\sigma $:
\begin{equation}
\sigma (x,y)={\frac 12}\sum_{i,k=1}^ng^{ik}(x_i-y_i)(x_k-y_k),\qquad g^{ik}=
\hbox{const},\qquad i,k=1,2,\ldots n \label{a3.1}
\end{equation}
\begin{equation}
\det \parallel g_{ik}\parallel =(\det \parallel g^{ik}\parallel )^{-1}\neq 0
\label{a3.2}
\end{equation}
\end{opred}
The function $\sigma $ is called the world function or simply $\sigma $
-function. $n$-dimensional Euclidean space $E_n$ is at the same time a $
\sigma$-space $E_n=\{\sigma , {\Bbb R}^n\}$.
{\it Remark.} The given definition is equivalent to the definition of $n$
-dimensional Euclidean space $E_{n}$ as $n$-dimensional linear space ${\Bbb R
}^n$ of vectors $x=\{x_1,x_2,\ldots ,x_n\}\in {\Bbb R}^n$ with given on it
the scalar product $(x.y)$ of vectors $x,y\in {\Bbb R}^n$
\begin{equation}
(x.y)=\sum^{n}_{i,k=1}g^{ik}x_{i}y_{k}=\sigma(0,x)+\sigma(0,y)-\sigma(x,y),
\qquad g^{ik}=\hbox{const},\qquad i,k=1,2,\ldots n \label{a3.3}
\end{equation}
where $\sigma$ is given by the relation (\ref{a3.1}).
The Euclidean space $E_n=\{\sigma ,\Omega \}$, $({\Omega ={\Bbb R}}^n)$,
considered as $\sigma$-space, have the following properties
\begin{equation}
\exists {\cal P}^n\subset\Omega,\qquad F_n({\cal P}^n)\ne 0,\qquad
F_{n+1}(\Omega ^{n+2})=0, \label{a3.4}
\end{equation}
\[
\sigma (P,Q)={\frac{1}{2}}\sum^{n}_{i,k=1}g^{ik}({\cal P}^{n}) [\Gamma
(P_{0},P_{i},P)-\Gamma (P_{0},P_{i},Q)]
\]
\begin{equation}
\times [\Gamma (P_{0},P_{k},P)-\Gamma (P_{0},P_{k},Q)], \qquad \forall
P,Q\in\Omega \label{a3.5}
\end{equation}
\begin{equation}
\Gamma (P_{0},P,Q)=\sum^{n}_{i,k=1}g^{ik}({\cal P}^{n})\Gamma
(P_{0},P_{i},P) \Gamma (P_{0},P_{k},Q),\qquad \forall P,Q\in\Omega ,
\label{a3.6}
\end{equation}
where ${\cal P}^n$ is some $(n+1)$-point tube $\sigma$-basis in $\Omega =
{\Bbb R}^n$ in the sense of the definition (\ref{a2.32}), i.e. $F_n({\cal P}
^n)\ne 0$, and the quantities $\Gamma (P_{0},P_{k},P)$ are defined by the
relations (\ref{a2.3}). $(n+1)$-point tube $\sigma$-basis ${\cal P}^n$
corresponds to the basis of $n$ vectors
\begin{equation}
{\bf e}_i={\bf P}_{0}{\bf P}_{i},\qquad P_i\in{\cal P}^n, \qquad i
=1,2,\ldots n \label{a3.7}
\end{equation}
and
\begin{equation}
x_{i}= x_{i}(P) = ({\bf P}_{0}{\bf P} .{\bf e}_{i})=\Gamma (P_{0},P,P_{i}),
\qquad i = 1,2,\ldots n,\qquad\forall P\in\Omega \label{a3.8}
\end{equation}
are covariant coordinates of the vector ${\bf P}_{0}{\bf P}$ in this basis.
The quantities
\begin{equation}
g_{ik}= g_{ik}({\cal P}^{n}) = ({\bf e}_{i}.{\bf e}_{k})= \Gamma
(P_{0},P_{i},P_{k}),\qquad i,k=1,2,\ldots n \label{a3.9}
\end{equation}
are covariant components of the metric tensor in this basis ${\cal P}^n$. As
far as ${\cal P}^n$ is the tube $\sigma$-basis, then according to (\ref
{a2.32}), (\ref{a2.33}) the following condition is fulfilled
\begin{equation}
F_n({\cal P}^n)\equiv \det \parallel g_{ik}({\cal P}^n)\parallel \neq 0,
\qquad i,k=1,2,\ldots n \label{a3.10}
\end{equation}
and one can determine the contravariant components $g^{ik}=g^{ik} ({\cal P}
^n)$ of the metric tensor by means of the relation
\begin{equation}
\sum^{n}_{k=1}g_{ik}({\cal P}^{n})g^{kl}({\cal P}^{n}) = \delta ^{l}_{i},
\qquad i,l=1,2,\ldots n \label{a3.11}
\end{equation}
Conditions (\ref{a3.5}) and (\ref{a3.6}) are equivalent, as it follows from (
\ref{a2.3}).
\begin{opred}
\label{d3.2.4}. $\sigma $-space $V=\{\sigma ,\Omega \}$ have the structure
of $n$-dimensional Euclidean space on $\Omega $, if there exists such a $
(n+1)$-point tube $\sigma $-basis ${\cal P}^n\subset \Omega $, that $\forall
P,$ $\forall Q\in \Omega $ the condition (\ref{a3.6}) is fulfilled.
\end{opred}
$\sigma $-space $V$, having the structure of $n$-dimensional Eucliden space
may be not Euclidean, because one-to-one correspondence between the points $
P\in \Omega $ and their coordinates $x\in {\Bbb R}^{n}$ may not exist. For
instance, two different points $P$ and $P^\prime$ may have similar
coordinates and be mapped on one point $x$ of the Euclidean space $
E_n=\{\sigma ,{\Bbb R}^n\}$.
Finally, the third property of the Euclidean space $E_n=\{\sigma , {\Bbb R}
^n\}$ is formulated as follows. The relation
\begin{equation}
\Gamma (P_0,P_i,P)=x_i, \qquad x_i\in {\Bbb R},\qquad i=1,2,\ldots n,
\label{a3.12}
\end{equation}
considered as equations for determination of $P\in\Omega ={\Bbb R}^n$,
always have one and only one solution.
Let us note that all three conditions are written in $\sigma $-immanent
form. They are necessary properties of the Euclidean space. In this
connection one can put the question whether these conditions are also
sufficient conditions for the $\sigma $-space $\{\sigma ,\Omega \}$ were the
Euclidean space. The following theorem answers this question.
\begin{theorem}
For the $\sigma $-space $\{\sigma ,\Omega \}$ were $n$-dimensional Euclidean
space, It is necessary and sufficient that the conditions (\ref{a3.4}), (\ref
{a3.5}) and (\ref{a3.12}) be fulfilled. \label{c2}
\end{theorem}
{\it Proof.} {\it Necessity} of conditions (\ref{a3.4}), (\ref{a3.5}) and (
\ref{a3.12}) is tested by the direct substitution of world function $\sigma$
for $n$-dimensional Euclidean space $E_n=\{\sigma , {\Bbb R}^n\}$.
{\it Sufficiency}. Let ${\cal P}^n$ be some $(n+1)$-point tube $\sigma$
-basis in $\sigma $-space $\{\sigma ,\Omega \}$ and $P,Q\in\Omega$ be two
arbitrary points. Let us introduce their covariant coordinates in ${\cal P}
^n $ by means of the relations of type (\ref{a3.8})
\begin{equation}
x_{i}= \Gamma (P_{0},P_{i},P),\qquad y_{i}= \Gamma (P_{0},P_{i},Q), \qquad i
= 1,2,\ldots n,\qquad\forall P,Q\in\Omega \label{a3.13}
\end{equation}
Then the relation (\ref{a3.6}) is rewritten in the form
\begin{equation}
(x.y)=\sum^{n}_{i,k=1}g^{ik}x_{i}y_{k}, \qquad g^{ik}=g^{ik}({\cal P}^{n})=
\hbox{const},\qquad i,k=1,2,\ldots n \label{a3.14}
\end{equation}
In virtue of the condition (\ref{a3.12}) any point $P\in\Omega$ corresponds
to one and only one point $x\in {\Bbb R}^n$ and vice versa. In other words,
the $\sigma $-space $\{\sigma ,\Omega \}$ is isometric to $n$-dimensional
Euclidean space $E_n=\{\sigma , {\Bbb R}^n\}$.
{\it Corollary of the theorem.} $n$-dimensional Euclidean space and all its
properties can be described $\sigma$-immanently (i.e. in terms of the world
function). In other words, the T-geomtry is rich and pithy enough to contain
Euclidean geometry as a special case, when the world function is restricted
by $\sigma$-immanent relation (\ref{a3.5}), or by equivalent relation (\ref
{a3.6}). The Riemanian geometry can be presented in the $\sigma$-immanent
form \cite{R90} also. This may be interpreted in the sense that T-geometry
contains the Riemannian geometry as a special case. T-geometry is
informative enough to contain other geometries. Apparently, it is rather
difficult to construct a geometry which would not be contained in
T-geometry. The fact is that that practically any geometry may be considered
as a result of a deformation (variation of the world function) of $\sigma$
-subspace of the Euclidean space of rather high dimensionality. At such a
variation the pithiness of geometry does not reduce, because the number of
tubes does not reduce. This number may only increase under deformation,
because any tube of the Euclidean space, determined by many $\sigma$-bases,
is splitted, in general, to several different tubes.
Pithiness of geometry (i.e. the number of geometric objects, suppositions
and theorems) depends not only on axioms of the geometry, it depends also on
development of the mathematical technique of the geometry. Conventionally
the metric geometry is considered as the geometry which is less pithy as
compared with the Euclidean geometry. One connects usually the pithiness of
the metric geometry with constraints (\ref{a1.3}), (\ref{a1.4}), which are
essential for construction of the shortest and geometric objects, connected
with it. Essentially, the pithiness of the metric geometry is connected with
its mathematical technique. Using more effective mathematical technique,
connected with the classification (\ref{a1.4}), the pithiness of metric
geometry increases even under removing constraints (\ref{a1.3}), (\ref{a1.4}
) on metric.
T-geometry has such a dignity as insensitivity of its mathematical technique
to continuity of the set, where the geometry is given. For instance, let us
take 100 points ${\cal P}^{99}$ of the three-dimensional Euclidean space and
try to study geometry of this set of points. Using usual way, one should
consider Euclidean space on the set ${\Bbb R}^3$, introduce a coordinate
system, remove all points except for ${\cal P}^{99}$ and begin to study the
way of embedding the set ${\cal P}^{99}$ in the Euclidean space, starting
from coordinates of its points. From point of view of T-geometry one should
study the set ${\cal P}^{99}$, imposing the constraint (\ref{a3.5}) on
metric and removing (\ref{a3.12}). Thus, approach of T-geometry appears to
be local in the sense that the geometry of the set ${\cal P}^{99}$ is
studied, but not the way of embedding the set in the Euclidean space.
Giving up of constraints (\ref{a3.12}) leads to a violation of the mapping $
\Omega \to {\Bbb R}^n $ reversibility. In particular, it is possible such a
case, when the $\sigma $-space $\{\sigma ,\Omega \}$ appears to be a $\sigma
$-subspace of the Euclidean space $E_n$.
\begin{opred}
\label{d3.2.2}. The Euclidean $\sigma $-space $E^{\prime }=\{\sigma ,\Omega
^{\prime }\}$ is the $\sigma $-space which can be isometrically embedded in
the Euclidean space. $n$-dimensional Euclidean $\sigma $-space $E_n^{\prime
}=\{\sigma ,\Omega ^{\prime }\}$ is $\sigma $-space which can be
isometrically embedded in $n$-dimensional Euclidean space $E_n=\{\sigma ,
{\Bbb R}^n\}$, but cannot be isometrically embedded in $(n-1)$-dimensional
Euclidean space $E_{n-1}=\{\sigma ,{\Bbb R}^{n-1}\}$.
\end{opred}
$n$-dimensional Euclidean $\sigma$-space is a $\sigma$-subspace of $n$
-dimensional Euclidean space $E_n=\{\sigma , {\Bbb R}^n\}$.
From the tube definition (\ref{a2.35}) and the condition (\ref{a3.4}) it
follows that $n$-dimensio\-nal Euclidean $\sigma$-space is the $n$th order
tube ${\cal T}({\cal P}^n)=\Omega$, generated by any $(n+1)$-point tube $
\sigma$-basis ${\cal P}^n\subset\Omega$, the condition of the tube section
minimality (\ref{a2.50}) being fulfilled. Then the following theorem takes
place.
\begin{theorem}
\label{t3.2.2}. Let ${\cal P}^n$ be $(n+1)$-point tube $\sigma $-basis in
the $\sigma $-space $V\{\sigma ,\Omega \}$. For the tube ${\cal T}({\cal P}
^n)$ be $n$-dimensional Euclidean $\sigma $-space it is necessary and
sufficient, that
\noindent (1)\quad $\sigma $-space ${\cal T}({\cal P}^n)$ have the structure
of $n$-dimensional Euclidean space on ${\cal T}({\cal P}^n)$,
\noindent (2)\quad Section of ${\cal T}({\cal P}^n)$ be minimal at any
point:
\[
{\cal S}_{n;P}({\cal T}({\cal P}^n))=\{P\},\qquad \forall P\in {\cal T}(
{\cal P}^n).
\]
\end{theorem}
\section{Triangle axiom as a condition of the first order tube degeneration}
Let us study constraints, imposed on $\sigma$-space by the triangle
inequality (\ref{a1.4}). Let us consider segment ${\cal T}_{[P_0P_1]}$ of
the tube ${\cal T}_{P_0P_1}$, contained between basic points $P_0,P_1$. It
is described by equations (\ref{c.1.23}), (\ref{a2.41}).
For continuous $\sigma$-space the tube segment ${\cal T}_{[P_0P_1]}$ is some
surface, containing points $P_0,P_1$. This surface ${\cal T}_{[P_0P_1]}$ and
the region outside the surface are described by the equation
\begin{equation}
S_2(P_0,R,P_1)\equiv \rho (P_0,R)+\rho (R,P_1 )- \rho (P_0,P_1 )\ge 0,
\label{a4.1}
\end{equation}
where $R$ is the running point. Thus, the triangle inequality is fulfilled
on the surface ${\cal T}_{[P_0P_1]}$ and outside it. The region inside the
surface ${\cal T}_{[P_0P_1]}$ associates with the inequality $
S_2(P_0,R,P_1)\le 0$, that corresponds to a violation of the triangle axiom.
In other words, in the metric space the first order tube segment ${\cal T}
_{[P_0P_1]}$ has no inner points. It means degeneration of the tube into a
line, or into a surface which has no inner points. In this sense the metric
geometry (i.e. geometry generated by the metric space) is degenerated
geometry.
In the case, when all first order tubes ${\cal T}_{P_0P_1}$ degenerate into
corresponding basic points $P_0,P_1$, the triangle inequality (\ref{a1.4})
takes the form of a strong inequality
\begin{equation}
\rho (P_0,R)+\rho (R,P_1)> \rho (P_0,P_1),\qquad P_0\ne R\ne P_1\ne P_0,
\qquad \forall P_0,P_1,R\in \Omega . \label{a4.2}
\end{equation}
In this case it seems to be reasonable to call the T-geometry
ultradegenerated.
{\it Example} 4.1. Let us consider two different $\sigma$-spaces (and two
T-geometries) on the unit sphere.
\begin{equation}
\Omega =\left\{ {\bf x}\left| |{\bf x}|^2\leq 1\right. \right\} \subset
{\Bbb R}^3,\qquad {\bf x}=\left\{ x^1,x^2,x^3\right\} \in {\Bbb R}^3,\qquad |
{\bf x|}^2\equiv \sum\limits_{i=1}^3\left( x^i\right) ^2 \label{a4.3}
\end{equation}
$\sigma$-space $V_E=\left\{ \sigma _E,\Omega \right\} $ generates the proper
Euclidean geometry
\begin{equation}
\sigma _E:\quad \Omega \times \Omega \rightarrow [0,\infty ) \subset {\Bbb R}
,\qquad \sigma _E({\bf x},{\bf x}^{\prime })=\frac 12|{\bf x}-{\bf x}
^{\prime }|^2,\qquad {\bf x},{\bf x}^{\prime }\in \Omega , \label{a4.4}
\end{equation}
$\sigma$-space $V=\left\{ \sigma ,\Omega \right\} $ generates T-geometry on
the same set $\Omega $ by means of relations
\begin{equation}
\sigma :\quad \Omega \times \Omega \rightarrow [0,\infty )\subset {\Bbb R},
\qquad \sigma ({\bf x},{\bf x}^{\prime })=2\left( \arcsin \sqrt{\frac{\sigma
_E({\bf x},{\bf x}^{\prime })}2}\right) ^2,\qquad {\bf x},{\bf x}^{\prime
}\in \Omega \label{a4.5}
\end{equation}
Along with the two $\sigma$-spaces in the sphere $\Omega $ one considers
their $\sigma$-subspaces $V_{Es}=\left\{ \sigma _E,\Sigma \right\} $ and $
V_s=\left\{ \sigma ,\Sigma \right\} $ on the sphere surface $\Sigma
=\partial \Omega =\left\{ {\bf x}\left| |{\bf x}|^2=1\right. \right\}
\subset \Omega $. As far as $\Sigma $ is a subset of the set $\Omega $, the
’-geometries $V_{Es}=\left\{ \sigma _E,\Sigma \right\} $ and $V_s=\left\{
\sigma ,\Sigma \right\} $ are generated by T-geometries $T_E$ ¨ $T$.
Let us design the tubes in $\sigma $-space $V_E$ by means of the symbol $
{\cal L}$, the tubes in the $\sigma $-space $V$ are denoted by the symbol $
{\cal T}$. The first order tubes ${\cal L}_{AB}\subset \Omega ,\;\;\left(
A,B\in \Sigma \right) $ are straight lines in $\Omega $, and they are formed
by two basic points $A,B$ in $\Sigma $. In other words, T-geometry $V_E$ is
degenerated in the first order in $\Omega $, and it is ultradegenerated in
the first order in $\Sigma $. The first order tubes ${\cal T}_{AB}\subset
\Omega ,\;\;\left( A,B\in \Sigma \right) $ are nondegenerated tubes in $
\Omega .$ They are surfaces, formed by a rotation of unit radius circles,
passing through points $A,B\in \Sigma $, around the axis ${\cal L}_{AB}$.
(see. Figure 4.1). These tubes tangent the sphere $\Sigma $ along the
circles of maximal radius. The segment ${\cal T}_{[AB]}$ of the tube between
the points $A,B\in \Omega $ is found inside the sphere $\Omega ,$ whereas
the remaining part of the tube ${\cal T}_{AB}$ is found outside the inner
part $\Omega \backslash \Sigma $ of the sphere. As a result the segment $
{\cal T}_{[AB]}$ of the tube ${\cal T}_{AB}\subset \Sigma ,\;\;\left( A,B\in
\Sigma \right) $ is the shortest in the $\sigma $-space $V_s=\left\{ \sigma
,\Sigma \right\} $. This shortest on the sphere surface $\Sigma $ connects
points $A,B\in \Sigma $. The remaining part of the tube ${\cal T}
_{AB}\subset \Omega $ is a continuation of the segment ${\cal T}
_{[AB]}\subset \Sigma $. In other words, $\sigma $-space $V_s=\left\{ \sigma
,\Sigma \right\} $ generates the degenerated in the first order T-geometry
on $\Sigma .$ Thus, T-geometry $V=\left\{ \sigma ,\Omega \right\} $ is
nondegenerated in $\Omega $ and it is degenerated in $\Sigma $. The second
order tube ${\cal T}_{ABC}\subset \Sigma $ consists of three points $
A,B,C\subset \Sigma $, and T-geometry in $V_s=\left\{ \sigma ,\Sigma
\right\} $ is ultradegenerated in the second order geometry in $\Omega $.
On the other hand, T-geometry in $V_s=\left\{ \sigma ,\Sigma \right\} $ on
the sphere surface $\Sigma $ can be constructed on the basis of Euclidedan
geometry in $V_E=\left\{ \sigma _E,\Omega \right\} $. To construct $
V_s=\left\{ \sigma ,\Sigma \right\} $ on the basis of $V_E=\left\{ \sigma
_E,\Omega \right\} $, one can use extremal properties of geodesics.
Let us consider the second order tube ${\cal L}_{ABC}\subset \Omega $, $
(A,B,C\in\Omega )$. This tube is a two-dimensional plane, passing through
the points $A,B,C\in \Omega $. In $V_{Es}=\left\{ \sigma _E,\Sigma \right\} $
the second order tube ${\cal L}_{ABC}\subset \Sigma $ has the form of a
circle, passing through points $A,B,C\in\Sigma $.
To construct internal geometry in $V_s=\left\{ \sigma ,\Sigma \right\} $, it
is necessary to determine the metric $\rho\left( A,B\right) = \sqrt{2\sigma
\left( A,B\right) },\;\;A,B\in \Sigma $ on $\Sigma \times\Sigma $, using the
following way.
\begin{equation}
\rho\left( A,B\right) =\inf\limits_{C\in\Sigma ,C\neq A,C\neq B}l_C\left(
A,B\right) ,\qquad A,B\in\Sigma , \label{a4.6}
\end{equation}
where $l_C\left( A,B\right)\subset [0,\infty ) $ is the length of the curve $
{\cal L}_{ABC}\subset\Sigma $ between the ponts $A,B$. Let us order the
points $R\in {\cal L}_{ABC}\subset\Sigma $ of the curve ${\cal L}
_{ABC}\subset\Sigma $, solving the equation
\begin{equation}
\rho _E\left( A,R\right) \left( 1-\tau \right) =\rho _E\left( B,R\right)
\tau ,\qquad R\in {\cal L}_{ABC}\subset \Omega ,\qquad \tau \in {\Bbb R}
\label{a4.7}
\end{equation}
Solution of this equation determines $R=R_{AB}\left( \tau ,C\right) \in
{\cal L}_{ABC}\subset\Sigma $ as a function of the parameter $\tau \in [0,1]$
and of the point $C\in\Sigma $. Therewith $A=R_{AB}\left(0,C\right)$, $
B=R_{AB}\left(1,C\right)$. The function $R=R_{AB}\left( \tau ,C\right)$ has
two branches $R=R_1\left( \tau ,C\right) $ and $R=R_2\left( \tau ,C\right)$.
It should change the branch with the less values of $\rho _E(A,R_{AB}\left(
\tau ,C\right))$ and determine $l_C\left( A,B\right) $ by means of the
relation
\begin{equation}
l_C\left( A,B\right) =\int\limits_0^1\left[ \frac{d\rho _E\left(R_{AB}\left(
\tau ,C\right) ,R_{AB}\left( \tau ^{\prime },C\right) \right) }{d\tau
^{\prime }}\right] _{\tau ^{\prime }=\tau }d\tau \label{a4.8}
\end{equation}
Substituting (\ref{a4.8}) in (\ref{a4.6}), one obtains
\begin{equation}
\rho \left( A,B\right) =2\arcsin \frac{S_E\left( A,B\right) }2,\qquad A,B\in
\Sigma \label{a4.9}
\end{equation}
that corresponds to (\ref{a4.5}).
This example shows that, using Euclidean geometry inside $\Omega $, one can
consrtuct internal metric (and T-geometry) on the surface $\Sigma $ of the
sphere $\Omega $. The extremal properties of geodesics (shortests) and the
second order tubes ${\cal L}_{ABC}\subset \Omega $ are used essentially for
construction of geometry on $\Sigma $. The second order tubes ${\cal L}
_{ABC}\subset \Omega $ generate a system of curves on the sphere surface $
\Sigma $. One chooses those among them which have the minimal length between
points $A,B$.
Apparently, with proper stipulations this procedure of constructing metric
space on the sphere surface with Euclidean space inside sphere can be
generalized on the case of arbitrary body and arbitrary metric space inside
it.
Thus, classification of finite metric spaces, using the series of mappings (
\ref{a1.5}), appears to be a very effective method of studying the metric
space. This method admits to describe the Euclidean geometry and the
Riemannian one in terms of only metric (world function). Geometries,
constructed on the basis of this classification do not use concept of
continuity. They are insensitive to discreteness or continuity of the space.
Concept of continuity may be introduced on the basis of metric (world
function) by means of a proper parametrization of extremal tubes \cite{R90}.
\end{document} | math |
हिंदी न्यूज़ बिजनेस शेयर बाजार में तेजी बरकरार, ३५३ अंक उछलकर सेंसेक्स ४१,१४२ पर बंद, निफ्टी भी १०९ अंक चढ़ा
शेयर बाजार में तेजी बरकरार, ३५३ अंक उछलकर सेंसेक्स ४१,१४२ पर बंद, निफ्टी भी १०९ अंक चढ़ा
वैश्विक बाजारों से मिल रहे पॉजिटिव संकेतों के बीच घरेलू शेयर बाजार में तेजी का रुख देखने को मिला। बुधवार को सेंसेक्स और निफ्टी में मंगलवार की तरह तेजी तो नहीं दिखी, लेकिन उछाल चौथे दिन भी जारी रहा। सेंसेक्स एक बार फिर ४१००० के पार पहुंच गया। वहीं निफ्टी भी १२००० अंकों के ऊपर चला गया। बुधवार को सेंसेक्स ३५३ अंक चढ़कर ४११४२ और निफ्टी १०९ अंक चढ़कर १२०८९ पर बंद हुआ।
कारोबार के दौरान ऊंचे में यह ४१,१५४.६६ के उच्च स्तर तक गया। इसी प्रकार, नेशनल स्टॉक एक्सचेंज का निफ्टी १०९.५० अंक यानी ०.९१ प्रतिशत मजबूत होकर १२,०89.१५ अंक पर बंद हुआ।
टाटा स्टील ५.१४ प्रतिशत की तेजी
सेंसेक्स के शेयरों में टाटा स्टील सर्वाधिक लाभ में रहा। इसमें ५.१४ प्रतिशत की तेजी आई। उसके बाद क्रमश: भारती एयरटेल, एचडीएफसी, टीसीएस, एल एंड टी, एसबीआई और रिलायंस इंडस्ट्रीज का स्थान रहा।
हीरो मोटो कार्प में सर्वाधिक ३.8३ प्रतिशत की गिरावट आयी। पावर ग्रिड, मारुति, एशियन पेंट्स और नेस्ले इंडिया भी नुकसान में रहे।
शेयर बाजार में तेजी के ५ कारण
१. विशेषज्ञों के अनुसार कोरोनो वायरस के इलाज में उल्लेखनीय सफलता मिलने से वैश्विक बाजारों में तेजी का रुख बना है।
२. भारत के सेवा क्षेत्र की गतिविधियों में सुधार और जनवरी में इसके सात साल के उच्च स्तर पर पहुंचने की रिपोर्ट से भी धारणा मजबूत हुई।
३. आईएचएस मार्किट इंडिया के सर्वे के अनुसार देश में सेवा कारोबार गतिविधि सूचकांक जनवरी में ५५.५ अंक पर रहा जबकि दिसंबर में यह ५३.३ अंक था।
४. एशिया के अन्य बाजारों में चीन में शंघाई, हांगकांग, तोक्यो और सोल १.२५ प्रतिशत तक मजबूत हुए।
५. यूरोप के प्रमुख बाजारों में भी शुरूआती कारोबार में तेजी दर्ज की गई।
स्टॉक शेयर प्राइस %
बता दें शुरुआती कारोबार में सेंसेक्स १०० अंकों की तेजी के साथ ४०८९१ के स्तर पर ट्रेड कर रहा था वहीं निफ्टी में भी तेजी दिखी । इससे पहले मंगलवार को भारतीय शेयर बाजार में बंपर उछाल देखने को मिला था। सेंसेक्स कल ९१७ अंक की जबरदस्त तेजी लेकर ४०,७८९.३८ पर बंद हुआ, वहीं निफ्टी २७१.७५ की तेजी के साथ ११,९७९.६५ के स्तर पर बंद हुआ। आज एशियाई बाजारों में तेजी देखने को मिली, जबकि मंगलवार को अमेरिकी बाजारों में भी शानदार तेजी रही।
यह भी पढ़ें: सोने-चांदी की कीमतों में आज भी बड़ी गिरावट, जानें ५ फरवरी २०२० का रेट
बीएसई का ३० कंपनियों का शेयर सूचकांक सेंसेक्स १०० अंक चढ़कर खुला और जल्द ही इसमें और तेजी देखी गयी। सुबह सवा दस बजे यह २०२.५६ अंक यानी ०.5० प्रतिशत बढ़कर 4०,९९१.९४ पर था। इसी तरह नेशनल स्टॉक एक्सचेंज का निफ्टी सुबह सवा दस बजे ५९.५५ अंक यानी ०.5० प्रतिशत की तेजी के साथ १२,०39.2० पर कारोबार कर रहा था। | hindi |
Data Auditing Quiz: Does Your Compliance Data System Prove Your Innocence?
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What are the Reasons that YOU Procrastinate? | english |
کونہٕ خاص کاروبار یا لیبارٹری خٲطرٕ چھ منٲسب کیمیٲئی حفظان صحتک منصوبک تعین کرنہٕ خٲطرٕ، معیارچ ضرورتہٕ سمجھن، موجودٕ حفاظت، صحت تہٕ ماحولیاتی طریقن ہنٛد تشخیص تہٕ خطرچ تشخیص ضروری۔ | kashmiri |
The text could address monitoring of the utilization of marine genetic resources of areas beyond national jurisdiction.
(a) How could the instrument address the monitoring of the utilization of marine genetic resources of areas beyond national jurisdiction?
References: Nagoya Protocol Art 15 (Compliance on ABS), Nagoya Protocol Art 17 (Monitoring), Explanatory Guide to the Nagoya Protocol. IUCN EPLP Series N°85.
See Implementation and Enforcement, section 6.3.3, page 38.
This page was last edited on 13 September 2018, at 14:57. | english |
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कॉमेडी वाइल्ड लाइफ फोटोग्राफी अवॉर्ड जीत चुकी ये तस्वीरें साल २०१८ की सबसे मज़ेदार तस्वीरें कहीं जा सकती हैं। इस प्रतियोगिता का ये चौथा वर्ष है। इस फोटोग्राफी पुरस्कार में विश्व भर से जानवरों...
बकरीद से पहले पेटा ने राज्यों को लिखा पत्र, पशुओं पर होने वाली क्रूरता को रोका जाए
नई दिल्ली(भाषा)। पशुओं के हितों के लिए काम करने वाले संगठन पेटा ने बकरीद से पहले राज्यों को पत्र लिखकर कुर्बानी के दौरान पशुओं पर होने वाली क्रूरता को रोकने की मांग की है। पेटा ने सात राज्यों को...
मथुरा। उत्तर प्रदेश के मथुरा में एक गधे और एक घोड़े में लाइलाज बीमारी ग्लैण्डर्स की पुष्टि हुई है। जनवरी में दिल्ली में तीन दर्जन से अधिक घोड़ों में इस बीमारी की पुष्टि हुई थी। इन पशुओं के खून के नमूने... | hindi |
ऑनलाईन धोखाधड़ी व सायबर फ्रॉड की रोकथाम हेतु, बैंको व पुलिस के बेहतर समन्वय के साथ कार्यवाही के लिये बैठक का आयोजन - इंडियास फास्टस्ट हिन्दी न्यूज पोर्तल
इन्दौर-दिनांक २२ अगस्त २०१८। वर्तमान समय में बढ़ते सायबर अपराधों एव ऑनलाईन ठगी की रोकथाम एवं बेहतर आपसी तालमेल व समन्वय के साथ कार्य करके, इन अपराधों पर नियत्रंण पाया जा सके इसी को ध्यान में रखते हुए, जिला इन्दौर के बैंको के नोडल अधिकारियों के साथ एक बैठक का आयोजन आज दिनांक २३.०८.१९ को पुलिस कंट्रोल रूम पर आयोजित की गयी।
जिसमें वरिष्ठ पुलिस अधीक्षक इन्दौर श्रीमती रूचि वर्धन मिश्र की विशेष उपस्थिति में अति. पुलिस अधीक्षक क्राईम श्री अमरेन्द्र सिंह, उप पुलिस अधीक्षक क्राईम श्री आलोक शर्मा, शहर के विभिन्न प्रमुख बैंको के नोडल अधिकारीगण एवं क्राईम ब्रांच इन्दौर की टेक्निकल टीम सहित अन्य पुलिस अधिकारीगण उपस्थित रहे।
उक्त बैठक में वर्तमान परिदृश्य में बढ़ते ऑनलाईन ट्रान्जेक्शन के कारण, सायबर अपराधियों द्वारा की जाने वाली ऑनलाईन ठगी व धोखाधड़ी की वारदातों की रोकथाम एवं इनसे बचने के लिये ध्यान में रखने वाली आवश्यक बातों पर चर्चा की गयी।
एसएसपी इन्दौर ने कहा कि पुलिस व बैंक बेहतर आपसी समन्वय व तालमेल के साथ काम करके, इन अपराधों पर बेहतर तरीके से नियंत्रण पाया जा सकता है, उन्होने इस संबंध में चर्चा के दौरान निम्नलिखित बातों को ध्यान रखने पर जोर दिया
बैंको की वेबसाईटों को समय-समय पर अपडेट करतें रहे, व इस पर बैंकिग फ्रॉड से बचने के लिए रखी जानें वाली सावधानियों को भी आम जनता के लिये अपलोड करें।
विभिन्न ई वॉलेट के माध्यम से किये जाने वाले ट्रांजेक्शन पर होने वाली ठगी से बचने के लिये पूरी सावधानी बरती जावें।
कॉल सेंटर के माध्यम से कॉल कर विभिन्न प्रकार के प्रलोभन दिये जाकर की जानें वाली ठगी को रोकनें के लिये भी आवश्यक कदम उठाये जाये।
बैंको की लिंक आदि भेजकर यूपीआई के माध्यम से की जाने वाली ठगी को रोकनें के लिए, इस प्रकार की वेबसाईटो की पहचान कर उन पर कार्यवाही की जावें एवं इस पर बैंको द्वारा अपनी वेबसाईट्स के सुरक्षा फीचर जोड़े जावें।
ऑनलाईन ट्रांजेक्शन के दौरान कुछ ऐसी व्यवस्था की जाये, जिससें रूपयें उसी समय ट्रासंफर ना होकर कुछ समय होल्ड होकर प्राप्त हो जिससें शिकायत प्राप्त हानें पर ट्रांजेक्शन को कैंसल किया जा सकें।
नया अकाउंट ओपन करतें समय केवाईसी का पूरा ध्यान रखा जाये जिससें फर्जी अकाउंट ओपन ना हो सकें।
सभी अकाउंट के केवाईसी अपडेट रखी जायें जिससें फ्रॉड होने पर फर्जी अकाउंट को आईडेटिंफाई किया जा सकें।
विदेशों मे करेंसी ट्रांसफर फ्रॉड मे शिकायतकर्ता द्वारा कहा जाता है कि फॉरेन मनी ट्रांसफर की सुविधा नही होने पर भी पैसा विदेश मे ट्रांसफर हो ग या। इस प्रकार व्यवस्था की जाये, जिससे विदेशों से होने वाले फ्रॉड को रोका जा सकें।
एटीएम के क्लोन बनाकर की जानें वाली ठगी न हो सके, इसके लिये एटीएम व बैंको के सुरक्षा फीचर सुव्यवस्थित किये जावें तथा समय-समय पर उनका अपडेशन भी किया जावे।
एटीएम मशीन मे फ्रॉड रोकनें के लिए एटीएम के अंदरूनी कैमरो को सही रखा जायें व एटीएम मे आवश्य क रूप से कैमरे लगाये जाकर गार्ड की नियुक्ति भी की जाये।
नेटबैकिंग मे इम्प्स, नेफ्ट के माध्यम से होनें वाली ट्रांजेक्शन मे कोई फ्रॉड ना हो, इसके लिए ट्रांजेक्शन से पहलें मोबाईल पर ओटीपी प्राप्त हो जिससें गलत ट्रांजेक्शन होने पर रोका जा सकें।
आयोजित बैठक में प्रमुख तौर पर आईसीआईसीआई बैंक, बैंक ऑफ महाराष्ट्र, पंजाब नेशनल बैंक, यूनियन बैंक ऑफ इण्डिया, यूको बैंक, आन्ध्रा बैंक, फिनो पेमेन्ट बैंक, कोटक बैंक, एक्सिस बैंक, केनेरा बैंक, जना स्मॉल फायनेन्स बैंक, यस बैंक, इलाहबाद बैंक, सेन्ट्रल बैंक ऑफ इण्डिया, बंधन बैंक, एच डी एफ सी बैंक, ओरिएन्टल बैंक ऑफ कोमर्स, बैंक ऑफ इण्डिया, सिंडिकेट बैंक, आईडीबीआई बैंक, इण्डियन बैंक, स्टेट बैंक ऑफ इण्डिया, बैंक ऑफ बड़ौदा, बैंक ऑफ इण्डिया, तथा एक्सिस बैंक सहित २० से अधिक बैंकों के नोडल अधिकारी उपस्थित हुये जिन्होंनें पुलिसकर्मियों तथा वरिष्ठ पुलिस अधिकारियों से संवाद के माध्यम से बैंक से संबंधित आने वाली तकनीकी समस्याओं को जाना तथा शीघ्र निराकरण का आशवासन दिया। इस दौरान पुलिस अधिकारियों के द्वारा सायबर फ्रॉड को रोकनें के लिए बैंको के नोडल अधिकारियों से सायबर अपराधों पर चर्चा कर फीडबेक लिये गये, जिससें भविष्य मे होने वाले सायबर अपराधों को रोकनें मे सफलता प्राप्त की जा सकें।
प्रेवियस पोस्टप्रिवियस एशेज सीरीज: हेजलवुड की घातक गेंदबाजी, इंग्लैंड ६७ रन पर ऑलआउट
नेक्स्ट पोस्टनेक्स्ट पूर्व वित्त मंत्री अरुण जेटली नहीं रहे, एम्स में हुआ निधन | hindi |
Take another little pizza my heart now baby.
It's finally happened: Domino's will now allow you to order a pizza by tweeting them the 🍕 emoji.
You'll have to grab yourself a Domino's Pizza Profile and sign up for their "Easy Order" system, but once that's done you'll be able to order a pizza on Twitter. It won't be available until Wednesday, May 20th - and probably only to our American friends - but it's nice that Domino's are embracing future pizza ordering technology with open arms.
🍕🍕'🍕 🍕🍕🍕🍕🍕🍕 🍕🍕🍕🍕. 🍕🍕🍕🍕🍕 🍕🍕🍕. | english |
شہزادس ٲس یہِ صلاحیت زِ سُہ اوس کانٛہہ تہِ مُشکل تہٕ کروٗٹھ تنازعہٕ أنٛزراوان | kashmiri |
/**
* Copyright Amazon.com, Inc. or its affiliates. All Rights Reserved.
* SPDX-License-Identifier: Apache-2.0.
*/
#pragma once
#include <aws/waf/WAF_EXPORTS.h>
#include <aws/core/utils/memory/stl/AWSString.h>
#include <aws/core/utils/memory/stl/AWSVector.h>
#include <aws/waf/model/XssMatchSetSummary.h>
#include <utility>
namespace Aws
{
template<typename RESULT_TYPE>
class AmazonWebServiceResult;
namespace Utils
{
namespace Json
{
class JsonValue;
} // namespace Json
} // namespace Utils
namespace WAF
{
namespace Model
{
/**
* <p>The response to a <a>ListXssMatchSets</a> request.</p><p><h3>See Also:</h3>
* <a
* href="http://docs.aws.amazon.com/goto/WebAPI/waf-2015-08-24/ListXssMatchSetsResponse">AWS
* API Reference</a></p>
*/
class AWS_WAF_API ListXssMatchSetsResult
{
public:
ListXssMatchSetsResult();
ListXssMatchSetsResult(const Aws::AmazonWebServiceResult<Aws::Utils::Json::JsonValue>& result);
ListXssMatchSetsResult& operator=(const Aws::AmazonWebServiceResult<Aws::Utils::Json::JsonValue>& result);
/**
* <p>If you have more <a>XssMatchSet</a> objects than the number that you
* specified for <code>Limit</code> in the request, the response includes a
* <code>NextMarker</code> value. To list more <code>XssMatchSet</code> objects,
* submit another <code>ListXssMatchSets</code> request, and specify the
* <code>NextMarker</code> value from the response in the <code>NextMarker</code>
* value in the next request.</p>
*/
inline const Aws::String& GetNextMarker() const{ return m_nextMarker; }
/**
* <p>If you have more <a>XssMatchSet</a> objects than the number that you
* specified for <code>Limit</code> in the request, the response includes a
* <code>NextMarker</code> value. To list more <code>XssMatchSet</code> objects,
* submit another <code>ListXssMatchSets</code> request, and specify the
* <code>NextMarker</code> value from the response in the <code>NextMarker</code>
* value in the next request.</p>
*/
inline void SetNextMarker(const Aws::String& value) { m_nextMarker = value; }
/**
* <p>If you have more <a>XssMatchSet</a> objects than the number that you
* specified for <code>Limit</code> in the request, the response includes a
* <code>NextMarker</code> value. To list more <code>XssMatchSet</code> objects,
* submit another <code>ListXssMatchSets</code> request, and specify the
* <code>NextMarker</code> value from the response in the <code>NextMarker</code>
* value in the next request.</p>
*/
inline void SetNextMarker(Aws::String&& value) { m_nextMarker = std::move(value); }
/**
* <p>If you have more <a>XssMatchSet</a> objects than the number that you
* specified for <code>Limit</code> in the request, the response includes a
* <code>NextMarker</code> value. To list more <code>XssMatchSet</code> objects,
* submit another <code>ListXssMatchSets</code> request, and specify the
* <code>NextMarker</code> value from the response in the <code>NextMarker</code>
* value in the next request.</p>
*/
inline void SetNextMarker(const char* value) { m_nextMarker.assign(value); }
/**
* <p>If you have more <a>XssMatchSet</a> objects than the number that you
* specified for <code>Limit</code> in the request, the response includes a
* <code>NextMarker</code> value. To list more <code>XssMatchSet</code> objects,
* submit another <code>ListXssMatchSets</code> request, and specify the
* <code>NextMarker</code> value from the response in the <code>NextMarker</code>
* value in the next request.</p>
*/
inline ListXssMatchSetsResult& WithNextMarker(const Aws::String& value) { SetNextMarker(value); return *this;}
/**
* <p>If you have more <a>XssMatchSet</a> objects than the number that you
* specified for <code>Limit</code> in the request, the response includes a
* <code>NextMarker</code> value. To list more <code>XssMatchSet</code> objects,
* submit another <code>ListXssMatchSets</code> request, and specify the
* <code>NextMarker</code> value from the response in the <code>NextMarker</code>
* value in the next request.</p>
*/
inline ListXssMatchSetsResult& WithNextMarker(Aws::String&& value) { SetNextMarker(std::move(value)); return *this;}
/**
* <p>If you have more <a>XssMatchSet</a> objects than the number that you
* specified for <code>Limit</code> in the request, the response includes a
* <code>NextMarker</code> value. To list more <code>XssMatchSet</code> objects,
* submit another <code>ListXssMatchSets</code> request, and specify the
* <code>NextMarker</code> value from the response in the <code>NextMarker</code>
* value in the next request.</p>
*/
inline ListXssMatchSetsResult& WithNextMarker(const char* value) { SetNextMarker(value); return *this;}
/**
* <p>An array of <a>XssMatchSetSummary</a> objects.</p>
*/
inline const Aws::Vector<XssMatchSetSummary>& GetXssMatchSets() const{ return m_xssMatchSets; }
/**
* <p>An array of <a>XssMatchSetSummary</a> objects.</p>
*/
inline void SetXssMatchSets(const Aws::Vector<XssMatchSetSummary>& value) { m_xssMatchSets = value; }
/**
* <p>An array of <a>XssMatchSetSummary</a> objects.</p>
*/
inline void SetXssMatchSets(Aws::Vector<XssMatchSetSummary>&& value) { m_xssMatchSets = std::move(value); }
/**
* <p>An array of <a>XssMatchSetSummary</a> objects.</p>
*/
inline ListXssMatchSetsResult& WithXssMatchSets(const Aws::Vector<XssMatchSetSummary>& value) { SetXssMatchSets(value); return *this;}
/**
* <p>An array of <a>XssMatchSetSummary</a> objects.</p>
*/
inline ListXssMatchSetsResult& WithXssMatchSets(Aws::Vector<XssMatchSetSummary>&& value) { SetXssMatchSets(std::move(value)); return *this;}
/**
* <p>An array of <a>XssMatchSetSummary</a> objects.</p>
*/
inline ListXssMatchSetsResult& AddXssMatchSets(const XssMatchSetSummary& value) { m_xssMatchSets.push_back(value); return *this; }
/**
* <p>An array of <a>XssMatchSetSummary</a> objects.</p>
*/
inline ListXssMatchSetsResult& AddXssMatchSets(XssMatchSetSummary&& value) { m_xssMatchSets.push_back(std::move(value)); return *this; }
private:
Aws::String m_nextMarker;
Aws::Vector<XssMatchSetSummary> m_xssMatchSets;
};
} // namespace Model
} // namespace WAF
} // namespace Aws
| code |
सरकारी नौकरी : जयपुर मेट्रो में भर्ती की अधिसूचना जारी, २३ दिसंबर से शुरू होगी आवेदन प्रक्रिया, - ऑफिशियल जॉब्स
होम जॉब्स सरकारी जॉब्स सरकारी नौकरी : जयपुर मेट्रो में भर्ती की अधिसूचना जारी, २३ दिसंबर से शुरू होगी आवेदन प्रक्रिया,
सरकारी नौकरी : जयपुर मेट्रो में भर्ती की अधिसूचना जारी, २३ दिसंबर से शुरू होगी आवेदन प्रक्रिया,
सरकारी नौकरी : जयपुर मेट्रो में भर्ती की अधिसूचना जारी, २३ दिसंबर से शुरू होगी आवेदन प्रक्रिया, यहां पढ़ें
सरकारी नौकरी: जयपुर मेट्रो रेल कॉर्पोरेशन ऑफ इंडिया में भर्ती के लिए अधिसूचना जारी की गई है। इच्छुक और पात्र उम्मीदवार नियत तिथि तक ऑनलाइन आवेदन कर सकते हैं। यह भर्ती मेंटेनर, जूनियर इंजीनियर और अन्य के रिक्त पदों पर निकाली गई है। इच्छुक व योग्य उम्मीदवार २३ जनवरी २०२० तक या उससे पहले निर्धारित प्रारूप के माध्यम से उक्त पदों पर आवेदन कर सकते हैं।
ऑनलाइन आवेदन प्रक्रिया शुरू: २३ दिसंबर २०१९
स्टेशन कंट्रोलर / ट्रेन ऑपरेटर - ४ पद
जूनियर इंजीनियर (इलेक्ट्रिकल) - ४ पद
जूनियर इंजीनियर (इलेक्ट्रॉनिक्स) - ३ पद
जूनियर इंजीनियर (सिविल) - ४ पद
ग्राहक संबंध सहायक (सीआरए) - ६ पद
मेंटेनर (इलेक्ट्रीशियन) - ६ पद
अनुरक्षक (इलेक्ट्रॉनिक्स) - ९ पद
मेंटेनर (फिटर) - १ पोस्ट
जयपुर मेट्रो भर्ती २०१९ नोटिफिकेशन के लिए यहां क्लिक करें
स्टेशन कंट्रोलर / ट्रेन ऑपरेटर पद के लिए उम्मीदवार का किसी मान्यता प्राप्त विश्वविद्यालय से किसी भी विषय में न्यूनतम ५०% अंकों के साथ स्नातक उत्तीर्ण होना जरुरी है। बारहवीं कक्षा गणित या भौतिकी के साथ उत्तीर्ण किया हो।
जूनियर इंजीनियर (सिविल / इलेक्ट्रिकल / इलेक्ट्रॉनिक्स) पद के लिए न्यूनतम ५०% अंकों के साथ किसी मान्यता प्राप्त विश्वविद्यालय से संबंधित में इंजीनियरिंग डिग्री।
क्रा- किसी मान्यता प्राप्त विश्वविद्यालय से न्यूनतम ५०% अंकों के साथ किसी भी विषय में स्नातक या समकक्ष।
मेंटेनर (इलेक्ट्रीशियन / इलेक्ट्रॉनिक्स / फिटर) - संबंधित ट्रेड में नेशनल ट्रेड सर्टिफिकेट या नेशनल अप्रेंटिसशिप सर्टिफिकेट या संबंधित ट्रेड में एनसीवीटी या स्टेट ट्रेड सर्टिफिकेट या एससीवीटी द्वारा जारी समकक्ष सर्टिफिकेट।
स्टेशन नियंत्रक / ट्रेन ऑपरेटर -रु ३७,८५६ / -
जूनियर इंजीनियर (सिविल / इलेक्ट्रिकल / इलेक्ट्रॉनिक्स) - रु ३७,८५६ / -
सीआरए- रु ३२,१४४ / -
अनुरक्षक (इलेक्ट्रीशियन / इलेक्ट्रॉनिक्स / फिटर) - रु २३,२९६ / -
उम्मीदवारों का चयन जूनियर इंजीनियर, सीआरए, मेंटेनर और स्टेशन कंट्रोलर पदों के लिए साक्षात्कार प्रक्रिया के माध्यम से किया जाएगा।
जयपुर मेट्रो भर्ती २०१९ के लिए आवेदन कैसे करें
इच्छुक उम्मीदवार जयपुर मेट्रो भर्ती २०१९ के लिए ऑनलाइन मोड से आवेदन कर सकते हैं। आवेदन २३ जनवरी २०१९ तक प्राप्त किए जाएंगे। | hindi |
# Cephalostigma diaguissae A.Chev. SPECIES
#### Status
ACCEPTED
#### According to
International Plant Names Index
#### Published in
null
#### Original name
null
### Remarks
null | code |
package e2etest
import (
"path/filepath"
"strings"
"testing"
"github.com/hashicorp/terraform/e2e"
)
func TestPackage_empty(t *testing.T) {
t.Parallel()
// This test reaches out to releases.hashicorp.com to download the
// template provider, so it can only run if network access is allowed.
// We intentionally don't try to stub this here, because there's already
// a stubbed version of this in the "command" package and so the goal here
// is to test the interaction with the real repository.
skipIfCannotAccessNetwork(t)
fixturePath := filepath.Join("testdata", "empty")
tfBundle := e2e.NewBinary(bundleBin, fixturePath)
defer tfBundle.Close()
stdout, stderr, err := tfBundle.Run("package", "terraform-bundle.hcl")
if err != nil {
t.Errorf("unexpected error: %s", err)
}
if stderr != "" {
t.Errorf("unexpected stderr output:\n%s", stderr)
}
if !strings.Contains(stdout, "Fetching Terraform 0.10.1 core package...") {
t.Errorf("success message is missing from output:\n%s", stdout)
}
if !strings.Contains(stdout, "Creating terraform_0.10.1-bundle") {
t.Errorf("success message is missing from output:\n%s", stdout)
}
if !strings.Contains(stdout, "All done!") {
t.Errorf("success message is missing from output:\n%s", stdout)
}
}
func TestPackage_manyProviders(t *testing.T) {
t.Parallel()
// This test reaches out to releases.hashicorp.com to download the
// template provider, so it can only run if network access is allowed.
// We intentionally don't try to stub this here, because there's already
// a stubbed version of this in the "command" package and so the goal here
// is to test the interaction with the real repository.
skipIfCannotAccessNetwork(t)
fixturePath := filepath.Join("testdata", "many-providers")
tfBundle := e2e.NewBinary(bundleBin, fixturePath)
defer tfBundle.Close()
stdout, stderr, err := tfBundle.Run("package", "terraform-bundle.hcl")
if err != nil {
t.Errorf("unexpected error: %s", err)
}
if stderr != "" {
t.Errorf("unexpected stderr output:\n%s", stderr)
}
if !strings.Contains(stdout, "Checking for available provider plugins on ") {
t.Errorf("success message is missing from output:\n%s", stdout)
}
// Here we have to check each provider separately
// because it's internally held in a map (i.e. not guaranteed order)
if !strings.Contains(stdout, `- Resolving "aws" provider (~> 0.1)...
- Downloading plugin for provider "aws" (0.1.4)...`) {
t.Errorf("success message is missing from output:\n%s", stdout)
}
if !strings.Contains(stdout, `- Resolving "kubernetes" provider (0.1.0)...
- Downloading plugin for provider "kubernetes" (0.1.0)...
- Resolving "kubernetes" provider (0.1.1)...
- Downloading plugin for provider "kubernetes" (0.1.1)...
- Resolving "kubernetes" provider (0.1.2)...
- Downloading plugin for provider "kubernetes" (0.1.2)...`) {
t.Errorf("success message is missing from output:\n%s", stdout)
}
if !strings.Contains(stdout, `- Resolving "null" provider (0.1.0)...
- Downloading plugin for provider "null" (0.1.0)...`) {
t.Errorf("success message is missing from output:\n%s", stdout)
}
if !strings.Contains(stdout, "Fetching Terraform 0.10.1 core package...") {
t.Errorf("success message is missing from output:\n%s", stdout)
}
if !strings.Contains(stdout, "Creating terraform_0.10.1-bundle") {
t.Errorf("success message is missing from output:\n%s", stdout)
}
if !strings.Contains(stdout, "All done!") {
t.Errorf("success message is missing from output:\n%s", stdout)
}
}
| code |
#ifndef SHARED_TYPES_DESTROY_H
#define SHARED_TYPES_DESTROY_H
#include <stdint.h>
#include <stddef.h>
#include <stdbool.h>
#include "GenericTaskDeclarations.h"
#include "GenericSharedDeclarations.h"
#include "GenericSyncDeclarations.h"
#ifdef __cplusplus
extern "C" {
#endif
int32_t shared_types_destroy_main(int32_t argc, char* argv[]);
void shared_types_destroy_initGlobalMutexesFor1Module_0(void);
void shared_types_destroy_initAllGlobalMutexes_0(void);
#ifdef __cplusplus
} /* extern "C" */
#endif
#endif
| code |
رامز جلال , اوس اوس , | kashmiri |
Mark the wind directions from which you would find acceptable. We have pre-filled the checkboxes with values typical for windsurfing, based on the orientation of La Arena. Your local knowledge and preferences may be different.
Specify the maximum allowed forecast average wind speeds in km/h, coming from 16 different directions. We have pre-filled the lists with values suggested by La Arena orientation and wind sheltering. Your local knowledge and preferences may be different.
Tide state for La Arena is approximated by data from Zierbena which is 3 km away. | english |
एलटी ग्रेड शिक्षक भर्ती के लिए ऑनलाइन आवेदन की प्रक्रिया १५ मार्च से प्रारंभ हो चुकी है,पहली बार एलटी ग्रेड शिक्षकों की भर्ती लिखित परीक्षा के माध्यम से होगी, उत्तर प्रदेश लोक सेवा आयोग द्वारा आयोजित इस परीक्षा के माध्यम से राजकीय स्कूलों में १५ विषयों में कुल १०७६८ शिक्षकों की भर्ती की जानी है|
जिनमें ५३६४ पुरुष और ५४०४ महिला शिक्षकों के पद शामिल हैं,हाल ही में प्राप्त जानकारी के अनुसार, हिंदी और कंप्यूटर विषय की अहर्ता में संशोधन होने की संभावना है, इसके बारें में आपको इस पेज पर विस्तार से बता रहें है |
कंप्यूटर और हिंदी पदों की अहर्ता में संशोधन
कंप्यूटर और हिंदी पदों की अहर्ता में संशोधन किये जाने की संभावना है, यूपी बोर्ड द्वारा हिंदी विषय की जो न्यूनतम शैक्षिक योग्यताएं निश्चित की है,उसमें बी ए हिंदी तथा संस्कृत के साथ इंटर या समकक्ष परीक्षा एवं बी एड व अन्य समकक्ष डिग्री अथवा बी ए हिंदी एवं संस्कृत के साथ समकक्ष परीक्षा एवं बी एड या अन्य डिग्री मान्य है, जबकि लोक सेवा आयोग द्वारा हिंदी विषय की शैक्षिक अहर्ता में इंटर में संस्कृत को अनिवार्य कर दिया है |
यूपी बोर्ड नें कंप्यूटर विषय के सहायक अध्यापको की योग्यता में बीएससी कंप्यूटर विज्ञान के साथ या बीएससी कंप्यूटर एप्लीकेशन के साथ, अथवा बेचलर ऑफ़ कंप्यूटर के साथ डीओएई से ओ लेवल कोर्स के साथ स्नातक या पीजी डिप्लोमा किसी भी मान्यता प्राप्त संस्थान से होना अनिवार्य है, जबकि लोक सेवा आयोग द्वारा कंप्यूटर में बीई अथवा बीटेक अथवा कंप्यूटर विज्ञान में विज्ञान स्नातक, कंप्यूटर एप्लीकेशन में विज्ञानं स्नातक अथवा एनआईईएलआईटी ए ग्रेड पाठ्यक्रम के साथ स्नातक के साथ बीएड की उपाधि अनिवार्य कर दिया है |
कला विषय में सशोधन की मांग
कला विषय के अहर्ता के संशोधन की मांग की जा रही है, अभ्यर्थियों के अनुसार, जिस योग्यता के आधार पर सहायता प्राप्त माध्यमिक विद्यालय में टीजीटी कला विषय के लिए निर्धारित है, उसी योग्यता के आधार पर एलटी गड़े शिक्षकों की भर्ती की जनि चाहिए, जबकि कुछ छात्रों द्वारा हाईकोर्ट में कला शिक्षक भर्ती की अहर्ता के विरुद्ध याचिका दर्ज करायी है |
यहाँ आपको हमनें एलटी ग्रेड शिक्षक भर्ती में हिंदी और कंप्यूटर विषय की अहर्ता में संशोधन होने के बारें में बताया | यदि इससे सम्बंधित आपके मन में कोई प्रश्न आ रहा है, तो कमेंट बाक्स के माध्यम से व्यक्त कर सकते है | हमें आपके द्वारा की गई प्रतिक्रिया का इंतजार है | | hindi |
میٛانہِ گوٚبرٕ مےٚ چھی ووٚنۍ نیرُن تہٕ پَتہٕ یِمہٕ نہٕ زانٛہہ واپَس | kashmiri |
1.修改配置 关闭aof
2.关闭redis
3.cp dump.rdb /var/lib/redis/dump.rdb
4.开启redis
5.redis-cli config set appendonly yes
6.修改配置打开aof
完毕
### 参考
* http://serverfault.com/a/692934/282459 | code |
class Celluloid::SMTP::Connection
attr_reader :socket, :automata
extend Forwardable
def_delegators :@socket, :close, :peeraddr, :print, :closed?
def_delegators :@automata, :transition
def initialize(socket, configuration)
@automata = Automata.new(self)
@configuration = configuration.dup
@socket = socket
@timestamps = {}
@context = nil
@behavior = configuration.fetch(:behavior, DEFAULT_BEHAVIOR)
transition :connection
end
def start!
@timestamps[:start] = Time.now
end
def finish!
@timestamps[:finish] = Time.now
end
def length
raise "Connection incomplete." unless @timestamps[:start] && @timestamps[:finish]
@timestamps[:finish].to_f - @timestamps[:start].to_f
end
def relaying?
@behavior == :relay
end
def delivering?
@behavior == :deliver
end
def print!(string)
print "#{string}\r\n"
end
def remote_ip
peeraddr(false)[3]
end
alias remote_addr remote_ip
def remote_host
# NOTE: This is currently a blocking operation.
peeraddr(true)[2]
end
end
| code |
<?php
/**
* Implements Special:UserLogin
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License along
* with this program; if not, write to the Free Software Foundation, Inc.,
* 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
* http://www.gnu.org/copyleft/gpl.html
*
* @file
* @ingroup SpecialPage
*/
/**
* Implements Special:UserLogin
*
* @ingroup SpecialPage
*/
class LoginForm extends SpecialPage {
const SUCCESS = 0;
const NO_NAME = 1;
const ILLEGAL = 2;
const WRONG_PLUGIN_PASS = 3;
const NOT_EXISTS = 4;
const WRONG_PASS = 5;
const EMPTY_PASS = 6;
const RESET_PASS = 7;
const ABORTED = 8;
const CREATE_BLOCKED = 9;
const THROTTLED = 10;
const USER_BLOCKED = 11;
const NEED_TOKEN = 12;
const WRONG_TOKEN = 13;
var $mUsername, $mPassword, $mRetype, $mReturnTo, $mCookieCheck, $mPosted;
var $mAction, $mCreateaccount, $mCreateaccountMail;
var $mLoginattempt, $mRemember, $mEmail, $mDomain, $mLanguage;
var $mSkipCookieCheck, $mReturnToQuery, $mToken, $mStickHTTPS;
var $mType, $mReason, $mRealName;
var $mAbortLoginErrorMsg = 'login-abort-generic';
/**
* @var ExternalUser
*/
private $mExtUser = null;
/**
* @param WebRequest $request
*/
public function __construct( $request = null ) {
parent::__construct( 'Userlogin' );
if ( $request === null ) {
global $wgRequest;
$this->load( $wgRequest );
} else {
$this->load( $request );
}
}
/**
* Loader
*
* @param $request WebRequest object
*/
function load( $request ) {
global $wgAuth, $wgHiddenPrefs, $wgEnableEmail, $wgRedirectOnLogin;
$this->mType = $request->getText( 'type' );
$this->mUsername = $request->getText( 'wpName' );
$this->mPassword = $request->getText( 'wpPassword' );
$this->mRetype = $request->getText( 'wpRetype' );
$this->mDomain = $request->getText( 'wpDomain' );
$this->mReason = $request->getText( 'wpReason' );
$this->mReturnTo = $request->getVal( 'returnto' );
$this->mReturnToQuery = $request->getVal( 'returntoquery' );
$this->mCookieCheck = $request->getVal( 'wpCookieCheck' );
$this->mPosted = $request->wasPosted();
$this->mCreateaccount = $request->getCheck( 'wpCreateaccount' );
$this->mCreateaccountMail = $request->getCheck( 'wpCreateaccountMail' )
&& $wgEnableEmail;
$this->mLoginattempt = $request->getCheck( 'wpLoginattempt' );
$this->mAction = $request->getVal( 'action' );
$this->mRemember = $request->getCheck( 'wpRemember' );
$this->mStickHTTPS = $request->getCheck( 'wpStickHTTPS' );
$this->mLanguage = $request->getText( 'uselang' );
$this->mSkipCookieCheck = $request->getCheck( 'wpSkipCookieCheck' );
$this->mToken = ( $this->mType == 'signup' ) ? $request->getVal( 'wpCreateaccountToken' ) : $request->getVal( 'wpLoginToken' );
if ( $wgRedirectOnLogin ) {
$this->mReturnTo = $wgRedirectOnLogin;
$this->mReturnToQuery = '';
}
if( $wgEnableEmail ) {
$this->mEmail = $request->getText( 'wpEmail' );
} else {
$this->mEmail = '';
}
if( !in_array( 'realname', $wgHiddenPrefs ) ) {
$this->mRealName = $request->getText( 'wpRealName' );
} else {
$this->mRealName = '';
}
if( !$wgAuth->validDomain( $this->mDomain ) ) {
$this->mDomain = 'invaliddomain';
}
$wgAuth->setDomain( $this->mDomain );
# When switching accounts, it sucks to get automatically logged out
$returnToTitle = Title::newFromText( $this->mReturnTo );
if( is_object( $returnToTitle ) && $returnToTitle->isSpecial( 'Userlogout' ) ) {
$this->mReturnTo = '';
$this->mReturnToQuery = '';
}
}
public function execute( $par ) {
if ( session_id() == '' ) {
wfSetupSession();
}
if ( $par == 'signup' ) { # Check for [[Special:Userlogin/signup]]
$this->mType = 'signup';
}
if ( !is_null( $this->mCookieCheck ) ) {
$this->onCookieRedirectCheck( $this->mCookieCheck );
return;
} elseif( $this->mPosted ) {
if( $this->mCreateaccount ) {
return $this->addNewAccount();
} elseif ( $this->mCreateaccountMail ) {
return $this->addNewAccountMailPassword();
} elseif ( ( 'submitlogin' == $this->mAction ) || $this->mLoginattempt ) {
return $this->processLogin();
}
}
$this->mainLoginForm( '' );
}
/**
* @private
*/
function addNewAccountMailPassword() {
global $wgOut;
if ( $this->mEmail == '' ) {
$this->mainLoginForm( wfMsgExt( 'noemail', array( 'parsemag', 'escape' ), $this->mUsername ) );
return;
}
$u = $this->addNewaccountInternal();
if ( $u == null ) {
return;
}
// Wipe the initial password and mail a temporary one
$u->setPassword( null );
$u->saveSettings();
$result = $this->mailPasswordInternal( $u, false, 'createaccount-title', 'createaccount-text' );
wfRunHooks( 'AddNewAccount', array( $u, true ) );
$u->addNewUserLogEntry( true, $this->mReason );
$wgOut->setPageTitle( wfMsg( 'accmailtitle' ) );
if( !$result->isGood() ) {
$this->mainLoginForm( wfMsg( 'mailerror', $result->getWikiText() ) );
} else {
$wgOut->addWikiMsg( 'accmailtext', $u->getName(), $u->getEmail() );
$wgOut->returnToMain( false );
}
}
/**
* @private
*/
function addNewAccount() {
global $wgUser, $wgEmailAuthentication, $wgOut;
# Create the account and abort if there's a problem doing so
$u = $this->addNewAccountInternal();
if( $u == null ) {
return;
}
# If we showed up language selection links, and one was in use, be
# smart (and sensible) and save that language as the user's preference
global $wgLoginLanguageSelector;
if( $wgLoginLanguageSelector && $this->mLanguage ) {
$u->setOption( 'language', $this->mLanguage );
}
# Send out an email authentication message if needed
if( $wgEmailAuthentication && User::isValidEmailAddr( $u->getEmail() ) ) {
$status = $u->sendConfirmationMail();
if( $status->isGood() ) {
$wgOut->addWikiMsg( 'confirmemail_oncreate' );
} else {
$wgOut->addWikiText( $status->getWikiText( 'confirmemail_sendfailed' ) );
}
}
# Save settings (including confirmation token)
$u->saveSettings();
# If not logged in, assume the new account as the current one and set
# session cookies then show a "welcome" message or a "need cookies"
# message as needed
if( $wgUser->isAnon() ) {
$wgUser = $u;
$wgUser->setCookies();
wfRunHooks( 'AddNewAccount', array( $wgUser, false ) );
$wgUser->addNewUserLogEntry();
if( $this->hasSessionCookie() ) {
return $this->successfulCreation();
} else {
return $this->cookieRedirectCheck( 'new' );
}
} else {
# Confirm that the account was created
$self = SpecialPage::getTitleFor( 'Userlogin' );
$wgOut->setPageTitle( wfMsgHtml( 'accountcreated' ) );
$wgOut->addWikiMsg( 'accountcreatedtext', $u->getName() );
$wgOut->returnToMain( false, $self );
wfRunHooks( 'AddNewAccount', array( $u, false ) );
$u->addNewUserLogEntry( false, $this->mReason );
return true;
}
}
/**
* @private
*/
function addNewAccountInternal() {
global $wgUser, $wgOut;
global $wgMemc, $wgAccountCreationThrottle;
global $wgAuth, $wgMinimalPasswordLength;
global $wgEmailConfirmToEdit;
// If the user passes an invalid domain, something is fishy
if( !$wgAuth->validDomain( $this->mDomain ) ) {
$this->mainLoginForm( wfMsg( 'wrongpassword' ) );
return false;
}
// If we are not allowing users to login locally, we should be checking
// to see if the user is actually able to authenticate to the authenti-
// cation server before they create an account (otherwise, they can
// create a local account and login as any domain user). We only need
// to check this for domains that aren't local.
if( 'local' != $this->mDomain && $this->mDomain != '' ) {
if( !$wgAuth->canCreateAccounts() && ( !$wgAuth->userExists( $this->mUsername )
|| !$wgAuth->authenticate( $this->mUsername, $this->mPassword ) ) ) {
$this->mainLoginForm( wfMsg( 'wrongpassword' ) );
return false;
}
}
if ( wfReadOnly() ) {
$wgOut->readOnlyPage();
return false;
}
# Request forgery checks.
if ( !self::getCreateaccountToken() ) {
self::setCreateaccountToken();
$this->mainLoginForm( wfMsgExt( 'nocookiesfornew', array( 'parseinline' ) ) );
return false;
}
# The user didn't pass a createaccount token
if ( !$this->mToken ) {
$this->mainLoginForm( wfMsg( 'sessionfailure' ) );
return false;
}
# Validate the createaccount token
if ( $this->mToken !== self::getCreateaccountToken() ) {
$this->mainLoginForm( wfMsg( 'sessionfailure' ) );
return false;
}
# Check permissions
if ( !$wgUser->isAllowed( 'createaccount' ) ) {
$wgOut->permissionRequired( 'createaccount' );
return false;
} elseif ( $wgUser->isBlockedFromCreateAccount() ) {
$this->userBlockedMessage( $wgUser->isBlockedFromCreateAccount() );
return false;
}
$ip = wfGetIP();
if ( $wgUser->isDnsBlacklisted( $ip, true /* check $wgProxyWhitelist */ ) ) {
$this->mainLoginForm( wfMsg( 'sorbs_create_account_reason' ) . ' (' . htmlspecialchars( $ip ) . ')' );
return false;
}
# Now create a dummy user ($u) and check if it is valid
$name = trim( $this->mUsername );
$u = User::newFromName( $name, 'creatable' );
if ( !is_object( $u ) ) {
$this->mainLoginForm( wfMsg( 'noname' ) );
return false;
}
if ( 0 != $u->idForName() ) {
$this->mainLoginForm( wfMsg( 'userexists' ) );
return false;
}
if ( 0 != strcmp( $this->mPassword, $this->mRetype ) ) {
$this->mainLoginForm( wfMsg( 'badretype' ) );
return false;
}
# check for minimal password length
$valid = $u->getPasswordValidity( $this->mPassword );
if ( $valid !== true ) {
if ( !$this->mCreateaccountMail ) {
if ( is_array( $valid ) ) {
$message = array_shift( $valid );
$params = $valid;
} else {
$message = $valid;
$params = array( $wgMinimalPasswordLength );
}
$this->mainLoginForm( wfMsgExt( $message, array( 'parsemag' ), $params ) );
return false;
} else {
# do not force a password for account creation by email
# set invalid password, it will be replaced later by a random generated password
$this->mPassword = null;
}
}
# if you need a confirmed email address to edit, then obviously you
# need an email address.
if ( $wgEmailConfirmToEdit && empty( $this->mEmail ) ) {
$this->mainLoginForm( wfMsg( 'noemailtitle' ) );
return false;
}
if( !empty( $this->mEmail ) && !User::isValidEmailAddr( $this->mEmail ) ) {
$this->mainLoginForm( wfMsg( 'invalidemailaddress' ) );
return false;
}
# Set some additional data so the AbortNewAccount hook can be used for
# more than just username validation
$u->setEmail( $this->mEmail );
$u->setRealName( $this->mRealName );
$abortError = '';
if( !wfRunHooks( 'AbortNewAccount', array( $u, &$abortError ) ) ) {
// Hook point to add extra creation throttles and blocks
wfDebug( "LoginForm::addNewAccountInternal: a hook blocked creation\n" );
$this->mainLoginForm( $abortError );
return false;
}
if ( $wgAccountCreationThrottle && $wgUser->isPingLimitable() ) {
$key = wfMemcKey( 'acctcreate', 'ip', $ip );
$value = $wgMemc->get( $key );
if ( !$value ) {
$wgMemc->set( $key, 0, 86400 );
}
if ( $value >= $wgAccountCreationThrottle ) {
$this->throttleHit( $wgAccountCreationThrottle );
return false;
}
$wgMemc->incr( $key );
}
if( !$wgAuth->addUser( $u, $this->mPassword, $this->mEmail, $this->mRealName ) ) {
$this->mainLoginForm( wfMsg( 'externaldberror' ) );
return false;
}
self::clearCreateaccountToken();
return $this->initUser( $u, false );
}
/**
* Actually add a user to the database.
* Give it a User object that has been initialised with a name.
*
* @param $u User object.
* @param $autocreate boolean -- true if this is an autocreation via auth plugin
* @return User object.
* @private
*/
function initUser( $u, $autocreate ) {
global $wgAuth;
$u->addToDatabase();
if ( $wgAuth->allowPasswordChange() ) {
$u->setPassword( $this->mPassword );
}
$u->setEmail( $this->mEmail );
$u->setRealName( $this->mRealName );
$u->setToken();
$wgAuth->initUser( $u, $autocreate );
if ( $this->mExtUser ) {
$this->mExtUser->linkToLocal( $u->getId() );
$email = $this->mExtUser->getPref( 'emailaddress' );
if ( $email && !$this->mEmail ) {
$u->setEmail( $email );
}
}
$u->setOption( 'rememberpassword', $this->mRemember ? 1 : 0 );
$u->saveSettings();
# Update user count
$ssUpdate = new SiteStatsUpdate( 0, 0, 0, 0, 1 );
$ssUpdate->doUpdate();
return $u;
}
/**
* Internally authenticate the login request.
*
* This may create a local account as a side effect if the
* authentication plugin allows transparent local account
* creation.
*/
public function authenticateUserData() {
global $wgUser, $wgAuth, $wgMemc;
if ( $this->mUsername == '' ) {
return self::NO_NAME;
}
// We require a login token to prevent login CSRF
// Handle part of this before incrementing the throttle so
// token-less login attempts don't count towards the throttle
// but wrong-token attempts do.
// If the user doesn't have a login token yet, set one.
if ( !self::getLoginToken() ) {
self::setLoginToken();
return self::NEED_TOKEN;
}
// If the user didn't pass a login token, tell them we need one
if ( !$this->mToken ) {
return self::NEED_TOKEN;
}
global $wgPasswordAttemptThrottle;
$throttleCount = 0;
if ( is_array( $wgPasswordAttemptThrottle ) ) {
$throttleKey = wfMemcKey( 'password-throttle', wfGetIP(), md5( $this->mUsername ) );
$count = $wgPasswordAttemptThrottle['count'];
$period = $wgPasswordAttemptThrottle['seconds'];
$throttleCount = $wgMemc->get( $throttleKey );
if ( !$throttleCount ) {
$wgMemc->add( $throttleKey, 1, $period ); // start counter
} elseif ( $throttleCount < $count ) {
$wgMemc->incr( $throttleKey );
} elseif ( $throttleCount >= $count ) {
return self::THROTTLED;
}
}
// Validate the login token
if ( $this->mToken !== self::getLoginToken() ) {
return self::WRONG_TOKEN;
}
// Load $wgUser now, and check to see if we're logging in as the same
// name. This is necessary because loading $wgUser (say by calling
// getName()) calls the UserLoadFromSession hook, which potentially
// creates the user in the database. Until we load $wgUser, checking
// for user existence using User::newFromName($name)->getId() below
// will effectively be using stale data.
if ( $wgUser->getName() === $this->mUsername ) {
wfDebug( __METHOD__ . ": already logged in as {$this->mUsername}\n" );
return self::SUCCESS;
}
$this->mExtUser = ExternalUser::newFromName( $this->mUsername );
# TODO: Allow some magic here for invalid external names, e.g., let the
# user choose a different wiki name.
$u = User::newFromName( $this->mUsername );
if( !( $u instanceof User ) || !User::isUsableName( $u->getName() ) ) {
return self::ILLEGAL;
}
$isAutoCreated = false;
if ( 0 == $u->getID() ) {
$status = $this->attemptAutoCreate( $u );
if ( $status !== self::SUCCESS ) {
return $status;
} else {
$isAutoCreated = true;
}
} else {
global $wgExternalAuthType, $wgAutocreatePolicy;
if ( $wgExternalAuthType && $wgAutocreatePolicy != 'never'
&& is_object( $this->mExtUser )
&& $this->mExtUser->authenticate( $this->mPassword ) ) {
# The external user and local user have the same name and
# password, so we assume they're the same.
$this->mExtUser->linkToLocal( $u->getID() );
}
$u->load();
}
// Give general extensions, such as a captcha, a chance to abort logins
$abort = self::ABORTED;
if( !wfRunHooks( 'AbortLogin', array( $u, $this->mPassword, &$abort, &$this->mAbortLoginErrorMsg ) ) ) {
return $abort;
}
global $wgBlockDisablesLogin;
if ( !$u->checkPassword( $this->mPassword ) ) {
if( $u->checkTemporaryPassword( $this->mPassword ) ) {
// The e-mailed temporary password should not be used for actu-
// al logins; that's a very sloppy habit, and insecure if an
// attacker has a few seconds to click "search" on someone's o-
// pen mail reader.
//
// Allow it to be used only to reset the password a single time
// to a new value, which won't be in the user's e-mail ar-
// chives.
//
// For backwards compatibility, we'll still recognize it at the
// login form to minimize surprises for people who have been
// logging in with a temporary password for some time.
//
// As a side-effect, we can authenticate the user's e-mail ad-
// dress if it's not already done, since the temporary password
// was sent via e-mail.
if( !$u->isEmailConfirmed() ) {
$u->confirmEmail();
$u->saveSettings();
}
// At this point we just return an appropriate code/ indicating
// that the UI should show a password reset form; bot inter-
// faces etc will probably just fail cleanly here.
$retval = self::RESET_PASS;
} else {
$retval = ( $this->mPassword == '' ) ? self::EMPTY_PASS : self::WRONG_PASS;
}
} elseif ( $wgBlockDisablesLogin && $u->isBlocked() ) {
// If we've enabled it, make it so that a blocked user cannot login
$retval = self::USER_BLOCKED;
} else {
$wgAuth->updateUser( $u );
$wgUser = $u;
// Please reset throttle for successful logins, thanks!
if( $throttleCount ) {
$wgMemc->delete( $throttleKey );
}
if ( $isAutoCreated ) {
// Must be run after $wgUser is set, for correct new user log
wfRunHooks( 'AuthPluginAutoCreate', array( $wgUser ) );
}
$retval = self::SUCCESS;
}
wfRunHooks( 'LoginAuthenticateAudit', array( $u, $this->mPassword, $retval ) );
return $retval;
}
/**
* Attempt to automatically create a user on login. Only succeeds if there
* is an external authentication method which allows it.
*
* @param $user User
*
* @return integer Status code
*/
function attemptAutoCreate( $user ) {
global $wgAuth, $wgUser, $wgAutocreatePolicy;
if ( $wgUser->isBlockedFromCreateAccount() ) {
wfDebug( __METHOD__ . ": user is blocked from account creation\n" );
return self::CREATE_BLOCKED;
}
/**
* If the external authentication plugin allows it, automatically cre-
* ate a new account for users that are externally defined but have not
* yet logged in.
*/
if ( $this->mExtUser ) {
# mExtUser is neither null nor false, so use the new ExternalAuth
# system.
if ( $wgAutocreatePolicy == 'never' ) {
return self::NOT_EXISTS;
}
if ( !$this->mExtUser->authenticate( $this->mPassword ) ) {
return self::WRONG_PLUGIN_PASS;
}
} else {
# Old AuthPlugin.
if ( !$wgAuth->autoCreate() ) {
return self::NOT_EXISTS;
}
if ( !$wgAuth->userExists( $user->getName() ) ) {
wfDebug( __METHOD__ . ": user does not exist\n" );
return self::NOT_EXISTS;
}
if ( !$wgAuth->authenticate( $user->getName(), $this->mPassword ) ) {
wfDebug( __METHOD__ . ": \$wgAuth->authenticate() returned false, aborting\n" );
return self::WRONG_PLUGIN_PASS;
}
}
$abortError = '';
if( !wfRunHooks( 'AbortAutoAccount', array( $user, &$abortError ) ) ) {
// Hook point to add extra creation throttles and blocks
wfDebug( "LoginForm::attemptAutoCreate: a hook blocked creation: $abortError\n" );
$this->mAbortLoginErrorMsg = $abortError;
return self::ABORTED;
}
wfDebug( __METHOD__ . ": creating account\n" );
$this->initUser( $user, true );
return self::SUCCESS;
}
function processLogin() {
global $wgUser;
switch ( $this->authenticateUserData() ) {
case self::SUCCESS:
# We've verified now, update the real record
if( (bool)$this->mRemember != (bool)$wgUser->getOption( 'rememberpassword' ) ) {
$wgUser->setOption( 'rememberpassword', $this->mRemember ? 1 : 0 );
$wgUser->saveSettings();
} else {
$wgUser->invalidateCache();
}
$wgUser->setCookies();
self::clearLoginToken();
// Reset the throttle
$key = wfMemcKey( 'password-throttle', wfGetIP(), md5( $this->mUsername ) );
global $wgMemc;
$wgMemc->delete( $key );
if( $this->hasSessionCookie() || $this->mSkipCookieCheck ) {
/* Replace the language object to provide user interface in
* correct language immediately on this first page load.
*/
global $wgLang, $wgRequest;
$code = $wgRequest->getVal( 'uselang', $wgUser->getOption( 'language' ) );
$wgLang = Language::factory( $code );
return $this->successfulLogin();
} else {
return $this->cookieRedirectCheck( 'login' );
}
break;
case self::NEED_TOKEN:
$this->mainLoginForm( wfMsgExt( 'nocookiesforlogin', array( 'parseinline' ) ) );
break;
case self::WRONG_TOKEN:
$this->mainLoginForm( wfMsg( 'sessionfailure' ) );
break;
case self::NO_NAME:
case self::ILLEGAL:
$this->mainLoginForm( wfMsg( 'noname' ) );
break;
case self::WRONG_PLUGIN_PASS:
$this->mainLoginForm( wfMsg( 'wrongpassword' ) );
break;
case self::NOT_EXISTS:
if( $wgUser->isAllowed( 'createaccount' ) ) {
$this->mainLoginForm( wfMsgExt( 'nosuchuser', 'parseinline', $this->mUsername ) );
} else {
$this->mainLoginForm( wfMsg( 'nosuchusershort', htmlspecialchars( $this->mUsername ) ) );
}
break;
case self::WRONG_PASS:
$this->mainLoginForm( wfMsg( 'wrongpassword' ) );
break;
case self::EMPTY_PASS:
$this->mainLoginForm( wfMsg( 'wrongpasswordempty' ) );
break;
case self::RESET_PASS:
$this->resetLoginForm( wfMsg( 'resetpass_announce' ) );
break;
case self::CREATE_BLOCKED:
$this->userBlockedMessage();
break;
case self::THROTTLED:
$this->mainLoginForm( wfMsg( 'login-throttled' ) );
break;
case self::USER_BLOCKED:
$this->mainLoginForm( wfMsgExt( 'login-userblocked',
array( 'parsemag', 'escape' ), $this->mUsername ) );
break;
case self::ABORTED:
$this->mainLoginForm( wfMsg( $this->mAbortLoginErrorMsg ) );
break;
default:
throw new MWException( 'Unhandled case value' );
}
}
function resetLoginForm( $error ) {
global $wgOut;
$wgOut->addHTML( Xml::element('p', array( 'class' => 'error' ), $error ) );
$reset = new SpecialChangePassword();
$reset->execute( null );
}
/**
* @param $u User object
* @param $throttle Boolean
* @param $emailTitle String: message name of email title
* @param $emailText String: message name of email text
* @return Status object
* @private
*/
function mailPasswordInternal( $u, $throttle = true, $emailTitle = 'passwordremindertitle', $emailText = 'passwordremindertext' ) {
global $wgServer, $wgScript, $wgUser, $wgNewPasswordExpiry;
if ( $u->getEmail() == '' ) {
return Status::newFatal( 'noemail', $u->getName() );
}
$ip = wfGetIP();
if( !$ip ) {
return Status::newFatal( 'badipaddress' );
}
wfRunHooks( 'User::mailPasswordInternal', array( &$wgUser, &$ip, &$u ) );
$np = $u->randomPassword();
$u->setNewpassword( $np, $throttle );
$u->saveSettings();
$userLanguage = $u->getOption( 'language' );
$m = wfMsgExt( $emailText, array( 'parsemag', 'language' => $userLanguage ), $ip, $u->getName(), $np,
$wgServer . $wgScript, round( $wgNewPasswordExpiry / 86400 ) );
$result = $u->sendMail( wfMsgExt( $emailTitle, array( 'parsemag', 'language' => $userLanguage ) ), $m );
return $result;
}
/**
* Run any hooks registered for logins, then HTTP redirect to
* $this->mReturnTo (or Main Page if that's undefined). Formerly we had a
* nice message here, but that's really not as useful as just being sent to
* wherever you logged in from. It should be clear that the action was
* successful, given the lack of error messages plus the appearance of your
* name in the upper right.
*
* @private
*/
function successfulLogin() {
global $wgUser, $wgOut;
# Run any hooks; display injected HTML if any, else redirect
$injected_html = '';
wfRunHooks( 'UserLoginComplete', array( &$wgUser, &$injected_html ) );
if( $injected_html !== '' ) {
$this->displaySuccessfulLogin( 'loginsuccess', $injected_html );
} else {
$titleObj = Title::newFromText( $this->mReturnTo );
if ( !$titleObj instanceof Title ) {
$titleObj = Title::newMainPage();
}
$redirectUrl = $titleObj->getFullURL( $this->mReturnToQuery );
global $wgSecureLogin;
if( $wgSecureLogin && !$this->mStickHTTPS ) {
$redirectUrl = preg_replace( '/^https:/', 'http:', $redirectUrl );
}
$wgOut->redirect( $redirectUrl );
}
}
/**
* Run any hooks registered for logins, then display a message welcoming
* the user.
*
* @private
*/
function successfulCreation() {
global $wgUser;
# Run any hooks; display injected HTML
$injected_html = '';
$welcome_creation_msg = 'welcomecreation';
wfRunHooks( 'UserLoginComplete', array( &$wgUser, &$injected_html ) );
//let any extensions change what message is shown
wfRunHooks( 'BeforeWelcomeCreation', array( &$welcome_creation_msg, &$injected_html ) );
$this->displaySuccessfulLogin( $welcome_creation_msg, $injected_html );
}
/**
* Display a "login successful" page.
*/
private function displaySuccessfulLogin( $msgname, $injected_html ) {
global $wgOut, $wgUser;
$wgOut->setPageTitle( wfMsg( 'loginsuccesstitle' ) );
if( $msgname ){
$wgOut->addWikiMsg( $msgname, $wgUser->getName() );
}
$wgOut->addHTML( $injected_html );
if ( !empty( $this->mReturnTo ) ) {
$wgOut->returnToMain( null, $this->mReturnTo, $this->mReturnToQuery );
} else {
$wgOut->returnToMain( null );
}
}
/**
* Output a message that informs the user that they cannot create an account because
* there is a block on them or their IP which prevents account creation. Note that
* User::isBlockedFromCreateAccount(), which gets this block, ignores the 'hardblock'
* setting on blocks (bug 13611).
* @param $block Block the block causing this error
*/
function userBlockedMessage( Block $block ) {
global $wgOut;
# Let's be nice about this, it's likely that this feature will be used
# for blocking large numbers of innocent people, e.g. range blocks on
# schools. Don't blame it on the user. There's a small chance that it
# really is the user's fault, i.e. the username is blocked and they
# haven't bothered to log out before trying to create an account to
# evade it, but we'll leave that to their guilty conscience to figure
# out.
$wgOut->setPageTitle( wfMsg( 'cantcreateaccounttitle' ) );
$block_reason = $block->mReason;
if ( strval( $block_reason ) === '' ) {
$block_reason = wfMsg( 'blockednoreason' );
}
$wgOut->addWikiMsg(
'cantcreateaccount-text',
$block->getTarget(),
$block_reason,
$block->getBlocker()->getName()
);
$wgOut->returnToMain( false );
}
/**
* @private
*/
function mainLoginForm( $msg, $msgtype = 'error' ) {
global $wgUser, $wgOut, $wgHiddenPrefs;
global $wgEnableEmail, $wgEnableUserEmail;
global $wgRequest, $wgLoginLanguageSelector;
global $wgAuth, $wgEmailConfirmToEdit, $wgCookieExpiration;
global $wgSecureLogin, $wgPasswordResetRoutes;
$titleObj = SpecialPage::getTitleFor( 'Userlogin' );
if ( $this->mType == 'signup' ) {
// Block signup here if in readonly. Keeps user from
// going through the process (filling out data, etc)
// and being informed later.
if ( wfReadOnly() ) {
$wgOut->readOnlyPage();
return;
} elseif ( $wgUser->isBlockedFromCreateAccount() ) {
$this->userBlockedMessage( $wgUser->isBlockedFromCreateAccount() );
return;
} elseif ( count( $permErrors = $titleObj->getUserPermissionsErrors( 'createaccount', $wgUser, true ) )>0 ) {
$wgOut->showPermissionsErrorPage( $permErrors, 'createaccount' );
return;
}
}
if ( $this->mUsername == '' ) {
if ( $wgUser->isLoggedIn() ) {
$this->mUsername = $wgUser->getName();
} else {
$this->mUsername = $wgRequest->getCookie( 'UserName' );
}
}
if ( $this->mType == 'signup' ) {
$template = new UsercreateTemplate();
$q = 'action=submitlogin&type=signup';
$linkq = 'type=login';
$linkmsg = 'gotaccount';
} else {
$template = new UserloginTemplate();
$q = 'action=submitlogin&type=login';
$linkq = 'type=signup';
$linkmsg = 'nologin';
}
if ( !empty( $this->mReturnTo ) ) {
$returnto = '&returnto=' . wfUrlencode( $this->mReturnTo );
if ( !empty( $this->mReturnToQuery ) ) {
$returnto .= '&returntoquery=' .
wfUrlencode( $this->mReturnToQuery );
}
$q .= $returnto;
$linkq .= $returnto;
}
# Pass any language selection on to the mode switch link
if( $wgLoginLanguageSelector && $this->mLanguage ) {
$linkq .= '&uselang=' . $this->mLanguage;
}
$link = '<a href="' . htmlspecialchars ( $titleObj->getLocalURL( $linkq ) ) . '">';
$link .= wfMsgHtml( $linkmsg . 'link' ); # Calling either 'gotaccountlink' or 'nologinlink'
$link .= '</a>';
# Don't show a "create account" link if the user can't
if( $this->showCreateOrLoginLink( $wgUser ) ) {
$template->set( 'link', wfMsgExt( $linkmsg, array( 'parseinline', 'replaceafter' ), $link ) );
} else {
$template->set( 'link', '' );
}
$resetLink = $this->mType == 'signup'
? null
: is_array( $wgPasswordResetRoutes ) && in_array( true, array_values( $wgPasswordResetRoutes ) );
$template->set( 'header', '' );
$template->set( 'name', $this->mUsername );
$template->set( 'password', $this->mPassword );
$template->set( 'retype', $this->mRetype );
$template->set( 'email', $this->mEmail );
$template->set( 'realname', $this->mRealName );
$template->set( 'domain', $this->mDomain );
$template->set( 'reason', $this->mReason );
$template->set( 'action', $titleObj->getLocalURL( $q ) );
$template->set( 'message', $msg );
$template->set( 'messagetype', $msgtype );
$template->set( 'createemail', $wgEnableEmail && $wgUser->isLoggedIn() );
$template->set( 'userealname', !in_array( 'realname', $wgHiddenPrefs ) );
$template->set( 'useemail', $wgEnableEmail );
$template->set( 'emailrequired', $wgEmailConfirmToEdit );
$template->set( 'emailothers', $wgEnableUserEmail );
$template->set( 'canreset', $wgAuth->allowPasswordChange() );
$template->set( 'resetlink', $resetLink );
$template->set( 'canremember', ( $wgCookieExpiration > 0 ) );
$template->set( 'usereason', $wgUser->isLoggedIn() );
$template->set( 'remember', $wgUser->getOption( 'rememberpassword' ) || $this->mRemember );
$template->set( 'cansecurelogin', ( $wgSecureLogin === true ) );
$template->set( 'stickHTTPS', $this->mStickHTTPS );
if ( $this->mType == 'signup' ) {
if ( !self::getCreateaccountToken() ) {
self::setCreateaccountToken();
}
$template->set( 'token', self::getCreateaccountToken() );
} else {
if ( !self::getLoginToken() ) {
self::setLoginToken();
}
$template->set( 'token', self::getLoginToken() );
}
# Prepare language selection links as needed
if( $wgLoginLanguageSelector ) {
$template->set( 'languages', $this->makeLanguageSelector() );
if( $this->mLanguage )
$template->set( 'uselang', $this->mLanguage );
}
// Give authentication and captcha plugins a chance to modify the form
$wgAuth->modifyUITemplate( $template, $this->mType );
if ( $this->mType == 'signup' ) {
wfRunHooks( 'UserCreateForm', array( &$template ) );
} else {
wfRunHooks( 'UserLoginForm', array( &$template ) );
}
// Changes the title depending on permissions for creating account
if ( $wgUser->isAllowed( 'createaccount' ) ) {
$wgOut->setPageTitle( wfMsg( 'userlogin' ) );
} else {
$wgOut->setPageTitle( wfMsg( 'userloginnocreate' ) );
}
$wgOut->disallowUserJs(); // just in case...
$wgOut->addTemplate( $template );
}
/**
* @private
*
* @param $user User
*
* @return Boolean
*/
function showCreateOrLoginLink( &$user ) {
if( $this->mType == 'signup' ) {
return true;
} elseif( $user->isAllowed( 'createaccount' ) ) {
return true;
} else {
return false;
}
}
/**
* Check if a session cookie is present.
*
* This will not pick up a cookie set during _this_ request, but is meant
* to ensure that the client is returning the cookie which was set on a
* previous pass through the system.
*
* @private
*/
function hasSessionCookie() {
global $wgDisableCookieCheck, $wgRequest;
return $wgDisableCookieCheck ? true : $wgRequest->checkSessionCookie();
}
/**
* Get the login token from the current session
*/
public static function getLoginToken() {
global $wgRequest;
return $wgRequest->getSessionData( 'wsLoginToken' );
}
/**
* Randomly generate a new login token and attach it to the current session
*/
public static function setLoginToken() {
global $wgRequest;
// Use User::generateToken() instead of $user->editToken()
// because the latter reuses $_SESSION['wsEditToken']
$wgRequest->setSessionData( 'wsLoginToken', User::generateToken() );
}
/**
* Remove any login token attached to the current session
*/
public static function clearLoginToken() {
global $wgRequest;
$wgRequest->setSessionData( 'wsLoginToken', null );
}
/**
* Get the createaccount token from the current session
*/
public static function getCreateaccountToken() {
global $wgRequest;
return $wgRequest->getSessionData( 'wsCreateaccountToken' );
}
/**
* Randomly generate a new createaccount token and attach it to the current session
*/
public static function setCreateaccountToken() {
global $wgRequest;
$wgRequest->setSessionData( 'wsCreateaccountToken', User::generateToken() );
}
/**
* Remove any createaccount token attached to the current session
*/
public static function clearCreateaccountToken() {
global $wgRequest;
$wgRequest->setSessionData( 'wsCreateaccountToken', null );
}
/**
* @private
*/
function cookieRedirectCheck( $type ) {
global $wgOut;
$titleObj = SpecialPage::getTitleFor( 'Userlogin' );
$query = array( 'wpCookieCheck' => $type );
if ( $this->mReturnTo ) {
$query['returnto'] = $this->mReturnTo;
}
$check = $titleObj->getFullURL( $query );
return $wgOut->redirect( $check );
}
/**
* @private
*/
function onCookieRedirectCheck( $type ) {
if ( !$this->hasSessionCookie() ) {
if ( $type == 'new' ) {
return $this->mainLoginForm( wfMsgExt( 'nocookiesnew', array( 'parseinline' ) ) );
} elseif ( $type == 'login' ) {
return $this->mainLoginForm( wfMsgExt( 'nocookieslogin', array( 'parseinline' ) ) );
} else {
# shouldn't happen
return $this->mainLoginForm( wfMsg( 'error' ) );
}
} else {
return $this->successfulLogin();
}
}
/**
* @private
*/
function throttleHit( $limit ) {
$this->mainLoginForm( wfMsgExt( 'acct_creation_throttle_hit', array( 'parseinline' ), $limit ) );
}
/**
* Produce a bar of links which allow the user to select another language
* during login/registration but retain "returnto"
*
* @return string
*/
function makeLanguageSelector() {
global $wgLang;
$msg = wfMessage( 'loginlanguagelinks' )->inContentLanguage();
if( !$msg->isBlank() ) {
$langs = explode( "\n", $msg->text() );
$links = array();
foreach( $langs as $lang ) {
$lang = trim( $lang, '* ' );
$parts = explode( '|', $lang );
if ( count( $parts ) >= 2 ) {
$links[] = $this->makeLanguageSelectorLink( $parts[0], $parts[1] );
}
}
return count( $links ) > 0 ? wfMsgHtml( 'loginlanguagelabel', $wgLang->pipeList( $links ) ) : '';
} else {
return '';
}
}
/**
* Create a language selector link for a particular language
* Links back to this page preserving type and returnto
*
* @param $text Link text
* @param $lang Language code
*/
function makeLanguageSelectorLink( $text, $lang ) {
global $wgUser;
$self = SpecialPage::getTitleFor( 'Userlogin' );
$attr = array( 'uselang' => $lang );
if( $this->mType == 'signup' ) {
$attr['type'] = 'signup';
}
if( $this->mReturnTo ) {
$attr['returnto'] = $this->mReturnTo;
}
$skin = $wgUser->getSkin();
return $skin->linkKnown(
$self,
htmlspecialchars( $text ),
array(),
$attr
);
}
}
| code |
مگر تِمَو کوٚر یہِ کارنامہٕ | kashmiri |
लखनऊ: लोगों की मदद को बस स्टैंड पहुंचे डप, कराई खाने और पानी की व्यवस्था
कोरोना वायरस में चलते पूरे देश में लॉक डाउन हो रखा है। जिसकी वजह से सैकड़ों लोग बस स्टैंड और रेलवे स्टेशन पर फंसे हुए हैं। ऐसे में यूपी पुलिस के डीजीपी हितेश चन्द्र अवस्थी समेत सभी पुलिस और प्रशासनिक अधिकारी लखनऊ के चारबाग बस स्टेशन पर लोगों की मदद कर रहे हैं। इतना ही नहीं प्रशासनिक अफसर बस स्टैंड पर फंसे लोगों की मदद को बस भी चलवा रहे हैं, ताकि किसी को कोई दिक्कत ना आने पाए।
डीजीपी उतरे फील्ड पर
जानकारी के मुताबिक, लॉक डाउन के दौरान के चारबाग से यात्रियों की सुविधा के लिये बस की व्यवस्था की गई है। ताकि हर कोई अपने गंतव्य तक पहुँच सके। कानपुर, बलिया, बनारस, गोरखपुर, आजमगढ़, फैजाबाद, बस्ती, प्रतापगढ़, सुल्तानपुर, अमेठी, रायबरेली, गोंडा, इटावा, बहराइच, श्रावस्ती, ऐसे कई जिलों की बसें यात्रियों को बैठाकर भेजी गई हैं। इतना ही नहीं डीजीपी खुद चारबाग जाकर लोगों की मदद के निर्देश दे रहे हैं
शनिवार की सुबह ही डीजीपी हितेश चन्द्र अवस्थी, लखनऊ पुलिस कमिश्नर सुजीत पांडेय के साथ चारबाग पहुंचे। जहां उन्होंने लोगो को खाने और पानी व्यवस्था भी की। डीजीपी ने खाने के पैकेट और पानी बोतलें लोगों को दिन, ताकि किसी को कोई दिक्कत ना होने पाए। और सभी लोग अपने घर सकुशल पहुंच जाएं।
लखनऊ पुलिस कर रही हर इंतजाम
कोरोना वायरस से जंग जीतने के लिए पूरे देश में लॉक डाउन किया गया है ताकि वायरस अपने पैर ना पसार सके। इस लॉक डाउन से सबसे ज्यादा फ़र्क मजदूरों और उन लोगों को हुए है, जोकि रोज की कमाई से घर चलाए हैं, सड़क किनारे खड़े ठेलों पर खाना खाकर अपना पेट भरते हैं। ऐसे लोगों की मदद को हर जिले में पुलिस आगे आती है। हम आपको दिखा रहे हैं लखनऊ की तस्वीरें जहां पुलिसकर्मी खुद आगे आकर बेसहारा लोगों का सहारा बन रहे हैं, उन्हें खाना खिला रहे हैं।
आपके किचन में ही है ब्लड प्रेशर का उपचार, इस प्रकार उठाएं लाभ
कोरोना: उप ब्जप ने गरीब और जरूरतमंदों के लिए जारी किया हेल्पलाइन नंबर, अध्यक्ष और महामंत्री ने बनाया मदद का प्लान
बांदा: प्र्व वाहन को तेज रफ्तार ट्रक ने मारी जोरदार टक्कर, बाहर गिरने पर सिपाही को रौंदा
मुरादाबाद: दुष्कर्म पीड़िता से सिपाही ले रहा था सात फेरे, आरोपी ने बीच में आकर रुकवाई शादी | hindi |
package org.se.lab;
import org.junit.Test;
public class SimpleTest
{
@Test
public void testSomething()
{
String s = "Hallo";
s.trim();
}
}
| code |
Published 04/19/2019 08:19:43 pm at 04/19/2019 08:19:43 pm in Tropical Landscape.
tropical landscape awesome tropical landscaping ideas tropical images with neon colors.
tropical meaning in hindi,tropical landscape meaning in urdu,tropical landscape concepts,cheap landscape trees,tropical landscape finance facility solutions,tropical images transparent background,tropical landscape finance facility maintenance,tropical meaning in urdu,beach landscape wallpaper travel,tropical landscapes images pictures,tropical landscapes images for painting. | english |
# Development Environment
> To take full advantage of TypeScript with autocomplete you would have to use an editor with the correct TypeScript plugins.
## Use a TypeScript-aware editor
I highly recommend [Visual Studio Code](https://code.visualstudio.com/) over Atom, or Sublime Text.
Even being a developer of a couple Atom packages and a contributor to Atom, and purchasing Sublime Text 3.
I recommend VS Code, because we are working with TypeScript, and VS Code is nearly an IDE for TypeScript.
For Visual Studio Code, I recommend using the following plugins:
* Color Picker - anseki
* EditorConfig for VS Code - EditorConfig
* TSLint - egamma
* CodeMetrics - Kiss Tamas
* Subword Navigation - ow
* Modified Seti Theme - Tarique Naseem
* Git Blame - Wade Anderson
## VS Code Configuration
### Fira Code Fontface
I highly recommend FiraCode-Retina as your choice fontface, since it has some pretty cool ligatures.
### Editor Preferences
Just a suggestion, modify at your own will.
```json
// Place your settings in this file to overwrite the default settings
{
"editor.tabSize": 2,
"editor.fontFamily": "Fira Code Retina, Operator Mono, 'Courier New', monospace",
"guides.active.color.light": "rgba(100, 100, 100, 0.75)",
"editor.renderIndentGuides": true,
"editor.fontLigatures": true,
"workbench.quickOpen.closeOnFocusLost": false,
// Enable word based suggestions.
"editor.wordBasedSuggestions": true,
// Controls the font size in pixels.
"editor.fontSize": 15,
// Controls the line height. Use 0 to compute the lineHeight from the fontSize.
"editor.lineHeight": 18,
"files.associations": {
"*.fbs": "C"
},
"typescript.check.tscVersion": false
}
```
### Keyboard shortcuts for `subwordNavigation`
```json
// Place your key bindings in this file to overwrite the defaults
[
{
"key": "alt+left",
"command": "subwordNavigation.cursorSubwordLeft",
"when": "editorTextFocus"
},
{
"key": "alt+right",
"command": "subwordNavigation.cursorSubwordRight",
"when": "editorTextFocus"
},
{
"key": "alt+shift+left",
"command": "subwordNavigation.cursorSubwordLeftSelect",
"when": "editorTextFocus"
},
{
"key": "alt+shift+right",
"command": "subwordNavigation.cursorSubwordRightSelect",
"when": "editorTextFocus"
}
]
```
| code |
In arguably the greatest, most enlightening discovery since the Rosetta Stone, cat ambassador Gemma Correll has managed to get hold of the secret diary of Tiddles, a three-year-old moggy, which she has transcribed and illustrated in order to explain to cat owners across the globe what their pet gets up to when left to its own devices. By reading through Tiddles’ journal, many of those great unanswered questions you have about your cat’s life will be answered, like what does he does for hours on end when he crawls under the bed just as you leave for work or where has she been when she barges her way through the cat flap and into the kitchen, yowling for attention? It’s a true insight into the secret world of cats and a must-have, eye-opening read for any cat owner. | english |
I was in the fourth grade when I fell in love with reading. It all started when my cousin told me to read the Harry Potter series, which is exactly what I did, eventually. At first, I was skeptical about reading them because if you’ve seen these books, you would know that they are huuuge. This made me hesitant because I thought there was no way I could ever finish that book, but I later decided to just get over my fears and read the books. And let me tell you, that was the best decision I have ever made in my prepubescent years. I mean what could be better than a few teenage wizards fighting against a scary guy with no nose in a cool world where magic actually existed. Its exciting, its fun, its a break from reality. From then on out, I was always the girl with her nose in a book. I read everything from fantasies to mysteries to horrors to romance. There was nothing I didn’t like. Well actually scratch that, I disliked most nonfiction novels because I just thought they were complete snoozers.
I thought it was just incredible how you could lose yourself in a great book and ignore everything that went on around you. It’s almost as if you were living a different life when you read. Books were a wonderful way to just forget and jump into a world where things that were stressing you out didn’t matter any more and you could just relax for a few hours. They illustrated things that you never thought could be possible: wizards, potions, magic, fairies, vampires, and the list goes on and on.
The sad thing is, as I began high school, I found myself reading less and less and less. Maybe its because of the larger workload or because I’m tired all the time. I don’t really know to be honest. Don’t get me wrong, I still love reading and I wish I would read more but there’s just no time for me to sit down and read a nice, long book anymore with school and other extracurricular activities. But that’s going to change starting now; I’m going to start reading more, even if that means I’m only gonna read 5 pages a day, so be it. It’s definitely a start.
Well thanks for reading all you lovely people! See you all next week.
I hope you don’t mind, but I nominated you for a Leibster Award. I really love your blog and wasn’t quite sure whether you had got one already but the questions are on my blog if you want to take a look at them.
Oh I definitely do not mind at all. Wow, I’m so flattered and you are such a lovely person, thank you so much! | english |
<?php
namespace TYPO3\Eel;
/* *
* This script belongs to the TYPO3 Flow package "TYPO3.Eel". *
* *
* It is free software; you can redistribute it and/or modify it under *
* the terms of the GNU Lesser General Public License, either version 3 *
* of the License, or (at your option) any later version. *
* *
* The TYPO3 project - inspiring people to share! *
* */
/**
* A general Eel exception
*/
class Exception extends \TYPO3\Flow\Exception {
}
| code |
It’s November 11th. I should have finished last night at 16,667 words. Instead, I went to bed with 25,151 under my belt.
The 50,000 word aim in 30 days has always been “do-able” for me.
I met the goal in 2009, though the ideas were very much stolen from another set of books.
In 2010, I wrote the ideas myself but had a lot of loose ends and half-written scenes. I didn’t finish the story, though a book was ‘done’.
In 2011, I used a lot of cheats to win.
But I’ve completed it 3 years in a row.
This year, I decided not to use any cheats. I use a # to denote the end of scene, as is common practise in manuscripts. I have chapters which adds 2 words every 5000 odd. I even have two-half scenes at the end; which I hope to include, but didn’t want to happen until later in the book. I wrote them in the 30 days, so I’m counting them.
I thought this year would be harder, and so far it is.
As Rachel Aaron pointed out on the forums, a sequel is harder.
As someone who’s currently finishing up the third book in her second series, I’m of the opinion that second books are the hardest, because you’re still figuring out the world and the characters to a large extent, only now, thanks to book 1, you can’t just change everything to suit your plot.
So I never expected to write 5,019 words on day 5. I didn’t expect to suddenly be ahead by three days. Yesterday, I wrote 5,925. And I’ve been keeping track of the actual time spent writing, to give me an average words per hour and and idea of how many hours it takes to write a novel.
I’ve been using a tag on twitter to do word sprints, where you set a time of say, 15 minutes, and see how many words you can write, and then compare it to others. I’ve begun running my own with friends and strangers who wanted to do extra ones.
This year I’ve also ended up with more people ahead of me (in terms of word-count) than behind me, which is rare. It’s an odd feeling to wake up and find people are 6-10k ahead of my own, also-ahead goal. Yesterday morning, one of my new NaNoBuddies in Aus (so it was the end of day 10 for them) was at 33k.
I thought after three years and having written the first novel, this would be more familiar. Not easier, but that I’d know what to expect. Instead, I’m wondering if I can meet a higher goal than the 50k in 30 days, and desperately trying to catch up with my friend who reached 30k as I passed 25.
If I can reach 30k tonight I might actually finish the book by the end of the month, reaching 85k. However, I’ve only got the next four scenes planned out and I’m aware of all the things I need to do about my new job, finding a flat, de-cluttering and packing for the move, writing my blog, practising driving, doing housework, working out finances and such.
NaNoWriMo is always one of those things I look forward to, although I’ve attempted just as many novels outside of November too. What I hadn’t expected was the connection with others, the ability to prove I can write 650 words in 15 minutes and the way a novel kind of makes itself once your make the world and characters real.
Just create characters, set them down in a place, and they will live.
It’s January. Last year, I did a 3/2/1 recap each month. For this first month, I’ll be talking a little more about my four focus attributes: space, legacy, emotion and practise.
If you missed my first post, head here to view my plans around Connecting with Space.
In July 2011, I was told that my Grandfather had suspected Dementia.
I won’t go into the massive details but he is the only male in my entire family I think has played his cards right, is a fully worthy human being and has done the best he can. He has the kindest heart and the gentlest manner, despite fighting for his beliefs; his mind has always been sharp and he loves to walk up mountains despite being quite old.
Legacy is finding out that I am like him, that I have some of that amazing greatness in my potential. I know of his life choices, his mistakes and especially the things he regrets but I think were the right choice.
As a Bardic student, I’m now learning the legacy of the Druids; the poems that taught of magic and emotions, and as a reader, the parables which told of the morals (Western) humans usually abide by. This has given me a glimpse of how much knowing the legacy and heritage of a belief, a label or an action can enrich an experience.
Legacy is important, and none more important to me than my own.
I want to know and understand my legacy – my family tree and the heritage of the family I’ve made for myself.
I’m looking at the genealogy within my bloodline; of the Irish great-grandparents, of the quintessentially English grandparents, of the French surname and of the Estuary English dialect I speak mixed with my mother’s odd northern pronunciation.
This is a delve into the past; of who I was made to be, how I got here and what possibilities lay within my veins.
I’m trying to bring that sense of welcome into my own personality; for nothing is more important than honour and hospitality in my grandfather’s eyes.
Unfortunately, I grew up miles from him and only one mile from other family members, who value educational terms, grades, money and… well. They said “That’s allowed then” when I told them my other half was studying Medicine.
This year I’m exploring the labels I’d always thought I couldn’t change, and deliberately adopting new ones through re-definition.
This includes the labels around being a student, a wife, a mother; all the things I hope to be. I want to really understand where my values are in how to treat differences (my family are generally quite judgemental), how to deal with emotions like anger and my views on non-violence.
This links to my third focus a little but essentially, emotional legacy covers my past labels around emotion like anger solves everything & weak people cry, and covers what kind of legacy I want to pass on to my friends, to my kids, to my friend’s kids.
I’m not yet sure how to go around this, except I’ve begun a journey along my thoughts of compassion over at The Phoenix Mind, which I’ll be updating each time I feel I’ve something to share. This will also cover the third aspect of my theme for the year; emotion a little too.
Finally, I want to continue the journey I began at 11 when I lost my faith in Christianity.
I want to really understand my choice not to continue with that path, to be able to explain why I’ve chosen Druidry, and to get my head around what I want to pass on to my children.
I want to understand the heritage of the land; this sacred isle surrounded by water and covered with history and mythology. And in exploring Celtic spirituality, reach out to the Germanic and Saxon paths which have influenced it.
Legacy is all about my past lessons and what I want to pass on.
Next time I’ll cover the aspect of Compassion and the emotional aspects touched on here in more detail.
It’s January. Last year, I did a 3/2/1 recap. This year, I’ll be doing a similar thing at the end of the month with my outcomes for month 1 and the plans for month 2.
For this first month though, I’ll be talking a little more about my four focus attributes: space, legacy, emotion and practise.
Practical goals such as budgeting, exercise, healthy food, exploring minimalism, keeping space for spirituality and sleep are interesting in terms of a student who straddles two bedrooms but no external house-space.
Connecting to Space is a major goal of mine this year, and covers my bedroom/living space, getting outside to explore the land and my bodily space; what I feed it, how I treat it and how I sleep.
My bedroom in my parents house is my only “me-space” – the only space which has my belongings in it. I think the cabinet in the living room has a statue I made when I was 3 and a porcelain doll which I wasn’t trusted to keep safe as a young child; but all my living items are upstairs; in this medium-sized room.
At university, it’s a similar story – one room holds a study, exercise space, ritual space and a bedroom, all encompassed within a room smaller than my home bedroom.
No wonder I can’t get rid of half the stuff… a lot of it would live in a living room if I had one.
So this year, I want to streamline, to minimalise, to find the right amount of stuff that will keep me happy but make my rooms pleasant and easy to find things in.
Again, being a student, I have a load of crap I won’t need once I’ve graduated (unless I get into the PhD in which case it’ll all be bundled up once more and taken to a new tiny study/bedroom –sigh–), but can’t yet part with as I’m still using it.
However, I’m looking at getting rid of the junk freebies from first year and clearing the floor space around my desk so that the carpet is clear and usable again (I like to dance and do exercise; both of which have caused injury as I’ve trod on a plug or broken something underfoot).
This process should then leave me a space to sit and meditate, a space to place candles and incense, and a space to just keep clear to make the room feel spacious. I can put down soft cushions or just put images on that part of wall and sit/lie beside it.
I’d like to set up a proper altar I can use for my pagan practise. And then use it regularly.
And I know nothing about this land; the plants which grow here, the animals which live here and the history of the areas.
I want to connect with the trees, to climb the hills and photograph the flowers. Once a fortnight will be my beginning goal; spend 30 minutes each two weeks outside, dedicating space to the outdoors.
Then we have my body; where food and exercise come in. I want to keep my brain going with knowledge and my mind healthy in terms of well-being; and then eat fresh foods and get the right amount of sleep and exercise. That alone can be a massive job, let alone as part of four separate foci this year.
I want to cut down on the processed foods; on chocolate and crisps, instead turning to healthier alternatives and getting my body used to gaps between meals. I want to get a good exercise routine going, and to keep my sleep and brain as healthy and effective as possible.
If you’ve been following my blog for long, you’ll know I’m both a fussy eater, and terrified of weight loss. I want to have energy and be healthy, but I mustn’t lose ANY weight as I’m still technically “underweight” according to the BMI measures and despite eating well over 2500 calories a day in chocolate, potatoes, marzipan, nuts and actual meals.
However, I am going to try and shift my unhealthy eating habits; but do so with a vigilant eye on the scales (once a fortnight). I’ll be making pasta salad and ham or marmite sandwhiches for lunches, re-heating rice (zomg, no!) and taking my own flasks of green tea into University.
– Protein every day, three forms a week (three types of meat or one meat, quorn & vegetable protein).
– Potatoes!!! At least once a week on average!
– Use up that crap in the cupboards!!!
Then in terms of sleep; get to bed by 10:30pm and TV off by 11:45pm. I have a 9am start on Thursdays this term; while my earliest was 11am last year; so those times may alter slightly as I’ll be waking at 7am.
So those are my goals in terms of space. Next time I’ll cover the aspect of Legacy and what heritage means to me this year.
As my word for 2012, connect, has four parts, I figured that gives me three specific goals for each; leading to twelve goals (either for the 12 in 2012, or for one a month).
In short, I want to connect in terms of physical space (room, body, land), legacy (habits, labels, potential), compassion (emotions, actions, biases) and practise (paganism/spirit, knowledge, energy).
As it turns out, there are thirteen; just as there are thirteen moons a year. I can handle that. I tried to put them into four groups, with three in each, but there were so many overlaps, I gave up. Here are my thirteen; together with the categories they come under.
I haven’t yet set myself time-frames, assigned months or moons to them; nor decided on how to go about all of them yet, but that will come in time.
Have a fantastic new year, good luck with your plans and I’ll see you on the other side.
Last post I spoke about Strength, my word for 2011 and how I’ve found the journey through it. Let’s get onto the coming year.
What do I plan for 2012?
Sonja posted last week about her plans for 2012: with four main focuses and some discussion about what has brought her to decide on those. I began the three subgroups of food, exercise and spiritual energy for my 2011 word of the year, strength, and that really worked for me, so I’m following this inspiration again this year.
I read Sonja’s words on minimalism, on how she keeps forgetting and losing things; feeling rushed. As I read her post, I felt that little ember in me alight. I still want that freedom.
I also remember one where she photographed and listed all 75 items she owned and lived on happily at camp.
And really, there’s nothing stopping me.
I’m at university in Brighton; the same room I’ve lived in for three years; thus it has a lot more than one years accumulation. I take home clothes, make-up, notes, books and electrical items; then I go home and have more than the same amount again in my room, ignoring the crap in the loft.
I did a major clothes purge around 2 years ago, back when I was writing at Pagan Wings. I got rid of over 150 items of clothing. Yet I still/now have probably 300 items of clothes and I except only 150 of them fit me properly/ actually get worn. The idea of trying to juggle selling them on ebay- having to lug them to campus then to the post office and weight them and buy packing tape and boxes/bags? It’s a lot to handle when you don’t have a simple 9-5 job that ends each evening.
I intend to read a few specific books in the hopes of lowering my book numbers… currently holding over 186 books in my bedroom at home not including library and borrowed books… and then a further 46 on my uni shelf and a few more dotted around my uni room. I’m currently part way through.. well. Probably 50 of the,. Dianne Sylvan went through all her part-read books and listed them, which I intend to do also as part of this aspect of the new year.
As I visited Maiden castle, Old Sarum andStonehenge I began to see the history of this ancient land. As I listen to Damh the Bard speaking in our rituals; as I take part in more ritual myself and continue through my bardic grade; I remember those years of my teens where I shunned everything on the principal that I didn’t know who I was, thus couldn’t be anyone; couldn’t be X’s friend or Y’s enemy..
The difference in my acceptance now; is that I create myself every day and that’s okay.
I intend to read a few key texts for this including: Spirits of the Sacred Grove, The Celestine Vision, Women Who Run With The Wolves, Pagan in the City, Wights and Ancestors, and The Bond Between Women.
These cover Compassion, Integration, Strength, Women, Paganism, Ancestors and Loyalty.
Again, inspired by the last six months where I ‘saw red’ for the first time in my life, began anger management, self-initiated on the druid path (the peacemakers…) and began to understand this feeling of nonviolence towards every human, who is suffering in their own way.
Much connected to the aspect of “no me without my anger” I’m exploring; I’ve begun to see this world which doesn’t require you to be an idiot – I see people running businesses based on love, who are learning to respect their capacity and to own their own experience; who are successful in this world I was always taught was cruel, dangerous and angry.
My family = angry/passive aggressive and doormat – my own experiences until I left home were danger/fear/dominated. I’m now 21, choose meditation and dance over karate and am no longer terrified that every man with blue eyes is him.
I intend to read a few key texts including: Nonviolent Communication, The Gentle Art of Verbal Self-Defence, A Complaint Free World, Start Where You Are and How to Practise.
These cover Buddhism, Complaining, Violence, Patience, Words and The Dalai Lama’s words.
This is for reaching the potential attributes I’ve wanted since I was a young child; the inspiration I found in Power Rangers, in Lara Croft and in the fascination with native American Medicine Women.
This connects knowledge with practise; all that scholarary seeking, the sensing of trees and the energy of my own aura; in connecting with me as Forestwife, Susan from the Hogfather, Lara Croft, Myrina in The Moon Riders, and Lyra in the Dark Materials Trilogy.
This will include watching a lot of knowledge-based shows like Horizon and QI and some exercise-based stuff like flexibility and arm strength focuses.
This strong, skilled, scholar of a woman who is knowledgeable; who has worth without needing to prove it.
A woman who can ‘practise herself’ – who casts magic and offers healing; who is kind, stable and understands the land and its plants and creatures. I can see my potential as advisor, healer and practitioner; but though I find myself less afraid and upset of being lost; I am lost nonetheless.
Connection with practise, with the land, with people and with other animals and plants is my final step.
Rewire Your Brain, The Quantum and the Lotus, The Tenth Insight, In Search of Schrodinger’s Cat, Irish Spirit, The Field, Oscar’s Books, The Quiet, and City Dharma.
One thing I’m learning about being a scholar is that it never ends. I come home and research, read or watch a programme where I’m learning. I love that aspect, but it does mean I can’t foresee being able to do other things ‘after hours’ as there’s almost no such thing.
I want to sing, to dance, to read, write, express. I want to write poems and songs, to dance and exercise; to be flexible and to be caring. I want to be kind. Inherently.
I want to CONNECT – to the people around me, to the land I live with, to the spirit I believe in. To Compassion.
Earth and land (nature/practise), water (emotion/compassion), and air (knowledge/scholar).
This year I want to connect. With my space (minimalism and spirit), with myself (legacy and land), with compassion (people and peace) and with practise (knowledge and worth).
Really, they’re all about me: my space and values, my heritage and spirituality, my emotions and labels and my potential.
And an important distinction in this is that connecting requires disconnecting. I can get back on my minimalist journey in order to connect more; losing items and objects. Despite counting online friends in my connecting – blogging, tweeting, facebook messages and online messenger conversations; sometimes the laptop will switch off and I will go for walks in nature.
And in spite of that, I’ll also sometimes turn to reading my Gwersi in my bedroom, alone; meditating and practising ritual.
On Saturday morning my partner’s mother commented on how “meek” I was five years ago when she first met me, and on how much of the world I’ve seen; how I’ve embraced it; how I’ve come out of my shell.
I intend to continue that in the coming year.
Merry Christmas, Blessed Yuletide and have a Wonderful New Year. | english |
موسلی چِھ گژھَن لٔڈکہ اؠوٛیر اِلؠونتَھس منٛز | kashmiri |
/**
* Copyright (C) 2011-2020 Red Hat, Inc. (https://github.com/Commonjava/indy)
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.commonjava.indy.koji.data;
import com.redhat.red.build.koji.KojiClient;
import com.redhat.red.build.koji.KojiClientException;
import com.redhat.red.build.koji.model.xmlrpc.KojiArchiveInfo;
import com.redhat.red.build.koji.model.xmlrpc.KojiBuildInfo;
import com.redhat.red.build.koji.model.xmlrpc.KojiSessionInfo;
import org.commonjava.atlas.maven.ident.ref.ArtifactRef;
import org.commonjava.atlas.maven.ident.ref.SimpleArtifactRef;
import org.commonjava.cdi.util.weft.DrainingExecutorCompletionService;
import org.commonjava.cdi.util.weft.ExecutorConfig;
import org.commonjava.cdi.util.weft.SingleThreadedExecutorService;
import org.commonjava.cdi.util.weft.WeftExecutorService;
import org.commonjava.cdi.util.weft.WeftManaged;
import org.commonjava.indy.IndyWorkflowException;
import org.commonjava.indy.audit.ChangeSummary;
import org.commonjava.indy.data.IndyDataException;
import org.commonjava.indy.data.StoreDataManager;
import org.commonjava.indy.koji.conf.IndyKojiConfig;
import org.commonjava.indy.koji.content.IndyKojiContentProvider;
import org.commonjava.indy.koji.content.KojiPathPatternFormatter;
import org.commonjava.indy.koji.model.KojiMultiRepairResult;
import org.commonjava.indy.koji.model.KojiRepairRequest;
import org.commonjava.indy.koji.model.KojiRepairResult;
import org.commonjava.indy.koji.util.KojiUtils;
import org.commonjava.indy.model.core.ArtifactStore;
import org.commonjava.indy.model.core.Group;
import org.commonjava.indy.model.core.RemoteRepository;
import org.commonjava.indy.model.core.StoreKey;
import org.commonjava.indy.model.core.StoreType;
import org.commonjava.maven.galley.event.EventMetadata;
import org.slf4j.Logger;
import org.slf4j.LoggerFactory;
import javax.enterprise.context.ApplicationScoped;
import javax.inject.Inject;
import java.net.MalformedURLException;
import java.util.ArrayList;
import java.util.List;
import java.util.Objects;
import java.util.Set;
import java.util.concurrent.ExecutionException;
import java.util.concurrent.locks.ReentrantLock;
import java.util.stream.Collectors;
import static org.commonjava.indy.core.ctl.PoolUtils.detectOverloadVoid;
import static org.commonjava.indy.koji.content.KojiContentManagerDecorator.CREATION_TRIGGER_GAV;
import static org.commonjava.indy.koji.model.IndyKojiConstants.KOJI_ORIGIN;
import static org.commonjava.indy.koji.model.IndyKojiConstants.KOJI_ORIGIN_BINARY;
import static org.commonjava.indy.model.core.StoreType.group;
import static org.commonjava.indy.model.core.StoreType.remote;
/**
* Component responsible for repair Koji remote repositories.
*
* @author ruhan
*/
@ApplicationScoped
public class KojiRepairManager
{
private final Logger logger = LoggerFactory.getLogger( getClass() );
public static final String METADATA_KOJI_BUILD_ID = "koji-build-id"; // metadata key for recording koji build id
@Inject
private IndyKojiConfig config;
@Inject
private StoreDataManager storeManager;
@Inject
private IndyKojiContentProvider kojiCachedClient;
@Inject
private KojiPathPatternFormatter kojiPathFormatter;
@Inject
private KojiUtils kojiUtils;
@Inject
@WeftManaged
@ExecutorConfig( named="koji-repairs", threads=50, priority = 3, loadSensitive = ExecutorConfig.BooleanLiteral.TRUE, maxLoadFactor = 100)
private WeftExecutorService repairExecutor;
private ReentrantLock opLock = new ReentrantLock(); // operations are synchronized
protected KojiRepairManager()
{
}
public KojiRepairManager( final StoreDataManager storeManager, final IndyKojiConfig config,
final KojiClient kojiClient )
{
this.storeManager = storeManager;
this.config = config;
this.kojiCachedClient = new IndyKojiContentProvider( kojiClient, null );
this.repairExecutor = new SingleThreadedExecutorService( "koji-repairs" );
}
public KojiMultiRepairResult repairAllPathMasks( final String user )
throws KojiRepairException, IndyWorkflowException
{
KojiMultiRepairResult result = new KojiMultiRepairResult();
if ( opLock.tryLock() )
{
try
{
List<RemoteRepository> kojiRemotes = getAllKojiRemotes();
DrainingExecutorCompletionService<KojiRepairResult> repairService =
new DrainingExecutorCompletionService<>( repairExecutor );
detectOverloadVoid( () -> kojiRemotes.forEach( r -> repairService.submit( () -> {
logger.info( "Attempting to repair path masks in Koji remote: {}", r.getKey() );
KojiRepairRequest request = new KojiRepairRequest( r.getKey(), false );
try
{
return repairPathMask( request, user, true );
}
catch ( KojiRepairException e )
{
logger.error( "Failed to execute repair for: " + r.getKey(), e );
}
return null;
} ) ) );
List<KojiRepairResult> results = new ArrayList<>();
try
{
repairService.drain( r -> {
if ( r != null )
{
results.add( r );
}
} );
}
catch ( InterruptedException | ExecutionException e )
{
logger.error( "Failed to repair path masks.", e );
}
result.setResults( results );
}
catch ( IndyDataException e )
{
throw new KojiRepairException( "Failed to list Koji remote repositories for repair. Reason: %s", e, e.getMessage() );
}
finally
{
opLock.unlock();
}
}
else
{
throw new KojiRepairException( "Koji repair manager is busy." );
}
return result;
}
public KojiRepairResult repairPathMask( KojiRepairRequest request, String user )
throws KojiRepairException
{
return repairPathMask( request, user, false );
}
public KojiRepairResult repairPathMask( KojiRepairRequest request, String user, boolean skipLock )
throws KojiRepairException
{
KojiRepairResult ret = new KojiRepairResult( request );
if ( skipLock || opLock.tryLock() )
{
try
{
ArtifactStore store = getRequestedStore(request, ret);
if ( store == null )
{
return ret;
}
store = store.copyOf();
StoreKey remoteKey = request.getSource();
if ( remoteKey.getType() == remote )
{
final String nvr = kojiUtils.getBuildNvr( remoteKey );
if ( nvr == null )
{
String error = String.format( "Not a koji store: %s", remoteKey );
return ret.withError( error );
}
try
{
KojiSessionInfo session = null;
KojiBuildInfo build = kojiCachedClient.getBuildInfo( nvr, session );
List<KojiArchiveInfo> archives = kojiCachedClient.listArchivesForBuild( build.getId(), session );
ArtifactRef artifactRef = SimpleArtifactRef.parse( store.getMetadata( CREATION_TRIGGER_GAV ) );
if ( artifactRef == null )
{
String error = String.format(
"Koji remote repository: %s does not have %s metadata. Cannot retrieve accurate path masks.",
remoteKey, CREATION_TRIGGER_GAV );
return ret.withError( error );
}
// set pathMaskPatterns using build output paths
Set<String> patterns = kojiPathFormatter.getPatterns( store.getKey(), artifactRef, archives, true );
logger.debug( "For repo: {}, resetting path_mask_patterns to:\n\n{}\n\n", store.getKey(),
patterns );
KojiRepairResult.RepairResult repairResult = new KojiRepairResult.RepairResult( remoteKey );
repairResult.withPropertyChange( "path_mask_patterns", store.getPathMaskPatterns(), patterns );
ret.withResult( repairResult );
store.setPathMaskPatterns( patterns );
final ChangeSummary changeSummary = new ChangeSummary( user,
"Repairing remote repository path masks to Koji build: "
+ build.getNvr() );
storeManager.storeArtifactStore( store, changeSummary, false, true, new EventMetadata() );
}
catch ( KojiClientException e )
{
String error = String.format( "Cannot getBuildInfo: %s, error: %s", remoteKey, e );
logger.debug( error, e );
return ret.withError( error, e );
}
catch ( IndyDataException e )
{
String error = String.format( "Failed to store changed remote repository: %s, error: %s", remoteKey, e );
logger.debug( error, e );
return ret.withError( error, e );
}
}
else
{
String error = String.format( "Not a remote koji repository: %s", remoteKey );
return ret.withError( error );
}
}
finally
{
if ( !skipLock )
{
opLock.unlock();
}
}
}
else
{
throw new KojiRepairException( "Koji repair manager is busy." );
}
return ret;
}
public KojiRepairResult repairVol( KojiRepairRequest request, String user, String baseUrl )
throws KojiRepairException, KojiClientException
{
boolean flag = opLock.tryLock();
if ( flag )
{
try
{
StoreKey storeKey = request.getSource();
KojiRepairResult ret = new KojiRepairResult( request );
ArtifactStore store;
try
{
store = storeManager.getArtifactStore( storeKey );
}
catch ( IndyDataException e )
{
String error = String.format( "Cannot get store: %s, error: %s", storeKey, e );
logger.warn( error, e );
return ret.withError( error );
}
if ( store == null )
{
String error = String.format( "No such store: %s.", storeKey );
return ret.withError( error );
}
if ( storeKey.getType() == group )
{
return repairGroupVol( request, (Group) store, user );
}
else if ( storeKey.getType() == remote )
{
return repairRemoteRepositoryVol( request, (RemoteRepository) store, user );
}
else
{
String error = String.format( "Not a group or remote koji store: %s", storeKey );
return ret.withError( error );
}
}
finally
{
opLock.unlock();
}
}
else
{
throw new KojiRepairException( "opLock held by other" );
}
}
private KojiRepairResult repairGroupVol( KojiRepairRequest request, Group group, String user )
throws KojiClientException
{
KojiRepairResult ret = new KojiRepairResult( request );
List<StoreKey> stores = group.getConstituents();
if ( stores.isEmpty() )
{
return ret.withNoChange( group.getKey() );
}
KojiSessionInfo session = null;
List<Object> args = new ArrayList<>( );
stores.forEach( storeKey -> {
String nvr = kojiUtils.getBuildNvr( storeKey );
if ( nvr != null )
{
args.add( nvr );
}
else
{
ret.withIgnore( storeKey );
}
} );
List<KojiBuildInfo> buildInfoList = kojiCachedClient.getBuildInfo( args, session );
buildInfoList.forEach( buildInfo -> {
try
{
KojiRepairResult.RepairResult repairResult =
doRepair( group.getPackageType(), null, buildInfo, user, request.isDryRun() );
ret.withResult( repairResult );
}
catch ( KojiRepairException e )
{
// we do not fail the whole attempt if one store failed
logger.debug( "Repair failed", e );
ret.withResult( new KojiRepairResult.RepairResult( e.getStoreKey(), e ) );
}
} );
return ret;
}
private KojiRepairResult repairRemoteRepositoryVol( KojiRepairRequest request, RemoteRepository repository,
String user ) throws KojiRepairException
{
StoreKey storeKey = repository.getKey();
KojiRepairResult ret = new KojiRepairResult( request );
final String nvr = kojiUtils.getBuildNvr( storeKey );
if ( nvr == null )
{
String error = String.format( "Not a koji store: %s", storeKey );
return ret.withError( error );
}
KojiBuildInfo buildInfo;
try
{
KojiSessionInfo session = null;
buildInfo = kojiCachedClient.getBuildInfo( nvr, session );
}
catch ( KojiClientException e )
{
String error = String.format( "Cannot getBuildInfo: %s, error: %s", storeKey, e );
logger.debug( error, e );
return ret.withError( error, e );
}
KojiRepairResult.RepairResult repairResult =
doRepair( repository.getPackageType(), repository, buildInfo, user,
request.isDryRun() );
return ret.withResult( repairResult );
}
/**
* Repair one remote repository.
* @param packageType
* @param repository repository to be repaired. If null, the repository name will be retrieved according to
* the buildInfo and the naming format in koji.conf.
* @param buildInfo koji build which this repository proxies
* @param user the user does the repair
* @param isDryRun
* @return
* @throws KojiRepairException
*/
private KojiRepairResult.RepairResult doRepair( String packageType, RemoteRepository repository,
KojiBuildInfo buildInfo, String user, boolean isDryRun )
throws KojiRepairException
{
StoreKey storeKey;
if ( repository != null )
{
storeKey = repository.getKey();
}
else
{
String name = kojiUtils.getRepositoryName( buildInfo );
storeKey = new StoreKey( packageType, StoreType.remote, name );
try
{
repository = (RemoteRepository) storeManager.getArtifactStore( storeKey );
}
catch ( IndyDataException e )
{
throw new KojiRepairException( "Cannot get store: %s. Reason: %s", e, storeKey,
e.getMessage() );
}
}
KojiRepairResult.RepairResult repairResult = new KojiRepairResult.RepairResult( storeKey );
String url = repository.getUrl();
String newUrl;
try
{
newUrl = kojiUtils.formatStorageUrl( config.getStorageRootUrl(), buildInfo ); // volume is involved
}
catch ( MalformedURLException e )
{
throw new KojiRepairException( "Failed to format storage Url: %s. Reason: %s", e, storeKey,
e.getMessage() );
}
boolean changed = !url.equals( newUrl );
if ( changed )
{
repairResult.withPropertyChange( "url", url, newUrl );
if ( !isDryRun )
{
ChangeSummary changeSummary = new ChangeSummary( user,
"Repair " + storeKey + " url with volume: " + buildInfo
.getVolumeName() );
repository.setUrl( newUrl );
repository.setMetadata( METADATA_KOJI_BUILD_ID, Integer.toString( buildInfo.getId() ) );
boolean fireEvents = false;
boolean skipIfExists = false;
try
{
storeManager.storeArtifactStore( repository, changeSummary, skipIfExists, fireEvents, new EventMetadata() );
}
catch ( IndyDataException e )
{
throw new KojiRepairException( "Failed to repair store: %s. Reason: %s", e, storeKey, e.getMessage() );
}
}
}
return repairResult;
}
public KojiMultiRepairResult repairAllMetadataTimeout( final String user, boolean isDryRun )
throws KojiRepairException, IndyWorkflowException
{
KojiMultiRepairResult result = new KojiMultiRepairResult();
if ( opLock.tryLock() )
{
try
{
List<RemoteRepository> kojiRemotes = getAllKojiRemotes();
DrainingExecutorCompletionService<KojiRepairResult> repairService =
new DrainingExecutorCompletionService<>( repairExecutor );
detectOverloadVoid( () -> kojiRemotes.forEach( r -> repairService.submit( () -> {
logger.info( "Attempting to repair path masks in Koji remote: {}", r.getKey() );
KojiRepairRequest request = new KojiRepairRequest( r.getKey(), isDryRun );
try
{
return repairMetadataTimeout( request, user, true );
}
catch ( KojiRepairException e )
{
logger.error( "Failed to execute repair for: " + r.getKey(), e );
}
return null;
} ) ) );
List<KojiRepairResult> results = new ArrayList<>();
try
{
repairService.drain( r -> {
if ( r != null )
{
results.add( r );
}
} );
}
catch ( InterruptedException | ExecutionException e )
{
logger.error( "Failed to repair metadata timeout.", e );
}
result.setResults( results );
}
catch ( IndyDataException e )
{
throw new KojiRepairException( "Failed to list Koji remote repositories for repair. Reason: %s", e, e.getMessage() );
}
finally
{
opLock.unlock();
}
}
else
{
throw new KojiRepairException( "Koji repair manager is busy." );
}
return result;
}
public KojiRepairResult repairMetadataTimeout( KojiRepairRequest request, String user, boolean skipLock ) throws KojiRepairException{
KojiRepairResult ret = new KojiRepairResult( request );
if ( skipLock || opLock.tryLock() )
{
try
{
ArtifactStore store = getRequestedStore(request, ret);
if ( store == null )
{
return ret;
}
store = store.copyOf();
StoreKey remoteKey = request.getSource();
if ( remoteKey.getType() == remote )
{
final String nvr = kojiUtils.getBuildNvr( remoteKey );
if ( nvr == null )
{
String error = String.format( "Not a koji store: %s", remoteKey );
return ret.withError( error );
}
try
{
final int NEVER_TIMEOUT_VALUE = -1;
if ( !request.isDryRun() )
{
( (RemoteRepository) store ).setMetadataTimeoutSeconds( NEVER_TIMEOUT_VALUE );
final ChangeSummary changeSummary = new ChangeSummary( user,
"Repairing remote repository path masks to Koji build: "
+ nvr );
storeManager.storeArtifactStore( store, changeSummary, false, true, new EventMetadata() );
}
KojiRepairResult.RepairResult repairResult = new KojiRepairResult.RepairResult( remoteKey );
repairResult.withPropertyChange( "metadata_timeout", ( (RemoteRepository) store ).getMetadataTimeoutSeconds(), NEVER_TIMEOUT_VALUE );
ret.withResult( repairResult );
}
catch ( IndyDataException e )
{
String error =
String.format( "Failed to store changed remote repository: %s, error: %s", remoteKey,
e );
logger.debug( error, e );
return ret.withError( error, e );
}
}
else
{
String error = String.format( "Not a remote koji repository: %s", remoteKey );
return ret.withError( error );
}
}
finally
{
if ( !skipLock )
{
opLock.unlock();
}
}
}
else
{
throw new KojiRepairException( "Koji repair manager is busy." );
}
return ret;
}
public KojiRepairResult repairMetadataTimeout( KojiRepairRequest request, String user )
throws KojiRepairException
{
return repairMetadataTimeout( request, user, false );
}
private List<RemoteRepository> getAllKojiRemotes()
throws IndyDataException
{
return storeManager.query()
.storeTypes( remote )
.stream( ( remote ) ->
KOJI_ORIGIN.equals( remote.getMetadata( ArtifactStore.METADATA_ORIGIN ) )
|| KOJI_ORIGIN_BINARY.equals(
remote.getMetadata( ArtifactStore.METADATA_ORIGIN ) ) )
.map( s -> (RemoteRepository) s )
.filter( Objects::nonNull )
.collect( Collectors.toList() );
}
private ArtifactStore getRequestedStore(final KojiRepairRequest request, final KojiRepairResult ret ){
StoreKey remoteKey = request.getSource();
ArtifactStore store = null;
try
{
store = storeManager.getArtifactStore( remoteKey );
}
catch ( IndyDataException e )
{
String error = String.format( "Cannot get store: %s, error: %s", remoteKey, e );
logger.warn( error, e );
ret.withError( error );
return null;
}
if ( store == null )
{
String error = String.format( "No such store: %s.", remoteKey );
ret.withError( error );
return null;
}
return store;
}
}
| code |
_.mixin({
getQueryString: function() {
var result = {}, queryString = location.search.substring(1),
re = /([^&=]+)=([^&]*)/g, m;
while (m = re.exec(queryString)) {
result[decodeURIComponent(m[1])] = decodeURIComponent(m[2]);
}
return result;
}
}); | code |
What Allen said next surprised a few members of the audience. “One of the roles of the government is picking winners and losers,” he explained. Allen expressed his belief that our government should “make choices based on which energy technologies it thinks are best, and the key is for the government to remain transparent.” One must wonder, however, that if the government is picking winners and losers, doesn’t that contradict his earlier contention that market competition should be the driver for change? | english |
<!doctype html>
<html lang="en">
<head>
<meta charset="UTF-8">
<title>Example - example-example24-production</title>
<script src="//ajax.googleapis.com/ajax/libs/angularjs/1.2.19/angular.min.js"></script>
</head>
<body ng-app="">
<button ng-mouseup="count = count + 1" ng-init="count=0">
Increment (on mouse up)
</button>
count: {{count}}
</body>
</html> | code |
I am using this module to run tasks and I am currently trying to make it run faster.
During my investigation, I discovered that running the task with the board connected to a usb 2.0 port makes it overall run faster than when I use a usb 3.0 port.
Can anyone explain this strange behaviour?
In addition, I have been testing the device on two identical machines. The speed is overall slightly slower on average compared to the other machine even though they are the same.
Same kernel version, usb driver version and system specifications etc.
I would like to discover the causes behind these different results in order to get faster and more consistent runtimes.
Please post your PipeTest runtimes.
You’ll also need to be more specific about what you mean by slower. What’s slower? Wire updates? In or out? Trigger updates? Pipe transfers? How big? Block size? Overall transfer size? | english |
शातिर 'मुत्तू' गिरफ्तार, मात्र १० से २० सेकेंड में खोल देता था बाइक का ताला - बाइक थीव अरेस्टेड इन गुरुग्राम
शातिर 'मुत्तू' गिरफ्तार, मात्र १० से २० सेकेंड में खोल देता था बाइक का ताला
गुरुग्राम (मोहित): गुरुग्राम पुलिस ने एक ऐसे शातिर बाईक चोर को गिरफ्तार किया है जो मात्र १० से २० सेकेंडों में ही किसी भी बाईक का ताला खोल सकता है। इस २२ साल के शातिर चोर का शाहिद उर्फ मुत्तू है जो दिल्ली एनसीआर के साथ-साथ राजस्थान और हरियाणा के अलग-अलग हिस्सों में बाइक चोरी की वारदातों को अंजाम दे चुका है। पुलिस की क्राइम यूनिट सोहना की टीम ने "मुत्तू" को सोहना मोड़ से गिरफ्तार किया है।
पुलिस ने बताया कि आरोपी बाईक चुराने के बाद मेवात में औने पौने दामो में बेच देता था। एसीपी क्राइम प्रीतपाल सिंह ने बताया कि आरोपी के कब्जे से ५ नई बाइक बरामद की गई हैं, जिसमें ३ बाइक दिल्ली के विभिन्न इलाकों से व २ गुरुग्राम के अलग-अलग इलाकों से चुराई गई थीं। आरोपी का पहले से आपराधिक रिकॉर्ड रहा है और बाइक चोरी के कई मामलों में आरोपी जेल की हवा खा चुका है।
पुलिस का कहना है कि क्राइम ब्रांच की टीम शाहिद उर्फ मुत्तू के नेटवर्क को भी खंगालने में जुटी है। आरोपी चोरी की बाइक मेवात में किसको बेचता था और चोरी की बाइक खरीदने-बेचने के काले कारोबार के पीछे किसका हाथ है, पुलिस इन सबकी खोजबीन में जुटी है। | hindi |
हाफ गर्लफ्रेंड चेतन भगत के इसी नाम से लिखे उपन्यास पर आधारित फिल्म है। अर्जुन ने इसमें माधव झा नाम के बिहारी लड़के का किरदार निभाया है।
जनसत्ता ऑनलाइन नई दिल्ली | अप्रैल २२, २०१७ १०:३२ आम
श्रद्धा कपूर और अर्जुन कपूर की फिल्म १९ मई को होगी रिलीज। (इमेज सोर्स: इंस्टाग्रम)
श्रद्धा कपूर और अर्जुन कपूर की फिल्म हाफ गर्लफ्रेंड १९ मई को भारत में १६०० स्क्रिन पर रिलीज होगी। एनएच स्टूडियोज जोकि नरेंद्र हीरावत एंड कंपनी का उपक्रम है उसने मोहित सूरी के निर्देशन में बनी इस फिल्म के डिस्ट्रीब्यूशन राइट्स को खरीद लिया है। एनएच स्टूडियोज के श्रेयंस हीरावत ने एक बयान जारी करते हुए कहा- चेतन भगत जैसे लेखक की कहानी पर आधारित फिल्में युवा पीढ़ी को काफी अपील करती हैं और यह हम अतीत में देख चुके हैं। हमने हाफ गर्लफ्रेंड के कंटेट को काफी अंदर तक देखा है और इसकी कहानी के साथ ही एक्टर्स की परफॉर्मेंस से हम काफी उत्साहित हैं। फिल्म की सफलता के लिए यह जरूरी है कि इसे सही दर्शकों तक ले जाया जाए। हमारे आखिरी वेंचर बेगम जान का कंटेंट काफी अलग और मजबूत था। हाफ गर्लफ्रेंड की तुलना में बेगम जान को काफी ज्यादा परिपक्व जनता के सामने पेश करना था इसीलिए हमने स्क्रिन की संख्या को घटाकर ९५६ कर दिया था।
गाने को अरिजीत सिंह शाशा तिरुपति ने आवाज दी है। इस गाने को लेकर फैन्स में गजब का क्रेज था। फैन्स की एक्साइटमेंट को देखते हुए श्रद्धा कपूर ने वीडियो रिलीज होने से पहले गाने के ऑडियो को ट्वीट कर दिया था। चेतन भगत के नॉवल हाफ गर्लफ्रेंड पर बनी फिल्म के इस गाने को अरिजीत सिंह ने अपनी आवाज दी है। फिल्म के ट्रेलर और पहले रिलीज हुए गाने को अच्छा रिस्पॉन्स मिला है। बारिश नाम से जो गाना पहले रिलीज किया गया था वह एक लव सॉन्ग था।
"हसीना" श्रद्धा कपूर और "डॉन" सिद्धांत कपूर का ये इंप्रेसिव लुक | hindi |
उप: किसानों के ३६३५९ करोड़ के कर्ज माफ
भाजपा सरकार ने लोक संकल्प पत्र में किया वादा पूरा किया, पहली केबिनेट बैठक में लिया कर्ज माफी का फैसला
लखन लखनऊ, ०४ अप्रैल, २०१७। (उ.प्र.समाचार सेवा)। उत्तर प्रदेश सरकार ने विधान सभा चुनाव के दौरान किये गए वादे के अनुरूप कैबिनेट की पहली बैठक में लघु और सीमान्त किसानों के ३६ हजार ३५९ करोड़ रुपये के ऋण माफ कर दिये हैं। इन कर्जों को प्रदेश सरकार ने अपने ऊपर ले लिया है। बैंकों को यह धनराशि प्रदेश सरकार अपने खजाने से अदा करेगी। करज माफी योजन के तहत प्रदेश सरकार किसानों के एक लाख रुपये तक के ऋण माफ करेगी। इस रकम को जुटाने के लिए सरकार ने किसान बाण्ड जारी करने का फैसला किया है।
लोकभवन सचिवालय में आयोजित कैबिनेट की पहली बैठक में मुख्यमंत्री योगी आदित्यनाथ योगी आदित्यनाथ की अध्यक्षता में किसानों के कर्जे माफ करने का निर्णय लिया गया। इस फैसले की जानकारी शाम को लोकभवन सभागार में आयोजित पत्रकार वार्ता में सरकार के प्रवक्ताओं सिद्धार्थनाथ सिंह और श्रीकांत शर्मा ने दी। प्रवक्ताओं ने बताया कि प्रदेश में २ करोड़ ३० लाख किसान हैं। इनमें 9२.५ प्रतिशत यानि कि २ करोड 1५ लाख सीमान्त और लघु किसान हैं। इनके द्वारा लिए गए फसली ऋण में एक लाख रुपये सीमा तक का ऋण माफ किया गया है। यह धनराशि ३० हजार 7२9 करोड़ रुपये है। सभी राष्ट्रीयकृत बैंकों तथा सहकारी बैंकों से लिए गए कर्ज को माफ किया गया है। माफी ३१ मार्च २017 तक के फसली ऋणों पर की गई है। किसानों के एक लाख रुपये तक के करज माफ किये गए हैं।
प्रवक्ता ने बताया कि इसके अलावा प्रदेश में ७ लाख और एेसे किसान हैं जिनके ऋण एनपीए ( नान परफार्मिंग एसेस्ट्स) में चले गए हैं। ऐसे किसानों की एनपीए धनराशि ५ हजार ६३० करोड़ रुपये है। इन किसानों का यह एनपीए भी सरकार द्वारा बैंकों को भुगतान किया जाएगा। इस धनराशि को सरकार ओटीएस स्कीम के तहत बैंकों को अदा करेगी। इस तरह प्रदेश सरकार ने किसानों को कुल ऋण माफी ३६ हजार 3५9 करोड़ की है। उन्होंने बताया कि इस धनराशि को सरकार अपने संसाधनों से जुटाएगी। इसके लिए किसान बाण्ड जारी किया जाएगा। इस बाण्ड से प्राप्त रकम को किसानों की कर्ज अदायगी में उपयोग किया जाएगा। प्रवक्ता के अनुसार चूंकि केन्द्र सरकार के एफआरबीएम एक्ट में यह प्रावधान है कि किसी भी राज्य का फिसकल डेफिसिट उसकी जीडीपी के तीन प्रतिशत से अधिक नहीं हो सकता। इसलिए बाण्ड से पैसा जुटाने की उपाय निकाला गया है।
फसली ऋण माफी योजना के तहत मुख्य सचिव की अध्यकता में आठ अधिकारियों की एक कमेटी बनायी गई है। जिसमें कृषि उत्पादन आयुक्त, अपर मुख्य सचिव, प्रमुख सचिव वित्त, प्रमुख सचिव संस्थागत वित्त एवं कृषि सहकारिता व राज्य की लीड बैंक के समन्वयक को सदस्य बनाया गया है। यह समिति सभी पहलुओं पर विचार करने के बाद विस्तृत फसली ऋण माफी योजना तैयार करेगी। योजना का मुख्यमंत्री से अनुमोदन लेने के बाद उसका क्रियान्वयन किया जाएगा। साथ ही समिति योजना के वित्त पोषण हेतु भी अपनी संस्तुतियां शासन को प्रस्तुत करेगी। जिसके आधार पर सरकार द्वारा इस हेतु वित्तीय व्यवस्थाएं सुनिश्चित की जाएंगी। क्रियान्वयन के अतिरिक्त समिति की इस योजना के सतत् अनुश्रवण में बी सक्रिय भूमिका होगी।
जातव्य है कि उत्तर प्रदेश ७८ प्रतिशत जनसंख्या ग्रामीण केत्रों में निवास करती है, जिसमें ६८ प्रतिशत परिवार कृषि पर निरभर हैं। प्रदेश में एक हेक्टेयर अरथात २.५ एकड़ तक क सभी किसान सीमान्त किसान की श्रेणी में आते हैं। जबकि २ हेक्टेयर यानि कि ५ एकड़ तक के किसान लघु किसान की श्रेणी में आते हैं। प्रदेश में ८६.६८ लाख किसानों ने बैंकों से फसली ऋण लिया हुआ है। | hindi |
\begin{document}
\title[Twisted Gauss sums and totally isotropic subspaces]
{Higher level quadratically twisted Gauss sums and totally isotropic subspaces}
\author{Lynne Walling}
\address{School of Mathematics, University of Bristol, University Walk, Clifton, Bristol BS8 1TW, United Kingdom;
phone +44 (0)117 331-5245}
\email{l.walling@bristol.ac.uk}
\keywords{Gauss sums, quadratic forms}
\begin{abstract} We consider a generalized Gauss sum supported on matrices over a number field. We evaluate this Gauss sum and relate it to the number of totally isotropic subspaces of related quadratic spaces. Then we consider a further generalization of such a Gauss sum, realizing its value in terms of numbers of totally isotropic subspaces of related quadratic spaces.
\end{abstract}
\maketitle
\def\arabic{footnote}{}
\footnote{2010 {\it Mathematics Subject Classification}: Primary
11L05, 11E08 }
\def\arabic{footnote}{\arabic{footnote}}
\section{Introduction}
Gauss sums and their numerous generalizations are ubiquitous in number theory.
When studying the action of Hecke operators on half-integral weight Hilbert-Siegel modular forms, the generalized Gauss sum we encounter is defined as follows.
Let ${\mathbb K}$ be a number field with $\mathcal O$ its ring of integers, ${\mathfrak P}$ a nondyadic prime ideal in $\mathcal O$, and $\mathbb F=\mathcal O/{\mathfrak P}$; we fix $\rho\in\partial^{-1}{\mathfrak P}^{-1}$ so that $\rho\mathcal O_{{\mathfrak P}}=\partial^{-1}{\mathfrak P}^{-1}\mathcal O_{{\mathfrak P}}$ (where $\partial$ is the different of ${\mathbb K}$).
Then
for $T\in\mathbb F^{n,n}_{\sym}$ (meaning that $T$ is a symmetric $n\times n$ matrix over $\mathbb F$), we set
$$\mathcal G^*_T({\mathfrak P})=\sum_{S\in\mathbb F^{n,n}_{\sym}}
\left(\frac{\det S}{{\mathfrak P}}\right)\e\{2TS\rho\}$$
where $\sigma$ denotes the matrix trace map,
$\e\{*\}=\exp(\pi i Tr^{{\mathbb K}}_{\mathbb Q}(\sigma(*)))$,
and $\left(\frac{*}{{\mathfrak P}}\right)$ is the Legendre symbol.
One sees that for $M,N\in\mathcal O^{n,n}_{\sym}$ with $M\equiv N\ (\text{mod }{\mathfrak P})$, we have
$\e\{2M\rho\}=\e\{2N\rho\}$; consequently, $\mathcal G^*_T({\mathfrak P})$ is well-defined, although it is dependent on the choice of $\rho$.
For our application to half-integral weight Hecke operators, we need to relate these Gauss sums to
$R^*(T\perp\big<1\big>,0_a)$, which is
the number of $a$-dimensional totally isotropic subspaces of the dimension $n+1$ $\mathbb F$-space $V$ whose quadratic form is given by $T\perp\big<1\big>$.
(A subspace $W$ of $V$ is totally isotropic if the quadratic form restricted to $W$ is 0, and $A\perp B$ denotes the block-diagonal matrix $\diag(A,B)$.)
In Theorem 1.1 we evaluate $\mathcal G^*_T({\mathfrak P})$, and in Corollary 1.2 we give
$\mathcal G^*_T({\mathfrak P})$ in terms of $R^*(T\perp\big<1\big>,0_a)$.
To state the theorem, set
$\varepsilon=\left(\frac{-1}{{\mathfrak P}}\right),$ and fix $\omega\in\mathbb F$ so that $\omega$ is not a square in $\mathbb F$; set $J_n=I_{n-1}\perp\big<\omega\big>$. For $T,S\in\mathbb F^{n,n}_{\sym}$, write $T\sim S$ if there is some $G\in GL_n(\mathbb F)$ so that $T=\,^tGSG$. Note that with $d=\rank T$, either $T\sim I_d\perp 0_{n-d}$ or $T\sim J_d\perp 0_{n-d}$.
With this notation, we have the following.
\begin{thm} Take $T\in\mathbb F^{n,n}_{\sym}$ where $n\in\mathbb Z_+$.
Suppose that $0\le d\le n$ and
$T\sim I_d\perp 0_{n-d}$ or $T\sim J_d\perp 0_{n-d}$. Take $c$ so that $d=2c$ or $d=2c+1$.
\begin{enumerate}
\item[(a)] Suppose that $n=2m$. Then with $N({\mathfrak P})$ the norm of ${\mathfrak P}$,
$$\mathcal G^*_T({\mathfrak P})=(-1)^c\varepsilon^m N({\mathfrak P})^{m^2}\cdot
\prod_{i=1}^{m-c}(N({\mathfrak P})^{2i-1}-1).$$
\item[(b)] Suppose that $n=2m+1$. When $d=2c$, $\mathcal G^*_T({\mathfrak P})=0$. When $d=2c+1$,
$$\mathcal G^*_T({\mathfrak P})=(-1)^c\varepsilon^{m+c}N({\mathfrak P})^{m^2+2m-c}\,\mathcal G^*_1({\mathfrak P})\cdot
\prod_{i=1}^{m-c}(N({\mathfrak P})^{2i-1}-1)$$
if $T\sim I_d\perp 0_{n-d}$, and
$\mathcal G^*_T({\mathfrak P})=-\mathcal G^*_{I_d\perp 0_{n-d}}({\mathfrak P})$
if $T\sim J_d\perp 0_{n-d}$.
\end{enumerate}
\end{thm}
\begin{cor} Take $T\in\mathbb F^{n,n}_{\sym}$ where $n\in\mathbb Z_+$.
Let $V$ be the dimension $n+1$ space over $\mathbb F$ with quadratic form given by $T\perp\big<1\big>$, and
let $R^*(T\perp\big<1\big>,0_a)$ be the number of $a$-dimensional totally isotropic subspaces of $V$.
We have
$$\left(\mathcal G_1^*({\mathfrak P})\right)^n\mathcal G^*_T({\mathfrak P})
=\sum_{a=0}^n (-1)^{n+a}N({\mathfrak P})^{n(n+1)/2+a(a-n)}
R^*(T\perp\big<1\big>,0_a).$$
\end{cor}
To prove Theorem 1.1, we perform a deconstruction to reduce $\mathcal G^*_T({\mathfrak P})$ to a sum in terms of Gauss sums $\mathcal G^*_Y({\mathfrak P})$ where the $Y$ are smaller than $T$. For this we repeatedly use the elementary fact that $\sigma(AB)=\sigma(BA)$, knowledge of representation numbers over finite fields, and elementary combinatorial methods. Then using induction, we prove Theorem 1.1; Corollary 1.2 then follows from Lemma 3.1 of \cite{half-int-aps}.
In Proposition 4.1, we consider the following generalized Gauss sum:
with $T\in\mathbb F^{n,n}_{\sym}$ and $0\le r\le n$, set
$$\mathcal G^*_T({\mathfrak P};r)=
\sum_{S\sim I_r\perp 0_{n-r}}\e\{2TS\rho\}-\sum_{S\sim J_r\perp 0_{n-r}}\e\{2TS\rho\}$$
(so for $r=n$, this is $\mathcal G^*_T({\mathfrak P})$).
We again deconstruct $\mathcal G^*_T({\mathfrak P};r)$ as a sum in terms of Gauss sums $\mathcal G^*_Y({\mathfrak P})$ with the $Y$ smaller than $T$, and then from this and Theorem 1.1, we describe $\mathcal G^*_T({\mathfrak P};r)$ in terms of numbers of totally isotropic subspaces of spaces of quadratic spaces related to $T$.
It is important for us to note that in \cite{S}, Saito studies analogues of these Gauss sums over finite fields, with an interest to applications to twists of Siegel modular forms.
Although his main interest is in twists by the quadratic and the trivial characters, he considers twists by all characters, making his arguments more complicated than ours. We note that Theorem 1.3 \cite{S} includes the results of our Theorem 1.1.
Saito also considers finite field analogues of the Gauss sums
$\mathcal G^*_T({\mathfrak P};r)$.
He develops relations between these Gauss sums, some of which are quite complicated. In Proposition 4.1 (a), we present a simple relation very similar to his relation in Proposition 1.12 \cite{S}; then in Proposition 4.1 (b) we present formulas for these Gauss sums in terms of numbers of totally isotropic subspaces.
The value of this paper is to present an approach simpler than that of \cite{S}, demonstrating our deconstruction technique,
and to relate these Gauss sums to representations of zeros.
Note that it is quite easy to modify our techniques to
generalized Gauss sums twisted by the trivial character, and to
Gauss sums over
a finite field $\mathbb F_q$ with odd characteristic $p$ where
$\e\{*\}$ is replaced by $\exp(\pi i Tr^{\mathbb F_q}_{\mathbb F_p}(\sigma(*))/p)).$
\section{Notation}
Besides the notation given in the introduction, we define the following.
For $t,s\in \mathbb Z_+$ with $s\le t$, and $X\in\mathbb F^{t,t}_{\sym}$, $Y\in\mathbb F^{s,s}_{\sym}$, define the
representation number $r(X,Y)$ to be
$$r(X,Y)=\#\{C\in\mathbb F^{t,s}:\ ^tCXC=Y\ \},$$
and define the primitive representation number $r^*(X,Y)$ to be
$$r^*(X,Y)=\#\{C\in\mathbb F^{t,s}:\ ^tCXC=Y,\ \rank C=s\ \}.$$
Let $o(X)$ denote the order of the orthogonal group of $X$; so $o(X)=r^*(X,X).$
We make great use of the following elementary functions, that help us encode formulas involving
representation numbers.
\begin{align*}
\boldsymbol \mu(t,s)&=\prod_{i=0}^{s-1}(N({\mathfrak P})^{t-i}-1),\
\boldsymbol \delta(t,s)=\prod_{i=0}^{s-1}(N({\mathfrak P})^{t-i}+1),\\
\boldsymbol \beta(t,s)&=\frac{\boldsymbol \mu(t,s)}{\boldsymbol \mu(s,s)},\
\boldsymbol \mathfrak Nu(t,s)=\prod_{i=s}^{t-1}(N({\mathfrak P})^t-N({\mathfrak P})^i),\
\boldsymbol \mathbf gamma(t,s)=\frac{\boldsymbol \mu\boldsymbol \delta(t,s)}{\boldsymbol \mu\boldsymbol \delta(s,s)}.
\end{align*}
We agree that when $s=0$, the value of any of these functions is 1; when $s<0$, we agree that $\boldsymbol \beta(t,s)=0$. Note that $\boldsymbol \beta(t,s)$ is the number of $s$-dimensional subspaces of a $t$-dimensional space over $\mathbb F$, and $\boldsymbol \mathfrak Nu(t,0)$ is the number of bases for a $t$-dimensional space.
Finally, for $d\in\mathbb Z_+$ and $i\in\mathbb Z$ with $0\le i\le d$, we set
$U_{d,i}=I_i\perp 0_{d-i}$ and $\overline U_{d,i}=J_i\perp 0_{d-i}.$
\section{Proofs of Theorem 1.1 and Corollary 1.2}
We begin by proving Theorem 1.1.
As ${\mathfrak P}$ is fixed, in this section we write $\mathcal G^*_T$ for $\mathcal G^*_T({\mathfrak P})$.
First notice that
$$\mathcal G^*_{0_n}=\sum_{Y\sim I_n}1-\sum_{Y\sim J_n}1
=\frac{|GL_n(\mathbb F)|}{o(I_n)}-\frac{|GL_n(\mathbb F)|}{o(J_n)};$$
so using Lemma 5.1, when $n$ is odd we get
$\mathcal G^*_{0_n}=0$, and when $n=2m$ we get $$\mathcal G^*_{0_n}=\varepsilon^mN({\mathfrak P})^{m^2}\frac{\boldsymbol \mu(2m,2m)}{\boldsymbol \mu\boldsymbol \delta(m,m)}.$$
For the rest of this section, take $d$ so that $0<d<n$.
With $G\in GL_n(\mathbb F)$, we have
$$\e\{2\,^tGI_nGU_{n,d}\cdot\rho\}
=\e\{2Y'\rho\},\
\e\{2\,^tGI_nG\overline U_{n,d}\cdot\rho\}
=\e\{2Y'J_d\rho\}$$
where $Y'$
is the upper left block of $^tGI_nG$; similarly,
$$\e\{2\,^tGJ_nGU_{n,d}\cdot\rho\}
=\e\{2Y'\rho\},\
\e\{2\,^tGJ_nG\overline U_{n,d}\cdot\rho\}
=\e\{2Y'J_d\rho\}$$
where $Y'$
is the upper left block of $^tGJ_nG$.
The number of $Y\sim I_n$ with upper left $d\times d$ block $Y'$ is
$\boldsymbol \mathfrak Nu(n,d) r^*(I_n,Y')/o(I_n),$
as for $C\in\mathbb F^{n,d}$ with $\rank C=d$, the number of ways to extend $C$ to an element of $GL_n(\mathbb F)$ is $\boldsymbol \mathfrak Nu(n,d)$.
Similarly, the number of $Y\sim J_n$ with upper left $d\times d$ block $Y'$ is $\boldsymbol \mathfrak Nu(n,d) r^*(J_n,Y')/o(J_n).$
Hence we have
\begin{align*}
\mathcal G^*_{U_{n,d}}
&=\sum_{Y\sim I_n} \e\{2YU_{n,d}\cdot\rho\}
-\sum_{Y\sim J_n} \e\{2YU_{n,d}\cdot\rho\}\\
&=\sum_{G\in GL_n(\mathbb F)}
\left( \frac{\e\{2\,^tGI_nGU_{n,d}\cdot\rho\}}{o(I_n)}
- \frac{\e\{2\,^tGJ_nGU_{n,d}\cdot\rho\}}{o(J_n)}\right)\\
&=\boldsymbol \mathfrak Nu(n,d)\sum_{Y'\in\mathbb F^{d,d}_{\sym}}
\left(\frac{r^*(I_n,Y')}{o(I_n)}-\frac{r^*(J_n,Y')}{o(J_n)}\right)\e\{2Y'\rho\}.
\end{align*}
Note that we can partition $\mathbb F^{d,d}_{\sym}$ into $GL_d(\mathbb F)$-orbits, and
in Lemma 5.1, we compute representation numbers $r^*(\cdot,\cdot)$.
We find that when $n$ is odd, we have $o(I_n)=o(J_n)$,
$r^*(I_n, U_{d,2k})-r^*(J_n,U_{d,2k})=0,$
and
$$r^*(I_n,U_{d,2k+1})-r^*(J_n,U_{d,2k+1})
=r^*(J_n,\overline U_{d,2k+1})-r^*(I_n,\overline U_{d,2k+1}).$$
Hence with $n=2m+1$, using Lemma 5.1 and then Lemma 5.3, we get
\begin{align*}
\mathcal G^*_{U_{n,d}}
&=\frac{\boldsymbol \mathfrak Nu(n,d)}{o(I_n)}
\sum_{k=0}^{d/2} 2\varepsilon^{m-k}N({\mathfrak P})^{2mk-k^2+m-k+(d-2k-1)(d-2k-2)/2}\\
&\mathfrak quad\cdot
\boldsymbol \mu\boldsymbol \delta(m,d-k-1)
\left(\sum_{Y\sim U_{d,2k+1}}\e\{2Y\rho\}
-\sum_{Y\sim\overline U_{d,2k+1}}\e\{2Y\rho\}\right)\\
&\mathfrak quad=\frac{\boldsymbol \mathfrak Nu(n,d)}{o(I_n)}
\sum_{k=0}^{d/2} 2\varepsilon^{m-k}N({\mathfrak P})^{2mk-k^2+m-k+(d-2k-1)(d-2k-2)/2}\\
&\mathfrak quad\cdot
\boldsymbol \mu\boldsymbol \delta(m,d-k-1)
\cdot\sum_{\cls Y\in\mathbb F^{2k+1,2k+1}_{\sym}}\frac{r^*(I_d,Y)}{o(Y)}\, \mathcal G^*_Y
\end{align*}
(where $\cls Y$ is the isometry class of $Y$, or equivalently, the $GL_d(\mathbb F)$-orbit of $Y$).
With $n=2m+1$, similar reasoning gives us
\begin{align*}
\mathcal G^*_{\overline U_{n,d}}
&=\boldsymbol \mathfrak Nu(n,d)\sum_{Y'\in\mathbb F^{d,d}_{\sym}}
\left(\frac{r^*(I_n,Y')}{o(I_n)}-\frac{r^*(J_n,Y')}{o(J_n)}\right)
\e\{2Y'J_d\rho\}\\
&\mathfrak quad=\frac{\boldsymbol \mathfrak Nu(n,d)}{o(I_n)}
\sum_{k=0}^{d/2} 2\varepsilon^{m-k}N({\mathfrak P})^{2mk-k^2+m-k+(d-2k-1)
(d-2k-2)/2}\\
&\mathfrak quad\cdot
\boldsymbol \mu\boldsymbol \delta(m,d-k-1)\cdot
\sum_{\cls Y\in\mathbb F^{2k+1,2k+1}_{\sym}}\frac{r^*(J_d,Y)}{o(Y)}\, \mathcal G^*_Y.
\end{align*}
Now suppose that $n=2m$. Then using Lemma 5.1 we have
\begin{align*}
&\frac{r^*(I_n,Y)}{o(I_n)}-\frac{r^*(J_n,Y)}{o(J_n)}\\
&\mathfrak quad=
\frac{1}{o(I_{n+1})}
(r^*(I_{2m+1},\big<1\big>\perp Y)-r^*(J_{2m+1},\big<1\big>\perp Y)).
\end{align*}
So following the above reasoning, for $n=2m$ we have
\begin{align*}
\mathcal G^*_{U_{n,d}}
&=\frac{\boldsymbol \mathfrak Nu(n,d)}{o(I_{n+1})}
\sum_{k=0}^{d/2} 2\varepsilon^{m-k}N({\mathfrak P})^{2mk-k^2+m-k+(d-2k)(d-2k-1)/2}\\
&\mathfrak quad\cdot
\boldsymbol \mu\boldsymbol \delta(m,d-k)\cdot
\sum_{\cls Y\in\mathbb F^{2k,2k}_{\sym}}\frac{r^*(I_d,Y)}{o(Y)}\, \mathcal G^*_Y,\\
\mathcal G^*_{\overline U_{n,d}}
&=\frac{\boldsymbol \mathfrak Nu(n,d)}{o(I_{n+1})}
\sum_{k=0}^{d/2} 2\varepsilon^{m-k}N({\mathfrak P})^{2mk-k^2+m-k+(d-2k)(d-2k-1)/2}\\
&\mathfrak quad\cdot
\boldsymbol \mu\boldsymbol \delta(m,d-k)\cdot
\sum_{\cls Y\in\mathbb F^{2k,2k}_{\sym}}\frac{r^*(J_d,Y)}{o(Y)}\, \mathcal G^*_Y.
\end{align*}
To evaluate $\mathcal G^*_{I_n}$ and $\mathcal G^*_{J_n}$, we make use of the (non-twisted) Gauss sums
\begin{align*}
\mathcal G_{I_n}&=\sum_{U\in\mathbb F^{n,n}}\e\{2I_n[U]\rho\},\
\mathcal G_{J_n}=\sum_{U\in\mathbb F^{n,n}}\e\{2J_n[U]\rho\},\\
\overline\mathcal G_{I_n}
&=\sum_{U\in\mathbb F^{n,n}}\e\{2I_n[U]J_n\rho\},\
\overline\mathcal G_{J_n}
=\sum_{U\in\mathbb F^{n,n}}\e\{2J_n[U]J_n\rho\}.
\end{align*}
For $Y\in\mathbb F^{n,n}$, by looking at the trace of the matrix $^tYY$, it is easy to check that
$\mathcal G_{I_n}=(\mathcal G_1^*)^{n^2}$.
Similarly, we have
\begin{align*}
\mathcal G_{J_n}&=(\mathcal G_1^*)^{n(n-1)}\cdot(\mathcal G^*_{\omega})^{n}=
\overline\mathcal G_{I_n},\
\overline\mathcal G_{J_n}=(\mathcal G^*_1)^{(n-1)^2}\cdot(\mathcal G^*_{\omega})^{2n-1}.
\end{align*}
Classical techniques give us $\mathcal G^*_{\omega}=-\mathcal G^*_1$
and $(\mathcal G^*_1)^2=\varepsilon N({\mathfrak P})$.
On the other hand, we have
\begin{align*}
\mathcal G_{I_{n}}&=\sum_{Y\in\mathbb F^{n,n}_{\sym}} r(I_{n}, Y) \e\{2Y\rho\},\
\mathcal G_{J_{n}}=\sum_{Y\in\mathbb F^{n,n}_{\sym}} r(J_{n}, Y) \e\{2Y\rho\},\\
\overline\mathcal G_{I_{n}}&=\sum_{Y\in\mathbb F^{n,n}_{\sym}} r(I_{n}, Y) \e\{2YJ_n\rho\},\
\overline\mathcal G_{J_{n}}=\sum_{Y\in\mathbb F^{n,n}_{\sym}} r(J_{n}, Y) \e\{2YJ_n\rho\}.
\end{align*}
Partitioning $\mathbb F^{n,n}_{\sym}$ into $GL_{n}(\mathbb F)$-orbits,
we get
\begin{align*}
&\frac{1}{o(I_n)}\,\mathcal G_{I_n}-\frac{1}{o(J_n)}\,\mathcal G_{J_n}\\
&\mathfrak quad=
\frac{r(I_n,0_n)}{o(I_n)}-\frac{r(J_n,0_n)}{o(J_n)}\\
&\mathfrak qquad+
\sum_{0<\ell\le n}\sum_{Y\sim U_{n,\ell}} \left(\frac{r(I_n,U_{n,\ell})}{o(I_n)}-
\frac{r(J_n,U_{n,\ell})}{o(J_n)}\right) \e\{2Y\rho\}\\
&\mathfrak qquad +
\sum_{0<\ell\le n}\sum_{Y\sim\overline U_{n,\ell}} \left(\frac{r(I_n,\overline U_{n,\ell})}{o(I_n)}-
\frac{r(J_n,\overline U_{n,\ell})}{o(J_n)}\right) \e\{2Y\rho\}.
\end{align*}
Notice that $r(I_n,I_{\ell}\perp 0_{d-\ell})=r^*(I_n,I_{\ell})r(I_{n-\ell},0_{d-\ell}).$
So using Lemmas 5.1 and 5.2, and then Lemma 5.3, when $n$ is odd we get
\begin{align*}
\mathcal G_{I_n}-\mathcal G_{J_n}
&\mathfrak quad=
\sum_{\substack{0\le \ell\le n\\ \ell\,\text{odd}}}
(r(I_{n}, U_{n,\ell})-r(J_{n}, U_{n,\ell}))\\
&\mathfrak qquad\cdot
\left(\sum_{Y\sim U_{n,\ell}}\e\{2Y\rho\}
-\sum_{Y\sim \overline U_{n,\ell}}\e\{2Y\rho\}\right)\\
&\mathfrak quad=
\sum_{\substack{0\le \ell\le n\\ \ell\,\text{odd}}}
(r(I_{n}, U_{n,\ell})-r(J_{n}, U_{n,\ell}))
\sum_{\cls Y\in\mathbb F^{\ell,\ell}_{\sym}}
\frac{r^*(I_{n},Y)}{o(Y)}\, \mathcal G^*_Y.
\end{align*}
Similarly, with $n$ odd,
\begin{align*}
\overline\mathcal G_{I_{2m+1}}-\overline\mathcal G_{J_{2m+1}}
&=\sum_{\substack{0\le \ell\le n\\ \ell\,\text{odd}}}
(r(I_{n}, U_{n,\ell})-r(J_{n}, U_{n,\ell}))
\sum_{\cls Y\in\mathbb F^{\ell,\ell}_{\sym}}
\frac{r^*(J_{n},Y)}{o(Y)}\, \mathcal G^*_Y.
\end{align*}
When $n$ is even, similar arguments give us
\begin{align*}
&r^*(I_{n+1},1)\,\mathcal G_{I_{n}}-r^*(J_{n+1},1)\,\mathcal G_{J_{n}}\\
&=
\sum_{\substack{0\le \ell\le n\\ \ell\,\text{even}}}
(r(I_{n+1}, \big<1\big>\perp U_{n,\ell})-r(J_{n+1}, \big<1\big>\perp U_{n,\ell}))
\sum_{\cls Y\in\mathbb F^{\ell,\ell}_{\sym}}
\frac{r^*(I_{n},Y)}{o(Y)}\, \mathcal G^*_Y,
\end{align*}
\begin{align*}
&r^*(I_{n+1},1)\,\overline\mathcal G_{I_{n}}-r^*(J_{n+1},1)\,\overline\mathcal G_{J_{n}}\\
&=
\sum_{\substack{0\le \ell\le n\\ \ell\,\text{even}}}
(r(I_{n+1}, \big<1\big>\perp U_{n,\ell})-r(J_{n+1}, \big<1\big>\perp U_{n,\ell}))
\sum_{\cls Y\in\mathbb F^{\ell,\ell}_{\sym}}
\frac{r^*(J_{n},Y)}{o(Y)}\, \mathcal G^*_Y.
\end{align*}
Now we argue by induction on $m$ to prove the theorem in the case that $n=2m+1$. For $m=0$, we have $\mathcal G^*_{U_{1,1}}=\mathcal G^*_1$ (by definition of $\mathcal G^*_1$), and as we have already noted, $\mathcal G^*_{\overline U_{1,1}}=-\mathcal G^*_1$.
So suppose that $m\mathbf ge1$ and that the theorem holds for all $\mathcal G^*_Y$ where
$Y\in\mathbb F^{2r+1,2r+1}_{\sym}$ and $0\le r<m$.
With $0<d<n$, we begin with the expression for $\mathcal G^*_{U_{n,d}}$ that we derived above.
By the induction hypothesis, for
$2k+1\le d$ and
$Y\in\mathbb F^{2k+1,2k+1}_{\sym}$, we have
$\mathcal G^*_Y=\varepsilon^k N({\mathfrak P})^{k^2+2k}\cdot h_Y$
where $h_Y$ is defined in Lemma 5.4.
So by Lemma 5.4 we have
$\mathcal G^*_{U_{n,d}}=0$ when $d$ is even, and
\begin{align*}
\mathcal G^*_{U_{n,d}}
&=\frac{\boldsymbol \mathfrak Nu(n,d)}{o(I_n)}
\sum_{k=0}^{d/2} 2\varepsilon^{m-k}N({\mathfrak P})^{2mk-k^2+m-k+(d-2k-1)(d-2k-2)/2}\\
&\mathfrak quad\cdot
\boldsymbol \mu\boldsymbol \delta(m,d-k-1)\cdot (-1)^k\varepsilon^{k+c} N({\mathfrak P})^{k^2+c}\,\boldsymbol \mathbf gamma(c,k)
\end{align*}
when $d$ is odd with $d=2c+1$.
So assume now that $d=2c+1$; then we have
\begin{align*}
\mathcal G^*_{U_{n,d}}
&= \frac{\boldsymbol \mathfrak Nu(n,d)}{o(I_n)}
\,2\varepsilon^{m+c} N({\mathfrak P})^{m+2c^2}\mathcal G^*_1\cdot A(c,0)
\text{ where }\\
A(t,q)&=\sum_{k=0}^t(-1)^k N({\mathfrak P})^{2k(m+k-2t-q)}\boldsymbol \mu\boldsymbol \delta(m,2t+q-k)
\boldsymbol \mathbf gamma(t,k).
\end{align*}
Since $\mathbf gamma(t,k)=N({\mathfrak P})^{2k}\boldsymbol \mathbf gamma(t-1,k)+\boldsymbol \mathbf gamma(t-1,k-1)$,
we have
\begin{align*}
A(t,q)
&=\sum_{k=0}^{t-1} (-1)^k N({\mathfrak P})^{2k(m+k+1-2t-q)}
\boldsymbol \mu\boldsymbol \delta(m,2t+q-k-1))\boldsymbol \mathbf gamma(t-1,k)\\
&\mathfrak quad\cdot
(\boldsymbol \mu\boldsymbol \delta(m,2t+q-k)-N({\mathfrak P})^{2(m-2t-q+k+1)})\\
&=-A(t-1,q+1)=(-1)^tA(0,t)=(-1)^t \boldsymbol \mu\boldsymbol \delta(m,t+q).
\end{align*}
Therefore, using that
$\boldsymbol \mathfrak Nu(n,d)=N({\mathfrak P})^{(n-d)(n+d-1)/2}\boldsymbol \mu(n-d,n-d)$,
\begin{align*}
\mathcal G^*_{U_{n,d}}
&=(-1)^c \varepsilon^{m+c} N({\mathfrak P})^{m^2+2m-c}\cdot
\frac{\boldsymbol \mu(2(m-c),2(m-c))}{\boldsymbol \mu\boldsymbol \delta(m-c,m-c)}\, \mathcal G^*_1,
\end{align*}
as claimed in the statement of the theorem.
A virtually identical argument gives us $\mathcal G^*_{\overline U_{n,d}}=-\mathcal G^*_{U_{n,d}}$.
Now, still taking $n=2m+1$ and beginning with our earlier expression for $\mathcal G_{I_n}-\mathcal G_{J_n},$
we use Lemmas 5.1 and 5.2 to give us
\begin{align*}
\mathcal G_{I_n}-\mathcal G_{J_n}&=2\sum_{k=0}^{m}\sum_{s=0}^{m-k}(-1)^{m-k-s}
\varepsilon^{m-k}N({\mathfrak P})^{2mk-k^2+m-k+(m-k)^2+s^2}\\
&\mathfrak quad\cdot
\boldsymbol \mu\boldsymbol \delta(m,k)
\boldsymbol \beta\boldsymbol \delta(m-k,s)
\cdot\sum_{\cls Y\in\mathbb F^{2k+1,2k+1}_{\sym}}
\frac{r^*(I_{2m+1},Y)}{o(Y)}\, \mathcal G^*_Y.
\end{align*}
Using that $o(I_n)=2N({\mathfrak P})^{m^2}\boldsymbol \mu\boldsymbol \delta(m,m)$,
and the induction hypothesis for $Y\in\mathbb F^{2k+1,2k+1}_{\sym}$ with $k<m$, we have \begin{align*}
\mathcal G_{I_n}-\mathcal G_{J_n}&=
o(I_{2m+1})\mathcal G^*_{I_{2m+1}}
-o(I_{2m+1})\varepsilon^mN({\mathfrak P})^{m^2+2m}\mathcal G^*_1\, h_{I_{2m+1}}+2\mathcal G^*_1B
\end{align*}
where
\begin{align*}B&=\sum_{k=0}^m\sum_{s=0}^{m-k}(-1)^{m-k-s}
\varepsilon^{m-k}N({\mathfrak P})^{m^2+m-k+s^2}\boldsymbol \mu\boldsymbol \delta(m,k)\boldsymbol \beta\boldsymbol \delta(m-k,s)\\
&\mathfrak quad\cdot
\varepsilon^kN({\mathfrak P})^{k^2+2k}
\sum_{\cls Y\in\mathbb F^{2k+1,2k+1}_{\sym}}
\frac{r^*(I_{2m+1},Y)}{o(Y)}\, h_Y.
\end{align*}
By Lemma 5.4, we get
\begin{align*}
B&=\sum_{k=0}^m\sum_{s=0}^{m-k}(-1)^{m-s}N({\mathfrak P})^{m^2+2m+k(k-1)+s^2}
\boldsymbol \mu\boldsymbol \delta(m,k)\boldsymbol \beta\boldsymbol \delta(m-k,s)\boldsymbol \mathbf gamma(m,k).
\end{align*}
Since
$\boldsymbol \mathbf gamma(m,k)\boldsymbol \beta\boldsymbol \delta(m-k,s)
=\boldsymbol \beta\boldsymbol \delta(m,s)\boldsymbol \mathbf gamma(m-s,k),$
we can sum on $0\le s\le m$, $0\le k\le m-s$. Then replacing $s$ by $m-s$ and using that $\boldsymbol \beta(m,m-s)=\boldsymbol \beta(m,s),$ we get
\begin{align*}
&B=\sum_{s=0}^m(-1)^sN({\mathfrak P})^{2m^2+2m-2ms+s^2}
\boldsymbol \delta(m,m-s)\boldsymbol \beta(m,s)\cdot C(s)
\text{ where}\\
&C(t)=\sum_{k=0}^t N({\mathfrak P})^{k(k-1)}\boldsymbol \mu\boldsymbol \delta(m,k)\boldsymbol \mathbf gamma(t,k).
\end{align*}
Since $\boldsymbol \mathbf gamma(t,k)=N({\mathfrak P})^{2k}\boldsymbol \mathbf gamma(t-1,k)+\boldsymbol \mathbf gamma(t-1,k-1),$ we have
\begin{align*}
C(t)
&=N({\mathfrak P})^{2m}C(t-1)=N({\mathfrak P})^{2mt}C(0)=N({\mathfrak P})^{2mt}.
\end{align*}
Therefore
\begin{align*}
&B=N({\mathfrak P})^{2m^2+2m}D(m,0)
\text{ where}\\
&D(t,q)=\sum_{s=0}^t (-1)^s N({\mathfrak P})^{s(s+q)}\boldsymbol \delta(t+q,t-s)\boldsymbol \beta(t,s).
\end{align*}
Since $\boldsymbol \beta(t,s)=N({\mathfrak P})^s\boldsymbol \beta(t-1,s)+\boldsymbol \beta(t-1,s-1),$ we have
\begin{align*}
D(t,q)
&=D(t-1,q+1)=D(0,t+q)=1.
\end{align*}
Therefore $B=N({\mathfrak P})^{2m^2+2m}$. Our earlier computations show that
$\mathcal G_{I_n}-\mathcal G_{J_n}=2 N({\mathfrak P})^{2m^2+2m}\mathcal G^*_1$;
so
$$\mathcal G^*_{I_{2m+1}}=\varepsilon^mN({\mathfrak P})^{m^2+2m}\cdot h_{I_{2m+1}}\mathcal G^*_1
=(-1)^mN({\mathfrak P})^{m^2+m}\,\mathcal G^*_1.$$
A virtually identical argument gives us $\mathcal G^*_{J_n}=-\mathcal G^*_{I_n}$.
Now we argue by induction on $m$ to prove the theorem in the case that $n=2m$.
Since the computation for $m=1$ is essentially identical to the induction step for $m>1$, we formally define $\mathcal G^*_{I_0}=\mathcal G^*_{J_0}=1$ (which is consistent with the formula claimed in the theorem). So now suppose that $m\mathbf ge1$ and that the theorem holds for all $\mathcal G^*_Y$ where $Y\in\mathbb F^{2r,2r}_{\sym}$ and $0\le r<m$.
With $0<d<n$, we begin with the expression for $\mathcal G^*_{U_{n,d}}$ that we derived above.
Take $c$ so that $d=2c$ or $d=2c+1$.
Using the induction hypothesis, Lemma 5.4, and arguing as we did when $n$ was odd, we get
$$\mathcal G^*_{U_{n,d}}=\frac{\boldsymbol \mathfrak Nu(n,d)}{o(I_{n+1})}
2\varepsilon^m N({\mathfrak P})^{m+d(d-1)/2}A(c,d-2c)$$
where $A(t,q)$ is as defined earlier in this proof; recall that
$$A(t,q)=(-1)^t\boldsymbol \mu\boldsymbol \delta(m,t+q).$$
Since $\boldsymbol \mu(2(m-c),1)=\boldsymbol \mu\boldsymbol \delta(m-c,1)$, for $d=2c$ or $2c+1$
we get
$$\mathcal G^*_{U_{n,d}}
=(-1)^c\varepsilon^mN({\mathfrak P})^{m^2}
\frac{\boldsymbol \mu(2(m-c),2(m-c))}{\boldsymbol \mu\boldsymbol \delta(m-c,m-c)}.$$
A virtually identical argument gives us $\mathcal G^*_{\overline U_{n,d}}=\mathcal G^*_{U_{n,d}}.$
Still assuming that $n=2m$ and beginning with our earlier expression for
$r^*(I_{n+1},1)\mathcal G_{I_n}-r^*(J_{n+1},1)\mathcal G_{J_n}$, we use Lemmas 5.1 and 5.2 and the induction hypothesis to get
\begin{align*}
&r^*(I_{n+1},1)\mathcal G_{I_n}-r^*(J_{n+1},1)\mathcal G_{J_n}\\
&\mathfrak quad=o(I_{n+1})\mathcal G^*_{I_n}-o(I_{n+1})\varepsilon^m N({\mathfrak P})^{m^2}
\cdot h_{I_n}
+2\varepsilon^m N({\mathfrak P})^{-m}B
\end{align*}
where $B$ is as in the case of $n$ odd. We saw that $B=N({\mathfrak P})^{2m^2+2m}$, and
$$r^*(I_{n+1},1)\mathcal G_{I_n}-r^*(J_{n+1},1)\mathcal G_{J_n}
=2\varepsilon^m N({\mathfrak P})^{2m+m};$$
so we get
$$\mathcal G^*_{I_{2m}}=\varepsilon^m N({\mathfrak P})^{m^2}\cdot h_{I_{2m}}
=(-1)^m\varepsilon^m N({\mathfrak P})^{m^2}.$$
The argument to evaluate $\mathcal G^*_{J_{2m}}$ is essentially identical to that of evaluating $\mathcal G^*_{J_{2m+1}}$, where for this we begin with the identity
$$r^*(J_{2m+1},1)\overline\mathcal G_{I_{2m}}-r^*(I_{2m+1},1)\overline\mathcal G_{J_{2m}}
=2\varepsilon^m N({\mathfrak P})^{2m^2+m}.$$
This proves the theorem.
To prove Corollary 1.2, we first note that by Theorem 1.1, $\left(\mathcal G^*_1\right)^m\mathcal G^*_T$ has no dependence on our choice of $\rho$. Thus we can follow the argument of Lemma 3.1 \cite{half-int-aps}, as the techniques are local. In \cite{half-int-aps}, all quadratic forms were assumed to be even; since $2$ is a unit in $\mathbb F$, we have $R^*(T\perp\big<1\big>,0_a)=
R^*(2T\perp\big<2\big>,0_a)$, and hence Corollary 1.2 follows.
\section{Variations on quadratically twisted Gauss sums}
For $T\in\mathbb F^{n,n}_{\sym}$ and $0\le r\le n$, here we consider
$\mathcal G_T^*({\mathfrak P};r)$, as defined in the introduction.
For $T=0_n$, we have $\mathcal G^*_{0_n}({\mathfrak P};r)=\mathcal G^*_{0_d}({\mathfrak P})$, so we only need to consider $T\mathfrak Not=0_n$.
\begin{prop} Take $n\in\mathbb Z_+$, $T\in\mathbb F^{n,n}_{\sym}$ and let $d=\rank T$.
\begin{enumerate}
\item[(a)] Suppose that $0\le 2t+1\le n$. When $d$ is even we have
$\mathcal G^*_T({\mathfrak P};2t+1)=0.$ When $d$ is odd with $d=2c+1$, we have
$\mathcal G^*_{\overline U_{n,d}}({\mathfrak P};2t+1)=-\mathcal G^*_{U_{n,d}}({\mathfrak P};2t+1),$ and
\begin{align*}
\mathcal G^*_{U_{n,d}}({\mathfrak P};2t+1)
&=\frac{\boldsymbol \mathfrak Nu(n,2c+1)}{\boldsymbol \mathfrak Nu(n-1,2c)}\varepsilon^cN({\mathfrak P})^c\mathcal G^*_1({\mathfrak P})
\mathcal G^*_{U_{n-1,2c}}({\mathfrak P};2t).
\end{align*}
\item[(b)] Suppose that $0\le 2t\le n$; set $s=n-2t$. Then with $c$ so that $d=2c$ or $2c+1$,
we have
\begin{align*}
\mathcal G^*_{T}({\mathfrak P};2t)
&=
\frac{\boldsymbol \mathfrak Nu(n,d)}{o(I_{2t+1}\perp0_s)}
\sum_{k=0}^c
(-1)^k\varepsilon^kN({\mathfrak P})^{s(2k+1)+2tk+t-k}\boldsymbol \mu\boldsymbol \delta(t,k) \\
&\mathfrak quad\cdot
\boldsymbol \mathbf gamma(c,k) A_s(t-k,c-k)
\end{align*}
where
\begin{align*}
A_s(x,y)=
(N({\mathfrak P})^x+\varepsilon^x) r^*(I_{2x}\perp0_s,0_{2y})
- (N({\mathfrak P})^x-\varepsilon^x)r^*(J_{2x}\perp0_s,0_{2y}).
\end{align*}
\end{enumerate}
\end{prop}
\mathfrak Noindent{\bf Remark:} As $\boldsymbol \mathfrak Nu(2y,0)$ is the number of bases for any $2y$-dimensional space,
$r^*(T',0_{2y})=\boldsymbol \mathfrak Nu(2y,0)\cdot R^*(T',0_{2y})$ for any symmetric $T'$.
\begin{proof}
Throughout this proof, we follow the lines of argument used in Section 3. In this way we get
\begin{align*}
\mathcal G^*_{U_{n,d}}({\mathfrak P};r)
&=\boldsymbol \mathfrak Nu(n,d)\sum_{Y\in\mathbb F^{d,d}_{\sym}}
\left(\frac{r^*(U_{n,r},Y)}{o(U_{n,r},Y)}
-\frac{r^*(\overline U_{n,r})}{o(\overline U_{n,r},Y)}\right) \e\{2Y/p\},
\end{align*}
\begin{align*}
\mathcal G^*_{\overline U_{n,d}}({\mathfrak P};r)
&=\boldsymbol \mathfrak Nu(n,d)\sum_{Y\in\mathbb F^{d,d}_{\sym}}
\left(\frac{r^*(U_{n,r},Y)}{o(U_{n,r},Y)}
-\frac{r^*(\overline U_{n,r})}{o(\overline U_{n,r},Y)}\right) \e\{2YJ_d/p\}.
\end{align*}
We have $o(0_{n-r})=\boldsymbol \mathfrak Nu(n-r,n-r),$ and
$$o(U_{n,r})=o(I_r)N({\mathfrak P})^{r(n-r)}o(0_{n-r}),\
o(\overline U_{n,r})=o(J_r)N({\mathfrak P})^{r(n-r)}\boldsymbol \mathfrak Nu(n-r,n-r).$$
First consider the case that $r=2t+1$. Then for even $\ell$ ($\ell\le d$), we have
$$r^*(U_{n,r},U_{d,\ell})-r^*(\overline U_{n,r},U_{d,\ell})
=0=r^*(U_{n,r}\overline U_{d,\ell})-r^*(\overline U_{n,r},\overline U_{d,\ell}),$$
and for odd $\ell$ we have
$$r^*(U_{n,r},U_{d,\ell})=r^*(\overline U_{n,r},\overline U_{d,\ell}),\
r^*(U_{n,r},\overline U_{d,\ell})=r^*(\overline U_{n,r},U_{d,\ell}).$$
Hence using Lemma 5.3, we have
\begin{align*}
\mathcal G^*_{U_{n,d}}({\mathfrak P};2t+1)
&=\frac{\boldsymbol \mathfrak Nu(n,d)}{o(U_{n,2t+1})}
\sum_{0\le 2k+1\le d}
\left(\sum_{\cls Y\in\mathbb F^{2k+1,2k+1}_{\sym}}\frac{r^*(I_d,Y)}{o(Y)} \mathcal G^*_Y\right)\\
&\mathfrak quad\cdot
(r^*(U_{n,2t+1},U_{d,2k+1})-r^*(\overline U_{n,2t+1},U_{d,2k+1})).
\end{align*}
So by Theorem 1.1 and Lemmas 5.1 and 5.4, when $d$ is even we get
$\mathcal G^*_{U_{n,d}}({\mathfrak P};2t+1)=0$, and when $d$ is odd with $d=2c+1$, we get
\begin{align*}
\mathcal G^*_{U_{n,d}}({\mathfrak P};2t+1)
&=\frac{\boldsymbol \mathfrak Nu(n,d)}{o(U_{n,2t+1})}
\sum_{k=0}^c (-1)^k\varepsilon^{k+c} N({\mathfrak P})^{k^2+c} \boldsymbol \mathbf gamma(c,k) \mathcal G^*_1\\
&\mathfrak quad\cdot
(r^*(U_{n,2t+1},U_{d,2k+1})-r^*(\overline U_{n,2t+1},U_{d,2k+1})).
\end{align*}
An almost identical argument gives us
$\mathcal G^*_{\overline U_{n,d}}({\mathfrak P};2t+1)=-\mathcal G^*_{U_{n,d}}({\mathfrak P};2t+1).$
Now consider the case that $r=2t$.
With $d=2c$ or $2c+1$, reasoning as in Section 3 gives us
\begin{align*}
\mathcal G^*_{U_{n,d}}({\mathfrak P};2t)
&=\frac{\boldsymbol \mathfrak Nu(n,d)}{o(U_{n+1,2t+1})}
\sum_{k=0}^c(-1)^k\varepsilon^k N({\mathfrak P})^{k^2} \boldsymbol \mathbf gamma(c,k)\\
&\mathfrak quad\cdot\left(r^*(U_{n+1,2t+1},U_{d+1,2k+1})
- r^*(\overline U_{n+1,2t+1},U_{d+1,2k+1})\right)\\
&=
\mathcal G^*_{\overline U_{n,d}}({\mathfrak P};2t).
\end{align*}
This gives us (a) and part of (b).
To finish proving (b), we begin with the above equation, taking $d=2c$.
It is easily seen that
$r^*(I_r\perp 0_s,1)=N({\mathfrak P})^s r^*(I_r,1),$
and consequently from Lemma 5.1 we get
\begin{align*}
&
r^*(U_{n+1,2t+1},U_{d+1,2k+1})
- r^*(\overline U_{n+1,2t+1},U_{d+1,2k+1})\\
&\mathfrak quad =
N({\mathfrak P})^{(n-2t)(2k+1)+2tk-k^2+t-k}\boldsymbol \mu\boldsymbol \delta(t,k) A_{n-2t}(t-k,c-k)
\end{align*}
with $A_s(x,y)$ as in the statement of the proposition.
\end{proof}
\section{Lemmas and their proofs}
\begin{lem} Take $t,d\in\mathbb Z_+$. We have:
\begin{align*}
r^*(I_{2t},1)&=N({\mathfrak P})^{t-1}(N({\mathfrak P})^t-\varepsilon^t)=r^*(I_{2t},\omega),\\
r^*(J_{2t},1)&=N({\mathfrak P})^{t-1}(N({\mathfrak P})^t+\varepsilon^t)=r^*(J_{2t},1),\\
r^*(I_{2t+1},1)&=N({\mathfrak P})^t(N({\mathfrak P})^t+\varepsilon^t)=r^*(J_{2t+1},\omega),\\
r^*(I_{2t+1},\omega)&=N({\mathfrak P})^t(N({\mathfrak P})^t-\varepsilon^t)=r^*(J_{2t+1},1);
\end{align*}
also,
\begin{align*}
r^*(I_{2t},0_d)&=N({\mathfrak P})^{d(d-1)/2}(N({\mathfrak P})^t-\varepsilon^t)\boldsymbol \mu\boldsymbol \delta(t-1,d-1)
(N({\mathfrak P})^{t-d}+\varepsilon^t),\\
r^*(J_{2t},0_d)&=N({\mathfrak P})^{d(d-1)/2}(N({\mathfrak P})^t+\varepsilon^t)\boldsymbol \mu\boldsymbol \delta(t-1,d-1)
(N({\mathfrak P})^{t-d}-\varepsilon^t),\\
r^*(I_{2t+1},0_d)&=N({\mathfrak P})^{d(d-1)/2}\boldsymbol \mu\boldsymbol \delta(t,d)=r^*(J_{2t+1},0_d).
\end{align*}
\end{lem}
\begin{proof}
The first collection of formulas
are from Theorems 2.59 and 2.60 of \cite{Ger}.
For the second collection of formulas,
we begin with Theorems 2.59 and 2.60 of \cite{Ger}, giving us formulas for $r(I_t,0)=r^*(I_t,0)+1$ and $r(J_t,0)=r^*(J_t,0)+1$.
Now consider the case that $V$ is a $2t$-dimensional space over $\mathbb F$ equipped with a quadratic form $Q_V$ given by $I_{2t}$ relative to some basis for $V$.
So $r^*(I_{2t},0_d)$ is the number of all (ordered) bases for $d$-dimensional, totally isotropic subspaces of $V$.
(Recall that a subspace $W$ of $V$ is totally isotropic if $Q_V$ restricts to 0 on $W$.)
Suppose that $d>1$; we construct all bases for $d$-dimensional, totally isotropic subspaces of $V$ as follows.
Choose an isotropic vector $x$ from $V$ (so $x\mathfrak Not=0$ and $Q_V(x)=0$; note that this is not possible if $t=1$ and $\varepsilon=-1$). Then as $V$ is a regular space, there is some $y\in V$ so that $y$ is not orthogonal to $x$; hence (by Theorem 2.23 \cite{Ger})
$x,y$ span a hyperbolic plane, and (by Theorem 2.17 \cite{Ger}), this hyperbolic plane splits $V$, giving us
$V=(\mathbb F x\oplus\mathbb F y)\perp V'$ where $V'$ is hyperbolic if and only if $V$ is.
We have $\disc V=\varepsilon \disc V'$ and so the quadratic form on $V'$ is given by $I_{2(t-1)}$ if $\varepsilon=1$, and by
$J_{2(t-1)}$ if $\varepsilon=-1$.
The number of all bases for $d$-dimensional, totally isotropic subspaces of $V$ with $x$ as the first basis element is
$N({\mathfrak P})^{d-1} r^*(I_{2(t-1)},0_{d-1})$ if $\varepsilon=1$, and
$N({\mathfrak P})^{d-1} r^*(J_{2(t-1)},0_{d-1})$ otherwise. The formula claimed now follows by induction on $d$.
Virtually identical arguments yield the formulas when $I_{2t}$ is replaced by $J_{2t}$ or $I_{2t+1}$ or $J_{2t+1}$.
\end{proof}
\begin{lem}
Suppose that $m\mathbf ge 0$. We have
\begin{align*}
&\sum_{s=0}^m (-1)^s N({\mathfrak P})^{(2m+1)(m-s)+s^2}\boldsymbol \beta\boldsymbol \delta(m,m-s)\\
&\mathfrak quad=r(I_{2m+1},0_{2m+1})
=r(J_{2m+1},0_{2m+1}),
\end{align*}
\begin{align*}
&\sum_{s=0}^m(-1)^s N({\mathfrak P})^{2m(m-s)+s(s-1)}\boldsymbol \beta(m,m-s)\boldsymbol \delta(m-1,m-s)\\
&\mathfrak quad=\begin{cases} r(I_{2m},0_{2m})&\text{if $\varepsilon^m=1$,}\\
r(J_{2m},0_{2m})&\text{if $\varepsilon^m=-1$,}\end{cases}\\
&\sum_{s=1}^m (-1)^{s+1} N({\mathfrak P})^{2m(m-s)+s(s-1)}\boldsymbol \beta(m-1,m-s)\boldsymbol \delta(m,m-s)\\
&\mathfrak quad=\begin{cases} r(J_{2m},0_{2m})&\text{if $\varepsilon^m=1$,}\\
r(I_{2m},0_{2m})&\text{if $\varepsilon^m=-1$.}\end{cases}
\end{align*}
\end{lem}
\begin{proof}
Suppose that $V$ is an $n$-dimensional vector space over $\mathbb F$ equipped with a quadratic form given by $Q_V=I_n$ or $J_n$.
Then $r(Q_V,0_n)$ is the number of (ordered) $x_1,\ldots,x_n\in V$ so that $\text{span}\{x_1,\ldots,x_n\}$ is totally isotropic.
As $\boldsymbol \mathfrak Nu(d,0)$ is the number of bases for any given dimension $d$ space over $\mathbb F$, the number of dimension $d$ totally isotropic subspaces of $V$ is
$$\varphi_d(V)=r^*(Q_V,0_d)/\boldsymbol \mathfrak Nu(d,0).$$
We treat the case that $V\simeq\mathbb H^m$, meaning that
$\dim V=2m$ and the quadratic form on $V$ is given by
$I_{2m}$ if $\varepsilon^m=1$, and by $J_{2m}$ otherwise
(analogous arguments treat the other cases). Slightly abusing notation, we write $(x_1,\ldots,x_{2m})\subseteq V$ to mean that $(x_1,\ldots,x_{2m})$ is an ordered $2m$-tuple of vectors from $V$. We set
$$\mathcal W_{m-s}=\{\text{dimension } m-s \text{ totally isotropic subspaces } W \text{ of }V\},$$
and we let
$${\mathbb 1}_W(x_1,\ldots,x_{2m})=
\begin{cases}1&\text{if $x_1,\ldots,x_{2m}\in W$,}\\
0&\text{otherwise}.\end{cases}$$
Thus for $(x_1,\ldots,x_{2m})\subseteq V$,
$\sum_{W\in\mathcal W_{m-s}} {\mathbb 1}_W(x_1,\ldots,x_{2m})$ is the number of elements of $\mathcal W_{m-s}$ containing $x_1,\ldots,x_{2m}$, and, noting that
$N({\mathfrak P})^{2m(m-s)}$ is the number of (ordered) $2m$-tuples of vectors in each $W\in\mathcal W_{m-s}$, we have
$$\sum_{(x_1,\ldots,x_{2m})\subseteq V} \left(\sum_{W\in\mathcal W_{m-s}}
{\mathbb 1}_W(x_1,\ldots,x_{2m})\right)
= N({\mathfrak P})^{2m(m-s)}\varphi_{m-s}(V).$$
So
\begin{align*}
\psi(V):=
&\sum_{s=0}^m (-1)^s N({\mathfrak P})^{s(s-1)+2m(m-s)}\varphi_{m-s}(V)\\
=&\sum_{(x_1,\ldots,x_{2m})\subseteq V}\left(\sum_{s=0}^m (-1)^s N({\mathfrak P})^{s(s-1)}
\sum_{W\in\mathcal W_{m-s}}
{\mathbb 1}_W(x_1,\ldots,x_{2m})\right).
\end{align*}
Fix $(x'_1,\ldots,x'_{2m})\subseteq V$; let $W'$ be the subspace spanned by $x'_1,\ldots,x'_{2m}$, and set $\ell=\dim W'$. If $W'$ is not totally isotropic then
${\mathbb 1}_W(x'_1,\ldots,x'_{2m})=0$ for all totally isotropic $W$.
So suppose that $W'$ is totally isotropic. Then repeatedly using Theorems 2.19, 2.23, 2.52 of \cite{Ger} and the assumption that $V$ is regular, we find that there is a dimension $\ell$ subspace $W''$ so that $W'\oplus W''\simeq\mathbb H^{\ell}$ and
$V=(W'\oplus W'')\perp V'$ where $V'\simeq \mathbb H^{m-\ell}$.
Hence the number of $W\in\mathcal W_{m-s}$ that contain $W'$ is $\varphi_{m-s-\ell}(V')$.
Therefore, using Lemma 5.1 and the above formula for $\varphi_{m-s-\ell}(V')$, we have
\begin{align*}
&\sum_{s=0}^{m-\ell} (-1)^s N({\mathfrak P})^{s(s-1)} \sum_{W\in\mathcal W_{m-s}}
{\mathbb 1}(x'_1,\ldots,x'_{2m})
=A(m-\ell,m-\ell-1)
\end{align*}
where
$$A(t,k)=\sum_{s=0}^t (-1)^sN({\mathfrak P})^{s(s+k-t)}\boldsymbol \delta(k,t-s)\boldsymbol \beta(t,t-s).$$
We argue by induction on $t$ to show that for any $k$ and $t\mathbf ge 0$,
we have $A(t,k)=1$.
Clearly $A(0,k)=1$ for all $k$.
So fix $t\mathbf ge0$ and suppose that $A(t,k)=1$ for all $k$.
Hence we have
\begin{align*}
1
&=(N({\mathfrak P})^k+1)\sum_{s=0}^t(-1)^sN({\mathfrak P})^{s(s+k-1-t)}
\boldsymbol \delta(k-1,t-s)\boldsymbol \beta(t,t-s)\\
&\mathfrak quad -N({\mathfrak P})^k\sum_{s=0}^t (-1)^s N({\mathfrak P})^{s(s+k-t)}\boldsymbol \delta(k,t-s)\boldsymbol \beta(t,t-s).
\end{align*}
Notice that in the first sum in the above equality, we can allow $s$ to vary from $0$ to $t+1$ (since $\boldsymbol \beta(t,-1)=0$), and in the second sum we can allow $s$ to vary from $-1$ to $t$ (since $\boldsymbol \beta(t,t+1)=0$).
Also, we know that
$$(N({\mathfrak P})^k+1)\boldsymbol \delta(k-1,t-s)=\boldsymbol \delta(k,t-s+1);$$
so replacing $s$ by $s-1$ in the second sum and using that
$$\boldsymbol \beta(t,t-s)+N({\mathfrak P})^{t+1-s}\boldsymbol \beta(t,t+1-s)=\boldsymbol \beta(t+1,t+1-s),$$
we find that $A(t+1,k)=1$ for all $k$.
Hence $\psi$ counts $(x_1,\ldots,x_{2m})\subseteq V$
zero times if
$\text{span}\{x_1,\ldots,x_{2m}\}$ is not totally isotropic, and once otherwise. Thus $\psi=r(Q_V,0_{2m}).$
\end{proof}
\begin{lem}
Fix $d\in\mathbb Z_+$ and $\ell\in\mathbb Z$ so that $0\le \ell<d$.
Then
$$\sum_{Y\sim U_{d,\ell}} \e\{2Y\rho\}
-\sum_{Y\sim \overline U_{d,\ell}} \e\{2Y\rho\}
=\sum_{\cls Y'\in\mathbb F^{\ell,\ell}_{\sym}} \frac{r^*(I_d,Y')}{o(Y')} \mathcal G^*_{Y'}({\mathfrak P} I_{\ell})$$
and
$$\sum_{Y\sim U_{d,\ell}} \e\{2YJ_d\rho\}
-\sum_{Y\sim \overline U_{d,\ell}} \e\{2YJ_d\rho\}
=\sum_{\cls Y'\in\mathbb F^{\ell,\ell}_{\sym}} \frac{r^*(J_d,Y')}{o(Y')} \mathcal G^*_{Y'}({\mathfrak P} I_{\ell}),$$
where $\cls Y'$ varies over a set of representatives for the $GL_{\ell}(\mathbb F)$-orbits in $\mathbb F^{\ell,\ell}_{\sym}$.
\end{lem}
\begin{proof}
We first consider the sum over $Y\sim \overline U_{d,\ell}$.
We know that for $G\in GL_{d}(\mathbb F)$ and $G'$ in the orthogonal group of
$\overline U_{d,\ell}$, we have
$^t(G'G)\overline U_{d,\ell}(G'G)=\,^tG\overline U_{d,\ell}G$, so when we let $G$ vary over $GL_{d}(\mathbb F)$, each element in the orbit of $\overline U_{d,\ell}$ appears exactly $o(\overline U_{d,\ell})$ times. Also,
recall that with $\sigma$ denoting the matrix trace map, we have
$\sigma(^tG\overline U_{d,\ell}G)=\sigma(\overline U_{d,\ell}GI_d\,^tG)$
and $\sigma(\overline U_{d,\ell}GI_d\,^tG)=\sigma(J_{\ell}Y')$ where $Y'$ is the upper left $\ell\times \ell$ block of $GI_d\,^tG$.
So we have
\begin{align*}
\sum_{Y\sim\overline U_{d,\ell}}\e\{2Y\rho\}
&=\frac{1}{o(\overline U_{d,\ell})}
\sum_{G\in GL_d(\mathbb F)} \e\{2\,^tG\overline U_{d,\ell}G\rho\}\\
&=\frac{\boldsymbol \mathfrak Nu(d,\ell)}{o(\overline U_{d,\ell})}
\sum_{Y'\in\mathbb F^{\ell,\ell}_{\sym}} r^*(I_d,Y') \e\{2 J_{\ell} Y'\rho\}
\end{align*}
since
$$r^*(I_d,Y')=\#\{C\in\mathbb F^{d,\ell}: \ ^tCC=Y',\ \rank C=\ell\ \},$$
and the number of ways to extend $C$ to an element of $GL_{d}(\mathbb F)$ is $\boldsymbol \mathfrak Nu(d,\ell)$.
Now, as $G$ varies over $GL_{\ell}(\mathbb F)$, $^tGY'G$ varies $o(Y')$ times over the elements in $\cls Y'$. Also, by Lemma 5.1, we have
$o(\overline U_{d,\ell})=o(J_{\ell})\boldsymbol \mathfrak Nu(d,\ell)$.
Hence
\begin{align*}
\sum_{Y\sim\overline U_{d,\ell}} \e\{2Y\rho\}
&= \frac{1}{o(J_{\ell})}
\sum_{G\in GL_d(\mathbb F)} \sum_{\cls Y'\in\mathbb F^{\ell,\ell}_{\sym}}
\frac{r^*(I_d,Y')}{o(Y')}\e\{2G J_{\ell}\,^tGY'\rho\}\\
&= \sum_{\cls Y'\in\mathbb F^{\ell,\ell}_{\sym}}\frac{r^*(I_d,Y')}{o(Y')}
\sum_{X\sim J_{\ell}}\e\{2XY'\rho\}
\end{align*}
where for the last equality we used that as $G$ varies over $GL_{\ell}(\mathbb F)$,
$GJ_{\ell}\,^tG$ varies $o(J_{\ell})$ times over the elements in the orbit of $J_{\ell}$.
The analysis of
$$\sum_{Y\sim U_{d,\ell}}\e\{2Y\rho\},\
\sum_{Y\sim U_{d,\ell}}\e\{2YJ_d\rho\},\text{ and }
\sum_{Y\sim \overline U_{d,\ell}}\e\{2YJ_d\rho\}$$
follow in a virtually identical manner. Then we note that
$$\sum_{X\sim I_{\ell}}\e\{2XY'\rho\}-\sum_{X\sim J_{\ell}}\e\{2XY'\rho\}
=\mathcal G^*_{Y'},$$
completing the proof.
\end{proof}
\begin{lem} Suppose that $0< \ell\le d$; take $c$ so that $d$ is $2c$ or $2c+1$.
Take $Y\in\mathbb F^{\ell,\ell}_{\sym}$, and take $b$ so that $\rank Y$ is $2b$ or $2b+1$.
\begin{enumerate}
\item[(a)] Suppose that $\ell=2k$; set
$$h_Y=(-1)^b\cdot\frac{\boldsymbol \mu(2(k-b),2(k-b))}{\boldsymbol \mu\boldsymbol \delta(k-b,k-b)}.$$
Then
$$\sum_{\cls Y\in\mathbb F^{\ell,\ell}_{\sym}}\frac{r^*(I_d,Y)}{o(Y)}\,h_Y=
\sum_{\cls Y\in\mathbb F^{\ell,\ell}_{\sym}}\frac{r^*(J_d,Y)}{o(Y)}\,h_Y=
(-1)^k\boldsymbol \mathbf gamma(c,k).$$
\item[(b)] Suppose that $\ell=2k+1$; set
$$h_Y=\begin{cases}
(-1)^b\varepsilon^b N({\mathfrak P})^{-b}\cdot\frac{\boldsymbol \mu(2(k-b),2(k-b))}{\boldsymbol \mu\boldsymbol \delta(k-b,k-b)}
&\text{if $Y\sim I_{2b+1}\perp 0_{2(k-b)}$,}\\
(-1)^{b+1}\varepsilon^b N({\mathfrak P})^{-b}\cdot\frac{\boldsymbol \mu(2(k-b),2(k-b))}{\boldsymbol \mu\boldsymbol \delta(k-b,k-b)}
&\text{if $Y\sim J_{2b+1}\perp 0_{2(k-b)}$,}\\
0&\text{if $\rank Y=2b$.}
\end{cases}$$
Then when $d=2c$,
$$\sum_{\cls Y\in\mathbb F^{\ell,\ell}_{\sym}}\frac{r^*(I_d,Y)}{o(Y)}\,h_Y=
\sum_{\cls Y\in\mathbb F^{\ell,\ell}_{\sym}}\frac{r^*(J_d,Y)}{o(Y)}\,h_Y=
0,$$
and when $d=2c+1$,
\begin{align*}
\sum_{\cls Y\in\mathbb F^{\ell,\ell}_{\sym}}\frac{r^*(I_d,Y)}{o(Y)}\,h_Y
&=
-\sum_{\cls Y\in\mathbb F^{\ell,\ell}_{\sym}}\frac{r^*(J_d,Y)}{o(Y)}\,h_Y\\
&=
(-1)^k\varepsilon^cN({\mathfrak P})^{c-2k}
\boldsymbol \mathbf gamma(c,k).
\end{align*}
\end{enumerate}
\end{lem}
\begin{proof}
We have
\begin{align*}
&\sum_{\cls Y\in\mathbb F^{\ell,\ell}_{\sym}}\frac{r^*(I_d,Y)}{o(Y)\,}h_Y
=\frac{r^*(I_d,0_{\ell})}{o(0_{\ell})}\\
&\mathfrak quad
+\sum_{a=1}^{\ell}\left(\frac{r^*(I_d,I_a\perp 0_{\ell-a})}{o(I_a\perp0_{\ell-a})}
\,h_{I_a\perp 0_{\ell-a}}
+\frac{r^*(I_d,J_a\perp 0_{\ell-a})}{o(J_a\perp0_{\ell-a})}
\,h_{J_a\perp 0_{\ell-a}}\right).
\end{align*}
\begin{align*}
&o(I_{2b+1})N({\mathfrak P})^{2bs}\boldsymbol \mathfrak Nu(s,0)=N({\mathfrak P})^{2b}\boldsymbol \mu(s,1)o(I_{2b+1}\perp 0_{s-1})\\
&\mathfrak quad=r^*(I_{2b+1},1)o(I_{2b}\perp0_s)
=r^*(J_{2b+1},1)o(J_{2b}\perp0_s).
\end{align*}
(a) Suppose that $\ell=2k$.
Then using Lemma 5.1, when $d=2c$ we get
\begin{align*}
&r^*(I_{2b+1},1)r^*(I_d,I_{2b}\perp 0_{2(k-b)})
+r^*(J_{2b+1},1)r^*(I_d,J_{2b}\perp 0_{2(k-b)})\\
&\mathfrak quad=
2N({\mathfrak P})^{(k-b)(2k-2b-1)+2cb-b^2}\\
&\mathfrak qquad\cdot
\boldsymbol \mu\boldsymbol \delta(c-1,2k-b-1)(N({\mathfrak P})^c-\varepsilon^c)(N({\mathfrak P})^{c-2(k-b)}+\varepsilon^c)
\end{align*}
and
\begin{align*}
&N({\mathfrak P})^{2b}(N({\mathfrak P})^{2(k-b)}-1)\\
&\mathfrak qquad\cdot
\left(r^*(I_d,I_{2b+1}\perp 0_{2(k-b)-1})
+r^*(I_d,J_{2b+1}\perp 0_{2(k-b)-1})\right)\\
&\mathfrak quad=
2N({\mathfrak P})^{(k-b)(2k-2b-1)+2cb-b^2+c-2(k-b)}\\
&\mathfrak qquad\cdot
\boldsymbol \mu\boldsymbol \delta(c-1,2k-b-1)(N({\mathfrak P})^c-\varepsilon^c)(N({\mathfrak P})^{2(k-b)}-1).
\end{align*}
So using that $\boldsymbol \mu\boldsymbol \delta(t,s+s')=\boldsymbol \mu\boldsymbol \delta(t,s)\boldsymbol \mu\delta(t-s,s'),$ we have
\begin{align*}
\sum_{\cls Y\in\mathbb F^{\ell,\ell}_{\sym}}\frac{r^*(I_{2c},Y)}{o(Y)}\,h_Y
&=
\sum_{b=0}^k(-1)^b\frac{N({\mathfrak P})^{2b(b+c-2k)}\boldsymbol \mu\boldsymbol \delta(c,2k-b)\boldsymbol \mu\boldsymbol \delta(k,b)} {\boldsymbol \mu\boldsymbol \delta(b,b)\boldsymbol \mu\boldsymbol \delta(k-b,k-b)\boldsymbol \mu\boldsymbol \delta(k,b)}\\
&=
\boldsymbol \mathbf gamma(k,c)\cdot S(k,c)
\end{align*}
where
$$S(k,c)=\sum_{b=0}^k(-1)^b N({\mathfrak P})^{2b(b+c-2k)}
\boldsymbol \mu\boldsymbol \delta(c-k,k-b)\boldsymbol \mathbf gamma(k,b).$$
Since $\boldsymbol \mathbf gamma(k,b)=N({\mathfrak P})^{2b}\boldsymbol \mathbf gamma(k-1,b)+\boldsymbol \mathbf gamma(k-1,b-1),$
we find that
\begin{align*}
S(k,c)
&=
-S(k-1,c-1)=(-1)^k S(0,c-k)=(-1)^k,
\end{align*}
proving one case of (a).
We follow this same line of argument when replacing $I_{2c}$ by $J_{2c}$, and when replacing $2c$ by $2c+1$.
(b) Suppose that $\ell=2k+1$.
Using the definition of $h_Y$, we have
\begin{align*}
\sum_{\cls Y\in\mathbb F^{\ell,\ell}_{\sym}}
\frac{r^*(I_d,Y)}{o(Y)}\, h_Y
&=\sum_{b=0}^k
\left(r^*(I_d,I_{2b+1}\perp0_{2(k-b)}) - r^*(I_d,J_{2b+1}\perp0_{2(k-b)})\right)\\
&\mathfrak quad\cdot
\frac{h_{I_{2b+1}\perp0_{2(k-b)}}}{o(I_{2b+1}\perp0_{2(k-b)})}.
\end{align*}
When $d$ is even, $r^*(I_d,I_{2b+1}\perp 0_{2(k-b)})
=r^*(I_d,J_{2b+1}\perp0_{2(k-b)}),$
so when $d$ is even the above sum on $\cls Y$ is 0. So suppose that $d=2c+1$.
Then with $S(k,c)$ as in case (a), we have
\begin{align*}
\sum_{\cls Y\in\mathbb F^{\ell,\ell}_{\sym}}
\frac{r^*(I_d,Y)}{o(Y)}\, h_Y
&=\varepsilon^c N({\mathfrak P})^{c-2k} \boldsymbol \mathbf gamma(c,k)
S(c,k)\\
&=(-1)^k\varepsilon^c N({\mathfrak P})^{c-2k} \boldsymbol \mathbf gamma(c,k).
\end{align*}
To evaluate the sum on $\cls Y$ when $I_d$ is replaced by $J_d$, we first note that
for any $s\mathbf ge0$, when $d$ is even we have
$r^*(J_d,I_{2b+1}\perp 0_{s})=r^*(J_d,J_{2b+1}\perp0_{s})$, and
when $d$ is odd we have
$r^*(J_d,I_{2b+1}\perp 0_{s})=r^*(I_d,J_{2b+1}\perp0_{s})$ .
So mimicking our above analysis, we find that when $d$ is even, the sum on $\cls Y =0$, and when $d$ is odd with $d=2c+1$,
we have
$$\sum_{\cls Y\in\mathbb F^{\ell,\ell}_{\sym}}
\frac{r^*(J_d,Y)}{o(Y)}\,h_Y
= -\sum_{\cls Y\in\mathbb F^{\ell,\ell}_{\sym}}
\frac{r^*(I_d,Y)}{o(Y)}\,h_Y.$$
\end{proof}
\end{document} | math |
/********
* This file is part of Ext.NET.
*
* Ext.NET is free software: you can redistribute it and/or modify
* it under the terms of the GNU AFFERO GENERAL PUBLIC LICENSE as
* published by the Free Software Foundation, either version 3 of the
* License, or (at your option) any later version.
*
* Ext.NET is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU AFFERO GENERAL PUBLIC LICENSE for more details.
*
* You should have received a copy of the GNU AFFERO GENERAL PUBLIC LICENSE
* along with Ext.NET. If not, see <http://www.gnu.org/licenses/>.
*
*
* @version : 1.2.0 - Ext.NET Pro License
* @author : Ext.NET, Inc. http://www.ext.net/
* @date : 2011-09-12
* @copyright : Copyright (c) 2006-2011, Ext.NET, Inc. (http://www.ext.net/). All rights reserved.
* @license : GNU AFFERO GENERAL PUBLIC LICENSE (AGPL) 3.0.
* See license.txt and http://www.ext.net/license/.
* See AGPL License at http://www.gnu.org/licenses/agpl-3.0.txt
********/
using System.ComponentModel;
using System.Web.UI;
namespace Ext.Net
{
/// <summary>
///
/// </summary>
[Description("")]
public partial class DesktopListeners : ComponentListeners
{
private ComponentListener shortcutClick;
/// <summary>
///
/// </summary>
[ListenerArgument(0, "id")]
[ListenerArgument(1, "e")]
[TypeConverter(typeof(ExpandableObjectConverter))]
[ConfigOption("shortcutclick", typeof(ListenerJsonConverter))]
[PersistenceMode(PersistenceMode.InnerProperty)]
[NotifyParentProperty(true)]
[Description("")]
public virtual ComponentListener ShortcutClick
{
get
{
if (this.shortcutClick == null)
{
this.shortcutClick = new ComponentListener();
}
return this.shortcutClick;
}
}
private ComponentListener ready;
/// <summary>
///
/// </summary>
[ListenerArgument(0, "el")]
[TypeConverter(typeof(ExpandableObjectConverter))]
[ConfigOption("ready", typeof(ListenerJsonConverter))]
[PersistenceMode(PersistenceMode.InnerProperty)]
[NotifyParentProperty(true)]
[Description("")]
public virtual ComponentListener Ready
{
get
{
if (this.ready == null)
{
this.ready = new ComponentListener();
}
return this.ready;
}
}
private ComponentListener beforeUnload;
/// <summary>
///
/// </summary>
[ListenerArgument(0, "el")]
[TypeConverter(typeof(ExpandableObjectConverter))]
[ConfigOption("beforeunload", typeof(ListenerJsonConverter))]
[PersistenceMode(PersistenceMode.InnerProperty)]
[NotifyParentProperty(true)]
[Description("")]
public virtual ComponentListener BeforeUnload
{
get
{
if (this.beforeUnload == null)
{
this.beforeUnload = new ComponentListener();
}
return this.beforeUnload;
}
}
}
} | code |
सिंघाना: घर वाले सो रहे थे, मकान का जंगला उखाड़ कर घुसे चोर, जेवरात और नकदी ले गए - नीमकथाना न्यूज
होम / नीमकथाना न्यूज / सिंघाना: घर वाले सो रहे थे, मकान का जंगला उखाड़ कर घुसे चोर, जेवरात और नकदी ले गए
सिंघाना: घर वाले सो रहे थे, मकान का जंगला उखाड़ कर घुसे चोर, जेवरात और नकदी ले गए
मोई भारू गांव में वारदात |रात ११ बजे तक जाग रहे थे घरवाले, उसके बाद हुई चोरी
सिंघाना- मोई भारू गांव में शुक्रवार देर रात एक मकान का जंगला उखाड़ कर चोर घर में घुसे और बक्से का ताला तोड़ कर ४५ हजार रुपए की नकदी समेत जेवरात चुरा ले गए। थानाधिकारी सतपाल यादव ने बताया कि चोरी की वारदात वीर सिंह जाट के घर हुई । शुक्रवार रात को वीर सिंह व उसके पिता बस्तीराम बैठक व कमरे में सो रहे थे।
वीर सिंह साइड के कैमरे में तथा उसका बुजुर्ग पिता बस्तीराम बैठक में सो रहे थे। चोरों ने शुक्रवार देर रात को कमरे के बाहर का जंगला उखाड़ कर मकान में घुस कर बक्से का ताला तोड़ दिया। चोरों ने उसमें रखे नकदी व जेवरात पर हाथ साफ कर दिया। सिंघाना थानाधिकारी सतपाल यादव पुलिस जाब्ते मौके पर पहुंचे। बुहाना डीएसपी रामप्रकाश मीणा ने भी घटना स्थल का जायजा लिया।
झुंझुनूं से एफएसएल की टीम को बुलाया गया। टीम ने घटना स्थल से पद मार्क उठाए। वीरसिंह ने बताया कि शुक्रवार रात वह और उसका पिता ११ बजे रात तक जाग रहे थे। उनके सोने के बाद देर रात यह वारदात हुई। चोर एक बक्से में रखे जरूरी कागजात, बही, बक्से में रखे ४५ हजार रुपए नकद व जेवरात ले गए। वीर सिंह की रिपोर्ट पर मामला दर्ज कर जांच शुरू कर दी।
सिंघाना: घर वाले सो रहे थे, मकान का जंगला उखाड़ कर घुसे चोर, जेवरात और नकदी ले गए रेवीव्द बाय एडमिन ऑन मई २७, २०१८ रेटिंग: ५ | hindi |
आरएसएमएसएसबी भर्ती २०१६ राजस्थान ४०२ कनिष्ठ अनुदेशक सरकारी नौकरी - सरकारी नौकरी इन हिंदी (२०१७)
बाय एडमिन | सिप्तंबर २२, २०१६ ० कमेंट आरएसएमएसएसबी भर्ती आरएसएमएसएसबी भर्ती २०१६: राजस्थान अधीनस्थ एवं मंत्रालयिक सेवा चयन बोर्ड, जयपुर द्वारा आरएसएमएसएसबी रिक्रूटमेंट नोटिफिकेशन जारी किया गया है | जिसके अंतर्गत आवेदकों को यह सूचित किया जाता है संस्थान द्वारा कनिष्ठ अनुदेशक के 4०2 पदों पर रिक्तियों कि उद्घोषणा कि गयी है | जो उम्मीदवार सरकारी नौकरी में अपना करियर बनाने कि इच्छा रखते है वे आरएसएमएसएसबी भर्ती के अंतर्गत आवेदन कर सकते है | आरएसएमएसएसबी रिक्रूटमेंट के लिए आवेदन करने वाले अभ्यर्थियों को अपना आवेदन २२ अक्टूबर २०१६ से पहले सुनिश्चित करना होगा |
आरएसएमएसएसबी भर्ती २०१६ कि पूरी जानकारी आप इसी पेज से प्राप्त कर सकते है जो सरकारिनौकरीन्हिंदी.कॉम द्वारा तैयार किया गया है |
राजस्थान अधीनस्थ एवं मंत्रालयिक सेवा चयन बोर्ड, जयपुर
आरएसएमएसएसबी वैकेंसी के लिए आवेदन करने वाले अभ्यर्थी के पास निम्नलिखित योग्यताएं होनी चाहिए
आवेदक का १२वीं कक्षा उत्तीर्ण होना अति आवश्यक है |
कंप्यूटर से जुडी पूरी जानकारी होनी चाहिए |
अभ्यर्थी के पास सम्बंधित व्यवसाय में राष्ट्रीय व्यवसाय प्रमाण पत्र अथवा राष्ट्रीय शिक्षुता प्रमाण पत्र हो साथ ही सम्बंधित क्षेत्र में निर्धारित वर्षों का अनुभव होना अनिवार्य है |
आवेदक, ०१ जनवरी 2०१7 को २० वर्ष कि आयु प्राप्त कर चुका हो तथा ३७ वर्ष का ना हुआ हो |
कनिष्ठ अनुदेशक के विभिन्न उक्त पदों का वेतनमान (९३०० ३४८००) ग्रेड पे रु ३६००/- रु देना निर्धारित किया है |
राजस्थान अधीनस्थ एवं मंत्रालयिक सेवा चयन बोर्ड द्वारा निम्नलिखित परीक्षा शुल्क तय किया गया है
सामान्य वर्ग व अन्य पिछड़ा वर्ग / विशेष पिछड़ा वर्ग ६५०/- रु
राजस्थान के नॉन क्रीमलेयर श्रेणी के पिछड़ा वर्ग / विशेष पिछड़ा वर्ग के आवेदक हेतु ४५०/- रु
राजस्थान के अनु० जाति / अनु० जनजाति के आवेदक हेतु ३५०/- रु
योग्य उम्मीदवार का चयन संस्थान के नियमानुसार किया जायेगा | लिखित परीक्षा / साक्षात्कार का आयोजन किया जा सकता है |
आरएसएमएसएसबी भर्ती २०१६ के लिए आवेदन प्रक्रिया:
सबसे पहले आप संस्थान कि वेबसाइट पर जायें |
राजस्थान अधीनस्थ एवं मंत्रालयिक सेवा चयन बोर्ड कि वेबसाइट है र्सम्सब.राजस्थान.गोव.इन
अब आपके सामने आरएसएमएसएसबी भर्ती अधिसूचना खुल जाएगी इसमें दी जानकारी का अध्ययन ध्यानपूर्वक करिए |
यदि अब आप आवेदन करने के इच्छुक है तो आप ऑनलाइन आवेदन पत्र भरिये तथा भरने के बाद उसकी पूरी जाँच भली भांति कीजिये |
अंत में आप सुब्मित बटन पर क्लिक करके अपना आवेदन कर सकते है |
आवेदन पत्र सबमिट करने के बाद आप भविष्य में प्रमाण हेतु उसकी प्रति अवश्य निकल लें |
आवेदन पत्र व आवेदन शुल्क भेजने कि अंतिम तिथि : २२ अक्टूबर २०१६
आरएसएमएसएसबी भर्ती २०१६ से सम्बंधित अन्य जानकारी आप अधिकारिक विज्ञप्ति पढ़ कर भी ज्ञात कर सकते है |
आरएसएमएसएसबी भर्ती २०१६ अधिकारिक विज्ञप्ति यहाँ क्लिक करें
टीएचडीसी इंडिया लिमिटेड भर्ती २०१६ थ्डक अधिसूचना मैनेजर रिक्त पद
राष्ट्रीय कोशिका विज्ञान केन्द्र भर्ती २०१६ नीअस साइंटिस्ट रिक्त पद | hindi |
इप्ल २०१९: ८वीं बार फाइनल में पहुंची चेन्नई सुपर किंग्स - अप्न लाइव हिंदी
होम खेल इप्ल २०१९: ८वीं बार फाइनल में पहुंची चेन्नई सुपर किंग्स
इप्ल २०१९: ८वीं बार फाइनल में पहुंची चेन्नई सुपर किंग्स
दिल्ली कैपिटल्स को छः विकेट से मात देकर चेन्नई सुपर किंग्स ने आठवीं बार आईपीएल के फाइनल में जगह बनाई है। चेन्नई आईपीएल के इतिहास में सबसे ज्यादा फाइनल खेलने वाली टीम है। अब इसका फाइनल मुकाबला १२ मई को मुंबई इंडियंस से होगा। दिल्ली कैपिटल्स पहले बल्लेबाजी करने उतरी और वह नौ विकेट पर १४७ रन ही बना पायी। चेन्नई सुपर किंग्स ने १९ ओवर में चार विकेट पर १५१ रन बनाकर आसानी से लक्ष्य हासिल कर दिय। उसकी तरफ से फाफ डुप्लेसिस (३९ गेंदों पर ५०) और शेन वाटसन (३२ गेंदों पर ५०) ने अर्धशतक जमाये तथा पहले विकेट के लिये ८१ रन जोड़कर टीम को अच्छी शुरुआत दिलायी।
महेंद्र सिंह धोनी ने मैच के बाद कहा, खिलाड़ियों ने आज जिस तरह का खेल दिखाया वह बेहतरीन था। स्पिनरों को कुछ टर्न मिल रहा था और हमने सही समय पर विकेट निकाले। उनके पास बायें हाथ के बल्लेबाज थे और हमारे बायें हाथ के स्पिनरों ने उनके सामने अच्छा प्रदर्शन किया। लगातार विकेट हासिल करना महत्वपूर्ण रहा। उन्होंने कहा, गेंदबाजों को श्रेय जाता है। कप्तान उन्हें यही कह सकता मैं यह चाहता हूं। इसके बाद उस हिसाब से गेंदबाजी करना उनका काम है। इस सत्र में हम अभी जहां पर हैं उसके लिए गेंदबाजी विभाग का आभार।
बता दें कि सुपर किंग्स बड़े मैचों में खेलने की आदी है, वह ३ बार आईपीएल का खिताब जीत चुकी है। जबकि ४ बार धोनी की सेना उप-विजेता रह चुकी है। चेन्नई सुपर किंग्स ने २००८, २०१०, २०११, २०१२, 201३, २०१५, २०१८ और २०१९ में फाइनल तक सफर तय किया है। जिसमें से २०१०, २०११ और २०१८ में उसे खिताबी जीत हासिल हुई है।
चेन्नई सुपर किंग्स के बाद मुंबई इंडियंस का नंबर आता है जिसने पांच बार आईपीएल के फाइनल में जगह बनाई है। मुंबई इंडियंस २०१०, २०१३, २०१५ और २०१७ में फाइनल तक सफर तय कर चुकी है। जिसमें से २०१० के आईपीएल फाइनल को छोड़ दें तो बाकी तीन बार उसे खिताबी जीत हासिल हुई है।
महेंद्र सिंह धोनी सबसे ज्यादा बार आईपीएल फाइनल मैच खेलने वाले खिलाड़ी है। महेंद्र सिंह धोनी रिकॉर्ड ९वीं बार आईपीएल फाइनल खेलेंगे। धोनी राइजिंग पुणे सुपरजायंट्स के खिलाड़ी के तौर पर एक बार फाइनल खेल चुके हैं जबकि वे ८वीं बार चेन्नई सुपर किंग्स के लिए आईपीएल फाइनल मैच खेलेंगे। धोनी के बाद सुरेश रैना आते हैं जिन्हें ७वीं बार आईपीएल फाइनल खेलने का मौका मिला है।
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\begin{document}
\title
{Induced subgraphs of product graphs and a generalization of Huang's theorem}
\author
{Zhen-Mu Hong\\
{\footnotesize School of Finance, Anhui University of Finance \& Economics, Bengbu, Anhui 233030, China} \\
\\
Hong-Jian Lai\thanks{Corresponding author. E-mail addresses: zmhong@mail.ustc.edu.cn
(Z.-M. Hong), hjlai@math.wvu.edu (H.-J. Lai), jl0068@mix.wvu.edu (J.-B. Liu).}, Jian-Bing Liu\\
{\footnotesize Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA}}
\date{}
\maketitle
\begin{center} {\bf Abstract} \end{center}
Recently, Huang showed that every $(2^{n-1}+1)$-vertex induced subgraph of the $n$-dimensional hypercube
has maximum degree at least $\sqrt{n}$ in [Annals of Mathematics, 190 (2019), 949--955].
In this paper, we discuss the induced subgraphs of Cartesian product graphs and semi-strong product graphs to generalize Huang's result.
Let $\Gamma_1$ be a connected signed bipartite graph of order $n$ and $\Gamma_2$
be a connected signed graph of order $m$. By defining two kinds of signed product
of $\Gamma_1$ and $\Gamma_2$, denoted by $\Gamma_1\widetilde{\Box}\Gamma_2$ and
$\Gamma_1\widetilde{\bowtie} \Gamma_2$, we show that if $\Gamma_1$ and $\Gamma_2$
have exactly two distinct adjacency eigenvalues $\pm\theta_1$ and $\pm\theta_2$ respectively,
then every $(\frac{1}{2}mn+1)$-vertex induced subgraph of
$\Gamma_1\widetilde{\Box}\Gamma_2$ (resp. $\Gamma_1\widetilde{\bowtie} \Gamma_2$)
has maximum degree at least $\sqrt{\theta_1^2+\theta_2^2}$ (resp. $\sqrt{(\theta_1^2+1)\theta_2^2}$).
Moreover, we discuss the eigenvalues of $\Gamma_1\widetilde{\Box} \Gamma_2$ and $\Gamma_1\widetilde{\bowtie} \Gamma_2$
and obtain a sufficient and necessary condition such that the spectrum of
$\Gamma_1\widetilde{\Box}\Gamma_2$ and $\Gamma_1\widetilde{\bowtie}\Gamma_2$
are symmetric, from which we obtain more general results on maximum degree of the induced subgraphs.
\vskip10pt
\indent{\bf Keywords:} Induced subgraph; Cartesian product; Semi-strong product; Signed graph; Eigenvalue
\vskip0.4cm \noindent {\bf AMS Subject Classification: }\ 05C22, 05C50, 05C76
\section{Introduction}
Let $Q_n$ be the $n$-dimensional hypercube, whose vertex set consists of vectors
in $\{0, 1\}^n$, and two vectors are adjacent if they differ in exactly one coordinate.
For a simple and undirected graph $G=(V,E)$, we use $\Delta(G)$ to denote the maximum degree of $G$.
The adjacency matrix of $G$ is defined to be a $(0,1)$-matrix
$A(G)=(a_{ij})$, where $a_{ij}=1$ if $v_i$ and $v_j$ are adjacent, and
$a_{ij}=0$ otherwise.
Recently, Huang \cite{Huan19} constructed a signed adjacency matrix of $Q_n$
with exactly two distinct eigenvalues $\pm \sqrt{n}$. Using eigenvalue interlacing, Huang
proceeded to prove that the spectral radius (and so, the maximum degree) of any $(2^{n-1} + 1)$-vertex induced
subgraph of $Q_n$, is at least $\sqrt{n}$. Combing this with the combinatorial equivalent
formulation discovered by Gotsman and Linial \cite{GoLi92},
Huang confirmed the Sensitivity Conjecture \cite{NiSz92} from theoretical computer science.
The main contribution of Huang is the following theorem.
\begin{thm}{(Huang \cite{Huan19})}\label{thm-Huang}
For every integer $n\geq 1$, let $H$ be an arbitrary $(2^{n-1}+1)$-vertex induced
subgraph of $Q_n$, then $\Delta(H)\geq \sqrt{n}$.
\end{thm}
The bound $\sqrt{n}$ (or more precisely, $\lceil\sqrt{n}\rceil$)
is sharp, as shown by Chung, Furedi, Graham, and Seymour \cite{CFGS88} in 1988.
Tao \cite{Tao19} also gave a great expository of Huang's work on his blog after Huang
announced the proof of the Sensitivity Conjecture.
Denote the Cartesian product of two graphs $G$ and $H$ by $G \Box H$.
It is known that the hypercube $Q_n$ can be constructed iteratively by Cartesian product, that is,
$Q_1=K_2$ and for $n\geq 2$, $Q_n=Q_1\Box Q_{n-1}$. Motivated by this fact,
in this paper, we generalize Huang's theorem to
Cartesian product graphs and semi-strong product graphs.
We introduce some necessary notations in the following.
A {\it signed graph} $\Gamma = (G, \sigma)$ is a graph $G = (V, E)$,
together with a sign function $\sigma : E \rightarrow \{+1,-1\}$ assigning
a positive or negative sign to each edge. An edge $e$ is positive if $\sigma(e) = 1$
and negative if $\sigma(e) = -1$. The unsigned graph $G$ is said to
be the {\em underlying graph} of $\Gamma$, while $\sigma$ is called the
{\em signature} of $G$. If each edge of $\Gamma$ is positive (resp. negative),
then $\Gamma$ is denoted by $\Gamma= (G,+)$ (resp. $\Gamma= (G,-)$).
A signed graph is connected if its underlying graph is connected.
The adjacency matrix of $\Gamma=(G,\sigma)$ is denoted by $A(\Gamma)=(a^{\sigma}_{ij})$, where
$a^{\sigma}_{ij} = \sigma(v_iv_j)$, if $v_i$ and $v_j$ are adjacent, and $a^{\sigma}_{ij}=0$ otherwise.
As $G$ is simple and undirected, the adjacency matrix $A(\Gamma)$ is a symmetric $(-1, 0, +1)$-matrix,
and $A(\Gamma)=A(G)$ if $\Gamma= (G,+)$, $A(\Gamma)=-A(G)$ if $\Gamma= (G,-)$.
Let $\lambda_1(\Gamma)\geq \lambda_2(\Gamma)\geq\cdots\geq\lambda_n(\Gamma)$ denote the eigenvalues of
$A(\Gamma)$, which are all real since $A(\Gamma)$ is real and symmetric.
If $\Gamma$ contains at least one edge, then $\lambda_1(\Gamma) > 0 > \lambda_n(\Gamma)$
since the trace of $A(\Gamma)$ is $0$. In general, the largest eigenvalue $\lambda_1(\Gamma)$ may not be equal to the spectral radius
$\rho(\Gamma) = \max\{|\lambda_i(\Gamma)| : 1 \leq i \leq n\} = \max\{\lambda_1(\Gamma),-\lambda_n(\Gamma)\}$ because
the Perron-Frobenius Theorem is valid only for nonnegative matrices.
The eigenvalues of the adjacency matrix of signed graph $\Gamma$ are called adjacency eigenvalues of $\Gamma$.
The spectrum of $A(\Gamma)$ is called the (adjacency) spectrum of $\Gamma$
and $A(\Gamma)$ is also called a signed adjacency matrix of $G$.
The spectrum of $\Gamma$ is {\it symmetric} if its adjacency eigenvalues
are symmetric with respect to the origin.
In this paper, all eigenvalues considered are adjacency eigenvalues.
For basic results in the theory of signed graphs, the reader is referred to Zaslavsky \cite{Zasl82}.
Recently, the spectra of signed graphs have attracted much attention,
as found in \cite{ABDN18,BCKW19,BiLi06,Gall16,GhFa17,GeHZ11,HoTW19,Rame15,Rame19,Stan20,Zasl10}, among others.
In \cite{BCKW19}, the authors surveyed some general results on the adjacency
spectra of signed graphs and proposed some spectral problems which are inspired by the
spectral theory of unsigned graphs. In particular, the signed graphs with exactly
two distinct eigenvalues have been greatly investigated in recent years,
see \cite{GhFa17,HoTW19,McSm07,Rame15,Rame19,Stan20}.
In \cite{HoTW19}, Hou et al. characterized all simple connected
signed graphs with maximum degree at most 4 and with just two distinct adjacency eigenvalues.
In this paper, we construct signed graphs with exactly two distinct eigenvalues
by two kinds of graph products, which generalizes Huang's result on the induced subgraph of the hypercube.
The {\it Kronecker product} $A\otimes B$ of matrices $A = (a_{ij})_{m\times n}$
and $B = (b_{ij})_{p\times q}$ is the $mp \times nq$ matrix obtained from $A$ by replacing each
element $a_{ij}$ with the block $a_{ij}B$.
Therefore the entries of $A\otimes B$ consist of all the $mnpq$ possible products of an entry
of $A$ with an entry of $B$. For matrices $A, B, C$ and $D$, we have
$(A \otimes B) \cdot (C \otimes D) = AC \otimes BD$ whenever the products $AC$ and $BD$ exist.
Note that, $(A \otimes B)^T=A^T\otimes B^T$.
The {\it Cartesian product} of two graphs $G_1$ and $G_2$ is a graph, denoted by $G_1 \Box G_2$, whose vertex set is
$V(G_1)\times V(G_2)$ and two vertices $(u_1, u_2)$ and $(v_1, v_2)$ being adjacent in $G_1 \Box G_2$
if and only if either $u_1=v_1$ and $u_2v_2\in E(G_2)$, or $u_1v_1 \in E(G_1)$ and $u_2=v_2$.
The {\it direct product} (or {\it Kronecker product}) of two graphs $G_1$ and $G_2$ is a graph, denoted by $G_1 \times G_2$,
whose vertex set is $V(G_1)\times V(G_2)$, and two vertices
$(u_1, u_2)$ and $(v_1, v_2)$ being adjacent to each other in $G_1 \times G_2$
if and only if both $u_1v_1 \in E(G_1)$ and $u_2v_2\in E(G_2)$.
The {\it semi-strong product} (or {\it strong tensor product} \cite{GaRW76})
of two graphs $G_1$ and $G_2$ is a graph, denoted by $G_1 \bowtie G_2$,
whose vertex set is $V(G_1)\times V(G_2)$, and two vertices
$(u_1, u_2)$ and $(v_1, v_2)$ being adjacent to each other in $G_1 \bowtie G_2$
if and only if either $u_1v_1 \in E(G_1)$ and $u_2v_2\in E(G_2)$, or $u_1=v_1$ and $u_2v_2\in E(G_2)$.
Then, by the definitions, the adjacency matrices of
$G_1\Box G_2$, $G_1\times G_2$ and $G_1\bowtie G_2$ are
$A(G_1\Box G_2) =A(G_1)\otimes I_m+I_n\otimes A(G_2)$,
$A(G_1\times G_2) =A(G_1)\otimes A(G_2)$ and $A(G_1\bowtie G_2) =A(G_1)\otimes A(G_2)+I_n\otimes A(G_2)$, respectively,
where $n=|V(G_1)|$, $m=|V(G_2)|$ and $I_n$ is the identity matrix of order $n$.
Let $\Gamma_1=(G_1,\sigma_1)$ be a connected signed bipartite graph of order $n$ with bipartition $(V_1,V_2)$,
where $|V_1|=s$ and $|V_2|=n-s$, and $\Gamma_2=(G_2,\sigma_2)$ be a connected signed graph of order $m$.
With suitable labeling of vertices, the adjacency matrix of $\Gamma_1$ can be represented as
$$
A(\Gamma_1)=\left[
\begin{array}{cc}
O_{s} & P \\
P^T & O_{n-s}
\end{array}
\right].
$$
The {\it signed Cartesian product} of signed bipartite graph $\Gamma_1$ and signed graph $\Gamma_2$,
denoted by $\Gamma_1\widetilde{\Box} \Gamma_2$, is the signed graph with adjacency matrix
\begin{equation}\label{e1.1}
A(\Gamma_1\widetilde{\Box} \Gamma_2) = A(\Gamma_1) \otimes I_m +
\left[
\begin{array}{cc}
I_s & O \\
O & -I_{n-s}
\end{array}
\right] \otimes A(\Gamma_2)=\left[
\begin{matrix}
I_s\otimes A(\Gamma_2) & P\otimes I_m \\
P^T\otimes I_m & -I_{n-s}\otimes A(\Gamma_2)
\end{matrix}
\right].
\end{equation}
The {\it signed semi-strong product} of signed bipartite graph $\Gamma_1$ and signed graph $\Gamma_2$,
denoted by $\Gamma_1 \widetilde{\bowtie} \Gamma_2$, is the signed graph with adjacency matrix
\begin{equation}\label{e1.2}
A(\Gamma_1\widetilde{\bowtie} \Gamma_2) = A(\Gamma_1)\otimes A(\Gamma_2) +
\left[
\begin{array}{cc}
I_s & O \\
O & -I_{n-s}
\end{array}
\right] \otimes A(\Gamma_2)=\left[
\begin{array}{cc}
I_s & P \\
P^T & -I_{n-s}
\end{array}
\right] \otimes A(\Gamma_2).
\end{equation}
As a generalization of Theorem \ref{thm-Huang}, we have the following theorem.
\begin{thm}\label{thm-DeltaH-2}
Let $\Gamma_1=(G_1,\sigma_1)$ be a signed bipartite graph of order $n$
with exactly two distinct eigenvalues $\pm \theta_1$ and
$\Gamma_2=(G_2,\sigma_2)$ be a signed graph of order $m$
with exactly two distinct eigenvalues $\pm \theta_2$.
If $H$ and $H'$ are arbitrary $(\frac{mn}{2}+1)$-vertex induced
subgraphs of $\Gamma_1\widetilde{\Box} \Gamma_2$ and
$\Gamma_1\widetilde{\bowtie} \Gamma_2$ respectively, then
$$
\Delta(H)\geq \sqrt{\theta_1^2+\theta_2^2}, \ \
\Delta(H')\geq \sqrt{(\theta_1^2+1)\theta_2^2}.
$$
\end{thm}
A direct proof of Theorem \ref{thm-DeltaH-2} is presented in Section 2.
In fact, from the proof we see that $\Gamma_1$ and $\Gamma_2$ in Theorem \ref{thm-DeltaH-2} are regular.
For more general graphs, we can obtain the following theorem.
A (signed) bipartite graph with bipartition $(V_1,V_2)$ is called {\it balanced}
if $|V_1|=|V_2|$.
\begin{thm}\label{thm-Delta-2-min}
Let $\Gamma_1=(G_1,\sigma_1)$ be a signed bipartite graph of order $n$
and $\Gamma_2=(G_2,\sigma_2)$ be a signed graph of order $m$,
and let $\lambda^2$ and $\mu^2$ be the minimum eigenvalues of $A(\Gamma_1)^2$ and $A(\Gamma_2)^2$, respectively.
Let $H$ and $H'$ be any $(\lfloor\frac{mn}{2}\rfloor+1)$-vertex induced
subgraph of $\Gamma_1\widetilde{\Box} \Gamma_2$ and
$\Gamma_1\widetilde{\bowtie} \Gamma_2$, respectively.
If $\Gamma_1$ is a balanced bipartite graph or the spectrum of $\Gamma_2$ is symmetric, then
$$
\Delta(H)\geq \sqrt{\lambda^2+\mu^2}, \ \ \Delta(H')\geq \sqrt{(\lambda^2+1)\mu^2}.
$$
\end{thm}
In Section 3, we display some preliminaries and examples.
In Section 4, we give a characterization of the eigenvalues of
$\Gamma_1\widetilde{\Box} \Gamma_2$ and $\Gamma_1\widetilde{\bowtie} \Gamma_2$
and obtain a sufficient and necessary condition such that the spectrum of
$\Gamma_1\widetilde{\Box}\Gamma_2$ and $\Gamma_1\widetilde{\bowtie}\Gamma_2$
are symmetric. In Section 5, we present the proof of Theorem \ref{thm-Delta-2-min}
and generalize the signed Cartesian product and signed semi-strong product of
two signed graphs to the products of $n$ signed graphs.
In the last section, we give some concluding remarks.
\section{A direct proof of Theorem \ref{thm-DeltaH-2}}
Using the idea that Shalev Ben-David contributed on July 3, 2019 to Scott Aaronson's blog,
Knuth \cite{knot19} gave a direct and nice proof of Huang's theorem in one page. Here,
arising from their ideas, we give a direct proof of Theorem \ref{thm-DeltaH-2}.
\noindent{\bf Proof of Theorem \ref{thm-DeltaH-2}.}
For simplicity, let $A_1:=A(\Gamma_1)$ and $A_2:=A(\Gamma_2)$.
Since $\Gamma_i$ has exactly two distinct eigenvalues $\pm\theta_i~(\neq 0)$ for $i=1,2$, we have
each eigenvalue of $A_i^2$ equals to $\theta_i^2$ and so there exist
orthogonal matrices $Q_1$ and $Q_2$ such that $A_1^2=Q_1(\theta_1^2I_n)Q_1^T=\theta_1^2 I_n$
and $A_2^2=Q_2(\theta_2^2I_m)Q_2^T=\theta_2^2I_m$.
The diagonal entries of $A_i^2$ are the degrees of vertices in $\Gamma_i$, so
$\Gamma_i$ is a $\theta_i^2$-regular graph for $i=1,2$.
Moreover, $|V_1|=s=\frac{n}{2}$ and $PP^T=P^TP=\theta_1^2I_{n/2}$.
(a) Let $\mathcal {A}:=A(\Gamma_1\widetilde{\Box} \Gamma_2)$ and define
$$
\mathcal{B}=\left[
\begin{matrix}
P\otimes (A_2+\sqrt{\theta_1^2+\theta_2^2}I_m) \\
\theta_1^2I_{n/2}\otimes I_m
\end{matrix}
\right]
$$
to be an $mn\times \frac{mn}{2}$ matrix.
Since $\theta_1\neq 0$, the rank of $\mathcal{B}$ is $\frac{mn}{2}$, and we have
\begin{align}
\mathcal{A}\cdot \mathcal{B}
= &\left[
\begin{matrix}
I_{n/2}\otimes A_2 & P\otimes I_m \\
P^T\otimes I_m & -I_{n/2}\otimes A_2
\end{matrix}
\right] \cdot \left[
\begin{matrix}
P\otimes (A_2+\sqrt{\theta_1^2+\theta_2^2}I_m) \\
\theta_1^2I_{n/2}\otimes I_m
\end{matrix}
\right] \nonumber\\
= & \left[
\begin{matrix}
P\otimes (A_2^2+\sqrt{\theta_1^2+\theta_2^2}A_2+\theta_1^2I_m) \\
P^TP\otimes (A_2+\sqrt{\theta_1^2+\theta_2^2}I_m)-\theta_1^2I_{n/2}\otimes A_2
\end{matrix}
\right] \nonumber\\
= & \left[
\begin{matrix}
P\otimes (\theta_2^2I_m+\sqrt{\theta_1^2+\theta_2^2}A_2+\theta_1^2I_m) \\
\theta_1^2I_{n/2}\otimes (A_2+\sqrt{\theta_1^2+\theta_2^2}I_m)-\theta_1^2I_{n/2}\otimes A_2
\end{matrix}
\right] \nonumber\\
= & \sqrt{\theta_1^2+\theta_2^2} \left[
\begin{matrix}
P\otimes (A_2+\sqrt{\theta_1^2+\theta_2^2}I_m) \\
\theta_1^2I_{n/2}\otimes I_m
\end{matrix}
\right]
= \sqrt{\theta_1^2+\theta_2^2} \mathcal{B}. \nonumber
\end{align}
Let $H$ be an arbitrary $(\frac{mn}{2}+1)$-vertex induced subgraph of $\Gamma_1\widetilde{\Box} \Gamma_2$.
Suppose $\mathcal{B}^*$ is the $(\frac{mn}{2}-1)\times \frac{mn}{2}$
submatrix of $\mathcal{B}$ whose rows corresponding to vertices not in $H$.
Then there exists a unit $\frac{mn}{2}\times 1$ vectors $x$ such that $\mathcal{B}^*x=0$,
since $\mathcal{B}^*x=0$ is a homogeneous system of $\frac{mn}{2}-1$ linear equations with $\frac{mn}{2}$ variables.
As ${\rm rank}(\mathcal{B})=\frac{mn}{2}$, $y=\mathcal{B}x$ is an $mn\times 1$
nonzero vector such that $y_v=0$ for any vertex $v\not\in H$, and
$\mathcal{A}y =\sqrt{\theta_1^2+\theta_2^2} y$.
Let $u$ be a vertex such that $|y_u|=\max\{|y_1|,\dots,|y_{mn}|\}$. Then $|y_u|>0$, $u\in V(H)$ and
$$
\sqrt{\theta_1^2+\theta_2^2} |y_u|=|(\mathcal{A}y)_{u}|=\left|\sum_{v=1}^{mn}\mathcal{A}_{uv}y_v\right|
=\left|\sum_{v\in H}\mathcal{A}_{uv}y_v\right|\leq \sum_{v\in H}|\mathcal{A}_{uv}||y_u|
\leq \Delta(H)|y_u|.
$$
Therefore, $\Delta(H)\geq \sqrt{\theta_1^2+\theta_2^2}$.
(b) Let $\mathcal{A}:=A(\Gamma_1\widetilde{\bowtie} \Gamma_2)$
and define
$$
\mathcal{B}=\left[
\begin{matrix}
P\otimes(\sqrt{\theta_1^2+1}A_2+\theta_2I_m) \\
\theta_1^2\theta_2I_{n/2}\otimes I_m
\end{matrix}
\right]
$$
to be an $mn\times \frac{mn}{2}$ matrix.
Since $\theta_1\neq 0$ and $\theta_2\neq 0$,
the rank of $\mathcal{B}$ is $\frac{mn}{2}$, and we have
\begin{align}
\mathcal{A}\cdot \mathcal{B}
= & \left[
\begin{matrix}
I_{n/2}\otimes A_2 & P\otimes A_2 \\
P^T\otimes A_2 & -I_{n/2}\otimes A_2
\end{matrix}
\right] \cdot \left[
\begin{matrix}
P\otimes(\sqrt{\theta_1^2+1}A_2+\theta_2I_m) \\
\theta_1^2\theta_2I_{n/2}\otimes I_m
\end{matrix}
\right] \nonumber\\
= & \left[
\begin{matrix}
P\otimes(\sqrt{\theta_1^2+1}A_2^2+\theta_2A_2+\theta_1^2\theta_2A_2) \\
P^TP\otimes (\sqrt{\theta_1^2+1}A_2^2+\theta_2A_2)-\theta_1^2\theta_2I_{n/2}\otimes A_2
\end{matrix}
\right] \nonumber\\
= & \left[
\begin{matrix}
P\otimes(\sqrt{\theta_1^2+1}\theta_2^2I_m+\theta_2(\theta_1^2+1)A_2) \\
\theta_1^2I_{n/2}\otimes (\sqrt{\theta_1^2+1}\theta_2^2I_m+\theta_2A_2)-\theta_1^2\theta_2I_{n/2}\otimes A_2
\end{matrix}
\right] \nonumber\\
= & \sqrt{(\theta_1^2+1)\theta_2^2} \left[
\begin{matrix}
P\otimes(\sqrt{\theta_1^2+1}A_2+\theta_2I_m) \\
\theta_1^2\theta_2I_{n/2}\otimes I_m
\end{matrix}
\right]
= \sqrt{(\theta_1^2+1)\theta_2^2} \mathcal{B}. \nonumber
\end{align}
Let $H'$ be an arbitrary $(\frac{mn}{2}+1)$-vertex induced subgraph of $\Gamma_1\widetilde{\bowtie} \Gamma_2$.
Suppose $\mathcal{B}^*$ is the $(\frac{mn}{2}-1)\times \frac{mn}{2}$ submatrix of $\mathcal{B}$
whose rows corresponding to vertices not in $H'$.
Then there exists a unit $\frac{mn}{2}\times 1$ vector $x$ such that $\mathcal{B}^*x=0$,
since $\mathcal{B}^*x=0$ is a homogeneous system of $\frac{mn}{2}-1$ linear equations with $\frac{mn}{2}$ variables.
As ${\rm rank}(\mathcal{B})=\frac{mn}{2}$, $y=\mathcal{B}x$ is an $mn\times 1$ nonzero vector
such that $y_v=0$ for any vertex $v\not\in H'$, and
$\mathcal{A}y =\sqrt{(\theta_1^2+1)\theta_2^2} y$.
Let $u$ be a vertex such that $|y_u|=\max\{|y_1|,\dots,|y_{mn}|\}$. Then $|y_u|>0$, $u\in V(H')$ and
$$
\sqrt{(\theta_1^2+1)\theta_2^2} |y_u|=|(\mathcal{A}y)_{u}|=\left|\sum_{v=1}^{mn}\mathcal{A}_{uv}y_v\right|
=\left|\sum_{v\in H'}\mathcal{A}_{uv}y_v\right|\leq \sum_{v\in H'}|\mathcal{A}_{uv}||y_u|
\leq \Delta(H')|y_u|.
$$
Therefore, $\Delta(H')\geq \sqrt{(\theta_1^2+1)\theta_2^2}$.
\rule{2mm}{2mm}
\section{Preliminaries}
In this section, we present some useful lemmas and examples.
\begin{lem}{(Hammack et al. \cite{HaIK11})}\label{lem-HaIK11}
Let $G_1$ and $G_2$ be nontrivial graphs. Then
(i) $G_1\Box G_2$ is connected if and only if $G_1$ and $G_2$ are connected,
and $G_1\Box G_2$ is bipartite if and only if $G_1$ and $G_2$ are bipartite;
(ii) $G_1\times G_2$ is connected if and only if $G_1$ and $G_2$ are connected and at most one of them is bipartite,
and $G_1\times G_2$ is bipartite if and only if at least one of $G_1$ and $G_2$ is bipartite.
\end{lem}
\begin{lem}{(Garman et al. \cite{GaRW76})}\label{lem-GaRW76}
Let $G_1$ and $G_2$ be nontrivial graphs. Then
(i) $G_1\bowtie G_2$ is connected if and only if $G_1$ and $G_2$ are connected;
(ii) $G_1\bowtie G_2$ is bipartite if and only if $G_2$ is bipartite;
(iii) The semi-strong product operation is neither associative nor commutative;
(iv) If $G_1$ is bipartite, then $G_1\bowtie K_2\cong G_1\Box K_2$.
\end{lem}
By Lemma \ref{lem-GaRW76} (iv), the following corollary can be obtained easily.
\begin{cor}{(Garman et al. \cite{GaRW76})}\label{cor-GaRW76}
(i) Let $G_1=K_2$, and for $n\geq 2$, $G_n=G_{n-1}\bowtie K_2$, then $G_n\cong Q_n$.
(ii) Let $G'_1=K_2$, and for $n\geq 2$, $G'_n=K_2\bowtie G'_{n-1}$, then $G'_n\cong K_{2^{n-1},2^{n-1}}$.
\end{cor}
\begin{pf}
By Lemma \ref{lem-GaRW76} (iv), $Q_{n-1}\bowtie K_2\cong Q_{n-1}\Box K_2=Q_n$. By induction, $G_n\cong Q_n$.
Let $V(K_2)=\{u,v\}$ and $(V_1,V_2)$ be the bipartition of $K_{2^{n-2},2^{n-2}}$.
Then there is an edge connecting any two vertices between $\{u,v\}\times V_1$ and $\{u,v\}\times V_2$
in $K_2\bowtie K_{2^{n-2},2^{n-2}}$. Hence, $K_2\bowtie K_{2^{n-2},2^{n-2}} = K_{2^{n-1},2^{n-1}}$.
By induction, $G'_n\cong K_{2^{n-1},2^{n-1}}$.
\end{pf}
By the definitions of Cartesian product, direct product and semi-strong product of graphs,
we can define the product of signed graphs
$\Gamma_1$ and $\Gamma_2$ by their adjacency matrices. That is,
$A(\Gamma_1\Box \Gamma_2)=A(\Gamma_1)\otimes I_m+I_n\otimes A(\Gamma_2)$,
where $n=|V(\Gamma_1)|$ and $m=|V(\Gamma_2)|$,
$A(\Gamma_1\times \Gamma_2)=A(\Gamma_1)\otimes A(\Gamma_2)$, and
$A(\Gamma_1\bowtie \Gamma_2)=(A(\Gamma_1)+I_n)\otimes A(\Gamma_2)$.
If $X$ and $Y$ are eigenvectors of $A_1=A(\Gamma_1)$ and $A_2=A(\Gamma_2)$
corresponding to eigenvalues $\lambda$ and $\mu$, respectively, then direct computation yields the following.
\begin{align*}
&A(\Gamma_1\Box \Gamma_2)(X\otimes Y) =(A_1\otimes I_m+I_n\otimes A_2)(X\otimes Y)=(\lambda+\mu) X\otimes Y,\\
&A(\Gamma_1\times \Gamma_2)(X\otimes Y) =(A_1\otimes A_2)(X\otimes Y)=A_1X\otimes A_2Y=\lambda\mu X\otimes Y,\\
&A(\Gamma_1\bowtie \Gamma_2)(X\otimes Y) =[(A_1+I_n)\otimes A_2](X\otimes Y)=(A_1+I_n)X\otimes A_2Y=(\lambda+1)\mu X\otimes Y.
\end{align*}
Thus, we can obtain the following theorem.
\begin{thm}\label{thm-Kronecker-eig}
If $\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_n$ and $\mu_1\geq\mu_2\geq\cdots\geq\mu_m$
are the adjacency eigenvalues of the signed graphs $\Gamma_1$ and $\Gamma_2$, respectively,
then, for $i=1,2,\dots,n$ and $j=1,2,\dots,m$,
(i) (Germina et al. \cite{GeHZ11}) $\lambda_i+\mu_j$ are the adjacency eigenvalues of $\Gamma_1\Box \Gamma_2$;
(ii) (Germina et al. \cite{GeHZ11}) $\lambda_i\mu_j$ are the adjacency eigenvalues of $\Gamma_1\times \Gamma_2$;
(iii) $(\lambda_i+1)\mu_j$ are the adjacency eigenvalues of $\Gamma_1\bowtie \Gamma_2$.
\end{thm}
By Lemma \ref{lem-HaIK11} (ii) and Theorem \ref{thm-Kronecker-eig} (ii), we have the following result immediately.
\begin{cor}\label{cor-kron-eig}
For $i=1,2$, let $\Gamma_i=(G_i,\sigma_i)$ be a connected signed graph
with exactly two distinct eigenvalues $\pm\theta_i$, respectively.
If at least one of $G_1$ and $G_2$ is non-bipartite, then $\Gamma_1\times \Gamma_2$ is a connected
signed graph with exactly two distinct eigenvalues $\pm\theta_1\theta_2$.
\end{cor}
In the following, we introduce some known results and examples which can be used to construct
signed graphs with exactly two distinct eigenvalues. First we give some definitions.
A {\it weighing matrix} of order $n$ and weight $k$ is an $n\times n$ matrix $W=W(n,k)$ with
entries $0$, $+1$ and $-1$ such that $WW^T=W^T W=kI_{n}$.
A weighing matrix $W(n,n)$ is a Hadamard matrix $H_n$ of order $n$.
A {\it conference matrix} $C$ of order $n$ is an $n\times n$ matrix with $0$'s on
the diagonal, $+1$ or $-1$ in all other positions and with the property
$CC^T=(n-1)I_n$. Thus, a conference matrix of order $n$ is a weighing matrix of order $n$ and weight $n-1$,
and a permutation matrix of order $n$ is a weighing matrix of order $n$ and weight $1$.
\begin{lem}\label{lem-K-nn}
For $n\geq 1$, let
$$
H_2=\left[\begin{matrix}
1 & 1 \\
1 & -1 \\
\end{matrix}
\right],
H_{2^{n+1}} = H_2\otimes H_{2^{n}},
A_n=\left[\begin{matrix}
0 & 1 \\
1 & 0
\end{matrix}
\right]\otimes H_{2^n}.
$$
Then $A_n$ is a signed adjacency matrix of $K_{2^n,2^n}$
and its eigenvalues are $\pm\sqrt{2^n}$, each with multiplicity $2^n$.
\end{lem}
\begin{pf}
Since $H_{2^n}$ is a symmetric matrix with entries $\pm1$, $A_n$ is a signed adjacency matrix of $K_{2^n,2^n}$.
Note that $H_{2^n}$ is a Hadamard matrix of order $2^n$ with eigenvalues $\pm\sqrt{2^n}$.
By the property of Kronecker product, the eigenvalues of $A_n$ are $\pm\sqrt{2^n}$, each with multiplicity $2^n$.
\end{pf}
\begin{lem}{(McKee and Smyth \cite{McSm07})}\label{lem-T2n}
Let $P$ be a permutation matrix of order $n$ such that $P+P^T$
is the adjacency matrix of the cycle $C_n$ and
$$
A_{n}=\left[\begin{array}{cc}
P+P^T & P-P^T \\
P^T-P & -(P+P^T) \\
\end{array}\right].
$$
Then $A_n$ is the adjacency matrix of the $2n$-vertex toroidal tessellation $T_{2n}$ (see Figure \ref{fig1}),
whose eigenvalues are $\pm 2$, each with multiplicity $n$.
\end{lem}
\begin{figure}
\caption{{\small The graphs $T_{2n}
\label{fig1}
\end{figure}
\begin{lem}{(McKee and Smyth \cite{McSm07})}\label{lem-S14}
Let $W(7,4)=(w_{ij})$ be the weighing matrix of order $7$ and weight $4$, where
$w_{ij} = w_{1,\ell}$ for $\ell\equiv j-i+1({\rm mod}\ 7)$ and
$
(w_{11},w_{12},w_{13},w_{14},w_{15},w_{16},w_{17}) = (-1,1,1,0,1,0,0).
$
Let
$$
W(14,4)=
\left[\begin{matrix}
0 & 1 \\
1 & 0 \\
\end{matrix}\right]\otimes W(7,4).
$$
Then $W(14,4)$ is the adjacency matrix of the $14$-vertex signed graph $S_{14}$ (see Figure \ref{fig1})
and its eigenvalues are $\pm 2$ with the same multiplicity $7$.
\end{lem}
\begin{exa}{(Stinson \cite{Stin04})}\label{exa-Kn}
{\rm
For each $n\in \{2,6,10,14,18,26,30\}$, there exists a symmetric conference matrices $W(n,n-1)$.
Then, $W(n,n-1)$ is a signed adjacency matrix of $K_n$ and its eigenvalues are $\pm\sqrt{n-1}$, each with multiplicity $n/2$.}
\end{exa}
By the property of Kronecker product of matrices, we have the following examples.
\begin{exa}\label{example-complete-k-part-graph}
{\rm
Let $W(k,k-1)$ be a symmetric conference matrix of order $k$ and
$H_n$ be a symmetric Hadamard matrix of order $n$. Then $W(k,k-1)\otimes H_n$
is a signed adjacency matrix of the complete $k$-partite graph
$K_{n,n,\dots,n}$ and its eigenvalues are $\pm\sqrt{(k-1)n}$ with the same multiplicity.
In particular, $W(6,5)\otimes H_2$ is a signed adjacency matrix of the complete $6$-partite graph
$K_{2,2,2,2,2,2}$. }
\end{exa}
\begin{exa}\label{example-Hn-AGamma}
{\rm
Let $\Gamma$ be a signed graph of order $m$ and $H_n$ be a symmetric Hadamard matrix of order $n$.
Then $H_n\otimes A(\Gamma)$ is an adjacency matrix of the signed
graph $\Gamma^{(n)}$ of order $mn$ obtained from $\Gamma$.
If $\Gamma$ has exactly two distinct eigenvalues $\pm\theta$,
then $\Gamma^{(n)}$ has exactly two distinct eigenvalues $\pm\theta\sqrt{n}$.
}
\end{exa}
\section{Eigenvalues of signed Cartesian product and signed semi-strong product graphs}
In this section, we discuss the adjacency eigenvalues of
$\Gamma_1\widetilde{\Box} \Gamma_2$ and $\Gamma_1\widetilde{\bowtie} \Gamma_2$
and obtain a sufficient and necessary condition such that the spectrums of
$\Gamma_1\widetilde{\Box}\Gamma_2$ and $\Gamma_1\widetilde{\bowtie}\Gamma_2$
are symmetric.
\begin{thm}\label{thm-eigenvalues}
Let $\Gamma_1=(G_1,\sigma_1)$ be a signed bipartite graph of order $n$ with bipartition $(V_1,V_2)$
and $\Gamma_2=(G_2,\sigma_2)$ be a signed graph of order $m$.
If $\lambda^2$ is an eigenvalue of $A(\Gamma_1)^2$ with multiplicity $p$
and $\mu^2$ is an eigenvalue of $A(\Gamma_2)^2$ with multiplicity $q$, then
each of the following holds.
(i) $\lambda^2 + \mu^2$ (resp. $(\lambda^2 + 1)\mu^2$) is an eigenvalue of
$A(\Gamma_1\widetilde{\Box} \Gamma_2)^2$ (resp. $A(\Gamma_1\widetilde{\bowtie} \Gamma_2)^2$)
with multiplicity $pq$.
(ii) If $\lambda=0$ and $\mu\neq 0$, then $\pm\mu$ are eigenvalues of
$\Gamma_1\widetilde{\Box} \Gamma_2$ (also $\Gamma_1\widetilde{\bowtie} \Gamma_2$)
with multiplicities $\frac{1}{2}pq \pm \frac{1}{2}(n-2|V_1|)(q-2t)$ respectively,
where $t$ is the multiplicity of eigenvalue $\mu$ of $A(\Gamma_2)$.
(iii) If $\lambda\neq 0$, then $\pm \sqrt{\lambda^2 + \mu^2}$ are eigenvalues of
$\Gamma_1\widetilde{\Box} \Gamma_2$, each with multiplicity $pq/2$.
(iv) If $\lambda \mu\neq 0$, then $\pm \sqrt{(\lambda^2 + 1)\mu^2}$ are eigenvalues of
$\Gamma_1\widetilde{\bowtie} \Gamma_2$, each with multiplicity $pq/2$.
\end{thm}
\begin{cor}\label{cor-eig-G1G2}
For $i=1,2$, let $\Gamma_i$ be a signed graph with exactly two distinct eigenvalues $\pm \theta_i$,
where $\Gamma_1$ is bipartite. Then $\Gamma_1\widetilde{\Box}\Gamma_2$ and $\Gamma_1\widetilde{\bowtie}\Gamma_2$
have exactly two distinct eigenvalues $\pm \sqrt{\theta_1^2+\theta_2^2}$ and $\pm \sqrt{(\theta_1^2+1)\theta_2^2}$, respectively.
\end{cor}
The following theorem gives a
sufficient and necessary condition such that the spectrums of
$\Gamma_1\widetilde{\Box}\Gamma_2$ and $\Gamma_1\widetilde{\bowtie}\Gamma_2$
are symmetric.
\begin{thm}\label{thm-symmetric}
Let $\Gamma_1$ be a signed bipartite graph and $\Gamma_2$ be a signed graph.
The spectrum of $\Gamma_1\widetilde{\Box}\Gamma_2$ (resp. $\Gamma_1\widetilde{\bowtie}\Gamma_2$)
is symmetric if and only if $\Gamma_1$ is balanced or the spectrum of $\Gamma_2$ is symmetric.
\end{thm}
In the following proofs of Theorem \ref{thm-eigenvalues} and Theorem \ref{thm-symmetric}, we always
assume that $A_1:=A(\Gamma_1)=\left[
\begin{matrix}
O_s & P \\
P^T & O_{n-s}
\end{matrix}
\right]$ and $A_2:=A(\Gamma_2)$, where $|V_1|=s$ and $P$ is an $s\times (n-s)$ matrix.
\noindent{\bf Proof of Theorem \ref{thm-eigenvalues} (i).}
By (\ref{e1.1}), we have
\begin{align}\label{e4.1}
A(\Gamma_1\widetilde{\Box} \Gamma_2)^2
= &\left(A_1 \otimes I_m +
\left[
\begin{matrix}
I_s & O \\
O & -I_{n-s}
\end{matrix}
\right] \otimes A_2\right)^2 \nonumber\\
= & A_1^2 \otimes I_m +
\left[
\begin{matrix}
I_s & O \\
O & I_{n-s}
\end{matrix}
\right] \otimes A_2^2 \nonumber\\%--------------
& + \left(\left[
\begin{matrix}
O & P \\
P^T & O
\end{matrix}
\right]\otimes I_m \right) \left( \left[
\begin{matrix}
I_s & O \\
O & -I_{n-s}
\end{matrix}
\right] \otimes A_2 \right) \nonumber\\%-------------------
& + \left( \left[
\begin{matrix}
I_s & O \\
O & -I_{n-s}
\end{matrix}
\right] \otimes A_2 \right)
\left(\left[
\begin{matrix}
O & P \\
P^T & O
\end{matrix}
\right]\otimes I_m \right) \nonumber\\%------------------------------------------------------
= & A_1^2 \otimes I_m + I_n \otimes A_2^2
+ \left[
\begin{matrix}
O & -P \\
P^T & O
\end{matrix}
\right]\otimes A_2
+ \left[
\begin{matrix}
O & P \\
-P^T & O
\end{matrix}
\right] \otimes A_2 \nonumber\\
= & A_1^2 \otimes I_{m} + I_{n} \otimes A_2^2.
\end{align}
For each $i=1,\dots,p$ and $j=1,\dots,q$, let $X_i$ and $Y_j$ be eigenvectors of $A_1^2$ and $A_2^2$
with respect to eigenvalues $\lambda^2$ and $\mu^2$, respectively.
Thus, by (\ref{e4.1}), we have
$$
A(\Gamma_1\widetilde{\Box} \Gamma_2)^2(X_i\otimes Y_j) = (A_1^2\otimes I_m+I_n\otimes A_2^2)(X_i\otimes Y_j)=(\lambda^2 + \mu^2) (X_i\otimes Y_j).
$$
Therefore, $\lambda^2 + \mu^2$ is an eigenvalue of
$A(\Gamma_1\widetilde{\Box} \Gamma_2)^2$ with multiplicity $pq$.
By (\ref{e1.2}), we have
\begin{align}\label{e4.2}
A(\Gamma_1\widetilde{\bowtie} \Gamma_2)^2
= &\left(\left[
\begin{matrix}
I_s & P \\
P^T & -I_{n-s}
\end{matrix}
\right] \otimes A_2\right)^2 \nonumber\\
= & \left(\left[
\begin{matrix}
O_s & P \\
P^T & O_{n-s}
\end{matrix}
\right]+\left[
\begin{matrix}
I_s & O \\
O & -I_{n-s}
\end{matrix}
\right]\right)^2 \otimes A_2^2 \nonumber\\
=& (A_1^2+I_n) \otimes A_2^2 + \left[
\begin{matrix}
O_s & -P+P \\
P^T-P^T & O_{n-s}
\end{matrix}
\right]\otimes A_2^2 \nonumber\\
=& (A_1^2+I_n) \otimes A_2^2.
\end{align}
For each $i=1,\dots,p$ and $j=1,\dots,q$, let $X_i$ and $Y_j$ be eigenvectors of $A_1^2$ and $A_2^2$
with respect to eigenvalues $\lambda^2$ and $\mu^2$, respectively.
Thus, by (\ref{e4.2}), we have
$$
A(\Gamma_1\widetilde{\bowtie} \Gamma_2)^2(X_i\otimes Y_j) =
[(A_1^2 +I_n)\otimes A_2^2](X_i\otimes Y_j)=(\lambda^2 +1) \mu^2 (X_i\otimes Y_j).
$$
Therefore, $(\lambda^2 +1) \mu^2$ is an eigenvalue of
$A(\Gamma_1\widetilde{\bowtie} \Gamma_2)^2$ with multiplicity $pq$.
\rule{2mm}{2mm}
\begin{lem}\label{lem-null-space}
Let $\Gamma$ be a signed bipartite graph of order $n$ with bipartition $(V_1,V_2)$, where $|V_1|=s$, and
$A=\left[
\begin{matrix}
O_s & P\\
P^T & O_{n-s} \\
\end{matrix}
\right]$ be the adjacency matrix of $\Gamma$.
Let $\{\mathbf{w}_1,\dots,\mathbf{w}_a\}$ be a basis of null space of $P^T$ and
$\{\mathbf{z}_1,\dots,\mathbf{z}_b\}$ be a basis of null space of $P$.
The following $a+b$ vectors of length $n$
$$
\left\{
\left[
\begin{matrix}
\mathbf{w}_1 \\
\bm{0} \\
\end{matrix}
\right], \dots,
\left[
\begin{matrix}
\mathbf{w}_a \\
\bm{0} \\
\end{matrix}
\right], \left[
\begin{matrix}
\bm{0} \\
\mathbf{z}_1 \\
\end{matrix}
\right], \dots,
\left[
\begin{matrix}
\bm{0} \\
\mathbf{z}_b \\
\end{matrix}
\right]
\right\},
$$
is a basis of null space of $A$.
\end{lem}
\begin{pf}
Since ${\rm rank}(A)={\rm rank}(P)+{\rm rank}(P^T)$, by Rank-Nullity Theorem
$$
n-{\rm rank}(A)=(n-s-{\rm rank}(P))+(s-{\rm rank}(P^T))=a+b.
$$
The result follows.
\end{pf}
\noindent{\bf Proof of Theorem \ref{thm-eigenvalues} (ii).}
Since the multiplicity of eigenvalue $\mu$ of $A_2$ is $t$,
the multiplicity of eigenvalue $-\mu$ of $A_2$ is $q-t$.
Assume that $A_2Y_j = \mu Y_j$ for each $1\leq j\leq t$ and $A_2Y'_k = -\mu Y'_k$ for each $1\leq k\leq q-t$.
In particular, if $t=0$, then $1\leq k\leq q$ and there exists no such $Y_j$;
if $t=q$, then $1\leq j\leq q$ and there exists no such $Y'_k$.
By the assumption, $\lambda=0$ is an eigenvalue of $A_1^2$ (and so $A_1$) with multiplicity $p$.
Hence, the rank of $A_1$ is ${\rm rank}(A_1)=n-p$ and
${\rm rank}(P)={\rm rank}(P^T)=(n-p)/2$. Thus,
the nullity of $P^T$ is
$$
r:= s - {\rm rank}(P^T)= p/2 - (n-2s)/2
$$
and the nullity of $P$ is $p-r = p/2 + (n-2s)/2$.
Suppose that $\{X_{11},\dots,X_{r1}\}$ is a basis of null space of $P^T$ and
$\{X_{12},\dots,X_{(p-r)2}\}$ is a basis of null space of $P$.
Let
$$
Z_i:=\left[
\begin{matrix}
X_{i1} \\
\bm{0} \\
\end{matrix}
\right],
Z'_{\ell}:=\left[
\begin{matrix}
\bm{0} \\
X_{\ell 2} \\
\end{matrix}
\right]
$$
be column vectors of length $n$ for each $1\leq i\leq r$ and $1\leq \ell\leq p-r$.
In particular, if $r=0$, then $1\leq \ell\leq p=n-2s$ and there is no such $Z_i$;
if $r=p$, then $1\leq i\leq p=2s-n$ and there is no such $Z'_{\ell}$.
By Lemma \ref{lem-null-space},
$\{Z_1,\dots,Z_r\}\cup \{Z'_1,\dots,Z'_{p-r}\}$ is a basis of
null space of $A_1$. Therefore, $A_1Z_i=A_1Z'_{\ell}=\bm{0}$ and
for every $1\leq i\leq r$, $1\leq j\leq t$ and $1\leq k\leq q-t$,
\begin{align*}
A(\Gamma_1\widetilde{\Box} \Gamma_2)(Z_i\otimes Y_j)&=\mu (Z_i\otimes Y_j)=A(\Gamma_1\widetilde{\bowtie} \Gamma_2)(Z_i\otimes Y_j),\\
A(\Gamma_1\widetilde{\Box} \Gamma_2)(Z_i\otimes Y'_k)&=-\mu (Z_i\otimes Y'_k)=A(\Gamma_1\widetilde{\bowtie} \Gamma_2)(Z_i\otimes Y'_k).
\end{align*}
For every $1\leq \ell\leq p-r$, $1\leq j\leq t$ and $1\leq k\leq q-t$,
\begin{align*}
A(\Gamma_1\widetilde{\Box} \Gamma_2)(Z'_{\ell}\otimes Y'_k)&=\mu (Z'_{\ell}\otimes Y'_k)
=A(\Gamma_1\widetilde{\bowtie} \Gamma_2)(Z'_{\ell}\otimes Y'_k),\\
A(\Gamma_1\widetilde{\Box} \Gamma_2)(Z'_{\ell}\otimes Y_j)&=-\mu (Z'_{\ell}\otimes Y_j)
=A(\Gamma_1\widetilde{\bowtie} \Gamma_2)(Z'_{\ell}\otimes Y_j).
\end{align*}
Note that all of $Z_i$, $Z'_{\ell}$, $Y_j$ and $Y'_k$ are nonzero vectors
for each $1\leq i\leq r$, $1\leq \ell\leq p-r$, $1\leq j\leq t$ and $1\leq k\leq q-t$.
Hence, the Kronecker products of them are also nonzero vectors.
By $(Z_i\otimes Y_j)^T (Z'_{\ell}\otimes Y'_k)=0$, we have $Z_i\otimes Y_j$
and $Z'_{\ell}\otimes Y'_k$ are
$$
rt+(p-r)(q-t) =pq/2 + (n-2s)(q-2t)/2
$$
eigenvectors of $A(\Gamma_1\widetilde{\Box} \Gamma_2)$ (resp. $A(\Gamma_1\widetilde{\bowtie} \Gamma_2)$)
with respect to eigenvalue $\mu$. By $(Z_i\otimes Y'_k)^T (Z'_{\ell}\otimes Y_j)=0$,
we know that $Z_i\otimes Y'_k$ and $Z'_{\ell}\otimes Y_j$ are
$$
r(q-t)+(p-r)t = pq/2 - (n-2s)(q-2t)/2
$$
eigenvectors of $A(\Gamma_1\widetilde{\Box} \Gamma_2)$
(resp. $A(\Gamma_1\widetilde{\bowtie} \Gamma_2)$) with respect to eigenvalue $-\mu$.
Thus, $\pm\mu$ are eigenvalues of
$\Gamma_1\widetilde{\Box} \Gamma_2$ (resp. $\Gamma_1\widetilde{\bowtie} \Gamma_2$)
with multiplicities $\frac{1}{2}pq\pm \frac{1}{2}(n-2s)(q-2t)$, respectively.
\rule{2mm}{2mm}
\noindent{\bf Proof of Theorem \ref{thm-eigenvalues} (iii).}
Suppose that $\lambda\neq 0$.
Since $\Gamma_1$ is bipartite, $\lambda$ and $-\lambda$ are
eigenvalues of $\Gamma_1$, each with multiplicity $p/2$.
Without loss of generality, assume that $\mu\geq 0$,
$A_2Y_j = \mu Y_j$ for each $j=1,\dots,t$ and $A_2Y'_k = -\mu Y'_k$ for each $k=1,\dots,q-t$.
In particular, if $t=0$, then $1\leq k\leq q$ and there exists no such $Y_j$;
if $t=q$, then $1\leq j\leq q$ and there exists no such $Y'_k$.
Note that if $\mu=0$, then $t=q$.
Now, for $i=1,\dots,p/2$, suppose that
$
X_i=\left[\begin{matrix}
X_{i1} \\
X_{i2}
\end{matrix}\right]
$
is the unit vector such that $A_1X_i=\lambda X_i$, where $X_{i1}$ and $X_{i2}$
are column vectors of length $s$ and $n-s$ respectively.
Then $PX_{i2}=\lambda X_{i1}$ and $P^TX_{i1}=\lambda X_{i2}$.
For each $i=1,\dots,p/2$, let
$X'_i=\left[\begin{matrix}
X_{i1} \\
-X_{i2}
\end{matrix}\right]
$, then $A_1X'_i=-\lambda X'_i$. Since $\lambda\neq 0$, we have
$X_i^TX'_i=0$ and so $X_{i1}^TX_{i1}=X_{i2}^TX_{i2}=\frac{1}{2}$,
which implies that $X_{i1}$ and $X_{i2}$ are nonzero vectors.
Based on eigenvalues $\pm\lambda, \pm\mu$ and the corresponding eigenvectors, we construct $pq/2$ vectors as follows
$$
Z_i\otimes Y_j=\left[
\begin{matrix}
(\sqrt{\lambda^2+\mu^2}+\mu) X_{i1} \\
\lambda X_{i2} \\
\end{matrix}
\right]\otimes Y_j, \, \,
W_i\otimes Y'_k=\left[
\begin{matrix}
\lambda X_{i1} \\
(\sqrt{\lambda^2+\mu^2}+\mu) X_{i2} \\
\end{matrix}
\right]\otimes Y'_k,
$$
for each $i=1,\dots,p/2$, $j=1,\dots,t$ and $k=1,\dots,q-t$, and construct $pq/2$ vectors as follows
$$
Z'_i\otimes Y_j=\left[
\begin{matrix}
-\lambda X_{i1} \\
(\sqrt{\lambda^2+\mu^2}+\mu)X_{i2} \\
\end{matrix}
\right]\otimes Y_j, \, \,
W'_i\otimes Y'_k=\left[
\begin{matrix}
(\sqrt{\lambda^2+\mu^2}+\mu) X_{i1} \\
-\lambda X_{i2} \\
\end{matrix}
\right]\otimes Y'_k,
$$
for $i=1,\dots,p/2$, $j=1,\dots,t$ and $k=1,\dots,q-t$.
Then, we have
\begin{align}\label{e4.3}
A(\Gamma_1\widetilde{\Box} \Gamma_2)\cdot(Z_i\otimes Y_j) = &
\left[
\begin{matrix}
I_s\otimes A_2 & P\otimes I_m \\
P^T\otimes I_m & -I_{n-s}\otimes A_2
\end{matrix}
\right] \cdot \left[
\begin{matrix}
(\sqrt{\lambda^2+\mu^2}+\mu) X_{i1} \otimes Y_j \\
\lambda X_{i2} \otimes Y_j
\end{matrix}
\right] \nonumber\\
= & \left[
\begin{matrix}
(\sqrt{\lambda^2+\mu^2}+\mu)X_{i1}\otimes \mu Y_j +\lambda^2 X_{i1}\otimes Y_j \\
(\sqrt{\lambda^2+\mu^2}+\mu)\lambda X_{i2}\otimes Y_j -\lambda X_{i2}\otimes \mu Y_j \\
\end{matrix}
\right] \nonumber\\
= & \sqrt{\lambda^2+\mu^2}\cdot \left[
\begin{matrix}
(\sqrt{\lambda^2+\mu^2}+\mu) X_{i1}\otimes Y_j \\
\lambda X_{i2}\otimes Y_j \\
\end{matrix}
\right] \nonumber\\
= & \sqrt{\lambda^2+\mu^2}\cdot (Z_i\otimes Y_j),
\end{align}
\begin{align}\label{e4.4}
A(\Gamma_1\widetilde{\Box} \Gamma_2)\cdot(W_i\otimes Y'_k) = &
\left[
\begin{matrix}
I_s\otimes A_2 & P\otimes I_m \\
P^T\otimes I_m & -I_{n-s}\otimes A_2
\end{matrix}
\right]\cdot \left[
\begin{matrix}
\lambda X_{i1}\otimes Y'_k \\
(\sqrt{\lambda^2+\mu^2}+\mu) X_{i2}\otimes Y'_k \\
\end{matrix}
\right] \nonumber\\
= & \left[
\begin{matrix}
-\lambda X_{i1}\otimes \mu Y'_k + (\sqrt{\lambda^2+\mu^2}+\mu)\lambda X_{i1}\otimes Y'_k\\
\lambda^2 X_{i2}\otimes Y'_k + (\sqrt{\lambda^2+\mu^2}+\mu) X_{i2}\otimes \mu Y'_k
\end{matrix}
\right]\nonumber\\
= & \sqrt{\lambda^2+\mu^2}\cdot \left[
\begin{matrix}
\lambda X_{i1}\otimes Y'_k \\
(\sqrt{\lambda^2+\mu^2}+\mu) X_{i2}\otimes Y'_k \\
\end{matrix}
\right] \nonumber\\
= & \sqrt{\lambda^2+\mu^2}\cdot (W_i\otimes Y'_k),
\end{align}
\begin{align}\label{e4.5}
A(\Gamma_1\widetilde{\Box} \Gamma_2)\cdot(Z'_i\otimes Y_j) = &
\left[
\begin{matrix}
I_s\otimes A_2 & P\otimes I_m \\
P^T\otimes I_m & -I_{n-s}\otimes A_2
\end{matrix}
\right] \cdot \left[
\begin{matrix}
-\lambda X_{i1}\otimes Y_j \\
(\sqrt{\lambda^2+\mu^2}+\mu)X_{i2}\otimes Y_j \\
\end{matrix}
\right] \nonumber\\
= & \left[
\begin{matrix}
-\lambda X_{i1}\otimes \mu Y_j + (\sqrt{\lambda^2+\mu^2}+\mu)\lambda X_{i1}\otimes Y_j\\
-\lambda^2 X_{i2}\otimes Y_j - (\sqrt{\lambda^2+\mu^2}+\mu) X_{i2}\otimes \mu Y_j
\end{matrix}
\right]\nonumber\\
= & -\sqrt{\lambda^2+\mu^2}\cdot \left[
\begin{matrix}
-\lambda X_{i1} \otimes Y_j \\
(\sqrt{\lambda^2+\mu^2}+\mu)X_{i2} \otimes Y_j \\
\end{matrix}
\right] \nonumber\\%---------------------------
= & -\sqrt{\lambda^2+\mu^2}\cdot (Z'_i\otimes Y_j),
\end{align}
\begin{align}\label{e4.6}
A(\Gamma_1\widetilde{\Box} \Gamma_2)\cdot(W'_i\otimes Y'_k) = &
\left[
\begin{matrix}
I_s\otimes A_2 & P\otimes I_m \\
P^T\otimes I_m & -I_{n-s}\otimes A_2
\end{matrix}
\right] \cdot \left[
\begin{matrix}
(\sqrt{\lambda^2+\mu^2}+\mu) X_{i1}\otimes Y'_k \\
-\lambda X_{i2}\otimes Y'_k \\
\end{matrix}
\right] \nonumber\\
= & \left[
\begin{matrix}
-(\sqrt{\lambda^2+\mu^2}+\mu) X_{i1} \otimes \mu Y'_k -\lambda^2 X_{i1}\otimes Y'_k\\
(\sqrt{\lambda^2+\mu^2}+\mu) \lambda X_{i2}\otimes Y'_k -\lambda X_{i2}\otimes \mu Y'_k
\end{matrix}
\right] \nonumber\\
= & -\sqrt{\lambda^2+\mu^2}\cdot \left[
\begin{matrix}
(\sqrt{\lambda^2+\mu^2}+\mu) X_{i1}\otimes Y'_k \\
-\lambda X_{i2}\otimes Y'_k \\
\end{matrix}
\right] \nonumber\\%---------------------------
= & -\sqrt{\lambda^2+\mu^2}\cdot (W'_i\otimes Y'_k).
\end{align}
Since $\lambda\neq 0$, and $X_{i1}$ and $X_{i2}$ are nonzero, we know that
all of $Z_i, W_i, Z'_i, W'_i, Y_j, Y'_k$ are nonzero vectors
for each $i\in \{1,\dots,p/2\}$, $j\in \{1,\dots,t\}$ and $k\in \{1,\dots,q-t\}$,
and the Kronecker products of them are also nonzero vectors.
As $(Z_i\otimes Y_j)^T (W_i\otimes Y'_k)=0$, by (\ref{e4.3}) and (\ref{e4.4}), we have $Z_i\otimes Y_j$
and $W_i\otimes Y'_k$ are $pq/2$ eigenvectors
of $A(\Gamma_1\widetilde{\Box} \Gamma_2)$ with respect to eigenvalue $\sqrt{\lambda^2+\mu^2}$. As
$(Z'_i\otimes Y_j)^T (W'_i\otimes Y'_k)=0$, by (\ref{e4.5}) and (\ref{e4.6}), we have $Z'_i\otimes Y_j$
and $W'_i\otimes Y'_k$ are $pq/2$ eigenvectors
of $A(\Gamma_1\widetilde{\Box} \Gamma_2)$ with respect to eigenvalue $-\sqrt{\lambda^2+\mu^2}$.
Therefore, $\pm\sqrt{\lambda^2+\mu^2}$ are adjacency eigenvalues of $\Gamma_1\widetilde{\Box} \Gamma_2$,
each with multiplicity $pq/2$.
\rule{2mm}{2mm}
\noindent{\bf Proof of Theorem \ref{thm-eigenvalues} (iv).}
Suppose that $\lambda\mu\neq 0$.
Since $\Gamma_1$ is bipartite, $\lambda$ and $-\lambda$ are
eigenvalues of $\Gamma_1$, each with multiplicity $p/2$.
Without loss of generality, assume that
$A_2Y_j = \mu Y_j$ for each $j=1,\dots,t$ and $A_2Y'_k = -\mu Y'_k$ for each $k=1,\dots,q-t$.
In particular, if $t=0$, then $1\leq k\leq q$ and there exists no such $Y_j$;
if $t=q$, then $1\leq j\leq q$ and there exists no such $Y'_k$.
Now, for each $i=1,\dots,p/2$, suppose that
$
X_i=\left[\begin{matrix}
X_{i1} \\
X_{i2}
\end{matrix}\right]
$
is the unit vector such that $A_1X_i=\lambda X_i$, where $X_{i1}$ and $X_{i2}$
are column vectors of length $s$ and $n-s$ respectively.
Then $PX_{i2}=\lambda X_{i1}$ and $P^TX_{i1}=\lambda X_{i2}$.
Let $
X'_i=\left[\begin{matrix}
X_{i1} \\
-X_{i2}
\end{matrix}\right]
$, then $A_1X'_i=-\lambda X'_i$ and so $X_i^TX'_i=0$. Thus
$
(X_{i1})^TX_{i1}=(X_{i2})^TX_{i2}=\frac{1}{2},
$
and so $X_{i1}$ and $X_{i2}$ are nonzero vectors.
Based on eigenvalues $\pm\lambda, \pm\mu$ and the corresponding eigenvectors, we construct $pq/2$ vectors as follows
$$
Z_i\otimes Y_j=\left[
\begin{matrix}
(\sqrt{\lambda^2+1}+1) X_{i1} \\
\lambda X_{i2} \\
\end{matrix}
\right]\otimes Y_j, \, \,
Z'_i\otimes Y'_k=\left[
\begin{matrix}
-\lambda X_{i1} \\
(\sqrt{\lambda^2+1}+1) X_{i2} \\
\end{matrix}
\right]\otimes Y'_k,
$$
for each $i=1,\dots,p/2$, $j=1,\dots,t$ and $k=1,\dots,q-t$, and construct $pq/2$ vectors as follows
$$
Z'_i\otimes Y_j=\left[
\begin{matrix}
-\lambda X_{i1} \\
(\sqrt{\lambda^2+1}+1)X_{i2} \\
\end{matrix}
\right]\otimes Y_j, \, \,
Z_i\otimes Y'_k=\left[
\begin{matrix}
(\sqrt{\lambda^2+1}+1) X_{i1} \\
\lambda X_{i2} \\
\end{matrix}
\right]\otimes Y'_k,
$$
for each $i=1,\dots,p/2$, $j=1,\dots,t$ and $k=1,\dots,q-t$.
Since
\begin{align*}
\left[
\begin{matrix}
I_s & P \\
P^T & -I_{n-s}
\end{matrix}
\right]Z_i = &
\left[
\begin{matrix}
I_s & P \\
P^T & -I_{n-s}
\end{matrix}
\right] \left[
\begin{matrix}
(\sqrt{\lambda^2+1}+1) X_{i1} \\
\lambda X_{i2} \\
\end{matrix}
\right] \\
= & \left[
\begin{matrix}
(\sqrt{\lambda^2+1}+1) X_{i1} + \lambda^2 X_{i1} \\
(\sqrt{\lambda^2+1}+1) \lambda X_{i2}-\lambda X_{i2} \\
\end{matrix}
\right] \\
= & \sqrt{\lambda^2+1}Z_i , \\
\left[
\begin{matrix}
I_s & P \\
P^T & -I_{n-s}
\end{matrix}
\right]Z'_i = &
\left[
\begin{matrix}
I_s & P \\
P^T & -I_{n-s}
\end{matrix}
\right] \left[
\begin{matrix}
-\lambda X_{i1} \\
(\sqrt{\lambda^2+1}+1)X_{i2} \\
\end{matrix}
\right] \\
= & \left[
\begin{matrix}
-\lambda X_{i1} + (\sqrt{\lambda^2+1}+1) \lambda X_{i1} \\
- \lambda^2 X_{i2} - (\sqrt{\lambda^2+1}+1) X_{i2} \\
\end{matrix}
\right] \\
= & -\sqrt{\lambda^2+1} Z'_i ,
\end{align*}
we can obtain the following equations
\begin{align}
A(\Gamma_1\widetilde{\bowtie} \Gamma_2)\cdot(Z_i\otimes Y_j) &= \sqrt{\lambda^2+1}Z_i\otimes A_2 Y_j
= \mu\sqrt{\lambda^2+1}\cdot (Z_i\otimes Y_j), \label{e4.7}\\
A(\Gamma_1\widetilde{\bowtie} \Gamma_2)\cdot(Z'_i\otimes Y'_k) &= -\sqrt{\lambda^2+1}Z'_i\otimes A_2 Y'_k
= \mu\sqrt{\lambda^2+1}\cdot (Z'_i\otimes Y'_k), \label{e4.8}\\
A(\Gamma_1\widetilde{\bowtie} \Gamma_2)\cdot(Z'_i\otimes Y_j) &= -\sqrt{\lambda^2+1}Z'_i \otimes A_2 Y_j
= -\mu\sqrt{\lambda^2+1}\cdot (Z'_i\otimes Y_j), \label{e4.9}\\
A(\Gamma_1\widetilde{\bowtie} \Gamma_2)\cdot(Z_i\otimes Y'_k) &= \sqrt{\lambda^2+1}Z_i \otimes A_2 Y'_k
= -\mu\sqrt{\lambda^2+1}\cdot (Z_i\otimes Y'_k).\label{e4.10}
\end{align}
Since $\lambda\mu\neq 0$, $X_{i1}$ and $X_{i2}$ are nonzero, we know that
all of $Z_i, Z'_i, Y_j, Y'_k$ are nonzero vectors
for each $i\in \{1,\dots,p/2\}$, $j\in \{1,\dots,t\}$ and $k\in \{1,\dots,q-t\}$, and
the Kronecker products of them are also nonzero.
As $(Z_i\otimes Y_j)^T (Z'_i\otimes Y'_k)=0$, by (\ref{e4.7}) and (\ref{e4.8}), we have $Z_i\otimes Y_j$
and $Z'_i\otimes Y'_k$ are $pq/2$ eigenvectors
of $A(\Gamma_1\widetilde{\bowtie} \Gamma_2)$ with respect to eigenvalue $\mu\sqrt{\lambda^2+1}$. As
$(Z'_i\otimes Y_j)^T (Z_i\otimes Y'_k)=0$, by (\ref{e4.9}) and (\ref{e4.10}), we have $Z'_i\otimes Y_j$
and $Z_i\otimes Y'_k$ are $pq/2$ eigenvectors
of $A(\Gamma_1\widetilde{\bowtie} \Gamma_2)$ with respect to eigenvalue $-\mu\sqrt{\lambda^2+1}$.
Therefore, $\pm\mu\sqrt{\lambda^2+1}$ are adjacency eigenvalues of $\Gamma_1\widetilde{\bowtie} \Gamma_2$,
each with multiplicity $pq/2$.
\rule{2mm}{2mm}
\noindent{\bf Proof of Theorem \ref{thm-symmetric}.}
Let $\lambda^2$ be any eigenvalue of $A(\Gamma_1)^2$ with multiplicity $p$
and $\mu^2$ be any eigenvalue of $A(\Gamma_2)^2$ with multiplicity $q$,
where $\mu$ is the eigenvalue of $\Gamma_2$ with multiplicity $t$.
(a) Consider $\Gamma_1\widetilde{\Box} \Gamma_2$.
By Theorem \ref{thm-eigenvalues} (i), $\lambda^2 + \mu^2$ is an eigenvalue of
$A(\Gamma_1\widetilde{\Box} \Gamma_2)^2$ with multiplicity $pq$.
Assume that $\Gamma_1$ is balanced or the spectrum of $\Gamma_2$ is symmetric.
Then the bipartition $(V_1,V_2)$ of $\Gamma_1$ satisfies
$|V_1|= \frac{n}{2}$ or the multiplicity of eigenvalue $\mu (\neq 0)$ of $\Gamma_2$ is equal to $t=\frac{q}{2}$, and so
$
(n-2|V_1|)(q-2t)=0.
$
It suffices to prove that the multiplicities of eigenvalues $\pm\sqrt{\lambda^2+\mu^2}$
of $\Gamma_1\widetilde{\Box} \Gamma_2$ are equal to $\frac{1}{2}pq$ when $\lambda^2+\mu^2\neq 0$.
If $\lambda\neq 0$, then by Theorem \ref{thm-eigenvalues} (iii),
the multiplicities of eigenvalues $\pm\sqrt{\lambda^2+\mu^2}$
of $\Gamma_1\widetilde{\Box} \Gamma_2$ are equal to $\frac{1}{2}pq$.
If $\lambda=0$ and $\mu\neq 0$, then by Theorem \ref{thm-eigenvalues} (ii),
the multiplicities of eigenvalues $\pm\mu$ of $\Gamma_1\widetilde{\Box} \Gamma_2$
are equal to $\frac{1}{2}pq$. Thus, the spectrum of $\Gamma_1\widetilde{\Box} \Gamma_2$ is symmetric.
Conversely, assume that the spectrum of $\Gamma_1\widetilde{\Box} \Gamma_2$ is symmetric.
If all the eigenvalues of $\Gamma_1$ are nonzero, then the rank of
$A(\Gamma_1)$ is $n$ and so ${\rm rank}(P)={\rm rank}(P^T)=\frac{n}{2}$.
This implies $|V_1|=|V_2|$ and so $\Gamma_1$ is balanced.
If $\lambda=0$ and $\mu\neq 0$, then the multiplicities of eigenvalues
$\pm\mu$ of $\Gamma_1\widetilde{\Box} \Gamma_2$ must be equal. By Theorem \ref{thm-eigenvalues} (ii),
we have
$$
pq + (n-2|V_1|)(q-2t) = pq - (n-2|V_1|)(q-2t),
$$
that is $(n-2|V_1|)(q-2t)=0$ and so $\Gamma_1$ is balanced or the spectrum of $\Gamma_2$ is symmetric.
(b) Consider $\Gamma_1\widetilde{\bowtie} \Gamma_2$.
By Theorem \ref{thm-eigenvalues} (i), $(\lambda^2 + 1)\mu^2$ is an eigenvalue of
$A(\Gamma_1\widetilde{\bowtie} \Gamma_2)^2$ with multiplicity $pq$.
Assume that $\Gamma_1$ is balanced or the spectrum of $\Gamma_2$ is symmetric.
Then the bipartition $(V_1,V_2)$ of $\Gamma_1$ satisfies
$|V_1|= \frac{n}{2}$ or the multiplicity of eigenvalue $\mu (\neq 0)$ of $\Gamma_2$ is equal to $t=\frac{q}{2}$, and so
$
(n-2|V_1|)(q-2t)=0.
$
It suffices to prove that the multiplicities of eigenvalues $\pm\sqrt{(\lambda^2+1)\mu^2}$
of $\Gamma_1\widetilde{\bowtie} \Gamma_2$ are equal to $\frac{1}{2}pq$ when $\mu^2\neq 0$.
If $\lambda\neq 0$ and $\mu\neq 0$, then by Theorem \ref{thm-eigenvalues} (iv),
the multiplicities of eigenvalues $\pm\sqrt{(\lambda^2+1)\mu^2}$
of $\Gamma_1\widetilde{\bowtie} \Gamma_2$ are equal to $\frac{1}{2}pq$.
If $\lambda=0$ and $\mu\neq 0$, then by Theorem \ref{thm-eigenvalues} (ii),
the multiplicities of eigenvalues $\pm\mu$ of $\Gamma_1\widetilde{\bowtie} \Gamma_2$
are equal to $\frac{1}{2}pq$. Thus, the spectrum of $\Gamma_1\widetilde{\bowtie} \Gamma_2$ is symmetric.
Conversely, assume that the spectrum of $\Gamma_1\widetilde{\bowtie} \Gamma_2$ is symmetric.
If all the eigenvalues of $\Gamma_1$ are nonzero, then the rank of
$A(\Gamma_1)$ is $n$ and so ${\rm rank}(P)={\rm rank}(P^T)=\frac{n}{2}$.
This implies $|V_1|=|V_2|$ and so $\Gamma_1$ is balanced.
If $\lambda=0$ and $\mu\neq 0$, then the multiplicities of eigenvalues
$\pm\mu$ of $\Gamma_1\widetilde{\bowtie} \Gamma_2$ must be equal. By Theorem \ref{thm-eigenvalues} (ii),
we have
$$
pq + (n-2|V_1|)(q-2t) = pq - (n-2|V_1|)(q-2t),
$$
that is $(n-2|V_1|)(q-2t)=0$ and so $\Gamma_1$ is balanced or the spectrum of $\Gamma_2$ is symmetric.
\rule{2mm}{2mm}
\section{Induced subgraphs of the signed product graphs}
In this section, we mainly give the proof of Theorem \ref{thm-Delta-2-min} and generalize it to
signed product of $n$ $(n\geq 3)$ graphs.
To establish Theorem \ref{thm-Delta-2-min}, we need the following lemmas.
\begin{lem}(Cauchy's Interlacing Theorem \cite{BrHa12})\label{lem-cauchy}
Let $A$ be an $n\times n$ symmetric matrix, and $B$ be an $m\times m$ principle submatrix of $A$, where $m<n$.
If the eigenvalues of $A$ are $\lambda_1\geq \lambda_2\geq \cdots\geq \lambda_n$, and the eigenvalues of $B$
are $\mu_1\geq \mu_2\geq \cdots\geq \mu_m$, then for all $1\leq i\leq m$,
$$
\lambda_i\geq \mu_i\geq \lambda_{n-m+i}.
$$
\end{lem}
\begin{lem}\label{lem-Delta-sign}
Suppose $\Gamma=(G,\sigma)$ is a signed graph of order $n$, and $A=(a^{\sigma}_{ij})$ is the adjacency matrix of $\Gamma$.
Let $\widetilde{A}=(\tilde{a}_{ij})$ be an $n\times n$ symmetric matrix with
$|\tilde{a}_{ij}|\leq |a^{\sigma}_{ij}|$ for any $1\leq i,j\leq n$. Then
$$
\Delta(\Gamma)\geq \lambda_1(\widetilde{A}).
$$
In particular, $\Delta(\Gamma)\geq \lambda_1(\Gamma)$ when $\widetilde{A}=A$.
\end{lem}
\begin{pf}
It suffices to consider that $\widetilde{A}$ is not an all zero matrix. Thus, $\lambda_1(\widetilde{A})>0$.
Suppose $X=(x_1,x_2,\dots,x_n)^T$ is an eigenvector corresponding to $\lambda_1(\widetilde{A})$.
Then $\lambda_1(\widetilde{A}) X = \widetilde{A} X$. Assume that
$|x_u|=\max\{|x_1|,|x_2|,\dots,|x_n|\}$. Then $|x_u|>0$ and
$$
|\lambda_1(\widetilde{A})x_u|=\left|\sum_{j=1}^n\tilde{a}_{uj}x_j\right| = \left|\sum_{j\sim u}\tilde{a}_{uj}x_j\right|
\leq\sum_{j\sim u}\left|\tilde{a}_{uj}\right||x_u|\leq\sum_{j\sim u}\left|a^{\sigma}_{uj}\right||x_u| \leq \Delta(\Gamma)|x_u|.
$$
Hence, $\Delta(\Gamma)\geq \lambda_1(\widetilde{A})$.
\end{pf}
\begin{lem}\label{lem-Delta-lambda}
Let $\Gamma$ be a connected signed graph of order $n$ with $k$ nonnegative adjacency eigenvalues
$\lambda_1(\Gamma)\geq \cdots\geq \lambda_k(\Gamma)\geq 0$. If $H$ is an $(n-k+1)$-vertex induced subgraph of $\Gamma$, then
$$\Delta(H)\geq \lceil\lambda_k(\Gamma)\rceil.$$
\end{lem}
\begin{pf}
Note that $A(H)$ is an $(n-k+1)\times (n-k+1)$ submatrix of $A(\Gamma)$.
By Lemma \ref{lem-cauchy}, $\lambda_1(H) \geq \lambda_k(\Gamma)$.
By Lemma \ref{lem-Delta-sign}, $\Delta(H)\geq \lambda_1(H) \geq \lambda_k(\Gamma)$.
Hence, $\Delta(H)\geq \lceil\lambda_k(\Gamma)\rceil$.
\end{pf}
\begin{exa}
{\rm
The Petersen graph $(PG,+)$ has spectrum $3^{(1)}, 1^{(5)}, -2^{(4)}$.
If $H$ is a $5$-vertex induced subgraph of $(PG,+)$, then by Lemma \ref{lem-Delta-lambda},
$\Delta(H)\geq 1$ and there exists a subgraph such that the bound is tight.
The signed Petersen graph $(PG,-)$ has spectrum $2^{(4)}, -1^{(5)}, -3^{(1)}$.
If $H$ is a $7$-vertex induced subgraph of $(PG,-)$, then by Lemma \ref{lem-Delta-lambda},
$\Delta(H)\geq 2$ and there exists a subgraph such that the bound is tight.
}
\end{exa}
\noindent{\bf Proof of Theorem \ref{thm-Delta-2-min}.}
Denote $N=mn$.
Let $\Gamma=\Gamma_1\widetilde{\Box}\Gamma_2$ and $\Gamma'=\Gamma_1\widetilde{\bowtie}\Gamma_2$.
Then $H$ (resp. $H'$) is a $(\lfloor\frac{N}{2}\rfloor+1)$-vertex
induced subgraph of $\Gamma$ (resp. $\Gamma'$). By Lemma \ref{lem-Delta-lambda},
$$
\Delta(H) \geq \lambda_{\lceil\frac{N}{2}\rceil}(\Gamma)\ \text{and} \
\Delta(H') \geq \lambda_{\lceil\frac{N}{2}\rceil}(\Gamma').
$$
By Theorem \ref{thm-eigenvalues} (i), $\lambda^2+\mu^2$
is the minimum eigenvalue of $A(\Gamma)^2$ and $(\lambda^2+1)\mu^2$
is the minimum eigenvalue of $A(\Gamma')^2$. Thus, by Theorem \ref{thm-symmetric},
the adjacency spectrums of $\Gamma$ and $\Gamma'$ are symmetric and so
$$
\lambda_{\lceil\frac{N}{2}\rceil}(\Gamma)=\sqrt{\lambda^2+\mu^2}\ \text{and} \
\lambda_{\lceil\frac{N}{2}\rceil}(\Gamma')=\sqrt{(\lambda^2+1)\mu^2}.
$$
Combining these (in)equalities, the results follow.
\rule{2mm}{2mm}
Now, we generalize the signed Cartesian product and signed semi-strong product
of two signed graphs to the product of $n$ signed graphs.
\begin{Def}\label{Def-product-n-graphs}
For $i=1,2,\dots,n-1$, let $\Gamma_i$ be a signed bipartite graph
and $\Gamma_n$ be a signed graph. Let $\Gamma_{\widetilde{\Box},R}^1=\Gamma_{\widetilde{\bowtie},R}^1=\Gamma_1$
and $\Gamma_{\widetilde{\Box},L}^1=\Gamma_{\widetilde{\bowtie},L}^1=\Gamma_n$. For $2\leq k\leq n$,
we define
(i) $\Gamma_{\widetilde{\Box},R}^k=\Gamma_{\widetilde{\Box},R}^{k-1}\widetilde{\Box} \Gamma_k$ and
$\Gamma_{\widetilde{\bowtie},R}^k=\Gamma^{k-1}_{\widetilde{\bowtie},R}\widetilde{\bowtie} \Gamma_k$;
(ii) $\Gamma_{\widetilde{\Box},L}^k=\Gamma_{n-k+1}\widetilde{\Box}\Gamma_{\widetilde{\Box},L}^{k-1}$ and
$\Gamma_{\widetilde{\bowtie},L}^k=\Gamma_{n-k+1}\widetilde{\bowtie}\Gamma_{\widetilde{\bowtie},L}^{k-1}$.
\end{Def}
To illustrate Definition \ref{Def-product-n-graphs}, one can consider $n=3$, that is
\begin{align*}
&\Gamma_{\widetilde{\Box},R}^3= ((\Gamma_1\widetilde{\Box}\Gamma_2)\widetilde{\Box}\Gamma_3) \text{\ and\ }
\Gamma_{\widetilde{\bowtie},R}^3= ((\Gamma_1\widetilde{\bowtie}\Gamma_2)\widetilde{\bowtie}\Gamma_3), \\
& \Gamma_{\widetilde{\Box},L}^3= (\Gamma_1\widetilde{\Box}(\Gamma_2\widetilde{\Box}\Gamma_3)) \text{\ and\ }
\Gamma_{\widetilde{\bowtie},L}^3= (\Gamma_1\widetilde{\bowtie}(\Gamma_2\widetilde{\bowtie}\Gamma_3)).
\end{align*}
By Lemma \ref{lem-HaIK11} and Lemma \ref{lem-GaRW76}, the Cartesian product and semi-strong product
of two bipartite graphs are still bipartite. Therefore, Definition \ref{Def-product-n-graphs} (i)
is well-defined. Since the Kronecker product of matrices is an associative operation, the underling graphs of
$\Gamma_{\widetilde{\Box},R}^n$ and $\Gamma_{\widetilde{\Box},L}^n$ are isomorphic.
However, by Lemma \ref{lem-GaRW76} (iii) and Corollary \ref{cor-GaRW76},
the underling graphs of $\Gamma_{\widetilde{\bowtie},R}^n$ and $\Gamma_{\widetilde{\bowtie},L}^n$ are not isomorphic.
By Definition \ref{Def-product-n-graphs}, Theorem \ref{thm-eigenvalues} (i) can be easily generalized to the following theorem.
\begin{thm}\label{thm-eigenvalues-n-graph}
For $i=1,2,\dots,n$, let $\Gamma_i=(G_i,\sigma_i)$ be a signed graph
and $\theta_i^2$ be an eigenvalue of $A(\Gamma_i)^2$ with multiplicity $p_i$,
where $\Gamma_1,\dots,\Gamma_{n-1}$ are bipartite. Then
$\sum_{i=1}^n\theta_i^2$, $\sum_{i=1}^n\theta_i^2$,
$\theta_n^2\prod_{i=1}^{n-1}(\theta_i^2+1)$
and $\sum_{k=1}^n\prod_{i=k}^{n}\theta_i^2$ are eigenvalues of
$\Gamma_{\widetilde{\Box},L}^n$, $\Gamma_{\widetilde{\Box},R}^n$,
$\Gamma_{\widetilde{\bowtie},L}^n$ and $\Gamma_{\widetilde{\bowtie},R}^n$
with multiplicity $p_1p_2\cdots p_n$, respectively.
\end{thm}
By Theorem \ref{thm-eigenvalues-n-graph}, we have the following corollary immediately.
\begin{cor}\label{cor-eig-G1Gn}
For $i=1,2,\dots,n$, let $\Gamma_i$ be a signed graph
with exactly two distinct eigenvalues $\pm \theta_i$, where $\Gamma_1,\dots,\Gamma_{n-1}$ are bipartite.
Then $\Gamma_{\widetilde{\Box},L}^n$, $\Gamma_{\widetilde{\Box},R}^n$, $\Gamma_{\widetilde{\bowtie},L}^n$ and $\Gamma_{\widetilde{\bowtie},R}^n$
have exactly two distinct eigenvalues $\pm \sqrt{\sum_{i=1}^n\theta_i^2}$, $\pm \sqrt{\sum_{i=1}^n\theta_i^2}$,
$\pm \sqrt{\theta_n^2\prod_{i=1}^{n-1}(\theta_i^2+1)}$
and $\pm \sqrt{\sum_{k=1}^n\prod_{i=k}^{n}\theta_i^2}$, respectively.
\end{cor}
\begin{exa}\label{example-Qn}
{\rm
Let $\Gamma_i=(K_2, +)$ for each $i=1,2,\dots,n$. Then
(i) each of $\Gamma_{\widetilde{\Box},R}^n$, $\Gamma_{\widetilde{\Box},L}^n$ and $\Gamma_{\widetilde{\bowtie},R}^n$
is a signed graph of $Q_n$ whose eigenvalues are $\pm \sqrt{n}$;
(ii) $\Gamma_{\widetilde{\bowtie},L}^n$ is a signed graph of $K_{2^{n-1},2^{n-1}}$
and its eigenvalues are $\pm \sqrt{2^{n-1}}$.
}
\end{exa}
\begin{figure}
\caption{{\small The signed graph $\Gamma_{\widetilde{\Box}
\label{fig2}
\end{figure}
\begin{exa}\label{example-Q2n}
{\rm
For each $i=1,2,\dots,n$, let $\Gamma_i=(K_{2,2},\sigma)$ be
the signed graph of $K_{2,2}$ with exactly one negative edge. Then
(i) $\Gamma_{\widetilde{\Box},R}^n$ and $\Gamma_{\widetilde{\Box},L}^n$
are signed graphs of $Q_{2n}$ whose eigenvalues are
$\pm \sqrt{2n}$;
(ii) the eigenvalues of $\Gamma_{\widetilde{\bowtie},L}^n$
are $\pm \sqrt{2\cdot3^{n-1}}$;
(iii) the eigenvalues of $\Gamma_{\widetilde{\bowtie},R}^n$
are $\pm \sqrt{\sum_{k=1}^n2^k}= \pm \sqrt{2^{n+1}-2}$.
}
\end{exa}
By Definition \ref{Def-product-n-graphs}, Theorem \ref{thm-symmetric} can be generalized to Theorem \ref{thm-sym-n}.
\begin{thm}\label{thm-sym-n}
For $n\geq 2$ and $i=1,2,\dots,n-1$, let $\Gamma_i$ be a signed bipartite graph
and $\Gamma_n$ be a signed graph.
(i) The spectrum of $\Gamma_{\widetilde{\bowtie},R}^n$ is symmetric if and only
if $\Gamma_{n-1}$ is balanced or the spectrum of $\Gamma_{n}$ is symmetric.
(ii) The spectrum of every graph in $\{\Gamma_{\widetilde{\Box},R}^n, \Gamma_{\widetilde{\Box},L}^n,
\Gamma_{\widetilde{\bowtie},L}^n \}$ is symmetric if and only if
there exists an integer $i\in\{1,\dots,n-1\}$ such that $\Gamma_i$ is balanced
or the spectrum of $\Gamma_{n}$ is symmetric.
\end{thm}
\begin{pf}
We only need to consider $n\geq 3$.
(i) By Theorem \ref{thm-symmetric}, the spectrum of $\Gamma_{\widetilde{\bowtie},R}^n
=\Gamma_{\widetilde{\bowtie},R}^{n-1}\widetilde{\bowtie}\Gamma_{n}$
is symmetric if and only if $\Gamma_{\widetilde{\bowtie},R}^{n-1}$ is balanced or
the spectrum of $\Gamma_{n}$ is symmetric.
Since the semi-strong product of a graph $G$ and a bipartite graph $H$ is
balanced if and only if $H$ is balanced, we have
$\Gamma_{\widetilde{\bowtie},R}^{n-1}
=\Gamma_{\widetilde{\bowtie},R}^{n-2}\widetilde{\bowtie} \Gamma_{n-1}$
is balanced if and only if $\Gamma_{n-1}$ is balanced. Thus, (i) is proved.
(ii) By Theorem \ref{thm-symmetric}, the spectrum of $\Gamma_{\widetilde{\Box},R}^n=
\Gamma_{\widetilde{\Box},R}^{n-1}\widetilde{\Box} \Gamma_n$ is symmetric if and only if
$\Gamma_{\widetilde{\Box},R}^{n-1}$ is balanced or the spectrum of $\Gamma_{n}$ is symmetric.
Since $\Gamma_{\widetilde{\Box},R}^{n-1}$ is balanced if and only if
there exists an integer $i\in\{1,\dots,n-1\}$ such that $\Gamma_i$ is balanced.
So the conclusion for $\Gamma_{\widetilde{\Box},R}^n$ is proved.
By Theorem \ref{thm-symmetric}, the conclusion holds for
$\Gamma_{\widetilde{\Box},L}^2=\Gamma_{n-1}\widetilde{\Box}\Gamma_n$
(resp. $\Gamma_{\widetilde{\bowtie},L}^2=\Gamma_{n-1}\widetilde{\bowtie}\Gamma_n$). By induction on $n$,
assume that the spectrum of $\Gamma_{\widetilde{\Box},L}^{n-1}$
(resp. $\Gamma_{\widetilde{\bowtie},L}^{n-1}$) is symmetric if and only if
there exists an integer $i\in\{2,\dots,n-1\}$ such that $\Gamma_i$ is balanced
or the spectrum of $\Gamma_{n}$ is symmetric.
By Theorem \ref{thm-symmetric}, the spectrum of $\Gamma_{\widetilde{\Box},L}^n=
\Gamma_1\widetilde{\Box}\Gamma_{\widetilde{\Box},L}^{n-1}$
(resp. $\Gamma_{\widetilde{\bowtie},L}^n=
\Gamma_1\widetilde{\bowtie}\Gamma_{\widetilde{\bowtie},L}^{n-1}$) is symmetric if and only if
$\Gamma_1$ is balanced or the spectrum of $\Gamma_{\widetilde{\Box},L}^{n-1}$
(resp. $\Gamma_{\widetilde{\bowtie},L}^{n-1}$) is symmetric.
By induction, the conclusion for $\Gamma_{\widetilde{\Box},L}^n$
(resp. $\Gamma_{\widetilde{\bowtie},L}^n$) is proved.
\end{pf}
Now, Theorem \ref{thm-Delta-2-min} is generalized to the following theorem.
\begin{thm}\label{thm-Delta-n-graph}
For $i=1,2,\dots,n$, let $\Gamma_i=(G_i,\sigma_i)$ be a signed graph of order $N_i$
and $\theta_i^2$ be the minimum eigenvalue of $A(\Gamma_i)^2$, where $G_1,\dots,G_{n-1}$ are bipartite.
Let $H_{\Box}$, $H_{\bowtie,L}$ and $H_{\bowtie,R}$ be any $(\lfloor\frac{1}{2}\prod_{i=1}^nN_i\rfloor+1)$-vertex induced
subgraph of $\Gamma_{\widetilde{\Box},L}^n$, $\Gamma_{\widetilde{\bowtie},L}^n$ and $\Gamma_{\widetilde{\bowtie},R}^n$, respectively.
(i) If there exists an integer $i\in\{1,2,\dots,n-1\}$ such that $\Gamma_i$ is balanced
or the spectrum of $\Gamma_n$ is symmetric, then
$\Delta(H_{\Box})\geq \sqrt{\sum_{i=1}^n\theta_i^2}$ and
$\Delta(H_{\bowtie,L})\geq \sqrt{\theta_n^2\prod_{i=1}^{n-1}(\theta_i^2+1)}$;
(ii) If $\Gamma_{n-1}$ is balanced or the spectrum of $\Gamma_n$ is symmetric, then
$\Delta(H_{\bowtie,R})\geq \sqrt{\sum_{k=1}^n\prod_{i=k}^{n}\theta_i^2}$.
\end{thm}
\begin{pf}
For simplicity, let $N=\prod_{i=1}^nN_i$.
Since $H_{\Box}$, $H_{\bowtie,L}$ and $H_{\bowtie,R}$ are $(\lfloor\frac{N}{2}\rfloor+1)$-vertex induced subgraphs of
$\Gamma_{\widetilde{\Box},L}^n$, $\Gamma_{\widetilde{\bowtie},L}^n$ and
$\Gamma_{\widetilde{\bowtie},R}^n$, respectively.
By Lemma \ref{lem-Delta-lambda},
$$
\Delta(H_{\Box})\geq \lambda_{\lceil\frac{1}{2}N\rceil}(\Gamma_{\widetilde{\Box},L}^n),
\ \Delta(H_{\bowtie,L})\geq \lambda_{\lceil\frac{1}{2}N\rceil}(\Gamma_{\widetilde{\bowtie},L}^n)\ \text{and} \
\Delta(H_{\bowtie,R})\geq \lambda_{\lceil\frac{1}{2}N\rceil}(\Gamma_{\widetilde{\bowtie},R}^n).
$$
By Theorem \ref{thm-eigenvalues-n-graph}, the minimum eigenvalues of
$A(\Gamma_{\widetilde{\Box},L}^n)^2$, $A(\Gamma_{\widetilde{\bowtie},L}^n)^2$
and $A(\Gamma_{\widetilde{\bowtie},R}^n)^2$ are obtained. Thus, by
Theorem \ref{thm-sym-n}, the spectrums of $\Gamma_{\widetilde{\Box},L}^n$,
$\Gamma_{\widetilde{\bowtie},L}^n$ and $\Gamma_{\widetilde{\bowtie},R}^n$ are symmetric and so
$\lambda_{\lceil\frac{1}{2}N\rceil}(\Gamma_{\widetilde{\Box},L}^n)$ $= \sqrt{\sum\nolimits_{i=1}^n\theta_i^2}$,
$\lambda_{\lceil\frac{1}{2}N\rceil}(\Gamma_{\widetilde{\bowtie},L}^n)=\sqrt{\theta_n^2\prod\nolimits_{i=1}^{n-1}(\theta_i^2+1)}$ and
$\lambda_{\lceil\frac{1}{2}N\rceil}(\Gamma_{\widetilde{\bowtie},R}^n)=\sqrt{\sum\nolimits_{k=1}^n\prod\nolimits_{i=k}^{n}\theta_i^2}.$
Combining these (in)equalities, the results follow.
\end{pf}
\begin{cor}\label{cor-DeltaH-n}
For $i=1,2,\dots,n$, let $\Gamma_i=(G_i,\sigma_i)$ be a signed graph of order $N_i$
with exactly two distinct eigenvalues $\pm\theta_i$, where $G_1,\dots,G_{n-1}$ are bipartite.
Let $H_{\Box}$, $H_{\bowtie,L}$ and $H_{\bowtie,R}$ be any $(\lfloor\frac{1}{2}\prod_{i=1}^nN_i\rfloor+1)$-vertex induced
subgraph of $\Gamma_{\widetilde{\Box},L}^n$, $\Gamma_{\widetilde{\bowtie},L}^n$ and $\Gamma_{\widetilde{\bowtie},R}^n$, respectively.
Then
$\Delta(H_{\Box})\geq \sqrt{\sum_{i=1}^n\theta_i^2}$,
$\Delta(H_{\bowtie,L})\geq \sqrt{\theta_n^2\prod_{i=1}^{n-1}(\theta_i^2+1)}$ and
$\Delta(H_{\bowtie,R})\geq \sqrt{\sum_{k=1}^n\prod_{i=k}^{n}\theta_i^2}$.
\end{cor}
When $\Gamma_i=(K_2,+)$ for each $i=1,2,\dots,n$ in Corollary \ref{cor-DeltaH-n},
$\Gamma_{\widetilde{\Box},L}^n$ is the signed graph of hypercube $Q_n$.
Therefore, Corollary \ref{cor-DeltaH-n} implies Huang's theorem.
\begin{exa}{\rm
For $i=1,2,\dots,n$, let $\Gamma_i=(K_{2^t,2^t},\sigma)$ be the signed graph $K_{2^t,2^t}$
with exactly two distinct eigenvalues $\pm \sqrt{2^t}$.
For any integer $n\geq 1$ and $t\geq 0$, let $H_{\Box}$, $H_{\bowtie,L}$ and $H_{\bowtie,R}$
be any $(2^{n(t+1)-1}+1)$-vertex induced subgraph of $\Gamma_{\widetilde{\Box},L}^n$,
$\Gamma_{\widetilde{\bowtie},L}^n$ and $\Gamma_{\widetilde{\bowtie},R}^n$ respectively.
Then
$\Delta(H_{\Box})\geq \sqrt{2^t \cdot n}$,
$\Delta(H_{\bowtie,L})\geq \sqrt{2^t(2^t+1)^{n-1}}$ and
$\Delta(H_{\bowtie,R})\geq \sqrt{\sum_{k=1}^n2^{kt}}$.
}
\end{exa}
\section{Concluding remarks}
{\bf I.} Corollary \ref{cor-kron-eig}, Corollary \ref{cor-eig-G1G2} and Corollary \ref{cor-eig-G1Gn}
provide product methods to construct signed graphs with exactly
two distinct eigenvalues of opposite signatures from factor graph $\Gamma_1$ and $\Gamma_2$.
There are many options for the factor graph, such as
the signed graphs of $Q_n$ and $K_{2^n,2^n}$ in Example \ref{example-Qn},
$T_{2n}$ in Lemma \ref{lem-T2n},
$S_{14}$ in Lemma \ref{lem-S14},
the signed graph of $K_n$ in Example \ref{exa-Kn},
signed graphs in Examples \ref{example-complete-k-part-graph},
\ref{example-Hn-AGamma}, \ref{example-Q2n} and so on.
{\bf II.} If the following conjecture is true, it would provide a way to construct an infinite
family of $d$-regular Ramanujan graphs by $2$-lift of graphs.
\begin{con}(Bilu-Linial \cite{BiLi06})\label{con-BiLi06}
Every connected $d$-regular graph $G$ has a signature $\sigma$ such that
$\rho(G,\sigma)\leq 2\sqrt{d-1}$.
\end{con}
Gregory considered the following Conjecture \ref{con-Greg12} without the regularity assumption on $G$.
\begin{con}(Gregory \cite{Greg12})\label{con-Greg12}
If $G$ is a nontrivial graph with maximum degree $\Delta>1$, then there exists
a signed graph $\Gamma=(G,\sigma)$ such that $\rho(\Gamma)\leq 2\sqrt{\Delta-1}$.
\end{con}
By Theorem \ref{thm-eigenvalues} (i), we have the following theorem.
\begin{thm}\label{thm-Ramanujan}
For $i=1,2$, let $G_i$ be a graph with maximum degree $\Delta_i$ and
$\Gamma_i=(G_i,\sigma_i)$ be a signed graph such that
$\rho(\Gamma_i)\leq 2\sqrt{\Delta_i-1}$. If $G_1$ is bipartite, then
$$\rho(\Gamma_1\widetilde{\Box} \Gamma_2)\leq 2\sqrt{\Delta_1+\Delta_2-2}.$$
\end{thm}
Since $\rho(\Gamma_1\widetilde{\Box} \Gamma_2)=\sqrt{\rho(\Gamma_1)^2+\rho(\Gamma_2)^2}$
and $\Delta(\Gamma_1\Box \Gamma_2)=\Delta(\Gamma_1)+\Delta(\Gamma_2)$,
Theorem \ref{thm-Ramanujan} shows that if Conjecture \ref{con-Greg12} holds for $\Gamma_1$ and $\Gamma_2$,
then Conjecture \ref{con-Greg12} also holds for the signed Cartesian product of them.
{\bf III.} The method which is utilized to construct a larger weighing matrix can
construct a larger signed graph with exactly two distinct eigenvalues $\pm\theta$ from small graphs.
Conversely, the ideas of signed Cartesian product and semi-strong product in our paper
can also be applied to construct a weighing matrix.
If for $i=1,2$, $W_i$ is a weighing matrix of order $n_i$ and weight $k_i$, then we can construct
weighing matrices as follows
\begin{equation*}
W(4n_1n_2, k_1+k_2) = \left[
\begin{array}{cc}
O_{n_1} & W_1 \\
W_1^T & O_{n_1}
\end{array}
\right] \otimes I_{2n_2} +
\left[
\begin{array}{cc}
I_{n_1} & O_{n_1} \\
O_{n_1} & -I_{n_1}
\end{array}
\right] \otimes \left[
\begin{array}{cc}
O_{n_2} & W_2 \\
W_2^T & O_{n_2}
\end{array}
\right],
\end{equation*}
\begin{equation*}
W(4n_1n_2, (k_1+1)k_2) = \left[
\begin{array}{cc}
I_{n_1} & W_1 \\
W_1^T & -I_{n_1}
\end{array}
\right] \otimes \left[
\begin{array}{cc}
O_{n_2} & W_2 \\
W_2^T & O_{n_2}
\end{array}
\right],
\end{equation*}
\begin{equation*}
W(2n_1n_2, (k_1+1)k_2) = \left[
\begin{array}{cc}
I_{n_1} & W_1 \\
W_1^T & -I_{n_1}
\end{array}
\right] \otimes W_2 .
\end{equation*}
Furthermore, if $W_2$ is symmetric, then we can construct weighing matrix
\begin{equation*}
W(2n_1n_2, k_1+k_2) = \left[
\begin{array}{cc}
I_{n_1}\otimes W_2 & W_1\otimes I_{n_2} \\
W_1^T\otimes I_{n_2} & -I_{n_1}\otimes W_2
\end{array}
\right].
\end{equation*}
}
\end{document} | math |
From WRAY, pressed down pleated front pant made with garment washed cotton and a metal button. Pant can be worn cuffed. Partial elastic band waist in back. Side pockets, zip down front. High on the waist. | english |
खरमास समाप्त, अब शुरू होगी सहालग, यहां देखें शुभ मुहूर्त पूर्वांचल२४
होम धर्म-कर्म खरमास समाप्त, अब शुरू होगी सहालग, यहां देखें शुभ मुहूर्त
अप्रैल १४, २०१९ कम्मंट ऑफ ऑन खरमास समाप्त, अब शुरू होगी सहालग, यहां देखें शुभ मुहूर्त २१५ वियूज
बलिया। एक माह से जारी खरमास के बाद १४-१५ अप्रैल से अच्छे दिनों की शुरुआत होने जा रही है। इसी के साथ विवाह व अन्य मांगलिक कार्यों के लिए सहालग की शुरुआत हो जाएगी। बैंड बाजा के साथ बारातों की धूम मचेगी। अप्रैल के दूसरे पखवाड़े से शुरू हो रही सहालग का दौर जुलाई में देवताओं केसोने से पहले तक जारी रहेगा। इस दौरान ५६ सहालग मिलेंगी।
गत १५ मार्च की सुबह से मीन राशि में सूर्य के जाने के बाद से खरमास चल रहा है। सारे शुभ काम रुके रहे। गोमतीनगर निवासी आचार्य प्रदीप ने बताया कि १४ अप्रैल की शाम ४:0४ बजे से सूर्य मेष राशि में आयेंगे और खरमास की समाप्ति होगी। इसी केसाथ अच्छे दिन शुरू हो जायेंगे।
अगले दिन १५ अप्रैल से विवाह आदि के मुहूर्त मिलेंगे। नव संवत्सर में खरमास समाप्त होने के बाद १५ अप्रैल से ही शादी विवाह के मुहूर्त मिलेंगे। उन्होंने बताया कि चूंकि इस बार शुक्र और गुरु अस्त का संयोग उन समय अवधि में ही हो रहा है, जिस समय या तो देव सो रहे होेंगे या खरमास चल रहा होगा। ऐसे में विवाह व मांगलिक कार्यों के लिये काफी लंबे समय बाद पूरे नव हिंदू संवत्सर में ११२ शादी विवाह के मुहूर्त मिल रहे हैं।
देवताओं के सोने तक शादी विवाह के लिए मिलेंगे इतने मुहूर्त
अप्रैल में १५, १६, १७, १८, १९, २०, २१, २२, २३, २४ और २६ अप्रैल।
मई में- १, २, ६, ७, ८, १२, १4, १5, १७, १८, १9, २0, २१, २3, २4, २८, २9 और ३० मई।
जून में ४, ८, ९, १०, ११, १२, १३, 1४, १५, १६, १७, 1८, 1९, २०, 2४, २५, २६, २७ और ३० जून।
जुलाई में १, ५, ६, ७, ८, ९, १0 और ११ जुलाई।
इसके बाद १२ जुलाई से देवताओं के शयन के चलते विवाह व मांगलिक कार्य नहीं होंगे।
८ नवंबर को देवताओं के जगने के चलते फिर से विवाह के मुहूर्त शुरू होंगे।
आचार संहिता के दौरान बैंड-बाजा के लिए अनुमति जरूरी
आम चुनावों के चलते आदर्श चुनाव आचार संहिता चल रही है। ऐसे में १५ अप्रैल से जब लग्न शुरू होंगी तो विवाह आयोजन चुनावी नियमों के साये में होंगे। आदर्श आचार संहिता के तहत सिर्फ अप्रैल-मई में ही लगभग ३० विवाह के मुहूर्त रहे हैं।
प्रेवियस तीन बेटियां जनने पर मां को मिली सजा-ए-मौत
नेक्स्ट छपरा से दिल्ली तक आज शुरू होगी समर स्पेशल ट्रेन | hindi |
\begin{document}
\title[Decidability Results for the Boundedness Problem]{Decidability Results for the Boundedness Problem}
\author[A.~Blumensath]{Achim Blumensath\rsuper a}
\address{{\lsuper{a,b}}Fachbereich Mathematik, Technische Universit\"at Darmstadt}
\email{\{blumensath,otto\}@mathematik.tu-darmstadt.de}
\author[M.~Otto]{Martin Otto\rsuper b}
\address{
}
\author[M.Weier]{Mark Weyer\rsuper c}
\address{{\lsuper c}
}
\email{mark@weyer-zuhause.de}
\begin{abstract}
We prove decidability of the boundedness problem for
monadic least fixed-point recursion based on
positive monadic second-order ($\MSO$) formulae over trees.
Given an $\MSO$-formula $\varphi(X,x)$ that is positive in~$X$, it is decidable
whether the fixed-point recursion based on~$\varphi$ is spurious
over the class of all trees in the sense that there is some uniform
finite bound for the number of iterations $\varphi$~takes
to reach its least fixed point, uniformly across all trees.
We also identify the exact complexity of this problem.
The proof uses automata-theoretic techniques.
This key result extends, by means of model-theoretic interpretations,
to show decidability of the boundedness problem for $\MSO$
and guarded second-order logic ($\GSO$) over the classes of structures
of fixed finite tree-width. Further model-theoretic transfer arguments
allow us to derive major known decidability results for boundedness
for fragments of first-order logic as well as new ones.
\end{abstract}
\maketitle
\section{Introduction}
\label{sect:intro}
In applications one frequently employs tailor-made logics
to achieve a balance between expressive power and algorithmic manageability.
Adding fixed-point operators to weak logics turned out to be a good way
to achieve such a balance.
Think, for example of the addition of transitive closure operators
or more general fixed-point constructs to database
query languages, or of various fixed-point defined reachability
or recurrence assertions to logics used in verification, like
linear or branching time temporal logics or the modal $\mu$-calculus.
Fixed-point operators introduce a measure of relational recursion and
typically boost expressiveness in the direction of more dynamic and
less local properties. They offer relational recursion based on the
iteration of relation updates that are definable in the underlying logic.
We here primarily consider monadic least fixed points, based on formulae
$\varphi(X,x)$ that are monotone (positive) in the
monadic recursion variable~$X$.
On a fixed structure~$\fA$,
any such~$\varphi$ induces a monotone operation
$F_\varphi : P \mapsto \set{ a \in \fA }{ \fA \models \varphi(P,a)}$
on monadic relations $P \subseteq A$.
The least fixed point of this operation over~$\fA$,
denoted as $\varphi^\infty(\fA)$, is also the first stationary point
of the monotone, ordinal-indexed iteration sequence of stages $\varphi^\alpha(\fA)$ starting from
$\varphi^0(\fA) := \emptyset$, with updates
$\varphi^{\alpha+1}(\fA) := F_\varphi( \varphi^{\alpha}(\fA))$
and unions in limits.
The least~$\alpha$ for which
$\varphi^{\alpha+1}(\fA) = \varphi^\alpha(\fA)$ is called the closure
ordinal for this fixed-point iteration on~$\fA$.
For a concrete fixed-point process it may be hard to tell
whether the recursion employed is crucial or whether it is
spurious and can be eliminated.
Indeed this question comes in two versions\?:
(a) one can ask whether a resulting fixed point is also
uniformly definable in the base logic without fixed-point recursion
(purely an expressiveness issue)\?;
(b) one may also be interested to know whether the given fixed-point
iteration terminates within a uniformly bounded finite number
of steps (an algorithmic issue, concerning the dynamics of the
fixed-point recursion rather than its result).
The boundedness problem $\BDD(L,\cC)$
for a class of formulae~$L$
and a class of structures~$\cC$
concerns question~(b)\?: to decide, for a given formula $\varphi \in L$,
whether there is a finite upper bound on its closure
ordinal, uniformly across all structures $\fA \in \cC$.
We call such fixed-point iterations, or~$\varphi$ itself,
\emph{bounded over~$\cC$.}
Interestingly, for first-order logic, as well as for many natural
fragments, the two questions concerning eliminability of least fixed
points coincide -- at least over the class of all structures.
By a classical theorem of
Barwise and Moschovakis~\cite{BarwiseMoschovakis78},
the only way that the fixed point $\varphi^\infty(\fA)$ can be
first-order definable for every~$\fA$, is that there is some finite~$\alpha$
for which $\varphi^\infty(\fA) = \varphi^\alpha(\fA)$
for all~$\fA$. The converse is clear from the fact that the unfolding
of the iteration to any fixed finite depth~$\alpha$ is easily mimicked
in $\FO$.
In other cases -- and even for $\FO$ over other, restricted classes of
structures, e.g., in finite model theory -- the two problems can
indeed be distinct, and of quite independent interest.
We here deal with the boundedness issue.
Boundedness (even classically, over the class of all structures, and
for just monadic fixed points as considered above) is
undecidable for most first-order fragments of interest (see, e.g.,~\cite{HillebrandEtAl95}).
Notable exceptions are monadic boundedness for positive existential
formulae (\textsc{Datalog})~\cite{CosmadakisGaKaVa88},
for modal formulae~\cite{Otto99}, and for
(a restricted class of) universal formulae without equality~\cite{Otto06}.
One common feature of these decidable cases of the boundedness problem
is that the fragments concerned have a kind of tree-model property
(not just for satisfiability in the fragment itself, but also for the
fixed points and for boundedness).
This is obvious for the modal fragment~\cite{Otto99}, but clearly
also true for positive existential $\FO$ (derivation trees for monadic
\textsc{Datalog} programs can be turned into models of bounded tree-width),
and similarly also for the restricted universal fragment in~\cite{Otto06}.
Motivated by this observation,
\cite{KOS}~has made a first significant step
in an attempt to analyse the boundedness problem from the opposite perspective,
varying the class of structures rather than the class of formulae. The hope
is that this approach could go beyond an ad-hoc exposition of the
decidability of the boundedness problem for individual syntactic fragments,
and offer a unified model-theoretic explanation instead.
\cite{KOS}~shows that boundedness is decidable for
\emph{all} monadic fixed points in $\FO$ over the class of all acyclic
relational structures.
Technically \cite{KOS}~expands on modal
and locality-based proof ideas and reductions to
the monadic second-order theory of trees from~\cite{Otto99,Otto06}
that also rest on the availability of a Barwise--Moschovakis
equivalence. These techniques do not seem to extend to either
the class of all trees (where Barwise--Moschovakis fails) or to
bounded tree-width (where certain simple locality criteria fail).
The present investigation offers another step forward in the alternative
approach to the boundedness problem, on a methodologically very
different note. Its most important novel feature may be that it deals
with a setting where neither locality nor
Barwise--Moschovakis are available. On the one hand, the class of formulae
considered is extended from first-order logic $\FO$
to full monadic second-order logic $\MSO$ -- a leap which greatly increases
the robustness of the results w.r.t.\ interpretations, and hence
their model-theoretic impact. On the other hand, automata are crucially used
and the underlying structures are restricted to trees.
Using $\MSO$-interpretations it follows that
the boundedness problem for $\MSO$ is decidable
over any $\MSO$-definable class of bounded tree-width, and similarly
even for guarded second-order logic $\GSO$ instead of $\MSO$.
These ramifications demonstrate the strength and unifying explanatory
power of our main decidability result in the wider context of the
boundedness issue.
One of our strongest concrete decidability results
concerns the boundedness problem for $\GSO$ over
$\GSO$-definable classes of bounded tree-width, cf.~Corollary~\ref{cor:decidability of BDD(GF), etc}.
This, in its turn,
encompasses all the major, previously known decidability results
for natural fragments of $\FO$ and, furthermore, settles decidability
of boundedness for the guarded fragment $\GF$.
Equally importantly it goes a long way
to explain the perceived dichotomy between the many undecidability
results, which may typically be understood in terms of reductions from
the tiling problem over suitably grid-like structures, and the
comparatively rare cases of decidability, which can now be systematically
linked to some generalised tree-model property.
Among the classical and previously known decidability results
for the boundedness of (systems of) monadic least fixed points,
which can be integrated into this new picture, are those for
\begin{itemize}
\item[--] monadic \textsc{Datalog,} or systems of monadic least fixed points
for the purely existential--positive fragment of first-order logic,
\cite{CosmadakisGaKaVa88}\?;
\item[--] dually, (systems of) monadic least fixed points
in the purely universal-negative fragment of first-order logic
(which may equivalently be phrased in terms of the
boundedness for greatest fixed points for \textsc{Datalog} or for
existential--positive first-order), \cite{Otto06}\?;
\item[--] modal logic, \cite{Otto99}\?;
\item[--] monadic least fixed points for unconstrained $\FO$
in restriction to the class of all acyclic relational structures, \cite{KOS}.
\end{itemize}
\noindent Our decidability results are based on a reduction of the monadic
boundedness problem to the \emph{limitedness problem} for
\emph{weighted parity automata,} whose decidability is due to
Colcombet and L\"oding \cite{ColcombetLoeding08}
(cf.~Theorems
\ref{thm:finite limitedness decidable}~and~\ref{thm:limitedness decidable}
below).
This reduction introduces a rather sophisticated annotation
(of ternary tree structures) that records dependencies between the stages
of a fixed-point iteration over these tree structures\?; it is
established that, subject to a limitedness condition on a related
cost function, these annotations can serve as certificates for boundedness.
The overall structure of the paper is as follows.
We divide the material into two major parts\?:
the first part, comprising Sections~\ref{sect:start I}--\ref{sect:end I},
is devoted to the development of the
new techniques and leads up to the core technical result\?:
the decidability of the boundedness problem for $\MSO$
on the class of all ternary trees
through reduction to the limitedness problem for a certain class of automata.
The ramifications of this result are investigated
in the second half of the paper.
Sections~\ref{sect:start II}--\ref{sect:end II} develop
transfer and reduction arguments that allow us to make links with previously
known decidability results and to derive several new concrete
decidability results.
Section~\ref{sect:complexity}, finally, discusses complexity issues.
\section{Preliminaries}
\label{sect:prelims}
We assume some familiarity with basic concepts of logic
as can be found, e.g., in \cite{EbbinghausFlum95}.
Throughout the paper we assume that all vocabularies are finite
and that they contain only relation symbols and constant symbols,
but no function symbols.
Consider a second-order formula $\varphi(X,\bar x)$
with free variables as indicated in an underlying vocabulary~$\tau$.
Suppose that $\varphi(X,\bar x)$ is positive in the $r$-ary
second-order variable~$X$ and $\bar x = (x_1,\dots,x_r)$ is a matching
tuple of free first-order variables.
Any $X$-positive formula of this format induces, over every
$\tau$-structure~$\fA$, an operation on the power set of $A^r$\?:
\begin{align*}
P \mapsto \varphi(\fA,P) := \set{ \bar{a} \in A^r }{ (\fA,P,\bar{a}) \models \varphi }\,.
\end{align*}
As $\varphi$~is $X$-positive, this operation is monotone
($P \subseteq P'$ implies $\varphi(\fA,P) \subseteq \varphi(\fA,P')$)
and hence possesses a unique least fixed point,
which we denote as $\varphi^\infty(\fA)$. This least fixed point
is obtained as the limit of the monotone sequence of inductive stages
$\varphi^\alpha(\fA)$
induced by~$\varphi$ on~$\fA$. These stages are defined by transfinite
induction, for all ordinals $\alpha$, according to\?:
\begin{align*}
\varphi^0(\fA) &:= \emptyset\,, \\
\varphi^{\alpha+1}(\fA) &:= \varphi(\fA,\varphi^\alpha(\fA))\,, \\
\varphi^\delta(\fA) &:= \bigcup_{\alpha < \delta} \varphi^\alpha(\fA)
\quad\text{for limits } \delta\,.
\end{align*}
The \emph{finite stages} $\varphi^\alpha(\fA)$, for $\alpha < \omega$,
are uniformly definable by formulae, which we also denote by~$\varphi^\alpha$,
obtained from $\varphi(X,\bar x)$ by iterated substitution
of~$\varphi$ for~$X$ in~$\varphi$.
Letting $\varphi[\psi(\bar x)/X]$ stand for the result of
replacing all free occurrences of~$X$ in atoms~$X\bar y$
in~$\varphi$ by~$\psi(\bar y)$, we obtain
formulae~$\varphi^\alpha$ for $\alpha < \omega$, by
\begin{align*}
\varphi^0 := \bot
\quad\text{and}\quad
\varphi^{\alpha+1} := \varphi[\varphi^\alpha(\bar x)/X]\,.
\end{align*}
Clearly, for finite $\alpha$, $\varphi^\alpha \in \MSO$ for $\varphi \in \MSO$,
and similarly for all natural fragments of first- and second-order logic that
are closed under this substitution operation. It is easy to see that
$\varphi^\alpha$~defines the stage $\varphi^\alpha(\fA)$
for finite $\alpha$, uniformly across all~$\fA$. We therefore need not
distinguish between the two readings of $\varphi^\alpha(\fA)$ for
\emph{finite}~$\alpha$. For infinite~$\alpha$, on the other hand,
we do not regard~$\varphi^\alpha$ as a formula (it would in general have to be
a formula in some infinitary extension of the base logic), but only
allow $\varphi^\alpha(\fA)$ as shorthand notation for the corresponding stage
of~$\varphi$ over~$\fA$.
Because of monotonicity,
$\varphi^\infty(\fA) = \bigcup_\alpha \varphi^\alpha(\fA) = \varphi^\gamma(\fA)$
for the least ordinal~$\gamma$ for which
$\varphi^{\gamma+1}(\fA) = \varphi^\gamma(\fA)$.
This ordinal~$\gamma$ is called the \emph{closure ordinal} for~$\varphi$
on~$\fA$, denoted $\cl{\varphi}{\fA}$.
The \emph{stage} of an individual $\bar a \in \varphi^\infty(\fA)$
is the least ordinal~$\alpha$ such that $\bar a \in \varphi^{\alpha+1}(\fA)$\?;
therefore, the closure ordinal could also be described as the least
ordinal greater than the stages of all members of the fixed point $\varphi^\infty(\fA)$.
The closure ordinal can in general only be bounded, for simple
cardinality reasons, by the successor cardinal of the cardinality
of~$\fA$, or by $\card{A}^r$ for finite~$\fA$.
For instance, the fixed-point induction based on $\varphi(X,x) = \forall y (Ryx \to Xy)$
yields as its fixed point over $\fA = (A,R)$ the set of elements
$a \in A$ that are well-founded w.r.t.\ $R$\?;
over the well-ordering $\fA = (\alpha,{<})$, the closure ordinal is~$\alpha$.
In fact, $\varphi^\infty(\fA) = \alpha$\?;
the stage of $\beta \in \alpha$ is~$\beta$.
The fixed-point induction based on~$\varphi$, or for simplicity\?: $\varphi$~itself,
is said to be bounded if, for some finite $\alpha < \omega$,
$\cl{\varphi}{\fA} \leq \alpha$ for all~$\fA$.
Similarly, $\varphi$~is bounded
over the class~$\cC$ if, for some $\alpha < \omega$,
$\cl{\varphi}{\fA} \leq \alpha$ for all $\fA \in \cC$.
\begin{defi}
{\normalfont (a)} Let $\varphi$~be a formula over~$\tau$, positive in~$X$,
and let $\alpha < \omega$.
We say that $\varphi$~is \emph{bounded by~$\alpha$} over a class~$\cC$
if $\varphi^{\alpha}(\fA) = \varphi^{\alpha+1}(\fA)$, for all $\fA \in \cC$.
We call~$\varphi$ \emph{bounded} over~$\cC$
if it is bounded by some $\alpha < \omega$.
{\normalfont (b)} The \emph{boundedness problem} for a logic~$L$
over a class~$\cC$ is the problem to decide,
given a formula $\varphi \in L$, whether $\varphi$~is bounded over~$\cC$.
We denote this decision problem as $\BDD(L,\cC)$.
The \emph{monadic boundedness problem} is
the corresponding problem where we only consider formulae~$\varphi$
with \emph{monadic} variables~$X$.
We denote it as $\BDDm(L,\cC)$.
If $\cC$~is the class of all structures,
we just write $\BDD(L)$ or $\BDDm(L)$.
\end{defi}
A vocabulary~$\tau$ is called a \emph{tree vocabulary,}
if $\tau$~consists of one binary relation symbol~$E$
and, otherwise, only of constant symbols and unary relation symbols.
A $\tau$-structure~$\fT$ is called a \emph{tree structure,}
or \emph{tree} for short,
if $E^\fT$ is a symmetric, acyclic, and connected relation on~$T$.
In particular, tree structures are undirected.
In Part~I of the paper,
we shall exclusively look at $\BDDm(\MSO,\cT)$
for the class of $\MSO$-formulae $\varphi(X,x)$ suitable for
monadic fixed points (positive in the monadic variable~$X$)
over the class~$\cT$ of all tree structures
and some of its subclasses. We refer to this core problem as
the boundedness problem for $\MSO$ over trees for short.
\begin{thm}[Main theorem]
$\BDDm(\MSO,\cT)$, the monadic boundedness problem for $\MSO$
over the class of all tree structures, is decidable.
\end{thm}
In Part~II we employ model-theoretic interpretations and similar transfer arguments
to deduce from this result the decidability of many other boundedness problems.
In particular, we obtain new proofs of many previous decidability results
for boundedness,
as well as some new results like the decidability for the guarded fragment
of first-order logic and for full guarded second-order logic over structures
of bounded tree-width.
\section*{Part I. The main result}
In this first part we prove the main technical result, the decidability
of the monadic boundedness problem for $\MSO$ on the class of all ternary trees.
The ramifications of this result will then be investigated
in the second half of the paper.
To help the reader through the later technicalities,
we start with a simplified outline of the proof idea
towards the main theorem.
The key idea is to derive, for every formula~$\varphi$, a
bound $N = N(\varphi)$ that provides a uniform strict upper bound on the
closure ordinals $\cl{\varphi}{\fT}$ over any tree structure~$\fT$ in case
$\varphi$ \emph{is} bounded.
Then boundedness of $\varphi$ is equivalent to the unsatisfiability
of $\varphi^N \wedge \neg \varphi^{N-1}$
(over the class of all tree structures~$\fT$).
In other words, a formula which (on the class of all trees)
is not bounded by this number~$N$ is not bounded at all.
To reason towards such a uniform bound~$N$,
assume that for some tree~$\fT$,
some node~$v$ enters the fixed point in stage~$N$.
Then $(\fT,\varphi^N(\fT),v) \models \varphi$
but $(\fT,\varphi^{N-1}(\fT),v) \not\models \varphi$.
Using a Feferman--Vaught style lemma (cf.~Proposition~\ref{prop:Feferman-Vaught}),
this change in the status of~$\varphi$ can be traced back to some other node~$w$
such that $(\fT,\varphi^N(\fT),w) \models Xx$
but $(\fT,\varphi^{N-1}(\fT),w) \not\models Xx$,
which means that $w$~entered the fixed point in stage $N-1$.
In this way we obtain a path of dependencies
which travels through the tree
and at places decreases the stage by~$1$.
In a chain of $N$~such jumps,
we conclude that, if $N$~is large
in comparison to the number of types used in the Feferman--Vaught style lemma,
then the path has repetitions and we can use a pumping argument
to produce trees where some node enters the fixed point at arbitrarily large stages.
Consequently, $\varphi$~is unbounded.
The actual proof has to deal with further difficulties,
so it does not exactly follow this outline.
One difficulty is that a pumping lemma essentially requires that
(in some very loose sense) we only use regular properties.
In particular, we have to weaken the counting of stages
and, consequently,
we will slightly relax the concept of a dependency.
Also, it is not sufficient to consider a single dependency path\?:
we have to do the pumping such that it works for all paths simultaneously.
Fortunately, there is already a suitable pumping theorem for a certain kind
of weighted automaton that we can reduce our problem to.
The main part of this paper describes this highly non-trivial reduction.
\paragraph*{\itshape Convention.}
For technical reasons we choose in the following not to distinguish
formally between (assignments to) free first and second-order variables
(and interpretations of) constant or relation symbols. For instance,
we shall often regard $x$~and~$X$, which in usual parlance occur
free in $\varphi(X,x)$, as part of the vocabulary, and think of
assignments $a \in A$ and $P \subseteq A$ over some~$\fA$
in terms of the expansion $(\fA,P,a)$ of~$\fA$.
\section{A Feferman--Vaught theorem for positive types}
\label{sect:Feferman-Vaught}
\label{sect:start I}
For a vocabulary~$\tau$, we denote by
$\MSO^n[\tau]$ the set of all $\MSO$-formulae
over~$\tau$ with quantifier rank at most~$n$
(we count both first- and second-order quantifiers).
If $X \in \tau$ is a unary predicate we write
$\Lpos n$ for the subset of all formulae
where the predicate~$X$ occurs only positively.
Recall that, for finite vocabularies~$\tau$, $\MSO^n[\tau]$,
and hence also $\Lpos n$,
is finite up to logical equivalence.
\begin{defi}
Let $\tau$~be a vocabulary and $X \in \tau$.
The \emph{$X$-positive $n$-type} of a $\tau$-structure~$\fA$ is the set
\begin{align*}
\mtype{n}{\fA} := \set{ \varphi\in\Lpos n }{ \fA\models\varphi }\,.
\end{align*}
We write $\mType{n}$
for the set of all $X$-positive $n$-types of $\tau$-structures.
\end{defi}
Let $\fT_1$ and $\fT_2$ be tree structures.
If $T_1$ and $T_2$ are disjoint,
and if furthermore no constant symbol is interpreted in both trees,
then we define a concatenation operation as follows\?:
let $c_1$~and~$c_2$ be constant symbols from the structures
$\fT_1$~and~$\fT_2$, respectively.
Then we denote by $\fT_1 +_{c_1,c_2} \fT_2$ the tree
obtained from the disjoint union of the trees $\fT_1$~and~$\fT_2$
by adding an edge between $c_1^{\fT_1}$ and $c_2^{\fT_2}$.
Note that every finite tree
can be constructed from one-element trees using this operation and reduct operations.
If $\fT$ is a tree and $vw$ an edge of~$\fT$,
then removing~$vw$ from~$\fT$ produces two disjoint trees.
Of these, we denote the one containing the vertex~$v$ by~$\fT_{vw}$.
Note that, if there are constants $c$~and~$d$ for $v$~and~$w$,
then $\fT = \fT_{vw} +_{c,d} \fT_{wv}$.
If $c$~is a constant symbol not interpreted by~$\fT_{vw}$,
then we set $\fT_{vw,c} := \parlr{\fT_{vw},v}$,
where the expansion interprets~$c$ by~$v$.
We will frequently need a derived operation\?:
let $\fT_1$~and~$\fT_2$ be trees
such that $T_1$ and $T_2$ are disjoint, and
suppose that there is exactly one constant symbol~$c$
that is interpreted both in $\fT_1$~and in~$\fT_2$.
Let $d$~be a constant symbol which is interpreted in neither.
Then we denote by $\fT_1 \add_c \fT_2$
the reduct of $\fT_1 +_{c,d} \subst{\fT_2}{d}{c}$
that expels~$d$ from the vocabulary ($\subst{\fT_2}{d}{c}$ denotes
the structure obtained from~$\fT_2$ by renaming the constant symbol~$c$
to~$d$).
Intuitively,
$c$~denotes the root of (directed versions of) the respective trees,
and $\add_c$~appends its second argument as a new subtree
below the root of its first argument.
For a more uniform treatment,
we allow the empty tree~$\triangle$
as a neutral second argument to~$\add_c$, and
we use~$\triangle$ also for its type.
\begin{prop}\label{prop:Feferman-Vaught}
For every $n < \omega$,
there is a binary operation~$\oplus^n_{c_1,c_2}$ on $X$-positive $n$-types
such that,
for all trees $\fT_1,\fT_2$ for which $\fT_1 +_{c_1,c_2} \fT_2$ is defined,
we have
\begin{align*} \mtype n{\fT_1 +_{c_1,c_2} \fT_2}
= \mtype n{\fT_1} \oplus^n_{c_1,c_2} \mtype n{\fT_2}\,.
\end{align*}
Furthermore, $\oplus^n_{c_1,c_2}$ is monotone\?:
\begin{align*}
t_1\subseteq t_1' \text{ and } t_2 \subseteq t_2'
\quad\text{implies}\quad
t_1 \oplus^n_{c_1,c_2} t_2 \subseteq t_1' \oplus^n_{c_1,c_2} t_2'\,.
\end{align*}
Finally, $t_1\oplus^n_{c_1,c_2}t_2$ is computable from $n$, $t_1$, and $t_2$.
\end{prop}
\begin{proof}
Computability of the operation will be evident,
once we show how to compute with types in an effective way.
For this sake,
note that we can represent an $n$-type by a finite
set of formulae where all maximal boolean combinations
are in disjunctive normal form without repetition of clauses
or of literals in clauses.
We proceed by induction on~$n$.
Assume that we already know how to compute~$\oplus^m_{c_1,c_2}$ for all $m<n$
and all vocabularies.
For convenience, we set
\begin{align*}
\fT:=\fT_1 +_{c_1,c_2} \fT_2\,,\quad
t_1:=\mtype n{\fT_1}\,,\quad
t_2:=\mtype n{\fT_2}\,,
\quad\text{and}\quad
t:=\mtype n{\fT}\,.
\end{align*}
We will describe~$t$ solely in terms of $n$,~$t_1$, $t_2$,
and the operations~$\oplus^m_{c_1,c_2}$ with $m < n$.
Each formula in an $X$-positive $n$-type
is a positive boolean combination of atoms, negated atoms,
and formulae of the form $\exists y\varphi$, $\forall y\varphi$,
$\exists Y\varphi$, and $\forall Y\varphi$,
where $y$~is a first-order variable and $Y$~is a set variable.
Whether the full formula belongs to~$t$ is clearly determined
by whether the individual formulae in the positive boolean combination do.
Also, as the boolean combinations are positive,
monotonicity is preserved.
Hence it suffices to consider subformulae of the above form.
In the following we explicitly treat the cases of
atomic and negated atomic formulae and of
$\exists y\varphi$ and $\forall Y\varphi$.
The remaining cases $\exists Y\varphi$ and $\forall y\varphi$
can be handled using combinations of the techniques
used in these cases.
First, we consider atoms and negated atoms.
Each (negated) atom that only uses constants from~$\fT_i$
occurs in~$t$ iff it occurs in~$t_i$.
It remains to consider (negated) atoms
involving constants from both $\fT_1$ and $\fT_2$.
As $E$~is the only relation symbol of arity more than~$1$,
such an atom must be of the form $c\seq d$ or $Ecd$ where,
without loss of generality, $c$~is from the vocabulary of $\fT_1$
and $d$~from the vocabulary of $\fT_2$.
In this case, we always have $c\seq d \notin t$
and, hence, $\neg(c\seq d) \in t$\?;
so
\begin{align*}
Ecd\in t \quad\text{iff}\quad \neg Ecd \notin t
\quad\text{iff}\quad
c\seq c_1\in t_1 \text{ and } d\seq c_2\in t_2\,.
\end{align*}
Next, let us consider a formula of the form $\exists y\varphi$
with $m := \qr(\varphi) < n$.
We make use of~$\oplus^m_{c_1,c_2}$.
Let $t_1'$ and $t_2'$ be the $X$-positive $m$-types of $\fT_1$ and $\fT_2$,
that is, $t_1' = t_1\cap\MSO^m_X$ and $t_2' = t_2\cap\MSO^m_X$.
Further, let $S_1$ be the set of $X$-positive $m$-types
of expansions of $\fT_1$ by some $a\in T_1$ interpreted for~$y$,
and let $S_2$ be the respective set of types of expansions of $\fT_2$.
Clearly, $\exists y\varphi\in t$ iff $\varphi\in\mtype m{\fT,a}$ for some $a\in T$.
For $a\in T_1$ and $t''_1:=\mtype m{\fT_1,a}$,
the inductive hypothesis implies that
\begin{align*}
\mtype m{\fT,a}
&= \mtype m{\fT_1+_{c_1,c_2}\fT_2,\ a} \\
&= \mtype m{\parlr{\fT_1,a}+_{c_1,c_2}\fT_2} \\
&= \mtype m{\fT_1,a}\oplus^m_{c_1,c_2}\mtype m{\fT_2}
= t''_1 \oplus^m_{c_1,c_2} t'_2\,.
\end{align*}
Note that $t''_1\in S_1$.
The case where $a\in T_2$ is similar.
It follows that
\begin{align*}
\exists y\varphi \in t
\quad\text{iff}\quad
& \varphi\in t''_1 \oplus^m_{c_1,c_2} t'_2\,, \text{ for some } t''_1\in S_1\,, \\
\text{or } & \varphi\in t'_1 \oplus^m_{c_1,c_2} t''_2\,, \text{ for some } t''_2\in S_2\,.
\end{align*}
As an artifact of positivity in~$X$,
the set~$S_1$ is not determined by~$t_1$.
The point is that, for instance, if
$\exists x ( Xx \wedge \chi(x)) \in t_1$,
then $S_1$~may or may not contain a type~$t'$
such that $\chi\in t'$ but $Xx\not\in t'$,
because we do not know
about the status of $\exists x(\neg Xx \wedge \chi(x))$.
Unlike $S_1$, the following superset of~$S_1$ is determined by~$t_1$\?:
\begin{align*}
S_1' := \bigset{ t''_1 \in \mType{m} }{ \textstyle\exists y\bigwedge t''_1 \in t_1 } \supseteq S_1\,.
\end{align*}
(Recall that representations of types are finite,
so $\bigwedge t''_1$ is in fact a formula.)
Hence it suffices to show that
\begin{align*}
\varphi\in t''_1 \oplus^m_{c_1,c_2} t'_2\,, \text{ for some } t''_1\in S_1
\quad\text{iff}\quad
\varphi\in t''_1 \oplus^m_{c_1,c_2} t'_2\,, \text{ for some } t''_1\in S_1'\,.
\end{align*}
(The corresponding statement for~$\fT_2$ then follows by symmetry.)
$(\Rightarrow)$ is trivial since $S_1\subseteq S_1'$.
For $(\Leftarrow)$,
assume that $t''_1$~is a type such that
$\exists y\bigwedge t''_1\in t_1$ and $\varphi\in t''_1\oplus^m_{c_1,c_2}t'_2$.
Let $a\in T_1$ be an element with $(\fT_1,a) \models \bigwedge t''_1$,
and set $t'''_1 := \mtype m{\fT_1,a}$.
Clearly, $t''_1 \subseteq t'''_1$.
Hence, monotonicity of~$\oplus^m_{c_1,c_2}$
implies that $\varphi \in t'''_1\oplus^m_{c_1,c_2}t'_2$, as desired.
It remains to show monotonicity of~$\oplus^n_{c_1,c_2}$
(as far as the formula $\exists y\varphi$ is concerned).
We need to establish that, if $\exists y\varphi\in t_1\oplus^n_{c_1,c_2}t_2$,
then this still holds after increasing $t_1$~or~$t_2$.
This follows from the fact
that the sets $S_1'$~and~$S_2'$ (defined analogously to~$S_1'$)
are monotone in $t_1$ and~$t_2$.
Finally, let us consider a formula of the form $\forall Y\varphi$
with $m := \qr(\varphi) < n$.
This time let $S_1$~be the set of $X$-positive $m$-types
of expansions of~$\fT_1$ by some unary predicate $P\subseteq T_1$ interpreted for~$Y$,
and let $S_2$~be the respective set for~$\fT_2$.
Using the equality
\begin{align*}
(\fT,P) = (\fT_1,P\cap T_1) +_{c_1,c_2} (\fT_2,P\cap T_2)
\end{align*}
we obtain, similarly to the case above, that
\begin{align*}
\forall Y\varphi\in t
\quad\text{iff}\quad
\varphi\in t''_1\oplus^m_{c_1,c_2}t''_2
\quad\text{for all } t''_1\in S_1 \text{ and } t''_2\in S_2\,.
\end{align*}
Let us call a pair $(S'_1,S'_2)$
\emph{good for $t_1,t_2$,} if the following conditions hold\?:
\begin{itemize}
\item $S_1'$~is a set of $X$-positive $m$-types
of the vocabulary used for expansions of $\fT_1$ by~$Y$ and
$S_2'$~is a corresponding set for of $\fT_2$.
\item $\forall Y\bigvee_{s_1\in S'_1}\bigwedge s_1 \in t_1$
and $\forall Y\bigvee_{s_2\in S'_2}\bigwedge s_2 \in t_2$.
\item For all $s_1\in S'_1$ and $s_2\in S'_2$
we have $\varphi\in s_1\oplus^m_{c_1,c_2}s_2$.
\end{itemize}
If $\forall Y\varphi\in t$, then $(S_1,S_2)$ is good, whence a good pair exists.
We claim that the converse also holds, i.e., that
the existence of a good pair implies $\forall Y\varphi \in t$.
Thus, we obtain a characterisation of whether $\forall Y\varphi\in t$
solely in terms of $t_1$,~$t_2$, and~$\oplus^m_{c_1,c_2}$.
Furthermore, being good for $t_1,t_2$
is clearly monotone in $t_1$~and~$t_2$.
To prove the claim, suppose that $\parlr{S'_1,S'_2}$ is a good pair
and let $t''_1\in S_1$ and $t''_2\in S_2$ be arbitrary.
We need to show that $\varphi\in t''_1\oplus^m_{c_1,c_2}t''_2$.
Fix a predicate~$P_1$ such that $t''_1=\mtype m{\fT_1,P_1}$.
By the second condition on good pairs, we have
$\parlr{\fT_1,P_1}\models\bigvee_{s_1\in S'_1}\bigwedge s_1$.
Hence, there is some $s_1\in S'_1$ such that $\parlr{\fT_1,P_1}\models\bigwedge s_1$.
This implies that $s_1 \subseteq t''_1$.
Analogously, we obtain some $s_2\in S'_2$ such that $s_2\subseteq t''_2$.
By the third condition on good pairs, we have $\varphi\in s_1\oplus^m_{c_1,c_2}s_2$.
Therefore, monotonicity of $\oplus^m_{c_1,c_2}$ implies that $\varphi\in t''_1\oplus^m_{c_1,c_2}t''_2$.
\end{proof}
The previous proof needed to consider different vocabularies.
From now on, a single vocabulary nearly suffices.
Let $\tau$ be a fixed tree vocabulary without any constant symbols.
Let $X$ be a unary relation symbol and $x$ a constant symbol
such that $x,X \notin \tau$.
We will consider fixed points with respect to $X$~and~$x$.
The fixed points are evaluated in trees of vocabulary~$\tau$.
Stages of the fixed-point induction are evaluated
in trees of vocabulary $\tauX$.
In order to determine whether a single tree node
belongs to some iteration for the fixed point,
we consider trees of vocabulary $\tauXx$.
If $x$ is present in the vocabulary,
its interpretation can be thought of as the root of the tree.
Let $\varphi$ be a $\tauXx$-formula positive in~$X$
and let $n$ be the quantifier rank of~$\varphi$.
\begin{cor}
Let $y \notin \tauXx$ be a new constant symbol.
We define a binary operation~$\add^n$
on $X$-positive $n$-types of $\tauXx$-structures by
\begin{align*}\textstyle
s \add^n t :=
\bigl(s \oplus_{x,y}^n \subst{t}{y}{x}\bigr) \cap \MSO^n_X[\tauXx]\,.
\end{align*}
The operation~$\add^n$ is monotone and satisfies
\begin{align*}
\mtype n{\fS \add_x \fT} = \mtype n{\fS} \add^n \mtype n{\fT}\,,
\end{align*}
for all non-empty tree structures $\fS$~and~$\fT$ of vocabulary~$\tauXx$.
\qed\end{cor}
We extend $\add^n$ by adjoining the $X$-positive $n$-type~$\triangle$ of the empty tree
as a right-neutral element.
This does not hurt monotonicity\?:
without loss of generality, assume that $n\geq 1$.
Then only~$\triangle$ contains $\forall y\bot$
and only this type does not contain $\exists y\top$,
so it is incomparable to any other type.
In the first part, which contains the technical heart of the article,
we will only consider ternary trees, that is,
undirected trees where each node has degree at most~$3$.
We assume that each such tree~$\fT$ is implicitly equipped
with an edge-colouring using $3$~colours $\{1,2,3\}$.
That means that, for every colour~$d$,
each vertex~$v$ of~$\fT$ has at most one neighbour
that is connected to~$v$ via an edge of colour~$d$.
We call this neighbour ``the neighbour of~$v$ in \emph{direction}~$d$''
and we denote it by~$v^d$.
If there is no such neighbour, we set $v^d := \triangle$.
To account for missing neighbours we extend the above
definition of~$\fT_{vw,x}$
by setting $\fT_{v\triangle,x} := (\fT,v)$
and letting $\fT_{\triangle w,x} := \triangle$.
Furthermore, let $\fT_{\{v\}} := (\fT \restriction \{v\},v)$.
With this notation we have
\begin{align*}
(\fT,v) = \fT_{\{v\}}
\add_x \fT_{v^1v,x} \add_x \fT_{v^2v,x} \add_x \fT_{v^3v,x}\,,
\end{align*}
where we assume that the operation~$\add_x$ is associative to the left.
We also need a variant of Proposition~\ref{prop:Feferman-Vaught}
that concerns a decomposition into a possibly infinite number of subtrees.
We omit the proof, which is similar to that of Proposition~\ref{prop:Feferman-Vaught}.
\begin{prop}\label{prop:Feferman-Vaught II}
Let $\fT$ be a $\tauXx$-tree and $\parlr{v_1,d_1},\parlr{v_2,d_2},\ldots$
a sequence of pairwise distinct pairs $\parlr{v_i,d_i}$,
such that $v_i\in T$ and $v_i^{d_i}=\triangle$.
Further, let $\fS_1,\fS_2,\ldots$ and $\fS'_1,\fS'_2,\ldots$
be sequences of $\tauXx$-trees
such that $\mtype n{\fS_i}=\mtype n{\fS'_i}$ for all~$i$.
Finally, let $\fU$~be the tree obtained from~$\fT$
by adding~$\fS_i$ as a child of~$v_i$
in direction~$d_i$ for all~$i$,
and define~$\fU'$ analogously using~$\fS'_i$ instead of~$\fS_i$.
Then, $\mtype n{\fU}=\mtype n{\fU'}$.
\qed\end{prop}
\section{Tilings}
\label{sect:tilings}
We are now in a position to provide a second, more precise proof outline.
Given a tree structure~$\fT$ of vocabulary~$\tau$,
we consider the fixed-point induction of~$\varphi$.
For every stage~$\alpha$ and every vertex~$v$ of~$\fT$
we consider the type $\tp^n_X(\fT,\varphi^\alpha(\fT),v)$.
We annotate~$\fT$ with all these types.
At each vertex~$v$ we write down the list of these types for all stages~$\alpha$.
These annotations can be used to determine the fixed-point rank of elements of~$\fT$.
A vertex~$v$ enters the fixed point at stage~$\alpha$ if the $\alpha$-th entry
of the list is the first one containing a type~$t$ with $Xx \in t$.
We can regard the annotation as consisting of several layers, one for each stage of the
induction. At a vertex~$v$ each change between two consecutive layers is caused
by some change at some other vertex in the previous step. In this way we can trace
back changes of the types through the various layers.
In order to determine whether the fixed-point inductions of the formula are bounded,
we construct a \emph{weighted automaton} (see Section~\ref{sect:automata} below)
that recognises (approximations of) such annotations and that computes
(an approximation of) the length of the longest path of changes in the annotation.
Actually, the annotations we use do not consist of single types but
of tuples of such types, called a \emph{tile.}
In this section we consider single layers of such tiles.
In the next section we will then introduce annotations
consisting of several such layers.
\begin{defi}
A \defn{letter} is a one-element $\taux$-tree.
\end{defi}
Observe that, for each letter~$\fL$,
there are exactly two $\tauXx$-expansions of~$\fL$\?:
one where the element belongs to~$X$ and one where it does not.
Let us denote their $X$-positive $n$-types
by $1_\fL$~and~$0_\fL$, respectively.
Note that $0_\fL \subseteq 1_\fL$ and that $Xx\in 1_\fL \smallsetminus 0_\fL$,
for every $\fL$. We omit the index~$\fL$ whenever it is irrelevant.
We can decompose a $\tauX$-tree~$\fT$ into its one-element
substructures~$\fT_{\{v\}}$, i.e., its letters.
Each of these letters~$\fT_{\{v\}}$ can be labelled with
its type and the types of the subtrees $\fT_{v^dv}$.
\begin{center}
\includegraphics{final-1.pdf}
\end{center}
For convenience, we will not only use the types $t_{\mi0}$ and $t_{\mi d}$
of $\fT_{\{v\}}$ and $\fT_{v^dv}$, $d = 1,2,3$, respectively,
but also the types~$t_{\mo d}$ of $\fT_{vv^d}$, $d = 1,2,3$
and the type~$t_{\mo4}$ of the whole tree $(\fT,v)$.
Our intuition regards the vertex~$v$ as a processing unit
that receives as its inputs the types $t_{\mi 0}$, $t_{\mi 1}$, $t_{\mi 2}$, $t_{\mi 3}$
and produces as output the types $t_{\mo 1}$, $t_{\mo 2}$, $t_{\mo 3}$, $t_{\mo 4}$.
The vertex~$v$ receives from its neighbours~$v^d$, $d = 1,2,3$,
the inputs~$t_{\mi d}$ and it passes back to~$v^d$ the outputs~$t_{\mo d}$.
\begin{defi}
\textup{(a)}
Let $\fL$ be a letter.
An \defn{$\fL$-tile} is an $8$-tuple
\begin{align*}
\parlr{t_{\mi0},\dots,t_{\mi3},t_{\mo1},\dots,t_{\mo4}}
\end{align*}
of $X$-positive $n$-types over $\tauXx$ where
\begin{itemize}
\item $t_{\mi0}\in\setlr{0_{\fL},1_{\fL}}$,
\item $t_{\mo1} = t_{\mi0} \add^n t_{\mi2} \add^n t_{\mi3}$,
\item $t_{\mo2} = t_{\mi0} \add^n t_{\mi1} \add^n t_{\mi3}$,
\item $t_{\mo3} = t_{\mi0} \add^n t_{\mi1} \add^n t_{\mi2}$, and
\item $t_{\mo4} = t_{\mi0} \add^n t_{\mi1} \add^n t_{\mi2} \add^n t_{\mi3}$.
\end{itemize}
If we do not want to mention the letter,
we refer to an $\fL$-tile simply as a \defn{tile.}
When $\gamma$~is a tile,
we denote its components by $\gamma_{\mi0}$ through $\gamma_{\mo4}$.
\textup{(b)}
Let $\fT$ be a $\tau$-tree.
A \defn{$\fT$-tiling} is a mapping~$c$ that assigns to each
vertex $v \in T$ a $\fT_{\setlr v}$-tile $c(v)$.
\textup{(c)}
Let $\fT$~be a $\tauX$-tree.
The \emph{canonical tiling}~$t_\fT$ of~$\fT$ is the function
assigning to a vertex~$v$ the tile
\begin{alignat*}{-1}
t_\fT(v)_{\mi 0} &:= \tp_X^n(\fT_{\{v\}})\,, \qquad
& t_\fT(v)_{\mi d} &:= \tp_X^n(\fT_{v^dv})\,,
&&\qquad\text{for } 1 \leq d \leq 3\,, \\
t_\fT(v)_{\mo 4} &:= \tp_X^n(\fT,v)\,, \qquad
& t_\fT(v)_{\mo d} &:= \tp_X^n(\fT_{vv^d})\,,
&&\qquad\text{for } 1 \leq d \leq 3\,.
\end{alignat*}
\end{defi}
Intuitively the $\mi d$-component of a tile
contains information \emph{incoming} from direction~$d$,
whereas the $\mo d$-component contains the information passed on in that direction.
Similarly, the $\mo 4$-component contains information passed on to the next stage.
The $\mi 0$-component is special, since it contains local information about the current vertex.
Note that the canonical tiling is indeed a tiling.
\begin{lem}\label{lem:tile is such}
Let $\fT$ be a $\tauX$-tree and $\fT_0$~its $\tau$-reduct.
Then $t_\fT$~is a $\fT_0$-tiling.
\end{lem}
\begin{proof}
Let $v \in T$.
Since $\fT_{\setlr v}$ is an expansion of $\fL := (\fT_0)_{\{v\}}$,
its type $t_\fT(v)_{\mi0}$ must be one of $0_\fL$ and~$1_\fL$.
For the equalities concerning $t_\fT(v)_{\mo d}$ with $1\leq d\leq 3$,
we may by symmetry assume that $d = 3$. Then
\begin{align*}
t_\fT(v)_{\mo3}
&= \mtype n{\fT_{vv^3}} \\
&= \mtype n{
\fT_{\setlr v}
\add_x \fT_{v^1v}
\add_x \fT_{v^2v}} \\
&= \mtype n{\fT_{\setlr v}}
\add^n \mtype n{\fT_{v^1v}}
\add^n \mtype n{\fT_{v^2v}} \\
&= t_\fT(v)_{\mi0}
\add^n t_\fT(v)_{\mi1}
\add^n t_\fT(v)_{\mi2}\,,
\end{align*}
as desired.
The equality for~$\mo4$ is obtained similarly.
\end{proof}
Not every tiling stems from an actual tree.
In the next definition we collect some simple consistency properties
a tiling should satisfy.
Note that these properties can be checked by an automaton.
\begin{defi}
Let $\fT$ be a $\tau$-tree and $v \in T$ a vertex.
\textup{(a)}
The \defn{orientation of~$\fT$ towards $v$}
is the mapping $o_v: T \to \setlr{1,\ldots,4}$
such that $\app{o_v}v=4$ and, for vertices $w\in T\smallsetminus\setlr v$,
we define $1\leq \app{o_v}w\leq 3$ such that the neighbour $w^{o_v(w)}$ is closer to~$v$ than~$w$.
\begin{center}
\includegraphics{final-2.pdf}
\end{center}
\textup{(b)}
A $\fT$-tiling~$c$ is \defn{locally consistent towards $v$}
if, for all $w\in T$ and all directions $1\leq d\leq 3$ with $d \neq o_v(w)$,
we have
\begin{align*}
c(w)_{\mi d} = \begin{cases}
c(w^d)_{\mo d} &\text{if } w^d \neq \triangle\,, \\
\triangle &\text{otherwise}\,.
\end{cases}
\end{align*}
\textup{(c)}
A $\fT$-tiling~$c$ is \defn{globally consistent towards $v$}
if, for all vertices $w\in T$ and all directions $1\leq d\leq 3$ with $d \neq o_v(w)$,
we have
\begin{align*}
c(w)_{\mi d} = \mtype n{(\fT,P)_{w^dw}}\,,
\end{align*}
where $(\fT,P)$ is the expansion of~$\fT$
by the set $P := \set{v\in T}{c(v)_{\mi0}=1}$ interpreted for~$X$.
\end{defi}
Of course, canonical tilings are globally consistent.
\begin{lem}\label{lem:tiling is such}
Let $\fT$~be a $\tau$-tree and $P \subseteq T$.
The $\fT$-tiling $t_{(\fT,P)}$ is
globally consistent towards each vertex $v\in T$.
\end{lem}
\begin{proof}
We have already seen in Lemma~\ref{lem:tile is such} that $t_{(\fT,P)}$ is a $\fT$-tiling.
Let $v \in T$. For global consistency, note that
\begin{align*}
P &= \bigset{v\in T}{ (\fT,P,v) \models Xx} \\
&= \bigset{v\in T}{\mtype n{(\fT,P)_{\setlr v}}=1} \\
&= \bigset{v\in T}{t_{\fT,P}(v)_{\mi0}=1}\,,
\end{align*}
as desired.
\end{proof}
Finally, let us show that global consistency implies
local consistency.
\begin{lem}\label{lem:global implies local for tiling}
Let $\fT$ be a $\tau$-tree and $v\in T$.
Every $\fT$-tiling that is globally consistent towards~$v$
is locally consistent towards~$v$.
\end{lem}
\begin{proof}
Let $c$ be a $\fT$-tiling globally consistent towards~$v$
and let~$\fT'$ be the $\tauX$-expansion of~$\fT$ by the set
$P := \set{v\in T}{c(v)_{\mi0}=1}$.
Let $w\in T$ and $d \neq o_v(w)$ be given.
Without loss of generality, we may assume that $d=3$.
If $w^3 = \triangle$, then $c(w)_{\mi 3}$ is the type of
$\fT'_{\triangle v} = \triangle$.
Otherwise, let $u := w^3 \neq \triangle$.
Since $c$~is a $\fT$-tiling, $c(u)_{\mi 0}$~is either $0_{\fT_{\{u\}}}$
or~$1_{\fT_{\{u\}}}$. By definition of~$\fT'$ it follows that
$c(u)_{\mi 0} = \mtype{n}{\fT'_{\{u\}}}$. Consequently,
\begin{align*}
c(w)_{\mi 3}
= \mtype n{\fT'_{uw}}
&= \mtype n{\fT'_{\setlr u}}
\add^n \mtype n{\fT'_{u^1u}}
\add^n \mtype n{\fT'_{u^2u}} \\
&= c(u)_{\mi0} \add^n c(u)_{\mi1} \add^n c(u)_{\mi2} \\
&= c(u)_{\mo 3}\,.
\end{align*}
\end{proof}
\section{Annotations}
\label{sect:annots}
Ideally we would like to annotate a given tree with one tiling for each stage
of the fixed-point induction. Since this is an infinite amount of data we have
to opt for something less\?: at each vertex of the tree we do not store the
full sequence of tiles for each stage, but only a shortened sequence
obtained by removing all duplicates. This is a finite amount of information
we can label the tree with.
The drawback of this method is that, by removing duplicates,
we lose synchronisation between the sequences from adjacent vertices.
Here are the formal definitions.
For a $\tau$-tree~$\fT$ and an ordinal~$\alpha$,
let $\fT^{\alpha} := \parlr{\fT,\app{\varphi^{\alpha}}{\fT}}$
be the $\tauX$-expansion of~$\fT$
by the $\alpha$th stage of the fixed-point induction.
Similarly, we set $\fT^\alpha_{vw,x} := (\fT^\alpha)_{vw,x}$
and $\fT^\alpha_{\{v\}} := (\fT^\alpha)_{\{v\}}$.
We extend the order $\subseteq$ on $X$-positive $n$-types
to tiles by requiring that $\subseteq$ holds component-wise.
\begin{defi}
\textup{(a)}
Let $\fL$ be a letter.
An \defn{$\fL$-history} is a strictly increasing sequence
$h=\parlr{h^0 \subsetneq \ldots \subsetneq h^m}$
of $\fL$-tiles such that
\begin{enumerate}
\item $h^0_{\mi0} = 0_{\fL}$ and
\item $h^{i+1}_{\mi0}=1_{\fL}$ iff $\varphi\in h^i_{\mo4}$, for $0\leq i<m$.
\end{enumerate}
The number~$m$ is the \defn{length} of the history, denoted $\len h$.
\textup{(b)}
Let $\fT$ be a $\tau$-tree and $v \in T$ a vertex.
\defn{The history} of~$\fT$ at~$v$
is the sequence~$h_\fT(v)$ of tiles $t_{\fT^{\alpha}}(v)$,
for all ordinals~$\alpha$, with duplicates removed.
\end{defi}
\begin{exa}\label{ex:paths}
For simplicity, we give an example of a fixed-point induction on
a path, instead of a tree, i.e., a tree where no vertex has a neighbour
in direction~$3$.
We consider the fixed-point of the formula $\varphi(X,x)$
stating that
\begin{align*}
x^1 = \triangle \quad\text{or}\quad
x^2 = \triangle \quad\text{or}\quad
T_{x^1x} \subseteq X \quad\text{or}\quad
T_{x^2x} \subseteq X\,.
\end{align*}
Figure~\ref{fig:annotated word} shows the histories of the
first $4$~elements of a finite path of length at least~$9$.
All further elements, except for the last two,
have the same history as the third and fourth elements.
Here, we assume that the edges are alternatingly labelled by $1$~and~$2$
and the tiles are drawn in the format
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$\mo4$&$\mo1$&$\mo2$&$\mo3$\\
\hline
$\mi0$&$\mi1$&$\mi2$&$\mi3$\\
\hline
\end{tabular}
\end{center}
where
\begin{itemize}\parskip=0pt\itemsep=0pt
\item $\triangle$~denotes the type of the empty tree,
\item $\varphi$~denotes any type containing $\varphi$,
\item $\times$~denotes any type not containing $\varphi$,
\item $\forall$~denotes any type containing the formula $\forall yXy$,
\item $\exists$~denotes any type containing $\exists yXy$,
but not $\forall yXy$, and
\item $-$ denotes any type not containing the formula $\exists yXy$.
\end{itemize}
\begin{figure}
\caption{Annotation for $\varphi(X,x)$\label{fig:annotated word}
\label{fig:annotated word}
\end{figure}
\end{exa}
Of course, the history of~$\fT$ at~$v$ is indeed a history.
\begin{lem}\label{lem:history is such}
Let $\fT$~be a $\tau$-tree and $v \in T$ a vertex.
Then $\app{h_{\fT}}v$ is a $\fT_{\setlr v}$-history.
\end{lem}
\proof
Let $h := \app{h_{\fT}}v$.
We have already seen in Lemma~\ref{lem:tile is such}
that each $h^i$ is a $\fT_{\setlr v}$-tile.
The sequence is increasing,
because we are considering positive (hence monotone) types
and the sequence $\app{\varphi^{\alpha}}{\fT}$ is increasing.
It is strictly increasing because we have removed duplicates.
For ordinals~$\alpha$, let $\app k{\alpha}$ be the index
at which the $\alpha$th stage appears in~$h$, i.e.,
$h^{\app k{\alpha}} = t_{\fT^{\alpha}}(v)$.
As the sequence is increasing, so is~$k$.
Since $\varphi^0(\fT) = \emptyset$, we have
\begin{align*}
h^0_{\mi0}
= h^{k(0)}_{\mi0}
= t_{\fT^0}(v)_{\mi0}
= \mtype n{\fT^0_{\setlr v}}
= 0_{\fL}\,.
\end{align*}
For $0\leq i<\len h$, let $\alpha$ be the minimal ordinal with $k(\alpha) = i+1$.
Then
\begin{align*}
h^{i+1}_{\mi0}=1
\quad\text{iff}\quad \mtype n{\fT^\alpha_{\setlr v}}=1
\quad\text{iff}\quad v\in \varphi^\alpha(\fT)\,.
\end{align*}
Since elements enter the fixed point only at successor stages, we have
\begin{align*}
v\in \varphi^\alpha(\fT)
&\quad\text{iff}\quad v \in \varphi^{\beta+1}(\fT) \quad\text{for some } \beta < \alpha\,, \\
&\quad\text{iff}\quad (\fT^\beta,v) \models \varphi \\
&\quad\text{iff}\quad \varphi\in\mtype n{\fT^\beta,v} \subseteq
h^i_{\mo4}\,.\rlap{\hbox to 144 pt{
\qEd}}
\end{align*}
\noindent We would like to annotate each vertex~$v$ of a tree~$\fT$ by the sequence
$(t_{\fT^\alpha}(v))_\alpha$. To obtain a finite object, we have to remove
duplicates and, therefore, we work with the history $h_\fT(v)$ instead.
For each~$\alpha$, we would like to have an automaton that can recover the
tiling~$t_{\fT^\alpha}$ from~$h_\fT$.
In general, this is not possible.
For instance, in Example~\ref{ex:paths} the `real' tilings $t_{\fT^\alpha}(v)$
for a path~$\fT$ of even length are words of the form $ux^ny^nv$
where $y^nv$ is the `mirror image' of $ux^n$.
This language is not regular.
Hence, we use an approximation.
For each vertex~$v$, each index~$i$ of $h_\fT(v)$, and each direction~$d$,
we record the index~$j$ of $h_\fT(v^d)$ such that
$h_\fT(v^d)^j$ and $h_\fT(v)^i$ belong to the same ordinal~$\alpha$.
Of course, given~$i$, there are several choices of~$\alpha$
and, hence, of~$j$, so we lose information. It will turn out that
these two pieces of data,
the function~$h$ and the function $(v,i,d) \mapsto j$,
are sufficient for our purposes.
\begin{defi}
\textup{(a)}
An \defn{annotated tree} is a tuple $\parlr{\fT,h,s}$, where
\begin{enumerate}
\item $\fT$ is a $\tau$-tree,
\item $h$ is a mapping that assigns to each vertex $v\in T$ a $\fT_{\setlr v}$-history $h(v)$, and
\item $s$ is a mapping assigning a natural number $s(v,i,d)$
to each vertex $v\in T$, each index $0\leq i\leq\len{\app hv}$, and every direction $1\leq d\leq 3$
with $v^d \neq \triangle$.
\end{enumerate}
We call $h$~the \defn{history map} and $s$~the \defn{synchronisation} of the annotated tree.
\textup{(b)}
Let $(\fT,h,s)$ be an annotated tree.
For $v\in T$ and $0\leq i\leq\len{\app hv}$,
the \defn{section} at $v,i$
is the tiling~$c$ defined inductively as follows\?:
\begin{enumerate}
\item $\app cv:=\app hv^i$.
\item For $w\in T\smallsetminus\setlr v$, let $u:=w^{\app{o_v}w}$.
We assume by induction that $\app cu$ is already defined.
Let $j$ be the index such that $\app cu=\app hu^j$.
Then we set $\app cw:=\app hw^{\app s{u,j,\app{o_v}w}}$.
\end{enumerate}
\end{defi}
\noindent Of course, not every annotated tree $(\fT,h,s)$
encodes the `real' fixed-point induction.
The next definition collects some necessary conditions.
\begin{defi}
Let $\cA = (\fT,h,s)$ be an annotated tree.
\textup{(a)}
$\cA$~is \defn{locally consistent}
if, for all vertices $v\in T$, indices $0\leq i\leq\len{\app hv}$, and directions $1\leq d\leq 3$
the following conditions are satisfied\?:
\begin{enumerate}
\item If $v^d=\triangle$,
then $\app hv^i_{\mi d} = \triangle$.
\item Otherwise, $\app s{v,i,d}\leq\len{\app h{v^d}}$
and $\app hv^i_{\mi d}=\app h{v^d}^{\app s{v,i,d}}_{\mo d}$.
\end{enumerate}
\textup{(b)}
$\cA$~is \defn{globally consistent}
if it is locally consistent
and if, for all $v,i$ as above, the section at $v,i$
is globally consistent towards~$v$.
\end{defi}
\begin{lem}\label{lem: sections are locally consistent}
Let $(\fT,h,s)$ be a locally consistent annotated tree.
Every section~$c$ at some $v,i$ is locally consistent towards~$v$.
\end{lem}
\begin{proof}
Let $v,w \in T$ be distinct vertices, $d \neq o_v(w)$, and let $i$~be the index
such that $c(w) = h(w)^i$.
Then we have $w^d=\triangle$ and $c(w)_{\mi d}=h(w)^i_{\mi d}=\triangle$, or
\begin{align*}
c(w)_{\mi d} = h(w)^i_{\mi d} = h(w^d)^{s(w,i,d)}_{\mo d} = c(w^d)_{\mo d}\,.
\end{align*}
\end{proof}
We have not yet defined the `real annotation' of a tree.
In fact, due to the choices involved in defining
the synchronisation there are several possible `real' annotations.
We obtain them by fixing an ordinal~$\beta$ and selecting
that synchronisation that
selects from among all possible choices
the stage that is closest to~$\beta$.
\begin{defi}\label{def: betasyndef}
Let $\fT$ be a $\tau$-tree and $\beta<\omega$.
We denote by $\cA_\beta(\fT)$
the annotated tree $\parlr{\fT,h_{\fT},s}$
where the synchronisation~$s$ is defined as follows.
For $v\in T$, $0\leq i\leq\len{\app{h_{\fT}}v}$,
and $1\leq d\leq 3$ with $w:=v^d \neq \triangle$,
we define $\app s{v,i,d}$ such that
\begin{align*}
\app{h_{\fT}}w^{\app s{v,i,d}}=t_{\fT^{\alpha}}(w)\,,
\end{align*}
where the ordinal~$\alpha$ is chosen as follows\?:
\begin{enumerate}
\item if $\app{h_{\fT}}v^i = t_{\fT^{\beta}}(v)$, then $\alpha=\beta$,
\item if $\app{h_{\fT}}v^i\subsetneq t_{\fT^{\beta}}(v)$,
then $\alpha\leq\beta$ is maximal
such that $t_{\fT^{\alpha}}(w)_{\mo d}=\app{h_{\fT}}v^i_{\mi d}$, and
\item if $\app{h_{\fT}}v^i\supsetneq t_{\fT^{\beta}}(v)$,
then $\alpha\geq\beta$ is minimal
such that $t_{\fT^{\alpha}}(w)_{\mo d}=\app{h_{\fT}}v^i_{\mi d}$.
\end{enumerate}
\end{defi}
We start with a technical lemma containing a monotonicity property
for the sections of an annotation.
\begin{lem}\label{lem:sections monotone}
Let $\fT$~be a tree with vertices $v,w \in T$,
let $c$~be the section of $\cA_\beta(\fT)$ at $v,i$,
and set $d := o_v(w)$.
\begin{enumerate}
\item[\normalfont(a)] If $c(w) = t_{\fT^\beta}(w)$,
then $c(u) = t_{\fT^\beta}(u)$, for all $u \in T_{ww^d}$.
\item[\normalfont(b)] Let $\alpha < \beta$. If $t_{\fT^\alpha}(w) \subseteq c(w)$, then
$t_{\fT^\alpha}(u) \subseteq c(u)$, for all $u \in T_{ww^d}$.
\end{enumerate}
(Here we set $T_{ww^4} := T$.)
\end{lem}
\begin{proof}
We only prove the claims for $w \neq v$. The argument for $w = v$ is similar.
We prove both claims by induction on the distance between $u \in T_{ww^d}$
and~$v$. The claims are immediate for $u=w$.
For the inductive step assume that the claims hold for~$u$
and let $u'$ be a neighbour of~$u$ which is further away from~$v$ than~$u$,
so that $u = (u')^{o_v(u')}$.
It follows that $u'=u^{d'}$ for some $d' \neq o_v(u)$.
Let $i$~be the index such that $c(u) = h_\fT(u)^i$.
By definition of $c$~and~$s$, respectively, we have
\begin{align*}
c(u') = h_{\fT}(u')^{s(u,i,d')} = t_{\fT^{\alpha'}}(u')\,,
\end{align*}
for some ordinal~$\alpha'$.
For (a), using the inductive hypothesis,
we have $h_\fT(u)^i = c(u) = t_{\fT^\beta}(u)$,
which implies that $\alpha' = \beta$.
Hence, $c(u') = t_{\fT^\beta}(u')$.
Similarly, for (b), we have $h_\fT(u)^i = c(u) \supseteq t_{\fT^\alpha}(u)$,
which implies that $\alpha' \geq \alpha$.
Hence, $c(u') \supseteq t_{\fT^\alpha}(u')$.
\end{proof}
Let us also show that $\cA_\beta(\fT)$ is always globally consistent.
\begin{lem}\label{lem:annotated tree is such}
For all $\tau$-trees $\fT$ and all $\beta<\omega$,
$\cA_\beta(\fT)$ is a globally consistent annotated tree.
\end{lem}
\proof
We have seen in Lemma~\ref{lem:history is such} that $h_\fT(v)$ is a
$\fT_{\{v\}}$-history.
Hence, $\cA_\beta(\fT)$ is an annotated tree.
For local consistency, fix $v,i,d$ and let $\alpha$~be the ordinal
from the definition of~$s$
at~$v$ in $\cA_\beta(\fT)$ (cf.~Definition~\ref{def: betasyndef}).
Then
\begin{align*}
h_\fT(v^d)^{s(v,i,d)}_{\mo d} = t_{\fT^\alpha}(v^d)_{\mo d} = h_\fT(v)^i_{\mi d}\,,
\end{align*}
as desired.
It remains to prove global consistency.
Fix a vertex $v\in T$ and an index $0\leq i\leq\len{\app{h_{\fT}}v}$,
and let $c$~be the section at~$v,i$.
For $w\in T$, let $\app{\alpha}w$ be the ordinal closest to~$\beta$
such that $\app cw = t_{\fT^{\app{\alpha}w}}(w)$.
Let $\fT'$ be the expansion of $\fT$
by the set $P := \set{w\in T}{\app cw_{\mi0}=1}$.
We need to show that
$\app cw_{\mi d}=\mtype n{\fT'_{w^dw}}$,
for all $w\in T$ and $d \neq \app{o_v}w$.
(Here, $\fT'_{v^4v}:=\fT'$.)
By local consistency of~$c$
(which holds by Lemma~\ref{lem: sections are locally consistent}),
it is sufficient to show that
$\app cw_{\mo d}=\mtype n{\fT'_{ww^d}}$,
for all $w\in T$ and $d:=\app{o_v}w$.
We do this by induction on the distance between $\alpha(w)$ and $\beta$.
First, suppose that $\alpha(w) = \beta$.
Then Lemma~\ref{lem:sections monotone}\,(a) implies that
$c(u) = t_{\fT^\beta}(u)$, for all $u \in T_{ww^d}$.
Consequently $\fT'_{ww^d} = \fT^\beta_{ww^d}$,
and hence $\mtype n{\fT'_{ww^d}} = t_{\fT^\beta}(w)_{\mo d} = c(w)_{\mo d}$, as desired.
It remains to consider the case that $\alpha(w) \neq \beta$.
By symmetry, we may assume that $\alpha(w) > \beta$.
Let $\fS$~be the maximal subtree of $\fT'_{ww^d}$
that contains the vertex~$w$ and
such that $\alpha(u) = \alpha(w)$ for all $u\in S$.
Let $\parlr{x_1,d_1},\parlr{x_2,d_2},\dots$ be the finite or infinite list
of all pairs $(x,d)$ such that $x \in S$ and $x^d \in T \smallsetminus S$.
Let $y_k := x_k^{d_k}$ be the missing neighbour.
Note that, by definition of~$s$, $\alpha(y_k)$ is the minimal ordinal~$\alpha$
such that
$\mtype n{\fT^\alpha_{y_kx_k}} = \mtype n{\fT^{\alpha(x_k)}_{y_kx_k}}$.
Hence, $\alpha(y_k) \leq \alpha(x_k)$ and it follows that
$\alpha(x_k) = \alpha(w)$ and $\alpha(y_k) < \alpha(w)$.
By local consistency and the inductive hypothesis, we have
\begin{align*}
c(x_k)_{\mi d_k} = c(y_k)_{\mo d_k} = \mtype n{\fT'_{y_kx_k}}\,,
\end{align*}
while, by definition of~$S$, we have
\begin{align*}
c(x_k)_{\mi d_k} = t_{\fT^{\alpha(w)}}(x_k)_{\mi d_k} = \mtype n{\fT^{\alpha(w)}_{y_kx_k}}\,.
\end{align*}
It follows that
$\mtype n{\fT'_{y_kx_k}}=\mtype n{\fT^{\app{\alpha}w}_{y_kx_k}}$
for each index~$k$.
As the subtrees of
$\fT'_{ww^d}$ and $\fT^{\app{\alpha}w}_{ww^d}$ induced by~$S$ agree,
we can use Proposition~\ref{prop:Feferman-Vaught II}
to deduce that
\[
c(w)_{\mo d} = \mtype n{\fT^{\alpha(w)}_{ww^d}} = \mtype n{\fT'_{ww^d}}\,.\eqno{\qEd}
\]
\section{Ranks}
\label{sect:ranks}
It remains to compute the length of the fixed-point iteration from
a given annotated tree.
The goal essentially is to obtain an estimate for the stage of a designated
element of the fixed point\?; this estimate is extracted from an annotation
in terms of the weight of an accepting run of a weighted
automaton which checks consistency of the annotation.
The appropriate kind of weighted automata for this purpose will be
presented in the next section.
\begin{defi}
\textup{(a)}
Let $(\fT,h,s)$ be an annotated tree and $v \in T$ a node.
We say that there is a \defn{jump} at~$v$
if there is some index~$i$ such that
$h(v)^i_{\mi0} = 0$ and $h(v)^{i+1}_{\mi0} = 1$.
Observe that this value of $i$~is uniquely determined.
We call the jump a \defn{base jump} if $i=0$.
\textup{(b)}
Suppose that there is a jump at~$v$ that is not a base jump.
We say that this jump \defn{depends} on another jump at a node~$w$
if $c(w)_{\mi0}=0$ where $c$~is the section at $v$ and $i-1$.
The \defn{rank} of a jump is the minimal number of jumps
on any dependency chain from this jump to some base jump.
\textup{(c)}
An annotated tree $(\fT,h,s)$ is \defn{jump-consistent,}
if the set of vertices with a jump equals $\app{\varphi^{\infty}}{\fT}$.
\end{defi}
The notion of \emph{dependency} in~(b) may warrant some comment, because
the terminology could easily be misunderstood. What the criterion is meant to
capture is not that there must be a (causal or temporal) dependence of the
appearance of~$v$ in the fixed point on the (prior) appearance of~$w$\?;
rather, it says that such a dependence cannot be ruled out.
At least any~$w$ that $v$~does \emph{not} depend on in the sense of the definition can
have had no influence on the appearance of~$v$.
In this sense our dependency relation provides a generous upper bound
on any intuitive `real' dependency\?:
it may be useful to think of $w$~as a \emph{potential} trigger for~$v$.
Let us compare the rank of a jump with the stage of the corresponding vertex
in the fixed point (the stage at which the vertex enters the fixed point).
First, we show that in every annotation the latter bounds the former.
\begin{lem}\label{lem:stage bounds min rank}
Let $\cA=\parlr{\fT,h,s}$
be a globally consistent and jump-consistent annotated tree,
let $v$~be a vertex with a jump in~$\cA$, and $\alpha < \omega$.
If $v\in\app{\varphi^{\alpha}}{\fT}$,
then the rank of~$v$ is at most~$\alpha$.
\end{lem}
\proof
We proceed by induction on~$\alpha$.
For $\alpha = 0$ there is nothing to do since
$\varphi^0(\fT) = \emptyset$.
Hence, we may assume that $\alpha > 0$ and that the claim already holds for smaller
ranks.
Let $i$ be the index such that $\app hv^i_{\mi0}=0$ and $\app hv^{i+1}_{\mi0}=1$.
If $i=0$, then there is a base jump at~$v$ and its rank is $1 \leq \alpha$.
For $i>0$, let $c$ be the section at $v,i-1$
and let $P:=\set{w\in T}{\app cw_{\mi0}=1}$.
From $\app hv^i_{\mi0}=0$ we conclude that $\varphi \notin \app cv_{\mo4}$.
As $c$ is globally consistent, it follows that $\varphi \notin \mtype n{\fT,P,v}$.
On the other hand, $\varphi\in\mtype n{\fT,\app{\varphi^{\alpha-1}}{\fT},v}$.
By monotonicity,
there must be some vertex $w \in \varphi^{\alpha-1}(\fT) \smallsetminus P$,
which, by jump-consis\-tency, has a jump.
As $w \notin P$, we have $\app cw_{\mi0}=0$. Consequently, $v$~depends on~$w$.
By inductive hypothesis, $w$~has rank at most $\alpha-1$.
Therefore, $v$~has rank at most~$\alpha$.
\qed
Some form of converse is true for annotations of the form $\cA_\beta(\fT)$.
\begin{lem}\label{lem:min rank bounds stage}
Let $\fT$ be a $\tau$-tree, $v \in \varphi^\infty(\fT)$, and $\alpha<\beta<\omega$.
If the rank of $v$ in $\cA_\beta(\fT)$ is at most $\alpha$,
then $v\in\app{\varphi^{\alpha}}{\fT}$.
\end{lem}
\proof
We proceed by induction on $\alpha$.
As all ranks are positive, $\alpha>0$.
Let $i$ be the index such that $\app{h_{\fT}}v^i_{\mi0}=0$ and $\app{h_{\fT}}v^{i+1}_{\mi0}=1$.
If $i=0$,
then $\varphi\in\app{h_{\fT}}v^0_{\mo4}=t_{\fT^0}(v)$.
Therefore, $v\in\app{\varphi^1}{\fT}\subseteq\app{\varphi^{\alpha}}{\fT}$ and we are done.
Hence we may assume that $i>0$, i.e., the jump at~$v$ is not a base jump.
Let $c$~be the section at $v,i-1$.
As the jump at~$v$ is not a base jump and its rank is finite,
there is some vertex~$w$ with a jump rank at most $\alpha-1$
such that $v$~depends on~$w$.
By the inductive hypothesis, we have $w\in \varphi^{\alpha-1}(\fT)$.
Consequently, $t_{\fT^{\alpha-1}}(w)_{\mi0}=1$.
On the other hand, we have $c(w)_{\mi0} = 0$ by choice of~$w$.
Therefore, $c(w) \subsetneq t_{\fT^{\alpha-1}}(w)$.
By Lemma~\ref{lem:sections monotone}, this implies that
\begin{align*}
\app{h_{{\fT}}}v^{i-1} = c(v) \subsetneq t_{\fT^{\alpha-1}}(v)\,.
\end{align*}
It follows that $\alpha'<\alpha-1<\beta<\omega$
for any~$\alpha'$ such that $\app{h_{\fT}}v^{i-1}=\app{t_{\fT^{\alpha'}}}v$.
Therefore, there exists a maximal such ordinal~$\alpha'$ and,
moreover, $\alpha'+2\leq\alpha$.
As $i$~is maximal such that $\app{h_{\fT}}v^i_{\mi0}=0$,
it follows that $i-1$ is maximal
such that $\varphi\notin\app{h_{\fT}}v^{i-1}_{\mo4}$.
Accordingly, $\alpha'$~is maximal such that
$\varphi\notin\app{t_{\fT^{\alpha'}}}v_{\mo4}$.
Thus, $\varphi\in\app{t_{\fT^{\alpha'+1}}}v_{\mo4}$
and $\app{t_{\fT^{\alpha'+2}}}v_{\mi0}=1$.
It follows that $v\in\app{\varphi^{\alpha'+2}}{\fT}\subseteq\app{\varphi^{\alpha}}{\fT}$.
\qed
It follows that boundedness of the fixed-point iteration
is equivalent to the existence of a finite bound on the ranks of all annotations.
\begin{defi}
A \defn{proposal} is a tuple $\parlr{\fT,h,s,v}$,
where $\parlr{\fT,h,s}$
is a globally consistent and jump consistent annotated tree
and $v\in\app{\varphi^{\infty}}{\fT}$.
The \defn{rank} of such a proposal is the rank of the jump at~$v$ in $\parlr{\fT,h,s}$.
\end{defi}
\begin{prop}\label{prop: boundeness and ranks}
A formula~$\varphi$ is bounded over the class of all ternary trees
if, and only if, there is some number $N < \omega$
such that the rank of each proposal is at most~$N$.
\end{prop}
\proof
$(\Leftarrow)$ Suppose there exists a bound $N < \omega$ on the ranks of proposals.
Let $\fT$ be some ternary tree, and $v\in\app{\varphi^{\infty}}{\fT}$.
By Lemma~\ref{lem:annotated tree is such},
$\parlr{\cA_{N+1}(\fT),v}$ is a proposal.
By choice of~$N$, the rank of~$v$ is at most~$N$.
Hence, Lemma~\ref{lem:min rank bounds stage} implies that $v\in\app{\varphi^N}{\fT}$.
As $v$ was arbitrary, it follows that $\app{\varphi^N}{\fT}=\app{\varphi^{\infty}}{\fT}$.
$(\Rightarrow)$ Suppose that $\varphi$~is bounded by some number $N < \omega$.
Let $\parlr{\fT,h,s,v}$ be an arbitrary proposal.
Then $v \in \varphi^\infty(\fT) = \varphi^N(\fT)$
and Lemma~\ref{lem:stage bounds min rank} implies
that the rank of the proposal is at most~$N$.
\qed
\section{Weighted automata}
\label{sect:automata}
\label{sect:end I}
In order to decide the boundedness problem for $\MSO$
we reduce it to the so-called \emph{limitedness problem}
for a certain kind of weighted automaton.
These automata have \defn{$\Sigma$-labelled directed trees} as inputs.
Such a tree is a triple $\parlr{T,E,\lambda}$
where $\lambda:T\to\Sigma$ is a labelling of~$T$
and $\parlr{T,E}$ is a directed tree
(meaning that $E\cap E^{-1}=\emptyset$,
$\parlr{T,E\cup E^{-1}}$ is a tree structure,
and there is some $r\in T$ called the \defn{root} of the tree
such that $r \mathrel{E^*} t$ for all $t\in T$).
\begin{defi}
\textup{(a)}
A \emph{weighted parity automaton} $\cA = (Q,\Sigma,\Delta,I,\Omega,w)$ consists
of a finite \emph{state space}~$Q$, a finite \emph{input alphabet}~$\Sigma$,
a set $I \subseteq Q$ of \emph{initial states,}
a finite \emph{transition relation}
$\Delta \subseteq \Sigma\times\omega^Q\times Q$,
a \emph{priority function} $\Omega : Q \to \omega$,
and a \emph{weight function} $w: \Delta\to\omega$.
A weighted parity automaton~$\cA$ takes as input
$\Sigma$-labelled directed trees $\parlr{T,E,\lambda}$.
Let $\pi_3 : \Sigma\times\omega^Q \times Q \to Q$ be the projection
to the third component.
A \emph{run} of $\cA$ on this tree is a mapping $\varrho:T\to\Delta$
satisfying, for all vertices $v \in T$, the following condition\?:
\begin{align*}
\varrho(v) = (c,f,q)
\quad\text{implies}\quad
&c = \lambda(v) \text{ and } f \text{ is the function mapping } p \in Q
\text{ to} \\
&\text{the number of children } u \text{ of } v \text{ with }
\pi_3(\varrho(u)) = p\,.
\end{align*}
A run~$\varrho$ is \emph{accepting,} if
\begin{itemize}
\item $\pi_3(\varrho(r)) \in I$ for the root~$r$ of~$\parlr{T,E,\lambda}$ and
\item for every branch $\beta$ of $(T,E)$, the limit
\begin{align*}
\liminf_{v \in \beta} \Omega(\pi_3(\varrho(v))) \quad\text{is even.}
\end{align*}
\end{itemize}
The \emph{language} $L(\cA)$ recognised by~$\cA$
is the set of all $\Sigma$-labelled directed trees~$\parlr{T,E,\lambda}$
on which there is an accepting run of~$\cA$.
For a run~$\varrho$ on some tree~$\parlr{T,E,\lambda}$
and a branch~$\beta$ of the tree, we set
\begin{align*}
w_\cA(\varrho,\beta) := \sum_{v\in\beta} w(\varrho(v))
\quad\text{and}\quad
w_\cA(\varrho) := \sup_\beta w_\cA(\varrho,\beta)\,,
\end{align*}
with values in $\omega \cup \{ \infty \}$.
The associated \emph{cost function} $w_\cA$ maps
$\parlr{T,E,\lambda} \in L(\cA)$ to
the minimum of $w_\cA(\varrho)$
taken over all accepting runs~$\varrho$ on~$\parlr{T,E,\lambda}$.
If $(T,E,\lambda) \notin L(\cA)$, $w_\cA$ returns~$\infty$.
\textup{(b)}
We say that the automaton~$\cA$ is \emph{limited,}
if there is some bound $N < \omega$
such that $w_\cA(T,E,\lambda) \leq N$ for all $(T,E,\lambda) \in L(\cA)$.
We say that $\cA$~is \emph{limited in the finite,}
if there is a bound $N < \omega$ such that
$w_\cA(T,E,\lambda) \leq N$ for all \emph{finite} $(T,E,\lambda) \in L(\cA)$.
\end{defi}
Note that, if we only consider finite trees as input, we can omit the priority
function~$\Omega$ from the automaton. Weighted automata as defined above
are a special case of so-called \emph{cost tree automata}
introduced in~\cite{ColcombetLoeding08}.
In that paper it is shown than the limitedness problem for cost tree automata
over finite trees is decidable.
Hence, the following is a direct
consequence of~\cite{ColcombetLoeding08}.
\begin{thm}[Colcombet and L\"oding]\label{thm:finite limitedness decidable}
It is decidable whether a weighted parity automaton~$\cA$ is limited in the
finite.
\qed\end{thm}
Colcombet and L\"oding have also announced a decidability result for the
general limitedness problem, but this result has not been published yet.
\begin{thm}[Colcombet and L\"oding]\label{thm:limitedness decidable}
It is decidable whether a weighted parity automaton~$\cA$ is limited.
\qed\end{thm}
Although the proof is still not published, its key arguments appear
in \cite{VandenBoom12,Colcombet13}.
The following sketch of how they fit together was communicated to the
authors by Colcombet and L\"oding.
A \emph{cost function} $f : \cT \to \omega \cup \{\infty\}$
associates with every tree a natural number or $\infty$.
We say that such a cost function~$f$ is \emph{dominated} by~$g$ if
$f$~is bounded over every subset $X \subseteq \cT$ over which~$g$ is bounded.
We denote this domination relation by $f \preceq g$.
We can state Theorem~\ref{thm:limitedness decidable} in terms of the
domination relation as follows.
Let $\cA$~be a weighted automaton and let $L$~be the language defined by~$\cA$
if we consider it as an ordinary parity automaton without weight function.
Let $f$~be the cost function $w_\cA$ associated with~$\cA$ and let $g$~be the
cost function that maps every tree in~$L$ to~$0$ and every other tree
to~$\infty$.
Then Theorem~\ref{thm:limitedness decidable} states that it is decidable
whether $f \preceq g$.
We would like to reduce this statement to Corollaire~8.11 of~\cite{Colcombet13},
which states -- in the terminology of~\cite{Colcombet13} --
that the domination relation $f \preceq g$ between cost functions
$f$~and~$g$ is decidable, provided that $f$~is given by a nondeterministic
$S$-Muller automaton and $g$~is given by a nondeterministic $B$-Muller
automaton.
The function~$g$ from above is given by a parity automaton without weight
function.
Such an automaton can trivially be converted into a $B$-Muller automaton.
Hence, to complete the proof it remains to find an $S$-Muller automaton
recognising~$f$.
This can be done in the same way as in the proof of Theorem~4.28
of~\cite{VandenBoom12}, where the author shows how to transform an
alternating $B$-B\"uchi automaton into a nondeterministic one.
This proof uses game-theoretic techniques.
One key argument is the fact that the games, which correspond to the
automata in question, are positionally determined.
To adapt the proof to our case, one needs positional determinacy for games
whose winning condition is a disjunction between an unboundedness condition
and a parity condition.
This can be shown as in Proposition~7.14 of~\cite{Colcombet13},
which treats winning conditions consisting of a conjunction of a boundedness
condition and a Rabin condition.
One further step of adaptation consists in the construction of a so-called
`history-deterministic' automaton that checks whether a given positional
strategy is winning.
For finite words, the underlying translation of nondeterministic automata into
history-deterministic ones can be found in an unpublished note
(cf.~Lemma~58 of~\cite{Colcombet09b}) on the author's web-page.
Using the results of the previous sections we can reduce
the boundedness problem for $\MSO$ on ternary trees
to Theorem~\ref{thm:limitedness decidable}.
To do so, we construct a weighted automaton computing the rank of a proposal
$(\fT,h,s,v)$.
In order to use $\parlr{\fT,h,s,v}$ as input for a tree automaton,
we encode it as a labelled directed tree with root~$v$.
The labelling contains information about the unary predicates in~$\tau$,
the histories, and the synchronisation.
As there is only a finite number of types,
there is a uniform bound on the length of histories
and we only need finitely many labels.
First we show that the set of all proposals is regular.
\begin{lem}\label{lem: proposals are regular}
Given a formula~$\varphi$,
we can effectively construct a parity automaton~$\cA$ recognising
the set of all proposals for~$\varphi$.
\end{lem}
\proof
Let $n$ be the quantifier rank of $\varphi$.
It is sufficient to show that the set of proposals can be defined in $\MSO$.
Being a locally consistent annotated tree can be expressed even in~$\FO$
since it is a purely local property.
For global consistency, note that we can encode a section~$c$
by a tuple of unary predicates~$\bar C$
(the precise number depends on the maximal length of a history)
such that there is an $\FO$-formula $\vartheta_i(v)$
stating that $\bar C$ encodes the section at $v,i$.
Thus, the section at $v,i$ is $\MSO$-definable and
the corresponding tiling is $\MSO$-interpretable.
In this tiling it is of course possible by means of $\MSO$
to determine the $\MSO$-type (of quantifier rank at most~$n$) of a subtree.
Consequently, we can express the global consistency of the tiling
and, hence, also the global consistency of the annotated tree.
As the set of jumps can be inferred from the tree labelling,
it is easy to check whether there is a jump at the root ($v \in \varphi^\infty$).
It remains to consider jump-consistency,
that is, it remains to define $\app{\varphi^{\infty}}{\fT}$
(where $\fT$ is the first component of the prospective proposal).
As $\varphi$ is positive in~$X$, this can be achieved by
\[
\psi(x) := \forall X[\forall y(\varphi(X,y) \to Xy) \to Xx] \,.\eqno{\qEd}
\]
\begin{lem}\label{lem: proposals and automata}
Given a formula~$\varphi$,
we can effectively construct a weighted parity automaton~$\cA$ such that
\begin{enumerate}
\item $L(\cA)$ is the set of proposals of finite rank\?;
\item if $P$ is a proposal and $r<\omega$~its rank,
then $\frac{1}{2} \log r \leq w_\cA(P) \leq r$.
\end{enumerate}
\end{lem}
\begin{proof}
Let $n$ be the quantifier rank of~$\varphi$
and let $\cA_1$~be the automaton from Lemma~\ref{lem: proposals are regular}.
We will construct the desired automaton~$\cA$
as a product of~$\cA_1$ and a weighted parity automaton~$\cA_2$,
where the weight function of~$\cA$ is that of $\cA_2$.
Recall that the rank of a proposal $P=\parlr{\fA,h,s,v}$
is the minimal number of jumps
on a dependency chain from the jump at~$v$ to some base jump.
By this minimality condition
we can restrict our attention to chains without cycles.
Each dependency in the chain, say from~$u$ on~$u'$,
corresponds to a path in the section at $u,i$ for a suitable~$i$.
By minimality again, we only need to consider
pairwise disjoint paths, one for each dependency in the chain.
(If two paths intersected, we could form a new path witnessing the dependency
of some former jump in the chain to a latter one. This could be used to shorten
the dependency chain.)
These paths can be concatenated to form a single path in the annotated tree.
For a dependency path~$p$ and a tree node~$u$,
we say that $u$~is \defn{active}
if there is at least one jump on~$p$ in the subtree rooted at~$u$.
Since the tree is ternary, we can encode dependency paths by a tuple of unary predicates.
We first construct a weighted parity automaton~$\cA_3$ that takes as input
a proposal together with such a path.
It checks that the path follows the synchronisation (except for the jumps),
and that it is indeed a single path.
Furthermore, $\cA_3$~is such that from its state at a node~$u$
one can deduce whether $u$~is active.
We define the weight function of $\cA_3$ such
that all transitions have weight $0$~or~$1$, where we assign
a weight of~$1$ if at least two children of the current node are active
or if there is a jump at the current node.
For a dependency path~$p$ in a proposal~$P$,
let us compare its number~$r$ of jumps
with the weight computed by~$\cA_3$.
Let $\varrho$~be any accepting run of~$\cA_3$
on the input $\parlr{P,p}$.
We claim that $\frac12\log r\leq\app{w_{\cA_3}}{\varrho}\leq r$.
For the second inequality,
let $\beta$~be a branch of $P$ which realizes the maximum for~$\varrho$,
that is, $\app{w_{\cA_3}}{\varrho}=\app{w_{\cA_3}}{\varrho,\beta}$.
With each node $u\in\beta$ such that $\app{w_{\cA_3}}{\app{\varrho}u}=1$
we associate a jump in~$p$ as follows\?:
if there is a jump at~$u$, we just take this jump.
Otherwise, $u$~has at least two active children,
so it has at least one active child not in~$\beta$.
We take some jump from the subtree rooted at that child.
It is clear that, for different $u\in\beta$, we have chosen different jumps.
Hence, $r\geq\app{w_{\cA_3}}{\varrho,\beta}=\app{w_{\cA_3}}{\varrho}$.
For the other inequality, we construct a branch~$\beta$ as follows\?:
the branch starts at the root
and, whenever we have constructed~$\beta$ up to some node~$u$
which is not a leaf, we extend~$\beta$ with a child~$u'$ of~$u$
such that the number of jumps on~$p$ in the subtree rooted at~$u'$
is at least as large as the respective number for any other child of~$u$.
Let us trace this number along~$\beta$. Initially, it is~$r$.
It never increases and, whenever it decreases,
the respective transition has weight~$1$
by construction of~$\cA_3$.
As we always descend into the fattest subtree,
the number cannot decrease indefinitely\?:
if it is~$m$ for some node, it is at least $\frac{m-1}3$ for its child
(recall that the original undirected tree is ternary,
so the directed tree has branching at most~$3$,
and even at most~$2$ apart from the root).
A very rough analysis gives that, if $r\geq 4^k$,
then at least~$k$ decreasing steps occur on~$\beta$.
Hence,
$\app{w_{\cA_3}}{\varrho}\geq\app{w_{\cA_3}}{\varrho,\beta}\geq\frac12\log r$.
Finally, we obtain the desired automaton~$\cA_2$ from~$\cA_3$
by nondeterministically guessing the extra component~$p$.
To see that the product automaton~$\cA$ has the claimed properties,
let~$P$ be an input for~$\cA$.
If $P$ is a proposal of finite rank,
then it is in particular a proposal. Hence, $\cA_1$ accepts~$P$.
As the rank is finite, there is some dependency path $p$ for~$P$.
Therefore, $\cA_3$ accepts $\parlr{P,p}$ and $\cA_2$ accepts~$P$.
Consequently, also $\cA$ accepts~$P$.
For the converse, assume that $\cA$ accepts~$P$.
Then $P$ is a proposal since $\cA_1$ accepts~$P$.
Furthermore, there is some~$p$ such that $\cA_3$ accepts $\parlr{P,p}$.
Thus, $p$~is a dependency path in~$P$ and $P$ has finite rank.
Now, assume that $P$ is a proposal of rank~$r$
and let~$p$ be a dependency path in~$P$ with $r'$~jumps.
Let $\varrho$ be the accepting run of $\cA_3$ on $\parlr{P,p}$
and let $\varrho'$ be the corresponding accepting run of $\cA$ on $P$.
For each accepting run of $\cA$ on $P$
there is such a $p$ by construction of~$\cA$.
If $p$ is such that $\app{w_{\cA}}{\varrho'}$ is minimal,
then we can deduce
$\frac12\log r\leq\frac12\log r'\leq\app{w_{\cA_3}}{\varrho}
=\app{w_{\cA}}{\varrho'}=\app{w_{\cA}}P$.
If, on the other hand, $p$~is such that $r'$ is minimal,
we obtain
$\app{w_{\cA}}P\leq\app{w_{\cA}}{\varrho'}=\app{w_{\cA_3}}{\varrho}
\leq r'=r$.
\end{proof}
Combining our results we obtain a proof of the following theorem.
\begin{thm}\label{thm:boundedness for ternary trees}
The boundedness problem for $\MSO$ on the class
of all ternary trees is decidable.
\end{thm}
\begin{proof}
Given an $\MSO$-formula~$\varphi$, we construct
the weighted automaton~$\cA$ from Lemma~\ref{lem: proposals and automata}.
By Proposition~\ref{prop: boundeness and ranks}, it follows
that $\varphi$~is bounded if, and only if, $\cA$ is limited.
The latter we can decide with the help of
Theorem~\ref{thm:limitedness decidable}.
\end{proof}
\section*{Part II. Ramifications}
The boundedness problem has long been of interest
both in classical model theory and in the study of
the algorithmic properties of various fragments,
which in turn is partly motivated by applications
in computer science. The seminal result in the classical
model theory of the boundedness problem
is the theorem of Barwise and Moschovakis~\cite{BarwiseMoschovakis78}
(see Theorem~\ref{thm:BMthm} below)\?;
the main interest in boundedness as a decision problem,
on the other hand, stems from an interest in \textsc{Datalog}
query optimisation as highlighted in the
first positive and negative results in
\cite{GaifmanMaSaVa93,HillebrandEtAl95}.
In both contexts, the natural emphasis
was on (not necessarily monadic) monotone inductions based on
first-order formulae or formulae in specific fragments of first-order
logic. Even in the study of rather weak fragments of first-order logic,
undecidability of the boundedness problem turned out to be the rule,
decidability the rare exception.
In this second part we link our new results to the wider setting
of the boundedness problem. After a short introduction to
this wider setting, we employ some rather more traditional tools
from model theory, like transfer results and interpretations,
to generalise the technical core results of Part~I and to
reap a number of further specific decidability results.
Some of these answer key open questions raised in the
more traditional setting, concerning, for instance, decidability
of boundedness for the guarded fragment or for the modal $\mu$-calculus.
To this end, we first review the shift in perspective
from boundedness for syntactically restricted fragments of $\FO$
to boundedness over restricted classes of structures\?;
a shift that was first explicitly proposed in \cite{KOS}
where boundedness for otherwise unconstrained monadic $\FO$
is treated over the class of acyclic structures.
The class $\cA$ of acyclic structures consists of those
structures whose Gaifman graph is acyclic.
\begin{thm*}[\cite{KOS}]
The boundedness problem for monadic least fixed points
of arbitrary $X$-positive $\FO$-formulae over the class of all
acyclic relational structures, $\BDDm(\FO,\cA)$, is decidable.
\qed\end{thm*}
The interest here was due to the observation
that reductions to settings involving tree-like structures seem to
be a common theme in most decidability results for boundedness.
On the other hand, availability of grid-like structures can
be widely used to show undecidability of boundedness issues via
reductions from tilings \cite{KOunpub}.
This suggested a rough dichotomy to
explain the borderline for decidability of (monadic)
boundedness problems for fragments of $\FO$.
On the positive side, our present results bring this approach
to fruition in the much wider and unifying setting of $\MSO$.
Part of this success draws on the above-mentioned
change of perspective, which allows us to re-chart the
relevant fragments with a decidable boundedness problem
into a taxonomy of relevant classes of structures
to which we can lift and extend our decidability results
from Part~I.
We link the more traditional approach to
the boundedness problem to this new perspective
in the following section\?: in particular, we discuss some of
the more prominent fragments that have featured in the quest
for decidability of boundedness so far, and review key results
from that tradition.
In Sections \ref{sect:transfer}~and~\ref{sect:interpretations} we discuss the natural model-theoretic
techniques that can be used to translate and extend our results\?:
transfer properties and reductions (Section~\ref{sect:transfer})
and interpretations (Sections~\ref{sect:interpretations}).
In view of the above discussion this yields results both
in terms of applicability of our key result to
wider classes of structures, and in terms of decidability results
for new fragments.
\paragraph*{\itshape Proviso.}
In this part all vocabularies are (finite and) purely relational.
\section{Boundedness in the classical setting}
\label{sect:classical boundedness}
\label{sect:start II}
The key result concerning boundedness from classical
model theory is the following.
\begin{thm}[Barwise--Moschovakis \cite{BarwiseMoschovakis78}]
\label{thm:BMthm}
The following are equivalent for least fixed points
based on any $X$-positive $\varphi(X,\bar{x}) \in \FO$\?:
\begin{enumerate}
\item $\varphi$ is bounded.
\item $\varphi^\infty$ is uniformly $\FO$-definable.
\item $\varphi^\infty(\fA)$ is $\FO$-definable in each $\fA$.
\qed\end{enumerate}
\end{thm}
The classical proof is based on compactness arguments
and works with $\aleph_0$-saturated models for the crucial
implication from~(3) to~(1). It is immediate that this argument
relativises to natural fragments of $\FO$.
For formulae $\varphi$ from some such fragment of $\FO$ we
may replace $\FO$-definability by definability in the
fragment if that fragment has the natural closure properties
that render the finite stages definable\?; for truly natural
fragments like those to be considered below, however,
$\FO$-definability will imply definability within the
fragment by classical preservation theorems.
While these considerations offer some guidelines as to
what the right candidates $L \subseteq \FO$ for
decidable $\BDD(L)$ might be, our results from Part~I take
us beyond the limitations of $\FO$ and compactness --
which also means that boundedness becomes divorced from
definability of the fixed point.
We start this section with a brief review of some logics and fragments
that feature prominently in connection with the boundedness problem --
be it in classical results or in new results flowing from our main theorem.
These may be grouped into three main categories\?:
\paragraph{\textit{Existential/universal fragments:}}
certain limited, purely existential/purely universal fragments
$\EFO \subseteq \FO$ and $\AFO\subseteq \FO$\?: these are the
natural candidates
for a decidable monadic boundedness problem $\BDDm(L)$ in terms of quantifier
prefix classes $L \subseteq \FO$
(cf.\ the classical decision problem, \cite{BGG}).
For decidability of the boundedness problem
extra restrictions on the polarities of
the given relations,
which are statically used in the fixed-point recursion,
and on equality, are necessary.
See Section~\ref{subsec:existforall} below.
\paragraph*{\textit{Modal fragments:}}
the modal fragments of first-order and monadic second-order logic\?:
basic modal logic $\ML \subseteq \FO$ and its monadic fixed-point extension
$\Lmu \subseteq \MSO$, the bisimulation invariant
fragments of $\FO$ and $\MSO$, respectively.
See Section~\ref{subsec:modal} below.
\paragraph*{\textit{Guarded fragments:}} the corresponding but more general guarded fragments\?:
the basic guarded fragment $\GF \subseteq \FO$ and its fixed-point extension
$\muGF \subseteq \GSO$. These correspond to the fragments of
$\FO$ and guarded second-order logic $\GSO$, respectively,
that are invariant under guarded bisimulation.
With these logics we also
extend the scope of our discussion beyond monadic fixed points.
See Section~\ref{subsec:guarded} below.
In relation to $\BDD(L)$ or $\BDD(L,\cC)$ it is useful
to have in mind the following observation, which severely limits
the expectations regarding decidability but also points to
natural candidates.
\begin{observation}\label{obs:SAT vs BDD}
Assume that $\BDD(L)$ is non-trivial in the sense
that there are unbounded formulae $\varphi \in L$.
Then simple closure properties of~$L$ -- as for instance
closure under monadic relativisation and under conjunctions --
imply that the satisfiability problem $\SAT(L)$
reduces to the boundedness problem $\BDD(L)$.
An analogous reduction applies w.r.t.\ to restricted classes of models, i.e.,
for $\SAT(L,\cC)$ and $\BDD(L,\cC)$ provided
$\cC$ also satisfies some simple closure requirements --
as for instance closure under disjoint unions and trivial expansions by unary
predicates.
\end{observation}
We sketch one typical argument to this effect.
Fix some $\varphi(X,x) \in L$ that is unbounded.
Then a sentence
$\psi \in L$ is unsatisfiable if, and only if,
the formula $\varphi(X,x)^Q \wedge \psi^P$ is bounded\?;
here $\varphi(X,x)^Q$ and $\psi^P$ stand for the relativisations
to two distinct unary predicates $P$~and~$Q$,
which do not occur in either formula.
Clearly, unsatisfiability of~$\psi$ implies that
$\varphi(X,x)^Q \wedge \psi^P$ is unsatisfiable and hence
has closure ordinal $0$. Conversely, if $\psi$ is satisfiable, then
structures obtained as the disjoint union of a $P$-coloured model
of $\psi$ and a $Q$-coloured part show $\varphi(X,x)^Q \wedge \psi^P$
to be unbounded. The basic idea can be
modified to suit various other situations. For instance, for modal logic,
where disjoint unions are not the right choice,
one could look at boundedness for $\varphi^Q \wedge
\Diamond (P \wedge \psi^P)$ to decide satisfiability of $\psi$.
We turn to the above-mentioned groups of logics.
\subsection{Purely existential and universal fragments}
\label{subsec:existforall}
$\EFO[\tau] \subseteq \FO[\tau]$ is the fragment of positive,
purely existential prenex first-order formulae (with equality), where
for $\BDD$ we also allow (positive occurrences of)
monadic second-order variables.
Dually, we let $\AFO[\tau] \subseteq \FO[\tau]$
be the fragment of prenex universal
first-order formulae that are negative in all
relation symbols from the underlying relational vocabulary
$\tau$ and equality, but of course we allow positive occurrences
of monadic second-order variables.
The first interest in boundedness as a decision problem
concerned the query language \textsc{Datalog}
corresponding to the evaluation of systems of least fixed
points of relational Horn clauses of the form
\begin{align*}
\textstyle
X\bar x \leftarrow \exists\bar y \bigwedge_i \alpha_i(\bar x,\bar y)
\end{align*}
with relational atomic formulae~$\alpha_i$.
This Horn clause translates into
\begin{align*}
\textstyle
\varphi(X,\bar x) = \exists\bar y \bigwedge_i \alpha_i(\bar x,\bar y) \in \EFO
\end{align*}
in our framework. In this connection
the first decidability results were obtained in
\cite{CosmadakisGaKaVa88}, and also the strict limitations for this
decidable case became apparent \cite{GaifmanMaSaVa93,HillebrandEtAl95}.
\begin{thm}
\begin{enumerate}[label=\({\alph*}]
\item
The monadic boundedness problem $\BDDm(\EFO)$ is decidable \textup{\cite{CosmadakisGaKaVa88}.}
\item
Boundedness for binary least fixed points in $\EFO$
is undecidable\?; so is boundedness even for monadic least fixed points
in the extension of\/ $\EFO$ that allows
negated equalities (or negative and positive
occurrences of some of the static relations)
\textup{\cite{GaifmanMaSaVa93,HillebrandEtAl95}.}
\qed\end{enumerate}
\end{thm}
\noindent As for $\BDDm(\AFO)$, whose decidability was established in \cite{Otto06},
it should be noted that the fragment $\AFO$ is strictly dual to
$\EFO$\?; but as duality of fixed points links least to greatest
fixed points, trivial dualisation of the \textsc{Datalog} result would
just cover boundedness for greatest fixed points over $\AFO$.
Indeed, the techniques employed in \cite{Otto06} for decidability
of $\BDDm(\AFO)$ owe more to a reduction inspired by the guarded
fragment (see Section~\ref{subsec:guarded} below) and also do not seem to carry over directly to
$\BDDm(\EFO)$ or vice versa.
\begin{thm}[\cite{Otto06}]
$\BDDm(\AFO)$ is decidable, and both the restriction to monadic
least fixed points and the polarity restriction built into $\AFO$ are
necessary for decidability.
\qed\end{thm}
W.r.t.\ polarity restrictions on the static predicates in~$\tau$,
it should be noted that, as long as we consider the class of all
$\tau$-structures, it does not matter which polarity is prescribed,
since we can replace each predicate by its complement to switch
between polarities (this does not carry over from to $\BDD(L)$ to
$\BDD(L,\cC)$ unless $\cC$ is closed under predicate complementation).
What does matter, even over the class of all
$\tau$-structures, however, is whether we allow some predicates to
appear both positively \emph{and} negatively in $\varphi$.
\subsection{Logics of modal character}
\label{subsec:modal}
For a relational vocabulary $\tau$ consisting of only unary and
binary relation symbols, $\ML[\tau] \subseteq \FO[\tau]$ stands for the
\emph{modal fragment} of first-order logic. $\ML[\tau]$ is obtained
as the closure of monadic atomic formulae (where we also allow
monadic second-order variables besides unary relation symbols
in $\tau$) in a single free first-order variable under boolean
connectives and modal quantification of the form
\begin{align*}
\psi(x) = \exists y ( Rxy \wedge \varphi(y))
\quad\text{and, dually,}\quad
\psi(x) = \forall y ( Rxy \to \varphi(y))
\end{align*}
for any $\varphi(y) \in \ML[\tau]$ and binary relation symbol $R \in \tau$.
The \emph{modal $\mu$-calculus} $\Lmu[\tau]$ is obtained as the natural
fixed-point extension of $\ML[\tau]$ through additional closure
under least fixed points\?: if $\varphi(X,x) \in \Lmu[\tau]$ is
positive in $X$, then $\psi(x) = \mu_X \varphi \in \Lmu[\tau]$
defines the least fixed-point $\varphi^\infty$.
Our definition of $\ML$ is the usual embedding of basic
modal logic into $\FO$ by means of
the standard translation $\varphi \mapsto \varphi^*$,
which translates the modal
formula $\Box_R \varphi$ into
$(\Box\varphi)^* (x) = \forall y ( Rxy \to \varphi^*(y))$.
By van~Benthem's classical result in~\cite{Ben83},
$\ML[\tau]$ provides equivalent syntax for exactly those first-order
formulae in a single
free element variable whose semantics is preserved under bisimulation
equivalence. In this sense $\ML$
\emph{is} the bisimulation invariant (read\?: modal)
fragment of first-order logic. (For these and other basic facts
in the model theory of modal logic compare e.g.~\cite{GOHBML2007}).
We have similarly translated the $\mu$-calculus
in a manner that in particular turns it into a fragment of $\MSO$. In fact
$\Lmu$ \emph{is} the modal fragment of $\MSO$, in just the sense that
$\ML$ is the modal fragment of $\FO$, by an important result
of Janin and Walukiewicz \cite{JaninWalukiewicz}.
For us it will be important that $\ML \subseteq \Lmu \subseteq \MSO$ and that
$\ML$ and $\Lmu$ are preserved under bisimulation, which entails the
tree-model property.
Decidability of $\BDDm(\ML)$ was first shown in \cite{Otto99}\?;
note, however, that although that paper shows more generally that
it is decidable for an arbitrary formula of $\Lmu$ whether it is
equivalent to any formula in plain modal logic (of which
$\BDDm(\ML)$ is a special case, by the modal variant of the
Barwise--Moschovakis Theorem), it does \emph{not} deal with $\BDDm(\Lmu)$.
As will be reviewed in Section~\ref{sect:classical transfer} below,
decidability of
$\BDDm(\ML)$ and $\BDDm(\Lmu)$ can be essentially attributed to the tree-model
property stemming from bisimulation invariance.
Decidability of $\BDDm(\Lmu)$ is new here\?;
see Corollary~\ref{cor:decidability of BDD(GF), etc} below.
This result obviously implies the result of \cite{Otto99}
concerning decidability of $\BDDm(\ML)$ (but not as far as the problem
of equivalence of a given $\Lmu$-formula to some $\ML$-formula is concerned).
\begin{thm}
$\BDDm(\Lmu)$ and hence $\BDDm(\ML) \subseteq \BDDm(\Lmu)$ are decidable.
\end{thm}
\subsection{Guarded logics}
\label{subsec:guarded}
The \emph{guarded fragment} $\GF \subseteq \FO$ of
first-order logic extends the idea of
the local, relativised quantification of modal logic to the setting
of higher-arity relations. Since its inception in~\cite{ABN} the guarded
fragment and its extensions have been shown to mirror many of the nice
model-theoretic properties of modal logic in this more general setting.
Just like $\ML$ and its fixed-point extension $\Lmu$,
$\GF$ as well as its fixed-point extension $\muGF$
are decidable for satisfiability, cf.\ \cite{ABN,Graedel99,GW}.
Their roles as the guarded bisimulation invariant fragments of $\FO$
and a suitable guarded second-order logic are strictly analogous to those
of $\ML$ and $\Lmu$ as
bisimulation invariant fragments of $\FO$ and $\MSO$\?:
$\GF \subseteq \FO$ captures precisely those
$\FO$ definable properties that are preserved under guarded bisimulation
\cite{ABN}, and similarly for $\muGF \subseteq \GSO$ w.r.t.\
the natural guarded second-order logic $\GSO$, \cite{GHO}.
Like $\ML$, $\GF$ still has the finite model property, and
both $\GF$ and $\muGF$ have a generalised tree-model
property~\cite{Graedel99,GW}, which implies
in particular that every satisfiable formula of
$\muGF[\tau]$ is satisfiable in a model whose tree-width
is bounded by the width of $\tau$ (maximal arity of relations in
$\tau$). But note that $\muGF$ does not have the finite model
property, in fact this is already true of the extension of $\Lmu$
that admits modal operators along backward edges (inverse or past
modalities).
$\GF$ has long been considered a good candidate for decidability of
$\BDD(\GF)$.
Let us define these logics and the concept of guardedness in more detail.
A subset of a $\tau$-structure $\fA$ is called \emph{guarded}
if it is a singleton set or a set of the form
$\set{ a }{ a \in \bar{a} }$ for some $\bar{a} \in R^\fA$,
$R \in \tau$. Clearly the cardinality of guarded subsets in
$\tau$-structures is bounded by the width of $\tau$.
A tuple is guarded if the set of its components is contained
in some guarded subset.
A subset $W \subseteq A^r$ is called a \emph{guarded relation} over $\fA$
if all tuples $\bar{a} \in W$ are guarded in $\fA$.
Syntactically, a \emph{guard} for variables $\bar{x}$ is an atomic
formula $\alpha(\bar{x}) \in \FO[\tau]$ (relational atom or equality)
in which precisely the variables $x \in \bar{x}$ occur (as free variables).
\emph{Guarded quantification} is relativised first-order quantification
of the form
\begin{align*}
\exists \bar y ( \alpha(\bar x) \land \varphi(\bar x) )
\quad\text{and, dually,}\quad
\forall \bar y ( \alpha(\bar x) \to \varphi(\bar x) )\,,
\end{align*}
where $\alpha$ is an atom (viz., a guard for $\bar{x}$),
$\mathrm{free}(\varphi) \subseteq \mathrm{free}(\alpha) = \set{x}{x \in \bar x}$
and $\bar y$ is any
tuple of (distinct) variables from $\mathrm{free}(\alpha)$.
\begin{defi}
\textup{(a)}
$\GF[\tau] \subseteq \FO[\tau]$, the \emph{guarded fragment}
of first-order logic, is obtained as the closure of
atomic $\FO[\tau]$-formulae under boolean connectives and
guarded quantification. We stress that, even if we admit a
second-order variable~$X$, $X$~may \emph{not} be used as
a guard for quantificational purposes.
\textup{(b)}
\emph{Guarded fixed-point logic} $\muGF$ is the natural extension
of $\GF$ that is additionally closed under the formation of
least fixed points over $X$-positive formulae.
Note again that second-order variables, which may
occur free or bound in formulae of $\muGF[\tau]$,
must not be used as guards.
\textup{(c)}
Also define \emph{strictly guarded} formulae of these logics to be those
formulae whose free first-order variables are explicitly guarded\?:
$\varphi(\bar x)$ is strictly guarded if it can only be satisfied by
guarded assignments to~$\bar x$ (a syntactic normal form can be
obtained with the help of the $\GF$-formula $\gdd(\bar x)$ below).
We denote these restrictions as $\GFs \subseteq \GF$ and $\muGFs \subseteq \muGF$.
\end{defi}
It is clear that $\ML \subseteq \GFs$ and $\Lmu \subseteq \muGFs$.
We also note in passing that there is, for every finite $\tau$ and arity $r$,
a $\GFs[\tau]$-formula
$\gdd(x_1,\dots, x_r)$ that uniformly defines the set of all
guarded $r$-tuples in $\tau$-structures $\fA$\?:
\begin{align*}
\set{ \bar{a} \in A^r }{ (\fA,\bar{a}) \models \gdd(\bar{x}) }
= \set{ \bar{a} \in A^r }{ \bar{a} \text{ guarded in } \fA }\,.
\end{align*}
Clearly these formulae can be used to restrict arbitrary
relations to their guarded parts.
For strictly guarded formulae we thus obtain a normal form of
\begin{align*}
\gdd(\bar x) \wedge \varphi(\bar x)
\end{align*}
where $\bar x$ is the tuple
of all the free first-order variables of $\varphi$.
For \emph{guarded second-order logic} there are several formalisations,
which were shown to be equally expressive \emph{in the absence of free
second-order variables} in~\cite{GHO}.
As we shall see
as a consequence of Theorems~\ref{thm:GFmuGF} and~\ref{thm:FOundec}
below, this equivalence breaks down
if free second-order variables (for the generation
of non-monadic least fixed points) are admitted.
Specifically, one can define $\GSO$ as the extension of either $\GF$
or $\FO$ by second-order quantifiers ranging over guarded relations.
This can be enforced syntactically by means
of the formulae $\gdd(\bar x)$ that uniformly
define the sets of all guarded $r$-tuples\?;
alternatively one can stick with ordinary
second-order syntax and modify the semantics
to admit just guarded relations as instantiations for
second-order variables (guarded semantics).
The equivalence between these two definitions according to
\cite{GHO} breaks down in the presence of free
second-order variables of arity greater than~$1$,
since such variables are not allowed to serve as guards.
Therefore, we introduce two variants of guarded second-order logic.
As we shall see below, the corresponding boundedness problems
are different\?: one is decidable for arbitrary fixed points,
while the other one is only decidable for monadic fixed points.
\begin{defi}
\emph{Guarded second-order logic} $\GSO[\tau]$ is the extension
of $\FO[\tau]$ by quantification over guarded relations.
We denote by $\GSOg[\tau]$ the fragment of $\GSO[\tau]$
where all first-order quantifications are guarded.
Again, we denote by $\GSOs$ and $\GSOgs$ the respective
fragments of \emph{strictly guarded} formulae,
in which the tuple of free first-order variables
is explicitly guarded.
\end{defi}
Clearly $\GF \subseteq \muGF \subseteq \GSOg \subseteq \GSO$.
Similar inclusions hold for the corresponding strict fragments.
Furthermore, $\MSO \subseteq \GSO$
since monadic relations are guarded (by the equality predicate).
We shall see that
the restriction to least fixed points that are guarded --
i.e., fixed points of formulae in the starred logics --
is the right counterpart, in the guarded world, for
monadic fixed points.
For the boundedness problem, moreover, we shall have
reductions from $\BDD(\GF)$ to $\BDD(\GFs)$ and
from $\BDD(\GSOg)$ to $\BDD(\GSOgs)$, see Section~\ref{GFredsec}.
The guarded fragment $\GF$ as well as its fixed-point extension $\muGF$
are preserved under \emph{guarded bisimulation,} the infinitary game
equivalence associated to the restricted quantification pattern of
guarded quantification. Guarded bisimulation equivalence plays a role for
guarded logics that is analogous to the role of ordinary bisimulation
for modal logics. In fact, just as modal logic is
the bisimulation-invariant fragment of first-order logic \cite{Ben83},
so $\GF$ corresponds to the fragment of first-order logic
that is invariant under guarded bisimulation \cite{ABN}\?;
and just as $\Lmu$~is the bisimulation-invariant fragment
of monadic second-order logic \cite{JaninWalukiewicz},
so $\muGF$ corresponds to the fragment of $\GSO$ that is
invariant under guarded bisimulation \cite{GHO}.
Note that, despite its name, $\GSO$ is not invariant under guarded bisimulation.
The model theory and crucial algorithmic properties of $\GF$ and $\muGF$
are discussed in \cite{Graedel99} and \cite{GW}. For both logics,
much of their well-behavedness is due to invariance under
guarded bisimulation equivalence, and, consequently, the
`generalised tree-model property' \cite{Graedel99}\?:
by means of a natural process of guarded tree unfolding, any structure
can be transformed into a guarded bisimilar structure that admits a
tree-decomposition based on guarded subsets. Hence any satisfiable
formula of $\GF$ or $\muGF$ has a model which is guarded tree-decomposable
so that its tree-width is bounded by the width of the underlying vocabulary.
Because of its vicinity to the modal fragment,
$\GF$ has been a promising candidate for
decidability of boundedness, even not just for
monadic least fixed points. Approaches to $\BDD(\GF)$
along those lines that worked for $\ML$ and even for $\AFO$
-- viz., the use of invariance under guarded bisimulation and
the guarded version of the Barwise--Moschovakis theorem --
have not been successful. Our present techniques do indeed
yield decidability of $\BDD(\GF)$,
see Corollary~\ref{cor:decidability of BDD(GF), etc},
and thus settle a major open problem in the classical
context. As we do not rely on either compactness or locality
criteria in our approach, we do get a much stronger result
in Theorem~\ref{thm: bdd for GSO over bounded twd},
concerning the decidability
of $\BDD(\GSOs,\cW_k)$, the boundedness problem for
least fixed points over $\GSOs$-formulae over the class
of all relational structures of tree-width up to~$k$.
This decidability is even uniform w.r.t.\ tree-width, so that
both the $X$-positive $\GSOs$-formula and the tree-width
parameter~$k$ may be regarded as input to a single algorithm.
\begin{thm}\label{thm:GFmuGF}
The following are decidable\?:
$\BDD(\GF)$, $\BDD(\muGF)$, $\BDD(\GSOg,\cW_k)$,
$\BDD(\GSOs,\cW_k)$, $\BDDm(\GSO,\cW_k)$.
\end{thm}
The transfer and reduction techniques to be discussed below
immediately show that decidability for $\BDD(\GFs)$ and
$\BDD(\muGFs)$ are an immediate
consequence of decidability for $\BDD(\GSOs,\cW_k)$.
These results essentially invoke the generalised tree-model
property of $\GF$.
As far as undecidability results are concerned,
we have the following fundamental result,
which follows from the proof given in \cite{GaifmanMaSaVa93}.
\begin{thm}\label{thm:FOundec}
$\BDD(\FO,\cP)$ is undecidable,
where $\cP$~is the class of all finite paths.
\qed\end{thm}
\begin{cor}
$\BDD(\GSO,\cW_k)$ is undecidable.
\end{cor}
In the same way, we obtain undecidability of $\BDD(L,\cC)$
for every logic $L \supseteq \FO$ and class $\cC \supseteq \cP$
in which the class of all finite paths is $L$-definable.
Examples include boundedness of $\MSO$ over the class of all trees, over
the class of all finite trees, or over the class of all structures
of tree-width~$k$.
The fragments discussed so far are closed under (at least)
positive boolean connectives and relativisation to unary
predicates. They are also closed under the substitution
operation used in defining the finite stages of fixed points.
So Observation~\ref{obs:SAT vs BDD}
applies to all of them and highlights the role of
$\EFO$, $\AFO$, $\ML$, $\Lmu$, $\GF$ and $\muGF$ as natural candidates
for decidability of $\BDD(L)$.
For $\FO$, $\MSO$ and $\GSO$ on the other hand,
not $\BDD(L)$ but at best
$\BDD(L,\cC)$ for suitably restricted classes $\cC$
can be decidable.
\section{Transfer properties for BDD}
\label{sect:transfer}
Model-theoretic transfer results involving special, restricted classes
of models are often useful. Key examples are provided by
the finite model property or the tree-model property, which, as
transfer results for satisfiability, can be useful
towards establishing decidability of $\SAT(L)$. The following
introduces a similar notion in connection with the boundedness
problem. The most far-reaching among these properties, which in the light
of our key result yields the strongest decidability consequences
for the boundedness problem, is the bounded-tree-width property.
We first define a general notion of transfer, then several
concrete specialisations that we need in the sequel.
\begin{defi}
A logic~$L$ allows \emph{$\cC$-to-$\cC'$ transfer for $\BDD$}
if, for all $\varphi \in L$,
$\varphi$~is bounded over~$\cC$ iff it is bounded over~$\cC'$\?:
$\BDD(L,\cC) = \BDD(L,\cC')$.
A logic~$L$ has the \emph{$\cC$-property for $\BDD$}
if it allows transfer from the class of all structures to $\cC$\?;
i.e., if $\BDD(L) = \BDD(L,\cC)$.
\end{defi}
Let $\cW_k$ stand for the class of all relational structures
of tree-width up to~$k$\?;
similarly $\cT_k$ stands for the class of tree models
of branching degree up to~$k$.
In accordance with the above, we say that
$L$~has the \emph{tree-width-$k$ property} for $\BDD$ for some concrete
bound~$k$ if $\BDD(L) = \BDD(L,\cW_k)$. In a similar spirit,
one could consider transfer properties from the class of all tree
models to the class of $k$-branching tree models, for concrete bounds $k$.
In both cases, however, our decidability arguments
require just a computable dependence of the width parameter
on the input $\varphi \in L$, rather than a uniform constant bound.
This motivates the following.
\begin{defi}
We say that $L$ has the \emph{bounded-tree-width property for $\BDD$}
if, for some computable function~$f$,
$\varphi \in L$ is bounded iff $\varphi$~is bounded over $\cW_{f(\varphi)}$
(transfer to models of bounded tree-width).
Similarly, $L$ has the \emph{bounded-branching property for
$\BDD$ over trees} if, for some computable function~$f$,
$\varphi \in L$ is bounded over the class of all tree models
iff it is bounded over $\cT_{f(\varphi)}$
(transfer to tree models of bounded branching).
\end{defi}
In all natural cases a $\cC$-model property (transfer for $\SAT(L)$)
implies a $\cC$-property for $\BDD$. This is clearly the
case if $L$ is closed under the kind of substitution used to define
the finite stages and under boolean connectives. In that case,
the finite stages $\varphi^\alpha$ for $\alpha < \omega$ and the finite
stage increments $\varphi^{\alpha +1} \wedge \neg \varphi^\alpha$ are
definable by formulae in~$L$ and $\varphi$~is unbounded iff
all these formulae are satisfiable.
Concerning the finite model property for $\BDD$, note that
(even for fragments $L \subseteq \FO$) it does not
imply decidability of $\BDD(L)$\?: one still would need to check
satisfiability for each member of the infinite family
$\varphi^{\alpha+1} \wedge \neg \varphi^\alpha$ (albeit just in finite models).
\subsection{Transfer results for classical fragments}
\label{sect:classical transfer}
We collect some transfer results for the fragments
and logics discussed in the last section.
\begin{observation}\label{obs:classical transfer results}
\begin{enumerate}[label=\({\alph*}]
\item
$\EFO$, $\AFO$, $\ML$, $\Lmu$ and\/ $\GF$
have the \emph{finite model property} for $\BDD$ just as for $\SAT$.
\item[\normalfont(b)]
$\ML$ and\/ $\Lmu$ have the \emph{tree-property} for
$\SAT$ and\/ $\BDD$\?;
$\ML$ even allows transfer to \emph{finite} tree-models
of bounded branching.
\item
$\EFO$, $\AFO$, $\ML$, $\Lmu$, $\GF$ and $\muGF$ all have
the \emph{bounded-tree-width property} for $\SAT$ and\/ $\BDD$.
Among these, the modal logics $\ML$, $\Lmu$ even allow transfer to
tree models of bounded branching\?;
$\EFO$, $\AFO$, $\GF$ and $\muGF$ allow transfer to models of
bounded tree-width, in the case of $\EFO$, $\AFO$, $\GF$ even
to \emph{finite} models of bounded tree-width.
\end{enumerate}
\end{observation}
\noindent More specifically, the necessary tree-width~$k$ in~(c) can be
bounded by the width of the underlying vocabulary~$\tau$
in the modal and guarded cases,
and (for a rough bound) by the size of the given formula~$\varphi$
in the case of $\EFO$, $\AFO$.
Most of these statements follow
from corresponding properties for $\SAT(L)$,
which are well known from the literature
(cf.~in particular Observation~\ref{obs:SAT vs BDD} above).
The bounded-tree-width property for $\BDD$ in the
case of $\GF$ and $\muGF$ is a direct consequence of
preservation of these logics under guarded bisimulation. Guarded
tree-unfoldings \cite{Graedel99,GHO} of arbitrary models yield
models possessing a tree decomposition whose bags are guarded subsets,
hence of width bounded by the width of~$\tau$.
For the assertions concerning the fragments $\EFO$ and $\AFO$, which
are not closed under negation, we prove the following lemma.
\begin{lem}
$\EFO[\tau]$ and $\AFO[\tau]$ allow transfer for $\BDDm$ to
finite models of bounded tree-width.\footnote{
Here tree-width can be bounded by
the size of the given prenex formula $\varphi(X,x)$\?; a better bound
would be the tree-width of the quantifier-free kernel formula.}
\end{lem}
\begin{proof}
We explicitly treat the case of $\EFO$\?; the argument for $\AFO$ is
strictly analogous.
For $X$-positive $\varphi(X,x) \in \EFO[\tau]$ and finite $\alpha < \omega$,
the stage increment
$\varphi^{\alpha+1}(\fA) \smallsetminus \varphi^\alpha(\fA)$
is uniformly definable by a conjunction of a purely existential
formula
$\varphi^{\alpha+1}(x) \in \EFO[\tau]$
and a purely universal
formula in $\AFO[\tau]$ equivalent to the negation of $\varphi^\alpha(x)$.
Formulae of this kind are known to have the finite
model property\?:\footnote{They fall in particular within
the Bernays--Sch\"onfinkel class of prenex $\FO$-formulae
with quantifier prefix $\exists^\ast\forall^\ast$, cf.~\cite{BGG},
but a more direct argument suffices here.}
from an arbitrary model $(\fA,a)$ of some conjunction of a prenex
$\exists^\ast$-formula $\psi_1(x)$ and a prenex
$\forall^\ast$-formula $\psi_2(x)$, one obtains a finite model
by restricting $\fA$ to $a$ together with any chosen instantiation for the
existentially quantified variables in $\psi_1$\?; this restriction
still satisfies $\psi_1$, and as an induced substructure
of $(\fA,a) \models \psi_2$ it also still satisfies the universal
formula~$\psi_2$.
To obtain suitable (finite) models of bounded tree-width,
though, we need to consider the stronger preservation properties of
the formulae $\varphi^{\alpha}(x) \in \EFO[\tau]$,
and to some extent use the polarity constraints in $\EFO$ and $\AFO$.
The following argument also makes an interesting connection
with $\GF$.
Let w.l.o.g.\ $\varphi$ be of the form
\begin{align*}
\varphi(X,x) =
\exists \bar{y} \bigvee_i
\bigl( \rho_i(x,\bar{y}) \wedge \bigwedge_{j \in s_i} X y_j \bigr)
\end{align*}
where $\bar{y} = (y_1,\dots, y_k)$,
the $\rho_i$ are conjunctions of relational $\tau$-atoms
(not involving $X$), and $s_i \subseteq \{ 1,\dots, k \}$.
For any $\tau$-structure $\fA$ let $\hat{\fA}$ be its expansion to
a $\hat{\tau}$-structure by new relations~$R_i$
of arity $k+1$, with $R_i$~defined by~$\rho_i$.
In $\hat{\fA}$, $\varphi$ is equivalent to the $\GFs$-formula
\begin{align*}
\hat{\varphi}(X,x) =
\exists \bar{y} \bigvee_i
\bigl( R_i x\bar{y} \wedge \bigwedge_{j \in s_i} X y_j \bigr)\,.
\end{align*}
An analogous equivalence obtains for formulae $\varphi^\alpha(x) \in \GF[\tau]$
and $\hat{\varphi}^\alpha(x) \in \GF[\hat{\tau}]$
defining the finite stages w.r.t.\ $\varphi$ and $\hat{\varphi}$.
Obviously
\begin{align*}
\bigwedge_i \forall x \forall \bar y \bigl( R_i x\bar y \to \rho_i(x,\bar y) \bigr)
\models \forall x \bigl( \hat{\varphi}^\alpha(x) \to \varphi^\alpha(x) \bigr)\,,
\tag{$\ast$}
\end{align*}
where the
formula on the left-hand side is in $\GF[\hat{\tau}]$.
Note, however, that implications of the form
$\forall x \forall \bar{y} \bigl( \rho_i(x,\bar{y}) \to R_i x\bar{y} \bigr)$,
which would be needed
towards the equivalence between $\varphi^\alpha$ and $\hat{\varphi}^\alpha$
cannot in general be expressed in $\GF$.
Let $\hat{\fA}^\ast$ be a guarded bisimilar unfolding of~$\hat{\fA}$.
Its tree-width is bounded by the maximum of
the width of~$\tau$ and $k+1$.
We also write $\fA^\ast$ for the $\tau$-reduct of $\hat{\fA}^\ast$.
Let $\pi : \hat{\fA}^\ast \to \hat{\fA}$ be the
projection from the unfolding onto the base structure\?;
$\pi$~is a homomorphism inducing the natural guarded bisimulation
between $\hat{\fA}^\ast$ and $\hat{\fA}$.
Preservation of $\GF[\hat{\tau}]$ under guarded bisimulations implies
that, for all $\alpha < \omega$,
\begin{align*}
\hat{\fA}^\ast, a \models \hat{\varphi}^\alpha
\quad\text{iff}\quad
\hat{\fA}, \pi(a) \models \hat{\varphi}^\alpha.
\end{align*}
Since $\fA,\pi(a) \models \varphi^\alpha$ implies $\hat\fA, \pi(a) \models \hat\varphi^\alpha$
and, therefore, also $\hat\fA^*,a \models \hat\varphi^\alpha$,
it follows with $(\ast)$ above that
$\fA,\pi(a) \models \varphi^\alpha$ implies
$\fA^*,a \models \varphi^\alpha$.
In the opposite direction,
since the~$\varphi^\alpha$, as existential positive
formulae, are preserved under homomorphisms,
the implication
$\fA^\ast, a \models \varphi^\alpha \Rightarrow
\fA, \pi(a) \models \varphi^\alpha$ is straightforward.
Therefore, for all $a \in \fA^\ast$ and all $\alpha < \omega$,
\begin{align*}
\fA^\ast, a \models \varphi^\alpha
\quad\text{iff}\quad
\fA, \pi(a) \models \varphi^\alpha,
\end{align*}
whence $\cl{\varphi}{\fA} = \cl{\varphi}{\fA^\ast}$.
Hence $\varphi$ is bounded iff it is bounded over structures
whose tree-width is bounded by the
maximum of the width of~$\tau$ and $k+1$.
In order to get back to finite models
of bounded tree-width, we may apply the simple
argument from above to find a finite induced substructure within
some $(\fA^\ast, a)$ that still satisfies the corresponding
$\exists^\ast/\forall^\ast$-conjunction
$\varphi^{\alpha+1}(x) \wedge \neg \varphi^\alpha(x)$.
\end{proof}
\subsection{\boldmath Transfer for $\MSO$ over trees}
At the level of $\MSO$ we obtain a bounded-branching property
for $\BDDm$ over trees. The availability of transfer at least
down to countable branching is essential to make a connection
via interpretations with our core result that was formulated over
ternary trees.
\begin{prop}\label{prop:countable branching property for MSO}
$\MSO$ has a countable branching property for monadic $\BDDm$ over trees.
\end{prop}
This statement follows immediately from
Proposition~\ref{prop:Loewenheim-Skolem for trees} below,
whose proof relies on
the availability of tree automata for $\MSO$ and involves, as a key step,
a L\"owenheim--Skolem property for $\MSO$-theories of trees.
We employ a certain kind
of tree automata introduced by Walukiewicz~\cite{Walukiewicz02}.
\begin{defi}
An \emph{$\MSO$-automaton} is a tuple $\cA = \langle Q,\Sigma,q_0,\delta,\Omega\rangle$
with a finite set of states~$Q$, an input alphabet~$\Sigma$,
an initial state~$q_0$, a parity function $\Omega : Q \to \omega$,
and a transition function $\delta : Q \times \Sigma \to \MSO$
that, given a state~$q$ and a letter~$c$, returns an $\MSO$-formula
$\delta(q,c)$ over the signature $\set{ P_q }{ q \in Q }$.
Such an automaton takes a $\Sigma$-labelled directed tree
$t=(T,E,\lambda)$ as input.
A \emph{run} of~$\cA$ on~$t$ is a function $\varrho : T \to Q$
with $\varrho(r) = q_0$ for the root $r$ of $(T,E)$
such that
\begin{align*}
\langle U_v,\bar P\rangle \models \delta(q,\lambda(v))\,,
\quad\text{for all } v \in T\,,
\end{align*}
where the universe~$U_v$ of the structure is the set of all children of~$v$
and the unary predicates are $P_p := \varrho^{-1}(p) \cap U_v$.
The run~$\varrho$ is \emph{accepting}
if, and only if, for all infinite branches $v_0v_1\dots$ of~$(T,E)$
\begin{align*}
\liminf_{n \to \infty} \Omega(\app{\varrho}{v_n}) \text{ is even.}
\end{align*}
The \emph{language recognised by~$\cA$} is the set $L(\cA)$
of all trees~$t$ such that there exists an accepting run
of~$\cA$ on~$t$.
\end{defi}
Over trees these automata have the same expressive power as monadic second-order logic.
\begin{thm}[Walukiewicz~\cite{Walukiewicz02}]
A class~$\cC$ of directed trees is definable
by an $\MSO$-sentence~$\varphi$ if, and only if,
it is recognised by some $\MSO$-automaton~$\cA$.
\qed\end{thm}
We use $\MSO$-automata to prove the following L\"owenheim-Skolem theorem.
\begin{prop}\label{prop:Loewenheim-Skolem for trees}
For every tree structure~$\fT$ there exists a countable tree structure $\fT_0 \subseteq \fT$
with the same $\MSO$-theory.
\end{prop}
\begin{proof}
We prove the proposition for directed trees.
Then the corresponding claim for undirected trees follows.
Suppose that $\fT$ is a directed tree with root~$r$.
Let us call a substructure $\fT_0 \subseteq \fT$ a
\emph{subtree} of~$\fT$ if $\fT_0$~is a tree and it contains the root~$r$.
To prove the claim, we construct a countable subtree $\fT_0 \subseteq \fT$
such that every $\MSO$-automaton accepting~$\fT$ also accepts~$\fT_0$.
Since every $\MSO$-formula is equivalent (on trees) to an $\MSO$-automaton
and since $\MSO$ is closed under complement, it follows that
$\fT$ and $\fT_0$ have the same $\MSO$-theory.
To construct~$\fT_0$ we proceed as follows.
For every $\MSO$-automaton~$\cA$ that accepts~$\fT$ and every vertex
$v \in T$, we fix a countable set $S_\cA(v) \subseteq T$ of children of~$v$
such that the following holds\?:
\begin{itemize}[label=$(*)$]
\item Every subtree $\fT_0 \subseteq \fT$ such that
\begin{align*}
v \in T_0 \quad\text{implies}\quad S_\cA(v) \subseteq T_0
\end{align*}
is accepted by~$\cA$.
\end{itemize}
Let us call a subtree~$\fT_0$ \emph{$S_\cA$-closed} if
$v \in T_0$ implies $S_\cA(v) \subseteq T_0$.
We take for~$T_0$ the minimal subset of~$T$ containing
the root~$r$ that is $S_\cA$-closed for every~$\cA$ accepting~$\fT$.
The subtree~$\fT_0$ induced by~$T_0$ is countable and has the desired property.
To define $S_\cA$ we fix an accepting run~$\varrho$ of~$\cA$ on~$\fT$.
Let $v \in T$ be a vertex with label~$c$ and let $U$~be the set of children
of~$v$ in~$T$.
For each state $q \in \varrho(v)$, $\varrho$~induces a structure $\langle U,\bar P\rangle$
satisfying the transition formula $\delta(q,c)$.
For every state $p \in Q$, we select a set $X^q_p \subseteq P_p = \varrho^{-1}(p) \cap U$ as follows.
If $P_p$~is countable, we set $X^q_p := P_p$.
Otherwise, we choose an arbitrary countably infinite subset $X_p \subseteq P_p$.
Then we set
\begin{align*}
S_\cA(v) := \bigcup_{q \in Q} \bigcup_{p \in Q} X^q_p\,.
\end{align*}
We claim that, for every $S_\cA$-closed subtree $\fT_0 \subseteq \fT$,
the restriction of~$\varrho$ to~$T_0$ is an accepting run of~$\cA$ on~$\fT_0$.
Obviously, every infinite branch of~$\fT_0$ is an infinite branch of~$\fT$
and, hence, satisfies the parity condition. So we only need to check
that the transition formulae hold at each vertex.
Let $v \in T$ be a vertex with label~$c$ and with set of children~$U$, and
let $\langle U,\bar P\rangle$ be the structure induced by~$\varrho$.
Since $\varrho$~is a run, we have
\begin{align*}
\langle U,\bar P\rangle \models \delta(\varrho(v),c)\,.
\end{align*}
Note that the structure $\langle U,\bar P\rangle$ has only unary relations.
There is a well-known Ehrenfeucht-Fra\"\i ss\'e argument showing that
an $\MSO$-sentence of quantifier rank~$m$ cannot distinguish
two such structures $\langle U,\bar P\rangle$ and $\langle U',\bar P'\rangle$,
provided that each quantifier-free $1$-type is realised the same
number of times in both structures, or it is realised at least $2^m$~times
in each structure.
By definition of~$S_\cA(v)$, it follows that,
for all subsets $U_0 \subseteq U$ containing $S_\cA(v)$,
the structures
\begin{align*}
\langle U,\bar P\rangle
\quad\text{and}\quad
\langle U_0,\bar P|_{U_0}\rangle
\end{align*}
have the same $\MSO$-theory.
Consequently,
\begin{align*}
\langle U_0,\bar P|_{U_0}\rangle \models \delta(\varrho(v),c)\,.
\end{align*}
In particular, this is the case for $U_0 := U \cap T_0$.
Therefore, $\varrho \restriction T_0$ is a run.
\end{proof}
\begin{rem}
If, instead of the full $\MSO$-theory, we are only interested in
the preservation of a single $\MSO$-sentence,
the construction of the theorem yields a tree that is finitely branching.
\end{rem}
\begin{proof}[Proof of Proposition~\ref{prop:countable branching property for MSO}]
Clearly, if an $\MSO$-formula $\varphi(X,x)$ is bounded over the class of all
trees, it is also bounded over the class of all countable trees.
Conversely, suppose that $\varphi(X,x)$ is unbounded over arbitrary
trees. Then we can find, for every $\alpha < \omega$, a tree~$\fT_\alpha$
satisfying the formula
$\psi_\alpha := \exists x[\varphi^{\alpha+1}(x) \land \neg\varphi^\alpha(x)]$.
By the above proposition, we can choose $\fT_\alpha$ to be countably branching.
Hence, $\varphi(X,x)$ is also unbounded over the class of all countably
branching trees.
\end{proof}
\section{Interpretations and reductions}
\label{sect:interpretations}
In the preceding section we have considered transfer of $\BDD(L,\cC)$
from one class~$\cC$ to a subclass $\cC_0 \subseteq \cC$.
In this section we will study more general reductions
of $\BDD(L,\cC)$ to $\BDD(L',\cC')$ where both the logic~$L$
and the class~$\cC$ may change.
\subsection{\boldmath A reduction for $\GF$}
\label{GFredsec}
We start by reducing $\BDD(\GSOg,\cC)$ to $\BDD(\GSOgs,\cC)$.
The following normal form for $\GSOg$-formulae is used in the proof
of the proposition below.
\begin{lem}\label{lem:normal form for GSOg}
Let $\varphi(\bar R,X,\bar x)$ be a $\GSOg$-formula with
free second-order variables $\bar R$,~$X$ and
free first-order variables~$\bar x$ that is positive in~$X$.
We can effectively construct
$\GSOg$-formulae $\psi^0_i$, $\psi^1_i$, for $i < n$,
such that
\begin{align*}
\varphi(\bar R,X,\bar x) \equiv \bigvee_{i < n}\bigl[\psi^0_i(X,\bar x) \land \psi^1_i(\bar R,X,\bar x)\bigr]\,,
\end{align*}
where
\begin{itemize}
\item the formulae~$\psi^0_i$ are quantifier-free and positive in~$X$,
\item the formulae~$\psi^1_i$ are positive in~$X$ and such that $X$~only appears in subformulae
of the form $\forall\bar y\vartheta$ and $\exists\bar y\vartheta$.
\end{itemize}
Furthermore, if $\varphi$~is a $\GF$-formula,
then so are the formulae $\psi^0_i,\psi^1_i$, $i < n$.
\end{lem}
\begin{proof}
We may assume that $\varphi$~is in negation normal form.
The claim follows by induction on the structure of~$\varphi$.
All other cases being trivial, we present only the case of second-order quantifiers.
Hence, let us assume that $\varphi = \mathsf{Q}Z\vartheta$, for $\mathsf{Q} \in \{\forall,\exists\}$.
By inductive hypothesis, we may assume that
\begin{align*}
\vartheta = \bigvee_{i < n}[\psi^0_i(X,\bar x) \land \psi^1_i(\bar R,Z,X,\bar x)]
\end{align*}
with $\psi^0_i$~and~$\psi^1_i$ as in the statement of the lemma.
In case of an existential quantifier, we are done since
\begin{align*}
\exists Z\vartheta \equiv \bigvee_{i < n}[\psi^0_i(X,\bar x) \land \exists Z\psi^1_i(\bar R,Z,X,\bar x)]\,.
\end{align*}
For a universal quantifier, note that
\begin{align*}
\forall Z\vartheta
&= \forall Z\bigvee_{i < n}\bigl[\psi^0_i(X,\bar x) \land \psi^1_i(\bar R,Z,X,\bar x)\bigr] \\
&\equiv \forall Z\bigwedge_{\sigma \in 2^n} \bigvee_{i < n} \psi^{\sigma(i)}_i \\
&\equiv \bigwedge_{\sigma \in 2^n} \forall Z
\Bigl[\bigvee_{i \in \sigma^{-1}(0)} \!\!\psi^0_i \ \lor\!\! \bigvee_{i \in \sigma^{-1}(1)} \!\!\psi^1_i\Bigr] \\
&\equiv \bigwedge_{\sigma \in 2^n}
\Bigl[\bigvee_{i \in \sigma^{-1}(0)} \!\!\psi^0_i \ \lor\ \forall Z\!\!\bigvee_{i \in \sigma^{-1}(1)} \!\!\psi^1_i\Bigr]\,.
\end{align*}
Hence, the claim follows by another application of the distributive law.
\end{proof}
\begin{prop}\label{prop: GSOg reduces to GSOgs}
For every formula $\varphi(X,\bar x) \in \GSOg[\tau]$,
we can effectively construct a formula $\varphi^g(X,\bar x) \in \GSOgs[\tau]$
such that $\varphi(X,\bar x)$ is bounded if, and only if, $\varphi^g(X,\bar x)$ is.
Furthermore, if $\varphi$~is a $\GF$-formula, then so is~$\varphi^g$.
\end{prop}
\begin{proof}
By the lemma we may assume that the formula $\varphi(X,\bar x)$ has the form
\begin{align*}
\varphi(X,\bar x) = \bigvee_{i < n}\bigl[\chi_i(X,\bar x) \land \psi_i(X,\bar x)\bigr]\,,
\end{align*}
where the formulae~$\chi_i$ are quantifier-free
and in the formulae~$\psi_i$ every occurrence of~$X$ is in a subformula
of the form $\forall\bar y\vartheta$ and $\exists\bar y\vartheta$.
Note that any occurrence of an atom~$X\bar y$ that is in the
scope of some (guarded\?!) first-order quantification may be replaced by the formula
$X^g\bar y := X\bar y \land \gdd(\bar y)$ without changing its semantics.
Therefore,
\begin{align*}
\varphi(X,\bar x) \equiv \bigvee_{i < n}\bigl[\chi_i(X,\bar x) \land \psi_i(X^g,\bar x)\bigr]\,,
\end{align*}
where $\psi_i(X^g,\bar x) := \psi_i(X^g/X,\bar x)$ is the formula obtained from~$\psi_i$
by replacing~$X$ by its guarded restriction~$X^g$ without affecting the semantics.
In the following a superscript $\rule{0pt}{1.5ex}^g$
is always used to indicate syntactic and/or semantic restriction to
the guarded part.
The fixed-point induction of $\varphi(X,\bar x)$
is closely related to the fixed-point induction of the strictly guarded formula
\begin{align*}
\varphi^g(X,\bar x) :=
\gdd(\bar x) \wedge \varphi(X,\bar x)
\equiv \bigvee_{i<n} \bigl[\gdd(\bar x) \wedge \chi_i(X,\bar x)
\wedge \psi_i(X^g,\bar x) \bigr]\,.
\end{align*}
Since $\varphi^g$~implies~$\varphi$ and both formulae are positive in~$X$,
it follows that the stages of~$\varphi^g$ are included in those for~$\varphi$.
In fact, it follows by a simple induction on~$\alpha$ that
$(\varphi^g)^\alpha(\fA) = (\varphi^\alpha(\fA))^g$.
Consequently, we have
\begin{align*}
\cl{\varphi^g}{\fA} \leq \cl{\varphi}{\fA}
\quad\text{and}\quad
(\varphi^\infty(\fA))^g = (\varphi^g)^\infty(\fA)\,.
\end{align*}
If we can show that there exists a constant $n < \omega$, depending only on~$\varphi$,
such that, for all structures~$\fA$,
\begin{align*}
\cl{\varphi}{\fA} \leq \cl{\varphi^g}{\fA} + n\,,
\end{align*}
then it follows that $\varphi$~is bounded if, and only if, $\varphi^g$~is bounded.
To find the constant~$n$, we consider the auxiliary formula
\begin{align*}
\xi(Z,X,\bar x) :=
\bigvee_{i<n} \bigl[\chi_i(X,\bar x) \wedge \psi_i(Z,\bar x)\bigr]
\end{align*}
in vocabulary $\tau \cup \{ Z \}$
(with a new second-order variable~$Z$ of the same arity~$r$ as~$X$,
which is regarded as a parameter) and
we consider its fixed-point induction in the expansion
$(\fA,P_0)$ of~$\fA$ where $Z$~is interpreted by the relation $P_0 := (\varphi^g)^\infty(\fA)$.
We claim that
\begin{align*}
\cl{\varphi}{\fA} \leq \cl{\varphi^g}{\fA} + \cl{\xi}{(\fA,P_0)}\,.
\end{align*}
Let $\gamma := \cl{\varphi^g}{\fA}$. We have shown above that
$P_0 = (\varphi^g)^\gamma(\fA) \subseteq \varphi^\gamma(\fA)$.
Using monotonicity, it follows by a simple induction on~$\alpha$ that
\begin{align*}
\varphi^\alpha(\fA) \subseteq \xi^\alpha(\fA,P_0) \subseteq \varphi^{\gamma+\alpha}(\fA)\,.
\end{align*}
The first inclusion implies that $\varphi^\infty(\fA) \subseteq \xi^\infty(\fA,P_0)$ while
the second inclusion implies that $\xi^\infty(\fA,P_0) \subseteq \varphi^\infty(\fA)$.
Setting $\beta := \cl{\xi}{(\fA,P_0)}$, it follows that
\begin{align*}
\varphi^\infty(\fA) = \xi^\beta(\fA,P_0) \subseteq \varphi^{\gamma+\beta}(\fA) \subseteq \varphi^\infty(\fA)\,.
\end{align*}
Hence, $\cl{\varphi}{\fA} \leq \gamma + \beta$, as desired.
We have shown that, for every structure~$\fA$,
\begin{align*}
\cl{\varphi^g}{\fA} \leq \cl{\varphi}{\fA} \leq \cl{\varphi^g}{\fA} + \cl{\xi}{(\fA,P_0)}\,.
\end{align*}
To conclude the proof it remains to prove that $\cl{\xi}{(\fA,P_0)}$ is uniformly bounded.
Note that $\xi$~treats~$Z$ as a static parameter, and only involves
its fixed-point variable~$X$ outside the scope of any quantifiers.
It follows that $\xi$~is trivially bounded (with a bound that is given by
the number of quantifier-free $r$-types in vocabulary $\tau \cup \{ Z \}$).
\end{proof}
Hence we may restrict attention
to fixed points over strictly guarded formulae.
This means that $\BDD(\GSOg)$ reduces to $\BDD(\GSOgs)$.
Let us remark that a corresponding result for $\GSO$ fails.
An argument analogous to the above
also applies to $\muGF$\?: according to \cite{GHO},
every $\muGF$-formula is equivalent to one where
every fixed-point operator is applied to a
strictly guarded formula.
For $\muGF$-formulae of this form,
a variant of Lemma~\ref{lem:normal form for GSOg} holds.
This is all we need for the proof of Proposition~\ref{prop: GSOg reduces to GSOgs}.
Consequently, $\BDD(\muGF)$ reduces to $\BDD(\muGFs)$.
\subsection{\boldmath $\MSO$-interpretations in trees}
In the first part we have obtained the decidability of $\BDDm(\MSO,\cT_3)$.
In this section, we use model-theoretic interpretations to
reduce the decidability of $\BDDm(\MSO,\cW_k)$ to this problem.
\begin{defi}
Let $\sigma$~and~$\tau$ be relational signatures.
\textup{(a)} A \emph{definition scheme} for an $\MSO$-interpretation
from~$\sigma$ to~$\tau$ is a list
\begin{align*}
\cI = \bigl\langle\chi, \delta(x),\varepsilon(x,y),(\varphi_R(\bar x))_{R \in \sigma}\bigr\rangle
\end{align*}
of $\MSO[\tau]$-formulae where
$\chi$~is a sentence,
$\delta(x)$~has one free variable, $\varepsilon(x,y)$ has two,
and the number of free variables of $\varphi_R(\bar x)$ equals the arity of the
relation symbol~$R$.
\textup{(b)} The \emph{operation defined} by a definition scheme~$\cI$
maps $\tau$-structures~$\fA$ to $\sigma$-structures $\cI(\fA)$.
A $\tau$-structure~$\fA$ such that
$\fA \models \chi$, $\delta[\fA] \neq \emptyset$
and such that $\varepsilon$~defines an equivalence relation~$\sim$
on $\delta[\fA]$,
is mapped to the $\sigma$-structure~$\fB$ with universe
\begin{align*}
B := \set{ [a]_\sim \in A/{\sim} }{ \fA \models \delta(a) }
\end{align*}
and, for each $n$-ary relation $R \in \sigma$, the relation
\begin{align*}
R^\fB := \set{ [\bar a]_\sim \in A^n/{\sim} }{ \fA \models \varphi_R(\bar a) }\,.
\end{align*}
For any other $\tau$-structure~$\fA$, we let $\cI(\fA)$ be undefined.
\textup{(c)} An \emph{$\MSO$-interpretation} is an operation defined
by a definition scheme~$\cI$.
If $\cC$~is a class of $\tau$-structures, we set
\begin{align*}
\cI(\cC) := \set{ \cI(\fA) }{ \fA \in \cC \text{ such that } \cI(\fA) \text{ is defined} }\,.
\end{align*}
\end{defi}
For the proof of Proposition~\ref{prop: boundedness and interpretations} below,
let us recall the following well-known lemma.
We include a proof, so that we may refer to a
precise format of the formulae~$\psi^\cI$ later.
\begin{lem}[Interpretation Lemma]\label{lem:interpretations}
Let $\cI = \bigl\langle\chi, \delta(x),\varepsilon(x,y),(\varphi_R(\bar x))_{R \in \sigma}\bigr\rangle$
be an $\MSO$-interpretation.
For every $\MSO[\sigma]$-formula~$\psi$,
there exists an $\MSO[\tau]$-formula~$\psi^\cI$ such that,
for all $\tau$-structures~$\fA$ and every tuple $\bar a$ in~$A$, we have
\begin{align*}
\fA \models \psi^\cI(\bar a)
\quad\text{iff}\quad
&\cI(\fA) \text{ is defined, }
\fA \models \delta(a_i) \text{ for all } i, \text{ and}\\
&\cI(\fA) \models \psi([\bar a]_\sim)\,.
\end{align*}
If $\psi$~is positive in a predicate~$X$ and the
formula $\varphi_X(\bar x)$ from~$\cI$ is positive
in a predicate~$Y$, then $\psi^\cI$~is also positive in~$Y$.
\end{lem}
\begin{proof}
First, we define a formula~$\psi^*$ by induction on~$\psi$ as follows\?:
\begin{alignat*}{-1}
(R\bar c)^* &:= \textstyle\exists\bar z\bigl[\bigwedge_i \varepsilon(z_i, c_i) \land \varphi_R(\bar z)\bigr]\,,
&\qquad
(\exists y\vartheta)^* &:= \exists y[\delta(y) \land \vartheta^*]\,, \\
(c\seq d)^* &:= \varepsilon(c,d)\,,
&\qquad
(\forall y\vartheta)^* &:= \forall y[\delta(y) \to \vartheta^*]\,,
\end{alignat*}
and the translation ${}\cdot{}^*$ commutes with boolean operations and set quantifiers.
Then we can set
\begin{align*}
\psi^\cI := \chi' \land \bigwedge_i \delta(x_i) \land \psi^*
\end{align*}
where
the conjunction is over all free variables of~$\psi$ and
$\chi' := \chi \land \eta$ is the conjunction of~$\chi$ with a formula~$\eta$ stating
that $\varepsilon$~defines an equivalence relation on~$\delta$.
\end{proof}
\begin{prop}\label{prop: boundedness and interpretations}
Let $\cI$~be an $\MSO$-interpretation and
$\cC$~a class of $\tau$-structures.
If\/ $\BDDm(\MSO,\cC)$ is decidable then so is $\BDDm(\MSO,\cI(\cC))$.
\end{prop}
\begin{proof}
We use the same notation as in the proof of Lemma~\ref{lem:interpretations}.
Suppose that $\cI = \langle\chi,\delta,\varepsilon,(\varphi_R)_R\rangle$
and let $\psi(x,X)$ be a formula over the signature $\sigma \cup \{x,X\}$.
We extend the notation~$\vartheta^*$ from above to formulae containing a free
set variable~$X$ by treating~$X$ as a relation defined by the formula~$Xx$, i.e.,
we set
\begin{align*}
(Xc)^* := \exists z[\varepsilon(z,c) \land Xz]\,.
\end{align*}
Let $\fA \in \cC$ be a structure such that $\cI(\fA)$ is defined.
Note that Lemma~\ref{lem:interpretations} implies that
\begin{align*}
\fA \models
\forall x\forall y\bigl[\varepsilon(x,y) \to (\vartheta^*(x) \leftrightarrow \vartheta^*(y))\bigr]\,,
\quad\text{for every formula } \vartheta(x)\,.
\end{align*}
For formulae $\vartheta(x)$~and~$\psi(X,x)$, it follows by induction on the structure of~$\psi$ that
\begin{align*}
\fA \models
\forall x\bigl[\delta(x) \to \bigl((\psi[\vartheta/X])^* \leftrightarrow \psi^*[\vartheta^*/X]\bigr)\bigr]\,.
\end{align*}
A simple induction on~$\alpha$ yields
\begin{align*}
\fA \models \forall x\bigl[\delta(x) \to \bigl((\psi^\alpha)^* \leftrightarrow (\psi^*)^\alpha\bigr)\bigr]\,.
\end{align*}
Since $\fA \models \chi'$, it follows by the definition of the mapping $\vartheta \mapsto \vartheta^\cI$
that, for every $\alpha < \omega$, we have
\begin{align*}
\cI(\fA) &\models \forall x(\psi^{\alpha+1} \leftrightarrow \psi^\alpha) \\
\text{iff}\quad
\fA &\models \forall x(\psi^{\alpha+1} \leftrightarrow \psi^\alpha)^\cI \\
\text{iff}\quad
\fA &\models \chi' \land \forall x\bigl[\delta(x) \to
((\psi^{\alpha+1})^* \leftrightarrow (\psi^\alpha)^*)\bigr] \\
\text{iff}\quad
\fA &\models \forall x\bigl[(\chi' \land \delta(x) \land (\psi^{\alpha+1})^*) \leftrightarrow
(\chi' \land \delta(x) \land (\psi^\alpha)^*)\bigr] \\
\text{iff}\quad
\fA &\models \forall x\bigl[(\chi' \land \delta(x) \land (\psi^*)^{\alpha+1}) \leftrightarrow
(\chi' \land \delta(x) \land (\psi^*)^\alpha)\bigr] \\
\text{iff}\quad
\fA &\models \forall x\bigl[(\chi' \land \delta(x) \land \psi^*)^{\alpha+1} \leftrightarrow
(\chi' \land \delta(x) \land \psi^*)^\alpha\bigr] \\
\text{iff}\quad
\fA &\models \forall x[(\psi^\cI)^{\alpha+1} \leftrightarrow (\psi^\cI)^\alpha\bigr]\,.
\end{align*}
Consequently, $\psi$~is bounded over $\cI(\cC)$
if and only if $\psi^\cI$ is bounded over~$\cC$.
\end{proof}
\begin{cor}\label{cor:decidability and subclasses}
Let $\cC$~be a class of $\tau$-structures and $\psi$~an $\MSO$-formula.
If\/ $\BDDm(\MSO,\cC)$ is decidable then so is
$\BDDm(\MSO,\cC_\psi)$ where
\begin{align*}
\cC_\psi := \set{ \fA \in \cC }{ \fA \models \psi }\,.
\end{align*}
\end{cor}
\proof
We can use the interpretation $\cI = \langle\chi,\delta,\varepsilon,(\varphi_R)\rangle$
with
\begin{alignat*}{-1}
\chi &:= \psi\,, &\qquad
\varepsilon(x,y) &:= x\seq y\,, \\
\delta(x) &:= x\seq x\,, &\qquad
\varphi_R(\bar x) &:= R\bar x\,.\rlap{\hbox to 143 pt{
\qEd}}
\end{alignat*}
For the application to boundedness below we will need the following
interpretation results. First, let us consider classes of trees.
The proof of the following lemma is straightforward.
For (a)~and~(b), we use the usual first-child/next-sibling encoding of a tree,
while for~(c) we use a marking of the root, which can be used to recover the
orientation of the edges since we can express reachability in $\MSO$.
\begin{lem}\label{lem: tree interpretations}
There exist $\MSO$-interpretations mapping
\begin{enumerate}
\item the class~$\cT_3$ of all ternary trees to the class~$\cT_{\aleph_0}$ of all countable trees\?;
\item the class of all finite ternary trees to the class of all finite trees\?;
\item the class of all undirected trees to the class of all directed trees.
\qed\end{enumerate}
\end{lem}
Next, we study structures of bounded tree-width.
\begin{defi}
Let $\tau$ be a relational vocabulary and $\fA$ a $\tau$-structure.
A \defn{tree-decomposition} of~$\fA$
is a $2^A$-labelled directed tree $D=\parlr{T,E,\lambda}$
satisfying the following conditions\?:
\begin{itemize}
\item $\bigcup_{t\in T} \lambda(t) = A$.
\item For all relation symbols $R\in\tau$ of arity~$n$ and all tuples
$(a_1,\dots, a_n) \in R^\fA$,
there is some $t\in T$ such that $a_1,\ldots,a_n \in \app{\lambda}t$.
\item For all $a\in A$,
the set $\set{ t\in T }{ a\in\app{\lambda}t }$
is connected in $\parlr{T,E}$.
\end{itemize}
The \defn{width} of~$D$ is $\max_{t\in T} {\card{\app{\lambda}t}} - 1$.
The \defn{tree-width} of~$\fA$ is the minimum width
of a tree decomposition of~$\fA$.
\end{defi}
\begin{lem}\label{lem: interpretation for bounded twd}
For every $k < \omega$ and all relational vocabularies~$\tau$,
there exists an $\MSO$-interpretation
mapping the class of all trees to the class $\cW_k[\tau]$
of all relational $\tau$-structures of tree-width at most~$k$,
and the class of all finite trees
to the class of all finite structures from $\cW_k[\tau]$.
\end{lem}
\proof
For finite trees, such an interpretation was first given
by Courcelle and Engelfriet~\cite{CourcelleEngelfriet95},
but using a slightly different notion of an interpretation.
We give a detailed proof, since the precise version needed here
does not appear in the literature.
We start by explaining how we can encode a structure $\fA \in \cW_k[\tau]$
into a tree.
Then we will construct an interpretation~$\cI$ performing the inverse translation.
Let $\fA$ be a $\tau$-structure of tree-width at most~$k$
and let $D=\parlr{T,E,\lambda}$ be a tree decomposition of~$\fA$ of width at most~$k$.
It is no restriction to assume that
no $\lambda(v)$ is empty
and that there is some injective function $\iota : A \to T$
with $a \in \lambda(\iota(a))$, for all~$a$.
For every $v\in T$, we
fix an enumeration $c^v_0,\dots,c^v_{\ell_v}$
of $\lambda(v)$ where $0 \leq \ell_v \leq k$
and $a = c^{\iota(a)}_0$ for all $a\in A$.
We obtain a tree structure
by turning $(T,E)$ into an undirected tree
expanded by the following monadic relations\?:
\begin{itemize}
\item a unary predicate~$R$ containing only the root\?;
\item unary relations~$E_{i,j}$, for $0\leq i,j\leq k$,
containing those $v\in T$ such that $v$~has a parent~$u$ and $c^v_i = c^u_j$\?;
\item unary predicates $P_\fC$, for every $\tau$-structure~$\fC$ with universe
$C = \{0,\ldots,\ell\}$, for some $0\leq\ell\leq k$,
where $P_\fC$~contains those $v\in T$ such that $\fC$ is isomorphic
to the substructure of $\fA$ induced by $\app{\lambda}v$
via the isomorphism $i \mapsto c^v_i$.
\end{itemize}
The interpretation $\cI = \langle\chi, \delta(x),\varepsilon(x,y),(\varphi_R(\bar x))_{R \in \tau}\rangle$
that reverses this encoding is defined as follows.
The formula $\chi$~states that
\begin{itemize}
\item $R$ is a singleton, unless the universe is empty\?;
\item for all $0\leq i,i',j,j' \leq k$ with $i \neq i'$ and $j \neq j'$,
$E_{i,j}$ is disjoint from $E_{i,j'}$ and from $E_{i',j}$\?;
\item the $P_{\fC}$ are disjoint\?;
\item if $v \in P_\fC$ and $0\leq i,j\leq k$ are such that $i\geq\card C$,
then there is no $u \in T$ with $(v,u) \in E_{i,j}$ or $(u,v) \in E_{j,i}$.
\end{itemize}
For all $0\leq i,j\leq k$,
there is an $\MSO$-formula $\app{\varepsilon_{i,j}}{x,y}$
which defines the set of pairs $\parlr{v,u}$ such that $c^v_i = c^u_j$.
We set
\begin{align*}
\delta(x) := x\seq x
\quad\text{and}\quad
\varepsilon := \varepsilon_{0,0}\,.
\end{align*}
Finally, for each relation symbol $R\in\tau$ of arity~$r$, we define
\[
\app{\varphi_R}{x_1,\ldots,x_r} :=
\bigvee_{0\leq i_1\leq k}\dots\bigvee_{0\leq i_r\leq k}
\bigvee_{\substack{\fC \\ \parlr{i_1,\ldots,i_r}\in R^{\fC}}}
\exists y
\biggl[
\bigwedge_{1\leq j\leq r}\app{\varepsilon_{0,i_j}}{x_j,y}
\wedge P_{\fC}y
\biggr]\,.\eqno{\qEd}
\]
\section{Decidability results for boundedness}
\label{sect:decidabilities}
\label{sect:end II}
Using the reduction techniques of the previous sections
we obtain a wealth of decidability results.
We start with $\BDDm(\MSO,\cT)$ and $\BDDm(\MSO,\cW_k)$.
\begin{prop}\label{prop:boundedness for all trees}
The monadic boundedness problem for $\MSO$ over the class of all trees is decidable.
\end{prop}
\begin{proof}
By Proposition~\ref{prop:Loewenheim-Skolem for trees}, an $\MSO$-formula
is bounded over the class of all trees if, and only if, it is bounded over
the class of all countable trees. Hence, it is sufficient to prove that
$\BDDm(\MSO,\cT_{\aleph_0})$ is decidable.
By Lemma~\ref{lem: tree interpretations},
there exists an $\MSO$-interpretation~$\cI$ mapping~$\cT_3$
to $\cT_{\aleph_0}$. Hence, the decidability of $\BDDm(\MSO,\cT_{\aleph_0})$
follows by Proposition~\ref{prop: boundedness and interpretations}
and Theorem~\ref{thm:boundedness for ternary trees}.
\end{proof}
\begin{thm}\label{theo:tree-width extension}
For every $k < \omega$ and all relational vocabularies~$\tau$,
$\BDDm(\MSO,\cW_k[\tau])$, the monadic boundedness problem for $\MSO$
over the class~$\cW_k[\tau]$ of all relational $\tau$-structures
of tree-width at most~$k$ is decidable.
\end{thm}
\begin{proof}
With the help of
Lemma~\ref{lem: interpretation for bounded twd}
and Proposition~\ref{prop: boundedness and interpretations}
we can reduce $\BDDm(\MSO,\cW_k[\tau])$ to $\BDDm(\MSO,\cT)$.
The latter is decidable by
Proposition~\ref{prop:boundedness for all trees}.
\end{proof}
\begin{cor}
$\BDDm(\EFO)$, $\BDDm(\AFO)$, and\/ $\BDDm(\ML)$ are decidable.
\end{cor}
\begin{proof}
For each of these logics,
Observation~\ref{obs:classical transfer results}
provides a transfer result
to (finite) structures of bounded tree-width.
\end{proof}
Using similar techniques as above, one can extend Theorem~\ref{theo:tree-width extension}
to the extension of $\MSO$ by counting quantifiers,
to guarded second-order logic $\GSO$, and to simultaneous fixed points.
Instead of replacing $\MSO$ by a stronger logic,
one can also replace tree-width by clique-width.
We only give a sketch of the proof.
Let us denote by $\MSO+\mathrm{C}$ and $\GSOs+\mathrm{C}$ the extension
of the respective logic by predicates of the form
$\lvert X\rvert < \aleph_0$ and $\lvert X\rvert \equiv k \pmod m$,
where $X$~is a second-order variable and $k,m < \omega$.
A \emph{simultaneous} fixed point is defined by a system of formulae
$\varphi_0(\bar X,\bar x_0),\dots,\varphi_{n-1}(\bar X,\bar x_{n-1})$
with first-order variables~$\bar x_i$ and $n$~second-order variables $X_0,\dots,X_{n-1}$.
\begin{thm}\label{thm: bdd for GSO over bounded twd}
For every $k < \omega$, the boundedness problem for simultaneous $(\GSOs + \mathrm{C})$-fixed points
over the class of all relational structures of tree-width at most~$k$
is decidable.
\end{thm}
\begin{proof}[Sketch]
Since structures of tree-width at most~$k$ are sparse,
we can find, for every $(\GSOs+\mathrm{C})$-formula, an equivalent $(\MSO+\mathrm{C})$-formula
(see \cite{Courcelle03,Blumensath10}).
Therefore, the boundedness problem reduces to the boundedness
of simultaneous $(\MSO+\mathrm{C})$-fixed points on that class.
Using the interpretation argument from above, we can reduce it further
to the boundedness for simultaneous $(\MSO+\mathrm{C})$-fixed points
on the class of all ternary trees. On ternary trees,
$\MSO+\mathrm{C}$ collapses to $\MSO$.
Therefore, we only need to decide boundedness for simultaneous
$\MSO$-fixed points.
Finally, using again an interpretation argument we can replace
a simultaneous fixed point by an ordinary one (by making several copies of each vertex
of the tree, one for each component of the simultaneous fixed point).
\end{proof}
\begin{cor}\label{cor:decidability of BDD(GF), etc}
The following problems are decidable\?:
$\BDD(\GF)$, $\BDD(\muGF)$, $\BDDm(\Lmu)$, $\BDD(\GSOg,\cW_k)$, and\/
$\BDDm(\GSO,\cW_k)$.
\end{cor}
\begin{proof}
By Observation~\ref{obs:classical transfer results},
$\GF$ and $\muGF$ have the bounded-tree-width property for $\BDD$.
Hence, $\BDD(\GF)$ and $\BDD(\muGF)$ reduce to $\BDD(\GF,\cW_k)$ and $\BDD(\muGF,\cW_k)$, respectively,
which in turn are subsumed by $\BDD(\GSOg,\cW_k)$.
According to Proposition~\ref{prop: GSOg reduces to GSOgs},
$\BDD(\GSOg,\cW_k)$ reduces to $\BDD(\GSOgs,\cW_k)$ which is decidable
by Theorem~\ref{thm: bdd for GSO over bounded twd}.
For $\BDDm(\GSO,\cW_k)$ note that, singletons being always guarded,
every $\GSO$-formula $\varphi(X,x)$
with a single free first-order variable~$x$ belongs
to $\GSOs$.
Hence, $\BDDm(\GSO,\cW_k)$ reduces to $\BDDm(\GSOs,\cW_k)$
and the claim follows again from Theorem~\ref{thm: bdd for GSO over bounded twd}.
\end{proof}
\section*{Part III. Complexity Results}
\section{Complexity}
\label{sect:complexity}
\label{sect:start III}
\label{sect:end III}
In connection with our decision procedures we have not been
specific about the algorithmic complexities involved.
The fact that we have to deal with $X$-positive $n$-types
as basic data has a major impact on all upper bounds that
can be derived from our approach.
Space $\exp^n(\Theta(\len{\tau}))$ is necessary to even
store such a type ($\exp^n$ denotes
the $n$-fold application of the exponentiation operation,
that is, a tower of height~$n$).
Overall it is straightforward to check that, on input~$\varphi$,
our decision procedure runs
in time $\exp^{\app{\qr}{\varphi} + O(1)}(\len{\varphi})$.
We now provide a corresponding lower bound,
even for monadic boundedness for first-order logic over just finite trees.
Note that, for most natural fragments of $\MSO$,
one can obtain a lower bound from the complexity of the
satisfiability problem of the fragment.
For instance, $\BDDm(\ML)$ is \textsc{Pspace}-hard
since satisfiability for $\ML$ is \textsc{Pspace}-complete.
For first-order logic over finite words \emph{with order,}
as well as over finite trees \emph{without order,}
we can similarly derive lower bounds from corresponding
bounds for the satisfiability problem.
\begin{thm}\label{theo:lower complexity bound}
\textup{(a)}
The boundedness problem $\BDDm(\FO,\cP)$ for
first-order logic over the class of finite words with order,
is complete for $\DSPACE(\exp^{\poly(n)}(1))$.
\textup{(b)}
The boundedness problem
$\BDDm(\FO,\cT_\fin)$ for first-order logic over the class
of all finite trees is hard for $\DTIME(\exp^{\app{\poly}n}(1))$.
\qed\end{thm}
Part~(a) follows from the corresponding
result for $\SAT(\FO,\cP)$\?;
see \cite{Reinhardt02} for a proof and exposition.
Part~(b) is a consequence of the following complexity bound for
$\SAT(\FO,\cT_\fin)$\?; although this is based on standard techniques,
we include a proof since this complexity bound does not seem to appear
in the literature.
\begin{prop}\label{Prop: SAT(FO,Tfin) hard}
$\SAT(\FO,\cT_\fin)$ is hard for $\DTIME(\exp^{\app{\poly}n}(1))$
under polynomial time reductions.
\end{prop}
\begin{proof}
We show that $\SAT(\FO,\cT_\fin)$ is hard for $\NTIME(\exp^{\app{\poly}n}(1))$,
which is the same as $\DTIME(\exp^{\app{\poly}n}(1))$.
We use the following tiling problem, which is complete for
$\NTIME(\exp^{\app{\poly}n}(1))$ (see~\cite{Harel85} for an overview)\?:
given a set~$D$ of tiles, two relations $H,V \subseteq D \times D$, and
a natural number~$n$ (in unary encoding), determine whether there exists a tiling
of the $(\exp^n(1)\times \exp^n(1))$-grid, i.e., a function
$\tau : \exp^n(1)\times \exp^n(1) \to D$ such that
\begin{align*}
(\tau(x,y),\tau(x+1,y)) \in H
\quad\text{and}\quad
(\tau(x,y),\tau(x,y+1)) \in V\,,
\quad\text{for all } x,y\,.
\end{align*}
For the reduction, we set $N := \exp^n(1)$.
One can show that the problem remains complete for
$\NTIME(\exp^{\app{\poly}n}(1))$
even if we require for convenience that there are at most~$N$ tiles.
Thus, we can represent tiles by numbers less than~$N$.
We construct a formula~$\psi$ that is satisfied by some finite tree
if, and only if, there exists a tiling of the $(N \times N)$-grid.
We use an encoding of numbers by directed trees
introduced in~\cite{FlumGrohe06} (see also~\cite{DawarGrKrSchw07})
where numbers from $\{0,\dots,N-1\}$ are encoded by trees of height at most~$n$.
The encoding is such that there are first-order formulae
$\varphi_N(x)$, $\varphi_{\min}(x)$, $\varphi_{\max}(x)$,
$\varphi_=(x,y)$ and $\varphi_{\mathrm{suc}}(x,y)$,
which can be constructed in time polynomial in~$n$,
with the property that
\begin{itemize}
\item a vertex~$v$ in a tree~$\fT$ satisfies $\varphi_N(x)$ if the
subtree rooted at~$v$ encodes a number from $\{0,\dots,N-1\}$\?;
\item a vertex~$v$ satisfies $\varphi_{\min}(x)$ or $\varphi_{\max}(x)$
if the subtree rooted at~$v$ encodes the number $0$~or $N-1$, respectively\?;
\item the formulae $\varphi_=(x,y)$ and $\varphi_{\mathrm{suc}}(x,y)$
similarly define equality and the successor relation for numbers
encoded in the subtrees rooted at~$x$ and~$y$.
\end{itemize}
\noindent We use this encoding to represent triples of numbers as follows.
The triple $(x,y,z)$ is encoded by a tree of the form
\begin{center}
\includegraphics{final-4.pdf}
\end{center}
where $\fT_x$, $\fT_y$, and $\fT_z$ are the trees representing
$x$, $y$,~and~$z$, respectively.
Then, a tiling can by represented by a set of triples $(x,y,z)$,
where $x$~and~$y$ are coordinates and $z$ is the tile at position $(x,y)$.
The respective set of trees is turned into a single tree
by making all these triples children of a new root.
To axiomatise the representation of a valid tiling,
we use a formula~$\psi$ based on the formulae
$\varphi_N$, $\varphi_{\min}$, $\varphi_{\max}$,
$\varphi_{\mathrm{suc}}$, and $\varphi_=$ from above.
The formula $\psi$ expresses the following\?:
\begin{itemize}
\item All children of the root encode triples of numbers.
\item There is some triple $(x,y,z)$ with $x = y = 0$.
\item Each triple has a neighbour to the right,
unless the $x$-coordinate already is $N-1$.
The tiles of the triple and its neighbour match.
\item Similarly, there is a neighbour above.
\item Each position occurs at most once.
\end{itemize}
Such a formula~$\psi$ is constructible
in time polynomial in $n$ and the size of the tile set.
Clearly, directed tree models of~$\psi$
correspond to valid tilings of an $(N\times N)$-grid.
Hence, $\psi$~is satisfiable by such a tree if, and only if,
such a tiling exists.
To work inside the class $\cT_\fin$,
we need to replace the directed trees by undirected ones.
Observe that every model of~$\psi$ is a tree of height at most $n+3$.
Hence, we can uniquely mark the root by attaching a path of length $n+4$ to it.
It is easy to modify~$\psi$ to work with such undirected trees instead.
\end{proof}
\begin{cor}
The following boundedness problems are $\app{\DTIME}{\exp^{\app{\poly}n}(1)}$-complete\?:
\begin{enumerate}
\item $\BDDm(\FO,\cT_{\mathrm{fin}})$ where $\cT_{\mathrm{fin}}$~is the class of all finite trees.
\item $\BDDm(\MSO,\cT)$ where $\cT$ is the class of all trees.
\item Boundedness for simultaneous $(\GSOs+\mathrm C)$-fixed points
over the class of structures of bounded tree-width.
\end{enumerate}
\end{cor}
\begin{proof}
(1) follows from Proposition~\ref{Prop: SAT(FO,Tfin) hard}.
As (2)~reduces to~(3), for which we already have a trivial non-elementary upper
bound, it is sufficient to provide a lower bound for $\BDDm(\MSO,\cT)$.
As we have seen, $\BDDm(\FO,\cT_{\mathrm{fin}})$ reduces to
$\BDDm(\MSO,\cT_3)$, which in turn reduces to $\BDDm(\MSO,\cT)$.
Hence, the lower bound follows from~(1).
\end{proof}
This lower bound shows that, for many cases, our algorithm is best possible.
Of course, there are important fragments of $\MSO$ to which the lower bound is not applicable.
For instance, the following upper bounds are known from the literature\?:
\begin{thm}\leavevmode
\begin{enumerate}
\item $\BDDm(\ML)$ is in \textsc{Exptime} \textup{\cite{Otto99}.}
\item $\BDDm(\EFO)$ is in \textsc{2-Exptime} \textup{\cite{CosmadakisGaKaVa88}.}
\qed\end{enumerate}
\end{thm}
Since it is not the main concern of this article, we leave the exact
complexity of $\BDDm(\ML)$, $\BDDm(\Lmu)$, $\BDDm(\EFO)$, $\BDDm(\AFO)$, $\BDD(\GF)$, and $\BDD(\muGF)$ open.
\end{document} | math |
# -*- coding: UTF-8 -*-
# Copyright 2014-2019 Rumma & Ko Ltd
# License: GNU Affero General Public License v3 (see file COPYING for details)
"""Adds functionality for managing households (i.e. groups of humans
who live together in a same house).
See :doc:`/specs/households`.
"""
from lino.api import ad, _
class Plugin(ad.Plugin):
"Extends :class:`lino.core.plugin.Plugin`."
verbose_name = _("Households")
person_model = "contacts.Person"
"""A string referring to the model which represents a human in your
application. Default value is ``'contacts.Person'`` (referring to
:class:`lino_xl.lib.contacts.models.Person`).
"""
adult_age = 18
"""The age (in years) a person needs to have in order to be considered
adult."""
# adult_age = datetime.timedelta(days=18*365)
def on_site_startup(self, site):
self.person_model = site.models.resolve(self.person_model)
super(Plugin, self).on_site_startup(site)
def post_site_startup(self, site):
if not site.is_installed('memo'):
return
rdm = site.plugins.memo.parser.register_django_model
rdm('household', site.models.households.Household)
def setup_main_menu(config, site, user_type, m):
mnugrp = site.plugins.contacts
m = m.add_menu(mnugrp.app_label, mnugrp.verbose_name)
m.add_action('households.Households')
def setup_config_menu(config, site, user_type, m):
mnugrp = site.plugins.contacts
m = m.add_menu(mnugrp.app_label, mnugrp.verbose_name)
# m.add_action(Roles)
m.add_action('households.Types')
def setup_explorer_menu(config, site, user_type, m):
mnugrp = site.plugins.contacts
m = m.add_menu(mnugrp.app_label, mnugrp.verbose_name)
m.add_action('households.MemberRoles')
m.add_action('households.Members')
| code |
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UA Plugs Sale -- What to do?
Help me find a kick sample!
pt freeze track...anyone see this? | english |
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BnB ToGo – Mobile Concierge assistance during your stay! | english |
कैराना उपचुनाव के रिजल्ट आ गए हैं. बीजेपी अपनी यह सीट बचा नहीं पाई और गठबंधन की प्रत्याशी तबस्सुम हसन ने बड़ी जीत दर्ज की.
कैराना उपचुनाव के रिजल्ट आ गए हैं. बीजेपी अपनी यह सीट बचा नहीं पाई और गठबंधन की प्रत्याशी तबस्सुम हसन ने बड़ी जीत दर्ज की. बीजेपी सांसद हुकुम सिंह के निधन के बाद खाली हुई सीट को बीजेपी किसी भी तरह खोने नहीं देना चाहती थी. इसके लिए पार्टी ने हुकुम सिंह की बेटी मृगांका को चुनाव मैदान में उतारा था, लेकिन उसे सफलता नहीं मिली. आइए जानते हैं बीजेपी के हार के कारण क्या हो सकते हैं अलसो रेड - सतारा लोक सभा बायपोल इलेक्शन २०१९ रेसल्ट लाइव उपकेट्स: शिवाजी के वंशज और पर्दे के पीछे पवार के बीच टक्कर
१. बीजेपी ने इस सीट पर हुकुम सिंह की बेटी मृगांका सिंह को उतारा था. मृगांका इससे पहले कैराना विधानसभा सीट पर साल 20१7 में चुनाव लड़ चुकी थीं. लेकिन उन्हें सफलता नहीं मिली थी. तबस्सुम हसन के बेटे ने उन्हें मात दी. ऐसे में जिस तबस्सुम हसन के बेटे नाहिद से मृगांका विधानसभा में हारी हों उसी तबस्सुम हसन के खिलाफ लोकसभा में वह कमजोर दिख रही थीं. प्रत्याशी का कमजोर होना बीजेपी के लिए बड़ी मुसीबत साबित हुआ. अलसो रेड - कैरानाः मृगांका सिंह को टिकट नहीं देना भाजपा को पड़ सकता है भारी, नाराज है गुर्जर समाज
२. मुजफ्फरनगर दंगे के बाद जाटों ने साल २014 और २017 के चुनाव में बीजेपी के पक्ष में वोट दिया था. लेकिन इसके बाद बदले समीकरण में बीजेपी जाटों को पूरी तरह से अपने पक्ष में करने में नाकामयाब साबित हुई. जो जाट एक समय राष्ट्रीय लोक दल के तगड़े समर्थक थे, उन्हीं को बीजेपी अपनी तरफ लाने में कामयाब हो गई थी. लेकिन, इस चुनाव में जयंती चौधरी ने जाटों को एकजुट करने में कामयाबी हासिल की. अलसो रेड - उपचुनाव नतीजों के बाद बीजेपी पर सहयोगियों ने तरेरी आंखें, कहा- फौरन सचेत होना जरूरी
३. कैराना में दलितों की नाराजगी भी बीजेपी की हार का बड़ा कारण साबित हुई. भीम आर्मी के उदय के साथ ही पश्चिमी उत्तर प्रदेश के दलितों में इसका प्रभाव बढ़ गया था. भीम आर्मी के संस्थापक रावण की गिरफ्तारी के बाद से उधर के दलित बीजेपी से नाराज चल रहे थे. कई टीवी चैनल की रिपोर्ट में ये बात सामने आई थी. ऐसे में बीजेपी उन्हें मनाने में कामयाब नहीं रही.
४. पश्चिमी यूपी में गन्ना किसानों की संख्या काफी है. लेकिन पिछले कई साल से चीनी मिलों की ओर से किसानों को उनके गन्ने का भुगतान नहीं हो रहा था. चुनाव में यह बड़ा मुद्दा था. किसानों में इसे लेकर नाराजगी थी. बीजेपी इसे भांप गई थी. सीएम आदित्यनाथ ने अपनी रैलियों में यह मुद्दा उठाया था और गन्ना किसानों की समस्या को दूर करने का वादा किया था. लेकिन परिणाम बता रहे हैं कि सीएम और बीजेपी के दूसरे नेता गन्ना किसानों को मनाने में नाकामयाब साबित हुए.
५. साल २०१४ के लोकसभा और २२०१७ के विधानसभा चुनाव में बीजेपी ने विकास की बात की थी. विकास के वादा पर दौड़ रही बीजेपी कैराना में लोगों को इस बात को समझा नहीं पाई. एक महीने पहले हुए जिन्ना प्रकरण से भी उसे नुकसान पहुंचा है. सोशल मीडिया पर भी इस बात का जोर था कि जिन्ना या गन्ना. इससे साफ होता है कि गन्ना किसानों का रोष उसे किसानों से तो दूर ले गया, वहीं दूसरी तरफ जिन्ना प्रकरण उन्हें ऐसे लोगों से दूर ले गया, जिन्हें बीजेपी से विकास की उम्मीद थी. | hindi |
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\begin{document}
\title{Extensions of Thomassen's Theorem to Paths of Length At Most Four: Part I}
\begin{center}\textbf{Abstract}\end{center} Let $G$ be a planar embedding with list-assignment $L$ and outer cycle $C$, and let $P$ be a path of length at most four on $C$, where each vertex of $G\setminus C$ has a list of size at least five and each vertex of $C\setminus P$ has a list of size at least three. This is the first paper in a sequence of three papers in which we prove some results about partial $L$-colorings $\phi$ of $C$ with the property that any extension of $\phi$ to an $L$-coloring of $\textnormal{dom}(\phi)\cup V(P)$ extends to $L$-color all of $G$, and, in particular, some useful results about the special case in which $\textnormal{dom}(\phi)$ consists only of the endpoints of $P$. We also prove some results about the other special case in which $\phi$ is allowed to color some vertices of $C\setminus\mathring{P}$ but we avoid taking too many colors away from the leftover vertices of $\mathring{P}\setminus\textnormal{dom}(\phi)$. We use these results in a later sequence of papers to prove some results about list-colorings of high-representativity embeddings on surfaces.
\section{Introduction and Motivation}\label{IntroMotivSec}
Given a graph $G$, a \emph{list-assignment} for $G$ is a family of sets $\{L(v): v\in V(G)\}$ indexed by the vertices of $G$, such that $L(v)$ is a finite subset of $\mathbb{N}$ for each $v\in V(G)$. The elements of $L(v)$ are called \emph{colors}. A function $\phi:V(G)\rightarrow\bigcup_{v\in V(G)}L(v)$ is called an \emph{$L$-coloring of} $G$ if $\phi(v)\in L(v)$ for each $v\in V(G)$, and, for each pair of vertices $x,y\in V(G)$ such that $xy\in E(G)$, we have $\phi(x)\neq\phi(y)$.
Given a set $S\subseteq V(G)$ and a function $\phi: S\rightarrow\bigcup_{v\in S}L(v)$, we call $\phi$ an \emph{ $L$-coloring of $S$} if $\phi(v)\in L(v)$ for each $v\in S$ and $\phi$ is an $L$-coloring of the induced graph $G[S]$. A \emph{partial} $L$-coloring of $G$ is a function of the form $\phi:S\rightarrow\bigcup_{v\in S}L(v)$, where $S$ is a subset of $V(G)$ and $\phi$ is an $L$-coloring of $S$. Likewise, given a set $S\subseteq V(G)$, a \emph{partial $L$-coloring} of $S$ is a function $\phi:S'\rightarrow\bigcup_{v\in S'}L(v)$, where $S'\subseteq S$ and $\phi$ is an $L$-coloring of $S'$. Given an integer $k\geq 1$, a graph $G$ is called \emph{$k$-choosable} if, for every list-assignment $L$ for $G$ such that $|L(v)|\geq k$ for all $v\in V(G)$, $G$ is $L$-colorable.
This paper is the first in a sequence of three papers. The motivation for the work of these papers is as follows. In a subsequent series of papers, we prove that, given a graph $G$ embedded on a surface of genus $g$, $G$ can be $L$-colored, where $L$ is a list-assignment for $G$ in which every vertex has a 5-list except for a collection of pairwise far-apart components, each precolored with an ordinary 2-coloring, as long as the face-width of $G$ is at leat $2^{\Omega(g)}$ and the precolored components are of distance at least $2^{\Omega(g)}$ apart. This provides an affirmative answer to a generalized version of a conjecture of Thomassen and also generalizes a result (\cite{DistPrecVertChoosePap}) of Dvo\v{r}\'ak, Lidick\'y, Mohar, and Postle about distant precolored vertices. The high-representativity result above can also be viewed as a generalization of the result in Thomassen’s 5-choosability proof (from \cite{AllPlanar5ThomPap}) in which a) arbitrarily many faces, rather than just one face, are permitted to have 3-lists and b) the embedding $G$ is not planar but rather locally planar in the sense that it has high representativity.
We prove the generalization of Thomassen's result described above by a minimal counterexample argument in which we color and delete a subgraph of a minimal counterexample $G$, where this subgraph consists mostly of a path between two of the specified faces of $G$ which have 3-lists. The trickiest part of this is dealing with the region near the boundaries, i.e near the designated facial cycles. To deal with this, we need a generalized ``cutting procedure", which is the purpose of this sequence of three papers. We stress that the results and proofs of these three papers only deal with planar graphs and they are self-contained, other than their reliance on several standard facts which are stated in Section \ref{BackgroundSect} below, but we use the results of these three papers as black boxes for subsequent papers.
Given a graph $G$ with list-assignment $L$, we very frequently analyze the situation where we begin with a partial $L$-coloring $\phi$ of a subgraph of $G$, and then delete some or all of the vertices of $\textnormal{dom}(\phi)$ and remove the colors of the deleted vertices from the lists of their neighbors in $G\setminus\textnormal{dom}(\phi)$. We thus make the following definition.
\begin{defn}\emph{Let $G$ be a graph, let $\phi$ be a partial $L$-coloring of $G$, and let $S\subseteq V(G)$. We define a list-assignment $L^S_{\phi}$ for $G\setminus (\textnormal{dom}(\phi)\setminus S)$ as follows.}
$$L^S_{\phi}(v):=\begin{cases} \{\phi(v)\}\ \textnormal{if}\ v\in\textnormal{dom}(\phi)\cap S\\ L(v)\setminus\{\phi(w): w\in N(v)\cap (\textnormal{dom}(\phi)\setminus S)\}\ \textnormal{if}\ v\in V(G)\setminus \textnormal{dom}(\phi) \end{cases}$$ \end{defn}
If $S=\varnothing$, then $L^{\varnothing}_{\phi}$ is a list-assignment for $G\setminus\textnormal{dom}(\phi)$ in which the colors of the vertices in $\textnormal{dom}(\phi)$ have been deleted from the lists of their neighbors in $G\setminus\textnormal{dom}(\phi)$. The situation where $S=\varnothing$ arises so frequently that, in this case, we simply drop the superscript and let $L_{\phi}$ denote the list-assignment $L^{\varnothing}_{\phi}$ for $G\setminus\textnormal{dom}(\phi)$. In some cases, we specify a subgraph $H$ of $G$ rather than a vertex-set $S$. In this case, to avoid clutter, we write $L^H_{\phi}$ to mean $L^{V(H)}_{\phi}$. Our main result for this set of three papers, which we prove in Paper III, is Theorem \ref{MainHolepunchPaperResulThm} below. The work of this paper and Paper II builds up to the proof of this result.
\begin{theorem}\label{MainHolepunchPaperResulThm} (Holepunch Theorem) Let $G$ be a planar embedding with outer cycle $C$. Let $P:=p_0q_0zq_1p_1$ be a subpath of $C$ whose three internal vertices have no common neighbor in $C\setminus P$, and let $L$ be a list-assignment for $V(G)$ such that the following hold.
\begin{enumerate}[label=\arabic*)]
\itemsep-0.1em
\item $|L(p_0)|+|L(p_1)|\geq 4$ and each of $L(p_0)$ and $L(p_1)$ is nonempty; AND
\item For each $v\in V(C\setminus P)$, $|L(v)|\geq 3$; AND
\item For each $v\in\{q_0, z, q_1\}\cup V(G\setminus C)$, $|L(v)|\geq 5$.
\end{enumerate}
Then there is a partial $L$-coloring $\phi$ of $V(C)\setminus\{q_0, q_1\}$, where $p_0, p_1, z\in\textnormal{dom}(\phi)$, such that each of $q_0, q_1$ has an $L_{\phi}$-list of size at least three, and furthermore, any extension of $\phi$ to an $L$-coloring of $\textnormal{dom}(\phi)\cup\{q_0, q_1\}$ also extends to $L$-color all of $G$.
\end{theorem}
In general, with $G, C, P, L$ as above, it is not true that there is an $L$-coloring $\phi$ of $\{p_0, z, p_1\}$ with the property that any extension of $\phi$ to an $L$-coloring of all of $V(P)$ extends to $L$-color all of $G$. Theorem \ref{MainHolepunchPaperResulThm} tells us that this is ``almost" true, i.e it becomes true if we relax the condition that we are only allowed to color $\{p_0, z, p_1\}$, but that this $L$-coloring of our larger domain in $C\setminus\{q_0, q_1\}$ behaves like an $L$-coloring of $\{p_0, z, p_1\}$ in the sense that each of $q_0, q_1$ still has at least three leftover colors. (Of course, in the special case where one of $q_0p_1, q_1p_0$ is a chord of $C$, then not every $L$-coloring of $\{p_0, z, p_1\}$ leaves each of $q_0, q_1$ with at least three colors, but this configuration is easy to deal with). However, it will take a lot of work to prove this, as there are many configurations to deal with. Theorem \ref{MainHolepunchPaperResulThm} is a statement about paths of lengths four in facial cycles of planar graphs. In order to prove this, we need some intermediate facts about paths of lengths 2, 3, and 4 in facial cycles of planar graphs. The purpose of this paper and Paper II is to prove these intermediate facts, which we then use as black boxes in Paper III, where Paper III consists entirely of the proof of Theorem \ref{MainHolepunchPaperResulThm}. In this paper, we focus on 2-paths in Sections \ref{2PathBWheelCaseSec}-\ref{ExtCol2PathAugColSec} and turn our attention to 3-paths in Sections \ref{LinkColoring3PathExCycleSec}. Each of Sections \ref{2PathBWheelCaseSec}-\ref{LinkColoring3PathExCycleSec} consists of one main result, as indicated in Table \ref{chapPmsiMaRe11}. These results are the black boxes which we take forward to Paper II and apply to prove Theorem \ref{MainHolepunchPaperResulThm} in Paper III.
\\
\begin{center}
\begin{tabular}{cc}
\begin{tabular}{ |c|c| }
\hline
\textbf{Section} & \textbf{Main Result} \\
\hline
\ref{2PathBWheelCaseSec} & Theorem \ref{BWheelMainRevListThm2}\\
\hline
\ref{2PathGenCaseSec} & Theorem \ref{EitherBWheelOrAtMostOneColThm} \\
\hline
\ref{ExtCol2PathAugColSec} & Theorem \ref{SumTo4For2PathColorEnds}\\
\hline
\end{tabular}
&
\begin{tabular}{ |c|c| }
\hline
\textbf{Section} & \textbf{Main Result} \\
\hline
\ref{LinkColoring3PathExCycleSec} & Theorem \ref{ThmFirstLink3PathForUseInHolepunch} \\
\hline
\ref{3ChordAnalogue2PathSec} & Theorem \ref{3ChordVersionMainThm1} \\
\hline
\end{tabular}
\end{tabular}
\captionof{table}{Sections and their main results}\label{chapPmsiMaRe11}\end{center}
We now introduce the following definition, which is our main object of study for all three papers.
\begin{defn} \emph{A \emph{rainbow} is a tuple $(G, C, P, L)$, where $G$ is a planar graph with outer cycle $C$, $P$ is a path on $C$ of length at least two, and $L$ is a list-assignment for $V(G)$ such that $|L(v)|\geq 3$ for each $v\in V(C\setminus P)$ and $|L(v)|\geq 5$ for each $v\in V(G\setminus C)$. We say that a rainbow is \emph{end-linked} if, letting $p ,p^*$ be the endpoints of $P$, each of $L(p)$ and $L(p^*)$ is nonempty and $|L(p)|+|L(p^*)|\geq 4$. }
\end{defn}
Given the statement of Theorem \ref{MainHolepunchPaperResulThm}, it is also natural to introduce the following terminology.
\begin{defn} \emph{Given a graph $G$ with list-assignment $L$, a subgraph $H$ of $G$, and a partial $L$-coloring $\phi$ of $G$, we say that $\phi$ is \emph{$(H,G)$-sufficient} if any extension of $\phi$ to an $L$-coloring of $\textnormal{dom}(\phi)\cup V(H)$ extends to $L$-color all of $G$.} \end{defn}
We also introduce the following notation.
\begin{defn}\label{GeneralAugCrownNotForLink} \emph{Let $G$ be a planar graph with outer cycle $C$, let $L$ be a list-assignment for $G$, and let $P$ be a path in $C$ with $|V(P)|\geq 3$. Let $pq$ and $p'q'$ be the terminal edges of $P$, where $p, p'$ are the endpoints of $P$. We let $\textnormal{Crown}_{L}(P, G)$ be the set of partial $L$-colorings $\phi$ of $V(C)\setminus\{q, q'\}$ such that}
\begin{enumerate}[label=\arabic*)]
\itemsep-0.1em
\item $V(P)\setminus\{q, q'\}\subseteq\textnormal{dom}(\phi)$ and, for each $x\in\{q, q'\}$, $|L_{\phi}(x)|\geq |L(x)|-2$; AND
\item\emph{$\phi$ is $(P, G)$-sufficient}
\end{enumerate}
\end{defn}
With the definitions above in hand, we have the following compact restatement of Theorem \ref{MainHolepunchPaperResulThm}.
\begin{thmn}[\ref{MainHolepunchPaperResulThm}] Let $(G, C, P, L)$ be an end-linked rainbow, where $P$ is a path of length four whose three internal vertices have no common neighbor in $C\setminus P$. Suppose further that each internal vertex of $P$ has an $L$-list of size at least five. Then $\textnormal{Crown}_{L}(P, G)\neq\varnothing$. \end{thmn}
\section{Background}\label{BackgroundSect}
In 1994, Thomassen demonstrated in \cite{AllPlanar5ThomPap} that all planar graphs are 5-choosable, settling a problem that had been posed in the 1970's. Actually, Thomassen proved something stronger.
\begin{theorem}\label{thomassen5ChooseThm}
Let $G$ be a planar graph with facial cycle $C$ and let $xy\in E(C)$. Let $L$ be a list assignment for $G$, where vertex of $G\setminus C$ has a list of size at least five and each vertex of $V(C)\setminus\{x,y\}$ has a list of size at least three, where $xy$ is $L$-colorable. Then $G$ is $L$-colorable.
\end{theorem}
Theorem \ref{thomassen5ChooseThm} has the following two useful corollaries, which we use frequently.
\begin{cor}\label{CycleLen4CorToThom} Let $G$ be a planar graph with outer cycle $C$ and let $L$ be a list-assignment for $G$ where each vertex of $G\setminus C$ has a list of size at least five. If $|V(C)|\leq 4$ then any $L$-coloring of $V(C)$ extends to an $L$-coloring of $G$. \end{cor}
\begin{cor}\label{2ListsNextToPrecEdgeCor}
Let $G$ be a planar graph with facial cycle $C$ and $p_0p_1\in E(C)$ and, for each $i=0,1$, let $u_i$ be the unique neighbor of $p_i$ on the path $C\setminus\{p_0, p_1\}$. Let $L$ be a list assignment for $G$, where each vertex of $G\setminus C$ has a list of size at least five and each vertex of $C\setminus\{p_0, p_1, u_0, u_1\}$ has a list of size at least three. Let $\phi$ be an $L$-coloring of $p_0p_1$, where $|L(u_i)\setminus\{\phi(p_i)\}|\geq 2$ for each $i=0,1$. Then $\phi$ extends to an $L$-coloring of $G$. \end{cor}
Because of Corollary \ref{CycleLen4CorToThom}, planar embeddings which have no separating cycles of length 3 or 4 play a special role in our analysis, so we give them a name.
\begin{defn} \emph{Given a planar graph $G$, we say that $G$ is \emph{short-separation-free} if, for any cycle $F\subseteq G$ with $|V(F)|\leq 4$, either $V(\textnormal{Int}_G(F))=V(F)$ or $V(\textnormal{Ext}_G(F))=V(F)$.} \end{defn}
In this paper, we also rely on the following very useful result from Postle and Thomas, which is an immediate consequence of Theorem 3.1 from \cite{2ListSize2PaperSeriesI} and is an is an analogue of Theorem \ref{thomassen5ChooseThm} where the precolored edge has been replaced by two lists of size two.
\begin{theorem}\label{Two2ListTheorem} (\cite{2ListSize2PaperSeriesI}) Let $G$ be a planar graph, let $F$ be the outer face of $G$, and let $v,w\in V(F)$. Let $L$ be a list-assignment for $V(G)$ where $|L(v)|\geq 2$, $|L(w)|\geq 2$, and furthermore, for each $x\in V(F)\setminus\{v,w\}$, $|L(x)|\geq 3$, and, for each $x\in V(G\setminus F)$, $|L(x)|\geq 5$. Then $G$ is $L$-colorable. \end{theorem}
We note that Theorem 3.1 from \cite{2ListSize2PaperSeriesI} does not require $G$ to be 2-connected for the statement above to hold, that is, possibly $F$ is a facial walk but not a cycle.
\section{Extending Colorings of 2-Paths: Broken Wheels}\label{2PathBWheelCaseSec}
The setting of the work of Sections \ref{2PathBWheelCaseSec}-\ref{ExtCol2PathAugColSec} is the following. Suppose we have a planar graph $G$ with an outer cycle $C$, a subpath $P$ of $C$ of length two, and a list-assignment $L$ for $V(G)$ such that each vertex of $G\setminus C$ has a list of size at least five and each vertex of $C\setminus P$ has a list of size at least three. In this case, a precoloring of $V(P)$ does not necessarily extend to an $L$-coloring of all of $G$ (although it does if $P$ has length at most one, this is precisely Theorem \ref{thomassen5ChooseThm})
\begin{defn}\emph{A \emph{broken wheel} is a graph $G$ with a vertex $p\in V(G)$ such that $G-p$ is a path $q_1\ldots q_n$ with $n\geq 2$, where $N(p)=\{q_1, \ldots, q_n\}$. The subpath $q_1pq_n$ of $G$ is called the \emph{principal path} of $G$.} \end{defn}
Note that, if $|V(G)|\leq 4$, then the above definition does not uniquely specify the principal path, although in practice, whenever we deal with broken wheels, we specify the principal path beforehand so that there is no ambiguity. Furthermore, since we frequently deal with colorings of paths, we introduce the following natural notation.
\begin{defn} \emph{Let $G$ be a graph with list-assignment $L$. Given an integer $n\geq 1$, a path $P:=p_1\ldots p_n$, and a partial $L$-coloring $\phi$ of $G$ with $V(P)\subseteq\textnormal{dom}(\phi)$, we denote the $L$-coloring $\phi|_P$ of $P$ as $(\phi(p_1), \phi(p_2),\ldots, \phi(p_n))$.} \end{defn}
To prove the results of subsequent sections, we need some facts about colorings of the principal path of a broken wheel which do not extend to the entire broken wheel. We also use these facts in subsequent papers. More generally, we frequently deal with the situation where, given a graph $G$ with a list-assignment $L$, a path $P$ of $G$ of length two, and an $L$-coloring of two of the vertices of $P$, we want to keep track of the colors used on the last vertex of $P$ by extensions of our precoloring to $L$-colorings of all of $G$.
\begin{defn} \emph{Let $G$ be a graph and let $L$ be a list-assignment for $V(G)$. Let $P:=p_1p_2p_3$ be a path of length two in $G$. For each $(c, c')\in L(p_1)\times L(p_3)$, we let $\Lambda_{G,L}^P(c, \bullet, c')$ be the set of $d\in L(p_2)$ such that there is an $L$-coloring of $G$ which uses $c,d,c'$ on the respective vertices $p_1, p_2, p_3$. Likewise, given a pair $(c, c')$ in either $L(p_1)\times L(p_2)$ or $L(p_2)\times L(p_3)$ respectively, we define the sets $\Lambda_{G,L}^P(c, c', \bullet)$ and $\Lambda_{G,L}^P(\bullet, c, c')$ analogously.} \end{defn}
Note that, in the setting above, we have $\Lambda_{G,L}^P(c, c, \bullet)=\varnothing$ for any $c\in L(p_1)\cap L(p_2)$. Likewise, $\Lambda_{G,L}(\bullet, d, d)=\varnothing$ for any $d\in L(p_2)\cap L(p_3)$. The use of the notation above always requires us to specify an ordering of the vertices of a given 2-path. That is, whenever we write $\Lambda_{G,L}^P(\cdot, \cdot, \cdot)$, where two of the coordinates are colors of two of the vertices
of $P$ and one is a bullet denoting the remaining uncolored vertex of $P$, we have specified beforehand which vertices the first, second, and third coordinates correspond to. Sometimes we make this explicit by writing $\Lambda^{p_1p_2p_3}_{G,L}(\cdot, \cdot, \cdot)$. Whenever any of $P, G, L$ are clear from the context, we drop the respective super- or subscripts from the notation above.
\begin{defn}\label{GUniversalDefinition} \emph{Let $G$ be a graph and let $P:= p_1p_2p_3$ be a subpath of $G$ of length two. Let $L$ be a list-assignment for $V(G)$. Given an $a\in L(p_3)$,}
\begin{enumerate}[label=\emph{\alph*)}]
\itemsep-0.1em
\item\emph{we say that $a$ is \emph{$(G, P)$-universal} if, for each $b\in L(p_2)\setminus\{a\}$, we have $\Lambda_G^P(\bullet, b, a)=L(p_1)\setminus\{b\}$.}
\item\emph{We say that $a$ is \emph{almost $(G,P)$-universal}, if, for each $b\in L(p_2)\setminus\{a\}$, we have $|\Lambda_G^P(\bullet, b, a)|\geq |L(p_1)|-1$.}
\end{enumerate}
\end{defn}
Note that, in the setting above, if $a$ is $(G,P)$-universal, then it is clearly also almost $(G,P)$-universal. Furthermore, if $p_1p_3\in E(G)$ and $L(p_3)\subseteq L(p_1)$, then there is no $(G, P)$-universal color in $L(p_3)$. That is, $a$ being a $(G, P)$-universal color of $L(p_3)$ is a stronger property than the property that any $L$-coloring of $V(P)$ using $a$ on $p_3$ extends to an $L$-coloring of $G$, unless either $a\not\in L(p_1)$ or $p_1p_3\not\in E(G)$. If the 2-path $P$ is clear from the context, then, given an $a\in L(p_3)$, we just say that $a$ is $G$-universal. Our main result for Section \ref{2PathBWheelCaseSec} is the following.
\begin{theorem}\label{BWheelMainRevListThm2} Let $G$ be a broken wheel with principal path $P=pp'p''$ and let $L$ be a list-assignment for $V(G)$ in which each vertex of $V(G)\setminus\{p, p'\}$ has a list of size at least three. Let $G-p'=pu_1\ldots u_tp''$ for some $t\geq 0$.
\begin{enumerate}[label=\arabic*)]
\item Let $\phi_0, \phi_1$ be a pair of distinct $L$-colorings of $pp'$. For each $i=0,1$, let $S_i:=\Lambda_G(\phi_i(p), \phi_i(p'), \bullet)$, and suppose that $|S_0|=|S_1|=1$. Then the following hold.
\begin{enumerate}[label=\alph*)]
\itemsep-0.1em
\item If $\phi_0(p)=\phi_1(p)$ and $S_0=S_1$, then $|E(G-p')|$ is even; AND
\item If $\phi_0(p)=\phi_1(p)$ and $S_0\neq S_1$, then $|E(G-p')|$ is odd and, for each $i=0,1$, $S_i=\{\phi_{1-i}(p')\}$; AND
\item If $\phi_0(p)\neq\phi_1(p)$ and $S_0=S_1$, then $|E(G-p')|$ is odd and $(\phi_0(p), \phi_0(p'))=(\phi_1(p'), \phi_1(p))$
\end{enumerate}
\item Let $\mathcal{F}$ be a family of $L$-colorings of $pp'$ and let $q\in\{p, p'\}$. Suppose that $|\mathcal{F}|\geq 3$ and $\mathcal{F}$ is constant on $q$. Then there exists a $\phi\in\mathcal{F}$ such that $|\Lambda_G(\phi(p), \phi(p'), \bullet)|\geq 2$ and in particular, if $G$ is not a triangle, then $L(p'')\setminus\{\phi(p')\}\subseteq\Lambda_G(\phi(p), \phi(p'), \bullet)$.
\item If $|V(G)|>4$ and there is an $a\in L(p)$ with $L(u_1)\setminus\{a\}\not\subseteq L(u_2)$, then $a$ is $G$-universal.
\item If $|V(G)|\geq 4$, then, letting $x$ be the unique vertex of distance two from $p$ on the path $G-p'$, the following holds: For any $a\in L(p)$ with $L(u_1)\setminus\{a\}\not\subseteq L(x)$, $a$ is almost $G$-universal.
\end{enumerate}
\end{theorem}
\begin{proof} Let $z$ be the unique vertex of $G$ which is a common neighbor to $p, p'$, i.e either $z=u_1$, or $H$ is a triangle and $z=p''$. We first prove 1). Since $|S_0|=|S_1|=1$, it follows that, for each $v\in V(G-p)$, we have $|L(v)|=3$ and $\{\phi_0(p'), \phi_1(p')\}\subseteq L(v)$. We first prove 1a) and 1b) together. Suppose that $\phi_0(p)=\phi_1(p)=c$ for some color $c$. Since $\phi_0, \phi_1$ are distinct, we have $\phi_0(p')\neq\phi_1(p')$, there is a lone color $r$ such that $L(p_3)=\{\phi_0(p'), \phi_1(p'), r\}$. Since $c\not\in\{\phi_0(p'), \phi_1(p')\}$ and $\{\phi_0(p'), \phi_1(p')\}\subseteq\bigcap (L(v): v\in V(G-p))$, it follows by symmetry that either $r\in S_0\cap S_1$ or $r\not\in S_0\cup S_1$. In particular, since $|S_0|=|S_1|=1$, we have $r\in S_0\cap S_1$ if and only if $S_0=S_1=\{r\}$. Likewise, $r\not\in S_0\cup S_1$ if and only if $S_i=\{\phi_{1-i}(p')\}$ for each $i=0,1$. Thus, if $S_0=S_1$, then we have $S_0=S_1=\{r\}$ and $G-p'$ is a path of even length, or else $\phi_{1-i}(p')\in S_i$ for each $i=0,1$. On the other hand, if $S_0\neq S_1$, then $S_i=\{\phi_{1-i}(p')\}$ for each $i=0,1$, and $G-p'$ is a path of odd length, or else $r\in S_0\cap S_1$. This proves 1a) and 1b).
Now we prove 1c). Suppose that $\phi_0(p)\neq\phi_1(p)$ and that $S_0=S_1=S$ for some singleton $S$.
\begin{Claim}\label{TwoPhisDifferentColP'CL} $\phi_0(p')\neq\phi_1(p')$ \end{Claim}
\begin{claimproof} Suppose that $\phi_0(p')=\phi_1(p')=c$ for some color $c$. Since $\phi_0, \phi_1$ are distinct $L$-colorings of $pp'$, we have $\phi_0(p)\neq\phi_1(p)$. Since $|S\cup\{c\}|=2$, there is an $r\in L(p'')\setminus (S\cup\{c\})$. Since $c\not\in\{\phi_0(p), \phi_1(p)\}$, it follows from Corollary \ref{2ListsNextToPrecEdgeCor} that there is an $i\in\{0,1\}$ such that the $L$-coloring $(\phi_i(p), c, r)$ of $pp'p''$ extends to an $L$-coloring of $G$, contradicting the fact that $r\not\in S$. \end{claimproof}
Since $\phi_0(p')\neq\phi_1(p')$, there is a color $r$ such that $L(p'')=\{\phi_0(p'), \phi_1(p'), r\}$ and $S=\{r\}$. Since $|S_0|=|S_1|=1$, we have $\{\phi_0(p), \phi_1(p)\}\cup\{\phi_0(p'), \phi_1(p')\}\subseteq L(z)$. By Claim \ref{TwoPhisDifferentColP'CL}, $|\{\phi_0(p'), \phi_1(p')\}|=2$. Since $|\{\phi_0(p), \phi_1(p)\}|=2$ by assumption, and since $|L(z)|=3$, we suppose without loss of generality that $\phi_1(p)=\phi_0(p')$. If $G-p_2$ has even length, then, since $\phi_1(p)=\phi_0(p')$ and $\{\phi_0(p'), \phi_1(p')\}\subseteq\bigcap (L(v): v\in V(G-p))$, it follows that $\phi_0(p')\in S_1$, since we can color $p, u_2, u_4, \ldots, u_{t-1}, p''$ with $\phi_0(p')$, which is false, since $S_1=\{r\}$. Thus, $G-p_2$ has odd length. We just need to check that $\phi_0(p)=\phi_1(p')$. Suppose not. Since $\{\phi_0(p), \phi_1(p)\}\cup\{\phi_0(p'), \phi_1(p')\}\subseteq L(z)$, there is a lone color $f$ with $L(z)=\{\phi_0(p'), \phi_1(p'), f\}$, where $f=\phi_0(p)$. If $G$ is a triangle, then $f=r$ and $f\not\in S_0$, which is false. Thus, $G$ is not a triangle. In particular, $G\setminus\{p, p'\}$ is a nonempty path of even length. Since $f\not\in\{\phi_0(p'), \phi_1(p')\}$ and $\{\phi_0(p'), \phi_1(p')\}\subseteq\bigcap (L(v): v\in V(G-p))$, it follows that $\phi_1(p')\in S_0$, because we can color $p$ with $f$ and color each of $u_1, u_3, \ldots, u_{t-1}, p''$ with $\phi_1(p')$. Since $\phi_1(p')\neq r$, we have a contradiction. This proves 1c) and completes the proof of 1) of Theorem \ref{BWheelMainRevListThm2}.
Now we prove 2). Let $q'$ be the vertex of $pp'$ which is distinct from $q$. Suppose toward a contradiction that there is no $\phi\in\mathcal{F}$ satisfying 3). Thus, we have $|L(z)|=3$, and, for each $\phi\in\mathcal{F}$, we have $\{\phi(q), \phi(q')\}\subseteq L(z)$. Letting $c\in L(q)$ be the unique color used on $q$ by all elements of $\mathcal{F}$, we have $c\in L(z)$ and $\{\phi(q'): \phi\in\mathcal{F}\}\subseteq L(z)$. The latter set consists of $|\mathcal{F}|$ colors, all distinct from $c$, contradicting the fact that $|L(z)|=3$.
Now we prove 3). Let $a\in L(p)$, where $L(u_1)\setminus\{a\}\not\subseteq L(u_2)$. Suppose toward a contradiction that $a$ is not $G$-universal. Since $|V(G)|>4$, we have $pp''\not\in E(G)$, so there is an $L$-coloring $\phi$ of $V(P)$ which does not extend to an $L$-coloring of $G$, where $\phi(p)=a$. Let $b=\phi(p')$. Since $|V(G)|>4$ and $\phi$ does not extend to an $L$-coloring of $V(P)$, we have $b\in L(u_1)\cap L(u_2)$ and $a\in L(u_1)$, and furthermore, $|L(u_1)|=|L(u_2)|=3$. By Theorem \ref{thomassen5ChooseThm}, the $L$-coloring $(\phi(p'), \phi(p''))$ of the edge $p'p''$ extends to an $L$-coloring $\psi$ of $V(G)\setminus\{p, u_1\}$, and $\psi\cup\phi$ is a proper $L$-coloring of its domain, which is $G-u_1$. By assumption, we have $L(u_1)\setminus\{a\}\not\subseteq L(u_2)$. Since $a,b\in L(u_1)$ and $b\in L(u_2)$, and since $|L(u_1)|=|L(u_2)|=3$, the lone color of $L(u_1)\setminus\{a, b\}$ does not lie in $L(u_2)$, so this color is distinct from $\psi(u_2)$. Thus, $\phi\cup\psi$ extends to an $L$-coloring of $G$, a contradiction. Thus, $a$ is $G$-universal.
Now we prove 4). If $|V(G)|>4$, then we are immediately done by applying 3), so suppose that $|V(G)|=4$. Thus, the unique vertex of distance two from $p$ on the path $G-p'$ is $p''$. By assumption, $|L(p'')|\geq 3$. Let $a\in L(p)$, where $L(u_1)\setminus\{a\}\not\subseteq L(p'')$. Suppose toward a contradiction that $a$ is not almost $G$-universal. Thus, there is a $b\in L(p')\setminus\{a\}$ such that $|\Lambda_G(a, b, \bullet)|<2$. It follows that $a\in L(u_1)$ and $b\in L(u_1)\cap L(p'')$. Since $L(u_1)\setminus\{a\}\not\subseteq L(p'')$, there is a $c\in L(u_1)\setminus\{a\}$ with $c\not\in L(p'')$, so $c\neq b$, and, coloring $u_1$ with $c$, each color of $L(p'')\setminus\{b\}$ is left over for $p''$, so $|\Lambda_G(a, b, \bullet)|\geq 2$, a contradiction. This proves 4) and completes the proof of Theorem \ref{BWheelMainRevListThm2}. \end{proof}
\section{Extending Colorings of 2-Paths: The General Case}\label{2PathGenCaseSec}
We begin this section by introducing the several pieces of notation, which we use throughout the remainder of this paper. Given a planar embedding $G$, a facial cycle $C$ of $G$, an integer $k\geq 1$ and a $k$-chord $Q$ of $C$, there is a natural way to talk about one or the other ``side" of $Q$ in $G$, which is made precise below.
\begin{defn}\label{ContractNatCQPartChordDefn} \emph{Let $G$ be a planar graph with outer cycle $C$, let $k\geq 1$ be an integer, and let $Q$ be a $k$-chord of $C$. Let $C_0$ and $C_1$ be the two cycles of $C\cup Q$ which contain $Q$. The unique \emph{natural $Q$-partition} of $G$ is the pair $\{G_0, G_1\}$ of subgraphs of $G$ such that $G_i=\textnormal{Int}_G(C_i)$ for each $i=0,1$. In particular, we note that $G=G_0\cup G_1$ and $G_0\cap G_1=Q$.}
\end{defn}
In this section and the next one, we prove some results about coloring 2-paths on facial cycles of planar graphs. Our man result for this section is Theorem \ref{EitherBWheelOrAtMostOneColThm}. In the statement of Theorem \ref{thomassen5ChooseThm}, it we replace the precolored edge on the outer cycle with a precolored 2-path, then $G$ is no longer necessarily $L$-colorable. In \cite{ExpManyColChooseThomPap}, Thomassen characterized the obstructions to a precoloring of a 2-path extending to an $L$-coloring of $G$ in the situation above. In Theorem \ref{EitherBWheelOrAtMostOneColThm}, we prove two useful facts about extending colorings of 2-paths, the first of which is is essentially a ``spanning subgraph" analogue to Lemma 1 from \cite{ExpManyColChooseThomPap}. In order to prove Theorem \ref{EitherBWheelOrAtMostOneColThm}, we first prove the following simple lemma, which we also use repeatedly in subsequent papers. To state this lemma, we first recall the following standard notation.
\begin{defn}\label{StandQD} \emph{Given a path $Q$ in a graph $G$, we let $\mathring{Q}$ denote the subpath of $Q$ consisting of the internal vertices of $Q$. In particular, if $|E(Q)|\leq 2$, then $\mathring{Q}=\varnothing$. Furthermore, for any $x,y\in V(Q)$, we let $xQy$ denote the unique subpath of $Q$ with endpoints $x$ and $y$. } \end{defn}
We also introduce the following notation, which we use frequently throughout this paper.
\begin{defn} \emph{Given a planar graph $G$ with outer face $C$ and an $H\subseteq G$, we let $C^H$ denote the outer face of $H$.} \end{defn}
\begin{lemma}\label{PartialPathColoringExtCL0}
Let $(G, C, P, L)$ be a rainbow and let $\phi$ be a partial $L$-coloring of $V(P)$ which includes the endpoints of $P$ in its domain and does not extend to an $L$-coloring of $G$. Then at least one of the following holds.
\begin{enumerate}[label=\arabic*)]
\itemsep-0.1em
\item There is a chord of $C$ with one endpoint in $\textnormal{dom}(\phi)$ and the other endpoint in $C\setminus P$; OR
\item There is a $v\in V(G\setminus C)\cup (V(\mathring{P})\setminus\textnormal{dom}(\phi))$ with $|L_{\phi}(v)|\leq 2$.
\end{enumerate}
\end{lemma}
\begin{proof} Suppose toward a contradiction that neither 1) nor 2) holds. Since $\phi$ does not extend to an $L$-coloring of $G$, it follows from Theorem \ref{thomassen5ChooseThm} that $|V(P)|\geq 3$, so $V(\mathring{P})\neq\varnothing$. Let $p,p'$ be the endpoints of $P$ and let $C\setminus\mathring{P}=pu_1\ldots u_tp'$ for some $t\geq 0$. Let $G':=G\setminus\textnormal{dom}(\phi)$. By assumption, every vertex of $C^{G'}\setminus C$ has an $L_{\phi}$-list of size at least three, and since there is no chord of $C$ with an endpoint in $\textnormal{dom}(\phi)$ and the other endpoint in $C\setminus P$, each internal vertex of the path $u_1\cdots, u_t$ has an $L_{\phi}$-list of size at least three as well. If $t=0$ then $G'$ is $L_{\phi}$-colorable by Theorem \ref{thomassen5ChooseThm}. Likewise, if $t=1$, then $|L_{\phi}(u_1)|\geq 1$ by our assumption on $\phi$, and thus, again applying Theorem \ref{thomassen5ChooseThm}, $G'$ is $L_{\phi}$-colorable. Finally, if $t\geq 2$, then $|L_{\phi}(u_1)|\geq 2$ and $|L_{\phi}(u_t)|\geq 2$, so, by Theorem \ref{Two2ListTheorem}, $G'$ is $L_{\phi}$-colorable. In any case, $\phi$ extends to an $L$-coloring of $G$, contradicting our assumption. \end{proof}
We now state and prove Theorem \ref{EitherBWheelOrAtMostOneColThm}.
\begin{theorem}\label{EitherBWheelOrAtMostOneColThm} Let $(G, C, P, L)$ be a rainbow, where $P=p_1p_2p_3$ is a 2-path. Suppose further that $G$ is short-separation-free and every chord of $C$ has $p_2$ as an endpoint. Then,
\begin{enumerate}[label=\arabic*)]
\itemsep-0.1em
\item either $G$ is a broken wheel with principal path $P$, or there is at most one $L$-coloring of $V(P)$ which does not extend to an $L$-coloring of $G$; AND
\item If $\phi$ is an $L$-coloring of $V(P)$ which does not extend to an $L$-coloring of $G$, then, for each $p\in\{p_1, p_3\}$, letting $u$ be the unique neighbor of $p$ on the path $C-p_2$, we have $L(u)\setminus\{\phi(p)\}|=2$.
\end{enumerate}
\end{theorem}
\begin{proof} We first prove 1). Let $G$ be a vertex minimal counterexample to 1) of Theorem \ref{EitherBWheelOrAtMostOneColThm}. Let $\phi, \phi'$ be two distinct $L$-colorings of $P$, neither of which extends to an $L$-coloring of $G$. Since neither $\phi$ nor $\phi'$ extends to an $L$-coloring of $G$, it follows from Corollary \ref{CycleLen4CorToThom} that $|V(C)|\geq 5$. Let $C=p_3p_2p_1u_1\ldots u_t$ for some $t\geq 2$.
\begin{Claim} $C$ is an induced subgraph of $G$. \end{Claim}
\begin{claimproof} By assumption, any chord of $C$ has $p_2$ as an endpoint, so suppose toward a contradiction that there is a chord $p_2u_j$ of $C$ for some $j\in\{1,\ldots, t\}$, and let $G^*\cup G^{**}$ be the natural $p_2u_j$-partition of $G$, where $p_1\in V(G^*)$ and $p_3\in V(G^{**})$. Let $P^*:=p_1p_2u_j$ and $P^{**}:=u_jp_2p_3$. Note that every chord of $C^{G^*}$ in $G^*$ has $p_2$ as an endpoint. If $G^*$ is a broken wheel with principal path $P^*$, and $G^{**}$ is a broken wheel with principal path $P^{**}$, then $G$ is a broken wheel with principal path $P$, contradicting our assumption, so suppose without loss of generality that $G^*$ is not a broken wheel with principal path $P^*$. Since $G$ is short-separation-free, $G^*$ is not a triangle, and $u_j\neq u_1$.
By Theorem \ref{thomassen5ChooseThm}, there are colors $r, r'\in L(u_j)$ such that the colorings $(\phi(p_3), \phi(p_2), r)$ and $(\phi'(p_3), \phi'(p_2), r')$ of $P^{**}$ extend to an $L$-coloring of $G$. If either of the colorings $(\phi(p_1), \phi(p_2), r)$, $(\phi'(p_1), \phi'(p_2), r')$ of $P^*$ extends to an $L$-coloring of $G^*$, then one of $\phi, \phi'$ extends to an $L$-coloring of $G$, contradicting our assumption. Since $G^*$ is not a broken wheel with principal path $P^*$, and every chord of $C^{G^*}$ in $G^*$ has $p_2$ as an endpoint, it follows from the minimality of $G$ that the colorings $(\phi(p_1), \phi(p_2), r)$ and $(\phi'(p_1), \phi'(p_2), r')$ of $P^*$ are not distinct. Thus, $\phi, \phi'$ use the same color on each of $p_1, p_2$ and $r=r'$. Since $\phi\neq\phi'$, it follows that $\phi, \phi'$ differ precisely on $p_3$. Let $a=\phi(p_1)=\phi'(p_1)$ and $b=\phi(p_2)=\phi'(p_2)$. By Theorem \ref{thomassen5ChooseThm}, there is an extension of the coloring $(a, b)$ of $p_1p_2$ to an $L$-coloring $\psi$ of $G^*$. Since the colors of $\{b, \phi(p_3), \phi'(p_3)\}$ are all distinct, it follows from Theorem \ref{thomassen5ChooseThm} that there is an extension of the coloring $(\psi(u_j), b)$ of $u_jp_2$ to an $L$-coloring of $G^{**}$ using one of $\phi(p_3), \phi'(p_3)$ on $p_3$. But then one of $\phi, \phi'$ extends to an $L$-coloring of $G$, contradicting our assumption. \end{claimproof}
Since there is no chord of $C$ in $G$ it follows from Lemma \ref{PartialPathColoringExtCL0} that $G\setminus C$ contains a vertex $v^*$ adjacent to all three vertices of $P$, and, since neither $\phi$ nor $\phi'$ extends to an $L$-coloring of $G$, we have $|L_{\phi}(v^*)|=|L_{\phi'}(v^*)|=2$. Let $L_{\phi}(v^*)=\{r,s\}$ and $L_{\phi'}(v^*)=\{r', s'\}$.
Since $G$ is short-separation-free, $G-p_2$ has outer cycle $p_1v^*p_3u_t\ldots u_1$, and there is no chord of $p_1v^*p_3u_t\ldots u_1$ which is not incident to $v^*$, or else there is a chord of $C$ in $G$ which is not incident to $p_2$. Thus, if $G-p_2$ is not a broken wheel with principal path $p_1v^*p_3$, then it follows from the minimality of $G$ that one of the two colorings $(\phi(p_1), r, \phi(p_3))$, $(\phi(p_1), s, \phi(p_3))$ extends to an $L$-coloring of $G-p_2$. If that holds, then $\phi$ extends to an $L$-coloring of $G$, contradicting our assumption. We conclude that $v^*$ is adjacent to each of $u_1, \ldots, u_t$, and $G$ is a wheel with central vertex $v^*$. Since neither $\phi$ nor $\phi'$ extends to an $L$-coloring of $G$, we have the following.
\begin{enumerate}[label=\roman*)]
\itemsep-0.1em
\item $\phi(p_1), \phi'(p_1)\in L(u_1)$ and $\phi(p_3), \phi'(p_3)\in L(u_t)$; \emph{AND}
\item For each $j=1,\ldots, t$, $|L(u_j)|=3$ and $\{r,s\}\cup\{r', s'\}\subseteq L(u_j)$.
\end{enumerate}
Now consider the following cases.
\textbf{Case 1:} $\phi, \phi'$ use the same color on $p_1$ and the same color on $p_3$
In this case, let $a=\phi(p_1)=\phi'(p_1)$ and $b=\phi(p_3)=\phi'(p_3)$. Since $L(v^*)=\{\phi(p_1), \phi(p_2), \phi(p_3), r,s\}=\{\phi'(p_1), \phi'(p_2), \phi'(p_3), r', s'\}$, we have $\{\phi(p_2), r, s\}=\{\phi'(p_2), r', s'\}$. Since $\phi, \phi'$ are distinct colorings of $P$, we have $\phi(p_2)\neq\phi'(p_2)$, so $|\{r,s,r', s'\}|\geq 3$. But then, by ii), $L(u_1)$ consists of three colors which are not $a,b$, contradicting i).
\textbf{Case 2:} $\phi, \phi'$ differ on at least one of $p_1, p_3$
In this case, suppose without loss of generality that $\phi(p_1)\neq\phi'(p_1)$. Let $\phi(p_1)=a$ and $\phi'(p_1)=b$. By i), we have $\{a,b\}\subseteq L(u_1)$. By ii), $\{r,s\}\subseteq L(u_1)$ and $|L(u_1)|=3$. Since $\{r,s\}=L_{\phi}(v^*)$, it follows that $b\in\{r,s\}$. Likewise, since $L_{\phi'}(v^*)=\{r',s'\}$ and $\{r', s'\}\subseteq L(u_1)$, we have $a\in\{r', s'\}$. Suppose without loss of generality that $a=r'$ and $b=r$. Since $|L(u_1)|=3$, it follows that $s=s'$, and, by ii), we have $L(u_j)=\{a,b,s\}$ for each $j=1,\ldots, t$. Since $\{r,s\}=L_{\phi}(v^*)$ and $\{r', s'\}=L_{\phi'}(v^*)$, and $\phi(p_3), \phi'(p_3)\in L(u_t)$, we have $\phi(p_3), \phi'(p_3)\in\{a,b\}$. Since $|L_{\phi}(v^*)=2$, we have $\phi(p_1)\neq\phi(p_3)$, so $\phi(p_3)=b$, contradicting the fact that $b\in L_{\phi}(v^*)$. This completes the proof of 1) of Theorem \ref{EitherBWheelOrAtMostOneColThm}.
Now we prove 2). Suppose that 2) does not hold and let $G$ be a vertex-minimal counterexample to 2). Thus, there is a $p\in\{p_1, p_3\}$ and an $L$-coloring $\phi$ of $V(P)$ which does not extend to an $L$-coloring of $G$, where, letting $u$ be the unique neighbor of $p$ on the path $C-p_2$, we have $\phi(p)\not\in L(u)$. Suppose without loss of generality that $p=p_1$.
\begin{Claim} $|V(C)|>4$. \end{Claim}
\begin{claimproof} Since $\phi$ is a $L$-coloring of $V(P)$ but $\phi$ does not extend to an $L$-coloring of $G$, it follows from Corollary \ref{CycleLen4CorToThom} that $V(C)\neq V(P)$. Now suppose that $|V(C)|=4$. Thus, $C=p_1p_2p_3u$ and $|L(u)|\geq 3$. Possibly $G$ is a broken wheel with principal path $P$, but, in any case, since $\phi(p_1)\not\in L(u)$, $\phi$ extends to an $L$-coloring of $V(C)$. Thus, again by Corollary \ref{CycleLen4CorToThom}, $\phi$ extends to an $L$-coloring of $G$, a contradiction. \end{claimproof}
Since $|V(C)|>4$, we have $u\neq p_3$. In particular, $|L(u)|\geq 3$.
\begin{Claim}\label{2BWThmChordIncidence} There is a chord of $C$. \end{Claim}
\begin{claimproof} Suppose not. Since $\phi$ does not extend to an $L$-coloring of $G$, it follows from Lemma \ref{PartialPathColoringExtCL0} that there is a $w\in V(G\setminus C)$ adjacent to each of $p_1, p_2, p_3$. Since $G$ is short-separation free, $G-p_2$ is bounded by outer cycle $(C-p_2)+\{p_1w, p_3w\}$. Since $|L_{\phi}(w)|\geq 2$ and every chord of the outer face of $G-p_2$ is incident to $w$, it follows from 1) that $G$ is a wheel with central vertex $w$. But then, since $\phi(p_1)\not\in L(u)$, it is mmediate that $\phi$ extends to an $L$-coloring of $G$, a contradiction. \end{claimproof}
Applying Claim \ref{2BWThmChordIncidence}, together with our assumption that every chord of $C$ is incident to $p_2$, we let $u^*$ be the unique neighbor of $p_2$ which is closest on the path $C\setminus\{p_1, p_2\}$. Let $G=G'\cup G''$ be the natural $u^*p_2$-partition of $G$, where $p_1\in V(G')$ and $p_3\in V(G'')$. Let $P':=p_1p_2u^*$ and $P'':=u^*p_2p_3$. Applying Theorem \ref{thomassen5ChooseThm}, we fix a $c\in\Lambda_{G''}^{P''}(\bullet, \phi(p_2), \phi(p_3))$. Consider the following cases.
\textbf{Case 1:} $u^*=u$
In this case, since $G$ is short-separation-free, $G'$ is a triangle and $G''$ is bounded by outer cycle $(C-p_1)+up_2$. Since $\phi(p_1)\not\in L(u)$, we have $c\neq \phi(p_1)$, so $\phi$ extends to an $L$-coloring of $G$ which uses $c$ on $u$, contradicting our assumption that $\phi$ does not extend to an $L$-coloring of $G$.
\textbf{Case 2:} $u^*\neq u$
In this case, as every chord of $C$ is incident to $p_2$, we have $p_1u^*\not\in E(G)$. Possibly $\phi(p_1)=c$, but, in any case, the $L$-coloring $(\phi(p_1), \phi(p_2), c)$ of $p_1p_2u^*$ is a proper $L$-coloring of $V(P')$. Since $\phi(p_1)\not\in L(u)$ and $|V(G')|<|V(G)|$, it follows from the minimality of $G$ that the $L$-coloring $(\phi(p_1), \phi(p_2), c)$ of $V(P')$ extends to an $L$-coloring of $G'$, so $\phi$ extends to an $L$-coloring of $G$, a contradiction. This proves 2) of Theorem \ref{EitherBWheelOrAtMostOneColThm}. \end{proof}
Theorem \ref{EitherBWheelOrAtMostOneColThm} has the following useful corollary, which we also use repeatedly.
\begin{cor}\label{CorMainEitherBWheelAtM1ColCor} Let $(G, C, P, L)$ be a rainbow, where $P:=p_1p_2p_3$ is a 2-path. Suppose further that $G$ is short-separation-free and every chord of $C$ has $p_2$ as an endpoint. Then,
\begin{enumerate}[label=\arabic*)]
\itemsep-0.1em
\item if $|V(C)|>4$ and $|L(p_3)|\geq 2$, then, letting $xyp_3$ be the unique 2-path of $C-p_2$ with endpoint $p_3$, either there is a $G$-universal $a\in L(p_3)$ or $G$ is a broken wheel with principal path $P$ such that $L(p_3)\subseteq L(x)\cap L(y)$; AND
\item If $|L(p_3)|\geq 3$ and either $|V(C)|>3$ or $G=C$, then
\begin{enumerate}[label=\roman*)]
\itemsep-0.1em
\item Either $G$ is a broken wheel with principal path $P$, or $|\Lambda_G^P(\phi(p_1), \phi(p_2), \bullet)|>1$ for any $L$-coloring $\phi$ of $p_1p_2$; AND
\item For each $a\in L(p_1)$, there are at most two $b\in L(p_2)\setminus\{a\}$ such that $|\Lambda_G^P(a, b, \bullet)|=1$; AND
\end{enumerate}
\item If $\phi$ is an $L$-coloring of $\{p_1, p_3\}$ and $S$ is a subset of $L_{\phi}(p_2)$ with $|S|\geq 2$ and $S\cap\Lambda_G^P(\phi(p_1), \bullet, \phi(p_3))=\varnothing$, then
\begin{enumerate}[label=\roman*)]
\itemsep-0.1em
\item $|S|=2$ and $G$ is a broken wheel with principal path $P$, and $G-p_2$ is a path of odd length; AND
\item For any $L$-coloring $\psi$ of $V(P)$ which does not extend to an $L$-coloring of $G$, either $\psi(p_1)=\psi(p_3)=s$ for some $s\in S$, or $\psi$ restricts to the same coloring of $\{p_1, p_3\}$ as $\phi$.
\end{enumerate}
\end{enumerate} \end{cor}
\begin{proof} We first prove 1). Suppose that $|V(C)|>4$ and suppose that $|L(p_3)|\geq 2$. Since every chord of $C$ is incident to $p_2$, we have $p_1p_3\not\in E(G)$. If $G$ is not a broken wheel with principal path $P$, then, since $|L(p_3)|\geq 2$, it follows from 1) of Theorem \ref{EitherBWheelOrAtMostOneColThm} that there is an $a\in L(p_3)$ such that any $L$-coloring of $V(P)$ using $a$ on $p_3$ extends to an $L$-coloring of $G$, and since $p_1p_3\not\in E(G)$, we then have $\Lambda_G^P(\bullet, b, a)=L(p_1)\setminus\{b\}$ for all $b\in L(p_2)$, so we are done in that case. Now suppose that $G$ is a broken wheel with principal path $P$. If $L(p_3)\not\subseteq L(y)$ then it is again immediate that there is a $G$-universal color of $L(p_3)$, so suppose that $L(p_3)\subseteq L(y)$. If $L(p_3)\subseteq L(x)$ as well, then we are done, so suppose that $L(p_3)\not\subseteq L(x)$. Since $L(p_3)\subseteq L(y)$ and $|L(p_3)|\geq 2$, but $L(p_3)\not\subseteq L(x)$, it follows that there is an $a\in L(p_3)$ such that $L(y)\setminus\{a\}\not\subseteq L(x)$. Since $|V(G)|>4$, it follows from 3) of Theorem \ref{BWheelMainRevListThm2} that $a$ is $G$-universal. This proves 1) of Corollary \ref{CorMainEitherBWheelAtM1ColCor}.
We now prove 2i). If $G$ is a triangle, then it is a broken wheel with principal path $P$, so we are done in that case, so suppose that $|V(C)|>3$. Since every chord of $C$ is incident to $p_2$, every $L$-coloring of the path $p_1p_2p_3$ is an $L$-coloring of $V(P)$, and since $|\Lambda_G^P(\phi(p_1), \phi(p_2), \bullet)|\leq 1$, it follows from 1) of Theorem \ref{EitherBWheelOrAtMostOneColThm} that $G$ is a broken wheel with principal path $P$. This proves 2i). If $G$ is not a broken wheel with principal path $P$, then 2ii) follows immediately from 2i), and, if $G$ is a broken wheel with principal path $P$, then 2ii) follows from 2) of Theorem \ref{BWheelMainRevListThm2}.
Now we prove 3). Since $|S|\geq 2$ and $S\cap\Lambda_G^P(\phi(p_1), \bullet, \phi(p_3))=\varnothing$, it follows from 1) of Theorem \ref{EitherBWheelOrAtMostOneColThm} that $G$ is a broken wheel with principal path $P$. Furthermore, $G$ is not a triangle, and, letting $G-p_2=p_1u_1\ldots u_tp_3$ for some $t\geq 1$, we have $|L(u_i)|=3$ for each $i=1,\ldots, t$. In particular, $S\subseteq L(u_1)\cap\ldots\cap L(u_t)$, and $L(u_1)=\{\phi(p_1)\}\cup S$ as a disjoint union and $L(u_t)=\{\phi(p_3)\}\cup S$ as a disjoint uniont, so $|S|=2$. Let $S=\{s, s'\}$.
If $t$ is odd, then we color $p_2$ with $s$ and color each of $u_1, u_3, \ldots, u_t$ with $s'$, which leaves a color over for each of $u_2, u_4, \ldots, u_{t-1}$, as each of these vertices has two neighbors using the same color. Thus, if $t$ is odd, then $s\in\Lambda(\phi(p_1), \bullet, \phi(p_3))$, contradicting our assumption, so $t$ is even and $G-p_2$ has odd length. This proves i) of 3).
Now we prove ii). Let $\psi$ be an $L$-coloring of $V(P)$ which does not extend to an $L$-coloring of $G$. If both $\psi(p_1)=\phi(p_1)$ and $\psi(p_3)=\phi(p_3)$, then we are done, so suppose without loss of generality that $\psi(p_1)\neq\phi(p_1)$. Since $L(u_1)=S\cup\{\phi(p_1)\}$ and $\psi(p_1)\in L(u_1)$, we have $\psi(p_1)\in S$. Suppose for the sake of definiteness that $\psi(p_1)=s$. We just need to show that $\psi(p_3)=s$ as well. Suppose not. Since $\psi(p_3)\in L(u_t)$ and $L(u_t)=\{\phi(p_3)\}\cup S$, we have $\psi(p_3)\in\{\phi(p_3), s'\}$. Consider the following cases.
\textbf{Case 1:} $\psi(p_3)=s'$
In this case, since $\psi(p_1)=s$, we have $\psi(p_2)\not\in S$. But since $G-p_2$ has odd length, we extend $\psi$ to an $L$-coloring of $G$ by first coloring each of $u_1, u_3,\ldots u_{t-1}$ with $s'$ and coloring $p_2$ with $\psi(p_2)$. Since $\psi(p_3)=s'$, each of $u_2, u_4, \ldots, u_t$ has two neighbors with the same color, so we extend $\psi$ to an $L$-coloring of $G$, contradicting our assumption.
\textbf{Case 2:} $\psi(p_3)=\phi(p_3)$
In this case, $\psi(p_3)\not\in S$. Since $\psi$ does not extend to an $L$-coloring of $G$, $\psi(p_2)\in L(u_1)\cap\ldots\cap L(u_t)$, and since $L(u_t)=S\cup\{\phi(p_3)$, we have $\psi(p_2)\in S$. Since $\psi(p_1)=s$, we have $\psi(p_2)=s'$. But now, since $\psi(p_3)\not\in S$ and $G-p_2$ has odd length, we extend $\psi$ to an $L$-coloring of $G$ by coloring each of $u_2, u_4, \ldots, u_t$ with $s$. Since $\psi(p_1)=s$, each of $u_1, u_3, \ldots, u_{t-1}$ has two neighbors of the same color, so $\psi$ extends to an $L$-coloring of $G$, a contradiction. \end{proof}
\section{Extending Colorings of the Endpoints of 2-Paths}\label{ExtCol2PathAugColSec}
To prove the result which makes up the Section \ref{ExtCol2PathAugColSec}, we first introduce the following notation.
\begin{defn}\label{EndNotationColor} \emph{Let $G$ be a graph and let $L$ be a list-assignment for $G$. Let $P$ be a path in $G$ with $|V(P)|\geq 3$, let $H$ be a subgraph of $G$ and let $\{p, p'\}$ be the endpoints of $P$. We let $\textnormal{End}_L(H, P, G)$ be the set of $L$-colorings $\phi$ of $\{p, p'\}\cup V(H)$ such that $\phi$ is $(P,G)$-sufficient.} \end{defn}
We usually drop the subscript $L$ if it is clear from the context. Furthemore, if $H=\varnothing$, then we just write $\textnormal{End}_L(P,G)$.That is, $\textnormal{End}_L(P, G)$ is the set of $L$-colorings $\phi$ of the endpoints of $P$ such that any extension of $\phi$ to an $L$-coloring of $V(P)$ extends to $L$-color all of $G$. Note that, in the case of a path of length either two or three, the notation of Definition \ref{GeneralAugCrownNotForLink} is a generalization of the definition above (in which we are allowed to color more than just the endpoints of $P$). In particular, we have the following observation, which is immediate from Definitions \ref{GeneralAugCrownNotForLink} and \ref{EndNotationColor}.
\begin{obs}\label{ObsStrengtheningCrownEnd} Let $G$ be a planar graph with outer cycle $C$, let $L$ be a list-assignment for $G$, and let $P$ be a path in $C$ with endpoints $p_0, p_1$, where $P$ has length either two or three. Let $H$ be a subgraph of $G$, where $V(H)\subseteq V(C\setminus P)$, and, for each $i=0,1$, $|V(H)\cap N(q_i)|\leq 1$. Then $\textnormal{End}_L(H, P, G)\subseteq\textnormal{Crown}_L(P,G)$. \end{obs}
In this section, we prove a result about colorings of the form given in Definition \ref{EndNotationColor} in the case where $P$ is a 2-path, and, in Section \ref{3ChordAnalogue2PathSec}, we prove an analogous result in the case where $P$ is a 3-path. The remainder of Section \ref{ExtCol2PathAugColSec} consists of the proof of Theorem \ref{SumTo4For2PathColorEnds}.
\begin{theorem}\label{SumTo4For2PathColorEnds} Let $(G, C, P, L)$ be an end-linked rainbow, where $P$ is a 2-path. Then $\textnormal{End}_{L}(P,G)\neq\varnothing$. \end{theorem}
\begin{proof} Suppose this does not hold, and let $G$ be a vertex-minimal counterexample to the theorem. Let $C, P, L$ be as in the statement of the theorem, where $P:=p_1p_2p_3$. If $V(C)=V(P)$, then, by Corollary \ref{CycleLen4CorToThom}, any $L$-coloring of $V(P)$ extends to an $L$-coloring of $G$, contradicting our assumption that $G$ is a counterexample. Thus, $|V(C)|>3$.
\begin{Claim} $p_1p_3\not\in E(G)$. \end{Claim}
\begin{claimproof} Suppose that $p_1p_3\in E(G)$. Since $|V(C)|>3$, $p_1p_3$ is a chord of $G$. Let $G_0\cup G_1$ be the natural $p_1p_3$-partition of $G$, where $p_2\in V(G_0)$. Since $|L(p_1)|+|L(p_3)|\geq 4$ and each of $L(p_1)$ and $L(p_3)$ is nonempty, there is an $L$-coloring $\phi$ of $\{p_1, p_3\}$. By Theorem \ref{thomassen5ChooseThm}, $\phi$ extends to an $L$-coloring of $G_1$. By Corollary \ref{CycleLen4CorToThom}, any $L$-coloring of $p_1p_2p_3$ extends to an $L$-coloring of $G_0$. Thus, any $L$-coloring of $V(P)$ using $\phi(p_1), \phi(p_3)$ on the respective vertices $p_1, p_3$ extends to an $L$-coloring of $G$, contradicting our assumption that $G$ is a counterexample.\end{claimproof}
Since $p_1p_3\not\in E(G)$ and $G$ is a counterexample, it follows that, for any $a\in L(p_1)$ and $b\in L(p_3)$, there is an $L$-coloring $\sigma_{ab}$ of $V(P)$ which uses $a,b$ on the respective colors $p_1, p_3$, where $\sigma_{ab}$ does not extend to an $L$-coloring of $G$. Let $\mathcal{F}:=\{\sigma_{ab}: (a,b)\in L(p_1)\times L(p_3)\}$.
\begin{Claim} $G$ is short-separation-free. \end{Claim}
\begin{claimproof} Suppose toward a contradiction that there is a separating cycle $D$ of length at most four in $G$. If necessary, we suppose that $D$ is an induced subgraph of $G$. This is permissible since if this does not hold, then there is a separating triangle $T$ in $G$ whose vertices lie in $V(D)$, so we replace $D$ with $T$, and $T$ is an induced subgraph of $G$. Since $D$ is a separating cycle, we have $|V(\textnormal{Ext}(D))|<|V(G)|$, and, by the minimality of $G$, there is a $\sigma\in\mathcal{F}$ which extends to an $L$-coloring $\sigma^*$ of $\textnormal{Ext}(D)$. Since $D$ is an induced cycle of $G$, $\tau$ is a proper $L$-coloring of the subgraph of $G$ induced by $V(\textnormal{Ext}(D))$. By Corollary \ref{CycleLen4CorToThom}, $\sigma^*$ extends to an $L$-coloring of $\textnormal{Int}(D)$, so $\sigma$ extends to an $L$-coloring of $G$, contradicting our assumption. \end{claimproof}
\begin{Claim}\label{evchrochordp2end}
$G$ is a broken wheel with principal path $P$. In particular, $V(G)=V(C)$.
\end{Claim}
\begin{claimproof} We first show that every chord of $C$ has $p_2$ as an endpoint. Suppose toward a contradiction that there is a chord $xy$ of $C$ without $p_2$ as an endpoint. Let $G=G'\cup G''$ be the natural $xy$-partition of $G$, where $P\subseteq G'$. By the minimality of $|V(G)|$, there is a $\sigma\in\mathcal{F}$ which extends to an $L$-coloring $\tau$ of $G'$, and, by Theorem \ref{thomassen5ChooseThm}, $G''$ is $L^{xy}_{\tau}$-colorable, so $\tau$ extends to an $L$-coloring of $G$. But then $\sigma$ extends to an $L$-coloring of $G$, which is false. Suppose now that $G$ is not a broken wheel with principal path $P$. As shown above, every chord of $C$ is incident to $p_2$. Since $|L(p_1)|+|L(p_3)|\geq 4$ and each of $L(p_1)$ and $L(p_3)$ is nonempty, it follows from Theorem 1) of Theorem \ref{EitherBWheelOrAtMostOneColThm} that there is a $\sigma\in\mathcal{F}$ which extends to an $L$-coloring of $G$, contradicting our assumption. \end{claimproof}
Since $|V(C)|>3$ and $G$ is a broken wheel with principal path $P$, we let $G\setminus P:=u_1\ldots u_t$ for some $t\geq 1$.
\begin{Claim}\label{SizeHClaim4} $|V(G)|>4$. \end{Claim}
\begin{claimproof} Suppose not. Thus, $|V(G)|=4$, and $G-p_2=p_1u_1p_3$. Since $|L(p_1)|+|L(p_3)|\geq 4$, there exist $c\in L(p_1)$ and $d\in L(p_3)$ with $|L(u_1)\setminus\{c,d\}|\geq 2$, so $\sigma_{cd}$ extends to an $L$-coloring of $G$, which is false. \end{claimproof}
\begin{Claim}\label{NotSubsetSideTwoAway} $L(p_1)\not\subseteq L(u_2)$, and furthermore, $L(p_3)\not\subseteq L(u_{t-1})$. \end{Claim}
\begin{claimproof} These two claims are symmetric, so it just suffices to prove that $L(p_1)\not\subseteq L(u_2)$. Suppose toward a contradiction that $L(p_1)\subseteq L(u_2)$. Let $G':=G\setminus\{p_1, u_1\}$. Now, $G'$ is a broken wheel with principal path $P':=u_2p_2p_3$. Furthermore, $|L(u_2)\cap L(p_1)|+|L(p_3)|=|L(p_1)|+|L(p_3)|$, so $|L(u_2)\cap L(p_1)|+|L(p_3)|\geq 4$. By the minimality of $G$, there is an $L$-coloring $\psi$ of $\{u_2, p_3\}$ with $\psi(u_2)\in L(u_2)\cap L(p_1)$, where any extension of $\psi$ to an $L$-coloring of $V(P')$ extends to an $L$-coloring of $G'$. Let $c=\psi(u_2)$ and $d=\psi(p_3)$. Since $c\in L(p_1)$, we have an $L$-coloring $\sigma_{cd}$ of $V(P)$ which does not extend to an $L$-coloring of $G$. Yet $\sigma_{cd}\cup\psi$ is a proper $L$-coloring of its domain which leaves the color $c$ for $u_2$, so $\sigma_{cd}\cup\psi$ extends to an $L$-coloring $\tau$ of $V(G')\cup\{p_1\}$ which uses $c$ on $u_2$. Now, $\tau$ is an $L$-coloring of $G-u_1$, and this $L$-coloring uses the same color on $p_1, u_2$, so $\tau$ extends to an $L$-coloring of $G$, and thus $\sigma_{cd}$ extends to an $L$-coloring of $G$, which is false. \end{claimproof}
Now we have enough to finish the proof of Theorem \ref{SumTo4For2PathColorEnds}. At least one of $p_1, p_3$ has an $L$-list of size greater than 1, so suppose without loss of generality that $|L(p_1)|>1$. By Claim \ref{NotSubsetSideTwoAway}, $L(p_1)\not\subseteq L(u_2)$. By Claim \ref{SizeHClaim4}, $|V(G)|>4$, and since $V(G)=V(C)$, it follows from 1) of Corollary \ref{CorMainEitherBWheelAtM1ColCor}, there is a $c\in L(p_1)$ such that any $L$-coloring of $V(P)$ using $c$ on $p_1$ extends to an $L$-coloring of $G$. Since $L(p_3)\neq\varnothing$, there is a $d\in L(p_3)$, so $\sigma_{cd}$ extends to an $L$-coloring of $G$, a contradiction. This completes the proof of Theorem \ref{SumTo4For2PathColorEnds}. \end{proof}
Theorem \ref{SumTo4For2PathColorEnds} has a simple corollary. To state this corollary, we first introduce the following natural notation.
\begin{defn}\label{DefnForColorSetsonVertex} \emph{Let $G$ be a graph with list-assignment $L$. Given a set $\mathcal{F}$ of partial $L$-colorings of $G$ and a vertex $x\in V(G)$ with $x\in\textnormal{dom}(\phi)$ for each $\phi\in\mathcal{F}$, we define $\textnormal{Col}(\mathcal{F}\mid x)=\{\phi(x): \phi\in\mathcal{F}\}$.} \end{defn}
\begin{cor}\label{GlueAugFromKHCor} Let $(G, C, P, L)$ be an end-linked rainbow and $pq$ be a terminal edge of $P$, where $p$ is an endpoint of $P$, and let $p'$ be the other endpoint of $P$. Let $qx$ be an edge with $x\not\in V(C-p')$. Let $H, K$ be subgraphs of $G$ bounded by respective outer faces $x(C\setminus\mathring{P})p'Pq$ and $p(C\setminus\mathring{P})xq$. Let $\mathcal{F}$ be a nonempty family of partial $L$-colorings of $H$, where each element of $\mathcal{F}$ has $x$ in its domain. Suppose further that either $x=p$ or $|\textnormal{Col}(\mathcal{F}\mid x)|\geq |L(p')|$. Then there is an $L$-coloring $\phi$ of $\{p, x\}$ and a $\psi\in\mathcal{F}$ such that $\phi(x)=\psi(x)$ and $\phi$ is $(q, K)$-sufficient. \end{cor}
\begin{proof} This is immediate if $x=p$, since $K$ is just the edge $pq$ in that case, so now suppose that $x\neq p$, so $qx$ is a chord of $C$ and $|L(x)|\geq 3$. If $|L(p)|\geq 3$, then it follows from Theorem \ref{SumTo4For2PathColorEnds} that, for each $c\in L(x)$, there is a $\phi\in\textnormal{End}(pqx, K)$ with $\phi(x)=c$, so we are done in that case. Now suppose that $1\leq |L(p)|\leq 2$. Since $|L(x)|\geq 3$, it follows from Theorem \ref{SumTo4For2PathColorEnds} that there is a set of $|L(p)|+|L(x)|-3$ elements of $\textnormal{End}(pqx, K)$, each using a different color on $x$. Since $|L(p)|+|L(p')|\geq 4$, there is a $\phi\in\textnormal{End}(pqx, K)$ and a $\psi\in\mathcal{F}$ with $\phi(x)=\psi(x)$. \end{proof}
\section{Extending Colorings of 3-Paths}\label{LinkColoring3PathExCycleSec}
This is the first of two sections in which we prove a result about partially coloring a 3-path in a facial cycle of a planar graph. We now state and prove our lone result for Section \ref{LinkColoring3PathExCycleSec}.
\begin{theorem}\label{ThmFirstLink3PathForUseInHolepunch} Let $(G, C, P, L)$ be an end-linked rainbow, where $P:=p_1p_2p_3p_4$ is a 3-path. Then
\begin{enumerate}[label=\arabic*)]
\itemsep-0.1em
\item $\textnormal{Crown}_L(P, G)\neq\varnothing$. Actually, something stronger holds. There is a subgraph $H$ of $G$ with $V(H)\subseteq V(C\setminus P)\cap N(p_1)\cup N(p_4))$, where $|V(H)\cap N(p)|\leq 1$ for each endpoint $p$ of $P$ and $\textnormal{End}_L(H,P,G)\neq\varnothing$; AND
\item If there is no chord of $C$ incident to a vertex of $\mathring{P}$, then $\textnormal{End}_{L}(P, G)\neq\varnothing$.
\end{enumerate}
\end{theorem}
\begin{proof} We first note that 2) of Theorem \ref{ThmFirstLink3PathForUseInHolepunch} is an immediate consequence of 1). To see this, suppose there is a $\phi\in\textnormal{Crown}_{L}(P,G)$ and that no chord of $C$ is incident to a vertex of $\mathring{P}$. Let $\phi'$ be the restriction of $\phi$ to the endpoints of $P$. We just need to show that $\phi'\in\textnormal{End}_{L}(P, G)$. Let $\psi$ be an arbitrary extension of $\phi'$ to an $L$-coloring of $V(P)$. Since no chord of $C$ is incident to either vertex of $\mathring{P}$, the union $\psi\cup\phi$ is a proper $L$-coloring of its domain, and this $L$-coloring extends to $L$-color all of $G$, as $\phi\in\textnormal{Crown}_{L}(P, G)$. Thus, $\phi'\in\textnormal{End}_{L}(P, G)$. So it suffices to show that 1) holds. Note that it follows from Observation \ref{ObsStrengtheningCrownEnd} that the second part of the statement of 1) implies the first part of the statement of 1). Suppose toward a contradiction that $(G, P, C, L)$ does not satisfy 1), where $G$ is vertex-minimal with respect to this property, and $P:=p_1p_2p_3p_4$. By adding edges to $G$ if necessary, we suppose further that every face of $G$, except possibly $C$, is bounded a triangle. By removing colors from the lists of some vertices if necessary, we also suppose for convenience that each vertex of $C\setminus P$ has a list of size precisely three, and that $L(p_1)$ and $L(p_4)$ are nonempty sets with $|L(p_1)|+|L(p_4)|=4$.
\begin{Claim}\label{SSFreeandChordEndpointRing} $G$ is short-separation-free, and any chord of $C$ has an endpoint in $\mathring{P}$. \end{Claim}
\begin{claimproof} It is immediate from Corollary \ref{CycleLen4CorToThom}, together with the minimality of $G$, that $G$ is short-separation-free. Now suppose toward a contradiction that there is a chord $xy$ of $C$, where $x,y\not\in V(\mathring{P})$. Likewise, it is immediate from Theorem \ref{thomassen5ChooseThm}, together with the minimality of $G$, that any chord of $C$ has an endpoint in $\mathring{P}$.\end{claimproof}
\begin{Claim}\label{AnyEndpointColClaExt3P} For any $L$-coloring $\phi$ of $\{p_1, p_4\}$, there is an extension of $\phi$ to an $L$-coloring of $V(P)$ which does not extend to $L$-color $G$. \end{Claim}
\begin{claimproof} Let $\phi$ be an $L$-coloring of $\{p_1, p_4\}$. Since $|\textnormal{dom}(\phi)|=2$, we have $|L_{\phi}(q)|\geq |L(q)|-2$ and $|N(q)\cap\textnormal{dom}(\phi)|\leq 2$ for each $q\in\{p_2, p_3\}$. Since $G$ is a counterexample, there is an extension of $\phi$ to an $L$-coloring of $V(P)$ which does not extend to $L$-color $G$. \end{claimproof}
\begin{Claim}\label{NoChorBothEndPClaimFirst3Path} $E(G)$ contains either $p_1p_3$ nor $p_2p_4$.Furthermore $|V(C)|>4$. \end{Claim}
\begin{claimproof} Suppose that $E(G)$ contains one of $p_1p_3, p_2p_4$, say without loss of generality that $p_1p_3\in E(G)$. Let $G=G_0\cup G_1$ be the natural $p_1p_3$-partition of $G$, where $p_2\in V(G_0)$. Since $G$ is short-separation-free by Claim \ref{SSFreeandChordEndpointRing}, $G_1$ is the triangle $p_1p_2p_3$. By Theorem \ref{SumTo4For2PathColorEnds}, since $|L(p_1)|+|L(p_4)|=4$, there is an $L$-coloring $\phi$ of $\{p_1, p_4\}$ such that any extension of $\phi$ to an $L$-coloring of $p_1p_2p_4$ extends to $L$-color all of $G$. Since $G_1=p_1p_2p_3$, any extension of $\phi$ to an $L$-coloring of $\textnormal{dom}(\phi)\cup\{p_2, p_3\}$ extends to $L$-color all of $G$, contradicting Claim \ref{AnyEndpointColClaExt3P}. We conclude that $E(G)$ contains either $p_1p_3$ nor $p_2p_4$. Now suppose that $|V(C)|\leq 4$. Thus, $V(C)=V(P)$, and it follows from Corollary \ref{CycleLen4CorToThom} that, for any $L$-coloring $\phi$ of $\{p_1, p_4\}$ and any extension of $\phi$ to an $L$-coloring $\phi'$ of $V(C)$, $\phi'$ extends to an $L$-coloring of $G$, contradicting Claim \ref{AnyEndpointColClaExt3P}. Thus, $|V(C)|>4$.
\end{claimproof}
Since $|V(C)|>4$, we let $C\setminus\mathring{P}=p_1u_1\ldots u_tp_4$ for some $t\geq 1$.
\begin{Claim}\label{First3PathCInducedMinCounterS} $C$ is an induced cycle. \end{Claim}
\begin{claimproof} Suppose not. By Claim \ref{SSFreeandChordEndpointRing}, every chord of $C$ has an endpoint in $\{p_2, p_3\}$, so suppose without loss of generality that there is a chord of $C$ incident to $p_2$. By Claim \ref{NoChorBothEndPClaimFirst3Path}, the other endpoint of this chord lies in $C\setminus P$, so $N(p_2)\cap\{u_1, \ldots, u_t\}\neq\varnothing$. Let $u_m$ be the neighbor of $p_2$ of maximal index. Let $G=G'\cup G''$ be the natural $p_2u_m$-partition of $G$, where $p_1\in V(G')$ and $p_4\in V(G'')$. Let $A=\textnormal{Col}(\textnormal{End}(p_1p_2u_m, G')\mid u_m)$. Since $|L(u_m)|=3$ and $1\leq |L(p_1)|\leq 3$, it follows from Theorem \ref{SumTo4For2PathColorEnds} that $|A|\geq |L(p_1)|$. In particular, $|A|+|L(p_4)|\geq 4$. Now, $G''$ contains the 3-path $P''=u_mp_2p_3p_4$. Since $|V(G'')|<|V(G)|$ and $A$ is a nonempty set $|A|+|L(p_4)|\geq 4$, it follows from the minimality of $G$ that there is a partial $L$-coloring $\psi$ of $V(C^{G''})\setminus\{p_2, p_3\}$, where $u_m, p_4\in\textnormal{dom}(\psi)$ and $\psi$ is $(P'', G'')$-sufficient, and furthermore, for each $q\in\{p_2, p_3\}$, $|N(q)\cap\textnormal{dom}(\psi)|\leq 2$. Since $p_2p_4\not\in E(G)$, it follows from the maximality of $m$ that $\textnormal{dom}(\psi)\cap N(p_2)=\{u_m\}$.
Since $\psi(u_m)\in A$, there is a $\phi\in\textnormal{End}(p_1p_2u_m, G')$ with $\phi(u_m)=\psi(u_m)$. The union $\phi\cup\psi$ is a proper $L$-coloring of its domain. Furthermore, $N(p_2)\cap\textnormal{dom}(\phi\cup\psi)=\{p_1, u_m\}$, and thus $|L_{\phi\cup\psi}(p_2)|\geq |L(p_2)|-2$. Since $G$ contains the chord $p_2u_m$ of $C$, we have $N(p_3)\cap\textnormal{dom}(\phi\cup\psi)=N(p_3)\cap\textnormal{dom}(\psi)$, so, by our choice of $\psi$, $|N(p_3)\cap\textnormal{dom}(\phi\cup\psi)|\leq 2$. In particular, $|L_{\phi\cup\psi}(p_3)|\geq |L(p_3)|-2$ as well. By our choice of $\phi$ and $\psi$, any extension of $\phi\cup\psi$ to an $L$-coloring of $V(P)\cup\{u_m\}$ extends to $L$-color both of $G'$ and $G''$ and thus extends to an $L$-coloring of $G$, contradicting our assumption that $G$ is a counterexample. \end{claimproof}
\begin{Claim}\label{First3PathNeighborShareClMCL} $N(p_1)\cap N(p_3)=\{p_2\}$, and likewise, $N(p_2)\cap N(p_4)=\{p_3\}$. \end{Claim}
\begin{claimproof} Suppose not, and suppose without loss of generality that there is a $w\in V(G\setminus C)$ which is adjacent to each of $p_2, p_4$. By Claim \ref{NoChorBothEndPClaimFirst3Path}, $p_2p_4\not\in E(G)$. Let $G^*=G-p_3$. Since $G$ is short-separation-free and $p_2p_4\not\in E(G)$, $G^*$ is bounded by outer cycle $C^{G^*}=p_1u_1\ldots u_tp_4wp_2$, and in particular, by our triangulation conditions, $N(p_3)=\{p_2, w, p_4\}$. Since $|V(G^*)|<|V(G)|$ and $C^{G^*}$ contains the 3-path $P^*=p_1p_2wp_4$, there is a partial $L$-coloring $\psi$ of $V(C^{G^*})\setminus\{p_2, w\}$ such that $p_1, p_4\in\textnormal{dom}(\psi)$, where $\psi$ is $(P^*, G^*)$-sufficient and furthermore, for each $q\in\{p_2, w\}$, $|N(q)\cap\textnormal{dom}(\psi)|\leq 2$. We have $V(C^{G^*})\setminus\{p_2, w\}\subseteq V(C)\setminus\{p_2, p_3\}$, so $\textnormal{dom}(\psi)\subseteq V(C)\setminus\{p_2, p_3\}$. By Claim \ref{First3PathCInducedMinCounterS}, $C$ has no chords. Thus, for each $q\in\{p_2, p_3\}$, we have $|N(q)\cap\textnormal{dom}(\psi)|\leq 2$. Since $p_1, p_4\in\textnormal{dom}(\psi)$ and $G$ is a counterexample, there is an extension of $\psi$ to an $L$-coloring $\psi^*$ of $\textnormal{dom}(\psi)\cup\{p_2, p_3\}$, where $\psi^*$ does not extend to an $L$-coloring of $G$. But since $|L_{\psi}(w)|\geq 3$, we have $|L_{\psi^*}(w)|\geq 1$, and since $\psi^*$ leaves a color left in $w$, it follows from our choice of $\psi$ that $\psi^*$ extends to an $L$-coloring of $G$, a contradiction. \end{claimproof}
Now let $\phi$ be an arbitrary $L$-coloring of $\{p_1, p_4\}$. By Claim \ref{AnyEndpointColClaExt3P}, $\phi$ extends to an $L$-coloring $\phi^*$ of $V(P)$, where $\phi^*$ does not extend to $L$-color $G$. By Claim \ref{First3PathCInducedMinCounterS}, $C$ is an induced subgraph of $G$. By Lema \ref{PartialPathColoringExtCL0}, there exists a $w\in V(G\setminus C)$ with at least three neighbors in $C$, contradicting Claim \ref{First3PathNeighborShareClMCL}. This competes the proof of Theorem \ref{ThmFirstLink3PathForUseInHolepunch}. \end{proof}
In the statement of Theorem \ref{ThmFirstLink3PathForUseInHolepunch}, we require that $p$ and $p'$ are colored by $\phi$ but also allow $\phi$ to possibly color many vertices of $C\setminus\mathring{P}$. What happens if we try to strengthen Theorem \ref{ThmFirstLink3PathForUseInHolepunch} so that $\phi$ is only permitted to color the endpoints of $P$? Can we find an $L$-coloring $\phi$ of $\{p, p'\}$ such that $\phi$ is $(P,G)$-sufficient? This is clearly a more difficult task than that of Theorem \ref{ThmFirstLink3PathForUseInHolepunch}, since the endpoints of the lone middle edge of $P$ may be incident to chords of $C$ whose other endpoint we are no longer allowed to color. That is, if we are allowed to color some more vertices of $C\setminus\mathring{P}$ than just the endpoints, we may block certain undesirable colors from being used on the middle edge of $P$, but this is no longer necessarily true if we restrict the domain of our coloring to consist only of the endpoints of $P$. In general, the answer to the question posed above is clearly ``no", and there are many ways that this can fail, as indicated by the counterexamples in Figures \ref{DropConditionCounterExFig}, \ref{DropConditionCounterExFig2}, and \ref{DropConditionCounterExFig3}. Although the answer to the question above is no in general, the answer is ``yes" under some restricted circumstances, which we investigate in Section \ref{3ChordAnalogue2PathSec} below.
\section{A 3-Chord Analogue to Theorem \ref{SumTo4For2PathColorEnds}}\label{3ChordAnalogue2PathSec}
The purpose of this section is to prove a useful 3-chord analogue to Theorem \ref{SumTo4For2PathColorEnds}, which is stated below in Theorem \ref{3ChordVersionMainThm1}. We note that we cannot drop Condition 2) from the statement of Theorem \ref{3ChordVersionMainThm1}, or else the result is clearly false, is indicated by the counterexample in Figure \ref{DropConditionCounterExFig}, where, no matter how we $L$-color, $\{p_1, p_4\}$, there is an extension of that $L$-coloring to an $L$-coloring of $\{p_1, p_2, p_3, p_4\}$ in which all of $a,b,c$ are used among the neighbors of $u_1$. It is also natural to wonder what happens if we drop Condition 1) and only require that $(G, C, P, L)$ is an end-linked rainbow, so that we only impose the weaker condition that $L(p_1)$ and $L(p_4)$ are nonempty sets with $|L(p_1)|+|L(p_4)|\geq 4$. Then the result becomes false, even if we retain the condition that $p_3$ is incident to no chords of $C$ (although it does become true if we insist that $p_2$ is also incident to no chords of $C$, this is precisely 2) of Theorem \ref{ThmFirstLink3PathForUseInHolepunch}). This is indicated by Figures \ref{DropConditionCounterExFig2} and \ref{DropConditionCounterExFig3}.
\begin{center}\begin{tikzpicture}
\node[shape=circle,draw=black] [label={[xshift=-0.0cm, yshift=-1.3cm]\textcolor{red}{$\{a\}$}}] (p1) at (0,0) {$p_1$};
\node[shape=circle,draw=black] [label={[xshift=-0.0cm, yshift=+0.0cm]\textcolor{red}{$\{a,b, c\}$}}] (p2) at (0,2) {$p_2$};
\node[shape=circle,draw=black] [label={[xshift=-0.0cm, yshift=-1.3cm]\textcolor{red}{$\{a, b, c\}$}}] (u1) at (2,0) {$u_1$};
\node[shape=circle,draw=black] [label={[xshift=0.0cm, yshift=-1.3cm]\textcolor{red}{$\{a,b, c\}$}}] (p4) at (4,0) {$p_4$};
\node[shape=circle,draw=black] [label={[xshift=0.00cm, yshift=-0.0cm]\textcolor{red}{$\{a, b, c\}$}}] (p3) at (4,2) {$p_3$};
\draw[-] (p1) to (u1);
\draw[-] (p1) to (p2);
\draw[-] (u1) to (p4);
\draw[-] (u1) to (p3);
\draw[-] (p3) to (p4);
\draw[-] (p3) to (p2);
\draw[-] (u1) to (p2);
\end{tikzpicture}\captionof{figure}{Theorem \ref{3ChordVersionMainThm1} is false if we drop the condition that $p_3$ is incident to no chord of $C$}\label{DropConditionCounterExFig}\end{center}
\begin{center}\begin{tikzpicture}
\node[shape=circle,draw=black] [label={[xshift=-0.0cm, yshift=-1.3cm]\textcolor{red}{$\{a, r, s\}$}}] (p1) at (0,0) {$p_1$};
\node[shape=circle,draw=black] [label={[xshift=-0.65cm, yshift=-0.5cm]\textcolor{red}{$\{b\}$}}] (p2) at (0,2) {$p_2$};
\node[shape=circle,draw=black] [label={[xshift=-0.0cm, yshift=-1.3cm]\textcolor{red}{$\{b, r, s\}$}}] (u1) at (2,0) {$u_1$};
\node[shape=circle,draw=black] [label={[xshift=0.0cm, yshift=-1.3cm]\textcolor{red}{$\{d\}$}}] (p4) at (6,0) {$p_4$};
\node[shape=circle,draw=black] [label={[xshift=0.65cm, yshift=-0.5cm]\textcolor{red}{$\{c\}$}}] (p3) at (6,2) {$p_3$};
\node[shape=circle,draw=black] [label={[xshift=0.00cm, yshift=-1.3cm]\textcolor{red}{$\{d, r, s\}$}}] (u2) at (4,0) {$u_2$};
\node[shape=circle,draw=black] (w) at (4,1) {$w$};
\node[shape=circle,draw=white] [label={[xshift=0.00cm, yshift=-0.50cm]\textcolor{red}{$\{b, c, d, r, s\}$}}] (wL) at (4,2.9) {};
\draw[-] (p1) to (u1) to (u2) to (p4) to (p3) to (p2) to (p1);
\draw[-] (p2) to (u1);
\draw[-] (u1) to (w);
\draw[-] (u2) to (w);
\draw[-] (p4) to (w);
\draw[-] (p3) to (w);
\draw[-] (p2) to (w);
\draw[dashed, -] (wL) to [out=-95, in=115] (w);
\end{tikzpicture}\captionof{figure}{Theorem \ref{3ChordVersionMainThm1} is false if $|L(p_1)|=3$ and $|L(p_4)|=1$}\label{DropConditionCounterExFig2}\end{center}
In Figure \ref{DropConditionCounterExFig2}, for any $L$-coloring $\phi$ of $\{p_1, p_4\}$, letting $\phi^*$ be the unique extension of of $\phi$ to an $L$-coloring of $p_1p_2p_3p_4$, each vertex of the triangle $u_1wu_2$ has an $L_{\phi^*}$-list which is a subset of $\{r, s\}$, so $\phi^*$ does not extend to an $L$-coloring of the entire graph. In Figure \ref{DropConditionCounterExFig3}, letting $0\leq i,j\leq 1$ be indices and letting $\phi$ be an $L$-coloring of $\{p_1, p_4\}$ using $c_i, d_j$ on the respective vertices $p_1, p_4$, there is an extension of $\phi$ to an $L$-coloring $\phi^*$ of $V(P)$, where $\phi^*$ uses $c_{1-i}, c_i$ on the respective vertices $p_2, p_3$, and $\phi^*$ does not extend to an $L$-coloring of the entire graph, because we are forced to use the color $s$ on $u_1$ and the endpoints of $wu_2$ are each left with one available color, which is the same color.
\begin{center}\begin{tikzpicture}
\node[shape=circle,draw=black] [label={[xshift=-0.0cm, yshift=-1.3cm]\textcolor{red}{$\{c_0, c_1\}$}}] (p1) at (0,0) {$p_1$};
\node[shape=circle,draw=black] [label={[xshift=-0.95cm, yshift=-0.5cm]\textcolor{red}{$\{c_0, c_1\}$}}] (p2) at (0,2) {$p_2$};
\node[shape=circle,draw=black] [label={[xshift=-0.0cm, yshift=-1.3cm]\textcolor{red}{$\{c_0, c_1, s\}$}}] (u1) at (2,0) {$u_1$};
\node[shape=circle,draw=black] [label={[xshift=0.0cm, yshift=-1.3cm]\textcolor{red}{$\{d_0, d_1\}$}}] (p4) at (6,0) {$p_4$};
\node[shape=circle,draw=black] [label={[xshift=0.95cm, yshift=-0.5cm]\textcolor{red}{$\{c_0, c_1\}$}}] (p3) at (6,2) {$p_3$};
\node[shape=circle,draw=black] [label={[xshift=0.00cm, yshift=-1.3cm]\textcolor{red}{$\{d_0, d_1, s\}$}}] (u2) at (4,0) {$u_2$};
\node[shape=circle,draw=black] (w) at (4,1) {$w$};
\node[shape=circle,draw=white] [label={[xshift=0.00cm, yshift=-0.50cm]\textcolor{red}{$\{c_0, c_1, d_0, d_1, s\}$}}] (wL) at (4,2.9) {};
\draw[-] (p1) to (u1) to (u2) to (p4) to (p3) to (p2) to (p1);
\draw[-] (p2) to (u1);
\draw[-] (u1) to (w);
\draw[-] (u2) to (w);
\draw[-] (p4) to (w);
\draw[-] (p3) to (w);
\draw[-] (p2) to (w);
\draw[dashed, -] (wL) to [out=-95, in=115] (w);
\end{tikzpicture}\captionof{figure}{Theorem \ref{3ChordVersionMainThm1} is false if we let $|L(p_1)|=|L(p_4)|=2$}\label{DropConditionCounterExFig3}\end{center}
We now state and prove Theorem \ref{3ChordVersionMainThm1}.
\begin{theorem}\label{3ChordVersionMainThm1} Let $(G, C, P, L)$ be an end-linked rainbow, where $P:=p_1p_2p_3p_4$ is a 3-path, and the following additional conditions are satisfied.
\begin{enumerate}[label=\arabic*)]
\itemsep-0.1em
\item $|L(p_1)|\geq 1$ and $|L(p_4)|\geq 3$; AND
\item $N(p_3)\cap V(C)=\{p_2, p_4\}$.
\end{enumerate}
Then $\textnormal{End}_{L}(P, G)\neq\varnothing$. \end{theorem}
\begin{proof} Suppose not and let $G$ be a vertex-minimal counterexample to Theorem \ref{3ChordVersionMainThm1}. Let $P, C, L$ be as in the statement of Theorem \ref{3ChordVersionMainThm1}, where $P:=p_1p_2p_3p_4$. Note that the conditions on $G$ do not specify anything about the number of colors available for $p_2, p_3$, since it does not matter how many colors are available for these vertices, and the result is vacuously true if $p_2, p_3$ is not $L$-colorable. By adding edges to $G$ if necessary, we suppose further that every face of $G$, except possibly $C$, is bounded by a triangle. This is permissible since it is possible to add edges to $G$ so as to triangulate the interior of $C$ without creating any chords of $C$ which are incident to $p_3$. By removing colors from the lists of some vertices if necessary, we also suppose for convenience that each vertex of $G\setminus C$ has a list of size precisely five, each vertex of $C\setminus P$ has a list of size precisely three, and furthermore, $|L(p_1)|=1$ and $|L(p_4)|=3$
\begin{Claim}\label{NoChordsinPClaim} $G$ is short-separation-free and $|V(C)|>4$, and there is no chord of $C$ with both endpoints in $P$. \end{Claim}
\begin{claimproof} It is immediate from Corollary \ref{CycleLen4CorToThom}, together with the minimality of $G$, that $G$ is short-separation-free. It follows from 1) of our conditions that $\{p_1, p_4\}$ admits an $L$-coloring $\psi$. If $|V(C)|\leq 4$, then $V(C)=V(P)$. If that holds, then it follows from Corollary \ref{CycleLen4CorToThom} that any extension of $\psi$ to an $L$-coloring of $V(P)$ extends to an $L$-coloring of $G$, contradicting our assumption that $G$ is a counterexample. Thus, $|V(C)|>4$. Now suppose toward a contradiction that $p_1p_4\in E(G)$. Since $|V(C)|>4$, $p_1p_4$ is a chord of $C$. Let $G=G'\cup G''$ be the natural $p_1p_4$-partition of $G$, where $p_2, p_3\in V(G'')$. Since $G$ is short-separation-free and $p_1p_4$ is a chord of $C$, we have $V(G)=V(G')\cup\{p_2, p_3\}$. Let $\psi'$ be an extension of $\psi$ to an $L$-coloring of $V(P)$. It follows from Corollary \ref{CycleLen4CorToThom} that $\psi'$ extends to an $L$-coloring $\psi''$ of $G''$, and, by Theorem \ref{thomassen5ChooseThm}, $G'$ is $L^{p_1p_4}_{\psi''}$-colorable, so $\psi''$ extends to an $L$-coloring of $G$, contradicting our assumption that $G$ is a counterexample. By assumption, $p_1p_3\not\in E(G)$, so, to finish, it suffices to prove that $p_2p_4\not\in E(G)$. Suppose toward a contradiction that $p_2p_4\in E(G)$. Let $G=G'\cup G''$ be the natural $p_2p_4$-partition of $G$, where $p_1\in V(G')$ and $p_3\in V(G'')$. Since $G$ is short-separation-free, $G''$ is the triangle $p_2p_3p_4$. By Theorem \ref{SumTo4For2PathColorEnds}, together with Condition 1), there is an $L$-coloring $\sigma$ of $\{p_1, p_4\}$ which is $(p_2, G')$-sufficient, so any extension of $\sigma$ to an $L$-coloring of $V(P)$ extends to an $L$-coloring of $G$, contradicting our assumption that $G$ is a counterexample. We conclude that $G$ contains no chord of $C$ with both endpoints in $P$, as desired. \end{claimproof}
It follows from Claim \ref{NoChordsinPClaim} that any chord of $C$ has at most one endpoint in $V(P)$. By assumption, i.e by Condition 4), no chord of $C$ is incident $p_3$, so applying Theorem \ref{thomassen5ChooseThm} and the minimality of $G$, we immediately have the following.
\begin{Claim}\label{EveryChordC3PathIncP2ClM0} Every chord of $C$ is incident to $p_2$. \end{Claim}
We now have the following.
\begin{Claim}\label{AtLeastOneP2ChordCLMClaim} $N(p_2)\cap V(C\setminus P)\neq\varnothing$ \end{Claim}
\begin{claimproof} Suppose toward a contradiction that $N(p_2)\cap V(C\setminus P)\neq\varnothing$. By Claim \ref{EveryChordC3PathIncP2ClM0}, every chord of $C$ is incident to $p_2$, so $C$ is an induced cycle in $G$. By 2) of Theorem \ref{ThmFirstLink3PathForUseInHolepunch}, $\textnormal{End}_{L}(P, G)\neq\varnothing$, contradicting our assumption that $G$ is a counterexample. \end{claimproof}
Since $|L(p_1)|=1$, we let $c$ be the lone color of $L(p_1)$. Since $G$ is a counterexample and $p_1p_4\not\in E(G)$, it follows that, for each $d\in L(p_4)$ (where possibly $d=c$), there is an $L$-coloring $\sigma^d$ of $V(P)$, where $\sigma^d$ uses $c,d$ on the respective vertices $p_1, p_4$ and $\sigma^d$ does not extend to an $L$-coloring of $G$. Applying Claim \ref{AtLeastOneP2ChordCLMClaim}, we now introduce the following additional notation, which we use throughout the remainder of the proof of Theorem \ref{3ChordVersionMainThm1}.
\begin{defn}
\textcolor{white}{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}
\begin{enumerate}[label=\emph{\arabic*)}]
\itemsep-0.1em
\item\emph{Since $C\setminus P\neq\varnothing$, we denote the path $C\setminus\{p_2, p_3\}$ by $p_1u_1\ldots u_tp_4$ for some $t\geq 1$.}
\item\emph{We fix an index $m\in\{1, \ldots, t\}$, where $u_m$ is the vertex of $N(p_2)\cap V(C\setminus\mathring{P})$ of maximal index among $u_1, \ldots, u_t$}
\item\emph{We let $G=G'\cup G''$ be the natural $p_2u_m$-partition of $G$, where $p_1\in V(G')$ and $p_3, p_4\in V(G'')$}
\item\emph{We let $P'$ be the 2-path $p_1p_2u_m$ and $P''$ be the 3-path $u_mp_2p_3p_4$}.
\end{enumerate}
\end{defn}
\begin{Claim}\label{EndG''MinimalityCL}
$C^{G''}$ is an induced subgraph of $G$, and $\textnormal{End}(P'', G'')\neq\varnothing$. In particular, for each $x\in\{u_m, p_4\}$ and each $s\in L(x)$, there is a $\phi\in\textnormal{End}(P'', G'')$ with $\phi(x)=s$.
\end{Claim}
\begin{claimproof} Since $p_2p_4\not\in E(G)$ and every chord of $C$ is incident to $P_2$, it is immediate from our choice of $u_m$ that $C^{G''}$ is an induced subgraph of $G$. Now, each of $u_m, p_4$ has an $L$-list of size three. Since $|V(G'')|<|V(G)|$ and $C^{G''}$ is an induced subgraph of $G$, it follows from the minimality of $G$ that, for each $x\in\{u_m, p_4\}$ and each $s\in L(x)$, there is a $\phi\in\textnormal{End}(P'', G'')$ with $\phi(x)=s$. \end{claimproof}
A tricky case we have to deal with is the case where such that $G''$ is a wheel with a central vertex adjacent to all the vertices on the cycle $C^{G''}$. We deal with this with Claim \ref{p2p4HaveNoCommonNbrG} below.
\begin{Claim}\label{p2p4HaveNoCommonNbrG} $N(p_2)\cap N(p_4)=\{p_3\}$. \end{Claim}
\begin{claimproof} Suppose toward a contradiction that $N(p_2)\cap N(p_4)\neq\{p_3\}$. By Claim \ref{EndG''MinimalityCL}, $C^{G''}$ is an induced subgraph of $G$, so it follows that there is a $w\in V(G''\setminus C^{G''})$ adjacent to each of $p_2, p_4$. Since $p_2p_4\not\in E(G)$ and $G$ is short-separation-free, it follows from our triangulation conditions that $w\in N(p_3)$. In particular, $N(p_3)=\{p_2, w, p_4\}$, and $G-p_3$ is bounded by outer cycle $C^{G-p_3}=p_1p_2wp_4u_t\ldots u_1$. Furthermore, $C^{G-p_3}$ contains the 3-path $p_1p_2wp_4$. Let $v$ be the unique neighbor of $w$ on the path $C\setminus\mathring{P}$ which is farthest from $p_4$ on this path. Possibly $v=p_4$ and no chord of $C^{G-p_3}$ is incident to $w$. Let $K$ be the subgraph of $G$ bounded by outer face $wp_4u_t\ldots v$. If $v=p_4$, then $K$ is just an edge.
\vspace*{-8mm}
\begin{addmargin}[2em]{0em}
\begin{subclaim}\label{wCentralVertexKSideSubCL} $v=u_m$. \end{subclaim}
\begin{claimproof} Suppose not. Let $H$ be the subgraph of $G$ bounded by outer cycle $C^H=p_1p_2wv\ldots u_1p_1$. Note that $H\cup K=G-p_3$, and furthermore, since $wp_1\not\in E(G)$, $H$ has no chords of $C^H$ which are incident to $w$. We have $|V(H)|<|V(G)|$, since $p_3\not\in V(H)$. By the minimality of $H$, there is a $\psi\in\textnormal{End}(p_1p_2wv, H)$. Consider the following cases.
\textbf{Case 1:} $K$ is an edge
In this case, we have $H=G-p_3$ and $v=p_4$, and $\psi(v)=d$ for some $d\in L(p_4)$. Note that $|L_{\sigma^{d}}|\geq 2$, since $wp_1\not\in E(G)$. Thus, $\sigma^{d}$ extends to an $L$-coloring of $G$, which is false.
\textbf{Case 2:} $K$ is not an edge
In this case, $v\neq p_4$. Since $v\neq u_m$ by assumption, we have $v=u_n$ for some $n\in\{m+1, \ldots, t\}$, and $K$ is bounded by outer cycle $wp_1u_t\ldots u_n$. Since $|L(p_4)|=3$, it follows from Theorem \ref{SumTo4For2PathColorEnds} that there is an $L$-coloring $\phi$ of $\{u_n, p_4\}$ which is $(w, K)$-sufficient, where $\phi(u_n)=\psi(u_n)$. In particular, $\psi\cup\phi$ is a proper $L$-coloring of $\{p_1, u_n, p_4\}$. Let $d=\phi(p_4)$. Possibly we have $\sigma^d(p_2)=\psi(u_n)$, but since $n\neq m$ by assumption, we have $u_n\not\in N(p_2)$, so the union $\sigma^d\cup\psi$ is a proper $L$-coloring of its domain. Since $N(w)\cap\textnormal{dom}(\sigma^d\cup\psi)|=4$, it follows that $\sigma^d\cup\psi$ extends to a proper $L$-coloring $\sigma'$ of $V(P)\cup\{w, u_n\}$. By our choice of colors for $u_n, p_4$, it follows that $\sigma'$ extends to an $L$-coloring of $G$, so $\sigma^d$ extends to an $L$-coloring of $G$, which is false. \end{claimproof}\end{addmargin}
Since $G$ is short-separation-free and $v=u_m$, we have $K:=G''\setminus\{p_2, p_3, p_4\}$, and $K$ is bounded by outer cycle $(u_m\ldots u_t)p_4w$. Let $T:=u_mwp_4$.
\vspace*{-8mm}
\begin{addmargin}[2em]{0em}
\begin{subclaim}\label{AtLeast6VerticesKSubCLM} $|V(G'')|\geq 6$ and furthermore, $L(p_4)=L(u_t)$, and $K$ is a broken wheel with principal path $T$. \end{subclaim}
\begin{claimproof} We first note that $|V(G'')|\geq 6$, or else $G'$ is a wheel with a central vertex $w$ adjacent to the vertices of a 4-cycle, and this 4-cycle separates $w, p_1$, contradicting short-separation-freeness. In particular, $K$ is not a triangle, and $m<t$. Thus, $p_4u_m\not\in E(G)$, since every chord of $G$ is incident to one of $p_2, p_3$. Now suppose that Subclaim \ref{AtLeast6VerticesKSubCLM} does not hold. Since $|L(p_4)|=|L(u_t)|=3$, we have precisely one of the following:
\begin{enumerate}[label=\emph{\alph*)}]
\itemsep-0.1em
\item$K$ is not a broken wheel with principal path $T$; \emph{OR}
\item $K$ is a broken wheel with principal path $T$ and $L(p_4)\setminus L(u_t)\neq\varnothing$
\end{enumerate}
If $K$ is a broken wheel with principal path $T$ and $d$ is a color of $L(p_4)\setminus L(u_t)$, the any $L$-coloring of $V(T)$ using $d$ on $p_1$ extends to an $L$-coloring of $K$. On the other hand, if $K$ is not a broken wheel with principal path $T$, then, since $|L(p_4)|>1$, it follows from 1) of Theorem \ref{EitherBWheelOrAtMostOneColThm} that there is a $d\in L(p_4)$ such that any $L$-coloring of $V(T)$ using $d$ on $p_4$ extends to an $L$-coloring of $K$. Thus, in any case, there exists a $d\in L(p_4)$ such that any $L$-coloring of $V(T)$ using $d$ on $p_4$ extends to an $L$-coloring of $K$. Let $r$ be a color of $\Lambda_{G'}^{P'}(c, \sigma^{d}(p_2), \bullet)$. Possibly $r=d$, but, in any case, since $u_mp_4\not\in E(G)$ and $|L(w)\setminus\{r, d, \sigma^{d}(p_2), \sigma^{d}(p_3)\}|\geq 1$, it follows that $\sigma^{d}$ extends to an $L$-coloring of $G$, which is false. \end{claimproof}\end{addmargin}
We now rule out the possibility that $G''$ is a 6-wheel.
\vspace*{-8mm}
\begin{addmargin}[2em]{0em}
\begin{subclaim}\label{G''WheelGreater6SubCL} $|V(G'')|>6$. \end{subclaim}
\begin{claimproof} Suppose not. Thus, by Subclaim \ref{AtLeast6VerticesKSubCLM}, $|V(G'')|=6$ and $m=t-1$, and $G''$ is a wheel whose central vertex is adjacent to all five vertices of the 5-cycle $u_{t-1}p_2p_3p_4u_t$. Consider the following caes.
\textbf{Case 1:} $L(p_4)=L(u_{t-1})$
In this case, it follows from Theorem \ref{SumTo4For2PathColorEnds} that there is a $\psi\in\textnormal{End}(P', G')$ with $\psi(u_{t-1})\in L(p_4)$. Let $d=\psi(u_{t-1})$. Since $d\in L(p_4)$, we consider $\sigma^{d}$, and we consider the following subcases.
\textbf{Subcase 1.1} $\sigma^{d}(p_2)=d$
In this case, we simply choose a color $r\in\Lambda_{G'}^{P'}(c, d, \bullet)$. The $L$-coloring $(r, d, \sigma^{d}(p_3), d)$ of $P''$ extends to $L$-color the wheel $G''$, since two neighbors of $w$ are using the same color. Thus, $\sigma^{d}$ extends to an $L$-coloring of $G$, which is false.
\textbf{Subcase 1.2} $\sigma^{d}(p_2)\neq d$
In this case, by our choice of $\psi$, we have $d\in\Lambda_{G'}^{P'}(c, \sigma^{d}(p_2), \bullet)$. The $L$-coloring $(d, \sigma^{d}(p_2), \sigma^{d}(p_3), d)$ of $P''$ extends to $L$-color the wheel $G''$, since two neighbors of $w$ are using the same color. Thus, $\sigma^{d}$ extend to an $L$-coloring of $G$, which is false.
\textbf{Case 2:} $L(p_4)\neq L(u_{t-1})$
Since $L(p_4)\not\subseteq L(u_{t-1})$ and $|L(p_4)|=3$, we let $L(p_4)=\{d, d_0, d_1\}$, where $d\not\in L(u_{t-1})$. By Subclaim \ref{AtLeast6VerticesKSubCLM}, we have $L(p_4)=L(u_t)$, so $d\in L(u_t)$. For each $j=0,1$, since $\sigma^{d_j}$ does not extend to an $L$-coloring of $G$, it follows that, for each $r\in\Lambda_{G'}^{P'}(c, \sigma^{d_j}(p_2), \bullet)$, we have $L(w)\setminus\{r, \sigma^{d_j}(p_2), \sigma^{d_j}(p_3), d_j\}=\{d\}$ and $r\in L(u_{t-1})\cap (L(u_t)\setminus\{d_j\})$. Since $L(u_t)=L(p_4)$ and $d\not\in L(u_{t-1})$, it follows that, for each $j\in\{0,1\}$, we have $\Lambda_{G'}^{P'}(c, \sigma^{d_j}(p_2), \bullet)=\{d_{1-j}\}$ and $L(w)\setminus\{\sigma^{d_j}(p_2), \sigma^{d_j}(p_3), d_j, d_{1-j}\}=\{d\}$. By 2) of Corollary \ref{CorMainEitherBWheelAtM1ColCor}, $G'$ is a broken wheel with principal path $P'$. But now, applying 1b) of Theorem \ref{BWheelMainRevListThm2} to $G', P'$, it follows that, for each $j=0,1$, we have $\sigma^{d_j}(p_2)=d_j$, so $\sigma^{d_j}$ uses the same color on two vertices of $N(w)\cap V(P)$, contradicting the fact that $|L(w)\setminus\{r, \sigma^{d_j}(p_2), \sigma^{d_j}(p_3), d_j\}|=1$. \end{claimproof}\end{addmargin}
\vspace*{-8mm}
\begin{addmargin}[2em]{0em}
\begin{subclaim}\label{Lp1AndLut-1NotSameListSubCL} $L(p_1)\neq L(u_{t-1})$ \end{subclaim}
\begin{claimproof} Suppose toward a contradiction that $L(p_1)=L(u_{t-1})$. Let $H:=(G\setminus\{p_4, u_t\})+u_{t-1}p_3$, where $H$ is bounded by outer cycle $C^H=p_1p_2p_3u_{t-1}\ldots u_1$, and $H$ contains the 3-path $Q:=p_1p_2p_3u_{t-1}$. Note that $N_H(p_3)=\{p_2, u_{t-1}\}$, so $H$ contains no chord of $C^H$ which is incident to $p_3$. Since $|V(H)|<|V(G)|$, there exists a $\psi\in\textnormal{End}(Q, H)$. Let $\psi(u_{t-1})=d$. By assumption, we have $d\in L(p_4)$. Since $L(p_1)=\{c\}$, we have $\psi(p_1)=\sigma^d(p_1)=c$. Now, since $\sigma^d(p_4)=d$ by definition, we have $\sigma^d(p_3)\neq d$. Furthermore, since $G''$ is a wheel with central vertex $w$, it follows from Subclaim \ref{G''WheelGreater6SubCL} that $u_{t-1}p_2\not\in E(H)$. Possibly, $G'$ is a triangle, but, in any case, since $\psi$ is a proper $L$-coloring of its domain in $H$, and since $\sigma^d(p_3)\neq d$ and $u_{t-1}p_2\not\in E(H)$, it follows that, in $H$, there is a proper $L$-coloring $\psi'$ of $V(Q)$ using the colors $c, \sigma^d(p_2), \sigma^d(p_3), d$ on the respective vertices $p_1, p_2, p_3, u_{t-1}$.
Since $\psi\in\textnormal{End}(Q, H)$, it follows that $\psi'$ extends to an $L$-coloring $\psi''$ of $V(H)$. Now, $\psi''$ is also a proper $L$-coloring of its domain in $G$, since we have only added edges to $G-\{u_t, p_4\}$ to form $H$. In particular, since $\psi''(p_3)\neq d$, $\psi''\cup\sigma^d$ is a proper $L$-coloring of its domain in $G$, which is $G-u_t$. Since $\psi''\cup\sigma_d$ uses the same color on $p_4, u_{t-1}$, it follows that $\psi''\cup\sigma_d$ extends to an $L$-coloring of $G$, which is false, since $\sigma^d$ does not extend to an $L$-coloring of $G$. \end{claimproof}\end{addmargin}
Since $|L(p_4)|=|L(u_{t-1})|=3$ and $L(p_4)\neq L(u_{t-1})$ by Subclaim \ref{Lp1AndLut-1NotSameListSubCL}, we let $L(p_4)=\{d, d_0, d_1\}$, where $d\not\in L(u_{t-1})$. Let $r\in\Lambda_{G'}^{P'}(c, \sigma^{d_0}(p_2), \bullet)$. Since $\sigma^{d_0}$ does not extend to an $L$-coloring of $G$, and since $G''$ is a wheel with $|V(G'')|>6$, we have $L(w)\setminus\{r, \sigma^{d_0}(p_2), \sigma^{d_0}(p_3), d_0\}\subseteq L(u_{t-1})\cap L(u_t)$. By Subclaim \ref{AtLeast6VerticesKSubCLM}, we have $L(u_t)=\{d, d_0, d_1\}$. Since $d\not\in L(u_{t-1})$, it follows that $L(w)\setminus\{r, \sigma^{d_0}(p_2), \sigma^{d_0}(p_3), d_0\}=\{d_1\}$. Coloring $w$ with $d_1$, the color $d$ is left over for $u_t$, and since $d\not\in L(u_{t-1})$, it follows that $\sigma^{d_0}$ extends to an $L$-coloring of $G$, which is false. This completes the proof of Claim \ref{p2p4HaveNoCommonNbrG}. \end{claimproof}
\begin{Claim}\label{AllColp2p3p4ExtendCL} Any $L$-coloring of $p_2p_3p_4$ extends to an $L$-coloring of $G''$. \end{Claim}
\begin{claimproof} $G''$ contains no chords of its outer face, and since $p_2, p_3, p_4$ have no common neighbor in $G''$ by Claim \ref{p2p4HaveNoCommonNbrG}, it follows from Lemma \ref{PartialPathColoringExtCL0} that any $L$-coloring of $p_2p_3p_4$ extends to an $L$-coloring of $G''$. \end{claimproof}
For each $d\in L(p_4)$, we now set $X^{d}$ to be the set of $f\in L(u_m)$ such that the restriction of $\sigma^{d}$ to $p_2p_3p_4$ extends to an $L$-coloring of $G''$ using $f$ on $u_m$.
\begin{Claim}\label{OnlyOneExtFromXcdCL}
Both of the following hold.
\begin{enumerate}[label=\arabic*)]
\itemsep-0.1em
\item For any $d\in L(p_4)$, $X^{d}$ and $\Lambda_{G'}^{P'}(c, \sigma^{d}(p_2), \bullet)$ are disjoint singletons; AND
\item $G'$ is a broken wheel with principal path $P'$, and $L(p_1)\subseteq L(u_1)$. Furthermore, for any $d\in L(p_4)$, we have $\sigma^{d}(p_2)\in\bigcap_{i=1}^mL(u_i)$, and $L(p_1)\subseteq L(u_1)$.
\end{enumerate}
\end{Claim}
\begin{claimproof} Let $d\in L(p_4)$. Since $\sigma^{d}$ does not extend to an $L$-coloring of $G$, it is immediate that $X^{d}$ and $\Lambda_{G'}^{P'}(c, \sigma^{d}(p_2), \bullet)$ are disjoint. Now suppose toward a contradiction that $|\Lambda_{G'}^{P'}(c, \sigma^{d}(p_2), \bullet)|\neq 1$. By Theorem \ref{thomassen5ChooseThm}, $\Lambda_{G'}^{P'}(c, \sigma^{d}(p_2), \bullet)$ is nonempty, so it has size at least two. Since $|L(u_m)|=3$ and $\sigma^{d}(p_2)\not\in\Lambda_{G'}^{P'}(c, \sigma^{d}(p_2), \bullet)$, we have $L(u_m)\setminus\{\sigma^{d}(p_2)\}=\Lambda_{G'}^{P'}(c, \sigma^{d}(p_2), \bullet)$. Since $\sigma^{d}(p_2)\not\in X^{d}$, it follows from Claim \ref{AllColp2p3p4ExtendCL} that $X^{d}\cap\Lambda_{G'}^{P'}(c, \sigma^{d}(p_2), \bullet)\neq\varnothing$, which is false. Thus, we have $|\Lambda_{G'}^{P'}(c, \sigma^{d}(p_2), \bullet)|=1$, and so $\sigma^{d}(p_2)\in L(u_m)$. It follows from Claim \ref{AllColp2p3p4ExtendCL} that $X^{d}\neq\varnothing$. Note that $\sigma^{d}(p_2)\not\in X^{d}$. Since $L(u_m)\setminus\{\sigma^{d}(p_2)\}|=2$ and $X^{d}$ is nonempty, we conclude that $X^{d}$ and $\Lambda_{G'}^{P'}(c, \sigma^{d}(p_2), \bullet)$ are disjoint singletons. This proves 1). Since $G$ is short-separation-free, either $G'$ is a triangle or the outer face of $G'$ is a cycle of length greater than three, and it follows from 2) of Corollary \ref{CorMainEitherBWheelAtM1ColCor} that $G'$ is a broken wheel with principal path $P'$. Since $G'$ is a broken wheel with principal path $P'$, it follows from 1) that, for each $d\in L(p_4)$, we have $\sigma^{d}(p_2)\in\bigcap_{i=1}^mL(u_i)$ and $c\in L(u_1)$, so $L(p_1)\subseteq L(u_1)$. \end{claimproof}
Applying Claim \ref{OnlyOneExtFromXcdCL}, we now have the following.
\begin{Claim}\label{ObstructionVertWStarCL} There exists a $w^{\star}\in V(G''\setminus C'')$ such that the following hold.
\begin{enumerate}[label=\arabic*)]
\itemsep-0.1em
\item $N(w^{\star})\cap V(P'')=\{u_m, p_2, p_3\}$, and, in particular $w^{\star}$ is the unique vertex of $G''\setminus C''$ with more than two neighbors in $P''$; AND
\item For any $d\in L(p_4)$ and $r\in\Lambda_{G'}^{P'}(c, \sigma^{d}(p_2), \bullet)$, we have $|L(w^{\star})\setminus\{r, \sigma^{d}(p_2), \sigma^{d}(p_3)\}|=2$; AND
\item Every chord of the outer face of $G''-p_2$ is incident to $w^{\star}$.
\end{enumerate}
\end{Claim}
\begin{claimproof} Let $d\in L(p_4)$ and let $r\in\Lambda_{G'}^{P'}(c, \sigma^{d}(p_2), \bullet)$. By 1) of Claim \ref{OnlyOneExtFromXcdCL}, we have $r\not\in X^{d}$, so the $L$-coloring $(r, \sigma^{d}(p_2), \sigma^d(p_3), \sigma^{d}(p_4))$ of $u_mp_2p_3p_4$ does not extend to an $L$-coloring of $G''$. By Claim \ref{p2p4HaveNoCommonNbrG}, $p_2, p_4$ have no common neighbor in $G''\setminus C''$, and since $C''$ is an induced subgraph of $G''$, it follows from Lemma \ref{PartialPathColoringExtCL0} that there is a $w^{\star}\in V(G''\setminus C'')$ with $N(w)\cap V(P'') =\{u_m, p_2, p_3\}$, and $w^{\star}$ is the unique vertex of $G''\setminus C''$ with more than two neighbors in $P''$. It also follows from Lemma \ref{PartialPathColoringExtCL0} that $|L(w^{\star})\setminus\{r, \sigma^d(p_2), \sigma^d(p_3)\}|=2$. Since $C''$ is an induced subgraph of $G$, every chord of the outer face of $G''-p_2$ is incident to $w^{\star}$. \end{claimproof}
Let $w^{\star}$ be as in Claim \ref{ObstructionVertWStarCL}. Since $G$ is short-separation-free, $G''-p_2$ is bounded by outer cycle $u_mw^{\star}p_3p_4u_t\ldots u_m$.
\begin{Claim}\label{eitherWStarChordorSecObsV}
Either $w^{\star}$ has a neighbor in $\{u_{m+1}, \ldots, u_t\}$ or, there exists a $v^{\star}\in V(G''\setminus C'')\cap N(w^{\star})$ such that both of the following hold.
\begin{enumerate}[label=\arabic*)]
\itemsep-0.1em
\item $N(v^{\star})\cap V(C'')=\{p_3, p_4\}$; AND
\item For each $d\in L(p_4)$ and $r\in\Lambda_{G'}^{P'}(c, \sigma^d(p_2), \bullet)$, the sets $L(w^{\star})\setminus\{r, \sigma^d(p_2), \sigma^{d}(p_3)\}$ and $\{\sigma^{d}(p_3), d\}$ are disjoint subsets of $L(v^{\star})$ of size two.
\end{enumerate}
\end{Claim}
\begin{claimproof} Note that $w^{\star}, p_2$ are the only common neighbors of $u_m, p_3$, or else $G''$ contains a copy of $K_{2,3}$, which contradicts the fact that $G$ is short-separation, as $V(G'')\neq V(G)$. Suppose that $w^{\star}$ has no neighbor in $\{u_{m+1}, \ldots, u_t\}$. Since $p_4\not\in N(w^{\star})$, we have $N(w^{\star})\cap V(C)=\{u_m, p_2, p_3\}$. Now, $G''-p_2$ is bounded by outer cycle $C^{G''-p_2}=u_mp_2p_3p_4u_t\ldots u_{m+1}$, and since every chord of $C$ is incident to $p_2$, it follows that $C^{G''-p_2}$ is an induced cycle of $G$. Let $S$ be the set of vertices of $V(G''-p_2)\setminus V(C^{G''-p_2})$ with at least three neighbors in $\{u_m, w^{\star}, p_3, p_4\}$. For each $d\in L(p_2)$ and $r\in\Lambda_{G'}^{P'}(c, \sigma^d(p_2), \bullet)$, we have $r\not\in X^d$ by 1) of Claim \ref{OnlyOneExtFromXcdCL}. Since $G''-p_2$ has no chords of its outer face, it follows from Lemma \ref{PartialPathColoringExtCL0} that $S\neq\varnothing$.
\vspace*{-8mm}
\begin{addmargin}[2em]{0em}
\begin{subclaim} $S\cap N(u_m)=\varnothing$ \end{subclaim}
\begin{claimproof} Suppose there is a $v\in S$ with $v\in N(u_m)$. As $N(u_m)\cap N(p_3)=\{w^{\star}, p_2\}$ and $v$ has at least three neighbors in $\{u_m, w^{\star}, p_3, p_4\}$, we have $u_m, w^{\star}, p_4\in N(v)$ and $p_3\not\in N(v)$. Thus, $G$ contains the 4-cycle $w^{\star}vp_4p_3$. Since $w^{\star}\not\in N(p_4)$, it follows from our triangulation conditions that $p_3\in N(v)$, a contradiction. \end{claimproof}\end{addmargin}
We conclude that, for each $v\in S$, we have $N(v)\cap\{u_m, w^{\star}, p_3, p_4\}=\{w^{\star}, p_3, p_4\}$. Since $S\neq\varnothing$ and $G$ is $K_{2,3}$-free, there exists a vertex $v^{\star}$ such that $S=\{v^{\star}\}$. For each $d\in L(p_2)$ and $r\in\Lambda_{G'}^{P'}(c, \sigma^d(p_2), \bullet)$, since $r\not\in X^d$, it follows from Lemma \ref{PartialPathColoringExtCL0} that $v^{\star}$ satisfies 2) as well. \end{claimproof}
As $|L(p_4)|=3$ and $|\Lambda(c, \sigma^d(p_2), \bullet)|=1$ for each $d\in L(p_4)$, it follows from 2) of Theorem \ref{BWheelMainRevListThm2} that there exist distinct $d, d'\in L(p_4)$ such that $\sigma^{d}(p_2)=\sigma^{d'}(p_2)=f$ for some color $f\in L(p_2)$. Let $d_*$ be the lone color of $L(p_4)\setminus\{d, d'\}$. Applying 1) of Claim \ref{OnlyOneExtFromXcdCL}, let $g, g_*\in L(u_m)$, where $\Lambda_{G'}^{P'}(c, f, \bullet)=\{g\}$ and $\Lambda_{G'}^{P'}(c, \sigma^{d_*}(p_2), \bullet)=\{g_*\}$.
\begin{Claim}\label{gAndg*AreDistinctCL} Both of the following hold.
\begin{enumerate}[label=\roman*)]
\itemsep-0.1em
\item $f\neq\sigma^{d_*}(p_2)$ and $g\neq g_*$; AND
\item $\sigma^{d}(p_3)\neq\sigma^{d'}(p_3)$.
\end{enumerate}
\end{Claim}
\begin{claimproof} By Claim \ref{EndG''MinimalityCL}, there is a $\psi\in\textnormal{End}(P'' G'')$ with $\psi(u_m)=g$. If $\psi(p_4)\in\{d, d'\}$, then one of $\sigma^{d}, \sigma^{d'}$ extends to an $L$-coloring of $G$, which is false. Thus, $\psi(p_4)=d_*$. If either $g=g_*$ or $f=\sigma^{d_*}(p_2)$, then, since $\Lambda_{G'}^{P'}(c, f, \bullet)=\{g\}$ and $\psi(p_4)=d_*$, it follows that $\sigma^{d_*}$ extends to an $L$-coloring of $G$, which is false. This proves i). Now we prove ii). Suppose toward a contradiction that $\sigma^{d}(p_3)\neq\sigma^{d'}(p_3)=h$ for some color $h$ and $L^{\dagger}$ be a list-assignment for $G''-p_2$ where $L^{\dagger}(p_4)=\{h, d, d'\}$ and otherwise $L^{\dagger}=L$. Recall that $G''-p_2$ is bounded by outer cycle $u_mw^{\star}p_3p_4u_t\ldots u_n$, and since $p_4\not\in N(w^{\star})$, we get that $G''-p_2$ is not a broken wheel with principal path $u_mw^{\star}p_3$. Since any chord of the outer face of $G''-p_2$ is incident to $w^{\star}$, it follows from 1) of Theorem \ref{EitherBWheelOrAtMostOneColThm} that there is at most one $L^{\dagger}$-coloring of $u_mw^{\star}p_3$ which does not extend to an $L^{\dagger}$-coloring of $G''$. Since $|L^{\dagger}(w^{\star})\setminus\{f, g, h\}|\geq 2$, there is an $L^{\dagger}$-coloring $\psi$ of $u_mw^{\star}p_3$ which extends to an $L^{\dagger}$-coloring $\tau$ of $G''-p_2$, where $\psi(u_m)=g$ and $\psi(p_3)=h$ and $\psi(w^{\star})\in L^{\dagger}(w^{\star})\setminus\{f, g, h\}$. Since $\psi(p_3)=h$, we have $\tau(p_4)\in\{d, d'\}$. In particular, $\tau$ is an $L$-coloring of $G''-p_2$, and $f$ is left over for $p_2$, so it follows that one of $\sigma^{d}, \sigma^{d'}$ extends to an $L$-coloring of $G$, which is false.\end{claimproof}
\begin{Claim}\label{G''minusp2InducedOuterFace} $G''-p_2$ has no chords of its outer face. \end{Claim}
\begin{claimproof} Suppose that there is such a chord in $G''-p_2$. Any such chord is incident to $w^{\star}$, and, since $w^{\star}\not\in N(p_4)$, let $u_n$ be the neighbor of $w^{\star}$ of minimal index among $\{m+1, \ldots, t\}$. Let $K$ be the subgraph of $G$ bounded by outer cycle $u_mw^{\star}u_n\ldots u_{m+1}$, and let $T:=u_mw^{\star}u_n$. Likewise, let $K^{\dagger}$ be the subgraph of $G$ bounded by outer cycle $u_nw^{\star}p_3p_4u_t\ldots u_n$ and let $T^{\dagger}:=u_nw^{\star}p_3p_4$. Note that every chord of the outer face of $K^{\dagger}$ (if there are any) is incident to $w^{\star}$. In particular, $K^{\dagger}$ has no chord of its outer cycle which is incident to $p_3$. By the minimality of $G$, we immediately have the following.
\vspace*{-8mm}
\begin{addmargin}[2em]{0em}
\begin{subclaim}\label{AugKDaggerTDaggerEachColSubCL} For each $s\in L(u_n)$, there is a $\psi\in\textnormal{End}(T^{\dagger}, K^{\dagger})$ with $\psi(u_n)=s$. \end{subclaim}\end{addmargin}
We now have the following.
\vspace*{-8mm}
\begin{addmargin}[2em]{0em}
\begin{subclaim} $K$ is a triangle. \end{subclaim}
\begin{claimproof} Suppose not. Thus, $n>m+1$, and it follows from the minimality of $n$ that $K$ is not a broken wheel with principal path $T$. By 1) of Theorem \ref{EitherBWheelOrAtMostOneColThm}, there is at most one $L$-coloring of $V(T)$ which does not extend to an $L$-coloring of $K$. Since $|L(u_n)|=3$, it follows from Subclaim \ref{AugKDaggerTDaggerEachColSubCL} that there is a $\psi\in\textnormal{End}(T^{\dagger}, K^{\dagger})$ such that any $L$-coloring of $V(T)$ using $\psi(u_n)$ on $u_n$ extends to an $L$-coloring of $K$. Let $\sigma:=\sigma^{\psi(p_4)}$ and let $r$ be a color of $\Lambda(c, \sigma(p_2), \bullet)$. Possibly $r=\psi(u_n)$, but since $n>m+1$ and every chord of $C$ is incident to $p_2$, there is a proper $L$-coloring of $\{u_m, u_n\}$ using $r, \psi(u_n)$ on the respective vertices $u_n, u_m$. Now, $\psi\cup\sigma$ is a proper $L$-coloring of its domain, and since $L(w^{\star})\setminus\{r, \sigma(p_2), \sigma(p_3), \psi(u_n)\}|\geq 1$, it follows from our choice of color for $u_n$ that $\psi\cup\sigma$ extends to an $L$-coloring $\sigma'$ of $V(P\cup G'\cup K)$. Since $\sigma'$ restricts to an element of $\textnormal{End}(T^{\dagger}, K^{\dagger})$ and $T^{\dagger}$ separates $V(G\setminus K^{\dagger})$ from $V(K^{\dagger})$, $\sigma'$ extends to an $L$-coloring of $G$, and thus $\sigma^{\psi(p_4)}$ extends to an $L$-coloring of $G$, which is false. \end{claimproof}\end{addmargin}
Since $K$ is a triangle, we have $n=m+1$. By Subclaim \ref{AugKDaggerTDaggerEachColSubCL}, for each $s\in L(u_{m+1})$, there exists a $\psi_s\in\textnormal{End}_(T^{\dagger}, K^{\dagger})$, where $\psi_s(u_{m+1})=s$. For each $s\in L(u_{m+1})$, we let $d_s=\psi_s(p_4)$.
\vspace*{-8mm}
\begin{addmargin}[2em]{0em}
\begin{subclaim}\label{distinctSMapToDistinctSigmaDsSubCL} For each $s\in L(u_{m+1})$, we have $\Lambda(c, \sigma^{d_s}(p_2), \bullet)=\{s\}$. \end{subclaim}
\begin{claimproof} Suppose there is an $s\in L(u_{m+1})$ for which this does not hold. Since $|\Lambda_{G'}^{P'}(c, \sigma^{d_s}(p_2), \bullet)|=1$, there is an $r\in\Lambda_{G'}^{P'}(c, \sigma^{d_s}(p_2), \bullet)$ with $r\neq s$. Since $L(w^{\star})\setminus\{r, s, \sigma^{d_s}(p_2), \sigma^{d_s}(p_3)\}|\geq 1$, the union $\psi_s\cup\sigma^{d_s}$ is a proper $L$-coloring of its domain which extends to an $L$-coloring $\sigma'$ of $V(P\cup G'\cup K)$. Since $\sigma'$ restricts to an element of $\textnormal{End}(T^{\dagger}, K^{\dagger})$, it follows that $\sigma'$ extends to $L$-color $K^{\dagger}$ as well, so $\sigma'$ extends to an $L$-coloring of $G$, and thus $\sigma^{d_s}$ extends to an $L$-coloring of $G$, which is false. \end{claimproof}\end{addmargin}
Let $Y:=\{\sigma^{d_s}(p_2): s\in L(u_{m+1})\}$. It follows from Subclaim \ref{distinctSMapToDistinctSigmaDsSubCL} that, for any distinct $s, r\in L(u_{m+1})$, we have $\sigma^{d_s}(p_2)\neq\sigma^{d_r}(p_2)$, so $|Y|=3$. But we also have $Y\subseteq\{\sigma^d(p_2): d\in L(p_4)\}$, and $\{\sigma^d(p_2): d\in L(p_4)\}=\{f, \sigma^{d_*}(p_2)\}$, a contradiction. This proves Claim \ref{G''minusp2InducedOuterFace}. \end{claimproof}
Since $G''-p_2$ has no chords of its outer face, it follows from Claim \ref{eitherWStarChordorSecObsV} that there is a $v^{\star}\in V(G''\setminus C'')$ with $v^{\star}\neq w^{\star}$, where $w^{\star}\in N(v^{\star})$ and $N(v^{\star})\cap V(P'')=\{p_3, p_4\}$.
\begin{Claim}\label{ColorBinBothVstarWStarLists}
Both of the following hold.
\begin{enumerate}[label=\arabic*)]
\itemsep-0.1em
\item There exists a color $b$ such that $L(w^{\star})=\{b, f, g, \sigma^{d}(p_3), \sigma^{d'}(p_3), \}$ and $L(v^{\star})=\{b, d, d', \sigma^{d}(p_3), \sigma^{d'}(p_3)\}$. Furthermore, $d, d'\not\in L(w^{\star})\setminus\{f, g\}$; AND
\item $\{g_*, \sigma^{d''}(p_2), \sigma^{d''}(p_3)\}$ is a subset of $L(w^{\star})$ of size three. Furthermore, $L(w^{\star})\setminus\{g_*, \sigma^{d''}(p_2), \sigma^{d''}(p_3)\}\subseteq L(v^{\star})$ and $\{\sigma^{d''}(p_3), d''\}\subseteq L(v^{\star})$.
\end{enumerate}
\end{Claim}
\begin{claimproof} By ii) of Claim \ref{gAndg*AreDistinctCL}, we have $\sigma^{d}(p_3)\neq\sigma^{d'}(p_3)$, so, applying Claim \ref{eitherWStarChordorSecObsV} to $\sigma^{d}$ and $\sigma^{d'}$, we have 1). Likewise, 2) follows from Claim \ref{eitherWStarChordorSecObsV} applied to $\sigma^{d''}$. \end{claimproof}
\begin{Claim}\label{fgFliptosigmag_*Claim} $\{f, g\}=\{\sigma^{d''}(p_2), g_*\}$ and $d''\in L(w^{\star})\setminus\{f, g\}$. \end{Claim}
\begin{claimproof} By i) of Claim \ref{gAndg*AreDistinctCL}, we have $g\neq g_*$. Recall that $\Lambda_{G'}^{P'}(c, f, \bullet)=\{g\}$ and $\Lambda_{G'}^{P'}(c, \sigma^{d''}(p_2), \bullet)=\{g_*\}$. Thus, it follows from 1c) of Theorem \ref{BWheelMainRevListThm2} that $\{f, g\}=\{\sigma(p_2), g_*\}$. Now suppose toward a contradiction that $d''\not\in L(w^{\star})\setminus\{f, g\}$. By 2) of Claim \ref{ColorBinBothVstarWStarLists}, we have $d''\in L(v^{\star})$. Since $d''\not\in L(w^{\star})\setminus\{f, g\}$ by assumption, it follows from Claim 1) of \ref{ColorBinBothVstarWStarLists} that $d''\in L(v^{\star})\setminus\{\sigma^{d}(p_3), \sigma^{d'}(p_3), b\}$. But then, again by 1) of Claim \ref{ColorBinBothVstarWStarLists}, we have $d''\in\{d, d'\}$, which is false. \end{claimproof}
Since $u_mp_3, p_2p_4\not\in E(G)$, it follows from Claim \ref{fgFliptosigmag_*Claim} that $\psi=(g_*, d'', \sigma^{d''}(p_3), d'')$ is a proper $L$-coloring of $u_mw^{\star}p_3p_4$, and $\sigma^{d''}(p_2)\in L_{\psi}(p_2)$. By Claim \ref{G''minusp2InducedOuterFace}, the outer face of $G''-p_2$ is an induced subgraph of $G''-p_2$. Since $|L_{\psi}(v^{\star})|\geq 3$, and $v^{\star}$ is the lone internal vertex of $V(G''-p_2)$ with at least three neighbors on $u_mw^{\star}p_3p_4$, it follows from Lemma \ref{PartialPathColoringExtCL0} that $\psi$ extends to an $L$-coloring of $G''-p_2$. Thus, $\psi$ extends to an $L$-coloring of $G''$ in which $p_2$ is colored with $\sigma^{d''}(p_2)$ and $u_m$ is colored with $g_*$. Since $\Lambda_{G'}^{P'}(c, \sigma^{d''}(p_2), \bullet)=\{g_*\}$, it follows that $\sigma^{d''}$ extends to an $L$-coloring of $G$, which is false. This completes the proof of Theorem \ref{3ChordVersionMainThm1}. \end{proof}
\end{document} | math |
A group of UK Russell Group Universities, led by Queen's University Belfast, has been selected to build major collaborations with the top ten engineering institutions in China.
The consortium is the first of its kind to partner to the E9 Chinese Excellence League -- the top ten engineering institutions in China. It will be led by Queen's, joined by University College London, The University of Manchester, Cardiff University, University of Warwick, and University of Birmingham. The group represents one quarter of the Russell Group of research intensive universities in the UK, with world leading research and education excellence in engineering.
The funding will see the successful universities working closely with E9, which will enable the UK universities to create a critical mass on a cluster of key engineering areas, such as energy and advanced manufacturing, and will help maintain the UK's global standing in light of increasing international competition. It will also allow China to make the transition from a big manufacturer to a manufacturing industry superpower.
Queen's University's Vice-Chancellor Professor Patrick Johnston recently visited China as part of a Russell Group delegation to explore collaboration with elite universities in mainland China. He commented: "The UK-China partnership is a strong example of Queen's University's commitment to making a positive impact in both research and education on a global scale.
"Queen's has long historic links with China, beginning with Sir Robert Hart who graduated from Queen's in 1853 and became Inspector-General of China's Imperial Maritime Custom Service. Our friendship with China is epitomised by this current partnership, led by Professor Mark Price and Professor Kang Li.
"This work will develop our institutional collaborations further, strengthening our links with both the UK and China. The award from the British Council is an indication of the international significance of this work, reaffirming our global reputation for excellence."
Pro-Vice-Chancellor for Engineering and Physical Sciences, Professor Mark Price, who is leading the partnership for Queen's University, commented: "I am delighted, and honoured, to be leading this international collaboration of some of the best engineering institutions in the world, here in the UK and in China.
"We have made many great friends in China and by working together we are now achieving great things in research and education. As a group, these 16 institutions have outstanding teams and we look forward to becoming even stronger by developing a more co-ordinated and synergised approach to our research and teaching. Together, we can tackle the major challenges facing the world and make a difference for everyone. We have exciting times ahead!"
Congratulating Queen's on their success was Jonathan Stewart, Deputy Director, British Council Northern Ireland. He said: "We at British Council Northern Ireland would like to congratulate Queen's University on their success. International research collaboration between Northern Ireland and countries like China are vital if we are to build understanding and nurture partnerships. The global challenges we now face are huge -- and there is an obvious and natural reason why we should strengthen relations between world-class universities like Queen's University and those in China, for the mutual benefit of both countries.
"Through this collaboration we can help to solve today's major challenges -- with the solutions increasingly requiring working across boundaries, disciplines and international borders. Engaging with and strengthening our links with China will continue to be an important element in the development and growth of our economy." | english |
The first is related to skills and expertise and the second is execution time.
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If you have any questions or if you would like to learn more about how we can support you in identifying potential TYPO3 business partners, we will be more than happy to do so! | english |
/* $Id: udp.h $ */
/** @file
* NAT - UDP protocol (declarations/defines).
*/
/*
* Copyright (C) 2006-2010 Oracle Corporation
*
* This file is part of VirtualBox Open Source Edition (OSE), as
* available from http://www.virtualbox.org. This file is free software;
* you can redistribute it and/or modify it under the terms of the GNU
* General Public License (GPL) as published by the Free Software
* Foundation, in version 2 as it comes in the "COPYING" file of the
* VirtualBox OSE distribution. VirtualBox OSE is distributed in the
* hope that it will be useful, but WITHOUT ANY WARRANTY of any kind.
*/
/*
* This code is based on:
*
* Copyright (c) 1982, 1986, 1993
* The Regents of the University of California. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
* 3. All advertising materials mentioning features or use of this software
* must display the following acknowledgement:
* This product includes software developed by the University of
* California, Berkeley and its contributors.
* 4. Neither the name of the University nor the names of its contributors
* may be used to endorse or promote products derived from this software
* without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*
* @(#)udp.h 8.1 (Berkeley) 6/10/93
* udp.h,v 1.3 1994/08/21 05:27:41 paul Exp
*/
#ifndef _UDP_H_
#define _UDP_H_
#define UDP_TTL 0x60
#define UDP_UDPDATALEN 16192
extern struct socket *udp_last_so;
/*
* Udp protocol header.
* Per RFC 768, September, 1981.
*/
struct udphdr
{
uint16_t uh_sport; /* source port */
uint16_t uh_dport; /* destination port */
int16_t uh_ulen; /* udp length */
uint16_t uh_sum; /* udp checksum */
};
AssertCompileSize(struct udphdr, 8);
/*
* UDP kernel structures and variables.
*/
struct udpiphdr
{
struct ipovly ui_i; /* overlaid ip structure */
struct udphdr ui_u; /* udp header */
};
AssertCompileSize(struct udpiphdr, 28);
#define ui_next ui_i.ih_next
#define ui_prev ui_i.ih_prev
#define ui_x1 ui_i.ih_x1
#define ui_pr ui_i.ih_pr
#define ui_len ui_i.ih_len
#define ui_src ui_i.ih_src
#define ui_dst ui_i.ih_dst
#define ui_sport ui_u.uh_sport
#define ui_dport ui_u.uh_dport
#define ui_ulen ui_u.uh_ulen
#define ui_sum ui_u.uh_sum
struct udpstat_t
{
/* input statistics: */
u_long udps_ipackets; /* total input packets */
u_long udps_hdrops; /* packet shorter than header */
u_long udps_badsum; /* checksum error */
u_long udps_badlen; /* data length larger than packet */
u_long udps_noport; /* no socket on port */
u_long udps_noportbcast; /* of above, arrived as broadcast */
u_long udps_fullsock; /* not delivered, input socket full */
u_long udpps_pcbcachemiss; /* input packets missing pcb cache */
/* output statistics: */
u_long udps_opackets; /* total output packets */
};
/*
* Names for UDP sysctl objects
*/
#define UDPCTL_CHECKSUM 1 /* checksum UDP packets */
#define UDPCTL_MAXID 2
extern struct udpstat udpstat;
extern struct socket udb;
struct mbuf;
void udp_init (PNATState);
void udp_input (PNATState, register struct mbuf *, int);
int udp_output (PNATState, struct socket *, struct mbuf *, struct sockaddr_in *);
int udp_attach (PNATState, struct socket *, int service_port);
void udp_detach (PNATState, struct socket *);
u_int8_t udp_tos (struct socket *);
void udp_emu (PNATState, struct socket *, struct mbuf *);
struct socket * udp_listen (PNATState, u_int32_t, u_int, u_int32_t, u_int, int);
int udp_output2(PNATState pData, struct socket *so, struct mbuf *m,
struct sockaddr_in *saddr, struct sockaddr_in *daddr,
int iptos);
#endif
| code |
One of the main accessories, without which we simply cannot imagine ourselves, is a sharif handbags.
During the day, lovely ladies may need countless things: from a comb to a wallet. And this whole world a woman places in her sharif handbags, from which, if necessary, you can extract a notebook, prismatic or lipstick, nail file, pin or centimeter. And, of course, the sharif handbags makes the image complete: it balances the dress and shoes, gives the necessary accent.
As a subject for wearing the necessary things, the history of the sharif handbags goes back to ancient times. Initially, primitive people still folded their unpretentious belongings into a well-dressed big skin, and, tying up a kind of bundle, they hooked it on a stick so that it was easier to carry. When they learned to weave, they began to use canvas instead of skins, and this was the birth of the first sacks, sharif handbags.
The history of creating a womens bag, sharif handbags, as an accessory began in the 12th century, when mirrors, prayer books, snuff salts, dices, and card decks began to be worn in such bags. | english |
\begin{document}
\maketitle
\begin{abstract}
After establishing the moderate deviation principle by the Classical Azencott method, we prove the Strassen's compact law of the iterated logarithm (LIL) for a class of stochastic partial differential equations (SPDEs). As an application, we obtain this type of LIL for two population models known as super-Brownian motion and Fleming-Viot process. In addition, the classical LIL is shown for the class of SPDEs and the two population models.
\end{abstract}
\section{Introduction}
Large deviations has noticeably become an active area of research with applications in queues, communication theory, exit problems and statistical mechanics. It is the study of very rare events that have probability tending to zero exponentially fast and its goal is to determine the exact form of this rate of convergence. Another closely related area of study is moderate deviations, which is proved for events that have probability going to zero at a rate slower than that of large deviations but faster than the rate for central limit theorem. An important application of large and moderate deviations is the law of the iterated logarithm (LIL). Beginning with J. Deuschel, D. Stroock \cite{Stroock} (Lemma 1.4.3), a notable number of authors have used this connection. For instance, P. Baldi \cite{Baldi2}, G. Divanji, K. Vidyalaxmi \cite{Divanji}, B. Jing, Q. Shao, Q. Wang \cite{Jing}, and A. Mogul'skii \cite{Mogulskii} applied their large deviation principle (LDP) to prove LIL; whereas, Y. Chen, L. Liu \cite{Y.Chen} and R. Wang, L. Xu \cite{Asymptotic} applied their result on moderate deviation principle (MDP).\\
LIL has useful applications in fields such as finance (see for example \cite{Gao, Zinchenko}). There are different forms of LIL in the literature: Classical LIL, Strassen's Compact LIL, Chover's type, and Chung's type, which inherited names from the authors who introduced them; namely, A. Khintchine \cite{Khintchine}, V. Strassen \cite{Strassen}, J. Chover \cite{Chover}, and K. Chung \cite{Chung}, respectively. In section two, we provide a description of each type of LIL. For a more detailed introduction and history on each type we recommend \cite{Bingham}. As one may observe from the literature, every type of LIL may be derived from large and moderate deviations by the use of the Borel-Cantelli lemma. The most common form for this application is the Stassen's compact LIL. Here we prove the MDP by Azencott method and as an application establish the Strassen's compact LIL. We note that LDP and MDP for the class of SPDEs and the population models considered here were achieved in \cite{large} and \cite{moderate}, respectively by the weak convergence approach introduced by \cite{Budhiraja2, BDM}. Here the MDP by the Azencott method provides us the Freidlin-Wentzell inequality, which plays a major role in proving the Strassen's compact LIL. \\
To achieve the Strassen's compact LIL, one needs to show that the centered process multiplied by $1/\sqrt{2\log \log t}$ is relatively compact and then specify the set of limit points. Since our process is real-valued, we obtain that it is relative compact by proving its tightness property by a classical method. Moreover, to determine the set of limit points, we apply the result introduced by P. Baldi \cite{Baldi1} and implemented in \cite{Eddhabi, Nzi, Ouahra}. Other methods have also been applied by some authors to establish that their process is relatively compact. A. Dembo, T. Zajic \cite{Dembo2} and L. Wu \cite{Wu} prove this condition by showing that their process is totally bounded. A. Schied \cite{Schied}, attains the Strassen's compact LIL for super-Brownian motion (SBM) in all dimensions, $d\geq 1$, as a corollary to its moderate deviation result also given in \cite{Schied}. Similar to many other results in LIL derived from LDP or MDP, A. Schied uses the rate function in MDP to form the set of limit points. Since his rate function is a good rate function, he applies the compactness property of its level sets and Lemma 1.4.3 in \cite{Stroock}. Different from results in \cite{Schied}, here our proof of LIL for SBM relies on the more recent presentation of SBM by a stochastic PDE given by \cite{J.Xiong}. To the best of our knowledge, LIL has not been proven for Fleming-Viot Process (FVP) in the literature. Also the Classical LIL for SBM and FVP would be new contributions. \\
We begin in section two with notations and spaces used throughout the paper and provide the main results. Also we offer some definitions and background on the Azencott method, LIL and the two population models. Section three is devoted to achieving the LDP, Strassen's compact LIL and Classical LIL for the class of SPDEs and in section four the results are applied to obtain the LDP and the two types of LIL for SBM and FVP.
\section{Notation and Main Results}
Following the notation given in \cite{large, moderate}, we introduce the space used here as follows. Suppose $(\Omega, \mathcal{F}, P)$ is a probability space and $\{\mathcal{F}_{t}\}_{t\geq 0}$ is a family of non-decreasing, right continuous sub-$\sigma$-fields of $\mathcal{F}$ such that $\mathcal{F}_{0}$ contains all $P$-null subsets of $\Omega$. We denote $\mathcal{C}_{b}(\mathbb{R})$ to be the space of continuous bounded functions on $\mathbb{R}$ and $\mathcal{C}_{c}(\mathbb{R})$ to be composed of continuous functions in $\mathbb{R}$ with compact support. For $0<\beta \in \mathbb{R}$, let $\mathcal{M}_{\beta}(\mathbb{R})$ denote the set of $\sigma$-finite measures $\mu$ on $\mathbb{R}$ such that,
\begin{equation}\label{Mbeta}
\int e^{-\beta |x|} d\mu(x) <\infty,
\end{equation}
and let $\mathcal{P}_{\beta}(\mathbb{R})$ be the set of probability measures satisfying \eqref{Mbeta}. We endow these spaces with the topology defined by a modification of the usual weak topology: $\mu^{n}\rightarrow \mu$ in $\mathcal{M}_{\beta}(\mathbb{R})$ (respectively in $\mathcal{P}_{\beta}(\mathbb{R})$) iff for every $f\in \mathcal{C}_{b}(\mathbb{R})$,
\begin{equation*}
\int_{\mathbb{R}}f(x)e^{-\beta |x|} \mu^{n}(dx) \rightarrow \int_{\mathbb{R}}f(x)e^{-\beta |x|}\mu(dx).
\end{equation*}
For $\alpha \in (0,1)$, consider the space $\mathbb{B}_{\alpha, \beta}$ composed of all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $m\in \mathbb{N}$, there exist $K_{1}, K_{2}>0$ with the following conditions:
\begin{eqnarray}
\left|f(y_{1})-f(y_{2})\right|&\leq& K_{1}e^{\beta m}|y_{1}-y_{2}|^{\alpha} , \hspace{.4cm} \forall |y_{1}|,|y_{2}| \leq m\\ \label{cond1}
|f(y)| &\leq& K_{2}e^{\beta |y|}, \hspace{.4cm} \forall y\in \mathbb{R},\label{cond2}
\end{eqnarray}
and with the metric,
\begin{equation*}
d_{\alpha,\beta}(u,v)=\sum^\infty_{m=1}2^{-m}(\|u-v\|_{m,\alpha,\beta}\wedge1),\qquad u,\;v\in\mathbb{B}_{\alpha,\beta},
\end{equation*}
where
\begin{equation*}
\|u\|_{m,\alpha,\beta}=\sup_{x\in\mathbb{R}}e^{-\beta|x|}|u(x)|+\sup_{y_{1}\neq y_{2}\\ \left|y_{1}\right|,\left|y_{2}\right|\leq m}\frac{|u(y_{1})-u(y_{2})|}{|y_{1}-y_{2}|^\alpha}e^{-\beta m}.
\end{equation*}
We refer to the collection of continuous functions on $\mathbb{R}$ satisfying \eqref{cond2} as $\mathbb{B}_{\beta}$, which is a Banach space with norm,
\begin{equation*}
\|f\|_{\beta}= \sup_{x\in \mathbb{R}} e^{-\beta |x|}|f(x)|.
\end{equation*}
The above space was used to match the setup in \cite{BDM}, where weak convergence approach was introduced to prove the large deviation principle and the technical difficulties in time discretization in classical Azencott method were avoided.\\
We now give a short introduction on the two population models under study. For more information we refer the reader to \cite{Dawson,Dynkin, Etheridge, J.Xiong2}. SBM is the continuous version of branching Brownian motion, the most classical and best known branching process where individuals reproduce according to Galton-Watson process. Since the population is set to evolve as a cloud in $\mathbb{R}^{d}$, it is a measure-valued process and because of its branching property, we associate a branching rate, denoted as $\varepsilon$. One of the common ways used to characterize SBM, denoted as $\{\mu_{t}^{\varepsilon}\}_{\varepsilon >0}$, is by the unique solution to the martingale problem given as: for all $f\in \mathcal{C}_{b}^{2}(\mathbb{R})$,
\begin{equation*}
M_{t}(f):= \left<\mu_{t}^{\varepsilon},f\right>-\left<\mu_{0}^{\varepsilon},f\right>
-\int_{0}^{t}\left<\mu_{s}^{\varepsilon},\frac{1}{2}\Delta f\right> ds,
\end{equation*}
is a square-integrable martingale with quadratic variation,
\begin{equation*}
\left<M(f)\right>_{t}= \varepsilon \int_{0}^{t} \left<\mu_{s}^{\varepsilon},f^{2}\right>ds,
\end{equation*}
(see \cite{Etheridge} Section 1.5 for this formulation). Recently, \cite{J.Xiong} offered the following SPDE to characterize SBM,
\begin{equation}\label{SBM}
u_{t}^{\varepsilon}(y)=F(y)+ \int_{0}^{t}\int_{0}^{u_{s}^{\varepsilon}(y)} W(dads) + \int_{0}^{t} \frac{1}{2} \Delta u_{s}^{\varepsilon}(y)ds,
\end{equation}
where,
\begin{equation}\label{SBM d}
u_{t}^{\varepsilon}(y)=\int_{0}^{y}\mu_{t}^{\varepsilon}(dx), \hspace{1cm} \forall y\in \mathbb{R},
\end{equation}
$F(y)=\int_{0}^{y}\mu_{0}(dx)$, and $W$ is an $\mathcal{F}_{t}$-adapted space-time white noise random measure on $\mathbb{R}^{+}\times \mathbb{R}$ with intensity measure $dsda$.
The other model studied here is FVP, which observes the evolution of the population based on the genetic type of individuals. It is the continuous version of the step-wise mutation model, in which individuals move in $\mathbb{Z}^{d}$ according to a continuous time sample random walk. In FVP, the population is fixed throughout time with each individual having a gene type and every time a mutation occurs the individual changes gene type and moves to a new location. Therefore, the distribution of gene types is observed, making FVP a probability-measure valued process with mutation rate given by $\varepsilon$. More background on FVP can be found in \cite{Etheridge, Kurtz1, Kurtz2}. Similar to SBM, FVP can be given by the unique solution to the following martingale problem: for $f\in \mathcal{C}_{c}^{2}(\mathbb{R})$,
\begin{equation*}
M_{t}(f)= \left<\mu_{t}^{\varepsilon},f\right>-\left<\mu_{0}^{\varepsilon},f\right>-\int_{0}^{t} \left<\mu_{s}^{\varepsilon},\frac{1}{2}\Delta f\right>ds,
\end{equation*}
is a continuous square-integrable martingale with quadratic variation,
\begin{equation*}
\left<M_{t}(f)\right>= \varepsilon \int_{0}^{t}\left(\left<\mu_{s}^{\varepsilon},f^{2}\right>-\left<\mu_{s}^{\varepsilon},f\right>^{2}\right)ds,
\end{equation*}
(see \cite{Etheridge} Section 1.11 for more details on this formulation). An SPDE characterization of FVP was also made in \cite{J.Xiong}. There by using,
\begin{equation}\label{FVP d}
u_{t}^{\varepsilon}(y)= \mu_{t}^{\varepsilon}((-\infty,y]),
\end{equation}
FVP was proved to be given by,
\begin{equation}\label{FVP}
u_{t}^{\varepsilon}(y)= F(y) + \int_{0}^{t}\int_{0}^{1} \left(1_{a\leq u_{s}^{\varepsilon}(y)}-u_{s}^{\varepsilon}(y)\right)W(dsda) + \int_{0}^{t} \frac{1}{2}\Delta u_{s}^{\varepsilon}(y)ds,
\end{equation}
with the same description for $F(y)$ and noise as for SBM in \eqref{SBM}. Note that the main difference between (\ref{SBM}) and (\ref{FVP}) is in the second term. Hence, as in \cite{large, moderate}, we consider the following class of SPDEs and have SBM and FVP as special cases,
\begin{equation}\label{SPDE}
u_{t}^{\varepsilon}(y)=F(y)+\sqrt{\varepsilon}\int_{0}^{t}\int_{U}G(a,y,u_{s}^{\varepsilon}(y))W(dads)
+\int_{0}^{t}\frac{1}{2}\Delta u_{s}^{\varepsilon}(y)ds,
\end{equation}
where $(U,\mathcal{U},\lambda)$ is a measure space such that $L^{2}(U,\mathcal{U},\lambda)$ is separable, $F$ is a function of $\mathbb{R}$ and $u_{1},u_{2},u,y\in \mathbb{R}$. In addition, $G:U\times \mathbb{R}^{2}\rightarrow \mathbb{R}$ satisfies the following conditions,
\begin{eqnarray}
\int_{U}\left|G(a,y,u_{1})-G(a,y,u_{2})\right|^{2}\lambda(da)&\leq& K|u_{1}-u_{2}|,\label{con1}\\
\int_{U}\left|G(a,y,u)\right|^{2}\lambda(da)&\leq& K\left(1+|u|^{2}\right)\label{con2}.
\end{eqnarray}
We note that since the well-posedness of \eqref{SPDE} achieved in \cite{J.Xiong} was in dimension one, our results are limited to dimension one only. For moderate deviation principle, we consider the centered process,
\begin{equation}\label{centered}
v_{t}^{\varepsilon}(y)= \frac{a(\varepsilon)}{\sqrt{\varepsilon}} \left(u_{t}^{\varepsilon}(y)-u_{t}^{0}(y)\right),
\end{equation}
where $a(\varepsilon)$ satisfies,
\begin{equation}\label{conditions}
0\leq a(\varepsilon)\rightarrow 0,\hspace{.2cm} \frac{a(\varepsilon)}{\sqrt{\varepsilon}} \rightarrow \infty \text{ as } \varepsilon \rightarrow 0.
\end{equation}
One may observe that the speed, $a(\varepsilon)$ of moderate deviations is less than $\sqrt{\varepsilon}$, the speed for large deviations; hence, the term moderate is used. For the rate of decay we need the following controlled PDE version of \eqref{centered}, also referred to as the skeleton equation given by,
\begin{equation}\label{controlled}
S_{t}(h,y)=\int_{0}^{t}\int_{U}G(a,y,u_{s}^{0}(y))h_{s}(a)\lambda(da)ds + \frac{1}{2}\int_{0}^{t}\Delta S_{s}(h,y)ds,
\end{equation}
where $h_{s}\in L^{2}\left([0,1]\times U, ds\lambda(da)\right)$. It is not difficult to show that for every $h_{s}(\cdot)$ there is a unique solution to (\ref{controlled}). For MDP, the first term of SPDE \eqref{SPDE}, $F(y)$, is assumed to be in space, $\mathbb{B}_{\alpha,\beta_{0}}$ where $\beta_{0}\in (0,\beta)$. By the classical Azencott method, we prove the following MDP result.
\begin{theorem}\label{them1}
If $F\in \mathbb{B}_{\alpha,\beta_{0}}$ for $\alpha \in \left(0,\frac{1}{2}\right)$, then family $\{v^{\varepsilon}_{t}(y)\}_{\varepsilon>0}$ satisfies the LDP in $\mathcal{C}([0,1];\mathbb{B}_{\beta})$ with speed $a(\varepsilon)$ and rate function,
\begin{equation}\label{rate1}
I(g)= \frac{1}{2}\inf\left\{\int_{0}^{1}\int_{U}\left|h_{s}(a)\right|^{2}\lambda(da)ds: g = S_{t}(h,y)\right\},
\end{equation}
which implies that family $\{u_{t}^{\varepsilon}(y)\}_{\varepsilon >0}$ obeys the MDP.
\end{theorem}
LDP by Azencott method was first introduced by \cite{Azencott, Priouret} and it may be described as follows. Suppose a family of random variables, $\{X_{1}^{\varepsilon}\}_{\varepsilon>0}$, on a Polish space, $E_{1}$, satisfies the LDP with rate function, $I_{1}: E_{1}\rightarrow [0,\infty]$. Then family $\{X_{2}^{\varepsilon}\}_{\varepsilon>0}$ on another Polish space, $E_{2}$, satisfies the LDP with rate function, $I_{2}(g):= \inf\{I_{1}(f):\Phi(f)=g\}$, if for any $R,\rho, a>0$, there exist $\eta>0$, and $\varepsilon_{0}>0$, such that for any $f\in E_{1}$ with $I_{1}(f)\leq a$ and any $\varepsilon \leq \varepsilon_{0}$,
\begin{equation}\label{exponentialinequality}
P\left(\|X_{2}^{\varepsilon}-\Phi(f)\|_{2}\geq \rho, \|X_{1}^{\varepsilon}-f\|_{1} <\eta\right) \leq \exp\left(-\frac{R}{\varepsilon^{2}}\right),
\end{equation}
where $\Phi: \{I_{1}\leq a\} \rightarrow E_{2}$ is continuous with respect to the topology of $E_{1}$ when restricted to sets $\{I_{1}\leq a\}$ for any $a>0$. Inequality \eqref{exponentialinequality} is referred to as the Freidlin-Wentzell inequality and in the setting of SPDEs, $\Phi(f)$ is the unique solution to the controlled PDE. For some examples of results on LDP for SPDEs by this method we refer the reader to \cite{Nualart, Burgers}. Below is the general definition of LDP. For more background on the large deviations theory we recommend \cite{Budhiraja1, Dembo, Dupuis}.
\begin{definition}[Large Deviation Principle (LDP)] The sequence $\left\{X^{\varepsilon}\right\}_{\varepsilon>0}$ satisfies the LDP on $\mathcal{E}$ with rate function $I$ if the following two conditions hold.\\
a. LDP lower bound: for every open set $U\subset \mathcal{E}$,
\begin{equation*}
-\inf_{x\in U} I(x) \leq \liminf_{\varepsilon \rightarrow 0} \varepsilon \log P(X^{\varepsilon} \in U),
\end{equation*}
b. LDP upper bound: for every closed set $C \subset \mathcal{E}$,
\begin{equation*}
\limsup_{\varepsilon \rightarrow 0} \varepsilon \log P(X^{\varepsilon} \in C) \leq -\inf_{x\in C}I(x).
\end{equation*}
\end{definition}
As for SBM and FVP, each model being a measure-valued process is denoted as $\{\mu_{t}^{\varepsilon}\}_{\varepsilon>0}$ with $\varepsilon$ being the branching rate or mutation rate based on context and is set to go to zero. For SBM, the Cameron-Martin space, $\mathcal{H}$ is used and for FVP we use $\tilde{\mathcal{H}}$, the space for which
conditions for $\mathcal{H}$ hold with $\mathcal{M}_\beta(\mathbb{R})$
replaced by the space of probability measures $\mathcal{P}(\mathbb{R})$, and with the additional assumption,
\begin{equation*}
\left<\mu_{t}^{0}, \frac{ d \left(\dot{\omega}_{t}-\frac{1}{2}\Delta^{*}\omega_{t}\right)}{d\mu_t^{0}}\right> = 0,
\end{equation*}
where for both population models the centered process for MDP is given by,
\begin{equation}\label{w}
\omega_{t}^{\varepsilon}(dy):= \frac{a(\varepsilon)}{\sqrt{\varepsilon}} \left(\mu_{t}^{\varepsilon}(dy)- \mu_{t}^{0}(dy)\right).
\end{equation}
With the above notation, we obtain the following theorems.
\begin{theorem}
If $\omega_{0} \in \mathcal{M}_{\beta}(\mathbb{R})$ such that $F\in \mathbb{B}_{\alpha, \beta_{0}}$ then super-Brownian motion, $\{\mu_{t}^{\varepsilon}\}_{\varepsilon>0}$, obeys the MDP in $\mathcal{C}([0,1];\mathcal{M}_{\beta}(\mathbb{R}))$ with speed $a(\varepsilon)$ and rate function,
\begin{equation}\label{rate2}
I(\omega)= \left\{\begin{array} {ll} \frac{1}{2} \displaystyle \int_{0}^{1}
\int_{\mathbb{R}}\left|\frac{d\left(\dot{\omega} -\frac{1}{2}\Delta^{*}\omega_{t}\right)}{d\mu_{t}^{0}}y\right|^2 \mu_{t}^{0}(dy) dt
& \mbox{\emph{if }} \mu_{t}^{0} \in \mathcal{H}_{\omega_{0}}, \\
\infty
& \mbox{\emph{otherwise}.} \end{array} \right.
\end{equation}
\end{theorem}
\begin{theorem}
Let $\mathcal{P}_{\beta}(\mathbb{R})$ be the probability measure analog of $\mathcal{M}_{\beta}(\mathbb{R})$. If $\omega_{0} \in \mathcal{P}_{\beta}(\mathbb{R})$ such that $F
\in \mathbb{B}_{\alpha,\beta_{0}}$, then, Fleming-Viot process, $\{\mu^{\varepsilon}\}_{\varepsilon>0}$,
satisfies the MDP on $\mathcal{C}([0,1];
\mathcal{P}_{\beta}(\mathbb{R}))$ with speed $a(\varepsilon)$ and rate function,
\begin{equation}\label{rate3}
I(\omega)= \left\{\begin{array} {ll} \frac{1}{2} \displaystyle \int_{0}^{1}
\int_{\mathbb{R}}\left|\frac{d \left(\dot{\omega_{t}}-\frac{1}{2}\Delta^{*}\omega_{t}\right)}{d\mu_{t}^{0}}
y \right|^2 \mu_{t}^{0}(dy) dt
& \mbox{\emph{if }} \mu_{t}^{0} \in \tilde{\mathcal{H}}_{\omega_{0}}, \\
\infty
& \mbox{\emph{otherwise.}} \end{array} \right.
\end{equation}
\end{theorem}
As mentioned in the introduction, there are different types of LIL that appear in the literature. Below we provide a definition for each type.
\begin{definition}[Law of the Iterated Logarithm (LIL)]
Let $\{X_{j}\}_{j\geq 1}$ be an i.i.d. sequence of random variables with $S_{n}:= \sum_{j=1}^{n}X_{j}$.
i. Classical LIL: $\{X_{j}\}_{j\geq 1}$ is said to satisfy the classical LIL, also referred to as the Khintchine's LIL, if
\begin{eqnarray}
\limsup_{n\rightarrow \infty} \frac{S_{n}-n\mu}{\sigma\sqrt{2n\log\log n}}&=&1 \text{ a.s. }\label{eq1}\\
\liminf_{n\rightarrow \infty} \frac{S_{n}-n\mu}{\sigma\sqrt{2n\log\log n}}&=&-1 \text{ a.s. }\label{eq2}
\end{eqnarray}
for common mean $\mu$ and variance $\sigma^{2}$. We note that this version is also given by (\ref{eq1}) and (\ref{eq2}) with $S_{n}-n\mu$ replaced by $X_{n}$ with $\mu=0$ and $\sigma^{2}=1$. For examples of this form see for instance \cite{Asymptotic, Jing}.
ii. Strassen's Compact LIL: A class of functions $\mathcal{F}$ satisfies Strassen's compact LIL with respect to $\{X_{j}\}_{j\geq 1}$ if there is a compact set $J$ in $\ell_{\infty}(\mathcal{F})$ such that $\{X_{j}\}_{j\geq 1}$ is a.s. relatively compact and its limit set is $J$. See for example \cite{Baldi1,Dembo2, Wu}.
iii. Chover-type LIL: $\{X_{j}\}_{j\geq 1}$ satisfies Chover-type LIL if
\begin{equation}\label{chover}
\limsup_{n\rightarrow \infty} \left(\frac{|S_{n}|}{n^{1/\alpha}}\right)^{\frac{1}{\log \log n}}= e^{1/\alpha} \text{ a.s.}
\end{equation}
for $0<\alpha <2$. For examples of this form see \cite{Peng,Yamamuro}.
iv. Chung-type LIL: Let $S_{n}^{*} = max_{k\leq n}|S_{k}|$. Chung-type LIL for $\{X_{j}\}_{j\geq 1}$ holds if
\begin{equation}\label{chung}
\liminf_{n\rightarrow \infty} \frac{S_{n}^{*} \sqrt{\log\log n}}{\sqrt{n}}= \frac{\pi}{\sqrt{8}} \text{ a.s. }
\end{equation}
For results of this type see for example, \cite{Y.Chen, Mogulskii}.
\end{definition}
We are now ready to give our results on LIL. For $0<\varepsilon <1$, let,
\begin{equation}
Z_{t}^{\varepsilon}(y):= \frac{1}{\sqrt{2\varepsilon \log \log \frac{1}{\varepsilon}}} \left(u_{t}^{\varepsilon}(y)-u_{t}^{0}(y)\right),
\end{equation}
more precisely,
\begin{equation}\label{Zprocess}
Z_{t}^{\varepsilon}(y)= \frac{1}{\sqrt{2 \log \log \frac{1}{\varepsilon}}} \int_{0}^{t}\int_{U}G_{s}^{\varepsilon}\left(a,y,Z_{s}^{\varepsilon}(y)\right)W(dads)+ \int_{0}^{t} \frac{1}{2} \Delta Z_{s}^{\varepsilon}(y)ds,
\end{equation}
where,
\begin{equation}
G_{s}^{\varepsilon}\left(a,y,Z_{s}^{\varepsilon}(y)\right):= G\left(a,y, \sqrt{2\varepsilon \log \log \frac{1}{\varepsilon}} Z_{s}^{\varepsilon}(y) + u_{s}^{0}(y)\right).
\end{equation}
Therefore, we have the process $\{v_{t}^{\varepsilon}(y)\}_{\varepsilon>0}$ from moderate deviations used in Theorem \ref{them1} with $a(\varepsilon)= 1/\sqrt{2\log \log (1/\varepsilon)}$. One can check that this fulfills the requirements of $a(\varepsilon)$ going to zero as $\varepsilon$ tends to zero, at a rate slower than $\sqrt{\varepsilon}$. Also based on conditions \eqref{con1} and \eqref{con2},
\begin{equation}\label{condition1}
\int_{U}\left|G_{s}^{\varepsilon}\left(a,y,Z^{\varepsilon}_{s,1}(y)\right)-
G_{s}^{\varepsilon}\left(a,y,Z^{\varepsilon}_{s,2}(y)\right)\right|^{2} \lambda(da) \leq
K_{3}\sqrt{2\varepsilon \log \log \frac{1}{\varepsilon}}\left|Z_{s,1}^{\varepsilon}(y)-Z_{s,2}^{\varepsilon}(y)\right|,
\end{equation}
\begin{equation}\label{condition2}
\int_{U}\left|G_{s}^{\varepsilon}\left(a,y,Z_{s}^{\varepsilon}(y)\right)\right|^{2}\lambda(da)\leq
K_{4}\left(1+ \left(2\varepsilon \log \log \frac{1}{\varepsilon}\right)Z_{s}^{\varepsilon}(y)^{2}+ e^{2\beta_{0}|y|}\right),
\end{equation}
where we have used the fact that $F\in \mathbb{B}_{\alpha,\beta_{0}}$, giving by condition \eqref{cond2},
\begin{equation}\label{u0con}
\left|u_{s}^{0}(y)\right|\leq K_{2}e^{\beta_{0}|y|}.
\end{equation}
We point out that the proof of the existence and uniqueness of solutions to SPDE, $\left\{u_{t}^{\varepsilon}(y)\right\}_{\varepsilon>0}$ given in \cite{J.Xiong} only relies on condition \eqref{con1}. Thus, we obtain the well-posedness of solutions to $Z_{t}^{\varepsilon}(y)$ and can use its mild solution given as,
\begin{equation}\label{mild}
Z_{t}^{\varepsilon}(y):= \frac{1}{\sqrt{2\log \log \frac{1}{\varepsilon}}}\int_{0}^{t}\int_{U}P_{t-s}G_{s}^{\varepsilon}(a,y,Z_{s}^{\varepsilon}(y))W(dads),
\end{equation}
where $P_{t-s}$ is the Brownian semigroup defined as $P_{t}f(y)=\int_{\mathbb{R}}p_{t}(x-y)f(x)dx$ with \\
$p_{t}(x-y)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{|x-y|^{2}}{2t}}$. We prove the following results using Theorems 1-3.
\begin{theorem}\label{theorem1}
Process $\{Z_{t}^{\varepsilon}(y)\}_{0<\varepsilon<1}$ is relatively compact in $\mathcal{C}([0,1];\mathbb{B}_{\beta})$ and its set of limit points is exactly $L_{1}:= \left\{g \in \mathcal{C}\left([0,1];\mathbb{B}_{\beta}\right): I(g)\leq 1\right\}$ where $I(g)$ is defined by (\ref{rate1}).
\end{theorem}
Similarly using the MDP result for SBM and FVP, let
\begin{equation}\label{Ztilde}
\tilde{Z}_{t}^{\varepsilon}:= \frac{1}{\sqrt{2\varepsilon \log \log \frac{1}{\varepsilon}}} \left(\mu_{t}^{\varepsilon}(dy)-\mu_{t}^{0}(dy)\right).
\end{equation}
\begin{theorem}\label{theorem2}
Process $\{\tilde{Z}_{t}^{\varepsilon}\}_{0<\varepsilon<1}$ formed by SBM process, $\{\mu_{t}^{\varepsilon}\}_{\varepsilon>0}$ in (\ref{Ztilde}) is relatively compact in $\mathcal{C}([0,1];\mathcal{M}_{\beta}(\mathbb{R}))$ with set of limit points being
$L_{2}:= \left\{\omega \in \mathcal{C}([0,1];\mathcal{M}_{\beta}(\mathbb{R})): I(\omega)\leq 1\right\}$, where $I(\omega)$ is given by (\ref{rate2}).
\end{theorem}
\begin{theorem}\label{theorem3}
Process $\{\tilde{Z}_{t}^{\varepsilon}\}_{0<\varepsilon<1}$ formed by FVP process, $\{\mu_{t}^{\varepsilon}\}_{\varepsilon>0}$ in (\ref{Ztilde}) is relatively compact in $\mathcal{C}([0,1];\mathcal{P}_{\beta}(\mathbb{R}))$ with set of limit points being
$L_{3}:= \left\{\omega \in \mathcal{C}([0,1];\mathcal{P}_{\beta}(\mathbb{R})): I(\omega)\leq 1\right\}$, where $I(\omega)$ is given by (\ref{rate3}).
\end{theorem}
Following the setup in \cite{Y.Chen, Wang,Asymptotic} we prove the classical LIL in our stochastic PDEs setting. We note that most results on classical LIL such as \cite{Davie, Jing} have been achieved for a sum of independent identically distributed random variables, where the Borel-Cantelli lemma is the main ingredient of the proof. We prove the classical LIL for the class of SPDEs and the population models by showing that for each respective family, $\{X^{\varepsilon}\}_{0<\varepsilon<1}$ of solutions,
\begin{eqnarray}
\limsup_{\varepsilon \rightarrow 0} \frac{\|X^{\varepsilon}-X^{0}\|_{\chi}}{\sqrt{2\varepsilon \log \log \frac{1}{\varepsilon}}}&=&1 \hspace{.4cm} \text{a.s.}, \label{limsup}\\
\liminf_{\varepsilon\rightarrow0} \frac{\|X^{\varepsilon}-X^{0}\|_{\chi}}{\sqrt{2\varepsilon \log \log \frac{1}{\varepsilon}}} &=&-1 \hspace{.4cm} \text{a.s.},
\label{liminf}
\end{eqnarray}
where $\chi$ is $\mathbb{B}_{\beta}$ for the class of SPDEs and it is $\mathcal{M}_{\beta}(\mathbb{R})$ and $\mathcal{P}_{\beta}(\mathbb{R})$ for SBM and FVP, respectively.
\section{LIL for Class of SPDEs}
We begin by proving Theorem 1, for which we derive the following Freidlin-Wentzell inequality based on our setting,
\begin{equation}\label{maininequality}
P\left(\left\|Z_{t}^{\varepsilon}-S_{t}(h,y)\right\|_{\beta}>\rho, \left\|\frac{1}{\sqrt{2\log \log \frac{1}{\varepsilon}}} W - h_{s}\right\|_{\infty}<\eta\right)
\leq \exp\left(-2R \log \log \frac{1}{\varepsilon}\right).
\end{equation}
Note that by Schilder's theorem, the LDP holds for Brownian sheet, $W$ with rate function denoted here by $\tilde{I}(\cdot)$.
\begin{lemma}
For every $a>0$, $S_{t}(\cdot,y):\{\tilde{I} \leq a\} \rightarrow \mathcal{C}([0,1];\mathbb{B}_{\beta})$ is continuous with respect to the uniform topology.
\end{lemma}
\begin{proof}
Let $a>0$ and $h_s,k_s\in L^{2}([0,1]\times U, ds\lambda(da))$ with $|h_{s}| \vee |k_{s}|\leq a$. By H\"older's inequality and \eqref{con2}, we have,
\begin{eqnarray}\label{Sinequality}
&&\|S_{t}(h_{s},x)-S_{t}(k_{s},x)\|_{\beta}^{2} \\
&=& \sup_{x\in \mathbb{R}} e^{-2\beta |x|} \left|\int_{0}^{t} \int_{U}P_{t-s}G(a,x,u_{s}^{0}(x))(h_{s}(a)-k_{s}(a))\lambda(da)ds\right|^{2}\nonumber\\
&\leq& \sup_{x\in \mathbb{R}} e^{-2\beta |x|} \left|\int_{0}^{t} \left(\int_{U}\left(P_{t-s}G(a,x,u_{s}^{0}(x))\right)^{2}\lambda(da)\right)^{1/2} \left(\int_{U}|h_{s}(a)-k_{s}(a)|^{2}\lambda(da)\right)^{1/2}ds\right|^{2}\nonumber\\
&\leq& t\sup_{x\in \mathbb{R}} e^{-2\beta |x|} \int_{0}^{t} \int_{\mathbb{R}} p_{t-s}^{2}(x-y)e^{2\beta |y|}dy \int_{\mathbb{R}}(1+|u^{0}_{s}(y)|^{2})e^{-2\beta |y|}dy \int_{U} |h_{s}(a)-k_{s}(a)|^{2}\lambda(da)ds.\nonumber
\end{eqnarray}
Observe that,
\begin{eqnarray}\label{pinequality}
\int_{0}^{t}\int_{\mathbb{R}} p_{t-s}^{2}(x-y)e^{2\beta |y|}dyds &=& \int_{0}^{t}\int_{\mathbb{R}} \frac{1}{2\sqrt{\pi (t-s)}}p_{\frac{t-s}{2}}(x-y)e^{2\beta |y|}dyds\nonumber\\
&\leq& \tilde{K}e^{2\beta |x|} \int_{0}^{t} \frac{1}{\sqrt{t-s}}ds= \tilde{K}\sqrt{t} e^{2\beta |x|}.
\end{eqnarray}
Thus, using \eqref{u0con} with the convention that $\beta_{0}<\beta$, we obtain,
\begin{equation*}
\|S_{t}(h_{s},x)-S_{t}(k_{s},x)\|_{\beta}^{2} \leq t^{3/2} \tilde{K} K_{2} \int_{U} \sup_{0\leq s\leq t} |h_{s}(a)-k_{s}(a)|^{2}\lambda(da),
\end{equation*}
where, by noting the domain $L^{2}([0,1]\times U, ds\lambda(da))$ of $h_{s}$ and $k_{s}$, we may apply the dominated convergence theorem to obtain the result.
\end{proof}
As shown in \cite{Nualart}, by an application of Girsanov's transformation theorem, to obtain inequality \eqref{maininequality}, it is sufficient to prove for all $h_{s}\in L^{2}\left([0,1]\times U, ds\lambda(da)\right)$, $R, \rho >0$, there exist $\eta>0, \varepsilon_{0}\in (0,1)$, such that for all $\varepsilon \in (0,\varepsilon_{0})$,
\begin{equation}\label{mainestimate}
P\left(\|Y^{\varepsilon}_{t}-S_{t}(h,y)\|_{\beta}>\rho, \left\|\frac{1}{\sqrt{2\log \log \frac{1}{\varepsilon}}}W\right\|_{\infty}<\eta \right)\leq \exp\left(-2R\log \log \frac{1}{\varepsilon}\right),
\end{equation}
with,
\begin{eqnarray}\label{Yepsilon}
Y_{t}^{\varepsilon}(y)&=& \frac{1}{\sqrt{2 \log \log \frac{1}{\varepsilon}}} \int_{0}^{t}\int_{U}G_{s}^{\varepsilon}(a,y,Y_{s}^{\varepsilon}(y))W(dads)+ \int_{0}^{t}\frac{1}{2}\Delta Y_{s}^{\varepsilon}(y)ds\nonumber \\
&&+ \int_{0}^{t}\int_{U}G_{s}^{\varepsilon}(a,y,Y_{s}^{\varepsilon}(y))h_{s}(a)\lambda(da)ds,
\end{eqnarray}
where the well-posedness of $Y^{\varepsilon}_{t}(y)$ may be verified following similar reasoning as in the proof of the well-posedness of $u_{t}^{\varepsilon}(y)$ in \cite{J.Xiong}. For our estimates, we use the following lemma, the proof of which is very similar to that of Lemma 1 in \cite{moderate} and it is thus omitted.
\begin{lemma}\label{Ylemma}
Suppose $Y_{t}^{\varepsilon}(y)$ is the unique solution to SPDE \eqref{Yepsilon}, then for every $p\geq 1, \varepsilon>0$ and $T>0$, there exists a positive constant $M_{1}$ such that,
\begin{equation}\label{M1}
\sup_{\varepsilon>0} \mathbb{E}\left(\sup_{0\leq t\leq T} \int_{\mathbb{R}}Y_{t}^{\varepsilon}(x)^{2}e^{-2\beta |x|}dx\right)^{p}\leq M_{1}.
\end{equation}
\end{lemma}
In order to obtain \eqref{mainestimate}, we apply a time discretization of $Y^{\varepsilon}_{t}$. For $n\in \mathbb{N}$, $i=0,1,...,n$, let $\Delta_{i}^{n}= \left[t_{i}^{n},t_{i+1}^{n}\right)$, where $t_{i}^{n}=iT/n$, then by the following two estimates we can achieve inequality \eqref{mainestimate}.
\begin{equation}\label{firstestimate}
P\left(\|Y^{\varepsilon}_{t}-Y^{\varepsilon}_{t_{i}^{n}}\|_{\beta}>\mu\right)\leq \exp\left(-2R\log \log \frac{1}{\varepsilon}\right),
\end{equation}
\begin{equation}\label{secondestimate}
P\left(\|Y^{\varepsilon}_{t}-S_{t}(h,y)\|_{\beta}>\rho, \left\|\frac{1}{\sqrt{2 \log \log \frac{1}{\varepsilon}}}W\right\|_{\infty}<\eta, \|Y^{\varepsilon}_{t}-Y^{\varepsilon}_{t_{i}^{n}}\|_{\beta}\leq \mu\right)
\leq \exp\left(-2R\log \log \frac{1}{\varepsilon}\right).
\end{equation}
\begin{lemma}\label{lemma1}
For all $R>0, \mu>0$ there exists $n_{0}\in \mathbb{N}$ such that for all $n\geq n_{0}$, and $\varepsilon \in (0,1)$,
\begin{equation}\label{lemma1r}
P\left(\|Y^{\varepsilon}_{t}-Y^{\varepsilon}_{t_{i}^{n}}\|_{\beta} >\mu \right)\leq \exp \left(-2R \log \log \frac{1}{\varepsilon}\right).
\end{equation}
\end{lemma}
\begin{proof}
For $n\in \mathbb{N}$, let $t\in \Delta_{i}^{n}$ and denote for $0<t_{1}<t_{2}$,
\begin{equation}\label{deltap}
\Delta p(t_{2},t_{1}):= p_{t_{2}-s}(x-y)-p_{t_{1}-s}(x-y).
\end{equation}
Then we have,
\begin{eqnarray*}
&&\sup_{0\leq t\leq 1}\sup_{y\in \mathbb{R}} e^{-\beta |y|}|Y^{\varepsilon}_{t}(y)-Y^{\varepsilon}_{t_{i}^{n}}(y)|\\
&\leq& \sup_{0\leq t\leq 1}\left|\sup_{y\in \mathbb{R}} \frac{e^{-\beta |y|}}{\sqrt{2\log \log \frac{1}{\varepsilon}}}\int_{t_{i}^{n}}^{t}\int_{U}P_{t-s}G_{s}^{\varepsilon}(a,y,Y_{s}^{\varepsilon}(y))W(dads)\right|\\
&&+ \sup_{0\leq t\leq 1}\left|\sup_{y\in \mathbb{R}}\frac{ e^{-\beta |y|}}{\sqrt{2 \log \log \frac{1}{\varepsilon}}} \int_{0}^{t_{i}^{n}}\int_{U} \int_{\mathbb{R}} \Delta p\left(t,t_{i}^{n}\right)G_{s}^{\varepsilon}(a,x,Y_{s}^{\varepsilon}(x))dxW(dads)\right|\\
&&+ \sup_{0\leq t\leq 1}\sup_{y\in \mathbb{R}} e^{-\beta |y|}\left|\int_{t_{i}^{n}}^{t}\int_{U} P_{t-s} G_s^{\varepsilon}(a,y,Y_{s}^{\varepsilon}(y))h_{s}(a)\lambda(da)ds \right|\\
&&+ \sup_{0\leq t\leq 1}\sup_{y\in \mathbb{R}} e^{-\beta |y|}\left|\int_{0}^{t_{i}^{n}}\int_{U}\int_{\mathbb{R}}\Delta p(t,t_{i}^{n}) G_s^{\varepsilon}(a,x,Y_{s}^{\varepsilon}(x))h_{s}(a)dx\lambda(da)ds\right|\\
&=& I_{1}+I_{2}+I_{3}+I_{4},
\end{eqnarray*}
leading to,
\begin{equation*}
P\left(\sup_{0\leq t\leq 1}\|Y_{t}^{\varepsilon}-Y_{t_{i}^{n}}^{\varepsilon}\|_{\beta}>\mu\right)
\leq \sum_{i=1}^{n}\sum_{j=1}^{4}P\left(\sup_{t \in \Delta_{i}^{n}}I_{j}(t)>\frac{\mu}{4}\right).
\end{equation*}
Similar to estimates in \eqref{Sinequality} and noting the domain $L^{2}\left([0,1]\times U, ds\lambda(da)\right)$ of $h_{s}(a)$, we have by Lemma 2,
\begin{eqnarray*}
P\left(\sup_{t\in \Delta_{i}^{n}}I_{3}(t)>\frac{\mu}{4}\right)&\leq& \frac{16}{\mu^{2}}\mathbb{E}\sup_{t\in \Delta_{i}^{n}}\left|I_{3}(t)\right|^{2}\leq M_{1}\tilde{K}_{1}\frac{16}{\mu^{2}}\sup_{t\in \Delta_{i}^{n}}\left|t-t_{i}^{n}\right|^{2},\\
\text{and } P\left(\sup_{t\in \Delta_{i}^{n}}I_{4}(t)>\frac{\mu}{4}\right)&\leq& \frac{16}{\mu^{2}}\mathbb{E}\sup_{t\in \Delta_{i}^{n}}\left|I_{4}(t)\right|^{2}\leq M_{1}\tilde{K}_{2}\frac{16}{\mu^{2}}\sup_{t\in \Delta_{i}^{n}}\left|t-t_{i}^{n}\right|^{2}.
\end{eqnarray*}
Then for any fixed $R>0, \varepsilon \in (0,1)$, there exists $n_{0}$ such that for all $n\geq n_{0}$,
\begin{equation*}
P\left(\sup_{t\in \Delta_{i}^{n}}I_{3}(t)>\frac{\mu}{4}\right)+ P\left(\sup_{t\in \Delta_{i}^{n}}I_{4}(t)>\frac{\mu}{4}\right)\leq \tilde{K}_{3}\frac{16}{\mu^{2}}\sup_{t\in \Delta_{i}^{n}}\left|t-t_{i}^{n}\right|^{2}\leq \exp\left(-2R\log \log \frac{1}{\varepsilon}\right).
\end{equation*}
Continuing, we obtain,
\begin{flalign*}
&P\left(\sup_{0\leq t\leq 1}\|Y^{\varepsilon}_{t}-Y^{\varepsilon}_{t_{i}^{n}}\|_{\beta}>\mu\right)\\
&\leq \sum_{i=1}^{n} \left(P\left(\sup_{t \in \Delta_{i}^{n}}I_{1}>\frac{\mu}{4}\right)+ P\left(\sup_{t \in \Delta_{i}^{n}}I_{2}>\frac{\mu}{4}\right)\right)\\
&\leq \sum_{i=1}^{n}\left[P\left(\sup_{t\in \Delta_{i}^{n}} \left|\sup_{y\in \mathbb{R}}\frac{e^{-\beta |y|}}{\sqrt{2\log \log \frac{1}{\varepsilon}}}\int_{t_{i}^{n}}^{t}\int_{U}
P_{t-s}G_{s}^{\varepsilon}(a,y,Y_{s}^{\varepsilon}(y))W(dads)\right|>\frac{\mu}{4}\right)\right.\\
& \left.+ P\left(\sup_{t\in \Delta_{i}^{n}} \left|\sup_{y\in \mathbb{R}}\frac{e^{-\beta |y|}}{\sqrt{2\log \log \frac{1}{\varepsilon}}}\int_{0}^{t_{i}^{n}}\int_{U}\int_{\mathbb{R}}\Delta p(t,t_{i}^{n})
G_{s}^{\varepsilon}(a,x,Y_{s}^{\varepsilon}(x))dxW(dads)\right|>\frac{\mu}{4}\right)\right]\\
&= \sum_{i=1}^{n}P\left(\sup_{t\in \Delta_{i}^{n}}J_{1}^{i}(t)>\frac{\mu\sqrt{2\log\log\frac{1}{\varepsilon}}}{4}\right)+ \sum_{i=1}^{n} P\left(\sup_{t\in \Delta_{i}^{n}}J_{2}^{i}(t)>\frac{\mu \sqrt{2\log\log \frac{1}{\varepsilon}}}{4}\right).
\end{flalign*}
Similar to \eqref{Sinequality} and \eqref{pinequality} and by Burkholder-Davis-Gundy inequality, \eqref{condition2} and Lemma 2,
\begin{eqnarray*}
\mathbb{E}\sup_{t\in \Delta_{i}^{n}} |J_{1}^{i}|^{2}&=& \mathbb{E} \sup_{t\in \Delta_{i}^{n}} \sup_{y\in \mathbb{R}}e^{-2\beta |y|} \left|\int_{t_{i}^{n}}^{t} \int_{U} P_{t-s}G_{s}^{\varepsilon}(a,y,Y_{s}^{\varepsilon}(y))W(dads)\right|^{2}\\
&\leq& \mathbb{E} \sup_{t\in \Delta_{i}^{n}} \sup_{y} e^{-2\beta |y|} \int_{t_{i}^{n}}^{t} \int_{U} \left(\int_{\mathbb{R}} p_{t-s}(x-y) G_{s}^{\varepsilon}(a,x,Y_{s}^{\varepsilon}(x))dx\right)^{2}\lambda(da)ds\\
&\leq& \mathbb{E}\sup_{t\in \Delta_{i}^{n}} \sup_{y} e^{-2\beta |y|} \int_{t_{i}^{n}}^{t} \int_{U}\int_{\mathbb{R}} p_{t-s}(x-y)^{2} e^{2\beta |x|}dx \int_{\mathbb{R}} G_{s}^{\varepsilon}(a,x,Y_{s}^{\varepsilon}(x))^{2}e^{-2\beta |x|} dx \lambda(da)ds\\
&\leq& K_{4}\tilde{K}_{4} \mathbb{E}\sup_{t\in \Delta_{i}^{n}} |t-t_{i}^{n}|^{1/2} \int_{\mathbb{R}} \left(1+ (2\varepsilon \log \log \frac{1}{\varepsilon})\sup_{t_{i}^{n}\leq s\leq t} Y_{s}^{\varepsilon}(x)^{2} + e^{2\beta_{0}|x|}\right)e^{-2\beta |x|}dx\\
&\leq& \tilde{K}_{5} M_{1} |t-t_{i}^{n}|^{1/2}.
\end{eqnarray*}
Following the same steps as above we find,
\begin{equation*}
\mathbb{E}\sup_{t\in \Delta_{i}^{n}}|J_{2}(t)|^{2} \leq \tilde{K}_{6} M_{1} |t_{i}^{n}|^{1/2}.
\end{equation*}
Notice that to obtain \eqref{lemma1r}, it is sufficient to prove,
\begin{equation}\label{k}
\sum_{i=1}^{n}\mathbb{E}\exp\left(\sup_{t\in \Delta_{i}^{n}}|J_{1}^{i}(t)|^{2}\right) + \sum_{i=1}^{n}\mathbb{E}\exp\left(\sup_{t\in \Delta_{i}^{n}}|J_{2}^{i}(t)|^{2}\right)\leq \tilde{C},
\end{equation}
for some positive constant, $\tilde{C}$. Inspired by the proof of Theorem 3.2 in \cite{Cerrai}, we write the left hand side of \eqref{k} as,
\begin{equation*}
\sum_{i=1}^{n}\mathbb{E} \lim_{k\rightarrow \infty} \sum_{p=0}^{k} \frac{1}{p!}\sup_{t\in \Delta_{i}^{n}}|J_{1}^{i}(t)|^{2p} + \sum_{i=1}^{n}\mathbb{E} \lim_{k\rightarrow \infty}\sum_{p=0}^{k} \frac{1}{p!} \sup_{t\in \Delta_{i}^{n}}|J_{2}^{i}(t)|^{2p}.
\end{equation*}
Then observing that \eqref{M1} holds for all $p\geq 1$, we may apply the Monotone convergence theorem to arrive at \eqref{k} and since the above estimates hold for any $\mu>0$, we obtain \eqref{lemma1r}.
\end{proof}
\begin{lemma}
For all $R>0, \rho>0, n\in \mathbb{N}$, there exist $\mu_{0}, \eta_{0}>0$ such that for all $\mu\leq \mu_{0}, \eta\leq \eta_{0}$, and $\varepsilon \in (0,1)$,
\begin{equation}\label{2ndequation}
P\left(\left\|Y^{\varepsilon}_{t}-S_{t}(h,y)\right\|_{\beta}>\rho, \left\|\frac{1}{\sqrt{2 \log \log \frac{1}{\varepsilon}}}W\right\|_{\infty}<\eta, \|Y^{\varepsilon}_{t}-Y^{\varepsilon}_{t_{i}^{n}}\|_{\beta}\leq \mu\right)
\leq \exp\left(-2R \log \log \frac{1}{\varepsilon}\right).
\end{equation}
\end{lemma}
\begin{proof}
For the simplicity of notation, we let,
\begin{equation*}
\Delta G_{s}^{\varepsilon}(v(x),w(x)):= G_{s}^{\varepsilon}(a,x,v(x))-G_{s}^{\varepsilon}(a,x,w(x)).
\end{equation*}
By the uniqueness of solutions of $S_{t}(h,y)$, we use its mild form and obtain,
\begin{eqnarray*}
\|Y_{t}^{\varepsilon}-S_{t}(h,y)\|_{\beta} &\leq& \sup_{y\in \mathbb{R}}\frac{e^{-\beta |y|}}{\sqrt{2\log \log \frac{1}{\varepsilon}}}\int_{0}^{t}\int_{U} P_{t-s}G_{s}^{\varepsilon}(a,y,Y_{s}^{\varepsilon}(y))W(dads) \\
&&+ \sup_{y\in \mathbb{R}} e^{-\beta |y|} \int_{0}^{t} \int_{U}P_{t-s} \Delta G_{s}^{\varepsilon}(Y_{s}^{\varepsilon}(y),0)h_{s}(a)\lambda(da)ds\\
&=& \frac{B_{1}^{\varepsilon}(t)}{\sqrt{2\log\log\frac{1}{\varepsilon}}} + B_{2}^{\varepsilon}(t).
\end{eqnarray*}
Thus, we may write,
\begin{eqnarray*}
&&P\left(\|Y_{t}^{\varepsilon}-S_{t}(h,y)\|_{\beta} >\rho, \left\|\frac{1}{\sqrt{2\log \log \frac{1}{\varepsilon}}} W\right\|_{\infty} <\eta, \|Y_{t}^{\varepsilon}-Y^{\varepsilon}_{t_{i}^{n}}\|_{\beta} \leq \mu\right)\\
&\leq& P\left(B_{1}^{\varepsilon}(t) >\frac{\rho \sqrt{2\log \log \frac{1}{\varepsilon}}}{2}, \left\|\frac{1}{\sqrt{2\log \log \frac{1}{\varepsilon}}} W\right\|_{\infty} <\eta\right) \\
&&+ P\left(B_{2}^{\varepsilon}(t)> \frac{\rho}{2}, \|Y_{t}^{\varepsilon}-Y^{\varepsilon}_{t_{i}^{n}}\|_{\beta} \leq \mu\right)\\
&=& I_{1}+I_{2}.
\end{eqnarray*}
By the same reasoning as in the proof of Lemma 3, there exists a constant $C>0$ such that $\mathbb{E}(\exp(B_{1}^{\varepsilon}(t)^{2}))\leq C$, which yields,
\begin{eqnarray}\label{B1}
I_{1} &=& P\left(\exp(B_{1}^{\varepsilon}(t)^{2}) > \exp\left(\frac{\rho^{2}(2\log \log \frac{1}{\varepsilon})}{4}\right), \left\|\frac{1}{\sqrt{2\log \log \frac{1}{\varepsilon}}} W\right\|_{\infty} <\eta\right)\nonumber\\
&\leq& C \exp\left(\frac{-\rho^{2}(2\log \log \frac{1}{\varepsilon})}{4}\right)\leq \exp(-2R\log \log \frac{1}{\varepsilon}),
\end{eqnarray}
since the first inequality in \eqref{B1} is true for any $\rho>0$. As for $I_{2}$, noting that $t\in \Delta_{i}^{n}$, we have,
\begin{eqnarray*}
B_{2}^{\varepsilon}(t)&=& \sup_{y\in \mathbb{R}} e^{-\beta |y|}\int_{0}^{t_{i}^{n}}\int_{U} P_{t_{i}^{n}-s}G_{s}^{\varepsilon}(a,y,Y_{s}^{\varepsilon}(y))h_{s}(a)\lambda(da)ds\\
&&+ \sup_{y}e^{-\beta |y|}\int_{t_{i}^{n}}^{t} \int_{U} P_{t-s}G_{s}^{\varepsilon}(a,y,Y_{s}^{\varepsilon}(y))h_{s}(a)\lambda(da)ds\\
&&- \sup_{y}e^{-\beta |y|} \int_{0}^{t_{i}^{n}} P_{t_{i}^{n}-s} G_{s}^{\varepsilon}(a,y,0)h_{s}(a)\lambda(da)ds\\
&&- \sup_{y}e^{-\beta |y|}\int_{t_{i}^{n}}^{t} P_{t-s}G_{s}^{\varepsilon}(a,y,0)h_{s}(a)\lambda(da)ds,
\end{eqnarray*}
which leads to,
\begin{eqnarray*}
\sup_{t\in \Delta_{i}^{n}}B_{2}^{\varepsilon}(t)&=& \sup_{t\in \Delta_{i}^{n}}\sup_{y}e^{-\beta |y|} \int_{0}^{t_{i}^{n}} \int_{U} P_{t_{i}^{n}-s} \Delta G_{s}^{\varepsilon}(Y_{s}^{\varepsilon}(y),Y_{s_{i}^{n}}^{\varepsilon}(y))h_{s}(a)\lambda(da)ds\\
&&+ \sup_{t\in \Delta_{i}^{n}}\sup_{y} e^{-\beta |y|} \int_{0}^{t_{i}^{n}} \int_{U} P_{t_{i}^{n}-s} \Delta G_{s}^{\varepsilon}(Y_{s_{i}^{n}}^{\varepsilon}(y),0)h_{s}(a)\lambda(da)ds\\
&&+\sup_{t\in \Delta_{i}^{n}} \sup_{y} e^{-\beta |y|} \int_{t_{i}^{n}}^{t} \int_{U} P_{t-s}\Delta G_{s}^{\varepsilon}(Y_{s}^{\varepsilon}(y),0)h_{s}(a)\lambda(da)ds\\
&=& \sup_{t\in \Delta_{i}^{n}}I_{21}+\sup_{t\in \Delta_{i}^{n}}I_{22}+\sup_{t\in \Delta_{i}^{n}}I_{23}.
\end{eqnarray*}
Using \eqref{con1} and condition $\|Y_{s}^{\varepsilon}(y)-Y_{s_{i}^{n}}^{\varepsilon}(y)\|_{\beta} <\mu$, we have $\mathbb{E}\exp(|I_{21}|^{2})\leq \mu^{2}\tilde{k}_{1}|t_{i}^{n}|^{1/2}$ and we may bound $\mathbb{E}\exp(|I_{22}|^{2})$ and $\mathbb{E}\exp(|I_{23}|^{2})$ by $\tilde{k}_{2}|t_{i}^{n}|^{1/2}$ and $\tilde{k}_{3}|t-t_{i}^{n}|^{1/2}$, respectively. Then for any fixed $R>0$ and $\varepsilon \in (0,1)$ using the fact that $t_{i}^{n}:= (Ti)/n$, we may choose an $n_{0}\in \mathbb{N}$ such that for any $n\geq n_{0}$ the following inequality holds.
\begin{eqnarray*}
I_{2}&\leq& P\left(\sup_{t\in \Delta_{i}^{n}}I_{21}>\frac{\rho}{6}, \left\|Y_{t}^{\varepsilon}-Y^{\varepsilon}_{t_{i}^{n}}\right\|_{\beta} \leq \mu\right)
+P\left(\sup_{t\in \Delta_{i}^{n}}I_{22}>\frac{\rho}{6}, \left\|Y_{t}^{\varepsilon}-Y^{\varepsilon}_{t_{i}^{n}}\right\|_{\beta} \leq \mu\right)\\
&&+P\left(\sup_{t\in \Delta_{i}^{n}}I_{23}>\frac{\rho}{6}, \left\|Y_{t}^{\varepsilon}-Y^{\varepsilon}_{t_{i}^{n}}\right\|_{\beta} \leq \mu\right)\\
&\leq& \exp(-2R\log \log \frac{1}{\varepsilon}).
\end{eqnarray*}
Hence, we obtain \eqref{2ndequation}.
\end{proof}
Notice that the above estimates hold with any $a(\varepsilon)$ instead of $1/\sqrt{2\log \log (1/\varepsilon)}$ satisfying \eqref{conditions} and thus we achieve the MDP for the class of SPDEs by Azencott method, where using the rate function from Schilder's theorem and letting $\Phi(h)=S_{t}(h,y)$, we obtain \eqref{rate1}. For the Strassen's compact LIL to prove the relative compactness of $Z_{t}^{\varepsilon}(y)$, we show its tightness property by following the well established theorem stated below.
\begin{theorem}[Theorem $12.3$ in \cite{Billingsley}]\label{T12.3}
The sequence $\{X^{\varepsilon}\}_{\varepsilon>0}$ is tight in $\mathcal{C}\left([0,1];\mathbb{R}\right)$, if\\
$(i)$ the sequence $\{X^{\varepsilon}(0)\}_{\varepsilon>0}$ is tight,\\
$(ii)$ there exist constants $\gamma \geq 0$ and $\alpha >1$ and a nondecreasing, continuous function $F$ on $[0,1]$ such that \begin{equation}\label{12.3}
P\left(\left|X^{\varepsilon}(t_{2})-X^{\varepsilon}(t_{1})\right|\geq \lambda\right) \leq \frac{1}{\lambda^{\gamma}}\left|F(t_{2})-F(t_{1})\right|^{\alpha},
\end{equation}
holds for all $t_{1},t_{2}$ and $n$ and all positive $\lambda$.
\end{theorem}
We need the following analogous result to Lemma 2 for the process $\{Z^{\varepsilon}_{t}(y)\}_{0<\varepsilon<1}$, where its proof is omitted due to its similarity with the proof of Lemma 1 in \cite{moderate}.
\begin{lemma}\label{Mlemma}
Let $Z_{t}^{\varepsilon}(y)$ be the unique solution to SPDE \eqref{SPDE}, then for any $p\geq 1, 0<\varepsilon<1$ and $T>0$, there exists a positive constant $M_{2}$ such that,
\begin{equation}\label{Mequation}
\sup_{\varepsilon >0}\mathbb{E}\left(\sup_{0\leq t \leq T} \int_{\mathbb{R}} Z_{t}^{\varepsilon}(x)^{2}e^{-2\beta|x|}dx\right)^{p}\leq M_{2}.
\end{equation}
\end{lemma}
\begin{theorem}\label{tight}
Family, $\{Z_{t}^{\varepsilon}\}_{0<\varepsilon<1}$ takes values in $\mathcal{C}\left([0,1];\mathbb{B}_{\beta}\right)$ and is tight.
\end{theorem}
\begin{proof}
It was shown in Lemma 3 of \cite{moderate} that $v_{t}^{\varepsilon}(y)$ defined by \eqref{centered} takes values in $\mathcal{C}\left([0,1];\mathbb{B}_{\beta}\right)$ and its solution is unique allowing us to use the mild solution,
\begin{equation}
Z_{t}^{\varepsilon}(y):= \frac{1}{\sqrt{2\log \log \frac{1}{\varepsilon}}}\int_{0}^{t}\int_{U}P_{t-s}G_{s}^{\varepsilon}(a,y,Z_{s}^{\varepsilon}(y))W(dads),
\end{equation}
where $P_{t-s}$ is the Brownian semigroup. Let $0<\varepsilon<1$ and $t_{1},t_{2}\in [0,1]$ be arbitrary with $t_{1}<t_{2}$. For $n>8$, we proceed as follows,
\begin{eqnarray*}
&&\mathbb{E}\left\|Z_{t_{2}}^{\varepsilon}(x)-Z_{t_{1}}^{\varepsilon}(x)\right\|_{\beta}^{n}\\
&=& \mathbb{E}\sup_{x\in \mathbb{R}}e^{-n\beta |x|}\left|\frac{1}{\sqrt{2\log \log \frac{1}{\varepsilon}}} \int_{0}^{t_{2}}\int_{U}\int_{\mathbb{R}}
\frac{1}{\sqrt{2\pi \left(t_{2}-s\right)}} e^{-\frac{|x-y|^{2}}{2\left(t_{2}-s\right)}}G_{s}^{\varepsilon}\left(a,y,Z_{s}^{\varepsilon}(y)\right)dyW(dads)\right.\\
&&\left.- \frac{1}{\sqrt{2\log \log \frac{1}{\varepsilon}}} \int_{0}^{t_{1}} \int_{U}\int_{\mathbb{R}} \frac{1}{\sqrt{2\pi \left(t_{1}-s\right)}} e^{-\frac{|x-y|^{2}}{2\left(t_{1}-s\right)}} G_{s}^{\varepsilon}\left(a,y,Z_{s}^{\varepsilon}(y)\right)dyW(dads)\right|^{n}.
\end{eqnarray*}
For better presentation, let $K_{0}:= \frac{1}{\sqrt{2 \log \log \frac{1}{\varepsilon}} \sqrt{2\pi}}$ and $\tilde{G}_{s}^{\varepsilon}(a,y):= G_{s}^{\varepsilon}\left(a,y,Z_{s}^{\varepsilon}(y)\right)$. We will call the first integral above $I(t_{2},t_{2})(x)$, where the first $t_{2}$ appears in the upper limit of the integral and the second is the time parameter in the Gaussian density. Similarly, the second integral is denoted as $I\left(t_{1},t_{1}\right)(x)$. Using this notation we have,
\begin{eqnarray*}
&&\mathbb{E}\left\|Z_{t_{2}}^{\varepsilon}(x)-Z_{t_{1}}^{\varepsilon}(x)\right\|_{\beta}^{n}
=\mathbb{E}\left\|I\left(t_{2},t_{2}\right)(x)-I\left(t_{1},t_{1}\right)(x)\right\|_{\beta}^{n}\\
&\leq& 2^{n-1}K_{0}^{n}\left[\mathbb{E}\left\|I\left(t_{2},t_{2}\right)(x)-I\left(t_{1},t_{2}\right)(x)\right\|_{\beta}^{n}+ \mathbb{E}\left\|I\left(t_{1},t_{2}\right)(x)-I\left(t_{1},t_{1}\right)(x)\right\|_{\beta}^{n}\right]\\
&\leq& 2^{n-1}\left(J_{1}+J_{2}\right).
\end{eqnarray*}
As for $J_{1}$, applying the Burkholder-Davis-Gundy inequality yields,
\begin{equation*}
J_{1}\leq K_{0}^{n}\mathbb{E}\sup_{x\in \mathbb{R}}e^{-n\beta |x|}\left|\int_{t_{1}}^{t_{2}} \int_{U}\left(\int_{\mathbb{R}} \frac{1}{\sqrt{t_{2}-s}}e^{-\frac{|x-y|^{2}}{2\left(t_{2}-s\right)}} \tilde{G}_{s}^{\varepsilon}(a,y)dy\right)^{2}\lambda(da)ds\right|^{\frac{n}{2}},
\end{equation*}
where by H\"older's inequality and condition \eqref{condition2},
\begin{eqnarray}\label{2}
&&\int_{U}\left(\int_{\mathbb{R}} \frac{1}{\sqrt{t_{2}-s}}e^{-\frac{|x-y|^{2}}{2\left(t_{2}-s\right)}}\tilde{G}_{s}^{\varepsilon}(a,y)dy\right)^{2}\lambda(da)\\
&\leq& \int_{U}\int_{\mathbb{R}} \frac{1}{t_{2}-s} e^{-\frac{|x-y|^{2}}{t_{2}-s}}e^{2\beta |y|}dy
\int_{\mathbb{R}} \tilde{G}_{s}^{\varepsilon}(a,y)^{2}e^{-2\beta|y|}dy\lambda(da)\nonumber\\
&\leq& K_{4}\int_{\mathbb{R}}\frac{1}{t_{2}-s} e^{-\frac{|x-y|^{2}}{t_{2}-s}} e^{2\beta |y|}dy \int_{\mathbb{R}}
\left(1+\left(2\varepsilon \log \log \frac{1}{\varepsilon}\right)Z_{s}^{\varepsilon}(y)^{2} + e^{2\beta_{0}|y|}\right) e^{-2\beta |y|}dy,\nonumber
\end{eqnarray}
and by our assumption, $\beta_{0}<\beta$ and $\beta>0$. Moreover, similar to \eqref{pinequality},
\begin{eqnarray*}
\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}}\frac{1}{t_{2}-s}e^{-\frac{|x-y|^{2}}{t_{2}-s}}e^{2\beta |y|}dyds
\leq k_{1}\int_{t_{1}}^{t_{2}}\int_{\mathbb{R}} p_{t_{2}-s}^{2}(x-y)e^{2\beta |y|}dy ds \leq k_{2}e^{2\beta |x|}\sqrt{t_{2}-t_{1}},
\end{eqnarray*}
therefore, using Lemma \ref{Mlemma},
\begin{eqnarray}\label{J1bound}
J_{1}&\leq & K_{0}^{n}K_{4}k_{3} \mathbb{E}\left|\sqrt{t_{2}-t_{1}}\int_{\mathbb{R}}
\left(2\varepsilon \log \log \frac{1}{\varepsilon}\right)\sup_{t_{1}\leq s\leq t_{2}}Z_{s}^{\varepsilon}(y)^{2} e^{-2\beta |y|}dy\right|^{\frac{n}{2}}\nonumber\\
&\leq& k_{4}M_{2}\left|t_{2}-t_{1}\right|^{\frac{n}{4}},
\end{eqnarray}
where $k_{4}M_{2}$ is independent of $\varepsilon$. Using notation \eqref{deltap}, we continue by estimating $J_{2}$,
\begin{eqnarray*}
J_{2}&=& K_{0}\mathbb{E}\left\|\int_{0}^{t_{1}}\int_{U}\int_{\mathbb{R}} \Delta p\left(t_{2},t_{1}\right)G_{s}^{\varepsilon}(a,y)dy W(dads)\right\|_{\beta}^{n}\\
&\leq& K_{0}^{n}\mathbb{E}\sup_{x\in \mathbb{R}}e^{-n\beta |x|}\left|\int_{0}^{t_{1}}\int_{U}\int_{\mathbb{R}}\left(\Delta p\left(t_{2},t_{1}\right)\right)^{2}e^{2\beta |y|}dy
\int_{\mathbb{R}} \tilde{G}_{s}^{\varepsilon}(a,y)^{2}e^{-2\beta |y|}dy\lambda(da)ds\right|^{\frac{n}{2}}\\
&\leq& k_{5}\sup_{x\in \mathbb{R}}e^{-n\beta |x|}\left|\int_{0}^{t_{1}} \int_{\mathbb{R}} \left(\Delta p\left(t_{2},t_{1}\right)\right)^{2}
e^{2\beta |y|}dyds \right|^{\frac{n}{2}},
\end{eqnarray*}
where steps similar to those taken for estimating $J_{1}$ were applied. It can be seen that for $0<\alpha\leq 1/2$,
\begin{eqnarray}\label{Deltap}
\left(\Delta p\left(t_{2},t_{1}\right)\right)^{2}&=&\left|p_{t_{2}-s}(x-y)-p_{t_{1}-s}(x-y)\right|^{\alpha}
\left|p_{t_{2}-s}(x-y)-p_{t_{1}-s}(x-y)\right|^{2-\alpha}\\
&\leq& 2^{1-\alpha} \left|p_{t_{2}-s}(x-y)-p_{t_{1}-s}(x-y)\right|^{\alpha}\left(p_{t_{2}-s}(x-y)^{2-\alpha}+
p_{t_{1}-s}(x-y)^{2-\alpha}\right).\nonumber
\end{eqnarray}
Also $\Delta p\left(t_{2},t_{1}\right)$ may be written as,
\begin{equation}\label{3}
\frac{1}{\sqrt{2\pi}} \frac{1}{\sqrt{t_{2}-s}} \left|e^{-\frac{|x-y|^{2}}{2\left(t_{2}-s\right)}}
-e^{-\frac{|x-y|^{2}}{2\left(t_{1}-s\right)}}\right|+\frac{1}{\sqrt{2\pi}}
e^{-\frac{|x-y|^{2}}{2\left(t_{1}-s\right)}}\left|\frac{1}{\sqrt{t_{2}-s}}-\frac{1}{\sqrt{t_{1}-s}}\right|=: I_{1}+I_{2}.
\end{equation}
Using this form in \eqref{Deltap} we obtain,
\begin{equation*}
\left(\Delta p\left(t_{2},t_{1}\right)\right)^{2}\leq
K\left|I_{1}+I_{2}\right|^{\alpha}\left(p_{t_{2}-s}(x-y)^{2-\alpha}+p_{t_{1}-s}(x-y)^{2-\alpha}\right),
\end{equation*}
hence,
\begin{eqnarray*}
J_{2}&\leq& k_{5}\sup_{x\in \mathbb{R}}e^{-n\beta |x|}\left|\int_{0}^{t_{1}}\int_{\mathbb{R}} \left|I_{1}\right|^{\alpha}p_{t_{2}-s}(x-y)^{2-\alpha}e^{2\beta |y|}dyds+ \int_{0}^{t_{1}}\int_{\mathbb{R}}\left|I_{2}\right|^{\alpha}p_{t_{2}-s}(x-y)^{2-\alpha}e^{2\beta |y|}dyds\right.\\
&&\left.+ \int_{0}^{t_{1}}\int_{\mathbb{R}} \left|I_{1}\right|^{\alpha}p_{t_{1}-s}(x-y)^{2-\alpha}e^{2\beta |y|}dyds+\int_{0}^{t_{1}}\int_{\mathbb{R}} \left|I_{2}\right|^{\alpha}p_{t_{1}-s}(x-y)^{2-\alpha}e^{2\beta |y|}dyds\right|^{\frac{n}{2}}\\
&=& k_{5}\sup_{x\in \mathbb{R}} e^{-n\beta |x|}\left|J_{2,1}+ J_{2,2}+ J_{2,3}+ J_{2,4}\right|^{\frac{n}{2}}.
\end{eqnarray*}
For $I_{1}$, we use the mean value theorem to obtain,
\begin{equation}\label{4}
I_{1}\leq k_{6}\frac{|x-y|^{2}}{2\sqrt{2\pi(t_{2}-s)}}\left|\frac{1}{t_{2}-s}-\frac{1}{t_{1}-s}\right|
=k_{6}\frac{|x-y|^{2}}{2\sqrt{2\pi(t_{2}-s)}}\frac{\left|t_{2}-t_{1}\right|}{\left(t_{2}-s\right)\left(t_{1}-s\right)}.
\end{equation}
In particular,
\begin{eqnarray*}
J_{2,1}&\leq& k_{6}\int_{0}^{t_{1}}\int_{\mathbb{R}} \frac{|x-y|^{2\alpha}}{2^{\alpha}\left(2\pi \left(t_{2}-s\right)\right)^{\frac{\alpha}{2}}}
\frac{\left|t_{2}-t_{1}\right|^{\alpha}}{\left(t_{2}-s\right)^{\alpha}\left(t_{1}-s\right)^{\alpha}}
p_{t_{2}-s}(x-y)^{2-\alpha}e^{2\beta |y|}dyds\\
&\leq&k_{7} \int_{0}^{t_{1}}\int_{\mathbb{R}} \frac{\left|t_{2}-t_{1}\right|^{\alpha}}{\left(t_{2}-s\right)
^{\frac{3\alpha}{2}}\left(t_{1}-s\right)^{\alpha}}|x-y|^{2\alpha} p_{t_{2}-s}(x-y)^{2-\alpha}e^{2\beta|y|}dyds\\
&\leq&k_{7}\int_{0}^{t}\int_{\mathbb{R}}\frac{\left|t_{2}-t_{1}\right|^{\alpha}}{\left(t_{2}-s\right)
^{\frac{3\alpha}{2}}\left(t_{1}-s\right)^{\alpha}} \frac{|x-y|^{2\alpha}}{\left(t_{2}-s\right)^{1-\frac{\alpha}{2}}}\sqrt{\frac{t_{2}-s}{2-\alpha}}
p_{\frac{t_{2}-s}{2-\alpha}}(x-y)e^{2\beta |y|}dyds\\
&\leq& k_{7}\int_{0}^{t_{1}}\int_{\mathbb{R}}\frac{\left|t_{2}-t_{1}\right|^{\alpha}}
{\left(t_{2}-s\right)^{\frac{1}{2}+\alpha}\left(t_{1}-s\right)^{\alpha}}|x-y|^{2\alpha}
p_{\frac{t_{2}-s}{2-\alpha}}(x-y)e^{2\beta |y|}dyds\\
&\leq& k_{8} e^{2\beta |x|}\int_{0}^{t_{1}} \frac{\left|t_{2}-t_{1}\right|^{\alpha}}{\left(t_{2}-s\right)^{\frac{1}{2}+\alpha}
\left(t_{1}-s\right)^{\alpha}}ds.
\end{eqnarray*}
Thus, noting the assumption $t_{1}<t_{2}$, we arrive at,
\begin{equation*}
J_{2,1} \leq k_{8}e^{2\beta|x|}\left|t_{2}-t_{1}\right|^{\alpha}\int_{0}^{t_{1}}
\left(t_{1}-s\right)^{-\left(\frac{1}{2}+2\alpha\right)}ds\leq k_{9}e^{2\beta |x|}\left|t_{2}-t_{1}\right|^{\alpha},
\end{equation*}
if $2\alpha < 1/2$. Similarly for $J_{2,3}$,
\begin{equation*}
J_{2,3}\leq k_{10}e^{2\beta |x|}\left|t_{2}-t_{1}\right|^{\alpha},
\end{equation*}
if $2\alpha < 1/2$. To determine bounds for $J_{2,2}$ and $J_{2,4}$, we have for $i,j=1,2$ with $i\neq j$,
\begin{eqnarray*}
&&\int_{\mathbb{R}} \left|\frac{1}{\sqrt{t_{1}-s}}-\frac{1}{\sqrt{t_{2}-s}}\right|^{\alpha}
p_{t_{i}-s}(x-y)^{2-\alpha}e^{2\beta |y|}dy\\
&\leq& k_{11}\int_{\mathbb{R}} \left|\frac{t_{2}-t_{1}}{\left(t_{1}-s\right)\sqrt{t_{2}-s}+\left(t_{2}-s\right)\sqrt{t_{1}-s}}\right|^{\alpha}
\frac{1}{\left(\sqrt{t_{i}-s}\right)^{1-\alpha}}p_{\frac{t_{i}-s}{2-\alpha}}(x-y)e^{2\beta |y|}dy\\
&\leq& k_{11} \frac{\left|t_{2}-t_{1}\right|^{\alpha}}{\left(t_{j}-s\right)^{\alpha}}\frac{e^{2\beta |x|}}{\left(t_{i}-s\right)^{\frac{1}{2}}},
\end{eqnarray*}
and
\begin{equation*}
k_{11}\int_{0}^{t_{1}}\frac{\left|t_{2}-t_{1}\right|^{\alpha}}{\left(t_{j}-s\right)^{\alpha}}
\frac{e^{2\beta |x|}}{\left(t_{i}-s\right)^{\frac{1}{2}}}ds
\leq k_{11}e^{2\beta |x|} \left|t_{2}-t_{1}\right|^{\alpha}\int_{0}^{t_{1}}\left(t_{1}-s\right)^{-\left(\alpha + \frac{1}{2}\right)}ds
\leq k_{11}e^{2\beta |x|} \left|t_{2}-t_{1}\right|^{\alpha},
\end{equation*}
for $\alpha < 1/2$. From values for $\alpha$ found above for each term of $J_2$, we require $0<\alpha < 1/4$ and obtain,
\begin{equation*}
J_{2}\leq k_{12}\left|t_{2}-t_{1}\right|^{\frac{\alpha n}{2}},
\end{equation*}
where $k_{12}$ is independent of $\varepsilon$. Furthermore, noting the bound for $J_{1}$ in \eqref{J1bound} we confirm our assumption of $n>8$ required to satisfy condition \eqref{12.3}.
\end{proof}
We now verify that the limit set for $\{Z^{\varepsilon}_{t}(y)\}_{0<\varepsilon<1}$ is $L_{1}$ given in Theorem 4. For better presentation, we let $\varepsilon = 1/(c^{j})$, where $c>1$ and $j\geq 1$.
\begin{lemma}\label{limit}
For any $g\in L_{1}$, $\varepsilon >0$, and $c>1$, there exists $j_{0}\in \mathbb{N}$ such that for every $j>j_{0}$, $P\left(\|Z^{\frac{1}{c^{j}}}-g\|_{\beta}\leq \varepsilon \text{ i.o.}\right)=1$.
\end{lemma}
\begin{proof}
Let $g\in L_{1}$ and $h_{s}\in L^{2}\left([0,1]\times U, ds\lambda(da)\right)$ such that $g=S_{t}(h_{s},y)$ and \\ $\frac{1}{2}\int_{0}^{t}\int_{U}|h_{s}(a)|^{2}\lambda(da)ds \leq 1$. Denote,
\begin{equation*}
F_{j}:=\left\{\|Z^{\frac{1}{c^{j}}}-g\|_{\beta}\leq \varepsilon\right\} \text{ and } G{j}:= \left\{\left\|\frac{1}{\sqrt{2\log \log c^{j}}}W-h\right\|_{\infty}\leq \eta \right\},
\end{equation*}
for some constant $\eta >0$. We need to prove that $P\left(\limsup_{j\rightarrow \infty}F_{j}\right)=1$. Based on the Strassen's compact LIL for Brownian sheets proved in Section 1.4 in \cite{Stroock}, $P\left(\limsup_{j}G_{j}\right)=1$. Let $R>1$, then by \eqref{maininequality} we have,
\begin{equation}\label{Fj}
P\left(F_{j}^{c}\cap G_{j}\right)\leq \exp\left(-2R\log \log c^{j}\right)= \frac{K_{R}}{j^{2R}}\leq \frac{K_{R}}{j^{2}},
\end{equation}
where we used the fact that for $k\in \mathbb{R}$,
\begin{equation*}
\exp\left(-k\log\log c^{j}\right)= \frac{K_{k}}{j^{k}}.
\end{equation*}
Now by the Borel-Cantelli lemma applied to (\ref{Fj}), we arrive at,
\begin{equation*}
P\left(\limsup_{j\rightarrow \infty}F_{j}^{c}\cap G_{j}\right)=0.
\end{equation*}
Thus, we obtain,
\begin{equation*}
1= P\left(\limsup_{j\rightarrow \infty }G_{j}\right)\leq P\left(\limsup_{j \rightarrow \infty}G_{j}\cap F_{j}\right)+ P\left(\limsup_{j\rightarrow \infty}G_{j}\cap F_{j}^{c}\right)\leq P\left(\limsup_{j\rightarrow \infty}F_{j}\right),
\end{equation*}
obtaining the result.
\end{proof}
As for the Classical LIL, note that it is sufficient to prove \eqref{limsup}, where \eqref{liminf} may be proved analogously. As above, we use the notation, $\varepsilon:=1/(c^{j})$. Then for every $\varepsilon >0$, \eqref{limsup} is equivalent to
\begin{equation*}
P\left(\left\| \frac{u_{t}^{\frac{1}{c^{j}}}(y)-u^{0}_{t}(y)}{\sqrt{\frac{2}{c^{j}}\log \log c^{j}}} - 1 \right\|_{\beta} >\varepsilon \text{ i.o.}\right)=0,
\end{equation*}
which may further be written as,
\begin{equation}\label{frac}
\lim_{j\rightarrow \infty} P\left(\sup_{k\geq j} \left\| \frac{u^{\frac{1}{c^{k}}}_{t}(y)-u^{0}_{t}(y)}{\sqrt{\frac{2}{c^{k}}\log \log c^{k}}}-1\right\|_{\beta}>\varepsilon\right)=0.
\end{equation}
By Chebyshev inequality,
\begin{eqnarray*}
&&\lim_{j\rightarrow \infty} P\left( \sup_{k\geq j}\sup_{y}e^{-\beta |y|} \frac{|u_{t}^{\frac{1}{c^{k}}}(y)-u^{0}_{t}(y)|}{\sqrt{\frac{2}{c^{k}}\log\log c^{k}}}>\varepsilon + \sup_{y\in \mathbb{R}}e^{-\beta |y|}\right) \\
&\leq& (\varepsilon + \sup_{y}e^{-\beta |y|})^{-2}\lim_{j\rightarrow \infty} \mathbb{E} \sup_{k\geq j} \sup_{y}e^{-2\beta |y|}
\frac{|u_{t}^{\frac{1}{c^{k}}}(y)-u^{0}_{t}(y)|^{2}}{\frac{2}{c^{k}}\log \log c^{k}}.
\end{eqnarray*}
Furthermore,
\begin{eqnarray}\label{righthand}
&&\mathbb{E} \sup_{k\geq j} \sup_{y} e^{-2\beta |y|} \left|u_{t}^{\frac{1}{c^{k}}}(y)-u^{0}_{t}(y)\right|^{2}\nonumber \\
&\leq& \mathbb{E} \sup_{k\geq j} \sup_{y}e^{-2\beta |y|}\frac{1}{c^{k}}\left|\int_{0}^{t}\int_{U}\int_{\mathbb{R}}p_{t-s}(x-y)G(a,x,u_{s}^{\varepsilon}(x))dxW(dads)\right|^{2},
\end{eqnarray}
which analogous to previous estimates in this section may be bounded above by $(1/c^{k})\tilde{M}$ for a positive constant, $\tilde{M}$. Hence, we arrive at,
\begin{eqnarray*}
&&\lim_{j\rightarrow \infty} P\left(\sup_{k\geq j} \sup_{y\in \mathbb{R}} e^{-\beta |y|} \frac{|u_{t}^{\frac{1}{c^{k}}(y)}-u^{0}_{t}(y)|}{\sqrt{\frac{2}{c^{k}}\log \log c^{k}}} >\varepsilon + \sup_{y} e^{-\beta |y|}\right)\\
&\leq& (\varepsilon + \sup_{y}e^{-\beta |y|})^{-2}\lim_{j\rightarrow \infty}\sup_{k\geq j}\frac{\tilde{M}}{(2\log \log c^{k})},
\end{eqnarray*}
which implies \eqref{frac}, noting that $\varepsilon >0$ was arbitrarily chosen.
\section{LIL for SBM and FVP}
In this section we apply the results from Section Three to derive the MDP and the two types of LIL for SBM and FVP. To achieve the MDP for these models, using relations \eqref{SBM d} and \eqref{FVP d}, we have by \eqref{w}, $v_{t}^{\varepsilon}(y)= \int_{0}^{y}\omega_{t}^{\varepsilon}(dx)$ for SBM and $v_{t}^{\varepsilon}(y)= \int_{-\infty}^{y}\omega_{t}^{\varepsilon}(dx)$ for FVP. In Lemma 6 in \cite{large} it was shown that for $\mathcal{A}$, the set of nondecreasing functions, the map, $\xi: \mathbb{B}_{\beta} \cap \mathcal{A} \rightarrow \mathcal{M}_{\beta}(\mathbb{R})$ defined as,
\begin{equation}\label{int}
\xi(u)(B)= \int 1_{B}(y)du(y),
\end{equation}
is continuous for all $B\in \mathbb{B}(\mathbb{R})$. Also the same reasoning may be applied to prove that the map $\tilde{\xi}:\mathbb{B}_{\beta} \cap \mathcal{A}\rightarrow \mathcal{P}_{\beta}(\mathbb{R})$ defined by \eqref{int} is continuous. Hence, SBM and FVP may be written as continuous functions of the solution of SPDE, $v_{t}^{\varepsilon}(y)$, and thus by the contraction principle, MDP follows for SBM and FVP. It is left to determine their exact form of rate function. In Section 4 of \cite{moderate}, for $h_{s}\in L^{2}\left([0,1]\times U, ds\lambda(da)\right)$, the following expression was derived for $\frac{1}{2}\inf \int_{0}^{1}\int_{U}\left|h_{s}(a)\right|^{2}\lambda(da)ds$ for both SBM and FVP by letting $a=u_{t}^{0}(y)$,
\begin{equation*}
\frac{1}{2}\int_{0}^{1}\int_{U}\left|h_{s}(a)\right|^{2}\lambda(da) ds= \frac{1}{2}\int_{0}^{1}\int_{\mathbb{R}}\left|\frac{d\left(\dot{\omega}-\frac{1}{2}\Delta^{*}\omega\right)}
{d\mu_{t}^{0}}y\right|\mu_{t}^{0}(dy)dt,
\end{equation*}
where $\mu_{t}^{0}\in \mathcal{H}_{\omega_{0}}$ in the case of SBM and $\mu_{t}^{0} \in \tilde{\mathcal{H}}_{\omega_{0}}$ for FVP and hence we obtain \eqref{rate2} and \eqref{rate3}. \\
For the two types of LIL, as stated in Section Two, based on the SPDE characterization of the two population models, SBM and FVP are given by the class of SPDE (\ref{SPDE}) with $G(a,y,u_{s}^{\varepsilon}(y)):=1_{0<a<u_{s}^{\varepsilon}(y)}+1_{u_{s}^{\varepsilon}(y)<a<0}$ and $G(a,y,u_{s}^{\varepsilon}(y):= 1_{a\leq u_{s}^{\varepsilon}(y)}-u_{s}^{\varepsilon}(y)$, respectively, where in both cases, $G(a,y,u_{s}^{\varepsilon}(y))$ satisfies conditions \eqref{con1} and \eqref{con2}. Using these SPDE characterizations we may form the analog process $Z_{t}^{\varepsilon}(y)$ and by estimates in Section Three obtain the Freidlin-Wentzell inequality \eqref{maininequality} and tightness of $\{Z_{t}^{\varepsilon}(y)\}_{0<\varepsilon<1}$ in $\mathcal{C}([0,1];\mathcal{M}_{\beta})$ and $\mathcal{C}([0,1];\mathcal{P}_{\beta})$, respectively. Furthermore, we have the continuity of their controlled PDE version $S_{t}(h,y)$ with respect to uniform topology when restricted on level sets $\{\tilde{I}\leq a\}$. Thus, we obtain the Strassen's compact LIL for SBM and FVP with limit sets being $L_{2}$ and $L_{3}$, respectively. The classical LIL may also be achieved in respective spaces following the same steps as in Section Three with no additional conditions required.
\end{document} | math |
\begin{document}
\title{$\mathbb{Z}
\begin{abstract}
We show that the homotopy groups of a Moore space $P^n(p^r)$, where $p^r \neq 2$, are $\mathbb{Z}/p^s$-hyperbolic for $s \leq r$. Combined with work of Huang-Wu, Neisendorfer, and Theriault, this completely resolves the question of when such a Moore space is $\mathbb{Z}/p^s$-hyperbolic for $p \geq 5$, or when $p=2$ and $r \geq 6$. We also give a criterion in ordinary homology for a space to be $\mathbb{Z}/p^r$-hyperbolic, and deduce some examples.
\end{abstract}
\section{Introduction}
Given a space $X$, one can ask about the behaviour of the partial sum of homotopy groups $$\bigoplus_{i=1}^m \pi_i(X) \textrm{ as } m \rightarrow \infty.$$ Rationally, deep results have been obtained, notably the famous dichotomy of F\'elix, Halperin and Thomas \cite[Chapter 33]{FHT}. Interpreted integrally, this dichotomy says that if $X$ is a simply connected finite $CW$-complex with finite rational category then either \begin{itemize}
\item the rank of $\bigoplus_{i=1}^\infty \pi_i(X)$ is finite, and $X$ is called \emph{rationally elliptic}, or
\item the rank of $\bigoplus_{i=1}^m \pi_i(X)$ grows exponentially with $m$, and $X$ is called \emph{rationally hyperbolic}.
\end{itemize} Study of the corresponding behaviour for the torsion parts of these groups, which is the subject of this paper, was initiated by Huang and Wu in \cite{HuangWu}.
Let $M$ be a $\mathbb{Z}$-module, let $p$ be a prime and let $t \in \mathbb{N}$. The \emph{$\mathbb{Z}/p^t$-dimension} or \emph{$\mathbb{Z}/p^t$-rank} of $M$, denoted $\dim_{\mathbb{Z}/p^t}(M)$, is the greatest $d \in \mathbb{N} \cup \{0, \infty \}$ such that there is an isomorphism $M \cong (\mathbb{Z}/p^t)^d \oplus C$ for some complementary module $C$. Said another way, $\dim_{\mathbb{Z}/p^t}(M)$ is the number of $\mathbb{Z}/p^t$-summands in $M$.
\begin{definition} Let $M$ be a graded $\mathbb{Z}$-module, Let $p$ be a prime, and let $S \subset \mathbb{N}$. We say that $X$ is $p$-\textit{hyperbolic concentrated in (the set of exponents)} $S$ if $$a_m := \sum_{t \in S} \dim_{\mathbb{Z}/p^t}(\bigoplus_{i=1}^m M_i)$$ grows exponentially, in the sense that $$\liminf_m\frac{\ln(a_m)}{m} > 0.$$ For a space $X$ we will say that $X$ is $p$-\textit{hyperbolic concentrated in} $S$ if $\pi_*(X)$ is $p$-hyperbolic concentrated in $S$. If $X$ is $p$-hyperbolic concentrated in $\mathbb{N}$ then we will say simply that $X$ is $p$-hyperbolic. \label{defdef}
\end{definition}
This definition generalises and interpolates between two definitions due to Huang and Wu \cite{HuangWu}. Namely, their \textit{$\mathbb{Z}/p^s$-hyperbolicity} is precisely our $p$-hyperbolicity concentrated in the singleton set $\{s\}$, and their \textit{$p$-hyperbolicity} is precisely our $p$-hyperbolicity concentrated in $\mathbb{N}$, as defined above.
\begin{definition} \label{MooreSpace} Let $P^n(\ell)$ denote the \textit{mod-$\ell$ Moore space}, which we take to be the cofibre $$S^{n-1} \xrightarrow{\ell} S^{n-1} \longrightarrow P^n(\ell)$$ of the degree $\ell$ map. \end{definition}
Huang and Wu show that for $p$ prime, $n \geq 3$, and $r \geq 1$ the Moore space $P^n(p^r)$ is $\mathbb{Z}/p^r$ and $\mathbb{Z}/p^{r+1}$-hyperbolic, and additionally that $P^n(2)$ is $\mathbb{Z}/8$-hyperbolic. In \cite{ZhuPan}, Zhu and Pan show that $P^n(p^r)$ is also $\mathbb{Z}/p$-hyperbolic. Our first main result fills in the gap between these exponents:
\begin{theorem} \label{MooreHyp} Let $p$ be a prime, and $r \in \mathbb{N}$ with $p^r \neq 2$. If $n \geq 3$, then $P^n(p^r)$ is $\mathbb{Z}/p^s$-hyperbolic for all $s \leq r$. \end{theorem}
The key is to show that the stable homotopy of $P^{n}(p^r)$ contains a $\mathbb{Z}/p^s$-summand for each $s \leq r$. This follows from work of Adams on the $J$-homomorphism \cite{AdamsII, AdamsIV}, which allows us to find such summands in the stable homotopy of spheres, and classical work of Barratt \cite{Barratt} allows us to transplant these summands to Moore spaces. Once this is done, the proof follows the same lines as those in \cite{HuangWu} and \cite{ZhuPan}.
For $p>3$ Huang and Wu's results and Theorem \ref{MooreHyp} together are best possible, in the following sense. In \cite{NeisendorferExponents1}, Neisendorfer shows that $\pi_*(P^n(p^r))$ contains no element of order $p^s$ for $s>r+1$. In fact, Neisendorfer claimed in \cite{NeisendorferExponents1} that this result also holds when $p = 3$, but later, with Brayton Gray, discovered some mistakes in the proof (see the unpublished \cite{NeisendorferExponents2}). These mistakes were repaired apart from when $p=3$. In \cite{NeisendorferExponents2}, Neisendorfer shows that the $3$-primary exponent of $P^n(3^r)$ is either $3^{r+1}$ or $3^{r+2}$.
Neisendorfer's result allows us to combine Huang and Wu's result with Theorem \ref{MooreHyp} to obtain the following (using Proposition \ref{CRT}):
\begin{corollary} \label{FunCor} For $p \neq 2, 3$ prime, $s,\ell \in \mathbb{N}$ and $n \geq 3$, the following are equivalent:
\begin{enumerate}
\item $P^n(\ell)$ is $\mathbb{Z}/p^s$-hyperbolic.
\item $\pi_*(P^n(\ell))$ contains a class of order $p^s$.
\item $p^{s-1} | \ell$. \qed
\end{enumerate}
\end{corollary}
Theriault \cite{Theriault2} has shown that for $n \geq 4$ and $r \geq 6$, $\pi_*(P^n(2^r))$ contains no element of order $2^{r+2}$. The result of Corollary \ref{FunCor} therefore holds also when $p=2$ and $\ell$ is divisible by $2^6=64$.
Our second main result is a homological criterion for hyperbolicity:
\begin{theorem} \label{HPrelim} Let $Y$ be a simply connected $CW$-complex, let $p \neq 2$ be prime, and let $s \leq r \in \mathbb{N}$. If there exists a map $$\mu: P^{n+1}(p^r) \longrightarrow Y$$ such that the induced map $$(\Omega \mu)_* : H_*(\Omega P^{n+1}(p^r);\mathbb{Z}/p^s) \longrightarrow H_*(\Omega Y;\mathbb{Z}/p^s)$$ is an injection, then $Y$ is $p$-hyperbolic concentrated in exponents $s, s+1, \dots, r$. In particular if $s=r$ then $Y$ is $\mathbb{Z}/p^r$-hyperbolic. \end{theorem}
We will see (using Proposition \ref{injection2}) that the hypotheses of Theorem \ref{HPrelim} simplify in the case that $Y=\Sigma X$ is a suspension, as follows:
\begin{theorem} \label{HCriterion} Let $X$ be a connected $CW$-complex, let $p \neq 2$ be prime, and let $s \leq r \in \mathbb{N}$. If there exists a map $$\mu: P^{n+1}(p^r) \longrightarrow \Sigma X$$ such that $$\mu_* : \widetilde{H}_*(P^{n+1}(p^r);\mathbb{Z}/p^s) \longrightarrow \widetilde{H}_*(\Sigma X;\mathbb{Z}/p^s)$$ is an injection, then $\Sigma X$ is $p$-hyperbolic concentrated in exponents $s, s+1, \dots, r$. In particular if $s=r$ then $\Sigma X$ is $\mathbb{Z}/p^r$-hyperbolic. \end{theorem}
Theorem \ref{HCriterion} is substantially more elementary than existing criteria for $\mathbb{Z}/p^r$-hyperbolicity: the criterion given in \cite{HuangWu} requires knowledge of a homotopy decomposition of $\Omega Y$, while that of \cite{Me} is given in terms of $K$-theory, and only gives $p$-hyperbolicity. Here, by contrast, we only need ordinary homology.
Together, Theorems \ref{MooreHyp} and \ref{HCriterion} may be thought of as doing for Moore spaces what \cite{Me} did for wedges of spheres. The main difference between the homological results of that paper and this is that the Hurewicz map is enough to detect $p^r$-torsion in the homotopy groups of the Moore space $P^n(p^r)$. In contrast, one needs more sophisticated machinery to see $p^r$-torsion in a wedge of spheres; \cite{Me} used Adams' $e$-invariant. This meant that the theorems of that paper had to be stated in terms of $K$-theory, rather than ordinary homology, and that the spaces under consideration had to be finite complexes.
This document is organized as follows. The proof of Theorem \ref{MooreHyp} may be read independently of the proof of Theorems \ref{HPrelim} and \ref{HCriterion}, and vice versa. Section \ref{ApplicationSection} contains applications of our results. Section \ref{CommonSection} contains definitions needed throughout. Sections \ref{MooreDecompSection}, \ref{CurlySection} and \ref{Proof1Section} prove Theorem \ref{MooreHyp}, while Sections \ref{ModuleSection}, \ref{LieSection}, \ref{FoundationSection} and \ref{proof2Section} prove Theorem \ref{HPrelim}, and Section \ref{SuspensionSection} shows that Theorem \ref{HPrelim} implies Theorem \ref{HCriterion}.
I would like to thank my PhD supervisor, Stephen Theriault, for many helpful conversations and much encouragement. Changes to the proof of Theorem \ref{MooreHyp} which make the result work for powers of $2$ are due to him.
\section{Applications} \label{ApplicationSection}
\subsection{Spaces containing a Moore space as a retract}
Various spaces have been shown to contain wedges of Moore spaces and spheres as $p$-local retracts after looping. This section collects some examples of this form.
\begin{example} Let $M$ be an (oriented) $(n-1)$-connected $(2n+1)$-manifold for $n \geq 2$. By Poincar\'e duality, the homology of $M$ is determined entirely by $$H_{n}(M) \cong \mathbb{Z}^r \oplus \bigoplus_{i=1}^\ell \mathbb{Z}/p_i^{r_i}.$$ When $r \geq 1$, Basu \cite[Theorem 5.4]{Basu} gives a decomposition of $\Omega M$, which shows in particular that $\Omega M$ contains a retract $\Omega (\bigvee_{r-1}S^n \vee \bigvee_{r-1} S^{n+1} \vee \bigvee_{i=1}^\ell P^{n}(p_i^{r_i}))$. By Theorem \ref{MooreHyp} and the work of Huang-Wu \cite{HuangWu} and Zhu-Pan \cite{ZhuPan}, it follows that $M$ is $\mathbb{Z}/p^s$-hyperbolic whenever $p^{s-1}$ divides the order of the torsion part of $H_n(M)$. In fact, if $r \geq 2$ then $\Omega M$ contains $\Omega (S^n \vee S^m)$ as a retract, so is $\mathbb{Z}/p^s$-hyperbolic for all $p$ and $s$ by \cite{Me}. Conversely, if $M$ is not $\mathbb{Z}/p^s$ hyperbolic for any $p$ and $s$ (and is not the sphere $S^{2n+1}$) then we must have $H_{n}(M) \cong \mathbb{Z}$. An example of such a manifold is $S^{n-1} \times S^{n}$, whose homotopy groups satisfy $\pi_i(S^{n-1} \times S^{n}) \cong \pi_i(S^{n-1}) \times \pi_i(S^{n})$. Determining hyperbolicity for these examples is therefore as difficult as determining hyperbolicity of $S^n$. \end{example}
In order to use Basu's result, we require that there be a $\mathbb{Z}$-summand in $H_n(M)$. In contrast, our next example has $H_n(M)$ a torsion group.
\begin{example} Let $p$ be an odd prime, let $r \in \mathbb{N}$, and let $M$ be a $5$-dimensional spin manifold with $H_2(M; \mathbb{Z})$ isomorphic to a direct sum of copies of $\mathbb{Z}/p^r$. In \cite{Theriault} Theriault notes that his Theorem 1.3, together with a classification of simply connected 5-dimensional Poincar\'e duality complexes by St\"ocker \cite{Stocker}, gives a decomposition of $\Omega M$. This decomposition shows that $\Omega M$ contains $\Omega P^{3}(p^r)$ as a retract. In particular, by Theorem \ref{MooreHyp}, $M$ is $\mathbb{Z}/p^s$-hyperbolic for all $1 \leq s \leq r$. \end{example}
\subsection{Suspensions}
This section deduces some examples of Theorem \ref{HCriterion}. As a first example, note that the identity map on the Moore space $P^n(p^r)$ satisfies the hypotheses of that theorem, and so we recover the $s=r$ case of Theorem \ref{MooreHyp} using purely homological methods.
Let $h: \pi_n(Y) \longrightarrow H_n(Y ; \mathbb{Z})$ be the Hurewicz map, which sends a homotopy class $f: S^n \longrightarrow Y$ to the image $f_*(\xi_n)$ of a generator $\xi_n$ of $H_n(S^n ; \mathbb{Z})$ under the map induced on homology by $f$.
\begin{corollary}[of Theorem \ref{HCriterion}] \label{last} Let $p$ be an odd prime and let $s \in \mathbb{N}$. Suppose that $H_{n-1}(\Sigma X;\mathbb{Z})$ contains a $\mathbb{Z}/p^s$-summand, generated by a class $z \in \textrm{Im}(h)$. Let $\nu : S^{n-1} \longrightarrow \Sigma X$ be a map with $h(\nu) = z$, and let $r \in \mathbb{N}$ be such that the order of $\nu$ is equal to $p^r c$, for $c$ prime to $p$. Then $\Sigma X$ is $p$-hyperbolic concentrated in exponents $s, s+1, \dots , r$. \end{corollary}
Before proving this Corollary, we note that by the Hurewicz Theorem it immediately implies the following.
\begin{corollary} \label{easyH} Let $n$ be the least natural number for which $\widetilde{H}_n(\Sigma X; \mathbb{Z})$ is nontrivial. If $\widetilde{H}_n(\Sigma X; \mathbb{Z})$ contains a $\mathbb{Z}/p^s$-summand, for $p$ an odd prime and $s \in \mathbb{N}$, then $\Sigma X$ is $\mathbb{Z}/p^s$-hyperbolic. \qed \end{corollary}
\begin{proof}[Proof of Corollary \ref{last}] By replacing $\nu$ with $c \nu$ (and $z$ with $cz$) we may assume without loss of generality that $c=1$. Since $\nu$ has order $p^r$, it extends to a map $\mu : P^{n}(p^r) \longrightarrow \Sigma X$.
Let $x$ generate $H_n(P^n(p^r);\mathbb{Z}/p^s)$, and let $y$ generate $H_{n-1}(P^n(p^r);\mathbb{Z}/p^s)$. The Bockstein $\beta$ satisfies $\beta(x)=y$. We have $\mu_*(y) = h(\nu) = z$, and $\beta(\mu_*(x)) = \mu_*(\beta(x)) = \mu_*(y) = z$. This implies that $\mu_*(x)$ and $\mu_*(y)$ must both have order $p^s$, hence that $$ \mu_* : H_*(P^n(p^r);\mathbb{Z}/p^s) \longrightarrow H_*(\Sigma X; \mathbb{Z}/p^s)$$ is an injection. Thus, by Theorem \ref{HCriterion}, $\Sigma X$ is $p$-hyperbolic concentrated in exponents $s, s+1, \dots , r$, as required. \end{proof}
A first example of this sort highlights how much bigger the homotopy of an Eilenberg-MacLane space becomes upon suspending.
\begin{example} The least-dimensional homology of $\Sigma K(\mathbb{Z}/p^s,n)$ is isomorphic to $\mathbb{Z}/p^s$, so Corollary \ref{easyH} implies that $\Sigma K(\mathbb{Z}/p^s,n)$ is $\mathbb{Z}/p^s$-hyperbolic for $p$ odd. \end{example}
More generally, we have:
\begin{example} Let $G$ be a finite group. Atiyah \cite[Theorem 13.1]{Atiyah} has shown that the cohomology of $G$ (which is the cohomology of $K(G,1)$) is nonvanishing in infinitely many degrees. Since the cohomology of $G$ is annihilated by multiplication by $\lvert G \lvert$ \cite[Corollary II.5.4]{AdemMilgram} the lowest-dimensional nontrivial cohomology $H^n(K(G,1); \mathbb{Z})$ must contain a $\mathbb{Z}/p^s$-summand for some $p^s$ dividing $\lvert G \lvert$. By the universal coefficient theorem, the least nontrivial homology is $H_{n-1}(K(G,1); \mathbb{Z})$, which must also contain such a summand. By the suspension isomorphism and Corollary \ref{easyH}, $\Sigma K(G,1)$ is $\mathbb{Z}/p^s$-hyperbolic, provided that $p \neq 2$. In particular, this means that if $\lvert G \lvert$ is odd, then $\Sigma K(G,1)$ is $\mathbb{Z}/p^s$-hyperbolic for some $p^s$ dividing the order of $G$.
If the (co)homology of $G$ is known in least nontrivial dimension, then we can be more precise. Algebraic interpretations exist for the first few nontrivial homology groups: $H_1(K(G,1),\mathbb{Z})$ is the abelianization $G_{\textrm{ab}}$, and $H_2(K(G,1),\mathbb{Z})$ is known as the \textit{Schur multiplier}. Consider the Alternating groups $A_n$. These are simple, hence have trivial abelianization, and the Schur multiplier is $\mathbb{Z}/2$ unless $n=6,7$, in which case it is $\mathbb{Z}/6$ \cite{Schur}. In particular, Corollary \ref{easyH} implies that the suspended Eilenberg-MacLane spaces of $A_6$ and $A_7$ are $\mathbb{Z}/3$-hyperbolic. Another example is the Suzuki group $\textrm{Suz}$, which is one of the sporadic simple groups, and has Schur Multiplier $\mathbb{Z}/6$ \cite{Griess}, so again $\Sigma K(\textrm{Suz}, 1)$ is $\mathbb{Z}/3$-hyperbolic. \end{example}
\section{Common preamble} \label{CommonSection}
This section collects some foundational material which will be used in the proofs of both main results. First, we have the following well-known proposition, which we use to deduce Corollary \ref{FunCor} from Theorem \ref{MooreHyp}.
\begin{proposition} \label{CRT} Let $n \geq 3$. If $\ell \in \mathbb{N}$ has a prime power factorization $\ell = p_1^{r_1}p_2^{r_2} \dots p_m^{r_m}$ then $$P^n(\ell) \simeq P^n(p_1^{r_1}) \vee P^n(p_2^{r_2}) \vee \dots \vee P^n(p_m^{r_m}),$$ and furthermore $P^n(p^r)$ is $q$-locally contractible for any prime $q \neq p$. \end{proposition}
\begin{proof} Define a map $f: P^n(p_1^{r_1}) \vee P^n(p_2^{r_2}) \vee \dots \vee P^n(p_m^{r_m}) \longrightarrow P^n(\ell)$ which is given on the wedge summand $P^n(p_i^{r_i})$ as degree 1 on the top cell and degree $\frac{\ell}{p_i^{r_i}}$ on the bottom cell; that is, according to the following diagram of defining cofibrations. \begin{center}
\begin{tabular}{c}
\xymatrix{
S^{n-1} \ar^{p_i^{r_i}}[r] \ar@{=}[d] & S^{n-1} \ar^{\frac{\ell}{p_i^{r_i}}}[d] \ar[r] & P^n(p_i^{r_i}) \ar^{f \lvert_{P^n(p_i^{r_i})}}[d] \\
S^{n-1} \ar^{\ell}[r] & S^{n-1} \ar[r] & P^n(\ell)
}
\end{tabular}
\end{center}
By the Chinese Remainder Theorem, $f$ induces an isomorphism on integral homology. Thus, by Whitehead's theorem \cite{Whitehead}, $f$ is in fact a homotopy equivalence.
To see that $P^n(p^r)$ is contractible after localization at $q \neq p$, note that the homology with coefficients in the integers localized at $q$, $H_*(P^n(p^r);\mathbb{Z}_{(q)})$, is trivial, and thus by Whitehead's theorem, the inclusion of the basepoint is a homotopy equivalence. \end{proof}
\subsection{The Witt Formula and the Hilton-Milnor Theorem} \label{WittSubs}
We will be interested in counting the dimension of various `weighted components' of free Lie algebras. These Lie algebras will be ungraded in the proof of Theorem \ref{MooreHyp} and will be graded for the proof of Theorems \ref{HPrelim} and \ref{HCriterion}. In both cases, the quantities we wish to count are determined by the Witt formula, which we now define.
Let $\mu: \mathbb{N} \longrightarrow \{-1, 0, 1\}$ be the \textit{M\"obius inversion function}, defined by $$\mu(s) = \begin{cases}
1 & s=1 \\
0 & s>1 \textrm{ is not square free} \\
(-1)^{\ell} & s>1 \textrm{ is a product of $\ell$ distinct primes.}
\end{cases}$$
The \textit{Witt Formula} $W_n(k)$ is then defined by
$$W_n(k)=\frac{1}{k} \sum_{d \lvert k} \mu(d) n^{\frac{k}{d}}.$$
The Witt formula feeds into the proof of Theorem \ref{MooreHyp} via Theorem \ref{ungradedWitt}, and into the proof of Theorems \ref{HPrelim} and \ref{HCriterion} via Theorem \ref{gradedWittCount}. The asymptotics of the Witt formula are as follows:
\begin{lemma}{\cite[Introduction]{Bahturin}} \label{WittAsymptotics} The ratio
$$\frac{W_n(k)}{\frac{1}{k}n^k}$$
tends to 1 as $k$ tends to $\infty$. \qed
\end{lemma}
We now introduce the Hilton-Milnor Theorem. Let $L$ be the free (ungraded) Lie algebra over $\mathbb{Z}$ on basis elements $x_1 , \dots, x_n$. For an iterated bracket $B$ of the elements $x_i$, let $k_i(B) \in \mathbb{N} \cup \{ 0 \}$ be the number of instances of the generator $x_i$ occurring in $B$. The sum $k(B) = \sum_{i=1}^n k_i(B)$ is called the \emph{weight} of $B$, following Hilton \cite{Hilton}. By induction on $k$, Hilton defines a subset $\mathscr{L}_k$ of the brackets of weight $k$, which he calls the set of \emph{basic products} of weight $k$. The basic products of weight 1 are precisely the $x_i$. The union $\mathscr{L} = \bigcup_{k=1}^\infty \mathscr{L}_k$ is a free basis for $L$ (see for example \cite[Theorem 5.3]{SerreBook}, but note that what we call basic products, Serre calls a \emph{Hall basis}).
\begin{theorem}{\cite[Theorems 3.2, 3.3]{Hilton}} \label{ungradedWitt} Let $L$ be the free Lie algebra over $\mathbb{Z}$ on basis elements $x_1 , \dots, x_n$. Then the cardinality $|\mathscr{L}_k|$ of the set of basic products of weight $k$ is equal to $W_n(k)$. \qed \end{theorem}
We are now ready to state the Hilton-Milnor Theorem. Write $X^{\wedge k }$ for the smash product of $k$ copies of the space $X$.
\begin{theorem}{\cite{Hilton,Milnor}} \label{HiltonMilnor} Let $X_1, X_2, \dots , X_n$ be connected $CW$-complexes. There is a homotopy equivalence $$ \Omega \Sigma ( X_1 \vee \dots \vee X_n) \simeq \prod_{B \in \mathscr{L}} \Omega \Sigma ( X_1^{\wedge k_1(B)} \wedge \dots \wedge X_n^{\wedge k_n(B)} ), $$ where the right hand side is the weak infinite product. \qed \end{theorem}
\section{Decompositions of Moore spaces} \label{MooreDecompSection}
In this section we make the first step in the proof of Theorem \ref{MooreHyp}. Namely, we will see that it follows from work of Cohen, Moore, and Neisendorfer that a Moore space $P^n(p^r)$ with $p^r \neq 2$ contains $P^{n_1}(p^r) \vee P^{n_2}(p^r)$ as a retract after looping, and so it suffices to prove that $P^{n_1}(p^r) \vee P^{n_2}(p^r)$ is $\mathbb{Z}/p^s$-hyperbolic. We will also record Corollary \ref{binomialNeisendorfer}, which describes the behaviour of Moore spaces under iterated smash products.
When $p$ is odd, the loop-decomposition of $P^n(p^r)$ depends on the parity of $n$. We have the following three theorems, which give the three cases of the decomposition.
\begin{theorem}{\cite[Theorem 1.1]{CMNTorsion}} \label{CMNEven} Let $p$ be an odd prime, and let $n>0$. Then \[
\pushQED{\qed}
\Omega P^{2n+2}(p^r) \simeq S^{2n+1}\{ p^r \} \times \Omega \bigvee_{m=0}^\infty P^{4n+2mn+3}(p^r).\qedhere
\popQED
\] \end{theorem}
\begin{theorem}{\cite{CMNExponents}} \label{CMNOdd} Let $p$ be an odd prime, and let $n>0$. Then there is a space $T^{2n+1}\{ p^r \}$ so that $$\Omega P^{2n+1}(p^r) \simeq T^{2n+1}\{ p^r \} \times \Omega \Sigma \bigvee_{\alpha} P^{n_\alpha}(p^r),$$ where $\bigvee_{\alpha} P^{n_\alpha}(p^r)$ is an infinite bouquet of mod-$p^r$ Moore spaces, and each $n_\alpha$ satisfies $n_\alpha \geq 4n-1$. \qed \end{theorem}
\begin{lemma}{\cite[Lemma 2.6]{Cohenp2}} \label{2decomp} Let $n \geq 3$ and $r \geq 2$. Then there exist spaces $T^n\{2^r\}$ such that $$ \Omega P^{n}(2^r) \simeq T^n\{2^r\} \times \Omega \bigvee_{\alpha} P^{m_\alpha}(2^r),$$ where $\bigvee_{\alpha} P^{m_\alpha}(2^r)$ is an infinite bouquet of mod-$2^r$ Moore spaces, and each $m_\alpha$ satisfies $m_\alpha \geq n$. \qed \end{lemma}
Theorems \ref{CMNEven} and \ref{CMNOdd}, together with Lemma \ref{2decomp} immediately imply the following corollary.
\begin{corollary} \label{hasAWedge} Let $p$ be prime and let $r \in \mathbb{N}$. Suppose that $p^r \neq 2$, and let $n \geq 3$. Then $\Omega P^n(p^r)$ has $\Omega ( P^{n_1}(p^r) \vee P^{n_2}(p^r))$ as a retract for some $n_1, n_2 \geq n$. \qed \end{corollary}
Smash powers of Moore spaces are well-understood, by means of the following Lemma.
\begin{lemma}{\cite{NeisendorferMemoir}} \label{Smashing} Let $p$ be prime, and let $r \in \mathbb{N}$, with $p^r \neq 2$. For $n,m \geq 2$, \[
\pushQED{\qed}
P^n(p^r) \wedge P^m(p^r) \simeq P^{m+n}(p^r) \vee P^{m+n-1}(p^r).\qedhere
\popQED
\]
\end{lemma}
For a space $X$, write $X^{\vee i}$ for the wedge sum of $i$ copies of $X$. Applying Lemma \ref{Smashing} repeatedly gives the following binomial-type formula.
\begin{corollary} \label{binomialNeisendorfer} Let $p$ be prime, and let $r \in \mathbb{N}$, with $p^r \neq 2$. For $n,m \geq 2$, and $k_1,k_2 \in \mathbb{N}$. Letting $k=k_1+k_2$, we have \[
\pushQED{\qed}
P^{n}(p^r)^{\wedge k_1} \wedge P^{m}(p^r)^{\wedge k_2} \simeq \bigvee_{i=0}^{k-1}(P^{k_1 n +k_2 m -i}(p^r))^{\vee \binom{k-1}{i}} .\qedhere
\popQED
\] \end{corollary}
\section{Classes in the homotopy groups of $P^n(p^r)$} \label{CurlySection}
In this section, we identify some stable classes in the homotopy groups of $P^n(p^r)$. The identification of these classes is the way in which we go beyond Huang and Wu's work. We will transfer known classes from the stable homotopy groups of spheres (Lemma \ref{sphereClasses}) into the stable homotopy groups of Moore Spaces by means of the stable homotopy exact sequence of the cofibration defining the Moore space. To show that the resulting classes have the correct order, we need assurances about the maximum order of the torsion in the stable homotopy groups of Moore spaces, and these assurances are provided by Corollary \ref{stableMooreAnnihilation}.
Cohen, Moore, and Neisendorfer have shown that the homotopy groups of $P^n(p^r)$ contain classes of order $p^{r+1}$ \cite{CMNTorsion}. However, these classes are all outside the stable range; the stable homotopy groups of $P^n(p^r)$ were already known to be annihilated by multiplication by $p^r$. The proof of this fact is due to Barratt.
\begin{lemma}{\cite{Barratt}} Let $A$ be $(n-1)$-connected, and let $p$ be a prime. Suppose that we have $p^s \textrm{id}_{\Sigma A} \simeq *$ in the group $[\Sigma A, \Sigma A]$, for some $s \in \mathbb{N}$. Then $p^s \pi_{n+j}(\Sigma A) = 0$ for $j \leq (p-1)n$. \label{stableAnnihilation} \qed \end{lemma}
\begin{corollary} Let $p$ be prime, and let $s \in \mathbb{N}$ such that $p^s \neq 2$. Then we have $p^s \pi_{n+j}(P^n(p^s)) = 0$ for $j \leq (p-1)(n-2) - 2$. \label{stableMooreAnnihilation} \end{corollary}
\begin{proof} By definition, $P^n(p^s) \simeq \Sigma P^{n-1}(p^s)$, and $P^{n-1}(p^s)$ is $(n-3)$-connected. By Lemma \ref{stableAnnihilation} the result therefore follows from the fact that the identity map on $P^n(p^s)$ has order $p^s$ \cite[Proposition 6.1.7]{NeisendorferBook}. \end{proof}
We continue in a similar vein. In general, the degree $\ell$ map on $S^n$ does not induce multiplication by $\ell$ on homotopy groups. However, it follows from the Hilton-Milnor Theorem (Theorem \ref{HiltonMilnor}) that it must do so in the stable range, as in the next lemma.
\begin{lemma} \label{stableHM} The degree $\ell$ map $S^n \xrightarrow{\ell} S^n$ induces multiplication by $\ell$ on $\pi_j(S^n)$ for $j \leq 2n-2$. \end{lemma}
\begin{proof} Write $n=m+1$ and $j=i+1$. By the adjoint isomorphism, it suffices to show that $\Omega \ell$ induces multiplication by $\ell$ on $\pi_{i}(\Omega S^{m+1})$ for $i < 2m$. The map $\ell$ is the composition $$ S^{m+1} \xrightarrow{c} \bigvee_{i=1}^\ell S^{m+1} \xrightarrow{\nabla} S^{m+1}$$ of the $\ell$-fold suspension comultiplication $c$ on $S^{m+1}$ with the fold map $\nabla$. Let $\mathscr{L}$ be the free Lie algebra on $\ell$ generators, as in Subsection \ref{WittSubs}. The Hilton-Milnor Theorem (Theorem \ref{HiltonMilnor}) gives a decomposition $$\Omega \bigvee_{i=1}^\ell S^{m+1} \simeq \Omega \prod_{B \in \mathscr{L}} S^{km+1},$$ where $k$ is the weight of $B \in \mathscr{L}$, so in particular is implicitly a function of $B$.
Let $f \in \pi_i(\Omega S^{m+1})$. Applying the above decomposition to $(\Omega \ell)_* (f) = (\Omega \ell) \circ f$ gives factorizations $\varphi$ and $\theta$ as in the following diagram.
\begin{center}
\begin{tabular}{c}
\xymatrix{
\Omega S^{m+1} \ar^{\Omega c}[r] & \Omega \bigvee_{i=1}^\ell S^{m+1} \ar^{\Omega \nabla}[r] & \Omega S^{m+1} \\
S^i \ar^{\varphi}[r] \ar^{f}[u] & \Omega \prod_{B \in \mathscr{L}} S^{km+1}. \ar^{\simeq}[u] \ar^{\theta}[ur] &
}
\end{tabular}
\end{center}
We must show that $\theta \circ \varphi \simeq \ell f$. Since $i < 2m$, cellular approximation tells us that $\varphi$ factors through the sub-product $\Omega \prod_{i=1}^\ell S^{m+1}$ consisting of those terms where $k=1$. Hilton \cite{Hilton} tells us that the restriction of the Hilton-Milnor map to these summands is given by the product under the loop multiplication of the looped wedge factor inclusions $\Omega S^{m+1} \longrightarrow \Omega \bigvee_{i=1}^\ell S^{m+1}$. Thus, the restriction of $\theta$ to these summands is the $\ell$-fold loop multiplication map $$m : \Omega \prod_{i=1}^\ell S^{m+1} \longrightarrow \Omega S^{m+1}.$$ Furthermore, this restriction of the Hilton-Milnor map is a left homotopy inverse to the looped inclusion $\Omega \iota: \Omega \bigvee_{i=1}^\ell S^{m+1} \longrightarrow \Omega \prod_{i=1}^\ell S^{m+1}$ of the wedge into the product, so $\theta \circ \varphi$ is homotopic to $\theta \circ \Omega \iota \circ \Omega c \circ f$.
To finish, we note that by the axiomatic definition of a comultiplication \cite{Arkowitz} we have that $\Omega \iota \circ \Omega c = \Delta$, the diagonal map into the $\ell$-fold product, and the composition $m \circ \Delta$ is by definition the map inducing multiplication by $\ell$ in the group structure on $[S^i, \Omega S^{m+1}] = \pi_i(\Omega S^{m+1})$ coming from the fact that $\Omega S^{m+1}$ is an $H$-group. But this group structure coincides with that of the homotopy group \cite{Arkowitz}, and so we are done. \end{proof}
Let $\pi^S_j$ denote the $j$-th stable homotopy group of spheres. Work of Adams on the $J$-homomorphism implies that any cyclic group of prime power order occurs as a summand in some $\pi^S_j$:
\begin{lemma}{\cite[Lemma 3.4]{Me}} \label{sphereClasses} For any prime $p$ and any $s \in \mathbb{N}$, there exists $j$ such that $\mathbb{Z}/p^s$ is a direct summand in $\pi^{S}_j$. That is, for a fixed choice of such a $j$, $\mathbb{Z}/p^s$ is a direct summand in $\pi_{n+j}(S^n)$ for all $n \geq j+2$.\qed \end{lemma}
These summands can be transplanted to $P^n(p^r)$ as in the next two corollaries.
\begin{corollary} \label{mooreClasses} Let $p$ be prime, and let $r \geq s \in \mathbb{N}$. If $p^s \neq 2$, then there exists $j$ such that $\mathbb{Z}/p^s$ is a direct summand in $\pi_{n+j}(P^n(p^r))$ for all $n > j+3$. \end{corollary}
Zhu and Pan \cite{ZhuPan} have already proven the case $s=1$, and Huang and Wu \cite{HuangWu} have already proven the case $s=r$.
\begin{proof} The cofibration $P^n(p^r) \longrightarrow S^n \xrightarrow{p^r} S^n$ gives a truncated long exact sequence on homotopy groups \cite{HiltonBook}: $$ \pi_{2n-3}(P^n(p^r)) \longrightarrow \pi_{2n-3}(S^n) \longrightarrow \pi_{2n-3}(S^n) \longrightarrow \pi_{2n-4}(P^n(p^r)) \longrightarrow \dots $$ $$ \dots \longrightarrow \pi_{n}(P^n(p^r)) \longrightarrow \pi_{n}(S^n) \longrightarrow \pi_{n}(S^n) \longrightarrow \pi_{n-1}(P^n(p^r)) \longrightarrow 0.$$
By Lemma \ref{sphereClasses}, there exists $j$ such that $\mathbb{Z}/p^s$ is a direct summand in $\pi_{n+j}(S^n)$ for all $n \geq j+2$. Fix $n \geq j+4$, and let $f : S^{n+j} \longrightarrow S^n$ generate a $\mathbb{Z}/p^s$-summand. By Lemma \ref{stableHM}, since we are in the stable range, the composite $p^s \circ f$ is homotopic to $p^s f$, and by assumption $f$ has order $p^s$. Thus, since $n \geq j+3$, the exact sequence applies, and taking $r=s$ we obtain a lift $\widetilde{f} \in \pi_{n+j}(P^n(p^s))$ making the following diagram commute.
\begin{center}
\begin{tabular}{c}
\xymatrix{
P^n(p^s) \ar[r] & S^{n} \ar[r]^{p^s} & S^{n}\\
& S^{n + j}. \ar[u]_{f} \ar[ur]_{\simeq *} \ar@{.>}[ul]^{\widetilde{f}}
}
\end{tabular}
\end{center}
We also have, for each $r \geq s$, a diagram \begin{center}
\begin{tabular}{c}
\xymatrix{
S^{n-1} \ar[r]^{p^r} & S^{n-1} \\
S^{n-1} \ar@{=}[u] \ar[r]^{p^s} & S^{n-1}. \ar[u]_{p^{r-s}}
}
\end{tabular}
\end{center}
Extending the rows of this diagram to cofibre sequences and combining with the previous one gives a diagram \begin{center}
\begin{tabular}{c}
\xymatrix{
S^{n-1} \ar[r] & P^n(p^r) \ar^{\underline{\rho}}[r] & S^{n} \ar[r]^{p^r} & S^{n} \\
S^{n-1} \ar[u]_{p^{r-s}} \ar[r] & P^n(p^s) \ar^{\underline{\rho}}[r] \ar[u]_{\varphi} & S^{n} \ar@{=}[u] \ar[r]^{p^s} & S^{n} \ar[u]_{p^{r-s}} \\
& & S^{n+j}, \ar[u]_{f} \ar[ur]_{\simeq *} \ar@{.>}[ul]^{\tilde{f}} &
}
\end{tabular}
\end{center}
We have that $\underline{\rho}_* (\varphi \circ \widetilde{f}) = f$, so the image of $\underline{\rho}_* : \pi_{n+j}(P^n(p^r)) \longrightarrow \pi_{n+j}(S^n)$ contains $f$. Since $f$ generates a $\mathbb{Z}/p^s$-summand, this gives a surjection $\pi_{n+j}(P^n(p^r)) \longrightarrow \mathbb{Z}/p^s$, and it suffices to argue that this surjection is split. From the diagram, it further suffices to do so in the case $r=s$.
By Corollary \ref{stableMooreAnnihilation}, since $n \geq j+4$ we have $p^s \pi_{n+j}(P^n(p^s)) = 0$. This means that the above surjection $\pi_{n+j}(P^n(p^s)) \longrightarrow \mathbb{Z}/p^s$ is a map of $\mathbb{Z}/p^s$-modules with free codomain, so is split, as required. \end{proof}
\begin{corollary} \label{mooreClasses2} Let $r \in \mathbb{N}$. For $n \geq 32$, the group $\pi_{n+28}(P^n(2^r))$ is isomorphic to $\mathbb{Z}/2$. \end{corollary}
This result has already been shown by Zhu and Pan \cite{ZhuPan}, but it is easy to give the more explicit argument below.
\begin{proof} We will take a similar approach to Corollary \ref{mooreClasses}. The argument differs slightly because Corollary \ref{stableMooreAnnihilation} fails when $p^s=2$; we compensate for this using knowledge of the $2$-components of the stable homotopy groups of spheres. Specifically, from \cite[Theorem 1.1.1 and Table 1.1.8]{MahowaldTangora}, we know that the $2$-localization of $\pi^S_{28}$ is isomorphic to $\mathbb{Z}/2$, while the $2$-localization of $\pi^S_{29}$ is trivial.
Let $n \geq 32$. As in the proof of Corollary \ref{mooreClasses}, consider the cofibration $P^n(2^r) \longrightarrow S^n \xrightarrow{2^r} S^n$. The truncated long exact sequence on homotopy groups contains the segment $$\pi_{n+29}(S^n) \longrightarrow \pi_{n+28}(P^n(2^r)) \longrightarrow \pi_{n+28}(S^n).$$ It follows that the $2$-localization of $\pi_{n+28}(P^n(2^r))$ is isomorphic to $\mathbb{Z}/2$. \end{proof}
\section{Proof of Theorem \ref{MooreHyp}} \label{Proof1Section}
In this section, we will prove Theorem \ref{MooreHyp}. In Section \ref{MooreDecompSection}, we reduced the problem to showing $\mathbb{Z}/p^s$-hyperbolicity of the wedge $P^n(p^r) \vee P^m(p^r)$. By the Hilton-Milnor Theorem (Theorem \ref{HiltonMilnor}) and Corollary \ref{binomialNeisendorfer}, we will see that each of the stable classes identified in Section \ref{CurlySection} will give exponentially many summands in the homotopy groups of $P^n(p^r) \vee P^m(p^r)$, which will suffice.
\begin{proof}[Proof of Theorem \ref{MooreHyp}] By Corollary \ref{hasAWedge}, it suffices to prove that if $n,m \geq 2$ then $\Omega(P^{n+1}(p^r) \vee P^{m+1}(p^r))$ is $\mathbb{Z}/p^s$-hyperbolic for all $s \leq r$. Let $\mathscr{L}$ be the free ungraded Lie algebra over $\mathbb{Z}$ on two generators. The Hilton-Milnor theorem (Theorem \ref{HiltonMilnor}) gives $$\Omega (P^{n+1}(p^r) \vee P^{m+1}(p^r)) \simeq \Omega \Sigma (P^{n}(p^r) \vee P^{m}(p^r)) \simeq \prod_{B \in \mathscr{L}} \Omega \Sigma P^{n}(p^r)^{\wedge k_1} \wedge P^{m}(p^r)^{\wedge k_2},$$ where we have written $k_i = k_i(B)$, leaving the fact that $k_i$ is a function of $B$ implicit. Applying Lemma \ref{binomialNeisendorfer} factor-wise, this last is homotopy equivalent to $$\Omega \prod_{B \in \mathscr{L}} \Sigma \bigvee_{i=0}^{k-1}(P^{k_1 n +k_2 m -i}(p^r))^{\vee \binom{k-1}{i}} \simeq \Omega \prod_{B \in \mathscr{L}} \bigvee_{i=0}^{k-1}(P^{k_1 n +k_2 m +1 -i}(p^r))^{\vee \binom{k-1}{i}},$$ where $k=k_1+k_2$ is also implicitly a function of $B$.
By Corollaries \ref{mooreClasses} and \ref{mooreClasses2}, let $j$ be such that $\pi_{N+j}(P^N (p^r))$ contains a $\mathbb{Z}/p^s$-summand for all $N > j+ 3$. For each $B \in \mathscr{L}$, the associated factor of the above decomposition contains $2^{k-1}$ Moore spaces. Supposing without loss of generality that $n \leq m$, the dimensions of these Moore spaces are at least $k(n-1)+2$. Thus, for $k > \frac{j+1}{n-1}$, the homotopy groups of each factor $$\bigvee_{i=0}^{k-1}(P^{k_1 n +k_2 m +1 -i}(p^r))^{\vee \binom{k-1}{i}}$$ contain $2^{k-1}$ summands isomorphic to $\mathbb{Z}/p^s$ in dimensions at most $km+1+j$.
The number of factors for which the weight of $B$ is $k$ is equal to $W_2(k)$ (Theorem \ref{ungradedWitt}), so we may conclude that $$\bigoplus_{i=1}^{km+1+j} \pi_i(P^{n+1}(p^r) \vee P^{m+1}(p^r))$$ contains at least $2^{k-1} W_2(k)$ summands isomorphic to $\mathbb{Z}/p^s$. The sequence $2^{k-1} W_2(k)$ certainly grows exponentially in $k$ (in fact, by Lemma \ref{WittAsymptotics}, it grows like $\frac{1}{2k} 4^k$) and this completes the proof. \end{proof}
\section{Modules over $\mathbb{Z}/p^s$} \label{ModuleSection}
The purpose of this section is to prove various elementary facts about modules over $\mathbb{Z}/p^s$ which we will use later. These facts are mostly intuitively clear, so we recommend that the reader skip this section on first reading, referring back only as necessary.
\subsection{Injections}
The main point of this subsection is to develop the `linear algebra' to prove Lemma \ref{summand}, which says that injections from free $\mathbb{Z}/p^s$-modules are split, and that therefore the `dimension' of the codomain must be at least the `dimension' of the domain.
Let $p$ be prime and let $s \in \mathbb{N}$. Let $M$ be a finitely generated module over $\mathbb{Z}/p^s$. By the structure theorem for finitely generated $\mathbb{Z}$-modules (for example as in \cite[Theorem 7.5]{Lang}) $M$ decomposes as a direct sum $$M \cong \bigoplus_{i=1}^n \mathbb{Z}/p^{s_i},$$ where each $s_i$ satisfies $1 \leq s_i \leq s$. Further, if we order the summands so that $s_{i+1} \geq s_i$, then the sequence $(s_i \ \lvert \ 1 \leq i \leq n)$ is uniquely determined. In particular, if we fix $t \in \mathbb{N}$, then the number of values of $i$ for which $s_i = t$ is uniquely determined. This number is then precisely the \emph{$\mathbb{Z}/p^t$-dimension} $\dim_{\mathbb{Z}/p^t}(M)$ of Definition \ref{defdef}. We will often use without comment the fact that a $\mathbb{Z}/p^s$-module is equivalently a $\mathbb{Z}$-module $M$ satisfying $p^sM = 0$.
We will wish to mimic the approach of ordinary linear algebra as far as possible. We will wish to be able to `change basis', and to do so we need a notion of basis, which must generalize the idea of a free basis in that our elements may have variable order.
\begin{definition} Let $M$ be a $\mathbb{Z}/p^s$-module. A \emph{basis} of $M$ is a list $$((e_i,s_i) \in M \times \mathbb{N} \ \lvert \ 1 \leq i \leq n ),$$ such that the following conditions are satisfied:
\begin{itemize}
\item Each $x \in M$ is expressible as $x = \sum_{i=1}^n \lambda_i e_i$ for $\lambda_i \in \mathbb{Z}/p^s$ (\emph{spanning}).
\item $\sum_{i=1}^n \lambda_i e_i = 0$ if and only if $p^{s_i} \lvert \lambda_i$ for each $i$ (\emph{linear independence}).
\end{itemize}
\end{definition}
\begin{lemma} Any finitely generated $\mathbb{Z}/p^s$-module has a basis. Conversely, if $((e_i,s_i) \ \lvert \ 1 \leq i \leq n )$ is a basis of $M$, then the map $$\bigoplus_{i=1}^n \mathbb{Z}/p^{s_i} \longrightarrow M$$ defined by sending the generator of the $i$-th summand to $e_i$ is an isomorphism.
\end{lemma}
\begin{proof} To see that $M$ has a basis write $M \cong \bigoplus_{i=1}^n \mathbb{Z}/p^{t_i}$, taking $e_i$ to be a generator of the $i$-th summand, and taking $s_i = t_i$. It follows immediately that this is a basis.
Conversely, let $\varphi: \bigoplus_{i=1}^n \mathbb{Z}/p^{s_i} \longrightarrow M$ be as in the theorem statement. By linear independence of the basis, $p^{s_i} e_i = 0$ for each $i$, so $\varphi$ is well-defined. Surjectivity of $\varphi$ follows immediately from the spanning condition, while injectivity follows immediately from linear independence. Thus, $\varphi$ is an isomorphism, as required. \end{proof}
\begin{lemma} \label{basisManeuvers} Let $((e_i,s_i) \ \lvert \ 1 \leq i \leq n )$ be a basis of $M$.
\begin{itemize}
\item If $\lambda$ is a unit in $\mathbb{Z}/p^s$, then replacing the basis element $(e_k,s_k)$ with $(\lambda e_k,s_k)$ again yields a basis.
\item If $j \neq k$ and $s_j \leq s_k$, then replacing the basis element $(e_k,s_k)$ with $(e_k + \mu e_j, s_k)$ for any $\mu \in \mathbb{Z}/p^s$ again yields a basis.
\end{itemize}
\end{lemma}
\begin{proof} We will show only that the basis obtained by the second replacement is linearly independent; the other parts are similar.
Write $(e_i', s_i)$ for the new basis, and suppose that $\sum_{i=1}^n \lambda_i e_i' = 0$. We must show that $p^{s_i}$ divides $\lambda_i$ for each $i$. Substituting in, we have $(\sum_{i \neq j,k} \lambda_i e_i) + \lambda_j e_j + \lambda_k (e_k + \mu e_j) = 0$. Since the original basis was linearly independent, we have that $p^{s_i} \lvert \lambda_i$ for $i \neq j$. In particular, $p^{s_k} \lvert \lambda_k$. We also have $p^{s_j} \lvert (\lambda_j + \mu \lambda_k)$. Since $s_j \leq s_k$ we have $p^{s_j} \lvert \lambda_k$, so $p^{s_j} \lvert \lambda_j$. Thus, $p^{s_i} \lvert \lambda_i$ for all $i$, and thus the $(e_i',s_i)$ form a basis, as required. \end{proof}
It is always true that a surjection onto a free module splits; over $\mathbb{Z}/p^s$, it is additionally true that an injection from a free module splits.
\begin{lemma} \label{summand} Let $M$ and $N$ be finitely-generated $\mathbb{Z}/p^s$-modules, with $M$ free. The image of any injection of $\mathbb{Z}/p^s$-modules $\varphi : M \longrightarrow N$ is a summand, and $\dim_{\mathbb{Z}/p^s}(N) \geq \dim_{\mathbb{Z}/p^s}(M)$. \end{lemma}
\begin{proof} Let $(x_1,t_1), \dots, (x_m,t_m)$ be a basis of $M$, and let $$(e_1,s_1), \dots , (e_n,s_n), (e'_1,s_1'), \dots (e'_{n'},s_{n'}')$$ be a basis of $N$, such that each $s_i=s$ and each $s'_i<s$.
Thus we have $f(x_1) = \sum_{i=1}^n \lambda_{i} e_i + \sum_{i=1}^{n'} \lambda_{i}' e_i'$ for some coefficients $\lambda_i$ and $\lambda_i'$. In particular, since $f(x_1)$ has order $p^s$, there must be some $\lambda_i$ which is not divisible by $p$.By repeated use of Lemma \ref{basisManeuvers} we may therefore change basis in $M$ by replacing $e_i$ by $\sum_{i=1}^n \lambda_{i} e_i + \sum_{i=1}^{n'} \lambda_{i}' e_i'$. After this change we have $f(x_1) = e_i$, and by renumbering we may assume that $i=1$.
We repeat this procedure inductively: at the $j$-th stage we have $f(x_i) = e_i$ for all $i < j$ and we wish to arrange that $f(x_j) = e_j$. We have that $f(x_j) = \sum_{i=1}^n \lambda_{i} e_i + \sum_{i=1}^{n'} \lambda_{i}' e_i'$ for some coefficients $\lambda_i$ and $\lambda_i'$, and the set $f(x_1), \dots , f(x_{j-1})$ spans the submodule $\langle e_1, \dots , e_{j-1} \rangle \subset M$. By changing basis according to Lemma \ref{basisManeuvers}, we may arrange that $\lambda_i = 0$ for $i<j$, and this does not change the fact that $f(x_i) = e_i$ for these values of $i$. Again, $f(x_j)$ has order $p^s$, so there must be $i \geq j$ with $\lambda_i$ not divisible by $p$, and by renumbering we may assume that $i=j$. By changing basis we may arrange that $f(x_j) = e_j$. This completes the inductive step, hence the proof that $\textrm{Im}(f)$ is a summand. Since after this procedure we have $f(x_i) = e_i$ for $i = 1, \dots , m$ we must have $n \geq m$, which is the other part of the theorem statement. \end{proof}
We also have the following technical lemma, which will be used in the proof of Proposition \ref{injection2}.
\begin{lemma} \label{factorTensor} Let $X$, $A$, $B$, and $Y$ be $\mathbb{Z}/p^s$-modules, with $X$ free and $p^{s-1} B = 0$. Let $f: X \longrightarrow A \oplus B$ and $g: A \oplus B \longrightarrow Y$ be homomorphisms. Let $i_A$ be the inclusion of $A$ in $A \oplus B$, and let $\pi_A$ be the projection $A \oplus B \longrightarrow A$. If $g \circ f$ is injective, then the composite $g \circ i_A \circ \pi_A \circ f$ is also injective.
\end{lemma}
\begin{proof} Since $X$ is free, a map defined on $X$ is an injection if and only if its restriction to $p^{s-1}X$ is an injection. It therefore suffices to show that if $g \circ i_A \circ \pi_A \circ f (p^{s-1}x) = 0$ then $p^{s-1} x = 0$.
Thus, suppose that $g \circ i_A \circ \pi_A \circ f (p^{s-1}x) = 0$. Write $f(x) = a+b \in A \oplus B$, for $a \in A$ and $b \in B$. Then $f(p^{s-1}x) = p^{s-1}a$, since $p^{s-1}B=0$. In particular, $f(p^{s-1}x) = i_A \circ \pi_A \circ f(p^{s-1}x)$. Thus, $g \circ f (p^{s-1}x) = 0$, and $g \circ f$ is an injection, so $p^{s-1}x=0$, as required. \end{proof}
\subsection{Surjections}
The main result of this subsection is Lemma \ref{surjectOnHype}, which is the basic algebraic scaffolding for the proof of Theorem \ref{HPrelim}.
\begin{lemma} \label{sCase} Let $\varphi: M \longrightarrow N$ be a surjection of $\mathbb{Z}/p^s$-modules. Then $$ \dim_{\mathbb{Z}/p^s}(M) \geq \dim_{\mathbb{Z}/p^s}(N).$$ \end{lemma}
\begin{proof} Write $N = F \oplus C$, where $F$ is free over $\mathbb{Z}/p^s$, and the complementary module $C$ satisfies $p^{s-1}C=0$. Let $\pi: N \longrightarrow F$ be the projection. The map $\pi \circ \varphi$ is a composite of surjections, hence a surjection, so is split by freeness of $F$. Thus, we have an isomorphism $M \cong F \oplus D$ for some complementary module $D$, so $$\dim_{\mathbb{Z}/p^s}(M) \geq \dim_{\mathbb{Z}/p^s}(F) = \dim_{\mathbb{Z}/p^s}(N),$$ as required. \end{proof}
\begin{lemma} \label{downDog} Let $A$ be a submodule of a $\mathbb{Z}/p^s$-module $N$, such that $A+pN=N$. Then $A=N$.
\end{lemma}
\begin{proof} Because $N$ is a $\mathbb{Z}/p^s$-module, we have $p^s N = 0$, so certainly $A \supset p^s N$. We will now show that if $A \supset p^k N$ then $A \supset p^{k-1} N$. By induction, this implies that $A \supset p^0 N = N$, which suffices.
Assume that $A \supset p^k N$, and let $z \in N$. We have by assumption that $z = x + py$ for $x \in A$ and $y \in N$. Thus, $p^{k-1} z = p^{k-1} x + p^k y$. But now, $p^k y \in p^k N$, which by induction is a subset of $A$, so $p^{k-1} z \in A$, and since $z$ is an arbitrary element of $N$, this implies that $p^{k-1} N \subset A$. This completes the inductive step, hence the proof. \end{proof}
\begin{lemma} \label{realSurjection} Let $M,M',N$ be $\mathbb{Z}$-modules. Let $p$ be prime and let $s \leq r \in \mathbb{N}$. Suppose that $p^r M = 0$, so $M$ may be regarded as a module over $\mathbb{Z}/p^r$, and that $p^s N = 0$. Let $\varphi: M \longrightarrow N$ be a surjection which admits a factorization \begin{center}
\begin{tabular}{c}
\xymatrix{
M' \ar^{\widetilde{\varphi}}[dr] & \\
M \ar^{\varphi}[r] \ar^{\iota}[u] & N.
}
\end{tabular}
\end{center}
Then $\sum_{t=s}^r \dim_{\mathbb{Z}/p^t}(M') \geq \dim_{\mathbb{Z}/p^s}(N).$
\end{lemma}
\begin{proof} We will first argue that we may assume $p^r M'=0$ without loss of generality. Write $M' = A \oplus B$, where $p^r A = 0$, and $B$ is a direct sum of copies of $\mathbb{Z}$, $\mathbb{Z}/q^t$ for various $q \neq p$ and $t \in \mathbb{N}$, and $\mathbb{Z}/p^t$ for $t>r$. This gives a decomposition $M = \iota^{-1}(A) \oplus \iota^{-1}(B)$. The restriction of $\iota$ to $\iota^{-1}(B)$ must have image contained in $pB$, so the same restriction of $\widetilde{\varphi} \circ \iota$ has image contained in $pN$. Furthermore, since $\widetilde{\varphi}$ is a surjection, we have that $\textrm{Im}(\widetilde{\varphi} \circ \iota \lvert_{\iota^{-1}(A)}) + \textrm{Im}(\widetilde{\varphi} \circ \iota \lvert_{\iota^{-1}(B)}) = N$, so in particular $\textrm{Im}(\widetilde{\varphi} \circ \iota \lvert_{\iota^{-1}(A)}) + pN = N$. By Lemma \ref{downDog} we then have $\textrm{Im}(\widetilde{\varphi} \circ \iota \lvert_{\iota^{-1}(A)}) = N$. We may therefore restrict $M'$ to $A$ and $M$ to $\iota^{-1}(A)$ in the diagram without affecting the hypotheses. In particular, since $p^{r}A=0$ it suffices to prove the lemma in the case that $p^r M'=0$.
We now tensor the diagram with $\mathbb{Z}/p^s$; since $p^s N=0$, we have $N \otimes \mathbb{Z}/p^s \cong N$. Since $p^r M' = 0$, we have $\dim_{\mathbb{Z}/p^s}(M' \otimes \mathbb{Z}/p^s) = \sum_{t=s}^r \dim_{\mathbb{Z}/p^t}(M')$. By Lemma \ref{sCase}, since $\widetilde{\varphi} \otimes \mathbb{Z}/p^s$ is a surjection we have $\dim_{\mathbb{Z}/p^s}(M' \otimes \mathbb{Z}/p^s) \geq \dim_{\mathbb{Z}/p^s}(N \otimes \mathbb{Z}/p^s)$, which completes the proof. \end{proof}
By applying Lemma \ref{realSurjection} in each degree we immediately obtain the following.
\begin{corollary}[The `Sandwich' Lemma] \label{surjectOnHype} Let $M,M',N$ be graded $\mathbb{Z}$-modules. Let $p$ be prime and let $r\geq s \in \mathbb{N}$. Suppose that $p^r M = 0$ and that $p^s N = 0$. Let $\varphi: M \longrightarrow N$ be a surjection which admits a factorization \begin{center}
\begin{tabular}{c}
\xymatrix{
M' \ar[dr] & \\
M \ar^{\varphi}[r] \ar[u] & N.
}
\end{tabular}
\end{center}
If $N$ is $\mathbb{Z}/p^s$-hyperbolic then $M'$ is $p$-hyperbolic concentrated in exponents $s, s+1, \dots, r$. \qed \end{corollary}
\begin{lemma} \label{reduceToField} Let $\varphi: M \longrightarrow N$ be a map of $\mathbb{Z}/p^s$-modules, with $N$ free. Then $\dim_{\mathbb{Z}/p^s}(\textrm{Im}(\varphi)) = \dim_{\mathbb{Z}/p}(\textrm{Im}(\varphi \otimes \mathbb{Z}/p))$. \end{lemma}
\begin{proof} Let $(e_1,s_1), \dots , (e_m,s_m), (e_1',s_1'), \dots , (e_{m'}',s_{m'}')$ be a basis of $M$, where $s_i = s$ and $s_i' < s$. Let $S$ be a maximal subset of the $e_i$ such that the restriction of $\varphi$ to the submodule of $M$ generated by $S$ is an injection. Denote this submodule by $\langle S \rangle$. By renumbering we may assume that $S=\{e_1, \dots e_k \}$ for some $k \leq n$. We clearly have $\textrm{Im}(\varphi \lvert_{\langle S \rangle}) \subset \textrm{Im}(\varphi)$, and we will now show that $\textrm{Im}(\varphi) \subset \textrm{Im}(\varphi \lvert_{\langle S \rangle}) + pN$.
Since $N$ is assumed free, and the elements $e_i'$ have order $p^{s_i}$ for $s_i < s$, we must have $\varphi(e_i') \in pN$. Now consider $e_j$, for $k+1 \leq j \leq m$. By construction of $S$, the restriction of $\varphi$ to $\langle S \cup \{ e_j \} \rangle$ is not injective, so there exist $\lambda_1 , \dots \lambda_k, \lambda \in \mathbb{Z}/p^s$ with $\lambda \neq 0$ such that $\varphi(\sum_{i=1}^k \lambda_i e_i + \lambda e_j) = 0$. This implies that $\lambda \varphi(e_j) \in \textrm{Im}( \varphi \lvert_{\langle S \rangle})$. Thus, $p^t \varphi(e_j) \in \textrm{Im}(\varphi \lvert_{\langle S \rangle})$ for some $t < s$. By Lemma \ref{summand} we may write $N = \textrm{Im}(\varphi \lvert_{\langle S \rangle}) \oplus C$ for some complementary module $C$, and under this correspondence we have $\varphi (e_j) = (\beta, \gamma)$ for $\gamma \in C$ and $\beta \in \textrm{Im}(\varphi \lvert_{\langle S \rangle})$. Since $p^t \varphi(e_j) \in \textrm{Im}(\varphi \lvert_{\langle S \rangle})$, we have $p^t \gamma = 0$, so by freeness of $N$, $t<s$ implies that $\gamma \in pN$, so $\varphi(e_j) \in \textrm{Im}(\varphi\lvert_{\langle S \rangle}) + pN$. We have now shown that all elements of the basis of $M$ are carried under $\varphi$ to $\textrm{Im}(\varphi\lvert_{\langle S \rangle}) + pN$, so $\textrm{Im}(\varphi) \subset \textrm{Im}(\varphi\lvert_{\langle S \rangle}) + pN$, as claimed.
Now, $\varphi\lvert_{\langle S \rangle}$ is split by Lemma \ref{summand}, so $\dim_{\mathbb{Z}/p^s}(\textrm{Im}(\varphi\lvert_{\langle S \rangle})) = k$. Furthermore, by taking the inclusion on each summand there is a surjection $\textrm{Im}(\varphi\lvert_{\langle S \rangle}) \oplus pN \longrightarrow \textrm{Im}(\varphi\lvert_{\langle S \rangle}) + pN \subset N$, and $pN$ is annihilated by multiplication by $p^{s-1}$, so by Lemma \ref{sCase} $\dim_{\mathbb{Z}/p^s}(\textrm{Im}(\varphi\lvert_{\langle S \rangle}) + pN) \leq k$. Since $\dim_{\mathbb{Z}/p^s}(\textrm{Im}(\varphi\lvert_{\langle S \rangle}) + pN) \geq \dim_{\mathbb{Z}/p^s}(\textrm{Im}(\varphi\lvert_{\langle S \rangle}))$, this implies that the former is equal to $k$. Thus, since $$\textrm{Im}(\varphi\lvert_{\langle S \rangle}) \subset \textrm{Im}(\varphi) \subset \textrm{Im}(\varphi\lvert_{\langle S \rangle}) + pN$$ applying Lemma \ref{summand} to the inclusions gives $$k= \dim_{\mathbb{Z}/p^s}(\textrm{Im}(\varphi\lvert_{\langle S \rangle})) \leq \dim_{\mathbb{Z}/p^s}(\textrm{Im}(\varphi)) \leq \dim_{\mathbb{Z}/p^s}(\textrm{Im}(\varphi\lvert_{\langle S \rangle})+pN) = k,$$ so $\dim_{\mathbb{Z}/p^s}(\textrm{Im}(\varphi))=k$.
To finish the proof we must show that $\dim_{\mathbb{Z}/p}(\textrm{Im}(\varphi \otimes \mathbb{Z}/p)) = k$. Since the images of $\varphi$ and $\varphi \lvert_{\langle S \rangle}$ differ only by at most $pN$, we have $\textrm{Im}(\varphi \otimes \mathbb{Z}/p) = \textrm{Im}(\varphi \lvert_{\langle S \rangle} \otimes \mathbb{Z}/p)$. Since $\varphi \lvert_{\langle S \rangle}$ is split injective, $\varphi \lvert_{\langle S \rangle} \otimes \mathbb{Z}/p$ is injective, so $\dim_{\mathbb{Z}/p}(\varphi \lvert_{\langle S \rangle} \otimes \mathbb{Z}/p) = k$, which completes the proof. \end{proof}
\subsection{\textrm{Tor} and the Universal Coefficient Theorem}
The purpose of this section is to prove that for $t<s$ a map inducing an injection on homology with $\mathbb{Z}/p^s$-coefficients also induces an injection on homology with $\mathbb{Z}/p^t$-coefficients (Lemma \ref{toTheFields}) provided that the domain is free. This follows straightforwardly from the Universal Coefficient Theorem for homology, where we regard $\mathbb{Z}/p^t$ as a module over $\mathbb{Z}/p^s$. The inclusion of the bottom cell of a Moore space provides an easy counterexample to the converse; the algebraic point being that the converse of Lemma \ref{oneway} is false.
\begin{lemma} \label{TorAnnihilation} For any finitely generated $\mathbb{Z}/p^s$-modules $M$, $N$ we have $$p^{s-1}\textrm{Tor}_{\mathbb{Z}/p^s}(M,N)=0,$$ and furthermore if $M$ or $N$ is free then $\textrm{Tor}_{\mathbb{Z}/p^s}(M,N)=0$. \end{lemma}
\begin{proof} For any ring $R$ and $R$-module $M$ we have $\textrm{Tor}_R(R,M)=0$, since $R$ is free as an $R$-module. If $1 \leq t<s$, then a free resolution of $\mathbb{Z}/p^t$ over $\mathbb{Z}/p^s$ is given by $$0 \longrightarrow \mathbb{Z}/p^s \xrightarrow{\cdot p^t} \mathbb{Z}/p^s \longrightarrow 0,$$ so, for any $\mathbb{Z}/p^s$-module $M$, $\textrm{Tor}_{\mathbb{Z}/p^s}(\mathbb{Z}/p^t, M) = \textrm{Ker}(M \xrightarrow{\cdot p^s} M)$, which is annihilated by multiplication by $p^t$, hence in particular is annihilated by multiplication by $p^{s-1}$. Since any $\mathbb{Z}/p^s$-module decomposes as a direct sum of modules isomorphic to $\mathbb{Z}/p^t$ for $1 \leq t \leq s$, both parts of the Lemma now follow by additivity of $\textrm{Tor}$. \end{proof}
\begin{lemma} \label{oneway} Let $\varphi: M \longrightarrow N$ be a map of $\mathbb{Z}/p^s$-modules, with $M$ free. Let $t<s$. If $\varphi$ is injective then $\varphi \otimes \mathbb{Z}/p^t : M \otimes \mathbb{Z}/p^t \longrightarrow N \otimes \mathbb{Z}/p^t$ is injective. \end{lemma}
\begin{proof} Note that $M \otimes \mathbb{Z}/p^t$ is a free $\mathbb{Z}/p^t$-module. Suppose that $\varphi \otimes \mathbb{Z}/p^t$ is not injective. Then there exists $x\in M$ which is not divisible by $p^t$ such that $\varphi(x)$ is divisible by $p^t$. By freeness of $M$, $p^{s-t} x$ is not divisible by $p^s$, hence is nonzero, but $\varphi(p^{s-t} x) = p^{s-t} \varphi (x)$ is divisible by $p^s$, hence is zero. That is, $\varphi$ is not injective. \end{proof}
\begin{lemma} \label{toTheFields} Let $t<s \in \mathbb{N}$. Let $f: X \longrightarrow Y$ be a map of spaces, and suppose that $H_*(X ; \mathbb{Z}/p^s)$ is a free $\mathbb{Z}/p^s$-module. If $$f_* : H_*(X ; \mathbb{Z}/p^s) \longrightarrow H_*(Y ; \mathbb{Z}/p^s)$$ is injective then $$f_* : H_*(X ; \mathbb{Z}/p^t) \longrightarrow H_*(Y ; \mathbb{Z}/p^t)$$ is injective. \end{lemma}
\begin{proof} Write $f_*^t$ for the induced map on homology with $\mathbb{Z}/p^t$-coefficients, and likewise $f_*^s$. Applying the universal coefficient theorem for the module $\mathbb{Z}/p^t$ over the ring $\mathbb{Z}/p^s$ we get a map of short exact sequences \begin{center}
\begin{tabular}{c}
\xymatrix{
0 \ar[r] & H_n(X; \mathbb{Z}/p^s) \otimes \mathbb{Z}/p^t \ar[r] \ar^{f^s_* \otimes \mathbb{Z}/p^t}[d] & H_n(X; \mathbb{Z}/p^t) \ar[r] \ar^{f^t_*}[d] & 0 \ar[d] \ar[r] & 0 \\
0 \ar[r] & H_n(Y; \mathbb{Z}/p^s) \otimes \mathbb{Z}/p^t \ar[r] & H_n(Y; \mathbb{Z}/p^t) \ar[r] & \textrm{Tor}(H_{n-1}(Y;\mathbb{Z}/p^s),\mathbb{Z}/p^t) \ar[r] & 0.
}
\end{tabular}
\end{center}
The $\textrm{Tor}$ term in the top row vanishes by the freeness hypothesis on $H_*(X; \mathbb{Z}/p^s)$. Since the first map in each exact sequence is an injection, $f^t_*$ is injective if and only if $f^s_* \otimes \mathbb{Z}/p^t$ is injective. By Lemma \ref{oneway}, if $f^s_*$ is injective, then $f^s_* \otimes \mathbb{Z}/p^t$ is injective, so $f^t_*$ is injective, as required. \end{proof}
\section{Free differential Lie algebras} \label{LieSection}
In this section we will show that the module of boundaries $BL(x,dx)$ in the free differential Lie algebra $L(x,dx)$ over $\mathbb{Z}/p^r$ is $\mathbb{Z}/p^r$-hyperbolic. In the situation of Theorem \ref{HPrelim} we will obtain a factorization of the tensor map $$BL(x,dx) \longrightarrow \pi_*(\Omega Y) \longrightarrow BL(x,dx) \otimes \mathbb{Z}/p^s,$$ which will imply by Corollary \ref{surjectOnHype} (The `Sandwich' Lemma) that $\Omega Y$ must be $p$-hyperbolic concentrated in exponents $s, s+1, \dots , r$. The desired $\mathbb{Z}/p^r$-hyperbolicity of $BL(x,dx)$ will follow from Cohen, Moore, and Neisendorfer's description of the homology of $L(x,dx)$, which is Proposition \ref{acyclicHomology}.
Throughout this section we work over a ground ring $R = \mathbb{Z}/p^r$ for $p \neq 2$. The next definitions are as in \cite{CMNTorsion}.
\begin{definition} \label{LieDef} A \textit{graded Lie algebra} is a graded $\mathbb{Z}/p^r$-module $L$, together with a $\mathbb{Z}/p^r$-bilinear pairing $$[ \phantom{x},\phantom{x} ]: L_n \times L_m \longrightarrow L_{n+m},$$ called a \textit{Lie bracket} which satisfies the relations of \begin{itemize}
\item (antisymmetry): $[x,y]=-(-1)^{\textrm{deg}(x)\textrm{deg}(y)}[y,x]$ for all $x$ and $y$ in $L$.
\item (the Jacobi identity): $[x,[y,z]] = [[x,y],z]+(-1)^{\textrm{deg}(x)\textrm{deg}(y)}[y,[x,z]]$ for all $x$, $y$, and $z$ in $L$.
\item $[x,[x,x]]=0$ for all $x$ of odd degree.
\end{itemize} \end{definition}
Let $V$ be a graded $\mathbb{Z}/p^r$-module. Denote by $L(V)$ the \textit{free Lie algebra} on $V$. There is a linear map $j: V \longrightarrow L(V)$ and $L(V)$ is characterized up to canonical isomorphism as follows. For any map $f: V \longrightarrow L$ where $L$ is a graded Lie algebra, there is a unique map $g:L(V) \longrightarrow L$ so that $g \circ j=f$. The Lie algebra $L(V)$ may be constructed as follows.
Let $L'(V)$ be the free nonassociative graded algebra on $V$, where we think of the operation as a bracket. Precisely, let $B_k$ be the set of bracketings of a string of $k$ symbols. Concatenation of bracketings gives an operation $B_{k_1} \times B_{k_2} \longrightarrow B_{k_1 + k_2}$, which makes $B = \bigcup_{i=1}^\infty B_i$ into a magma. As a module, $$L'(V)=\bigoplus_{k=1}^\infty (\bigoplus_{b \in B_k} V^{\otimes k}),$$ where we think of each copy of $V^{\otimes k}$ as being bracketed according to $b$. The bracket operation on $L'(V)$ is obtained by extending the operation on $B$ bilinearly.
The free Lie algebra $L(V)$ is obtained as the quotient of $L'(V)$ by the relations of Definition \ref{LieDef}, and automatically has the desired universal property. Denote by $\theta$ the quotient map $L'(V) \longrightarrow L(V)$. It follows that for $s<r$, we have $L(V \otimes \mathbb{Z}/p^s)=L(V) \otimes \mathbb{Z}/p^s$.
Note also that any map from $V$ into a graded $\mathbb{Z}/p^r$-module $A$ with a bilinear operation (that is to say, a nonassociative $\mathbb{Z}/p^r$-algebra) extends uniquely to a map of graded nonassociative algebras $L'(V) \longrightarrow A$. The map $\theta$ is a map of nonassociative algebras, hence is uniquely determined by its effect on $V$, and we may therefore call it the \emph{natural quotient}.
\begin{definition} A \textit{differential Lie algebra} is a graded Lie algebra together with an $\mathbb{Z}/p^r$-linear map $d:L \longrightarrow L$ of degree $-1$, which \begin{itemize}
\item is a differential: $d^2(x)=0$ for all $x$ in $L$.
\item is a derivation: $d[x,y]=[dx,y]+(-1)^{\textrm{deg}(x)}[x,dy]$ for all $x$ and $y$ in $L$.
\end{itemize}
\end{definition}
If $V$ carries a differential $d$, then we may define a differential on $L'(V)$ which is the unique derivation extending $d$. This differential can be seen to satisfy the relations of Definition \ref{LieDef}, and therefore descends to give a differential on $L(V)$, which makes $L(V)$ into a differential Lie algebra.
When $p=3$, Samelson products in $\pi_*(\Omega X; \mathbb{Z}/3^r)$ fail to satisfy the Jacobi identity, so $L'(V)$ will also serve as a version of $L(V)$ which does not satisfy the Jacobi identity. For $p \neq 3$, $L'(V)$ may be replaced with $L(V)$ everywhere in this paper, which slightly simplifies things \cite[Remark 6.3]{CMNTorsion}.
Write $L(V)^k$ for the weight-$k$ component of $L(V)$, that is, the submodule generated by brackets of length $k$ in the elements of $V$. It follows from our construction of the free Lie algebra $L(V)$ that $L(V) \cong \bigoplus_{k=1}^\infty L(V)^k$, so weight gives a second grading on $L(V)$, and we shall write $\textrm{wt}(x)=k$ whenever $x \in L(V)^k$. We will use subscripts (as in $L(V)_i$) for ordinary grading, and superscripts (as in $L(V)^k$) for weight. The dimension of the weighted components is given by the Witt formula, which we defined in Section \ref{CommonSection}.
\begin{theorem}{\cite[Theorems 3.2, 3.3]{Hilton}} \label{gradedWittCount} Let $V$ be a free graded $\mathbb{Z}$- or $\mathbb{Z}/p^s$-module of total dimension $n$. Then the total dimension of $L(V)^k$ is $W_n(k)$. \qed \end{theorem}
\subsection{Homology and boundaries}
Let $x$ be an even-dimensional class in a graded Lie algebra $L$ over $\mathbb{Z}/p^r$ for $p \neq 2$. Let $$\tau_k(x) = \textrm{ad}^{p^k-1}(x)(dx),$$ $$\textrm{so } \deg(\tau_k(x))=p^k\deg(x)-1$$ and let $$\sigma_k(x) = \frac{1}{2} \sum_{j=1}^{p^k-1} \frac{1}{p}{p^k \choose j}[\textrm{ad}^{j-1}(x)(dx), \textrm{ad}^{p^k-1-j}(x)(dx)],$$ $$\textrm{so } \deg(\sigma_k(x))=p^k\deg(x)-2,$$ where we understand the coefficients $\frac{1}{p}{p^k \choose j}$ to be computed in the integers and then reduced mod $p$.
\begin{proposition}{\cite[Proposition 4.9]{CMNTorsion}} \label{acyclicHomology} Let $V$ be an acyclic differential $\mathbb{Z}/p$-vector space. Write $L(V) \cong HL(V) \oplus K$, for an acyclic module $K$. If $K$ has an acyclic basis, that is, a basis $$\{x_\alpha, y_\alpha, z_\beta, w_\beta\},$$ where $\alpha$ and $\beta$ range over index sets $\mathscr{I}$ and $\mathscr{J}$ respectively, and we have $$d(x_\alpha) = y_\alpha, \textrm{ deg}(x_\alpha) \textrm{ even,}$$ $$d(z_\beta) = w_\beta, \textrm{ deg}(z_\beta) \textrm{ odd,}$$ then $HL(V)$ has a basis $$\{\tau_k(x_\alpha), \sigma_k(x_\alpha)\}_{\alpha \in \mathscr{I}, k \geq 1}.$$ \end{proposition}
\begin{remark} \label{basisChoice} An acyclic basis for $K$ may always be chosen, by the following inductive procedure. Write $K_i$ for the $i$-th graded component of $K$. Then $d: K_{i+1} \longrightarrow K_i$, and since $K$ is acyclic we have $\textrm{Im}(d)=\textrm{Ker}(d)$ in each $K_i$. Assume that we have a basis of $\textrm{Ker}(d) \subset K_i$. Because $\textrm{Ker}(d) = \textrm{Im}(d)$, $d$ induces an isomorphism $\faktor{K_{i+1}}{\textrm{Ker}(d)} \longrightarrow \textrm{Im}(d)$. Choose representatives of this basis in $K_{i+1}$, and choose a basis of $\textrm{Ker}(d) \subset K_{i+1}$. Combining these two sets gives a basis of $K_{i+1}$, and the subset which forms a basis of $\textrm{Ker}(d)$ is precisely what we need to continue the induction. The induction can be started using the fact that $K_{-1}=0$. \end{remark}
Recall that we write $L(V)^k$ for the weight-$k$ component of $L(V)$, that is, the submodule generated by brackets of length $k$ in the elements of $V$, and recall also that weight defines a grading. Note that the differential $d$ preserves weight. The operations $\tau_k$ and $\sigma_k$ satisfy $$\textrm{wt}(\tau_k(x))=p^k\textrm{wt}(x),$$ $$\textrm{wt}(\sigma_k(x))=p^k\textrm{wt}(x).$$
We will use weight to produce a modified dimension function which makes precise the idea that `most' of the decomposition of $L(V)$ in Proposition \ref{acyclicHomology} consists of the summand $K$; the summand $HL(V)$ is comparatively small.
\begin{definition} Let $M$ be a $\mathbb{Z}/p^r$-module, together with a grading $\textrm{wt}$, which we think of as a weight, such that each weight-component $M^i$ is free and finitely generated. Define $\dim^k(M) \in \mathbb{R}$ by setting $$\dim^k(M) = \sum_{i=1}^k \frac{\dim(M^i)}{i}.$$ \end{definition}
It follows immediately from the definition that $$\dim^k(A \oplus B) = \dim^k(A)+\dim^k(B).$$
We will be concerned with evaluating the functions $\dim^k$ on submodules of the free Lie algebra $L(V)$. We write $BM$ for the module $\textrm{Im}(d)$ of boundaries in a differential module $(M,d)$.
\begin{lemma} \label{weightInequalities} Let $V$ be an acyclic differential $\mathbb{Z}/p$-vector space. For all $k \in \mathbb{N}$ we have: \begin{itemize}
\item $\dim^k(HL(V)) < \frac{1}{p} \dim^k(L(V))$, and
\item $\dim^k(BL(V)) > \frac{p-1}{2p} \dim^k(L(V))$.
\end{itemize} \end{lemma}
\begin{proof} Decompose $L(V) \cong HL(V) \oplus K$ as in Proposition \ref{acyclicHomology}, and choose a basis $\{x_\alpha, y_\alpha, z_\beta, w_\beta\}$ of $K$ as in Remark \ref{basisChoice}, where $\alpha$ and $\beta$ run over indexing sets $\mathscr{I}$ and $\mathscr{J}$ respectively. The differential preserves weight, so by choosing such a basis in each weighted component separately, we may assume that the basis vectors are homogenous in weight. Let $S_k$ be the set of those $\alpha \in \mathscr{I}$ with $\textrm{wt}(x_\alpha) \leq k$. Proposition \ref{acyclicHomology} gives that $$\dim^k(HL(V)) < \sum_{\alpha \in S_k} \sum_{j=1}^\infty \frac{1}{ \textrm{wt}(\tau_j(x_\alpha))} + \frac{1}{\textrm{wt}(\sigma_j(x_\alpha))} = \sum_{\alpha \in S_k} \sum_{j=1}^\infty \frac{1}{p^j \textrm{wt}(x_\alpha)} + \frac{1}{p^j \textrm{wt}(x_\alpha)}$$ $$= \sum_{\alpha \in S_k} \frac{2}{\textrm{wt}(x_\alpha)} \sum_{j=1}^{\infty} \frac{1}{p^j} = \frac{1}{p-1} \sum_{\alpha \in S_k} \frac{2}{\textrm{wt}(x_\alpha)}.$$ On the other hand, the contribution of the $x_\alpha$ and $y_\alpha$ to the dimension of $K$ gives that $$\dim^k(K) \geq \sum_{\alpha \in S_k} \frac{2}{\textrm{wt}(x_\alpha)},$$ so $$\dim^k(K) > (p-1) \dim^k(HL(V)).$$ Since $L(V) \cong HL(V) \oplus K$, we have that $\dim^k(L(V)) = \dim^k(K) + \dim^k(HL(V))$, so $$\dim^k(L(V)) > p \dim^k(HL(V)),$$ which proves the first inequality. This also implies that $\dim^k(K) > \frac{p-1}{p} \dim^k(L(V))$, and since $K$ is acyclic, we must have $$\dim^k(BL(V)) \geq \frac{1}{2} \dim^k(K).$$ Combining these proves the second inequality and completes the proof. \end{proof}
All we will require for our application is the case when $V$ is the free $\mathbb{Z}/p^r$-module on two generators $x$ and $y$ satisfying $d(x)=y$. In this case we will write $L(x,dx)=L(V)$ and $L'(x,dx)=L'(V)$. Note that $L(x,dx) \otimes \mathbb{Z}/p^s$ is the free Lie algebra on $V \otimes \mathbb{Z}/p^s$, which is the free differential module over $\mathbb{Z}/p^s$ on $x$ and $y$ with $dx=y$.
\begin{lemma} \label{ManyBoundaries} Let $V$ be a graded acyclic $\mathbb{Z}/p^r$-module, free and finitely generated in each dimension, of total dimension at least 2. Then the module of boundaries $BL(V)$ is $\mathbb{Z}/p^r$-hyperbolic. In particular, the module of boundaries $BL(x,dx)$ in the free differential Lie algebra $L(x,dx)$ is $\mathbb{Z}/p^r$-hyperbolic. \end{lemma}
\begin{proof} Since it has the correct universal property, $L(V) \otimes \mathbb{Z}/p$ is the free Lie algebra over $\mathbb{Z}/p$ on $V \otimes \mathbb{Z}/p$. Thus, by Lemma \ref{reduceToField} applied to the differential $d$ it suffices to prove the $r=1$ case, for which we can use Proposition \ref{acyclicHomology}, in the guise of Lemma \ref{weightInequalities}.
By Lemma \ref{weightInequalities}, we know that $$\dim^k(BL(V)) > \frac{p-1}{2p} \dim^k(L(V)).$$ Thus, $$\sum_{i=1}^k \dim(BL(V)^i) \geq \sum_{i=1}^k \frac{\dim(BL(V)^i)}{i} > \frac{p-1}{2p} \sum_{i=1}^k \frac{\dim(L(V)^i)}{i} \geq \frac{p-1}{2p} \sum_{i=1}^k \frac{\dim(L(V)^i)}{k}.$$
Let $n$ be the maximum $i$ for which $V_i \neq 0$. The leftmost term is equal to $\dim(\bigoplus_{i=1}^k BL(V)^i)$, and $BL(V)^i \subset L(V)^i \subset L(V)_{ni}$, so we have $$\dim(\bigoplus_{j=1}^{nk} BL(V)_j) > \frac{p-1}{2pk} \sum_{i=1}^k \dim(L(V)^i) \geq \frac{p-1}{2pk} \dim(L(V)^k) = \frac{p-1}{2pk} W_\ell(k),$$ by Theorem \ref{gradedWittCount}, where we let $\ell = \dim(V)$, so $$\dim(\bigoplus_{i=1}^{k} BL(V)_i) > \frac{p-1}{2p \lfloor \frac{k}{n} \rfloor} W_\ell(\lfloor \frac{k}{n} \rfloor) \sim \frac{p-1}{2p \lfloor \frac{k}{n} \rfloor^2} \ell^{\lfloor \frac{k}{n} \rfloor}$$ by Lemma \ref{WittAsymptotics}. Now, $\ell$ is assumed greater than $1$, so
$$\frac{p-1}{2p \lfloor \frac{k}{n} \rfloor^2} \ell^{\lfloor \frac{k}{n} \rfloor} \geq \frac{p-1}{2p (\frac{k}{n})^2} \ell^{\frac{k}{n} -1},$$ so for any $\varepsilon > 0$, once $k$ is large enough we have $\dim(\bigoplus_{i=1}^{k} BL(V)_i) > (\ell^{\frac{1}{n}} - \varepsilon)^{k}$. That is, $\dim(\bigoplus_{i=1}^{k} BL(V)_i)$ grows faster than an exponential in any base smaller than $\ell^{\frac{1}{n}}$. In particular, if $\dim(V) = \ell \geq 2$, then $BL(V)$ is $\mathbb{Z}/p$-hyperbolic, as required. \end{proof}
Since $\theta : L'(V) \longrightarrow L(V)$ is surjective and commutes with $d$, we immediately obtain the following corollary.
\begin{corollary} \label{mbCor} The submodule $\textrm{Im}(\theta \circ d)$ in the free differential Lie algebra $L(x,dx)$ is $\mathbb{Z}/p^r$-hyperbolic. \qed \end{corollary}
\section{Loop-homology of Moore spaces} \label{FoundationSection}
In this section we will study the question `what part of $H_*(\Omega P^{n+1}(p^r); \mathbb{Z}/p^r)$ can be shown to consist of Hurewicz images?' The answer is `the module of boundaries in a differential sub-Lie algebra isomorphic to $L(x,dx)$'. In Section \ref{LieSection} we have seen that such a module is $\mathbb{Z}/p^r$-hyperbolic. The hypotheses of Theorem \ref{HPrelim} are really conditions under which the image of this module under the map $(\Omega \mu)_*$ remains $\mathbb{Z}/p^s$-hyperbolic, and we thus obtain a $\mathbb{Z}/p^s$-hyperbolic submodule of the image of the Hurewicz map.
We follow the notation from Neisendorfer's book \cite{NeisendorferBook}. Let $p$ be a prime and let $s \leq r \in \mathbb{N}$. For a space $Y$, recall that the \textit{homotopy groups of $Y$ with coefficients in $\mathbb{Z}/p^s$}, denoted $\pi_n(Y;\mathbb{Z}/p^s)$ are the based homotopy sets $[P^n(p^s),Y]$, which are groups for $n \geq 3$. There are a number of useful operations relating the integral and mod-$p^s$ homotopy groups, which we introduce next.
Let $\underline{\beta}^s :S^{n-1} \longrightarrow P^n(p^s)$ be the inclusion from the cofibration sequence of Definition \ref{MooreSpace}. This defines a map of degree $-1$ $$\beta^s: \pi_n(Y;\mathbb{Z}/p^s) \longrightarrow \pi_{n-1}(Y)$$ $$f \longmapsto f \circ \underline{\beta}^s.$$
Similarly, let $\underline{\rho}^s :P^n(p^s) \longrightarrow S^n$ be the pinch map, which is obtained by extending the cofibration sequence of Definition \ref{MooreSpace} to the right. Again, this defines a map of degree $0$ $$\rho^s: \pi_n(Y) \longrightarrow \pi_n(Y;\mathbb{Z}/p^s)$$ $$f \longmapsto f \circ \underline{\rho}^s.$$
Lastly, let $\underline{\textrm{red}}^{r,s}: P^n(p^s) \longrightarrow P^n(p^r)$ be the map defined by the diagram of cofibrations \begin{center}
\begin{tabular}{c}
\xymatrix{
S^{n-1} \ar^{p^s}[r] \ar@{=}[d] & S^{n-1} \ar[r] \ar^{p^{r-s}}[d] & P^{n}(p^s) \ar^{\underline{\textrm{red}}^{r,s}}[d] \\
S^{n-1} \ar^{p^r}[r] & S^{n-1} \ar[r] & P^{n}(p^r),
}
\end{tabular}
\end{center} and let $$\textrm{red}^{r,s}: \pi_n(Y;\mathbb{Z}/p^r) \longrightarrow \pi_n(Y;\mathbb{Z}/p^s)$$ $$f \longmapsto f \circ \underline{\textrm{red}}^{r,s}.$$
It follows from the definitions that $\beta^s$, $\rho^s$ and $\textrm{red}^{r,s}$ are all natural in $Y$.
We will now use these operations to produce elements $u$ and $v$ of $\pi_*(\Omega P^{n+1}(p^r); \mathbb{Z}/p^s)$. The Hurewicz images of $v$ and $u$ will play the roles of the elements $x$ and $dx$ of Section \ref{LieSection}. Although these elements are easily described in terms of things we already have, we will give them new names for clarity.
Let $$v': P^n(p^s) \longrightarrow P^n(p^r)$$ be equal to $\underline{\textrm{red}}^{r,s}$.
Let $$u': P^{n-1}(p^s) \longrightarrow P^n(p^r)$$ be the composite $$P^{n-1}(p^s) \xrightarrow{\underline{\rho}^s} S^{n-1} \xrightarrow{\underline{\beta}^r} P^{n}(p^r).$$
Recall that for any space $X$ there is a natural map $\eta: X \longrightarrow \Omega \Sigma X$, which is the unit of the adjunction $\Sigma \dashv \Omega$ and sends $x \in X$ to the loop $\gamma_x = (t \longmapsto \langle t,x \rangle)$ on $\Sigma X$. Let $v = \eta \circ v' : P^n(p^s) \longrightarrow \Omega P^{n+1}(p^r)$, and let $u = \eta \circ u': P^{n-1}(p^s) \longrightarrow \Omega P^{n+1}(p^r)$.
Now let $G$ be an $H$-group, and suppose that the prime $p$ is odd. As in the integral setting, the homotopy groups with coefficients $\pi_*(G; \mathbb{Z}/p^s)$ carry a \textit{Samelson product}; a bilinear operation which resembles a Lie bracket \cite{CMNTorsion}. In particular, loop spaces are $H$-groups, so we have Samelson products in $\pi_*(\Omega X;\mathbb{Z}/p^s)$ for any $X$.
\begin{lemma} \label{notQuiteGLA} Let $p$ be an odd prime. The map $$\pi_*(\Omega X;\mathbb{Z}/p^s) \xrightarrow{\beta^s} \pi_*(\Omega X) \xrightarrow{\rho^s} \pi_*(\Omega X;\mathbb{Z}/p^s)$$ is a differential (that is, $(\rho^s \circ \beta^s)^2=0$) of degree $-1$, which satisfies the Leibniz identity relative to Samelson products. \end{lemma}
\begin{proof} By \cite[Section 7]{CMNTorsion}, we have the Leibniz identity. To see that it is a differential, note that $\beta^s \circ \rho^s = 0$, so $(\rho^s \circ \beta^s)^2= \rho^s \circ (\beta^s \circ \rho^s) \circ \beta^s = 0$. \end{proof}
By construction of $u$ and $v$ we have $(\rho^s \circ \beta^s)(v)=p^{r-s}u$ in $\pi_*(\Omega P^{n+1}(p^r); \mathbb{Z}/p^s)$. Let $L'(x,dx)$ and $L(x,dx)$ be as in Section \ref{LieSection}, where we let $\deg(x) = n$ and $\deg(y)=n-1$. Let $\langle x, dx \rangle$ be the free graded $\mathbb{Z}/p^r$-module of dimension 2 on basis $\{x, dx\}$, so that $L(x,dx) = L( \langle x,dx \rangle)$, and with this notation note that $L'(x,dx) \otimes \mathbb{Z}/p^s = L'(\langle x,dx \rangle \otimes \mathbb{Z}/p^s)$, the analogous construction over $\mathbb{Z}/p^s$.
We define a map of $\mathbb{Z}/p^s$-modules $\phi_\pi^{r,s} : \langle x, dx \rangle \otimes \mathbb{Z}/p^s \longrightarrow \pi_*(\Omega P^{n+1}(p^r); \mathbb{Z}/p^s)$ by sending $x \longmapsto v$ and $dx \longmapsto u$. Samelson products in $\pi_*(\Omega P^{n+1}(p^r); \mathbb{Z}/p^s)$ are bilinear, so by the universal property of $L'(x,dx) \otimes \mathbb{Z}/p^s$, $\phi_\pi^{r,s}$ extends to a map $$\Phi_\pi^{r,s} : L'(x,dx) \otimes \mathbb{Z}/p^s \longrightarrow \pi_*(\Omega P^{n+1}(p^r); \mathbb{Z}/p^s)$$ of graded (nonassociative) $\mathbb{Z}/p^s$-algebras.
The following lemma relates $\Phi_\pi^{r,s}$ to $\Phi_\pi^{r,r}$.
\begin{lemma} \label{daFactor} If $s \leq r$ then $p^{r-s} \Phi_\pi^{r,s} \circ d = \rho^s \circ \beta^s \circ \Phi_\pi^{r,s}$. In particular, if $s=r$, then $\Phi_\pi^{r,s} = \Phi_\pi^{r,r}$ is a map of differential Lie algebras. \end{lemma}
\begin{proof} It suffices to show that the composites $p^{r-s} \Phi_\pi^{r,s} \circ d$ and $\rho^s \circ \beta^s \circ \Phi_\pi^{r,s}$ agree on brackets of length $k$ in $L'(x,dx) \otimes \mathbb{Z}/p^s$ for each $k \in \mathbb{N}$. We will do this by induction.
In the case $k=1$, the restriction of $\Phi_\pi^{r,s}$ to brackets of length 1 is $\phi_\pi^{r,s}$. By construction of $u$ and $v$ we have $(\rho^s \circ \beta^s)(v)=p^{r-s}u$ in $\pi_*(\Omega P^{n+1}(p^r); \mathbb{Z}/p^s)$, so $\phi_\pi^{r,s}$ satisfies $p^{r-s} \phi_\pi^{r,s} \circ d = \rho^s \circ \beta^s \circ \phi_\pi^{r,s}$, as required.
Now let $a \in L'(x,dx) \otimes \mathbb{Z}/p^s$ be a bracket of length $k>1$. We have $a=[b,c]$ for brackets $b$, $c$ of lengths $i$ and $j$ respectively with $i+j=k$, $i<k$, $j<k$. Thus $$ \rho^s \circ \beta^s \circ \Phi_\pi^{r,s} (a) = \rho^s \circ \beta^s \circ \Phi_\pi^{r,s} ([b,c]) = \rho^s \circ \beta^s ([\Phi_\pi^{r,s} (b), \Phi_\pi^{r,s} (c)]) $$ $$ = [\rho^s \circ \beta^s \circ \Phi_\pi^{r,s}( b), \Phi_\pi^{r,s} (c)] +(-1)^{\deg b } [ \Phi_\pi^{r,s} (b), \rho^s \circ \beta^s \circ \Phi_\pi^{r,s} (c)], $$ where the last equality is by Lemma \ref{notQuiteGLA}. By induction we have $\rho^s \circ \beta^s \circ \Phi_\pi^{r,s}( b) = p^{r-s} \Phi_\pi^{r,s} \circ d(b)$ and $\rho^s \circ \beta^s \circ \Phi_\pi^{r,s}( c) = p^{r-s} \Phi_\pi^{r,s} \circ d(c)$, so the above is equal to $$ [p^{r-s} \Phi_\pi^{r,s} \circ d( b), \phi_\pi^{r,s} (c)] +(-1)^{\deg b } [ \Phi_\pi^{r,s} (b), p^{r-s} \Phi_\pi^{r,s} \circ d (c)] = p^{r-s} \Phi_\pi^{r,s}( [ d( b), c] +(-1)^{\deg b } [ b, d (c)])$$ $$ = p^{r-s} \Phi_\pi^{r,s} \circ d( [ b, c]). $$
This completes the induction, and hence the proof. \end{proof}
Lemma \ref{daFactor} identifies a factor of $p^{r-s}$. The next lemma makes precise the idea that this factor comes from the map $\beta^s$, rather than the map $\rho^s$, by relating each $\Phi_\pi^{r,s}$ to $\Phi_\pi^{r,r}$.
\begin{lemma} \label{PhiPiDiffs} The following diagram commutes: \begin{center}
\begin{tabular}{c}
\xymatrix{
L'(x,dx) \ar_{d}[dd] \ar^{\Phi_\pi^{r,r}}[r] & \pi_*(\Omega P^{n+1}(p^r); \mathbb{Z}/p^r) \ar^{\beta^s}[d] \\
& \pi_*(\Omega P^{n+1}(p^r)) \ar^{\rho^s}[d] \\
L'(x,dx) \otimes \mathbb{Z}/p^s \ar^{\Phi_\pi^{r,s}}[r] & \pi_*(\Omega P^{n+1}(p^r); \mathbb{Z}/p^s). \\
}
\end{tabular}
\end{center} In particular, $\textrm{Im}(\rho^s) \supset \textrm{Im}(\Phi_\pi^{r,s} \circ d)$. \end{lemma}
\begin{proof} By Lemma \ref{daFactor}, the top face of the following diagram commutes, and the bottom face commutes up to a factor of $p^{r-s}$, in the sense that $p^{r-s} \Phi_\pi^{r,s} \circ d = \rho^s \circ \beta^s \circ \Phi_\pi^{r,s}$:
\begin{center}
\begin{tabular}{c}
\xymatrix@C=0.1cm{
& \phantom{\Omega P^{n+1}(p^r)} & L'(x,dx) \ar^{\textrm{quotient}}[dd] \ar^{\Phi_\pi^{r,r}}[ddrr] \ar_{d}[ddll] & & \\
& & & & \\
L'(x,dx) \ar^{\textrm{quotient}}[dd] \ar^(.3){\Phi_\pi^{r,r}}[ddrr] & & L'(x,dx) \otimes \mathbb{Z}/p^s \ar'[dl]_{d}[ddll] \ar'[dr]^{\Phi_\pi^{r,s}}[ddrr]& & \pi_*(\Omega P^{n+1}(p^r);\mathbb{Z}/p^r) \ar^{\textrm{red}^{r,s}}[dd] \ar_{\beta^r}[dl] \\
& \phantom{\ell} & & \pi_*(\Omega P^{n+1}(p^r)) \ar_{\rho^r}[dl] \ar@{=}[dd] & \\
L'(x,dx) \otimes \mathbb{Z}/p^s \ar^{\Phi_\pi^{r,s}}[ddrr] & & \pi_*(\Omega P^{n+1}(p^r); \mathbb{Z}/p^r) \ar^{\textrm{red}^{r,s}}[dd] & & \pi_*(\Omega P^{n+1}(p^r); \mathbb{Z}/p^s) \ar_{\beta^s}[dl] \\
& & & \pi_*(\Omega P^{n+1}(p^r)) \ar_{\rho^s}[dl] & \\
& & \pi_*(\Omega P^{n+1}(p^r);\mathbb{Z}/p^s). & & \\
}
\end{tabular}
\end{center}
Commutativity of the back left face is clear. We now check commutativity of the front left and back right faces, which are identical. Since the reduction map $\textrm{red}$ is a map of Lie algebras, both composites are maps of nonassociative algebras, and by the uniqueness part of the universal property of $L'(x,dx)$, it suffices to show that the restrictions to $\langle x, dx \rangle$ agree, and this is easily seen by direct calculation.
We now turn to the front right face. The square involving $\rho^s$ commutes, since the composite $$P^m(p^s) \xrightarrow{\underline{\textrm{red}}^{r,s}} P^m(p^r) \xrightarrow{\underline{\rho}^r} S^m $$ is equal to $\underline{\rho}^s: P^m(p^s) \longrightarrow S^m$. For the square involving $\beta^s$, we have that the composite $$S^{m-1} \xrightarrow{\underline{\beta}^s} P^m(p^s) \xrightarrow{\underline{\textrm{red}}^{r,s}} P^m(p^s)$$ is equal to $p^{r-s} \underline{\beta}^r : S^{m-1} \longrightarrow P^m(p^r)$.
Putting all of this together, we have that $$ \Phi_\pi^{r,s} \circ d \circ \textrm{quotient} = \textrm{red}^{r,s} \circ \Phi_\pi^{r,r} \circ d = \textrm{red}^{r,s} \circ \rho^r \circ \beta^r \circ \Phi_\pi^{r,r} = \rho^s \circ \beta^r \circ \Phi_\pi^{r,r},$$ as required. \end{proof}
Let $s \leq r$. The homology $\widetilde{H}_*(P^m(p^r);\mathbb{Z}/p^s)$ is free over $\mathbb{Z}/p^s$; in particular we have $$\widetilde{H}_i(P^m(p^r);\mathbb{Z}/p^s) = \begin{cases} \mathbb{Z}/p^s & i = m,m-1, \\
0 & \textrm{ otherwise.}
\end{cases} $$
Write $e_m$ for a choice of generator of $H_m(P^m(p^r);\mathbb{Z}/p^s)$, and $s_{m-1} = \beta(e_m)$, where $\beta$ is the homology Bockstein. The group $H_{m-1}(P^m(p^r);\mathbb{Z}/p^s)$ is generated by $s_{m-1}$.
The Pontrjagin product makes $\widetilde{H}_*(\Omega P^{n+1}(p^r);\mathbb{Z}/p^s)$ into a $\mathbb{Z}/p^s$-algebra. Any graded associative algebra carries a Lie bracket, defined by setting $[x,y]=xy-(-1)^{\deg(x)\deg(y)}yx$, and this is what will be meant by `the bracket on $H_*(\Omega P^{n+1}(p^r);\mathbb{Z}/p^s)$'.
Recall that an element of $\pi_m(Y; \mathbb{Z}/p^r)$ is a homotopy class of maps $P^m(p^r) \longrightarrow Y$. Let $h: \pi_*(Y; \mathbb{Z}/p^s) \longrightarrow H_*(Y;\mathbb{Z}/p^s)$ be the \emph{Hurewicz map}, which sends $f \in \pi_*(Y; \mathbb{Z}/p^s)$ to $f_*(e_m) \in H_*(Y;\mathbb{Z}/p^s)$. By \cite[Proposition 6.4]{CMNTorsion}, the generators $e_m$ may be chosen so that $h$ carries Samelson products to commutators; that is, so that $h([f,g]) = [h(f),h(g)] \in H_*(\Omega P^{n+1}(p^r); \mathbb{Z}/p^s)$.
Thus, the composition $h \circ \Phi_\pi^{r,s}$ respects brackets, and the codomain, $H_*(\Omega P^{n+1}(p^r); \mathbb{Z}/p^s)$ carries a (genuine) Lie algebra structure. We therefore obtain a factorization of $h \circ \Phi_\pi^{r,s}$ through $\theta$ to give a map of Lie algebras $\Phi_H^{r,s}$ which satisfies the following lemma:
\begin{lemma} \label{hurewiczTriangle} The following diagram commutes: \begin{center}
\begin{tabular}{c}
\xymatrix{
L'(x,dx) \otimes \mathbb{Z}/p^s \ar^{\Phi_\pi^{r,s}}[r] \ar^{\theta}[d] & \pi_*(\Omega P^{n+1}(p^r); \mathbb{Z}/p^s) \ar^{h}[d] \\
L(x,dx) \otimes \mathbb{Z}/p^s \ar^{\Phi_H^{r,s}}[r] & H_*(\Omega P^{n+1}(p^r); \mathbb{Z}/p^s).
}
\end{tabular}
\end{center} \qed \end{lemma}
\subsection{Tensor algebras and the Bott-Samelson Theorem} \label{TBSSubsection}
The purpose of this section is to introduce some notation for dealing with tensor algebras, and to recall the Bott-Samelson Theorem (Theorem \ref{BS}). We define the \textit{tensor algebra} on a graded $R$-module $V$ to be $T(V) = \bigoplus_{k=1}^\infty V^{\otimes k}$, where $V^{\otimes k}$ is the tensor product of $k$ copies of $V$. In particular, this definition is `reduced' since we do not insert a copy of $R$ in degree 0. The multiplication is given by concatenation of tensors, and makes $T(V)$ into the free graded associative algebra on $V$. Let $A$ be an algebra and let $\varphi : V \longrightarrow A$ be a homomorphism. We write $\widetilde{\varphi} : T(V) \longrightarrow A$ for the map of algebras induced by $\varphi$. Let $$\iota_i : V^{\otimes i} \longrightarrow T(V)$$ be the inclusion, and let $$\zeta_i : T(V) \longrightarrow V^{\otimes i}$$ be the projection.
Bott and Samelson first proved their theorem in \cite{BottSamelson}; we give the formulation from Selick's book \cite{Selick}.
\begin{theorem}[Bott-Samelson] \label{BS} Let $R$ be a PID, and let $X$ be a connected space with $\widetilde{H}_*(X;R)$ free over $R$. Then $\widetilde{H}_*(\Omega \Sigma X;R) \cong T(\widetilde{H}_*(X;R))$ and $\eta: X \longrightarrow \Omega \Sigma X$ induces the canonical map $\widetilde{H}_*(X;R) \longrightarrow T(\widetilde{H}_*(X;R))$. \end{theorem}
The Bott-Samelson Theorem immediately allows us to find a free Lie algebra in the loop-homology of a Moore space.
\begin{lemma} \label{injection1} The map $\Phi_H^{r,s} : L(x,dx) \otimes \mathbb{Z}/p^s \longrightarrow H_*(\Omega P^{n+1}(p^r); \mathbb{Z}/p^s)$ is an injection. \end{lemma}
\begin{proof} Since $r \geq s$, the module $H_*( P^{n}(p^r); \mathbb{Z}/p^s)$ is free over $\mathbb{Z}/p^s$. By the Bott-Samelson Theorem \ref{BS}, $\widetilde{H}_*(\Omega P^{n+1}(p^r); \mathbb{Z}/p^s) \cong T(x,dx) \otimes \mathbb{Z}/p^s$, and this isomorphism identifies $\Phi_H^{r,s}$ with the natural map $L(x,dx) \otimes \mathbb{Z}/p^s \longrightarrow T(x,dx) \otimes \mathbb{Z}/p^s$. But this latter map is an injection by Proposition 2.9 and Corollary 2.7 of \cite{CMNTorsion}. \end{proof}
We have the following corollary, which will be the main ingredient in the proof of Theorem \ref{HPrelim}.
\begin{corollary} \label{combining} Let $Y$ be a simply connected $CW$-complex, let $p$ be an odd prime, and let $r \in \mathbb{N}$. Let $\mu: P^{n+1}(p^r) \longrightarrow Y$ be a continuous map. If the induced map $$(\Omega \mu)_* : H_*(\Omega P^{n+1}(p^r);\mathbb{Z}/p^s) \longrightarrow H_*(\Omega Y;\mathbb{Z}/p^s)$$ is an injection, then the module $\textrm{Im}((\Omega \mu)_* \circ \Phi_H^{r,s} \circ \theta \circ d)$ is $\mathbb{Z}/p^s$-hyperbolic. \qed \end{corollary}
\begin{proof} By Lemma \ref{injection1}, $\Phi_H^{r,s}$ is an injection, and by Corollary \ref{mbCor}, the module $\textrm{Im}(\theta \circ d)$ is $\mathbb{Z}/p^s$-hyperbolic. It follows that $(\Omega \mu)_* \circ \Phi_H^{r,s} (\textrm{Im}(\theta \circ d)) = \textrm{Im}((\Omega \mu)_* \circ \Phi_H^{r,s} \circ \theta \circ d)$ is also $\mathbb{Z}/p^s$-hyperbolic. \end{proof}
\section{The suspension case} \label{SuspensionSection}
The purpose of this section is to show that Theorem \ref{HPrelim} implies Theorem \ref{HCriterion}. This will be accomplished by means of Proposition \ref{injection2}, whose proof is the goal of this section. The main point is that even if $\widetilde{H}_*(X; \mathbb{Z}/p^s)$ is not free over $\mathbb{Z}/p^s$, the canonical map of the Bott-Samelson Theorem (Theorem \ref{BS}) is still an injection. That is, the homology $\widetilde{H}_*(\Omega \Sigma X ; \mathbb{Z}/p^s)$ always contains the tensor algebra on $\widetilde{H}_*(X; \mathbb{Z}/p^s)$, but if $\widetilde{H}_*(X; \mathbb{Z}/p^s)$ is not free then it will contain other things too.
In Subsection \ref{StructureSubs}, we recall the James splitting $\Sigma \Omega \Sigma X \simeq \bigvee_{k=1}^\infty \Sigma X^{\wedge k}.$ This gives us Proposition \ref{BSGen}, which describes the structure of the Pontrjagin algebra $\widetilde{H}_*(\Omega \Sigma X ; \mathbb{Z}/p^s)$, in particular identifying the tensor algebra $T(\widetilde{H}_*(X; \mathbb{Z}/p^s))$ as a subalgebra. Subsection \ref{evSubs} proves Lemma \ref{evproj}, which describes the effect of the evaluation map on $H_*(\Omega \Sigma X ; \mathbb{Z}/p^s)$. Subsection \ref{FinnickySubs} draws these ingredients together to prove Proposition \ref{injection2}.
Let $\sigma : \widetilde{H}_*(Y) \longrightarrow \widetilde{H}_{*+1}(\Sigma Y)$ denote the suspension isomorphism. For a space $X$, let $X^k$ denote the product of $k$ copies of $X$, and let $X^{\wedge k}$ denote the smash product. Let $\sim$ be the relation on $X^k$ defined by $$(x_1, \dots ,x_{i-1}, * , x_{i+1}, x_{i+2}, \dots x_{k}) \sim (x_1, \dots , x_{i-1} , x_{i+1}, * , x_{i+2} , \dots x_{k}).$$
Let $J_k(X)$ be the space $\faktor{X^k}{\sim}$. There is a natural inclusion $$J_k(X) \longrightarrow J_{k+1}(X)$$ $$(x_1, \dots , x_k) \mapsto (x_1, \dots, x_k, *).$$
The \emph{James construction} $JX$ is defined to be the colimit of the diagram consisting of the spaces $J_k(X)$ and the above inclusions. Notice that $JX$ carries a product given by concatenation, which makes it into the free topological monoid on $X$, and that a topological monoid is in particular an $H$-space.
The adjunction isomorphism $[\Sigma X, Y] \cong [X, \Omega Y]$ will be written in both directions as $f \longmapsto \overline{f}$. Recall that $\eta$ denotes the unit of the adjunction, which is the map $X \longrightarrow \Omega \Sigma X$ sending $x \in X$ to $(t \mapsto \langle t,x \rangle) \in \Omega \Sigma X$. We will write $\textrm{ev}$ for the \textit{evaluation map}; the counit $\Sigma \Omega Y \longrightarrow Y$, which sends $\langle t, \gamma \rangle \in \Sigma \Omega Y$ to $\gamma(t) \in Y$.
\subsection{The tensor algebra inside $H_*(\Omega \Sigma X)$} \label{StructureSubs}
In this section we will generalise the Bott-Samelson theorem to suit our purpose. Specifically, the map $\eta : X \longrightarrow \Omega \Sigma X$ induces a map $\eta_* : \widetilde{H}_*(X) \longrightarrow \widetilde{H}_*(\Omega \Sigma X)$ on homology. By the universal property of the tensor algebra, $\eta_*$ extends to a map of algebras $$ \widetilde{\eta_*} : T(\widetilde{H}_*(X)) \longrightarrow \widetilde{H}_*(\Omega \Sigma X).$$
The Bott-Samelson Theorem (Theorem \ref{BS}) says that if the homology $H_*(X;\mathbb{Z}/p^s)$ is free then $\widetilde{\eta_*}$ is an isomorphism. We will show that even if $H_*(X;\mathbb{Z}/p^s)$ is not free, the map $\widetilde{\eta_*}$ is still an injection. This is by no means new, but follows reasonably easily from better-known results, so we shall derive it in this way. In this section homology is taken with $\mathbb{Z}/p^s$-coefficients (unless otherwise stated).
\begin{lemma} \label{crossCoker} The cross product map $\widetilde{H}_*(X)^{\otimes k} \xrightarrow{\times} \widetilde{H}_*(X^{\wedge k})$ is injective, split (although not naturally) and its cokernel $C$ satisfies $p^{s-1} C = 0$. \end{lemma}
\begin{proof} For spaces $A$ and $B$ The K\"unneth Theorem gives an exact sequence $$0 \longrightarrow H_*(A) \otimes H_*(B) \xrightarrow{\times} H_*(A \times B) \longrightarrow \textrm{Tor}(H_*(A), H_{*-1}(B)) \longrightarrow 0,$$ where the $\textrm{Tor}$ is taken over $\mathbb{Z}/p^s$, and this sequence is (unnaturally) split. By Lemma \ref{TorAnnihilation} we have $p^{s-1}\textrm{Tor}(H_*(A), H_{*-1}(B)) = 0$.
Let $a_0: pt \longrightarrow A$ denote the inclusion of the basepoint of $A$ and let $b_0$ denote the inclusion of the basepoint of $B$. Let $j: H_*(A) \oplus H_*(B) \longrightarrow H_*(A) \otimes H_*(B)$ be the composite $$H_*(A) \oplus H_*(B) \xrightarrow{\cong} H_*(A) \otimes H_*(pt) \oplus H_*(pt) \otimes H_*(B) \xrightarrow{(id_A)_* \otimes (b_0)_* \oplus (a_0)_* \otimes (id_B)_*} H_*(A) \otimes H_*(B).$$ To relate the reduced and unreduced situations we have the following diagram (which we take to define the reduced cross product) where $i$, $i_1$ and $i_2$ are the inclusions and $p$ is the quotient. \begin{center}
\begin{tabular}{c}
\xymatrix{
0 \ar[r] & H_*(A) \oplus H_*(B) \ar@{^{(}->}^j[d] \ar^(.55){(i_1)_* \oplus (i_2)_*}[r] & H_*(A \vee B) \ar@{^{(}->}^{i_*}[d] \ar[r] & 0 \ar[d] \ar[r] & 0 \\
0 \ar[r] & H_*(A) \otimes H_*(B) \ar^(.55){\times}[r] \ar[d] & \ar[r] H_*(A \times B) \ar^{p_*}[d] & \textrm{Tor}(H_*(A), H_{*-1}(B)) \ar@{=}[d] \ar[r] & 0 \\
0 \ar[r] & \widetilde{H}_*(A) \otimes \widetilde{H}_*(B) \ar^(.55){\times}[r] & \widetilde{H}_*(A \wedge B) \ar[r] & \textrm{Tor}(H_*(A), H_{*-1}(B)) \ar[r] & 0.
}
\end{tabular}
\end{center} The bottom row is obtained from the other two by taking cokernels, so is automatically exact, and it therefore suffices to check that the top two squares commute. The top right square commutes because the map $(i_1)_* \oplus (i_2)_*$ is an isomorphism, so the composite of $i_*$ with the map into the $\textrm{Tor}$ term factors through two terms of an exact sequence, hence is zero, as required.
We now check that the top left square commutes. It suffices to check commutativity on each summand of the domain individually. We will do so for $H_*(A)$; the case of $H_*(B)$ is analogous. Identifying $H_*(A)$ with $H_*(A) \otimes H_*(pt)$, the restriction of $j$ becomes $(id_A)_* \otimes (b_0)_*$. The composite with the cross product is written $(id_A)_* \times (b_0)_*$, and by bilinearity of cross product this is the same as $(id_A \times b_0)_*$, where now the product is taken in spaces. But under the identification $A \cong A \times \{pt\}$, this is just the inclusion $A \longrightarrow A \times B$, which is the map obtained by going the other way round the square, as required.
To finish, we note that since the middle row is split, the bottom row is also split. \end{proof}
The understanding of the cross product from Lemma \ref{crossCoker} allows us to understand part of the homology of $JX$, by constructing a map $\varphi$ as in the following lemma.
\begin{lemma} \label{JamesHomology} The maps $$\widetilde{H}_*(X)^{\otimes k} \longrightarrow H_*(X)^{\otimes k} \xrightarrow{\times} H_*(X^k) \longrightarrow H_*(J_k(X)) \longrightarrow H_*(J(X))$$ define an injection of algebras $T(\widetilde{H}_*(X)) \xrightarrow{\varphi} \widetilde{H}_*(J(X)).$ Furthermore, $\textrm{Im}(\varphi)$ is a direct summand, and we may write $\widetilde{H}_*(J(X)) \cong T(\widetilde{H}_*(X)) \oplus C$ such that the complementary module $C$ satisfies $p^{s-1} C = 0$. \end{lemma}
\begin{proof} We use a modified version of the argument in \cite[Proposition 3C.8]{Hatcher}. First, $\varphi$ is a ring homomorphism, because the product in $J(X)$ descends from the natural map $X^i \times X^j \longrightarrow X^{i+j}$. To see that we have an injection, we consider the following diagram, where we follow Hatcher's notation and set $T_k(M) = \bigoplus_{i=1}^k M^{\otimes i}$: \begin{center}
\begin{tabular}{c}
\xymatrix{
0 \ar[r] & T_{k-1}(\widetilde{H}_*(X)) \ar^{\varphi}[d] \ar[r] & T_{k}(\widetilde{H}_*(X)) \ar^{\varphi}[d] \ar[r] & (\widetilde{H}_*(X))^{\otimes k} \ar^{\times}[d] \ar[r] & 0 \\
0 \ar[r] & \widetilde{H}_*(J_{k-1}(X)) \ar[r] & \widetilde{H}_*(J_k(X)) \ar[r] & \widetilde{H}_*(X^{\wedge k}) \ar[r] & 0.
}
\end{tabular}
\end{center}
Commutativity of the diagram follows from the definition of $\varphi$. Exactness of the top row is clear. The bottom row is obtained from the long exact sequence of the pair $(J_k(X), J_{k-1}(X))$, applying excision to pass to the quotient $J_k(X)/J_{k-1}(X) \simeq X^{\wedge k}$. This sequence is split because the quotient $X^k \longrightarrow X^{\wedge k}$ factors through the map $J_k(X) \longrightarrow X^{\wedge k}$, and the former map is split after suspending. Thus we get that $\widetilde{H}_*(J_k(X)) \cong \widetilde{H}_*(J_{k-1}(X)) \oplus \widetilde{H}_*(X^{\wedge k})$. Lemma \ref{crossCoker} tells us that $\widetilde{H}_*(X^{\wedge k}) \cong (\widetilde{H}_*(X))^{\otimes k} \oplus C$ with $p^{s-1} C = 0$, so the result follows immediately by inducting over $k$. \end{proof}
Our next job is to translate this understanding of $JX$ into an understanding of $\Omega \Sigma X$. It is well-known that the two are homotopy equivalent, but we wish to be precise about the maps. For a based space $Y$, let $\Omega' Y$ denote the space of loops of any length in $Y$, so that $\Omega Y$ is the subspace of $\Omega' Y$ consisting of loops of length 1. We will write $\gamma_1 \# \gamma_2$ for the concatenation of loops $\gamma_1$ and $\gamma_2$. For $\gamma \in \Omega' Y$ and $\ell \in \mathbb{R}_{>0}$, let $\gamma^{\ell}$ denote the linear reparameterization of $\gamma$ which has length $\ell$. Note that $\gamma \longmapsto \gamma^1$ is a continuous map $\Omega' Y \longrightarrow \Omega Y$, which is a retraction for the inclusion $\Omega Y \subset \Omega' Y$. For $x \in X$, let $\gamma_x \in \Omega \Sigma X$ be the loop defined by $\gamma_x(t) = \langle t, x \rangle$, which is equal to $\eta(x)$.
Now let $X$ be a connected $CW$-complex, which we take without loss of generality to have a single 0-cell, which is the basepoint. Let $d: X \longrightarrow [0,1]$ be any continuous map such that $d^{-1}(0)=\{*\}$. Define a map $$\lambda : J(X) \longrightarrow \Omega \Sigma X$$ $$(x_1, \dots x_k) \longmapsto (\gamma_{x_1}^{d(x_1)} \# \gamma_{x_2}^{d(x_2)} \# \dots \# \gamma_{x_k}^{d(x_k)})^1.$$
The reparameterization is necessary so that $\lambda$ is well-defined when some $x_i = *$.
Hatcher proves the following as \cite[Theorem 4J.1]{Hatcher}.
\begin{lemma} \label{LAMBDA} The map $\lambda$ is a weak homotopy equivalence for any connected $CW$-complex $X$. Furthermore, $\lambda$ is an $H$-map, so it induces a map of algebras on homology. \qed \end{lemma}
The following lemma is immediate from the definition of $\lambda$.
\begin{lemma} \label{describeLambda} The composite $$X^k \rightarrow J_k(X) \rightarrow J(X) \xrightarrow{\lambda} \Omega \Sigma X$$ is homotopy equivalent to $m \circ \eta^k$, where $m$ is any choice of $k$-fold loop multiplication on $\Omega \Sigma X$. \qed \end{lemma}
Recall from Subsection \ref{TBSSubsection} that $\iota_k : V^{\otimes k} \longrightarrow T(V)$ is the inclusion.
We are now ready to prove the main result of this subsection, which is what we will use later.
\begin{proposition} \label{BSGen} The map $$\widetilde{\eta_*} : T(\widetilde{H}_*(X)) \longrightarrow H_*(\Omega \Sigma X)$$ is an injection onto a summand, each restriction $\widetilde{\eta_*} \circ \iota_k$ is equal to $$ \widetilde{H}_*(X)^{\otimes k} \xrightarrow{\times} \widetilde{H}_*(X^k) \xrightarrow{(\eta^k)_*} \widetilde{H}_*((\Omega \Sigma X)^k) \xrightarrow{m_*} \widetilde{H}_*(\Omega \Sigma X),$$ and we may write $$H_*(\Omega \Sigma X) \cong T(\widetilde{H}_*(X)) \oplus C$$ such that the complementary module $C$ satisfies $p^{s-1} C=0$.
\end{proposition}
\begin{proof} By Lemmas \ref{JamesHomology}, \ref{LAMBDA} and \ref{describeLambda}, it suffices to show that $\lambda_* \circ \varphi = \widetilde{\eta_*}$. Since both maps are algebra maps, by the universal property of the tensor algebra it further suffices to show that the composite $$\widetilde{H}_*(X) \xrightarrow{\iota_1} T(\widetilde{H}_*(X)) \xrightarrow{\varphi} \widetilde{H}_*(JX) \xrightarrow{\lambda_*} \widetilde{H}_*(\Omega \Sigma X) $$ is equal to $\eta_*$.
To see this, first note that the composite $\widetilde{H}_*(X) \xrightarrow{\iota_1} T(\widetilde{H}_*(X)) \xrightarrow{\varphi} \widetilde{H}_*(JX)$ is equal to the map induced by the inclusion $X \longrightarrow J_1(X) \subset J(X)$ which carries $x \in X$ to the equivalence class of $x$ in $J(X)$. By definition of $\lambda$ we then have $\lambda(x) = \gamma_x$, which by definition is $\eta(x)$, as required. \end{proof}
\subsection{The effect of the evaluation map} \label{evSubs}
The goal of this section is to prove Lemma \ref{evproj}, which says that up to suspension isomorphisms, the evaluation map $\textrm{ev}: \Sigma \Omega \Sigma X \longrightarrow \Sigma X$ induces the projection onto the tensors of length 1. Our strategy is to first prove Lemma \ref{niceSquare}, the point of which is that when one evaluates a concatenation of $k$ loops at some time $t$, the result only depends on one of the loops - this is the $i$ appearing in the proof of Lemma \ref{niceSquare}. We will then see that this, together with simple formal properties of the cross product, is enough to prove Lemma \ref{evproj}.
In this section, for a co-$H$-space $Y$, $c: Y \longrightarrow Y \vee Y$ denotes the comultiplication, and for a product $\prod_{i=1}^k X_i$, the map $\pi_i$ is the projection onto the $i$-th factor. In the next lemma we take the iterated comultiplication $c$ and the iterated multiplication $m$ to be parameterized so as to spend equal time on each component - this does not change anything up to homotopy.
\begin{lemma} \label{niceSquare} The following diagram commutes. \begin{center}
\begin{tabular}{c}
\xymatrix{
\Sigma X^k \ar^{c}[d] \ar^{\Sigma \eta^k}[r] & \Sigma (\Omega \Sigma X)^k \ar^{\Sigma m}[r] & \Sigma \Omega \Sigma X \ar^{\textrm{ev}}[d]\\
(\Sigma X^k)^{\vee k} \ar^{\bigvee_{i=1}^k \Sigma \pi_i}[d] & & \Sigma X \\
(\Sigma X)^{\vee k} \ar^{\textrm{fold}}[r] & \Sigma X \ar^{\Sigma \eta}[r] & \Sigma \Omega \Sigma X. \ar_{\textrm{ev}}[u] \\
}
\end{tabular}
\end{center} \end{lemma}
\begin{proof} We will evaluate both composites. A point of $\Sigma X^k$ may be written in suspension coordinates as $\langle t, x_1, x_2, \dots, x_k \rangle$, for $t \in I$ and $x_i \in X$. There exists some integer $i$ with $1 \leq i \leq k$ so that $\frac{i-1}{k} \leq t \leq \frac{i}{k}$.
For the top right composite, $$\textrm{ev} \circ \Sigma m \circ \Sigma \eta^k \langle t, x_1, \dots, x_k \rangle = \textrm{ev} \langle t, m(\gamma_{x_1}, \dots , \gamma_{x_k}) \rangle = (\gamma_{x_1} \# \dots \# \gamma_{x_k})(t)= \gamma_{x_i}(kt - (i-1)).$$
For the bottom left composite, we first introduce some notation. For a point $y$ of a space $Y$, we write $(y)_i$ for the image of $y$ under the inclusion of the $i$-th wedge summand in $Y \longrightarrow Y^{\vee k}$. With this notation, taking $Y = \Sigma X^k$, we have $c \langle t, x_1, \dots, x_k \rangle = (\langle kt-(i-1), x_1, \dots, x_k \rangle)_i $. Therefore, $$\textrm{ev} \circ \Sigma \eta \circ \textrm{fold} \circ (\bigvee_{i=1}^k \Sigma \pi_i) \circ c \langle t, x_1, \dots, x_k \rangle = \textrm{ev} \circ \Sigma \eta \circ \textrm{fold}(\langle kt-(i-1), x_i \rangle)_i$$ $$=\textrm{ev} \circ \Sigma \eta \langle kt-(i-1), x_i \rangle = \gamma_{x_i}(kt-(i-1)),$$ as required. \end{proof}
\begin{lemma} \label{crossWorks} Let $X$ be a space. The composite $$\widetilde{H}_*(X)^{\otimes k} \xrightarrow{\times} \widetilde{H}_*(X^k) \xrightarrow{(\pi_i)_*} \widetilde{H}_*(X)$$ of the cross product with any projection is trivial for $k \geq 2$. \end{lemma}
\begin{proof} Up to homeomorphism, $X$ may be regarded as the space $\prod_{j=1}^k Y_j$, where $Y_j = *$ for $j \neq i$ and $Y_i = X$. Under this identification, $\pi_i$ is identified with the map $\prod_{j=1}^k f_j : X^k \longrightarrow \prod_{j=1}^k Y_j$, where $f_j$ is the identity on $X$ when $j=i$, and is the trivial map otherwise.
The composite of maps $(\prod_{j=1}^k f_j)_* \circ \times$ is the cross product of homomorphisms $(f_1)_* \times (f_2)_* \times \dots \times (f_k)_*$. Cross product of homomorphisms is $k$-multilinear, and since $k \geq 2$ there is at least one $j$ with $f_j$ equal to the constant map, hence $(f_j)_*=0$. This means that $(\prod_{j=1}^k f_j)_* \circ \times$ is trivial for $k \geq 2$, as required. \end{proof}
\begin{corollary} \label{crossPlus} Let $X$ be a space. The composite $$\widetilde{H}_*(X)^{\otimes k} \xrightarrow{\times} \widetilde{H}_*(X^k) \xrightarrow{\sigma} \widetilde{H}_*(\Sigma X^k) \xrightarrow{c_*} \widetilde{H}_*((\Sigma X^k)^{\vee k}) \xrightarrow{(\bigvee_{i=1}^k \Sigma \pi_i)_*} \widetilde{H}_*((\Sigma X)^{\vee k})$$ is trivial for $k \geq 2$. \end{corollary}
\begin{proof} For a space $Y$, let $p_i : Y^{\vee k} \longrightarrow Y$ be the projection onto the $i$-th wedge summand. The comultiplication $c$ satisfies $p_i \circ c \simeq id_{\Sigma X^k}$ for each $i$, so on homology we have $$c_* : \widetilde{H}_*(\Sigma X^k) \longrightarrow \widetilde{H}_*((\Sigma X^k)^{\vee k}) \cong \bigoplus^{k}_{i=1} \widetilde{H}_*(\Sigma X^k)$$ $$x \longmapsto (x, x, \dots, x).$$
That is, $c_*$ may be identified with the diagonal map $\Delta : \widetilde{H}_*(\Sigma X^k) \longrightarrow \bigoplus^{k}_{i=1} \widetilde{H}_*(\Sigma X^k)$.
Thus, $$(\bigvee_{i=1}^k \Sigma \pi_i)_* \circ c_* \circ \sigma \circ \times (x_1 \otimes \dots \otimes x_k) = (\bigvee_{i=1}^k \Sigma \pi_i)_* \circ c_*( \sigma (x_1 \times \dots \times x_k))$$ $$ = \bigoplus_{i=1}^k (\Sigma \pi_i)_* \circ \Delta (\sigma (x_1 \times \dots \times x_k))=0,$$ since by Lemma \ref{crossWorks} we have $$(\Sigma \pi_i)_* (\sigma (x_1 \times \dots \times x_k)) = \sigma \circ (\pi_i)_* (x_1 \times \dots \times x_k)=0.$$
This completes the proof. \end{proof}
\begin{lemma} \label{evproj} The composite $$T(\widetilde{H}_*(X)) \xrightarrow{\widetilde{\eta_*}} \widetilde{H}_*(\Omega \Sigma X) \xrightarrow{\sigma} \widetilde{H}_*(\Sigma \Omega \Sigma X) \xrightarrow{\textrm{ev}_*} \widetilde{H}_*(\Sigma X) \xrightarrow{\sigma^{-1}} \widetilde{H}_*(X)$$ is equal to the projection $\zeta_1$.
\end{lemma}
\begin{proof} Write $\Gamma$ for the above composite. We must show that $\Gamma \circ \iota_k$ is the identity map on $H_*(X)$ when $k=1$, and is $0$ otherwise.
For the $k=1$ statement, note that $\widetilde{\eta_*} \circ \iota_1 = \eta_*$ (this is the definition of $\widetilde{\eta_*}$). We may therefore write $$ \Gamma \circ \iota_1 = \sigma^{-1} \circ \textrm{ev}_* \circ \sigma \circ \widetilde{\eta_*} \circ \iota_1 = \sigma^{-1} \circ \textrm{ev}_* \circ \sigma \circ \eta_* = \sigma^{-1} \circ \textrm{ev}_* \circ (\Sigma \eta)_* \circ \sigma,$$ and by the triangle identities for the adjunction $\Sigma \dashv \Omega$ we have a commuting diagram \begin{center}
\begin{tabular}{c}
\xymatrix{
\Sigma X \ar^{id_{\Sigma X}}[dr] \ar^{\Sigma \eta}[r] & \Sigma \Omega \Sigma X \ar^{\textrm{ev}}[d] \\
& \Sigma X.
}
\end{tabular}
\end{center}
Thus, $\Gamma \circ \iota_1 = \sigma^{-1} \circ \sigma = id_{H_*(X)}$, as we required.
Now let $k>1$. Juxtaposing the diagram of Lemma \ref{niceSquare} (after taking homology) with the result of Corollary \ref{crossPlus} gives a commuting diagram \begin{center}
\begin{tabular}{c}
\xymatrix{
\widetilde{H}_*(X)^{\otimes k} \ar^{\times}[r] \ar^{0}[ddrr] & \widetilde{H}_*(X^k) \ar^{\sigma}[r] & \widetilde{H}_*(\Sigma X^k) \ar^{c_*}[d] \ar^{(\Sigma \eta^k)_*}[r] & \widetilde{H}_*(\Sigma (\Omega \Sigma X)^k) \ar^{(\Sigma m)_*}[r] & \widetilde{H}_*(\Sigma \Omega \Sigma X) \ar^{\textrm{ev}_*}[d] \\
& & \widetilde{H}_*((\Sigma X^k)^{\vee k}) \ar^{(\bigvee_{i=1}^k \Sigma \pi_i)_*}[d] & & \widetilde{H}_*(\Sigma X) \\
& & \widetilde{H}_*((\Sigma X)^{\vee k}) \ar^{\textrm{fold}_*}[r] & \widetilde{H}_*(\Sigma X) \ar^{(\Sigma \eta)_*}[r] & \widetilde{H}_*(\Sigma \Omega \Sigma X). \ar_{\textrm{ev}_*}[u] \\
}
\end{tabular}
\end{center}
The description of $\widetilde{\eta_*} \circ \iota_k$ of Proposition \ref{BSGen} implies that the top-right route round the diagram is equal to $\sigma \circ \Gamma \circ \iota_k$. The diagram shows that this factors through the zero map, so $\sigma \circ \Gamma \circ \iota_k = 0$, and since $\sigma$ is an isomorphism, this implies that $\Gamma \circ \iota_k$ is itself zero, which completes the $k>1$ case and hence the proof. \end{proof}
\subsection{Loops on homology injections} \label{FinnickySubs}
The goal of this section is to prove Proposition \ref{injection2}. We first prove two lemmas. Recall that $\iota_i : V^{\otimes i} \longrightarrow T(V)$ is the inclusion, and that $\zeta_i : T(V) \longrightarrow V^{\otimes i}$ is the projection. Similarly, let $\iota_{\leq k}$ and $\zeta_{\leq k}$ be the inclusion and projection associated to the submodule $\bigoplus_{i=1}^k V^{\otimes i}$ of $T(V)$.
\begin{lemma} \label{leadingTrims} Let $a_1, a_2, \dots a_k$ be elements of a tensor algebra $T(V)$. We have that $$\zeta_i (a_1 \otimes \dots \otimes a_k) = \begin{cases} \zeta_1(a_1) \otimes \dots \otimes \zeta_1(a_k) & i=k \\
0 & i<k.
\end{cases} $$ \end{lemma}
\begin{proof} By definition, $T(V) \cong \bigoplus_{i=1}^{\infty} V^{\otimes i}$, and the maps $\zeta_i$ are precisely the projections onto these summands. Further, the multiplication in $T(V)$ restricts to maps $V^{\otimes i} \otimes V^{\otimes j} \longrightarrow V^{\otimes (i+j)}$, which is to say that it is additive in weight. This gives the formula $\zeta_k(a \otimes b) = \sum_{i=1}^{k-1} \zeta_i(a) \otimes \zeta_{k-i}(b)$, which we will use to induct.
When $k=1$ the result is automatic. Assuming the result for $k-1$, we have $$\zeta_j(a_1 \otimes \dots \otimes a_k) = \sum_{i=1}^{j-1} \zeta_i(a_1 \otimes \dots \otimes a_{k-1}) \otimes \zeta_{j-i}(a_k).$$
By induction $\zeta_i(a_1 \otimes \dots \otimes a_{k-1}) = 0$ for $i<k-1$, so the above is $0$ when $j<k$ and when $j=k$ it becomes $$\zeta_{k-1}(a_1 \otimes \dots \otimes a_{k-1}) \otimes \zeta_1(a_k) = \zeta_1(a_1) \otimes \dots \otimes \zeta_1(a_{k-1}) \otimes \zeta_1(a_k),$$ by induction, as required. \end{proof}
The following lemma does not depend on the algebra structure in the tensor algebras; only on the fact that tensor algebras are graded by weight. Nonetheless, we will state it only for tensor algebras because we already have the necessary notation. It formalizes the sort of `leading terms' argument that we wish to make in proving Proposition \ref{injection2}.
\begin{lemma} \label{leadingTerms} Let $f: T(A) \longrightarrow T(B)$ be a homomorphism of $\mathbb{Z}/p^s$-modules (not necessarily of algebras) with $A$ free. Suppose that $p^{s-1}\zeta_j \circ f \circ \iota_k = 0$ whenever $j<k$ and that for each $k \in \mathbb{N}$, the map $\zeta_k \circ f \circ \iota_k$ is an injection. Then $f$ is also an injection. \end{lemma}
\begin{proof} Firstly, since $T(A)$ is a free $\mathbb{Z}/p^s$-module, it suffices to show that if $f(p^{s-1} x) = 0$, for $x \in T(A)$, then $p^{s-1} x = 0$. This is precisely showing injectivity of the restriction of $f$ to $p^{s-1}T(A)$. The module $T(A)$ is filtered by the submodules $\bigoplus_{i=1}^k A^{\otimes i}$ for $k \in \mathbb{N}$, so it further suffices to show that each map $$\zeta_{\leq k} \circ f \circ \iota_{\leq k} : p^{s-1} \bigoplus_{i=1}^k A^{\otimes i} \longrightarrow p^{s-1} \bigoplus_{i=1}^k B^{\otimes i}$$ is injective.
We proceed by induction. The case $k=1$ is immediate, so assume that the result is known for $k-1$. Write $\bigoplus_{i=1}^k A^{\otimes i} \cong \bigoplus_{i=1}^{k-1} A^{\otimes i} \oplus A^{\otimes k}$, so that $\iota_{\leq k}$ is identified with $\iota_{\leq (k-1)} \oplus \iota_k$. Suppose that $f(y)=0$ for $y \in p^{s-1} \bigoplus_{i=1}^k A^{\otimes i}$, so that there exists $x \in \bigoplus_{i=1}^k A^{\otimes i}$ with $y=p^{s-1}x$. We must show that $y=0$. Write $x = x' + x_k$, for $x' \in \bigoplus_{i=1}^{k-1} A^{\otimes i}$ and $x_k \in A^{\otimes k}$. Now, $$\zeta_{\leq (k-1)} \circ f \circ \iota_{\leq k} (y) = p^{s-1} \zeta_{\leq (k-1)} \circ f \circ \iota_{\leq k}(x) $$ $$= p^{s-1} \zeta_{\leq (k-1)} \circ f( \iota_{\leq (k-1)}x' + \iota_k(x_k)) = \zeta_{\leq (k-1)} \circ f(p^{s-1} \iota_{\leq (k-1)}x'),$$ since $p^{s-1}\zeta_j \circ f \circ \iota_k = 0$ for $j<k$. By inductive hypothesis, this implies that $p^{s-1} x' = 0$, so $y = x_k$, and $$\zeta_k \circ f \circ \iota_{\leq k} (y) = \zeta_k \circ f \circ \iota_{ k} (p^{s-1} x_k).$$
By assumption, $\zeta_k \circ f \circ \iota_{ k}$ is an injection, so $p^{s-1} x_k = 0$, and therefore $y=0$, as required. \end{proof}
\begin{proposition} \label{injection2} Let $X$ be a connected $CW$-complex, let $p$ be an odd prime, and let $s \leq r \in \mathbb{N}$. Let $\mu: P^{n+1}(p^r) \longrightarrow \Sigma X$ be a continuous map. If the induced map $$\mu_* : H_*(P^{n+1}(p^r);\mathbb{Z}/p^s) \longrightarrow H_*(\Sigma X;\mathbb{Z}/p^s)$$ is an injection, then $$(\Omega \mu)_* : H_*(\Omega P^{n+1}(p^r);\mathbb{Z}/p^s) \longrightarrow H_*(\Omega \Sigma X;\mathbb{Z}/p^s)$$ is also an injection. \end{proposition}
The principal difficulty in the proof is that $\textrm{Im}(\mu_*)$ might not be contained in the tensor algebra $T(\widetilde{H}_*( X;\mathbb{Z}/p^s))$ inside $\widetilde{H}_*(\Omega \Sigma X ; \mathbb{Z}/p^s)$. We navigate this using the condition $p^{s-1} C = 0$ of Proposition \ref{BSGen}, which prevents the complementary part $C$ from interfering too much. This proposition is much simpler to prove if one assumes that the map $\mu$ is a suspension, but this assumption is not necessary.
\begin{proof} Homology is taken with $\mathbb{Z}/p^s$-coefficients throughout. By the Bott-Samelson theorem (Theorem \ref{BS}), we have an isomorphism $$\widetilde{\eta_*} : T(\widetilde{H}_*(P^n(p^r))) \longrightarrow H_*(\Omega P^{n+1}(p^r)),$$ so it suffices to show that $(\Omega \mu)_* \circ \widetilde{\eta_*}$ is an injection. By definition, $(\Omega \mu)_* \circ \widetilde{\eta_*}$ is the unique map of algebras extending $$(\Omega \mu)_* \circ \eta_* : \widetilde{H}_*(P^n(p^r)) \longrightarrow H_*(\Omega \Sigma X),$$ and by the triangle identities for the adjunction $\Sigma \dashv \Omega$, we have that $(\Omega \mu) \circ \eta = \overline{\mu}$. Thus, $(\Omega \mu)_* \circ \widetilde{\eta_*}$ is the unique map of algebras extending $\overline{\mu}_*$.
The other triangle identity tells us that we have a commuting diagram \begin{center}
\begin{tabular}{c}
\xymatrix{
P^{n+1}(p^r) \ar^{\Sigma\overline{\mu}}[r] \ar_{\mu}[dr] & \Sigma \Omega \Sigma X \ar^{\textrm{ev}}[d] \\
& \Sigma X.
}
\end{tabular}
\end{center} By assumption, $\mu$ induces an injection on homology, so $\textrm{ev} \circ (\Sigma \overline{\mu})$ must also induce an injection on homology.
The next step is to turn the problem into one about tensor algebras. Proposition \ref{BSGen} gives a module decomposition of $\widetilde{H}_*(\Omega \Sigma X)$ as the direct sum $T(\widetilde{H}_*(X)) \oplus C$ with $p^{s-1} C = 0$. Under this decomposition, the inclusion associated to the factor $T(\widetilde{H}_*(X))$ is $\widetilde{\eta_*}$. Write $\tau$ for the projection. Consider the diagram \begin{center}
\begin{tabular}{c}
\xymatrix{
& T(\widetilde{H}_*(X)) \ar@/^/^{\sigma \circ \widetilde{\eta_*}}[d] \\
\widetilde{H}_*(P^{n+1}(p^r)) \ar^{(\Sigma\overline{\mu})_*}[r] \ar_{\mu_*}[dr] & \widetilde{H}_*(\Sigma \Omega \Sigma X) \ar^{\textrm{ev}_*}[d] \ar@/^/^{\tau \circ \sigma^{-1}}[u] \\
& \widetilde{H}_*(\Sigma X).
}
\end{tabular}
\end{center}
The maps $\sigma \circ \widetilde{\eta_*}$ and $\tau \circ \sigma^{-1}$ differ from $\widetilde{\eta_*}$ and $\tau$ only up to suspension isomorphisms, so they are the inclusion and projection associated to the decomposition of $\widetilde{H}_*(\Sigma \Omega \Sigma X)$ obtained by suspending that of Proposition \ref{BSGen}. Lemma \ref{factorTensor} (with $g = \textrm{ev}_*$, $f = (\Sigma \overline{\mu})_*$, $i_A = \sigma \circ \widetilde{\eta_*}$, and $\pi_A = \tau \circ \sigma^{-1}$) then tells us that the whole composite $\textrm{ev}_* \circ (\sigma \circ \widetilde{\eta_*}) \circ (\tau \circ \sigma^{-1}) \circ (\Sigma \overline{\mu})_*$ is an injection. Furthermore, by Lemma \ref{evproj}, the composite $\textrm{ev}_* \circ (\sigma \circ \widetilde{\eta_*})$ is identified via suspension isomorphisms with the projection $\zeta_1 : T(\widetilde{H}_*(X)) \longrightarrow \widetilde{H}_*(X)$, so the composite $\zeta_1 \circ \tau \circ \overline{\mu}_*$ is an injection.
Let $a$ and $b$ form a basis of the free $\mathbb{Z}/p^s$-module $\widetilde{H}_*(P^{n}(p^r))$. By Lemma \ref{summand}, the images of $a$ and $b$ under $\zeta_1 \circ \tau \circ \overline{\mu}_*$ generate a summand isomorphic to $(\mathbb{Z}/p^s)^2$ inside $\widetilde{H}_*(X)$.
Since $\widetilde{H}_*(P^{n}(p^r))$ is free on $a$ and $b$, a basis of $T(\widetilde{H}_*(P^{n}(p^r)))$ consists of the elements $x_1 \otimes \dots \otimes x_k$, for $k \in \mathbb{N}$, where each $x_i$ is equal to $a$ or $b$. We will show that the image of this basis under $(\Omega \mu)_* \circ \widetilde{\eta_*}$ is the basis of a free $\mathbb{Z}/p^s$-submodule of $H_*(\Omega \Sigma X)$, which will imply the result. Firstly, since $(\Omega \mu)_* \circ \widetilde{\eta_*}$ is the unique map of algebras extending $\overline{\mu}_*$, we have
$$p^{s-1} \zeta_j \circ \tau \circ (\Omega \mu)_* \circ \widetilde{\eta_*} (x_1 \otimes \dots \otimes x_k) = p^{s-1} \zeta_j \circ \tau (\overline{\mu}_*(x_1) \otimes \dots \otimes \overline{\mu}_*(x_k))$$ $$= \zeta_j \circ \tau ( p^{s-1} (\overline{\mu}_*(x_1) \otimes \dots \otimes \overline{\mu}_*(x_k))).$$
By Proposition \ref{BSGen}, we may write each $\overline{\mu}_*(x_i)$ as $\widetilde{\eta_*}(t_i)+c_i$, for $t_i = \tau(\overline{\mu}_*(x_i)) \in T(\widetilde{H}_*(X))$ and some $c_i$ with $p^{s-1} c_i = 0$. The above is therefore equal to $$\zeta_j \circ \tau ( p^{s-1} ((\widetilde{\eta_*}(t_1)+c_1) \otimes \dots \otimes (\widetilde{\eta_*}(t_k)+c_k)) = \zeta_j \circ \tau ( p^{s-1} (\widetilde{\eta_*}(t_1) \otimes \dots \otimes \widetilde{\eta_*}(t_k))$$ $$= \zeta_j ( p^{s-1} t_1 \otimes \dots \otimes t_k) = \begin{cases} p^{s-1} \zeta_1(t_1) \otimes \dots \otimes \zeta_1(t_k) & j=k \\ 0 & j<k \end{cases}$$ by Lemma \ref{leadingTrims}. Since $t_i = \tau(\overline{\mu}_*(x_i))$, we have $$p^{s-1} \zeta_1(t_1) \otimes \dots \otimes \zeta_1(t_k) = p^{s-1} (\zeta_1 \circ \tau(\overline{\mu}_*(x_1))) \otimes \dots \otimes (\zeta_1 \circ \tau(\overline{\mu}_*(x_k))).$$
Now, each $x_i$ is equal to $a$ or $b$, and we have seen that the images of $a$ and $b$ under $\zeta_1 \circ \tau \circ \overline{\mu}_*$ generate a $(\mathbb{Z}/p^s)^2$-summand inside $\widetilde{H}_*(X)$. It follows that the elements $\zeta_1 \circ \tau(\overline{\mu}_*(x_1)) \otimes \dots \otimes \zeta_1 \circ \tau(\overline{\mu}_*(x_k))$ generate a copy of $T((\mathbb{Z}/p^s)^2)$ inside $T(\widetilde{H}_*(X))$.
The above calculation therefore tells us that the map $\zeta_k \circ \tau \circ (\Omega \mu)_* \circ \widetilde{\eta_*} \circ \iota_k$ carries $p^{s-1}$ times a basis of $\widetilde{H}_*(P^{n}(p^r))^{\otimes k} \subset T(\widetilde{H}_*(P^{n}(p^r)))$ to $p^{s-1}$ times a basis of $((\mathbb{Z}/p^s)^2)^{\otimes k} \subset T((\mathbb{Z}/p^s)^2)$ inside $T(\widetilde{H}_*(X))$. This implies that the restriction of $\zeta_k \circ \tau \circ (\Omega \mu)_* \circ \widetilde{\eta_*} \circ \iota_k$ to $p^{s-1} \widetilde{H}_*(P^{n}(p^r))^{\otimes k}$ is an injection, so $\zeta_k \circ \tau \circ (\Omega \mu)_* \circ \widetilde{\eta_*} \circ \iota_k$ must itself be an injection and we have also seen that $p^{s-1}\zeta_j \circ \tau \circ (\Omega \mu)_* \circ \widetilde{\eta_*} \circ \iota_k = 0$ for $j<k$
Thus, by Lemma \ref{leadingTerms}, $\tau \circ (\Omega \mu)_* \circ \widetilde{\eta_*}$ is an injection, so $(\Omega \mu)_* \circ \widetilde{\eta_*}$ is an injection, as required. \end{proof}
\section{Proof of Theorems \ref{HPrelim} and \ref{HCriterion}} \label{proof2Section}
In this section we will prove Theorem \ref{HPrelim}, and then from that, together with Proposition \ref{injection2}, deduce Theorem \ref{HCriterion}.
\begin{proof}[Proof of Theorem \ref{HPrelim}] By Lemma \ref{toTheFields} it suffices to prove the theorem when $t=s$. Combining Lemmas \ref{PhiPiDiffs}, \ref{hurewiczTriangle}, and naturality of the maps $\beta^r$, $\rho^s$, and $h$ with respect to the map of spaces $\Omega \mu$, we obtain the following commuting diagram: \begin{center}
\begin{tabular}{c}
\xymatrix{
L'(x,dx) \ar_{d}[dd] \ar^(0.4){\Phi_\pi^{r,r}}[r] & \pi_*(\Omega P^{n+1}(p^r); \mathbb{Z}/p^r) \ar^{\beta^r}[d] \ar^(0.57){(\Omega \mu)_*}[r] & \pi_*(\Omega Y; \mathbb{Z}/p^r) \ar^{\beta^r}[d] \\
& \pi_*(\Omega P^{n+1}(p^r)) \ar^{\rho^s}[d] \ar^(0.57){(\Omega \mu)_*}[r] & \pi_*(\Omega Y) \ar^{\rho^s}[d] \\
L'(x,dx) \otimes \mathbb{Z}/p^s \ar^(0.45){\Phi_\pi^{r,s}}[r] \ar_{\theta}[d] & \pi_*(\Omega P^{n+1}(p^r); \mathbb{Z}/p^s) \ar^{h}[d] \ar^(0.57){(\Omega \mu)_*}[r] & \pi_*(\Omega Y; \mathbb{Z}/p^s) \ar^{h}[d] \\
L(x,dx) \otimes \mathbb{Z}/p^s \ar^(0.45){\Phi_H^{r,s}}[r] & H_*(\Omega P^{n+1}(p^r); \mathbb{Z}/p^s) \ar^(0.57){(\Omega \mu)_*}[r] & H_*(\Omega Y; \mathbb{Z}/p^s). \\
}
\end{tabular}
\end{center}
By Corollary \ref{combining}, $\textrm{Im}((\Omega \mu)_* \circ \Phi_H^{r,s} \circ \theta \circ d)$ is $\mathbb{Z}/p^s$-hyperbolic. By commutativity of the diagram, $(\Omega \mu)_* \circ \Phi_H^{r,s} \circ \theta \circ d = h \circ \rho^s \circ (\Omega \mu)_* \circ \beta^r \circ \Phi_\pi^{r,r}$, so the image of the latter map is also $\mathbb{Z}/p^s$-hyperbolic.
We thus obtain a diagram \begin{center}
\begin{tabular}{c}
\xymatrix{
\pi_*(\Omega Y) \ar^{h \circ \rho^s}[dr] & \\
\textrm{Im}((\Omega \mu)_* \circ \beta^r \circ \Phi_\pi^{r,r}) \ar^(.4){h \circ \rho^s}[r] \ar@{^{(}->}[u] & \textrm{Im}((h \circ \rho^s) \circ ((\Omega \mu)_* \circ \beta^r \circ \Phi_\pi^{r,r})).
}
\end{tabular}
\end{center}
The bottom map is a surjection by choice of codomain, and we have shown above that this codomain is $\mathbb{Z}/p^s$-hyperbolic. The domain of $(\Omega \mu)_* \circ \beta^r \circ \Phi_\pi^{r,r}$ is $L'(x,dx)$, which is a $\mathbb{Z}/p^r$-module, hence is automatically annihilated by multiplication by $p^r$. Therefore, the group in the bottom left, $\textrm{Im}((\Omega \mu)_* \circ \beta^r \circ \Phi_\pi)$, is also annihilated by multiplication by $p^r$. The group in the bottom right, $\textrm{Im}((\Omega \mu)_* \circ \beta^r \circ \Phi_\pi)$, is contained in $H_*(\Omega Y; \mathbb{Z}/p^s)$, hence is annihilated by multiplication by $p^s$. This means that we can apply Corollary \ref{surjectOnHype} (The `Sandwich' Lemma) to see that $\pi_*(\Omega Y) \cong \pi_{*+1}(Y)$ is $p$-hyperbolic concentrated in exponents $s, s+1, \dots, r$, so by definition $Y$ is $p$-hyperbolic concentrated in exponents $s, s+1, \dots, r$, which completes the proof. \end{proof}
Theorem \ref{HCriterion} now follows.
\begin{proof}[Proof of Theorem \ref{HCriterion}] By Proposition \ref{injection2}, $(\Omega \mu)_*$ is an injection, so by Theorem \ref{HPrelim}, $\Sigma X$ is $p$-hyperbolic concentrated in exponents $s, s+1, \dots, r$, as required. \end{proof}
\printbibliography
\end{document} | math |
जन्मदिन २५ नवंबर: शनिवार को करें यह कार्य, मिलेगी सफलता
(आज जिन जातकों का जन्मदिन है, उन्हें जन्मदिन की हार्दिक शुभकामनाएं।)
नया वर्ष रजत पाद से प्रवेश कर रहा है। सितंबर शेष में राजनीतिक क्षेत्र में लाभ मिल सकता है। नवंबर में ग्रहचाल मध्यम है, गुप्त शत्रुओं के षड़यंत्र से बचें। दिसंबर जनवरी पूर्वापेक्षा शुभ रहेगा। व्यय से अधिक लाभ होगा। कुशल जातकों के भाग्य का उदय फरवरी २०२० से लेकर मार्च तक होगा। बुध के सहयोग से अधिकांश जातकों की बौद्धिक, धार्मिक और व्यवसायिक प्रयास सार्थक होंगे।
अप्रैल और मई में कुछ अनावश्यक प्रपंचों के कारण परेशानी हो सकती है। बौद्धिक कार्यों से जुड़े महानुभाव इस वर्ष जून में अपने कामकाज का विस्तार करेंगे। जो जातक नौकरी पेशा हैं, उनका मनोवांछित जगह पर बदलाव भी जुलाई औप अगस्त तक मामूली प्रयासों से हो जाएगा। सितंबर १९ से लेकर ३० अगस्त तक सभी स्थितियां सामान्य सुखद रहेंगी।
महिलाओं के लिए वर्ष अधिक लाभप्रद है। विद्यार्थियों को नियमित सहज परिश्रम से ही सफलता प्राप्त हो जायेगी। मंगल आपकी राशि के अनुकूल नहीं है अतः श्री सुंदरकांड पाठ एक नित्य रात्रि में ८ बजे के उपरांत शुद्ध धृत का दीपक जलाकर ४० दिन पर्यन्त करें। शनिवार को सायंकाल पीपल के नीचे सरसों का दीया जलाएं।
जन्मदिन २५ नवंबर
जन्मदिन १८ जनवरी: कुशलता और एकाग्रता से सभी काम हो जाएंगे सरलजन्मदिन १७ जनवरी: कठीन कार्य सरल करने के लिए करें यह कार्यजन्मदिन १६ जनवरी: पराक्रम में वृद्धि के योग के साथ रचनात्मक कार्यों में प्रगति होगीजन्मदिन १५ जनवरी: आजीविका के क्षेत्र में विशेष उपलब्धियां प्राप्त होंगीजन्मदिन १४ जनवरी: इनके पूजन से मिलेगा रूका हुआ धनजन्मदिन १३ जनवरी: जानिए, कैसा रहेगा आपके लिए यह सालजन्मदिन १२ जनवरी: महिलाओं को कार्यक्षेत्र में उन्नति के साथ यश प्राप्ति का योगजन्मदिन ११ जनवरी: आमदनी बढ़ने के साथ पर्यटन का भी मिलेगा आनंदजन्मदिन १० जनवरी: ४० दिन तक करें यह उपाय, मिलेगी हर जगह सफलताजन्मदिन ०९ जनवरी: विद्यार्थियों के लिए साल रहेगा उत्तमजन्मदिन ८ जनवरी: इस उपाय व्यवसाय में आ रही बाधा होगी दूरजन्मदिन ७ जनवरी: शनिवार को करें यह पाठ, होगी सभी मनोकामना पूरीजन्मदिन ०५ जनवरी: विद्यार्थियों के लिए अत्यंत शुभ है यह सालजन्मदिन ०४ जनवरी: सतर्क रहें, सोच-समझकर करें भरोसा, विश्वासघात हो सकता हैजन्मदिन ०३ जनवरी: जानें, यह साल आपके लिए क्या कुछ लेकर आया है अपनी झोली में? | hindi |
"आर्थिक आदमी" वोट नहीं देंगे, लेकिन मनुष्य क्या करेंगे मनोविज्ञान दुनिया
प्रेरणा विवेक सरकार स्वास्थ्य
चुपके मतदाताओं की जीत के लिए ट्रम्प पोल वॉल्ट की सुविधा अपने कसरत के दौरान रोकें बेहतर परिणाम प्राप्त करने के लिए हम अपने बच्चों को छुट्टी कार्ड के रूप में क्यों भेजें? प्रधान और पूर्वाग्रह: हम सब थोड़ी बिट जातिवादी क्यों हैं प्रारंभिक विषाक्त तनाव परिवर्तन मस्तिष्क संरचना धन्यवाद: स्वस्थ से खुश क्यों अधिक महत्वपूर्ण है? # १ रास्ता आप इसे जानने के बिना अपने साथी को शत्रुतापूर्ण कर सकते हैं स्कूल निशानेबाज़ कौन सफेद नर नहीं हैं सिमुलेशन उत्तेजना अपने बच्चों के साथ सबसे बुरी लड़ाई, अब क्या? समानता के बारे में क्या अच्छा है? कुत्ते की आवश्यकता के एक पदानुक्रम: इब्राहीम मास्लोव मॉट्स को मिलता है जब मैं आत्मघाती था तब साइलेंट चर्च ने मुझे विफल कर दिया था हम शुरू से ही अपने सर्वश्रेष्ठ मित्रों के साथ क्यों क्लिक करते हैं बच्चों का ओसीडी प्रभाव पूरे परिवार कैसे करता है
डेढ़ दशक पहले के बारे में, मैकआर्थर फाउंडेशन ने अर्थशास्त्र और अन्य सामाजिक विज्ञानों में विद्वानों के ढीले संगठित नेटवर्क का समर्थन करना शुरू किया, जिसका लक्ष्य मानवीय प्रेरणा के बारे में धारणाओं पर चर्चा करना था, जिस पर आर्थिक मॉडल आमतौर पर बाकी है। यह विचार यह देखना था कि उन मान्यताओं को विस्तारित करने की आवश्यकता है, दोनों महत्वपूर्ण मानवीय कार्यों के बेहतर खाते और सामाजिक समस्याओं को संबोधित करने के लिए बॉक्स के बाहर सोचने के लिए। मुझे याद है कि नेटवर्क के लिए लिखे गए एक शुरुआती पोजिशन पेपर को याद किया गया है जिसमें मुख्य उदाहरण सूचीबद्ध किए गए हैं, जहां मानक धारणा है कि लोगों को उनकी स्वयं की भौतिक भलाई के बारे में केवल हर रोज़ अनुभव से इनकार कर दिया जाता है। इन प्रमुख उदाहरणों में कुछ लाखों अमेरिकियों कल कल करेंगे: वोट करने के लिए एक मतदान स्थान पर जा रहे हैं।
हम विशेष रूप से इस वर्ष की तरह एक साल में मतदान के बारे में सोचते हैं, जैसा कि आर्थिक प्रेरणा के साथ अच्छी तरह से शामिल किया गया है। दरअसल, बिल क्लिंटन की बीसवीं वर्षगांठ की यह सफल है "यह अर्थव्यवस्था है, बेवकूफ" चुनाव अभियान, और "व्याख्याता-इन-चीफ" क्लिंटन फिर से आगे बढ़ रहे हैं। राष्ट्रपति ओबामा के साथ-साथ, वह अमेरिकियों से आग्रह कर रहे हैं कि डेमोक्रेट के सामने आने वाले आर्थिक दृष्टिकोण पर वापस जाने के लिए न तो हम संकट से बाहर निकलने के लिए संघर्ष कर रहे हैं। रिपब्लिकन चैलेंजर मिट रोमनी ने मतदाताओं को याद दिलाया और कहा कि उन्हें एक व्यवसाय चलाने का तरीका पता है और इसलिए हमें गड़बड़ी से बाहर ले जाने का अधिकार है। *
लेकिन अगर आप एक अर्थशास्त्री से पूछते हैं जो आर्थिक सिद्धांतों में घिरा हुआ है, तो वह आपको बताएगी कि चुनाव में जाने के लिए तर्कसंगत विकल्प के सिद्धांतों का एक कट्टरपंथी उल्लंघन है। ऐसा इसलिए है क्योंकि हमेशा कुछ खर्च होता है, हालांकि, मामूली, किसी भी समय कुछ भी करने के बजाय चुनाव में स्वयं को प्राप्त करने में, यह आपको जो भी समय लेता है। और तर्कसंगतता के लिए आवश्यक है कि हम ऐसी लागतों को केवल तभी ले लें जब एक ऑफसेटिंग लाभ होता है यहां तक कि अगर ओबामा या रोमनी का चुनाव आपकी पॉकेटबुक के लिए अच्छा होगा, तो वो वोटिंग में शामिल लागतों को सही नहीं ठहराएगा, क्योंकि संभावना है कि आपके व्यक्तिगत वोट का नतीजा पर असर पड़ेगा, यह बहुत कम है। पिछली बार जब आपने १ वोट के मार्जिन के आधार पर चुनाव के बारे में सुना था? अमेरिकी राष्ट्रमंडल और कांग्रेस दौड़ में, कुछ सौ वोटों के मार्जिन को रेज़र पतली माना जाता है, लेकिन ऐसी तंग प्रतियोगिताओं में भी कोई भी मतदाता अपने उम्मीदवार के प्रति नतीजों का शाब्दिक सुझाव नहीं देता है उसी उम्मीदवार ने उसके वोट के साथ या उसके बिना जीत हासिल कर ली होती। यहां तक कि यदि उम्मीदवार जेन की जीत वोटर जो की पॉकेटबुक के लिए अच्छा है, तो उसके साथ या उसके बिना (या नहीं हुआ) हुआ होता, और उसने गैसोलीन या बस किराए पर बचत करके और समय व्यतीत करने से अपनी पॉकेटबुक को अधिक मदद दी हो। अतिरिक्त पैसा, बेहतर बिक्री ढूंढना, या अपनी चेकबुक को संतुलित करना
तो हम वोट क्यों करते हैं? यदि आप एक अर्थशास्त्री हैं, लेकिन एक अर्थशास्त्री नहीं हैं, तो आपका "आंतरिक कांत" शायद ऊपर और नीचे कूद रहा है, बस इसके बारे में, विरोध करते हुए कि आखिरी अनुच्छेद में यह सब गलत है अगर सभी ने तर्क दिया कि उनके मत में कोई फर्क नहीं पड़ता है, तो कोई भी वोट नहीं देगा, जिसमें कुछ सौ शामिल हैं जो रेज़र-पतली मार्जिन प्रदान करते हैं। क्या सही है, हमारे कांतियन विवेक कहते हैं, हमारे लिए वह करना है जो हम चाहते हैं कि दूसरों को करना चाहिए और यदि हर कोई तदनुसार कार्य करता है, तो गली से, हमारे पास चुनाव होगा और सबसे पसंदीदा उम्मीदवार जीतेंगे
यह तर्क अपील कर रहा है, लेकिन होमो इकोनॉमिकस या "स्वार्थी आर्थिक आदमी" के संकीर्ण अर्थों में हमारे स्व-हित के लिए नहीं। बल्कि यह हमारे समूह के पशु या सामाजिक नस्लों के लिए अपील करता है, जैसा ईओ विल्सन, फ्रांस डी वाल , जोनाथन हैड, और मेरी किताब, द गुड, द बैड, और द इकोनॉमी में हम खुद को समूह के सदस्यों के रूप में देखते हैं, और हम समूह के मामलों में खुद को शामिल करने के लिए तैयार हैं, कभी-कभी तो तर्कशास्त्र के सबसे अधिक व्यक्तिपरक के विपरीत।
हमारे सामाजिक नस्लों के बिना, लोकतांत्रिक रूप से जवाबदेह सरकार असंभव होगी न केवल मतदान के द्वारा समय और पैसा बचाएगा और कभी भी राजनीतिक अभियानों में मदद करने के लिए स्वयंसेवा नहीं करेगा वे राजनैतिक मुद्दों और उम्मीदवारों के विचारों के बारे में सीखने में समय बर्बाद भी नहीं करना चाहते थे, और पत्रकारिता के लिए कोई वित्तीय स्थिति नहीं थी, क्योंकि अखबारों और पत्रिकाओं के लिए कोई पाठक नहीं होगा, रेडियो और टेलीविजन समाचार के लिए कोई दर्शक नहीं होगा । सरकारी भ्रष्टाचार पर एकमात्र संभव जांच का भुगतान किया जाएगा जो काउंटर मॉनिटरिंग की परत पर परत के बिना भ्रष्ट होगा। हिरन कहीं भी नहीं रोकेंगे, क्योंकि कोई उस मकसद से कोई भी मकसद नहीं लगा सकता है, जो उस सर्वशक्तिमान हिरण के पीछा के अलावा
दुर्भाग्य से, आर्थिक कैलकुस मार्जिन पर मतदान को प्रभावित करता है। अधिक आर्थिक रूप से सफल अधिक आसानी से चुनावों में आने के लिए परिवहन के समय और साधनों को आसानी से उठा सकते हैं, यहां तक कि मुद्दों पर अध्ययन करने का समय भी है, और यह कुछ भी बताता है कि वे बड़ी संख्या में वोट क्यों देते हैं। यह भी बताता है कि राजनीतिक अधिकार मतदान को और अधिक कठिन बनाने के लिए क्यों काम करता है, और क्यों छोड़ दिया इसके "ग्राउंड गेम" पर इतना जोर डालता है।
कल अपने मतदान स्थान पर जाएं, अगर आपने पहले ही मतदान नहीं किया है और जब आप करते हैं, तब भी अगर आप उम्मीदवार के लिए वोट करने की योजना बना रहे हैं, तो आप अपने पॉकेटबुक के लिए सबसे अच्छा सोचते हैं, अपने भाग्यशाली सितारों का धन्यवाद करें कि आप सामाजिक रूप से दिमाग की जातियों का हिस्सा हैं और हमारे बारे में कुछ ऐसा है जो हमें हमारे बारे में ध्यान दिलाता है एक समूह के सदस्यों के रूप में भूमिकाएं, और अकेले दिमाग में नहीं और स्वयं के बारे में ही। हम अपने अधिकारों, स्वास्थ्य, सुरक्षा और संपत्ति की रक्षा के लिए सरकार का उपयोग करने में सक्षम हैं, क्योंकि हम समूहों के सदस्यों के रूप में सोचने के लिए इच्छुक हैं, चाहे हम अपनी प्रवृत्ति का प्रयोग कर रहे हैं या फिर पहले से हमारे सुधारों को बेहतर बनाने के लिए हमारे समाज का
* मैंने कुछ तरीकों पर चर्चा की जिसमें हाल ही में पोस्टिंग में गैर-आर्थिक या गैर-स्वार्थी विचारों से मतदान प्रभावित हुआ, अमेरिकी राजनीति और निष्पक्षता के दो चेहरे | hindi |
By Who’s Your? Beauty Contributor, Claire Vyverberg.
Philosophy Kiss Me – A gentle sugar-based exfoliator that instantly works to scrub and smooth rough, flaky and dehydrated lips. Infused with a blend of oils and a hint of peppermint for an instant breath-freshening effect.
Kit Cosmetics Multipurpose Wonder Balm – There is nothing that this balm cannot do. Certified organic and containing five of the finest essential oils, this hydrating and antioxidant powerhouse balm will heal cracks in your lips and your cuticles. A winter weather handbag essential!
Smashbox O-Plump Intuitive Lip Plumper - An ‘intuitive’ lip plumper that goes on clear and then reacts with your personal chemistry to turn lips your own custom shade of pink, while instantly plumping to perfection. Packed with botanicals and revitalizing ingredients, it is the perfect companion to complete your winter lip maintenance routine. | english |
پیٚپَر فرٛای پؠٹھٕ کٟتیس چھُ مےٚ خٕریدٲری کرٕنۍ میٲنۍ کوپن ایکٹویٹ کرنہٕ خٲطرٕ | kashmiri |
'कागद हो तो हर कोई बांचे.: चांद तो गांव में उगता है
चांद तो गांव में उगता है
गांव से फोन आयाचांद निकल आया हैव्रत खोल लोसमाचार पा करपांचवी मंजिल केपूर्वी पिछवाड़े की औरदिया चंद्रदेव को अर्ध्यकैलेण्डर में देखा चांदव्रत खोल-पाया भोजनसोते वक्तबच्यों को सुनाईचंदा मामा की कहानीबच्चों ने पूछामां,कैसा होता है चांदहमें दिखाओ नाखिड़की खोल कर !मां बोलीयहां कहां उगता है चांदचांद तो गांव में उगता हैइस बार गांव चलेंगेतब दिखाउंगीअब खिड़की बंद करोबाहर पॉल्यूशन हैटीवी ऑन कर लोआज करवा चौथ हैदिखा रहा होगाकोई न्यूज चैनलफिलहाल उसी मेंदेख लो चांद ! | hindi |
<?php
defined('_JEXEC') or die('Restricted access');
?>
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echo $this->generalInfo;
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Related <b>News</b>
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If you use <?php echo @$this->attr['label']?>, please post a rating and a review at the Joomla! Extensions Directory. With this small gesture you will help the community a lot. Thank you very much!
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For inoculation of nutrient media. Its high flexibility permits gentle streaking without damaging the surface of a nutrient medium. Made of polystyrene with loop at one end and needle at the other end. The loops are gamma sterilized, for single use with 20 pieces in a bag, 50 bags in a carton. | english |
Infertility books are a key player in the journey to parenthood when the journey becomes complicated, confusing and difficult.
Not being able to become pregnant or carry a child full term is one of the hardest things a woman will ever face. I personally suffered three miscarriages, but consider myself blessed to have eventually birthed three healthy babies, 2 of them after a MTHFR diagnosis.
Infertility was SO DIFFICULT and such a raw time in my life.
The following are 9 powerful infertility books that can help bring you the results you desire – a family.
The following two infertility books should be read before seeing a doctor or an infertility specialist.
One of my favorite books of all time is Taking Charge of Your Fertility (TCOYF). It is an eye opening book that gives insight into the basics of the female body. TCOYF demystified my fertility and how everything worked (or should work) together. My body suddenly made sense to me. It was as if a lightbulb turned on – brightly!
The first step in your infertility journey is to read Taking Charge of Your Fertility. It will teach you the basics, and help you to overcome any potential preliminary obstacles to becoming pregnant.
The Sperm Meets Egg Plan is another book aimed at solving basic problems becoming pregnant. By using this book you may discover you are not actually infertile, but possibly just misinformed (or uninformed).
This book will help the reader create a “baby making schedule” of sorts, demystify ovulation test strips and give great tips to increase fertile cervical fluid.
This is a quick read, and free to read on a Kindle or the free Kindle app. Check it out!
So… you’ve been diagnosed with infertility. Is your head spinning and you’re wondering what to do next?
Reading Navigating the Land of IF will be as if a close friend, that went through the same thing a few years ago, took you aside and gave you all the nitty, gritty details, tips, tricks and hacks on everything infertility. Melissa Ford writes as a friend in your corner.
Melissa will guide you through the acronyms, the procedures and the medications. She’ll help you to know what to expect so that there’s a little bit less of a learning curve throughout your journey. If you want to get a sense of her writing style, check out her blog.
Many LOVE the book described above (Navigating the Land of IF), but sometimes the facts, and not the life story is what’s needed. If this sounds familiar, you’ll love this book. What to Do When You Can’t Get Pregnant is extremely thorough!
Polycystic Ovary Syndrome, or PCOS, is a common infertility diagnosis. If you have been diagnosed with PCOS this workbook is for you.
This is a workbook and like any other workbook out there, prepare yourself to… work! In this guide you will find a lot of amazing information about PCOS and will be challenged to apply it to you and your specific situation.
The workbook is written by a registered dietitian because, as you’ll discover, there’s a huge link to the symptoms of PCOS and diet. One of my favorite things about this workbook is that the author embraces a whole body approach – she not only discusses diet, but emotional health too.
Empty Womb, Aching Heart could easily have been titled “Chicken Soup for the Infertile Christian”. Like the famous Chicken Soup series, this book contains a collection of honest, heartwarming stories, that focus on infertility and faith.
If you’re feeling alone and isolated, but are looking for a Christian take on your experience, this book will leave you feeling validated but also encouraged on your journey of infertility and your walk with the Lord.
If you are considering In Vitro Fertilization and are in need of a personal guide, The Couple’s Guide to In Vitro Fertilization is your book! Much of the infertility information out there is dry, medical knowledge, but this book makes things simple, informative and personal.
Believe it or not, this book is a page turner that will appeal to both partners – not just mom!
The Couple’s Guide covers the preparation, financial aspects, medications, the emotions and the actual procedures. Author Liza Charlesworth didn’t speak only to her own experience, but the varied experiences of many other couples as well. As we know, each infertility journey is unique, so you will find these other stories very helpful.
If you have used non-traditional avenues such as surrogacy or a donor to become a parent, you might want to check out Mommies, Daddies, Donors, Surrogates. This is a thought provoking book that explores the dynamics of creating and raising such a child.
Before choosing surrogacy or using a donor, I think it’s important to think about various aspects of this path. How will you answer your child’s question about how they were made? What will you tell extended family members? How will using a donor impact your relationship with your partner?
If you’re ready to dive in and think about these things, this book will be an invaluable tool.
If you are battling infertility and still trying to conceive (TTC) I would caution you against reading Silent Sorority. However, if you find yourself at the end of your infertility journey and ultimately child-free, this book is for you.
This book is also helpful to friends and family members to better understand the difficult and intense emotional aspects of infertility.
My favorite part about Silent Sorority is that ultimately the reader will no longer feel alone and isolated, but validated in everything she’s gone through – frustration, sadness, anger, rage, and the funny parts too.
What Are Your Favorite Infertility Books?
Leave me a comment and let me know what your favorite infertility books are. Thank you so much! | english |
# Spring Data JPA - JPA 2.1 example
This project contains samples of JPA 2.1 specific features of Spring Data JPA.
## Support for declarative Fetch Graphs customization
You can customize the loading of entity associations via EntityGraphs. JPA 2.1 provides the `NamedEntityGraph` annotation
that allows you define fetching behavior in a flexible way.
In Spring Data JPA we support to specify which fetch-graph to use for a repository query method via the `EntityGraph` annotation.
You can refer to a fetch graph by name like in the following example.
```java
@EntityGraph("product-with-tags")
Product findOneById(Long id);
```
We also offer an alternative and more concise way to declarativly specify a fetch graph for a repository query method in an
ad-hoc manner:
```java
@EntityGraph(attributePaths = "tags")
Product getOneById(Long id);
```
By explicitly specifying which associations to fetch via the `attributePaths` attribute you don't need to specify a
`NamedEntityGraph` annotation on your entity :)
## Support for stored procedure execution
You can execute stored procedures either predefined using the JPA 2.1 mapping annotations or dynamically let the stored procedure definition be derived from the repository method name.
Stored procedure declaration in the database (schema.sql):
```sql
DROP procedure IF EXISTS plus1inout
/;
CREATE procedure plus1inout (IN arg int, OUT res int)
BEGIN ATOMIC
set res = arg + 1;
END
/;
```
JPA 2.1 stored procedure declaration:
```java
@Entity
@NamedStoredProcedureQuery(name = "User.plus1", procedureName = "plus1inout", parameters = {
@StoredProcedureParameter(mode = ParameterMode.IN, name = "arg", type = Integer.class),
@StoredProcedureParameter(mode = ParameterMode.OUT, name = "res", type = Integer.class) })
public class User {
@Id @GeneratedValue//
private Long id;
}
```
Spring Data JPA repository declaration to execute procedures:
```java
public interface UserRepository extends CrudRepository<User, Long> {
// Explicitly mapped to named stored procedure {@code User.plus1} in the {@link EntityManager}.
// By default, we would've try to find a procedure declaration named User.plus1BackedByOtherNamedStoredProcedure
@Procedure(name = "User.plus1")
Integer plus1BackedByOtherNamedStoredProcedure(@Param("arg") Integer arg);
// Directly map the method to the stored procedure in the database (to avoid the annotation madness on your domain classes).
@Procedure
Integer plus1inout(Integer arg);
}
```
Calling `UserRepository.plus1BackedByOtherNamedStoredProcedure(…)` will execute the stored procedure `plus1inout` using the meta-data declared on the `User` domain class.
`UserRepository.plus1inout(…)` will derive stored procedure metadata from the repository and default to positional parameter binding and expect a single output parameter of the backing stored procedure.
## Support for custom SqlResultSetMapping with ConstructorResult
Sometimes, e.g. for analytics, it is handy to be able to return a different entity result type from a Repository query method than the base Repository entity type or an interface based projection.
In those cases one can leverage JPAs `SqlResultSetMapping` feature to map the columns of the result of a query to different fields.
JPA 2.1 introduced the new `SqlResultSetMapping` type `ConstructorResult` which allows to map columns of a result set row to a constructor invocation
which can be nicely used in combination with Value Objects.
This example shows how to define a custom `SqlResultSetMapping` for the result of an analytical native query that reports the usage summary for a set of Subscriptions.
`SqlResultSetMapping` definition on the `Subscription` entity class:
```java
@Entity
@NoArgsConstructor
@SqlResultSetMapping(
name = "subscriptionSummary",
classes = @ConstructorResult(
targetClass = SubscriptionSummary.class,
columns = {
@ColumnResult(name = "productName", type = String.class),
@ColumnResult(name = "subscriptions", type = long.class)
}))
@NamedNativeQuery(
name = "Subscription.findAllSubscriptionSummaries",
query = "select product_name as productName, count(user_id) as subscriptions from subscription group by product_name order by productName",
resultSetMapping = "subscriptionSummary"
)
@Data
public class Subscription {
…
}
```
`SubscriptionSummary` is modelled as a value object:
```java
@Value
public class SubscriptionSummary {
String product;
Long usageCount;
}
```
The `SubscriptionRepository` declares the custom query method `findAllSubscriptionSummaries` which is backed by the named native query declared on the `Subscription` entity.
```java
interface SubscriptionRepository extends CrudRepository<Subscription,Long> {
@Query(nativeQuery = true)
List<SubscriptionSummary> findAllSubscriptionSummaries();
}
``` | code |
वृहद वृक्षारोपण अंतर्गत जिला एवं अनुभाग स्तरीय पर्यवेक्षण समिति गठित
शासन के निर्देशानुसार ०२ जुलाई को नर्मदा बेसिन स्थित जिलों में एक दिवसीय वृहद वृक्षारोपण कार्य किया जाना है। वृहद वृक्षारोपण कार्यक्रम में शासन द्वारा जिले के लिये ३० लाख पौधों का लक्ष्य निर्धारित किया गया है। जिसमें वनमंडल उत्तर सामान्य वनमंडल द्वारा१८ लाख, मंडल प्रबंधक बरघाट परियोजना मंडल सिवनी २ लाख, पंचायत एवं ग्रामीण विकास विभाग ६ लाख, किसान कल्याण एवं कृषि विकास विभाग ६ लाख, उद्यानिकी १ लाख, शिक्षा विभाग १ लाख के पौधा रोपण कार्य वनक्षेत्र, सामुदायिक भवनों, निजी भूमियों, शासकीय कार्यालय, आंगनबाडी एवं शासकीय स्कूलों में किया जायेगा।
वृहद वृक्षारोपण कार्य हेतु कलेक्टर श्री धनराजू एस द्वारा जिला स्तरीय अनुश्रवण एवं पर्यवेक्षण समिति (ट्रास्क फोर्स) एवं अनुभाग स्तरीय अनुश्रवण एवं पर्यवेक्षण समिति (ट्रास्क फोर्स) का गठन किया गया है। जिला स्तरीय समिति का गठन जिला पंचायत मुख्य कार्यपालन अधिकारी अध्यक्षता में किया गया है। जिसमें वनमंडलाधिकारी दक्षिण (सामान्य) एवं वनमंडलाधिकारी उत्तर (सामान्य) वनमंडल, मंडल प्रबंधकबरघाट परियोजना मंडल, अपर कलेक्टर, किसान कल्याण एवं कृषि विकास विभाग उपसंचालक, आत्मा परियोजना परियोजना संचालक, सहायक संचालक उद्यान, जिला शिक्षा अधिकारी, सहायक आयुक्त आदिवासी विकास, परियोजना समन्वयक जिला शिक्षा केन्द्र, जिला कार्यक्रम अधिकारी एकीकृत बाल विकास एवं मुख्य चिकित्सा एवं स्वास्थ्य अधिकारी को सदस्य के रूप में नियुक्त किया गया है।
इसी प्रकार अनुभाग स्तरीय समिति का गठन अनुविभागीय अधिकारी राजस्व की अध्यक्षता में किया गया है जिसमें मुख्य कार्यपालन अधिकारी जनपद पंचायत, अनुविभागीय अधिकारी वन, अनुविभागीय अधिकारी कृषि, तहसीलदार, नायब तहसीलदार, उद्यान विकास विस्तार अधिकारी, वरिष्ठ कृषि विकास एवं विस्तार अधिकारी, ब्लाक विकासखंड अधिकारी, मंडल संयोजक, परियोजना अधिकारी, एकीकृत बाल विकास, विकासखंड स्त्रोत समवन्यक, खंड चिकित्सा अधिकारी को सदस्य के रूप में नियुक्त किया गया।
यह समितियां एक दिवसीय वृहद वृक्षारोपण हेतु विभागीय लक्ष्यानुसार वन क्षेत्रों, सामुदायिक एवं निजी भूमियों में वृक्षारोंपण की आवश्यक तैयारियों का शासन के निर्देशानुसार जारी मापदंडों के अनुसार वृक्षारोपण की कार्यवाही संपादित करने की उत्तरदायी होगी। यह समितियां सप्ताह में एक बार वृक्षारोपण की प्रगति, समस्याओं, समस्याओं के निराकरण के संबंध में बैठक कर आवश्यक कार्यवाही करेगी। समिति के सदस्य अपने-अपने कार्य क्षेत्रान्तर्गत वृक्षारोपण के स्थल का निर्धारण, पौधों की सुरक्षा एवं वृक्षारोपण के समस्त कार्यो को सम्पन्न कराये जाने हेतु उत्तरदायी होंगे। | hindi |
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Amongst probably the most widespread electrical wiring issues is regarding how to wire a swap. Even though applying switches at your home is kind of easy, wiring 1 might not be that easy for everybody. An ON-OFF switch is in fact quite simple to wire. You will find various kinds of switches, but for this instance, let's say you will be installing a single-pole toggle switch, an exceedingly popular swap (and also the most basic).
You will find three colors of wires in the typical single-pole toggle switch: black, white, and eco-friendly. Splice the black wire in two and join them about the terminal screws - a person on top rated as well as the other about the base screw with the swap. The white wire serves to be a resource of uninterrupted energy and is also ordinarily related to a mild coloured terminal screw (e.g., silver). Hook up the eco-friendly wire into the ground screw within your swap.
These actions would frequently be ample to generate a normal change operate and not using a trouble. Nevertheless, should you are not self-confident you can accomplish the task adequately and properly you far better allow the professionals get it done alternatively. Right after all, there's a rationale why this job is without doubt one of the commonest electrical wiring thoughts questioned by many people.
For many reason, tips on how to wire a ceiling fan can also be certainly one of probably the most popular electrical wiring inquiries. To simplify this task, you should utilize a single swap for a single ceiling fan. To wire the enthusiast, it is really simply a make any difference of connecting the black wire of the ceiling fan for the black wire of the swap. When there is a lightweight, the blue wire needs to be linked for the black wire in the switch likewise.
You'll find good reasons why these are definitely the most typically requested electrical wiring issues. 1, several think it really is basic to accomplish, and two, these are the typical electrical jobs at home. But then you definitely should not set your security in danger in your aim to save money. The stakes could even be considerably higher in the event you try and cut costs and do an electrical wiring career without adequate expertise or working experience. | english |
I whipped up a little batik mug rug last night that is made from the scraps from my Intertwined by the River quilt. I know I’ve said it before but I can’t throw those beautiful little scraps away so I just start making fabric and see what ends up being created.
If you missed this quilt just click on the photo to find out more about it.
I finished one more blue string block for my quilt as you go (QAYG) that I’ll put together in December. Be sure and check out what everyone else is creating at Scraphappy Saturday.
As soon as the blocks were on the floor Simon had to come over and lay on them. Looks like he is getting ready to say something!
Today Karen at Sew Many Ways is having a Thread Organizing linking party to show how you store organize your threads.
As you can see mine is all over the place! Since I use my 15-91 Singer which is 61 years old for 90% of my quilting I am fortunate that I can use just about any thread. Those old machines will use anything!
It is true that some of the threads I have produce a lot of lint but I always have my lint brush close by so I don’t worry. How do you store your threads??
So many beautiful pictures, candy for my eye! I love how your Rainbow String blocks look together.
Love both blue blocks this month. Thread….. Mine is all over the place. I really do need to organize it up a bit. Such a pretty rainbow of colors.
I really like your mug rug idea—I was in the quilt shop planning a project today, and the clerk kept asking me if I had scraps I could use. When I really think about it, I don't have many scraps. When I finish a project, I usually make make-up bags, notebook covers, etc. until the scraps are gone. Plus, I make allot of fat quarter quilts, and most of them don't seem to leave allot of scraps. I'm making a Yellow Brick Road quilt now, and I was disappointed to see that there aren't even enough scraps for one single little zipper pouch.
All your strings together look yummy – looking forward to the final quilt assembly!
and i'm still wondering when i will use it all.
Great blue scrap string blocks. Love all the thread. I should show off mine some day.
All three items are beautiful…the mug rug, love it! The quilt is gorgeous…wish it were mine and the quilt as you go strings….yum! Give Simon a scratch for me, hes such a purdy boy!
Oh my gosh, your threads look so amazing! I just love looking at threads and fabrics and how people organize them. My fabric is organize now, I should do my thread now!
beautiful…I DO love thread…LOL. Love the new button!
I love your little mug rug, and I'm glad to see I am not the only one with TONS of thread!!! Have a great weekend!
After I saw your post with your ruler holder, I figured out something for holding mine, too. I bought a steel shelf from Ikea that has s-hooks, and I hang them all there.
It's nice you can use all different kind of threads. My Janome is tempermental and I like Connecting Threads the best, and Gutermann if I need to match something perfectly so I can go to Joanns.
What a cute mug rug. You always have the prettiest creations!
Cute mug rug…there's no reason to throw out batik scraps when you can make something useful with them!!!
Love your mugrug a perfect way to use those scraps 🙂 I organize my threads pretty much the same way you do and I have a lot of it .
What a great blog post! I just love your scrappy challenge blocks and think it will be a spectacular quilt!
I will hop over and check out the thread party…love your thread collection! | english |
using FluentValidation;
using System;
using System.Collections.Generic;
using System.Linq;
using System.Net;
using System.Net.Http;
using System.Web;
using System.Web.Http;
using System.Web.Http.Filters;
using TestCase.Service.Infrastructure.Exceptions;
namespace TestCase.WebApi.Infrastructure.Filters
{
/// <summary>
/// Exception filter.
/// </summary>
/// <seealso cref="System.Web.Http.Filters.ExceptionFilterAttribute" />
public class ExceptionFilter : ExceptionFilterAttribute
{
public override void OnException(HttpActionExecutedContext context)
{
if (context.Exception is HttpResponseException)
{
return;
}
var statusCode = HttpStatusCode.InternalServerError;
if (context.Exception is ValidationException)
{
statusCode = HttpStatusCode.BadRequest;
}
else if (context.Exception is UnauthorizedAccessException)
{
statusCode = HttpStatusCode.Unauthorized;
}
else if (context.Exception is NotFoundException)
{
statusCode = HttpStatusCode.NotFound;
}
context.Response = context.Request.CreateErrorResponse(statusCode, context.Exception);
}
}
} | code |
ییٚلہِ سٟمنہِ ژکھہِ سان پرٛژھُس زِ ژےٚ کیٛازِ کوٚرُتھ یہِ أمۍ ووٚنس مےٚ اوس فقیرَن یہِ ہدایت دِتمُت | kashmiri |
राजधानी दिल्ली में धुंध के कारण सांस लेना दूभर हो गया है। इसी को मद्देनज़र रखते हुए नेशनल ग्रीन ट्रिब्यूनल में गुरुवार सुबह सुनवाई हुई। इस सुनवाई में ङट ने राज्य सरकार, एमसीडी और पड़ोसी राज्यों को कड़ी फटकार लगाई। ऑड इवन पर बड़ा फैसला लिया है। राज्य में १३ से १७ तक ऑड इवन लागू कर दिया गया है। गौरतलब है कि ऑड इवन का यह तीसरा चरण होगा।राजधानी दिल्ली में धुंध के कारण सांस लेना दूभर हो गया है। इसी को मद्देनज़र रखते हुए नेशनल ग्रीन ट्रिब्यूनल में गुरुवार सुबह सुनवाई हुई। इस सुनवाई में ङट ने राज्य सरकार, एमसीडी और पड़ोसी राज्यों को कड़ी फटकार लगाई। ऑड इवन पर बड़ा फैसला लिया है। राज्य में १३ से १७ तक ऑड इवन लागू कर दिया गया है। गौरतलब है कि ऑड इवन का यह तीसरा चरण होगा।
सुनवाई के दौरान ङट ने कहा कि आज सुनवाई होनी है इसलिए कल ही आदेश जारी कर दिया गया था। आप सभी पक्षों के लिए ये शर्मनाक है कि आप आने वाली पीढ़ी को क्या दे रहे हो। ङट ने फटकार लगाते हुए कहा कि खुलेआम निर्माण कार्य चल रहा है लेकिन आप लोग रोक नहीं लगा पा रहे हैं, ऐसे हालात बनते हैं तभी आप कहते हैं कि कार्रवाई कर रहे हैं।
दिल्ली के लोधी रोड पर गुरुवार को प्रदूषण पीएम २.५ और पीएम १० का लेवल ५00 के पर जा चुका है। वही पंजाबी बाग में तो पीएम २.५ और पीएम १० का लेवल ६०० तक पहुंच चुका है। दिल्ली विश्वविद्यालय की बात करें तो यहां भी पीएम २.५ और पीएम १० का लेवल ५00 से ज़्यादा ही है।
नोएडा में उड़ा प्रशासन का मजाक
दिल्ली से सटे नोएडा में गुरुवार को भी स्कूल खुले हैं। जबकि प्रशासन ने साफ तौर पर स्कूल खोलने पर रोक लगाई है। नोएडा के सेक्टर १२७ स्थित ज्ञानश्री स्कूल और सेक्टर १४१ स्थित स्टैनफोर्ट स्कूल समेत कुछ स्कूल खुले रहे। बता दें कि दिल्ली में बढ़ते प्रदूषण को देखते हुए कक्षा पांच तक के स्कूलों में रविवार तक छुट्टी कर दी गई है। | hindi |
\begin{document}
\begin{abstract}
Various connections between the theory of permutation groups and the theory of topological groups are described. These connections are applied in permutation group theory and in the structure theory of topological groups.
\end{abstract}
\begin{classification}
Primary 22D05, 20B07, 05C25; Secondary 20E06, 20E08.
\end{classification}
\begin{keywords}
Totally disconnected locally compact groups, permutation groups, graphs.
\end{keywords}
\maketitle
\mbox{\bf s}ection*{Introduction}
The aim of this paper is to discuss various links between permutation groups, graphs and topological groups. The action of a group on a set can be used to define a topology on the group, called the {\em permutation topology}. The earliest references for this topology are the paper \cite{Maurer1955} by Maurer and the paper \cite{KarrassSolitar1956} by Karrass and Solitar. This topology opens up the possibility of applying concepts and results from the theory of topological groups in permutation group theory. One can also go the other way and apply ideas from permutation group theory to problems about topological groups. In particular, some simple constructions of graphs, that are commonly used in permutation group theory, can be applied.
In the first section we discuss the languages we use, when working with graphs and permutation groups.
In Section 2 we look at the definition of the permutation topology and
consider applications of the theory of topological groups to questions
about permutation groups. The main result in this section is a
theorem of Schlichting from \cite{Schlichting1980}. This is a theorem
about permutation groups, but Schlichting's proof uses notions from
functional analysis and result of Iwasawa \cite{Iwasawa1951} about
topological groups. Here we present a proof using concepts from
permutation group theory and Iwasawa's Theorem.
In the third and fourth sections we discuss applications of techniques and ideas from permutation groups theory and graph theory to the theory of topological groups.
In Section 3 Willis' structure theory of totally disconnected locally compact groups is in the limelight. Willis' paper \cite{Willis1994} helped spark a new interest in totally disconnected locally compact groups. Later work by Willis and others has shown that the concepts of the theory have many applications and are open to various interpretations. Most of the material in this section comes from the paper \cite{Moller2002}, where the basics of Willis' theory are given a graph theoretic interpretation.
In the fourth and last section we discuss an analogue of a Cayley graph, called a rough Cayley graph, that one can construct for a compactly generated, totally disconnected, group. The rough Cayley graph is defined in Section~\ref{SDefRough}. In that section, it is also shown that this graph is a quasi-isometry invariant of the group. In the latter parts of Section 4, it is shown how one uses rough Cayley graphs, by developing an analogue of the theory of ends of groups and the theory of groups with polynomial growth.
There are various other topics, that should be discussed in a survey like this. The study of random walks on groups and graphs is another place where graphs, permutations and topological groups meet. The book by Woess \cite{Woess2001} is an excellent introduction to this field. Another meeting place for graphs, permutations and topological groups is the theory of groups acting on trees. In particular, one could mention the theory of harmonic analysis and representation theory of groups acting on trees, see the book by Fig{\`a}-Talamanca and Nebbia \cite{Figa-TalamancaNebbia1991} and the theory of tree lattices, see the book by Bass and Lubotzky \cite{BassLubotzky2001}.
Then there is the topic of generic elements and subgroups, see the papers [7, 8] by Bhattacharjee and the paper \cite{AbertGlasner2008} by Abert and Glasner. And then I have not even mentioned the manifold appearances of our trio of graphs, permutations and topology in model theory. Describing all these topics would have meant a book length paper.
\mbox{\bf s}ection{Languages for graphs and permutation groups}
\mbox{\bf s}ubsection{A language for graphs}\label{SGraphs}
We will discuss both {\em undirected graphs}, or just {\em
graphs}, and {\em directed graphs}, called in this paper {\em
digraphs}.
Our undirected graphs are without loops and multiple edges. Thus
one can think of a (undirected)
graph $\Gamma$ as an ordered pair $(V\Gamma, E\Gamma)$ where
$V\Gamma$ is a set and $E\Gamma$ is a set of two element subsets of
$V\Gamma$. The
elements of $V\Gamma$ are called {\em vertices} and the elements of
$E\Gamma$ are called {\em edges}.
Vertices $\alpha$ and $\beta$ in a graph $\Gamma$ are said to be
{\em neighbours}, or {\em adjacent}, if $\{\alpha, \beta\}$ is an edge
in $\Gamma$. The {\em valency} of a vertex is the number of its
neighbours. A graph, all of whose vertices have finite valency,
is called {\em locally finite}.
For vertices $\alpha$ and $\beta$ of the graph, a \emph{walk} of length $n$
from $\alpha$ to $\beta$ is a sequence $\alpha=\alpha_0,\alpha_1,\ldots,\alpha_n=\beta$
of vertices, such that $\alpha_i$
and $\alpha_{i+1}$ are adjacent for $i=0, 1, \ldots, n-1$. A walk, all
of whose vertices are distinct, is called a {\em path}.
A {\em ray} in a graph is a sequence $\alpha_0, \alpha_1, \ldots$ of
distinct vertices such that $\alpha_i$ is adjacent to $\alpha_{i+1}$
for all $i$.
A graph
is connected if for any two vertices $\alpha$ and $\beta$ there is a
walk from $\alpha$ to $\beta$. Let $d(\alpha,\beta)$ denote the length of
a shortest walk from a vertex $\alpha$ to a vertex $\beta$. For a connected
graph, the function $d$ is a metric on its set of vertices.
Let $A$ be a set of vertices of $\Gamma$. The
{\em subgraph of $\Gamma$ spanned by $A$} is the graph whose vertex set is $A$,
and whose edge set is the set of all edges in $\Gamma$ whose end vertices are
both in $A$. We say that a set of vertices $A$ is {\em
connected}, if the subgraph spanned by $A$ is connected. The {\em
connected components} (or just {\em components}) of a graph are the
maximal connected sets of vertices.
We define a {\em digraph} $\Gamma$
to be a pair $(V\Gamma, E\Gamma)$ where
$V\Gamma$ is a set and $E\Gamma$ is a set of ordered pairs of distinct
elements of $V\Gamma$, i.e. $E\Gamma\mbox{\bf s}ubseteq (V\Gamma\times V\Gamma)
\mbox{\bf s}etminus\{(\alpha, \alpha)\mid \alpha\in V\Gamma\}$. An edge
$(\alpha, \beta)$ may be thought of as an \lq\lq arrow\rq\rq\ starting
in $\alpha$ and
ending in $\beta$.
The
{\em in-valency} of a vertex $\alpha$ in $\Gamma$ is the number of
edges of the type $(\beta, \alpha)$ (number of edges going \lq\lq in
to\rq\rq\ $\alpha$) and the {\em out-valency} of a vertex $\alpha$
is the number of edges $(\alpha, \beta)$ that go \lq\lq out of\rq\rq\
$\alpha$.
A digraph is said to be {\em locally finite} if every vertex has
finite in- and out-valencies.
One can \lq\lq forget\rq\rq\ the directions of edges in $\Gamma$ and define an undirected graph
$\overline{\Gamma}$ with the same vertex set and two vertices $\alpha$
and $\beta$ adjacent if and only if $(\alpha, \beta)$ or $(\beta,
\alpha)$ is an edge in the digraph $\Gamma$.
A {\em walk} in a digraph $\Gamma$ is a sequence $\alpha_0,
\ldots, \alpha_n$ such that $(\alpha_i, \alpha_{i+1})$ or
$(\alpha_{i+1}, \alpha_i)$ is an edge in $\Gamma$ for $i=0, \ldots,
n-1$. (That is to say, $\alpha_0,
\ldots, \alpha_n$ is a walk in the undirected graph
$\overline{\Gamma}$ associated to $\Gamma$.) An {\em arc}, more specifically an $n$-{\em arc},
is a sequence $\alpha_0,\ldots, \alpha_n$ of distinct vertices such that
$(\alpha_i, \alpha_{i+1})$ is an edge in $\Gamma$ for $i=0, \ldots,
n-1$. Arcs are sometimes referred to as directed paths. The set of {\em descendants} of a vertex $\alpha$ is defined as
the set
$$\mbox{${\rm deg}$}s(\alpha)=\{\beta\in V\Gamma\mid \mbox{there exists an arc
from }\alpha\mbox{ to }\beta\}.$$
The set $\mbox{${\rm deg}$}s_k(\alpha)$ is defined as the set of all vertices $\beta$
such that the shortest arc from $\alpha$ to $\beta$ has
length $k$.
The set of {\em ancestors} of a vertex $\alpha$, denoted
$\mbox{${\rm anc}$}(\alpha)$, is defined as the set
of all vertices $\beta$ such that $\alpha \in \mbox{${\rm deg}$}s(\beta)$, i.e.\
$\beta\in \mbox{${\rm anc}$}(\alpha)$ if and only if there exists an arc
from $\beta$ to $\alpha$.
Finally, we review the definition of a \emph{Cayley graph} of a group. Let $G$
be a group and $S$ a subset of $G$. The (undirected) Cayley graph
$\mbox{${\rm Cay}$}(G, S)$ of $G$ with
respect to $S$ has $G$ as the vertex set and $\{g, h\}$ is an edge if
$h=gs$ or $h=gs^{-1}$ for some $s$ in $S$. The Cayley graph
$\mbox{${\rm Cay}$}(G, S)$ is connected if and only if $S$ generates $G$. The left regular
action of $G$ on itself gives a transitive action of $G$ as a group of
graph automorphisms on $\mbox{${\rm Cay}$}(G, S)$.
It is common to define the Cayley graph of a group $G$ with respect to a subset
$S$ of $G$ as the digraph with vertex set $G$ and set of directed edges
the collection of all ordered pairs $(g,gs)$ for $g$ in $G$ and $s$ in $S$.
In these notes, however, it is convenient to
think of Cayley graphs as being undirected.
\mbox{\bf s}ubsection{A language for permutation groups}\label{Sperm}
In this section $G$ is a group acting on a set $\Omega$.
The image of a point $\alpha\in \Omega$ under an
element $g\in G$ will be written $g\alpha$.
The group of all permutations of a set $\Omega$ is called the {\em
symmetric group on} $\Omega$ and is denoted by $\mbox{${\rm Sym}$}(\Omega)$.
The action of $G$ on $\Omega$ induces a natural homomorphisms
$G\rightarrow \mbox{${\rm Sym}$}(\Omega)$.
If this homomorphism is injective, then the group $G$ is said to act {\em faithfully} on $\Omega$.
A group $G$ which acts faithfully on a set $\Omega$ can be regarded as a
subgroup of $\mbox{${\rm Sym}$}(\Omega)$, and then we say that $G$ is
a {\em permutation group} on $\Omega$.
An action is said to be {\em
transitive}, if for every two points $\alpha, \beta\in \Omega$ there is
some element $g\in G$ such that $g\alpha=\beta$. For a point $\alpha\in
\Omega$, the subgroup
$$G_\alpha=\{g\in G\mid g\alpha =\alpha\}$$
is called the {\em stabilizer} in $G$
of the point $\alpha$.
The {\em pointwise
stabilizer} $G_{(\Delta)}$
of a subset $\Delta$ of $\Omega$ is defined as the subgroup of all the elements in $G$ that fix
every element of $\Delta$, that is,
$$G_{(\Delta)}=\{g\in G\mid g\mbox{${\rm deg}$}lta =\mbox{${\rm deg}$}lta \mbox{ for every }\mbox{${\rm deg}$}lta\in
\Delta\}=\bigcap_{\mbox{${\rm deg}$}lta\in \Delta}G_\mbox{${\rm deg}$}lta.$$
The {\em setwise stabilizer} $G_{\{\Delta\}}$ of $\Delta$ is defined as
the subgroup consisting of all elements of $G$ that leave $\Delta$
invariant, that is,
$$G_{\{\Delta\}}=\{g\in G\mid g\Delta=\Delta\}.$$
The {\em $G$-orbit}, or, simpler {\em orbit}, of a
point $\alpha$ is the set $\{g\alpha\mid g\in G\}$.
Suppose $U$ is a subgroup of $G$. The group $G$ acts on the set
$G/U$ of right cosets of $U$, and this action is transitive. The image of a
coset $hU$ under an element $g\in G$ is $(gh)U$. Conversely, if $G$
acts transitively $\Omega$ and $\alpha$ is a point in
$\Omega$, then $\Omega$ can be identified with
$G/G_\alpha$. Here \lq\lq identified\rq\rq\ means that there is a
bijective map $\theta:\Omega\rightarrow G/G_\alpha$ such that for
every $\omega\in \Omega$ and every element $g\in G$ we have
$\theta(g\omega)=g\theta(\omega)$.
The orbits of stabilizers of points in $\Omega$ are called {\em suborbits},
that is, the suborbits are sets of the form $G_\alpha\beta$ where $\alpha,
\beta\in \Omega$. Orbits of $G$ on the set of ordered pairs of elements from
$\Omega$ are called {\it orbitals}. When $G$ is transitive on
$\Omega$ one can, for a fixed point
$\alpha\in\Omega$, identify the suborbits of $G_\alpha$ with
the orbitals: the suborbit $G_\alpha\beta$ is identified with the
orbital $G(\alpha, \beta)$.
The
number of elements in a suborbit $G_\alpha\beta$, often called the
{\em length} of the suborbit, is given by the
index $|G_\alpha: G_\alpha\cap G_\beta|$.
The number of elements in the orbit $U\beta$ of a
subgroup $U$ is equal to
the index $|U:U\cap G_\beta|$.
Next, we define a digraph, whose vertex set is $\Omega$ and whose edge set is
the union of $G$-orbitals, and call it the \emph{directed orbital
graph of the action}.
The action of $G$ on its vertex set $\Omega$ induces an action of $G$ as a
group of automorphisms on the directed orbital graph $\Gamma$, because if
$(\alpha,\beta)$ is an edge in $\Gamma$ then $(g\alpha,g\beta)$ is
in the same orbital as $(\alpha,\beta)$ and therefore also an edge in $\Gamma$.
Similarly, we define the {\em undirected orbital graph} of a $G$-action on a
set $\Omega$ as a graph whose vertex set is $\Omega$ and whose edge set is the union of
$G$-orbits on the set of two element subsets of $\Omega$.
A {\em block of imprimitivity} for $G$ is a subset $\Delta$ of $\Omega$
such that for every $g\in G$, either $g\Delta=\Delta$ or $\Delta\cap
(g\Delta)=\emptyset$. The existence of a non-trivial proper block of
imprimitivity $\Delta$ ({\em non-trivial} means that $|\Delta|>1$ and
{\em proper} means that $\Delta\neq \Omega$)
is equivalent to the existence of a
non-trivial proper $G$-invariant equivalence relation $\mbox{\bf s}im$ on
$\Omega$. If there is no non-trivial
proper $G$-invariant equivalence relation on $\Omega$ we say that $G$
acts primitively on $\Omega$. In most books on permutation
groups it is shown that, if $G$ acts transitively on $\Omega$
then $G$ acts primitively on $\Omega$ if and only if $G_\alpha$ is a maximal
subgroup of $G$ for every $\alpha\in \Omega$. Part of the proof of
this fact is to show that if $G_\alpha<H<G$ then $H\alpha$ is a
non-trivial proper block of imprimitivity. A further useful fact
is that if $N$ is a
normal subgroup of $G$, then the orbits of $N$
on $\Omega$ are blocks of imprimitivity for $G$.
Recent books covering this material are
\cite{BMMN1997}, \cite{Cameron1999} and \cite{DixonMortimer1996}.
\mbox{\bf s}ection{The permutation topology}
\mbox{\bf s}ubsection{Definition of the permutation topology}
Let $G$ be a group acting on a set $\Omega$. The action
can be used to introduce a topology on $G$. The topology of a
topological group is completely determined by a neighbourhood basis of
the identity element. The {\em permutation topology} on $G$ is defined
by choosing as a neighbourhood basis of the identity the family of
pointwise stabilizers of finite subsets of $\Omega$, i.e.\ a neighbourhood
basis of the identity is given by the family of subgroups
\[\{G_{(\Phi)}\mid \Phi\mbox{ is a finite subset of }\Omega\}.\]
A sequence $(g_i)$ of elements in $G$ has an element
$g\in G$ as a limit if and only if for every
$\alpha\in \Omega$ there is a number $N$ (depending on $\alpha$) such
that $g_n\alpha =g\alpha$ for every $n\geq N$.
There are other ways to define the permutation topology.
Think of $\Omega$ as having the discrete topology and elements of
$G$ as maps $\Omega\rightarrow \Omega$.
Then the permutation
topology is equal to the topology of pointwise convergence, and it is
also the same as the compact-open topology.
The basic idea is that two permutations $g$ and $h$ are \lq\lq
close\rq\rq\
to each other if they agree on \lq\lq many\rq\rq\ points.
If the set $\Omega$ is
countable, then the permutation topology can be defined by a metric, here is one way to do that.
\mbox{\bf s}mallskip
{\it Enumerate
the points in
$\Omega$ as $\alpha_1, \alpha_2, \ldots$. Take two elements $g, h\in G$.
Let $n$ be the smallest number such that $g\alpha_n\neq h\alpha_n$
or $g^{-1}\alpha_n\neq
h^{-1}\alpha_n$. Set $d(g,h)=1/2^n$. Then $d$ is a metric on $G$
that induces the permutation topology.}
\mbox{\bf s}mallskip
From the definition of the permutation topology
we can immediately characterize open subgroups in $G$:
\mbox{\bf s}mallskip
{\em A subgroup of $G$ is open if and only if it contains the pointwise
stabilizer of some finite set of points.}
\mbox{\bf s}mallskip
Various properties of the action of $G$ on $\Omega$ are reflected by
properties of this topology on $G$.
\mbox{\bf s}mallskip
{\em The permutation topology on $G$ is
Hausdorff if and only if the action of $G$ on $\Omega$ is faithful.
Moreover, $G$ is totally
disconnected if and only if the action is faithful.}
{\em Remark.} In general we do not assume that our topological groups
are Hausdorff, but note that totally disconnected groups are always
Hausdorff.
\mbox{\bf s}mallskip
We say that a group $G$ acting on $\Omega$ is
a {\em closed} if it is image in $\mbox{\bf s}ym(\Omega)$ is a closed subgroup,
where $\mbox{\bf s}ym(\Omega)$ has the permutation topology. Closed
permutation groups can be characterized in the following way, see
\cite[Section 2.4]{Cameron1990}.
\begin{proposition}
A permutation group $G$ on a set $\Omega$ is closed if and only if $G$ is
the full automorphism group of some first order structure on $\Omega$.
\end{proposition}
\noindent
A first order structure on $\Omega$ is a collection of:
\begin{itemize}
\item constants that belong to $\Omega$,
\item functions defined on $\Omega$ and taking their values in $\Omega$,
\item relations defined on $\Omega$.
\end{itemize}
It is easy to show that the
automorphism group of such a structure is closed. To prove the converse,
one uses the concept of the {\em canonical relational structure} on $\Omega$
such that for $n=1, 2, \ldots$ we get one $n$-ary relation
for each orbit of $G$ on $n$-tuples of $\Omega$. If $G$ is a closed permutation group
on $\Omega$, then $G$ is the full automorphism group
of this structure.
For those actions that are not faithful,
we say that $G$ is closed in the permutation
topology if the image of $G$ in $\mbox{${\rm Sym}$}(\Omega)$ under the natural
homomorphism is closed. This condition is
equivalent to the condition that
stabilizers of points are closed subgroups of $G$.
Compactness (note that in this work the term {\em compact} does not
include the Hausdorff condition) has a natural
interpretation in the permutation topology.
A subset of a topological space is said to be {\em
relatively compact} if it has compact closure.
The following lemma slightly generalizes a result by Woess
for automorphism groups of locally finite connected
graphs, and the proof is the same as Woess' proof.
\begin{lemma}{\rm (\cite[Lemma 1 and Lemma 2]{Woess1992})}\label{LCompact}
Let $G$ be a group acting transitively on a set $\Omega$ and endow
$G$ with the permutation topology.
Assume that $G$ is closed in the permutation topology and that all suborbits are finite.
(i) The stabilizer $G_\alpha$ of a point $\alpha \in \Omega$ is
compact.
(ii) A subset $A$ of $G$ is
relatively compact
in $G$ if and only if the set $A\alpha$ is finite for every $\alpha$
in $\Omega$.
Furthermore, if $A$ is a subset of $G$ and $A\alpha$
is finite for some $\alpha\in\Omega$ then $A\alpha$ is finite for every
$\alpha$ in $\Omega$.
\end{lemma}
\begin{proof} (i) Let $K$ denote the kernel of the action of $G$ on
$\Omega$. Every open neighbourhood of the identity will contain $K$
and thus $K$ is compact. Because $G_\alpha$ is closed we see that
$K\cap G_\alpha$ is compact. In order to show that $G_\alpha$
is compact it is thus enough to show that $G_\alpha/(K\cap
G_\alpha)$ is compact. The group $G_\alpha/(K\cap
G_\alpha)$ acts faithfully on $\Omega$ and the quotient topology on
$G_\alpha/(K\cap G_\alpha)$ is the same as the permutation topology
induced by the action on $\Omega$. We may thus assume that
$G_\alpha$ acts faithfully on $\Omega$.
Let $(\Omega_i)_{i\in I}$ denote the family of $G_\alpha$-orbits on
$\Omega$. Let $H_i$ denote the permutation group induced by
$G_\alpha$ on $\Omega_i$. Since, each $\Omega_i$ is finite, then each
group $H_i$ is finite and discrete in permutation topology.
Set $H=\prod_{i\in I} H_i$, the full Cartesian product. Note that
$\Omega$ is the disjoint union of the $\Omega_i$'s and that $H$ has a
natural action on $\Omega$. The
permutation topology on $H$ induced by the action on $\Omega$ is the
same as the product topology. Thinking of $G_\alpha$ and $H$ as
subgroups of $\mbox{\bf s}ym(\Omega)$ we see that $G_\alpha$ is a closed
subgroup of the compact subgroup $H$ and is thus compact.
(ii) Suppose that ${A}^-$, the closure of $A$ in $G$,
is compact. Then for a point
$\alpha\in \Omega$ there is a finite open
covering of ${A}^-$ by sets of the type $gG_\alpha$; that is to
say,
we can find $g_1,\ldots,g_n\in G$ such that
${A}^-\mbox{\bf s}ubseteq\bigcup_{i=1}^n g_iG_\alpha$. Then $A\alpha\mbox{\bf s}ubseteq
\{g_1\alpha, \ldots,g_n\alpha\}$.
Conversely, suppose that $A\alpha=\{\alpha_1,\ldots,\alpha_n\}$.
Let $g_i$ be an element in $A$
such that $g_i\alpha=\alpha_i$. Then
$A\mbox{\bf s}ubseteq\bigcup_{i=1}^n g_iG_\alpha$.
The latter set is compact, so the closure of $A$ is compact.
\end{proof}
Lemma \ref{LCompact}
implies that if $G$ is a closed transitive permutation group
on a countable set $\Omega$ such that all
suborbits are finite then $G$, with the permutation topology, is a
locally compact, totally
disconnected group. In particular, the automorphism group of a locally finite,
transitive graph is a locally compact, totally disconnected group.
A subgroup $H$ in a topological group $G$ is said to be cocompact if
$G/H$ is a compact space. This concept has also a natural
interpretation in terms of the permutation topology.
\begin{lemma}\label{Lcocompact}
{\rm (\cite[Proposition~1]{Nebbia2000}, cf.~\cite[Lemma~7.5]{Moller2002})}
Let $G$ be a group acting transitively on a set $\Omega$. Assume
that $G$ is closed in the permutation topology and all suborbits are
finite.
Then a subgroup $H$ of $G$ is
cocompact if and only if $H$ has finitely many orbits on
$\Omega$.
\end{lemma}
\begin{proof} Suppose first that $H$ is cocompact. This means that
both the spaces of right and left cosets of $H$ in $G$ are compact.
Let
$X$ denote the set of right cosets of $H$ in $G$. The quotient map
$\pi:G\rightarrow X$ is open. The family of cosets $\{gG_\alpha\}_{g\in
G}$ is an open covering of $G$ and hence
$\{\pi(gG_\alpha)\}_{g\in G}$ is an open covering of $X$. Since $X$ is compact, there is a finite subcovering
$\pi(g_1G_\alpha),\ldots, \pi(g_nG_\alpha)$ of $X$. Then
$G=Hg_1G_\alpha\cup\ldots\cup Hg_nG_\alpha$ and therefore
$\Omega=H(g_1\alpha)\cup\ldots\cup H(g_n\alpha)$.
Conversely, suppose that $H$ has only finitely many orbits on $\Omega$, say
there are elements $g_1, \ldots, g_n$ such that
$\Omega=H(g_1\alpha)\cup\ldots\cup H(g_n\alpha)$. Then
$G=Hg_1G_\alpha \cup\ldots\cup Hg_nG_\alpha$ and
$X=\pi(g_1G_\alpha)\cup \ldots\cup \pi(g_nG_\alpha)$. Each of the
sets $\pi(g_1G_\alpha)$ is compact. Hence $X$, the set of right
cosets of $H$ in $G$, is
compact, because it is a union of finitely many compact
sets. \end{proof}
Ideas from
permutation group theory and the permutation topology can be
applied to the study of a topological group $G$. For an
open subgroup
$U$ we set $\Omega=G/U$, the space of left
cosets. The stabilizers in $G$ of points in $\Omega$ are conjugates
of $U$, and thus also open subgroups of $G$. The stabilizer of a finite set
$\Phi=\{\alpha_1, \ldots, \alpha_n\}$ of points is just the
intersection of the open subgroups $G_{\alpha_1}, \ldots,
G_{\alpha_n}$ and is thus open in $G$. From this we see that the
permutation topology coming from the action of $G$ on $\Omega$ is
contained in the topology on $G$. If the topological group
$G$ is assumed to be
a totally disconnected and
locally compact, then we can choose $U$ to
be a compact open subgroup of $G$
(by a theorem of van Dantzig \cite{Dantzig1936}).
In particular, the stabilizers in $G$ of
points in $\Omega$ are all compact (they are conjugates of $U$).
The second part of Lemma~\ref{LCompact}
above also holds for the action of $G$ on $\Omega$, so a subset $A$ of
$G$ has compact closure if and only if $A\alpha$ is finite for every
$\alpha \in \Omega$. This
implies that $|G_\alpha\beta|< \infty$ for all points
$\alpha$ and $\beta$ in $\Omega$. This is because
$|G_\alpha\beta|=|G_\alpha:G_\alpha\cap G_\beta|$ and this index is
finite because $G_\alpha\cap G_\beta$ is an open subgroup of the
compact group $G_\alpha$.
\mbox{\bf s}ubsection{Suborbits and the modular function}
In this section the general assumption will be that $G$ is a closed
permutation group acting on a set $\Omega$ and that
all suborbits of the group $G$ are finite.
Under the above assumptions the group $G$ is a locally compact, totally
disconnected group. In this section we want to interpret the modular
function on $G$ in terms of the action of $G$ on $\Omega$.
The connection between suborbits and the modular function can be seen from
the following argument due to Schlichting \cite{Schlichting1979}, see
also \cite{Trofimov1985a}.
Let $\mu$ be a right
Haar-measure on $G$. Define the modular function $\Delta$
so that if
$A$ is a measurable set then $\mu(gA)=\Delta(g)\mu(A)$.
\begin{lemma} \label{LModular}
{\rm (\cite[Lemma 1]{Schlichting1979}, cf. \cite[Theorem 1]{Trofimov1985a})})
Let $G$ be a closed, transitive permutation
group on a set $\Omega$. Assume furthermore that all
suborbits of $G$ are finite. Let $\Delta$ denote the modular function
on $G$. If $h$ is an element in $G$ with $h\alpha=\beta$ then
$$\Delta(h)=\frac{|G_\beta\alpha|}{|G_\alpha\beta|}=
\frac{|G_\alpha:G_\alpha\cap h^{-1}G_\alpha h|}
{|G_\alpha:G_\alpha\cap hG_\alpha h^{-1}|}.$$
\end{lemma}
\noindent
\begin{proof}
Then, with $\mu$ denoting the right
Haar-measure on $G$,
\begin{align*}
\mid G_\beta\alpha\mid & = \mid G_\beta:G_\alpha\cap G_\beta\mid \\
& = \mu(G_\beta)/\mu(G_\alpha\cap G_\beta) \\
& = \mu(hG_\alpha h^{-1})/
\mu(G_\alpha\cap G_\beta) \\
& = \Delta(h)\mu(G_\alpha )/
\mu(G_\alpha\cap G_\beta) \\
& = \Delta(h)
\mid G_\alpha:G_\alpha\cap G_\beta\mid \\
& = \Delta(h)\mid G_\alpha \beta\mid.
\end{align*}
And we see that $\Delta(h)={|G_\beta \alpha|}/{|G_\alpha\beta|}$.
\end{proof}
{\em Remark.} Let $G$ be a locally compact, Hausdorff group with
modular function $\Delta$. Assume $G$ acts transitvely on a set $\Omega$
such that the stabilizers of points are compact open subgroups of $G$.
The calculation in the proof of Lemma~\ref{LModular} is also valid in
this case and thus the conclusion in Lemma~\ref{LModular} holds also.
\mbox{\bf s}mallskip
For an orbital $A=G(\alpha, \beta)$ we define the {\em paired orbital}
as the orbital $A^*=G(\beta, \alpha)$. This pairing of orbitals also
gives us a pairing of suborbits, where the
suborbit $G_\alpha\beta$ is paired to a suborbit $G_\alpha\gamma$
where $\gamma$ is a point in $\Omega$ such that $(\alpha,\gamma)$ is
in $G(\beta, \alpha)$. Denote by $\Gamma$ the directed orbital graph for the
orbital $G(\alpha, \beta)$. The size of the suborbit $G_\alpha\beta$
is the out-valency of $\Gamma$ and the size of the paired suborbit
$G_\alpha\gamma$ is the in-valency of $\Gamma$.
Using Lemma~\ref{LModular}, we obtain the following proposition.
\begin{proposition}{\rm (\cite[Theorem 1]{Trofimov1985a})}
Let $G$ be a closed, transitive permutation
group on a set $\Omega$. Assume that
all suborbits of $G$ are finite. Then the lengths of paired suborbits
are always equal if and only if ${G}$ is unimodular.
\end{proposition}
Using that the modular function is a homomorphism, we obtain the following.
\begin{corollary}
Let $G$ be a closed, transitive permutation
group on a set $\Omega$.
Let $K=\ker(\Delta)$. Then
(a) If $\alpha\in \Omega$ then $G_\alpha\leq K$.
(b) Let $G'$ be the derived group of $G$. Then $G'\leq K$.
(c) If $g$ is an torsion element of $G$ then $g\in K$.
\end{corollary}
\noindent
\begin{proof} Follows directly from the definitions and the fact
that $G/K$
is isomorphic to a multiplicative subgroup of the positive real numbers. \end{proof}
\begin{corollary}
Let $G$ be a closed, transitive permutation
group on a set $\Omega$. Assume that
all suborbits of $G$ are finite.
If $G$ is also
primitive, then paired suborbits have equal length.
\end{corollary}
\noindent
\begin{proof} Let $K=\ker(\Delta)$. Then, because $G$ is primitive
and $K$ is normal in $G$,
we know that either $K=\{e\}$ or $K$ is transitive. If $K=\{e\}$
then $G$ is abelian and must in fact be trivial.
If $K$ is transitive
then the result of Lemma~\ref{LModular} implies that lengths of paired
suborbits are equal. \end{proof}
For a long time it was an open question, posed by Peter M.~Neumann,
whether one could have a group
acting primitively on a set $\Omega$ with a finite suborbit
paired to an infinite one. Such examples were constructed by David M.~Evans in \cite{Evans2001}.
Consider a connected graph $\Gamma$ and assume that $G$ is a closed
subgroup of $\mbox{${\rm Aut}$}ga$ that acts transitively on $\Gamma$ and that all
suborbits are finite. For convenience we
think of each undirected
edge $\{\alpha,\beta\}$
in $\Gamma$ as consisting of two directed edges $(\alpha,\beta)$ and
$(\beta,\alpha)$. Each directed edge $e=(\alpha,\beta)$
will be labeled by a number
$$\Delta_e=\frac{|G_\beta \alpha|}{|G_\alpha \beta|}.$$
Observe that $\Delta_{(\alpha,\beta)}=\Delta_{(\beta,\alpha)}^{-1}$.
Furthermore, note that if $g$ is an element of $G$
such that $g\alpha=\beta$ and $e=(\alpha, \beta)$ then
$\Delta(g)=\Delta_e$. Suppose $g$ is an element of $G$
and $g\alpha=\gamma$ and
that there is a vertex
$\beta$ in $X$ such that $(\alpha,\beta)$ and $(\beta,\gamma)$ are
edges in $X$. Find elements $g_1$ and $g_2$ in $G$ such that
$g_1\alpha=\beta$ and $g_2\beta=\gamma$.
Then $g\alpha=g_2g_1\beta$ and from the
formula above for the modular function we can deduce that
$\Delta(g)=\Delta(g_2g_1)$. We also find that
$$\Delta(g)=\Delta(g_2g_1)=\Delta(g_2)\Delta(g_1)=\Delta_{(\beta,\gamma)}
\Delta_{(\alpha,\beta)}.$$
This can be extended to directed walks
of arbitrary length, so that if $g\alpha=\beta$
then we take a directed walk from $\alpha$ to $\beta$, enumerate the edges in
the walk as $e_1, \ldots, e_k$ and then
$$\Delta(g)=\Delta_{e_1}\cdots \Delta_{e_k}.$$
Hence the labeled graph completely describes the modular function on
$G$. This idea can be found in the paper \cite{BassKulkarni1990} by Bass and Kulkarni.
Suppose now not only that all the suborbits of $G$ are finite
but also that there is a finite upper
bound $m$ on their length.
In that case, Lemma \ref{LModular} implies that the image of the modular
function is a bounded set. The image of the modular function is a bounded subgroup
of the multiplicative group of positive
real numbers and is thus the trivial subgroup. Hence $G$ is unimodular.
We have deduced the following unpublished result of Praeger.
\begin{corollary}
Let $G$ be a closed, transitive permutation
group on a set $\Omega$ such that
all suborbits of $G$ are finite.
If there is a finite bound on the length of suborbits, then paired suborbits
have equal length.
\end{corollary}
The next result, also due to Praeger \cite{Praeger1991},
uses the modular function to infer information about graph structure.
The {\em directed integer graph}
$\mbox{\bf Z}$ has the set of integers as a vertex set and the edge set is the set of
all ordered pairs $(n, n+1)$.
\begin{theorem} {\rm (\cite{Praeger1991})} \label{Tepimorphism}
Let $\Gamma$ be an infinite, connected, vertex and edge transitive directed
graph with finite but unequal in- and out-valence. Then there is a
graph epimorphism $\varphi$ from $\Gamma$ to the directed integer graph
$\mbox{\bf Z}$. For each $i\in \mbox{\bf Z}$, the inverse image $\varphi^{-1}(i)$ is
infinite.
\end{theorem}
\begin{proof} Write $G=\mbox{${\rm Aut}$}ga$. Let $q=d^-/d^+$ where $d^+$ is the
out-valence of $\Gamma$ and $d^-$ is the in-valence of $\Gamma$.
Consider an edge $(\alpha,\beta)$ in $\Gamma$. Because $G$ acts
transitively on the edges of $\Gamma$ we can conclude that
$|G_\alpha \beta|=d^+$
and $|G_\beta \alpha|=d^-$ and hence if $g\alpha=\beta$
then $\Delta(g)=d^-/d^+$. Using the graph $\Gamma$ to calculate the
modular function in a similar way as described above we
conclude that for every element $g\in G$ there is an integer $i$ such
that $\Delta(g)=q^i$. Hence, if $K$ denotes the kernel of the modular
function then $G/K=\mbox{\bf Z}$. Fix a vertex $\alpha_0$ in $\Gamma$ and define a map
$\varphi:V\Gamma\rightarrow \mbox{\bf Z}$ so that $\varphi(\beta)=i$ if there is an
element $g$ in $G$ such that $g\alpha_0=\beta$ and $\Delta(g)=q^i$.
It is clear that the choice of $g$ is immaterial. From the way
one uses $\Gamma$ to calculate the modular function we see that if
$(\alpha,\beta)$ is
an edge in $\Gamma$ then $\varphi(\beta)=\varphi(\alpha)+1$ which implies that
$\varphi$ is a homomorphism from $\Gamma$ to the directed integer
graph.
Assume now, seeking contradiction, that $\varphi^{-1}(i)$ is finite for some $i$.
Note that the fibers of $\varphi$ are just the orbits of the kernel of
the modular homomorphism and are thus blocks of imprimitivity for
$G$. Hence, all the fibers of $\varphi$ have the same cardinality,
say $k$. The number of edges going out of $\varphi^{-1}(0)$ is
$d^+k$ and the number of edges going into $\varphi^{-1}(1)$ is
$d^-k$. But, both these numbers should be equal to the number of
edges going from $\varphi^{-1}(0)$ to $\varphi^{-1}(1)$ and because we
are assuming that $d^-\neq d^+$ we have a contradiction.
(A similar proof of Praegers result is in a paper by Evans
\cite{Evans1997}.)\end{proof}
{\em Remark.} In the next section {\em highly arc transitive}
digraphs are discussed. A digraph $\Gamma$ satisfying the conditions
in Theorem~\ref{Tepimorphism}
need not be highly arc transitive, but it is easy to show that if
$d^+$ and $d^-$ are coprime, then $\Gamma$ must be highly arc
transitive.
The next result we discuss, is a remarkable theorem of
Schlichting \cite{Schlichting1980}.
\begin{theorem}\label{TSchlichting}{\rm (\cite{Schlichting1980})}
Let $G$ be a group acting transitively on a set $\Omega$. Then
there is a finite bound on the sizes of suborbits of $G$ if and only
if there is a $G$-invariant equivalence relation $\mbox{\bf s}im$ on $\Omega$ with finite
classes, such that the stabilizers of points in the action of $G$ on
$\Omega/\mbox{\bf s}im$ are finite.
\end{theorem}
While this is a theorem about
permutation groups, Schlichting's proof utilizes various concepts
from functional analysis and a theorem of Iwasawa
\cite[Theorem~1]{Iwasawa1951}.
Later Theorem~\ref{TSchlichting} was rediscovered by
Bergman and Lenstra \cite{BergmanLenstra1989}, who gave a group
theoretical/combinatorial proof.
The theorem of Iwasawa that Schlichting uses in his proof of
Theorem~\ref{TSchlichting} is about the
relationship between the classes [{\em IN}] and [{\em SIN}] of
topological groups.
\begin{definition}
A locally compact group is said to be in the class [{\em IN}] if there is
a compact neighbourhood $K$ of the identity (i.e.~$K$ contains an open
set containing the identity) that is invariant under
conjugation by elements in $G$.
A locally compact group is said to be in the class [{\em SIN}] if every
neighbourhood of the identity contains
a compact neighbourhood $K$ of the identity that is invariant under
conjugation by elements in $G$.
\end{definition}
Let us start by relating these two properties to the permutation
topology.
\begin{proposition}\label{PIN}
Let $G$ be a transitive permutation group on a set $\Omega$ and
assume that all suborbits are finite. Then $G$ with the permutation
topology is in the class [{\em IN}] if and only if there is a finite bound
on the sizes of suborbits.
\end{proposition}
\begin{proof} First assume that $G$ is in the class [{\em IN}].
Suppose $\alpha,\beta\in \Omega$. We want to find a constant upper bound,
independent of
$\alpha$ and $\beta$, for the size of the suborbit $G_\alpha \beta$.
Let $K$ be a compact neigh\-bourhood of the identity that is invariant
under conjugation. Since $K$ contains an open neigh\-bourhood of the
identity we can find a finite set $\Phi$ such that $G_{(\Phi)}\mbox{\bf s}ubseteq
K$.
Choose a point $\gamma$ in $\Omega$.
Because $K$ is compact,
$|K\gamma|=m<\infty$. Let $k$ be an upper bound for the indices
$|G_\mbox{${\rm deg}$}lta:G_{(\Phi)}|\leq k$ with $\mbox{${\rm deg}$}lta\in \Phi$.
Find an element
$f\in G$ such that $f\beta=\gamma$ and set $\alpha'=f\alpha$. Whence
$|G_\alpha \beta|=|G_{\alpha'} \gamma|$. Then we find an element $h$
such that $\alpha'\in
h\Phi$. Note that $G_{(h\Phi)}=hG_{(\Phi)}h^{-1}$. Since $K$ is invariant
under conjugation and $G_{(\Phi)}\mbox{\bf s}ubseteq K$,
we conclude that $G_{(h\Phi)}\mbox{\bf s}ubseteq K$. There\-fore
$|G_{(h\Phi)}\alpha|\leq
|K\alpha|=m$. We also know that $|G_{\alpha'}:G_{(h\Phi)}|\leq k$ and thus
$|G_{\alpha} \beta|=|G_{\alpha'} \gamma|\leq k|G_{(h\Phi)}\gamma|\leq km$. Hence $km$
is an upper bound for the size of suborbits of $G$.
Conversely, assume that there is an upper bound $m$ on the sizes of
suborbits. For a finite subset $\Phi$ of $\Omega$ we let $m(\Phi)$ denote the
size of the largest orbit of $G_{(\Phi)}$. We choose a finite subset
$\Phi$ such that $m_0=m(\Phi)$ is as small as possible. Define
$A=\bigcup_{g\in G} gG_{(\Phi)}g^{-1}$. Clearly, the set $A$ is open
and invariant under conjugation by elements of $G$.
We claim that $A$ is relatively
compact. By Lemma~\ref{LCompact}(ii),
we only need to show that $A\alpha$ is finite for some
$\alpha\in \Omega$. Choose $\alpha\in \Omega$ such that
$|G_{(\Phi)}\alpha|=m(\Phi)$. For this $\alpha$ we will show that
$|A\alpha|\leq m$. Arguing by contradiction, suppose
$|A\alpha|\geq n>m(\Phi)$. Take $f_1, \ldots, f_n\in A$ such that the
elements $f_1\alpha, \ldots, f_n\alpha$ are all distinct. Since $f_1, \ldots,
f_n\in A=\bigcup_{g\in G} gG_{(\Phi)}g^{-1}$, we can find elements $g_1,
\ldots, g_n\in G$ such that $f_i\in g_iG_{(\Phi)}g_i^{-1}=G_{(g_i\Phi)}$.
Write $\Phi_i=g_i\Phi$ and set
$E=\{f_1\alpha, \ldots, f_n\alpha\}\cup \Phi_1\cup\cdots\cup
\Phi_n$. Then $m(E)=m_0$. Let $\Delta$ denote a $G_{(E)}$-orbit of size $m_0$. Note that $\Delta$ is also a $G_{(\Phi_i)}$-orbit.
Choose an element $\mbox{${\rm deg}$}lta\in \Delta$. There is for each
$i=1, \ldots, n$ an element $h_i\in G_{(E)}$ such that
$h_i\mbox{${\rm deg}$}lta=f_i\mbox{${\rm deg}$}lta$ and therefore
$h_i^{-1}f_i\in G_\mbox{${\rm deg}$}lta$. But
$h_i^{-1}f_i\alpha=f_i\alpha$ and we can conclude that $|G_\mbox{${\rm deg}$}lta
\alpha|\geq n>m$
contrary to assumptions. (The above argument is related to the proof
of Theorem~\ref{TSchlichting} by Bergmann and Lenstra
\cite{BergmanLenstra1989}, but this version is from a lecture course
given by Peter M.~Neumann in Oxford 1988--1989.) \end{proof}
\begin{proposition}\label{PSIN}
Let $G$ be a transitive permutation group on a set $\Omega$ and
assume that all suborbits are finite. Then $G$ with the permutation
topology is in the class [{SIN}] if and only $G$ is discrete
(i.e. the stabilizers of points are finite).
\end{proposition}
\begin{proof} It is obvious that if $G$ is discrete then $G$ is
in [{\em SIN}].
Let us now assume that $G$ is in [{\em SIN}]. Then, if $\alpha$
denotes a point in $\Omega$, the open subgroup $G_\alpha$ contains a compact
neighbourhood $K$ of the identity that is invariant under conjugation.
Then for $g\in G$ we see that $K=gKg^{-1}\mbox{\bf s}ubseteq gG_\alpha
g^{-1}=G_{g\alpha}$ and
thus $K\mbox{\bf s}ubseteq \bigcap_{g\in G}gG_\alpha g^{-1}=\bigcap_{g\in G}
G_{g\alpha}$.
Because $G$ is
assumed to be transitive we conclude that $K$ fixes every point of
$\Omega$. But we are also assuming that $G$ acts faithfully on $\Omega$ so
$K=\{e\}$
and $G$ is discrete. \end{proof}
The theorem of Iwasawa mentioned above says that if a locally compact
group is in the class [{\em IN}] then there is a compact normal subgroup $N$
such that $G/N$ is in the class [{\em SIN}].
\begin{proof} (Theorem~\ref{TSchlichting}) Assume that there is a
finite upper bound on the sizes of suborbits of $G$.
By Proposition~\ref{PIN} the
group $G$ is in the class [{\em IN}]. Iwasawa's Theorem gives us a
compact, normal subgroup $N$ of $G$ such that $G/N$ is in [{\em
SIN}]. The orbits of the normal subgroup $N$ are the classes of a
$G$-invariant equivalence relation $\mbox{\bf s}im$ on $\Omega$
and because $N$ is compact
these classes are all finite. The group $H=G/N$ certainly
is in [{\em SIN}], but it is not certain that $H$ acts faithfully on
$\Omega'=\Omega/\mbox{\bf s}im$ so we can not apply Proposition~\ref{PSIN} directly.
Note that if $\alpha\in\Omega'$ then $H_\alpha$ is an open subgroup of
$H$. Since $H$ is in [{\em SIN}] there is a compact invariant
neighbourhood neighbourhood $K$ contained in $H_\alpha$. As in the
proof of Proposition~\ref{PSIN} we conclude that $K$ is contained in
the kernel of the action of $H$ on $\Omega'$. Thus the kernel $N'$ of the
action of $H$ on $\Omega'$ is an open subgroup of $H$. The group
$H/N'$ can be regarded as a permutation group on $\Omega'$. From this we
conclude that the permutation topology on $H/N'$ is discrete, which
implies that the stabilizer in $H/N'$ of a point $\alpha\in \Omega'$
must be finite.
The proof of the other direction is left to the reader. \end{proof}
Schlichting's Theorem implies the following general result about
totally disconnected, locally compact groups.
\begin{corollary} Let $G$ be a totally disconnected locally compact group.
(i) The group $G$ has a compact open normal subgroup if and only if
there is a compact open subgroup $U$ and a number $m_U$ such that
$|U:U\cap gUg^{-1}|\leq m_U$ for all $g\in G$.
(ii) If there is such a number $m_U$ for one compact open subgroup
$U$ then there is a number $m_V$ for any compact open subgroup
$V$ such that $|V:V\cap gVg^{-1}|\leq m_V$ for all $g\in G$.
\end{corollary}
\begin{proof} (i) If $G$ contains a compact open normal subgroup $N$, then
we can take $U=N$ and
$m_N=1$.
Conversely, assume that $U$ is a compact open subgroup and there is a number
$m_U$ such that $|U:U\cap gUg^{-1}|\leq m_U$ for all $g \in G$.
Put $\Omega=G/U$. Take a point $\alpha$ in $\Omega$
such that $U=G_\alpha$. Note that if $g\in G$ and $\beta=g\alpha$
then $|G_\alpha\beta|=|G_\alpha:G_\alpha\cap G_\beta|=|U:U\cap
gUg^{-1}|\leq m_U$. Thus there is a finite upper bound on the sizes of
suborbits and we can apply Schlichting's Theorem, which
provides us with a
$G$-invariant equivalence relation $\mbox{\bf s}im$ on $\Omega$ with finite classes
such that the stabilizer of a $\mbox{\bf s}im$-class acts like a finite group
on $\Omega'=\Omega/\mbox{\bf s}im$. This in turn implies that $N$, the kernel
of the action of $G$ on $\Omega'$, is a open normal subgroup of $G$.
If $\alpha\in \Omega$ then $N\alpha$ is contained in the $\mbox{\bf s}im$-class
of $\alpha$ and thus $N\alpha$ is finite. Therefore $N$ is also
compact.
(ii) From the proof of statement (i) we get the existence of a compact open normal
subgroup $N$. Consider the action of $G$ on the set $\Omega=G/V$. The
orbits of $N$ on $\Omega$ are all finite and give us the equivalence
classes of a $G$-invariant equivalence relation $\mbox{\bf s}im$. Let $k$ be the
number of element in a $\mbox{\bf s}im$-class. The group $G$
acts on $\Omega'=\Omega/\mbox{\bf s}im$ and the stabilizers of points in
$\Omega'$ act like finite groups on $\Omega'$. Thus the sizes of
suborbits in the action of $G$ on $\Omega'$ are bounded above by some
number $l$. Let $\tilde{\alpha}$
denote the $\mbox{\bf s}im$-class of an element $\alpha\in\Omega$. For
elements $\alpha$ and $\beta$ in $\Omega$ we see that
$|G_\alpha\beta|\leq |G_{\tilde{\alpha}}\beta|\leq
k|G_{\tilde{\alpha}}\tilde{\beta}|=kl$. From this it follows that
$|V:V\cap gVg^{-1}|\leq kl$. \end{proof}
\mbox{\bf s}ubsection{The theorems of Trofimov}\label{STrofimov}
As already mentioned, the automorphism group of a locally finite, connected
graph with the permutation topology is locally compact and totally
disconnected. In this section we will discuss three theorems of
Trofimov (see \cite{Trofimov1984},
\cite{Trofimov1985}, \cite{Trofimov1987}).
The conclusions in them all resemble the conclusion in Schlichting's
Theorem, but none of the theorems is proved by referring to
Schlichting's Theorem.
We start with the earliest of these three theorems. First, we
explain the terminology used.
An automorphism $g$ of a connected graph
$\Gamma$
is said to be {\em bounded} if there is a constant $c$ such that
$d_\Gamma(\alpha,g\alpha)\leq c$ for all vertices $\alpha$ in
$\Gamma$.
\begin{theorem}{\rm (\cite{Trofimov1984})}\label{TTrofimovBounded}
Let $\Gamma$ be a locally finite, transitive graph and $B(\Gamma)$ be the
subgroup of
bounded automorphisms. The following assertions are equivalent:
(i) The subgroup $B(\Gamma)$ is transitive.
(ii) There is an equivalence relation $\mbox{\bf s}im$ with finite equivalence
classes on the vertex set of $\Gamma$
such that $B(\Gamma)$ acts on $\Gamma$ like a finitely generated free abelian
group.
\end{theorem}
In a connected graph $\Gamma$ we define {\em the ball of
radius} $n$ {\em with
center} in a vertex $\alpha$ as the set $B_n(\alpha)=\{\beta\in
V\Gamma\mid d(\alpha, \beta)\leq n\}$.
The bounded
automorphisms of $\Gamma$ are related to topological properties of $\mbox{${\rm Aut}$}ga$ via the
following result of Woess. With a later application in mind,
Woess' result is stated for metric spaces rather than
just graphs. A (closed) {\em ball of radius $r$ with center }
$\alpha$ in a metric space $X$
as the set $B_r(\alpha)=\{\beta\in
X\mid d_X(\alpha, \beta)\leq r\}$.
An isometry $g$ of a metric space $X$
is said to be {\em bounded} if there is a constant $c$ such that
$d_X(\alpha,g\alpha)\leq c$ for all points $\alpha$ in
$X$.
Recall also that an element $g$ in a
topological group $G$ is called an FC\/$^-$-element if the conjugacy
class of $g$ has compact closure.
\begin{lemma}
{\rm (Cf. \cite[Lemma 4]{Woess1992})}\label{LWoessBounded}
Suppose $G$ is a topological group acting transitively
by isometries on a metric space $X$. Assume furthermore that the
stabilizer in $G$ of a point in $X$ is a compact open subgroup and
that for every value of $n$ the ball
$B_n(\alpha)$ is finite.
An element $g\in G$ is bounded if and only if $g$ is an
FC\/$^-$-element of $G$.
\end{lemma}
\begin{proof} Suppose $g\in G$ acts as a bounded isometry on $X$.
Find a number $M$ such that $d(g\alpha,\alpha)\leq M$ for every
$\alpha\in X$.
For $h\in G$, write $g^h=hgh^{-1}$. Set $g^G=\{g^h\mid h\in G\}$.
It is clear that
$d(g^h\alpha ,\alpha)=d(gh^{-1}\alpha , h^{-1}\alpha)\leq M$
for every $\alpha\in X$.
We see that the set
$g^G\alpha$ is finite and by Lemma~\ref{LCompact}(ii)
the conjugacy class $g^G$ has compact
closure.
Conversely, suppose that the conjugacy class
$g^G$ has compact closure. Then, for every $\alpha\in X$
the set $g^G\alpha$ is finite. Take a number $M$
such that $d(g^h\alpha,\alpha)\leq M$
for every $h\in G$. Take some $\beta\in X$.
Choose $h\in G$ so that $\beta =h^{-1}\alpha$. Then
$d(g\beta,\beta)=d(gh^{-1}\alpha, h^{-1}\alpha)=
d(g^h\alpha,\alpha)\leq M$. So $g$ acts on $\Gamma$ as a
bounded automorphism. \end{proof}
This connection with topological notions
can be used to give a short proof of Theorem~\ref{TTrofimovBounded},
where only elementary results from the
theory of topological groups are used, see \cite{Moller1998}.
A locally finite graph $\Gamma$ is said to have {\em polynomial
growth} if the number of vertices of $\Gamma$ in
$B_n(\alpha)$ is bounded above by a polynomial in $n$.
It is easy to see that this property does not depend on the choice of
the vertex $\alpha$.
A finitely generated group $G$ is said to have polynomial growth if
its Cayley graph with respect to a finite generating set
has polynomial growth (the choice of
generating sets is immaterial, since having polynomial growth is a
quasi-isometry invariant).
The second theorem of Trofimov related to Schlichting's Theorem
is the following:
\begin{theorem} \label{TTrofimovPoly}
{\rm (\cite[Theorem~2]{Trofimov1985})}
Suppose $\Gamma$ is a connected, locally finite graph with polynomial
growth, and $G$ is a group that acts transitively on $\Gamma$.
Then there is a $G$-invariant equivalence relation
$\mbox{\bf s}im$ with finite classes on the vertex set of $\Gamma$
such that the quotient of $G$ by the kernel of the induced
action on $\Gamma/\mbox{\bf s}im$ is a finitely generated, virtually
nilpotent group with finite stabilizers for vertices of
$\Gamma/\mbox{\bf s}im$.
\end{theorem}
It should be noted that Trofimov proves an even stronger result
\cite[Theorem~1]{Trofimov1985}, since
he shows that it is possible to find an equivalence relation
$\mbox{\bf s}im$ as described in Theorem~\ref{TTrofimovPoly} such that
the stabilizer of a vertex in $\mbox{${\rm Aut}$}(\Gamma/\mbox{\bf s}im)$ is
finite.
The theorem of Trofimov can be seen as a graph theoretical version of
Gromov's celebrated theorem characterizing finitely generated groups
with polynomial growth, see \cite{Gromov1981}. Indeed, Trofimov
uses Gromov's Theorem in his proof.
A version of
Gromov's theorem for topological groups has been proved by Losert
in \cite{Losert1987}.
Woess in \cite{Woess1992} used Losert's version of Gromov's Theorem
from \cite{Losert1987}
to give a short proof of Theorem~\ref{TTrofimovPoly}.
We will be returning to
polynomial growth and Trofimov's result in Section~\ref{SPolynomial}.
There is a third theorem of Trofimov's with a similar feel to it as the
two theorems stated above. This theorem involves the concept of
an $o$-automorphisms of a graph. An automorphism $g$ of a connected graph
$\Gamma$ is called an {\em $o$-automorphism} if
$$\max\{d(\beta, g\beta)\mid \beta\in V\Gamma, d(\alpha,\beta)\leq n\}=o(n),$$
where $\alpha$ is a fixed vertex. It is easy to show that this property
does not depend on the choice of the vertex $\alpha$. It is also easy to
prove that the $o$-automorphisms form a normal subgroup of $\mbox{${\rm Aut}$}ga$.
\begin{theorem} \label{TTrofimovoauto}
{\rm (\cite[Corollary~1]{Trofimov1987})}
Suppose $\Gamma$ is a connected, locally finite graph
and $G$ is a group that acts transitively on $\Gamma$. Then
the following are equivalent:
(i) $G\leq o(\mbox{${\rm Aut}$}ga)$
(ii) There is a $G$-invariant equivalence relation $\mbox{\bf s}im$ on the vertex
set of $\Gamma$ such that the equivalence classes of $\mbox{\bf s}im$ are
finite and
if $K$ denotes the kernel of the action of $G$ on the equivalence
classes then $G/K$ is a finitely generated nilpotent group acting
regularly on $\Gamma/\mbox{\bf s}im$.
\end{theorem}
Trofimov's proofs of these three theorems are long and
difficult. The proofs mentioned above of the first two theorems,
are
short, but admittedly, in the proof of Theorem~\ref{TTrofimovPoly}
the results from the theory of topological groups
used are highly non-trivial. It would be interesting to find
a topological interpretation of the concept
of an $o$-automorphism. Possibly that could lead to a shorter proof of
Theorem~\ref{TTrofimovoauto}.
\mbox{\bf s}ection{The scale function and tidy subgroups}\label{STidyScale}
\indent
The theory of locally compact groups is the part of the theory of
topological groups that has widest appeal and most applications in
other branches of mathematics. When
looking at locally compact groups there are the connected
groups on one end of the spectrum
and the totally disconnected groups on the other
end.
The fundamental result in the theory of locally compact totally
disconnected groups is the theorem of van Dantzig \cite{Dantzig1936} that
such a group must always contain a compact open subgroup.
A big step towards a general theory was taken in
the paper \cite{Willis1994} by Willis.
The fundamental concepts of Willis's theory are
the scale function and tidy subgroups.
\begin{definition}\label{DTidyscale}
Let $G$ be a locally compact totally
disconnected group and $x$ an
element in $G$. For a compact open subgroup $U$ in $G$ define
$$U_+=\bigcap_{i=0}^\infty x^iUx^{-i}\qquad\mbox{and}\qquad
U_-=\bigcap_{i=0}^\infty x^{-i}Ux^{i}.$$
Say $U$ is {\em tidy} for $g$ if\\
{(TA)} $U=U_+U_-=U_-U_+$\\
and \\
{(TB)} $U_{++}=\bigcup_{i=0}^\infty x^iU_+x^{-i}$ and
$U_{--}=\bigcup_{i=0}^\infty x^{-i}U_- x^{i}$ are both closed in $G$.
\par
Let $G$ be a locally compact totally
disconnected group.
The {\em scale function} on $G$ is defined as
$$\mbox{\bf s}(x)=\min\{|U:U\cap x^{-1}Ux|: U \mbox{\rm\ a compact open subgroup of }
G\}.$$
\end{definition}
The connection between the scale function and tidy subgroups is
described in the following theorem due to Willis.
\begin{theorem}
\label{TScaletidy}
{\rm (\cite[Theorem 3.1]{Willis2001a})}
Let $G$ be a totally disconnected, locally compact group and $g\in G$.
Then $\mbox{\bf s}(g)=|U:U\cap g^{-1}Ug|$ if and only if $U$ is tidy for $g$.
\end{theorem}
{\em Remark.} Instead of stating our results for totally disconnected, locally compact groups, we could phrase our results for locally compact groups, that contain a compact, open subgroup.
Now on to something completely different.
\mbox{\bf s}ubsection{Highly arc transitive digraphs}\label{Shat}
\begin{definition}
A digraph $\Gamma$ is called $s$-{\em arc transitive} if the
automorphism group acts transitively on the set of $s$-arcs. If $\Gamma$ is $s$-arc transitive for all numbers $s\geq 0$
then $\Gamma$ is said to be {\em highly arc transitive}.
\end{definition}
We also say that a group $G\leq \mbox{${\rm Aut}$}ga$ acts {\em highly arc transitively}
on $\Gamma$ if $G$ acts transitively on the $s$-arcs in $\Gamma$ for
all $s$.
The definition of highly arc transitive digraphs occurs first in the
paper \cite{CPW1993} by Cameron, Praeger and
Wormald. Similar conditions, both for directed and undirected
graphs, have been studied by various authors in various contexts.
Let us start by looking at several examples.
{\em Examples.} (i) Let $\Gamma$ be a directed tree with constant
in- and out-valencies. Clearly $\Gamma$ is highly arc transitive.
(ii) Let $\Gamma$ be a digraph with the set $\mbox{\bf Q}$ of rational numbers
as a vertex set and $(\alpha, \beta)$ an edge in $\Gamma$ if and only
if $\alpha >\beta$. Again it is clear that $\Gamma$ is a highly arc
transitive digraph.
(iii) (Cf.~\cite[Example 1]{Moller2002})
Let $T_1$ denote the regular directed tree in which every
vertex has in-valency 1 and out-valency $q$. Let $L=\ldots,
\alpha_{-1},\alpha_0,\alpha_1,\alpha_2,\ldots$ be a directed line in $T_1$.
Define
$$H=\{h\in\mbox{${\rm Aut}$}(T_1)\mid \mbox{there is a number }i \mbox{ such that }
h\alpha_i=\alpha_i\}.$$ If $h\in H$ and $h$ fixes some $\alpha_i$ then $h$
also fixes all vertices $\alpha_j$ with $j<i$. One can also see that the
orbits of $H$ are infinite and each orbit contains precisely one vertex
from $L$. The orbits of $H$ are called {\em horocycles}. The
horocycles could also be defined without reference to
the automorphism group. Then we could define two vertices $\alpha$ and
$\beta$ to be in the same {\em horocycle} if there is a number $n$
such that the unique path in $\Gamma$ from $\alpha$ to $\beta$
starts by going backwards along
$n$ arcs and then going forward along $n$ arcs.
Let $C_i$
denote the horocycle containing $\alpha_i$.
For each $i\in\mbox{\bf Z}$ take $r-1$ copies $S^1_i, \ldots, S^{r-1}_i$ of $T_1$
and let $\psi^j_i:S^j_i\rightarrow T_1$ be an isomorphism. The preimage
of the horocycle $C_i$ is a horocycle $B^j_i$ in $S^j_i$. When restricted,
$\psi^j_i$ defines an isomorphism between the digraphs spanned by
$\mbox{${\rm deg}$}s(B^j_i)$ and $\mbox{${\rm deg}$}s(C_i)$. Use
this partial isomorphism to identify the vertices in $\mbox{${\rm deg}$}s(B^j_i)$ with the vertices in
$\mbox{${\rm deg}$}s(C_i)$.
Do this for every $i$ and you get a new digraph $T_2$. The
digraph $T_2$ is far from being a tree, but if $\alpha$ is a vertex in
$T_2$ then the digraph spanned by
$\mbox{${\rm deg}$}s(\alpha)$ is a rooted infinite directed $q$-ary tree. The
vertices in $T_2$ that did belong to $T_1$ now all have out-valency
equal to $q$,
and in-valency equal to $r$.
Look at the part of $S^j_i$ that did not get identified with vertices
in $T_1$. This part is a union of horocycles, at each horocycle in it we
glue $r-1$ new copies of $T_1$ in the same fashion. Do this for each $i$ and
each horocycle in $S^j_i$, not belonging to $T_1$, and
get a digraph $T_3$. Continuing in the same fashion we construct a
sequence $T_1\mbox{\bf s}ubseteq T_2\mbox{\bf s}ubseteq T_3\mbox{\bf s}ubseteq\ldots$ of digraphs. In
the end we get a digraph $DL(q,r)=\bigcup T_i$. In this digraph every
vertex has
in-valency equal to $r$ and out-valency equal to $q$.
If $\alpha$ is a vertex in
$DL(q,r)$, then the subdigraph spanned by $\mbox{${\rm deg}$}s(\alpha)$
is an infinite rooted directed $q$-ary tree and the subgraph spanned
by $\mbox{${\rm anc}$}(\alpha)$ is a rooted tree, such that all edges are directed
towards the root and the in-valency of every vertex is $r$ and the
out-valency is $1$.
Clearly $DL(q,r)$ is highly arc transitive.
The digraphs $DL(q,r)$ are a directed versions of the Diestel-Leader
graphs (defined in \cite{DiestelLeader2001}) that have been studied by
various authors. Woess \cite{Woess1992} asked if every locally finite
transitive graph is quasi-isometric to some Cayley graph.
It was conjectured by Diestel and Leader that if
$q\neq r$ then the graph $DL(q,r)$ is
not quasi-isometric to any Cayley graph.
This conjecture was proved by Eskin,
Fisher and Whyte in \cite{EskinFisherWhyte2005}.
An optimist would hope to find a general classification of
locally finite,
highly arc transitive graphs, but
it seems very implausible that any such
classifications is possible.
But, there is a particular class of highly arc transitive digraphs
where one can give a precise description of their
structure. Surprisingly enough this particular class can be used to
probe the secrets of Willis' theory.
First, we state two simple lemmata from the paper \cite{CPW1993}
by Cameron, Praeger and
Wormald. We prove the second one, because it is natural to apply the
permutation topology on \mbox{${\rm Aut}$}ga\ in the proof.
\begin{lemma}\label{LCPW}
{\rm (\cite[Proposition~3.10]{CPW1993})} Let $\Gamma$ be a connected,
highly arc transitive digraph with finite out-valency. Suppose
$\Gamma$ is not a directed cycle. If $\alpha$ and $\beta$ are vertices
in $\Gamma$ and there is a directed path of length $n$ from $\alpha$
to $\beta$, then every directed path from from $\alpha$ to $\beta$ has
length $n$. Furthermore, $\Gamma$ has no directed cycles.
\end{lemma}
\begin{lemma}\label{LTransitveLines}
Let $\Gamma$ be a locally finite, highly arc transitive digraph.
Take two directed lines
$L_1=\ldots, \alpha_{-1}, \alpha_0, \alpha_1, \alpha_2\ldots$
and $L_2=\ldots, \beta_{-1}, \beta_0, \beta_1, \beta_2\ldots$ in $\Gamma$ then there is a an automorphism $g$ of
$\Gamma$ such that $g\alpha_i=\beta_i$ for all $i$.
\end{lemma}
\begin{proof} Write $G=\mbox{${\rm Aut}$}ga$ and note that $G$ is locally compact.
Using the property that $\Gamma$ is highly arc transitive,
we can find an element $g_i\in \mbox{${\rm Aut}$}ga$ such that $g_i\alpha_j=\beta_j$
for all $j\in\{-i,\ldots,i\}$. The sequence $(g_i)_{i\in {\bf N}}$ is contained in
the set $g_1G_{\alpha_0}$, which is compact in the permutation
topology on $\mbox{${\rm Aut}$}ga$. Hence this sequence has a convergent
subsequence that converges to an element $g$ in $\mbox{${\rm Aut}$}ga$ which has
the desired property. \end{proof}
\begin{proposition} {\rm (\cite[Lemma~3]{Moller2002a})}
Let $\Gamma$ be a locally finite, highly arc transitive digraph and
$L$ a directed line in $\Gamma$. Then the subdigraph $\Gamma_L$
spanned by
$\mbox{${\rm deg}$}s(L)$, is highly arc transitive and has more than one end.
\end{proposition}
\begin{proof} Write $L=\ldots, \alpha_{-1}, \alpha_0, \alpha_1,
\alpha_2\ldots$. Consider $s$-arcs $\beta_0,\ldots, \beta_s$ and
$\gamma_0,\ldots, \gamma_s$ in $\Gamma_L$.
The vertices $\beta_0$ and $\gamma_0$
will have a common ancestor $\alpha_{i_0}$ on the line $L$. Now we can
extend the $s$-arcs to infinite lines $L_\beta=\ldots, \beta_{-1},
\beta_0, \beta_1, \beta_2\ldots$ and $L_\gamma=\ldots, \gamma_{-1},
\gamma_0, \gamma_1, \gamma_2\ldots$ that both contain the directed ray
$\ldots, \alpha_{i_0-2},\alpha_{i_0-1}, \alpha_{i_0}$.
Then we can find an element
$g\in G$ such that $g\beta_i=\gamma_i$ for all $i$ and because $g$
maps the ray $\ldots, \alpha_{i_0-1}, \alpha_{i_0}$ into $L$ we can
see that $g(\mbox{${\rm deg}$}s(L))=\mbox{${\rm deg}$}s(L)$, i.e. the subdigraph $\Gamma_L$ is
invariant under $g$. Whereupon we conclude that $\Gamma_L$ is highly
arc transitive.
Let $\beta'$ be a vertex in $\Gamma_L$. Since $\mbox{${\rm in}$}_{\Gamma_L}(\beta')$ is
finite, there
clearly is a number $i$ such that $\mbox{${\rm in}$}_{\Gamma_L}(\beta')
\mbox{\bf s}ubseteq\mbox{${\rm deg}$}s(\alpha_i)$. Let $k$ be the length of a directed path
from $\alpha_i$ to $\beta'$ (by Lemma~\ref{LCPW}
all directed paths from $\alpha_i$ to $\beta'$ have the same length).
Making use
of arc transitivity we conclude that
if $\beta\in\mbox{${\rm deg}$}s_k(\alpha_0)$, then there is an element
$g\in\mbox{${\rm Aut}$}(\Gamma_L)$ such that
$g(\alpha_i)=\alpha_0$ and $g(\beta')=\beta$.
Therefore we see that if $(\gamma,\beta)$ is an
arc in $\Gamma_L$ (i.e.\ $\gamma\in\mbox{${\rm in}$}_F(\beta)$)
then $\gamma\in \mbox{${\rm deg}$}s(\alpha_0)$. More precisely,
$\gamma\in \mbox{${\rm deg}$}s_{k-1}({\alpha_0})$. This is so because, if
$\alpha_0,\gamma_1,\ldots,\gamma_l,\gamma$
is a directed path from $\alpha_0$ to
$\gamma$ then $\alpha_0,\gamma_1,\ldots,\gamma_l,\gamma,\beta$ is a
directed path from $\alpha_0$ to $\alpha$ and thus has length $k$.
Set $A=\bigcup_{i\geq k}\mbox{${\rm deg}$}s_i(\alpha_0)$ and $A^*=VF\mbox{\bf s}etminus A$. Suppose
$(\gamma,\beta)$ is an arc from $A^*$ to $A$.
Now $\beta\in\mbox{${\rm deg}$}s_l(\alpha_0)$ for some
$l\geq k$. Then $\gamma\in \mbox{${\rm deg}$}s_{l-1}(\alpha_0)$, by the choice of $k$.
Obviously $l=k$ and $(\gamma,\beta)$
is an arrow from $\mbox{${\rm deg}$}s_{k-1}(\alpha_0)$ to $\mbox{${\rm deg}$}s_{k}(\alpha_0)$.
{\em A priori}, there is also the possibility that some
arc $(\beta,\gamma)$ in
$F$ goes from $A$ to $A^*$. But on closer look, this is impossible,
because then $\beta$ would be in $\mbox{${\rm deg}$}s_l(\alpha_0)$ for some
$l\geq k$ and thus $\gamma\in\mbox{${\rm deg}$}s_{l+1}(\alpha_0)\mbox{\bf s}ubseteq A$, and
therefore $\gamma\in A$, contradicting the assumption that $\gamma\in
A^*$.
The only arcs
between $A$ and $A^*$ are going from $\mbox{${\rm deg}$}s_{k-1}(\alpha_0)$ to
$\mbox{${\rm deg}$}s_{k}(\alpha_0)$. The set of such arcs is clearly finite (because
$\mbox{${\rm deg}$}s_{k-1}(\alpha_0)$ is finite and the
out-valency of vertices in $\Gamma$ is
finite), and by removing them,
we split $\Gamma$ up into components. The two sets
$\{\ldots,\alpha_0,\alpha_1,\ldots,\alpha_{k-1}\}$ and $\{\alpha_k,\alpha_{k+1},\ldots\}$ will
belong to different components, so we have at least two infinite components.
Hence $\Gamma$ has more than
one end. \end{proof}
The structure of digraphs like $\Gamma_L$ in the above
proposition is described in the following theorem.
\begin{theorem}
{\rm (\cite[Theorem 1]{Moller2002a})}\label{THATtree}
Let $\Gamma$ be a locally finite, highly arc
transitive digraph. Suppose that there
is a line $L= \ldots, \alpha_{-1}, \alpha_0, \alpha_1,\ldots$
such that $V\Gamma={\mbox{${\rm deg}$}s}(L)$.
Then there exists a surjective homomorphism
$\phi:\Gamma\rightarrow T$ where $T$
is a directed tree
with in-valency 1 and finite out-valency.
The automorphism group of $\Gamma$ has a natural action on $T$
as a group of automorphisms such that
$\phi(g\alpha)=g\phi(\alpha)$ for every
$g\in \mbox{${\rm Aut}$}ga$
and every vertex $\alpha$ in $\Gamma$.
This action of $\mbox{${\rm Aut}$}ga$ on $T$ is highly arc
transitive. Furthermore, the
fibers $\phi^{-1}(\alpha)$, $\alpha\in VT$,
are finite and all have the same number
of elements.
\end{theorem}
Let $\alpha$ be a vertex in a highly arc transitive digraph $\Gamma$
and denote with $c_k$ the number of vertices in $\mbox{${\rm deg}$}s_k(\alpha)$.
Cameron, Praeger and Wormald in \cite[Definition~3.5]{CPW1993}
define the {\em out-spread} of a vertex
in $\Gamma$ as $\limsup_{k\rightarrow\infty} c_k^{1/k}$. One can
define the {\em in-spread} of a highly arc transitive digraph in
a similar way. Theorem~\ref{THATtree} implies the following
\begin{theorem}{\rm (\cite[Theorem~2]{Moller2002a})} The out-spread of
a locally finite, highly arc transitive digraph is an integer.
\end{theorem}
The in-spread can be used to characterize the highly arc
transitive digraphs treated in Theorem~\ref{THATtree}.
\begin{theorem}{\rm (\cite[Theorem~2.6]{MMMSTZ2005})} Let $\Gamma$ be a
locally finite, highly arc transitive digraph. The in-spread of
$\Gamma$ is 1 if and only if there is a line $L$ in $\Gamma$ such
that $\mbox{${\rm deg}$}s(L)=V\Gamma$.
\end{theorem}
\mbox{\bf s}ubsection{Tidy subgroups and highly arc transitive digraphs}\label{STidy}
Now we turn our attention back to totally disconnected, locally compact
groups. The following notation will be used extensively in what
follows. Let $G$ be a totally disconnected, locally compact
group and $x$ a fixed element in $G$.
Take a compact open sub\-group $U$.
We set $\Omega=G/U$ and let $\alpha_0$ denote the point
in $\Omega$ that has $U$ as stabilizer. Then define a digraph
$\Gamma=\Gamma_U$ that has $\Omega$ as a vertex set and edge set
$G(\alpha_0, x\alpha_0)$ -- the $G$-orbit of the ordered pair
$(\alpha_0, x\alpha_0)$. Note that $\Gamma$ need not be connected.
For an integer $i$ set
$\alpha_i=x^i\alpha_0$. The vertices $\alpha_i$ form a line $L$ in
$\Gamma$. Observe that $x^iUx^{-i}$ is the stabilizer of $\alpha_i$
in $G$ and $U_+$ is the stabilizer of the
vertices $\alpha_0, \alpha_1, \ldots$ and $U_-$ is the stabilizer
of the vertices $\alpha_0, \alpha_{-1}, \ldots$.
\begin{proposition}\label{PTAHAT}
{\rm (Cf.~\cite[Theorem~2.1]{Moller2002})}
The subgroup $U$ satisfies condition (TA) in Definition~\ref{DTidyscale}
if and only if $G$ acts highly arc transitively on the
digraph $\Gamma$.
\end{proposition}
\begin{proof}
Let us start by looking at what happens when the digraph $\Gamma$ is
highly arc transitive.
Let $g\in U=G_{\alpha_0}$. Since $G$ is assumed to act highly arc
transitively on $\Gamma$ and $G$ is a closed in the permutation
topology we deduce from Lemma~\ref{LTransitveLines} that $G$ acts
transitively on the set of lines in $\Gamma$. For $i\geq 1$ we set
$\beta_i=g\alpha_i$ and let $L_1$ denote the line $\ldots,
\alpha_{-1}, \alpha_0, \beta_1, \beta_2, \ldots$.
We find an element $g_-$ that moves the line $L$ to
the line $L_1$ such that $g_-\alpha_i=\alpha_i$ for $i\leq 0$ and
$g_-\alpha_i=\beta_i$ for $i\geq 1$. Note that $g_-\in U_-$.
Set $g_+=g_-^{-1}g$ and note that
$g_+$ fixes all the vertices $\alpha_0, \alpha_1, \ldots$ and thus
$g_+\in U_+$. Therefore $g\in U_-U_+$. From this we
deduce that $U$ satisfies condition TA.
Conversely, assume that $U$ satisfies condition TA. Take a vertex $\beta$ in
$\mbox{${\rm out}$}(\alpha_0)$. Then there must be an element $g\in U$ such that
$g\alpha_1=\beta$. Write $g=g_-g_+\in U_-U_+$ and we see that
$g_-\alpha_1=\beta$. Thus $U_-$ acts transitively on
$\mbox{${\rm out}$}(\alpha_0)$.
Now we use induction over $s$ to show that $G$
acts transitively on the set of $s$-arcs. Suppose we are given an
$(s+1)$-arc $\beta_0, \ldots, \beta_s, \beta_{s+1}$. Use the induction
hypothesis to find an element $h\in G$ such that $h\alpha_0=\beta_s,
h\alpha_{-1}=\beta_{s-1},\ldots, h\alpha_{-s}=\beta_0$. Then by the above,
$hU_+h^{-1}$ acts transitively on $\mbox{${\rm out}$}(h\alpha_0)=\mbox{${\rm out}$}(\beta_s)$. We
pick an element $h'$ from $hU_+h^{-1}$ such that
$h'(h\alpha_1)=\beta_{s+1}$. Now we have found an element $h'h$ that
moves the $(s+1)$-arc $\alpha_0, \ldots, \alpha_s, \alpha_{s+1}$
to the $(s+1)$-arc $\beta_0, \ldots, \beta_s, \beta_{s+1}$ and can
conclude that $G$ acts transitively on the $(s+1)$-arcs in $\Gamma$ and
also that $G$ acts highly arc transitively on $\Gamma$.
\end{proof}
Condition (TB) can also be translated in to a condition about the graph
$\Gamma$ defined at the start of the section.
We use the following lemma.
\begin{lemma}{\rm (\cite[Lemma 3]{Willis1994})}
Let $G$ be a totally disconnected,
locally compact group and $x\in G$. Suppose that $U$ is a compact,
open subgroup of $G$ that satisfies condition (TA). Then
(a) $U_{++}$ is closed if and only if $U_{++}\cap U=U_+$.
(b) $U_{++}$ is closed if and only if $U_{--}$ is closed.
\end{lemma}
In our setting $U_{++}$ is the set of all elements $g$ in $G$ such that
there exists a number $k$ such that $g$ fixes $\alpha_k,
\alpha_{k+1},\ldots$. The condition that $U_{++}\cap U=U_+$ says that
an element in $G$ that fixes $\alpha_0$ and also $\alpha_k,
\alpha_{k+1}, \ldots$ for some $k\geq 0$ must also fix $\alpha_1,
\ldots, \alpha_{k-1}$. If we assume that $G$ acts highly arc
transitively on $\Gamma$ then this implies that $\alpha_0, \ldots,
\alpha_k$ is the unique path in $\Gamma$ from $\alpha_0$ to
$\alpha_k$ and we conclude that the subgraph spanned by
$\mbox{${\rm deg}$}s(\alpha_0)$ is a tree.
On the other hand, if the subgraph spanned
by $\mbox{${\rm deg}$}s(\alpha_0)$ is a tree then clearly a group element that fixes
$\alpha_0$ and $\alpha_k$ must fix $\alpha_1,
\ldots, \alpha_{k-1}$ since these vertices lie on the directed
path from $\alpha_0$ to $\alpha_k$. Hence $U_{++}\cap U=U_+$.
Thus we have shown the following result.
\begin{proposition}
Suppose $U$ satisfies condition (TA). Then $U$ satisfies condition (TB)
if and only if the subgraph spanned by $\mbox{${\rm deg}$}s(\alpha_0)$ is a tree.
\end{proposition}
Putting these observation together as a theorem we get.
\begin{theorem}
{\rm (Cf.~\cite[Theorem~3.4]{Moller2002})}
Let $G$ be a totally disconnected, locally compact group and $U$ a
compact, open subgroup. Let $x$ be an element in $G$ and define a
graph $\Gamma$ such that the vertex set is $G/U$ and the edge set is
$G(\alpha_0, x\alpha_0)$ where $\alpha_0$ is the vertex in $\Gamma$ such
that $U=G_\alpha$. Suppose furthermore that the orbit of $\alpha_0$ under $x$ is infinite. Then $U$ is tidy for $x$ if and only if $G$ acts
highly arc transitively on $\Gamma$ and the subgraph spanned by
$\mbox{${\rm deg}$}s(\alpha_0)$ is a tree.
\end{theorem}
\mbox{\bf s}ubsection{Using the connection}
In this section $G$ denotes a totally disconnected, locally compact group.
From the definition of tidy subgroup it is far from obvious that there
always is a compact, open subgroup of $G$ that is tidy for a given
element $x$ in $G$. Our first task is thus to construct
a compact, open subgroup $U$ that is tidy for $x$.
First, the case where $x$ is periodic (i.e.~the subgroup generated
by $x$ is relatively compact). Let $U$ be a compact, open
subgroup. Put $\Omega=G/U$. Let $\alpha$ be a point in $\Omega$ such
that $G_\alpha=U$. Define $A$ as the closure (in the given topology
on $G$) of the subgroup generated by $x$. By assumption $A$ is
compact. Since the permutation topology induced by the action of $G$
on $\Omega$ is contained in the original topology on $G$ we conclude
that $A$ is also compact in the permutation topology. Hence,
by Lemma~\ref{LCompact}(ii), all the
orbits of the subgroup generated
by $x$ are finite. So there is a number $N$ such that
$x^N\alpha=\alpha$ and, therefore, $x^N\in G_\alpha=U$.
The subgroup $U\cap xUx^{-1}\cap \cdots\cap x^{N-1}Ux^{-(N-1)}$ is compact and
open and normalized by $x$ and thus tidy for $x$. Hence we will
assume in what follows that $x$ is not periodic.
Let $V$ be some compact, open subgroup of $G$. Construct a graph
$\Gamma=\Gamma_V$ as done at the start of the last section. From the proof of
Proposition~\ref{PTAHAT} we see that $G$ acts highly arc transitively
on $\Gamma$ if and only if $V_-$ acts transitively on $\mbox{${\rm out}$}(\alpha_0)$.
Look at the group $V_n=\bigcap_{i=0}^{n}x^{-i}Vx^i=G_{\alpha_0,
\alpha_{-1},\ldots, \alpha_{-n}}$. We claim that there is a number $n$
such that $V_n\alpha_1=V_-\alpha_1$. Otherwise one could find an
element $g_i\in V_i$ for each $i$ such that
$g_i\alpha_1\mbox{$\ \not\in\ $} V_-\alpha_1$. The
sequence $(g_i)_{i\in{\bf N}}$
has a convergent subsequence converging to an element
$g$ and clearly this element is in $V_-$, but $g\alpha_1\mbox{$\ \not\in\ $}
V_-\alpha_1$, so we have reached a contradiction. Now set $W=V_n$.
Note that $W_+=V_+$. We can use a similar argument as
in the first part of Proposition~\ref{PTAHAT} to show that $W$
satisfies condition (TA). Using this compact, open subgroup $W$ to get
a compact, open subgroup that also satisfies condition (TB) is more
involved, and the details will be left out. By finding a compact, open
subgroup $W$ satisfying (TA),
we have ensured that $G$ acts highly arc transitively on the digraph
$\Gamma_W$. What is missing is condition (TB), which would mean that the
subgraph spanned by the descendants of a vertex is a tree. To achieve
that, Theorem~\ref{THATtree} is used to produce a highly arc
transitive digraph, wherein the graph spanned by the descendants of a
vertex is a tree. This will then prove the following theorem of Willis.
\begin{theorem}{\rm (\cite[Theorem 1]{Willis1994},
see also \cite[Theorem 4.1]{Moller2002})}
Let $G$ be a totally disconnected, locally compact group and $x$ an
element of $G$. Then there is a compact, open subgroup $U$ of $G$ that
is tidy for $x$.
\end{theorem}
Now we have ensured that there is something to talk about.
We next use digraphs to deduce further facts about tidy subgroups
and the scale function.
First, we use Lemma~\ref{LModular} to deduce
the following.
\begin{theorem}{\rm (\cite[Corollary 1]{Willis1994})}\label{Tscale-modular}
Let $G$ be a totally disconnected, locally compact group. Denote by
$\Delta$ the modular function on $G$ and by ${\bf s}$ the scale
function on $G$. Then, for every $x\in G$,
$$\Delta(x)=\frac{{\bf s}(x)}{{\bf s}(x^{-1})}.$$
\end{theorem}
\begin{proof} Let $U_1$ and $U_2$ be compact open subgroups of $G$ such
that
$$|U_1:U_1\cap x^{-1}U_1x|={\bf s}(x)\qquad\mbox{and}
\qquad |U_2:U_2\cap xU_2x^{-1}|={\bf s}(x^{-1}).$$
Note that
$$|U_1:U_1\cap xU_1x^{-1}|\geq {\bf s}(x^{-1})
\qquad\mbox{and}\qquad
|U_2:U_2\cap x^{-1}U_2x|\geq {\bf s}(x).$$
Now we use the Remark following Lemma~\ref{LModular} and get
$$\frac{{\bf s}(x)}{{\bf s}(x^{-1})}\geq
\frac{|U_1:U_1\cap x^{-1}U_1x|}{|U_1:U_1\cap xU_1x^{-1}|}
=\Delta(x)=\frac{|U_2:U_2\cap x^{-1}U_2x|}{|U_2:U_2\cap xU_2x^{-1}|}
\geq \frac{{\bf s}(x)}{{\bf s}(x^{-1})}.$$
Hence $\Delta(x)={\bf s}(x)/{\bf s}(x^{-1}).$ \end{proof}
This also implies the following corollary.
\begin{corollary}{\rm (\cite[Corollary 3.11]{Willis2001a})}
\label{Cequal}
Let $x$ be an element of a totally disconnected, locally compact group $G$,
and $U$ a compact, open subgroup of $G$.
Then $|U:U\cap x^{-1}Ux|={\bf s}(x)$ if and
only if $|U:U\cap xUx^{-1}|={\bf s}(x^{-1})$.
\end{corollary}
Tidy subgroups are related to the scale function as described in
Theorem~\ref{TScaletidy}. The proof of Theorem~\ref{TScaletidy} is
involved, and we will only have a look at the proof that compact, open
subgroup $U$ such that ${\bf s}(x)=|U:U\cap x^{-1}Ux|$ must be tidy.
Consider a compact, open subgroup $U$, a fixed element $x\in G$, and the
digraph $\Gamma$ defined above. By the above, when trying to minimize
$|U:U\cap x^{-1}Ux|$ in order to find $\mbox{\bf s}(x)$, we could equally try to
minimize $|U:U\cap xUx^{-1}|$. The latter index is just the
out-valency in the digraph $\Gamma_U$. When constructing a tidy
subgroup for $x$, we start with an arbitrary compact, open subgroup $V$
and next find a compact, open subgroup $V$ satisfying (TA). The
out-valency in $\Gamma_V$ is at most the out-valency of $\Gamma_U$.
In the second step, we ensure that condition (TB) is satisfied, and
in the process the out-valency does not increase. Thus we can be sure
that if $U$ minimizes $|U:U\cap xUx^{-1}|$ then $U$ must be tidy for
$x$.
Again we look at a compact open subgroup $U$ and the graph $\Gamma$ as
above. Note
that $U\alpha_n=|U:U\cap x^nUx^{-n}|$. If $\Gamma$ is highly arc
transitive, then this is precisely the number $b_n$
of vertices $\beta$ such that
there is a directed path of length $n$ from $\alpha_0$ to $\beta$.
The out-valency $d_+$ of $\Gamma$ is equal to $|U:U\cap xUx^{-1}|$.
The subgraph spanned by $\mbox{${\rm deg}$}s(\alpha_0)$ is a tree if and only if
$b_n=d_+^n=|U:U\cap xUx^{-1}|^n$ for all natural numbers $n$. But $U$ is tidy
if an only if $\Gamma$ is highly arc transitive and the subgraph
spanned by $\mbox{${\rm deg}$}s(\alpha_0)$ is a tree. Thus we derive
the following result.
\begin{theorem}
{\rm (\cite[Corollary~3.5]{Moller2002})}
Let $G$ be a totally disconnected, locally compact group and $x$ an
element in $G$. Then a compact, open subgroup $U$ is tidy for $x$ if
and only if
$$|U:U\cap x^nUx^{-n}|=|U:U\cap xUx^{-1}|^n$$
for all $n\geq 1$.
\end{theorem}
\begin{corollary}
Let $G$ be a totally disconnected locally compact group and $x$ an
element in $G$. Then $\mbox{\bf s}(x^n)=\mbox{\bf s}(x)^n$.
\end{corollary}
\begin{proof} Let $U$ be a compact, open subgroup of $G$ that is tidy
for $x$.
It is easy to check that if a subgroup $U$ is tidy for
$x$ then $U$ is also tidy for $x^n$ for every integer $n$.
Hence
$$\mbox{\bf s}(x^n)=|U:U\cap x^{-n}Ux^{n}|=|U:U\cap x^{-1}Ux|^n=\mbox{\bf s}(x)^n. \qedhere$$
\end{proof}
If $\Gamma$ is highly arc transitive, the index $|U:U\cap x^{-n}Ux^{n}|$
is the number of vertices $\beta$ such that
$\alpha_0$ is in $\mbox{${\rm deg}$}s_n(\beta)$. This observation suggests that we compare
the scale function and the in-spread of the associated digraph $\Gamma$. The following
theorem describes their relationship.
\begin{theorem} {\rm (\cite[Theorem~7.7]{Moller2002})}
Let $G$ be a totally disconnected, locally compact group and $x$ an
element of $G$. If $V$ is some compact, open subgroup of $G$, then
$$\mbox{\bf s}(x)=\lim_{n\rightarrow\infty}|V:V\cap x^{-n}Vx^n|^{1/n}.$$
\end{theorem}
For a different formulation and a proof see
\cite[Lemma~4]{BaumgartnerWillis2006}.
This line of thought also gives us information about the case
$\mbox{\bf s}(x)=1$.
\begin{theorem}
{\rm (\cite[Corollary~7.8]{Moller2002})}
Let $G$ be a totally disconnected, locally compact group and $x$ an
element of $G$ such that $\mbox{\bf s}(x)=1$.
If $V$ is some compact, open subgroup of $G$, then there is a constant
$C$ such that
$|V:V\cap x^{-n}Vx^n|\leq C$ for all $n\geq 0$.
\end{theorem}
These two results can also be formulated as results about
permutation groups.
\begin{theorem}\label{TExponential}
Let $G$ be a group acting transitively on a set $\Omega$. Assume that
all sub\-orbits of $G$ are finite. Let $x$ be an element in $G$ and
$\alpha_0$ a point in $\Omega$. Set $\alpha_i=x^i\alpha_0$. Then
either there is a constant $C$ such that $|G_{\alpha_0}\alpha_n|\leq
C$ for all $n$ or the numbers $|G_{\alpha_0}\alpha_n|$ grow
exponentially with $n$ and
$\lim_{n\rightarrow\infty}|G_{\alpha_0}\alpha_n|^{1/n}=s$
for some integer $s$.
\end{theorem}
{\em Remark.}
In
\cite{Trofimov2007} Trofimov studies {\em
generalized $x$-tracks}, which
are similar to the directed line $...,
\alpha_{-1}, \alpha_0, \alpha_1, \alpha_2, ...$ that is fundamental to
the graph-theoretical interpretation of Willis' theory in
\cite{Moller2002}. Theorem~\ref{TExponential} is
clearly related to \cite[Theorem~4.1, part 3]{Trofimov2007}.
The final illustration of the uses of graphs in Willis' structure
theory is a proof of the
following theorem.
\begin{theorem} {\rm (\cite[Theorem 2]{Willis1995})}
Let $G$ be a totally disconnected, locally compact group.
The set $P(G)$ of periodic elements in $G$ is closed.
(An element $x\in G$ is {\em periodic} if and only if $\overline{\langle
x\rangle}$ is compact.)
\end{theorem}
\begin{proof} The trick is to use the fact that
that a connected infinite and locally finite highly arc
transitive digraph has no directed cycles, see Lemma~\ref{LCPW}.
Suppose $x$ is not periodic, but is in the closure of $P(G)$.
Let $U$ be a compact, open subgroup of $G$, that is tidy for
$x$. Define a digraph $\Gamma$ as at the start of
Section~\ref{STidy}. If $x$ is not periodic, then the orbit of
$\alpha_0$ under $x$ is infinite, and the connected
component of $\Gamma$ that contains $\alpha_0$ is infinite. (It
must contain the line $\ldots,\alpha_{-1}, \alpha_0, \alpha_1,
\alpha_2,\ldots$.)
The set $xU$ is an open neighbourhood of $x$, and must therefore
contain some periodic element $g$. The fact that $g\in xU=xG_{\alpha_0}$
implies
$g\alpha_0=x\alpha_0=\alpha_1$. The element $g$ is periodic, hence
the orbit of $\alpha_0$ under $g$ is finite, and therefore there is an
integer $n$ such that $g^n(\alpha_0)=\alpha_0$. The sequence
$\alpha_0, \alpha_1=g\alpha_0, \beta_2=g^2\alpha_0,
\ldots, \beta_n=g^n\alpha_0=\alpha_0$
is a directed cycle in
$\Gamma$. This contradicts Lemma~\ref{LCPW} mentioned above.
Hence we
conclude that it is impossible that the closure of $P(G)$ contains any
elements that are not periodic. Thus $P(G)$ is closed. \end{proof}
\mbox{\bf s}ection{Rough Cayley graphs}\label{SRough}
Most of the material in this section is taken from a paper by Kr\"on
and M\"oller \cite{KronMoller2008}. Let $G$ be a compactly
generated, totally disconnected, locally compact group. In
\cite{KronMoller2008} the authors construct a locally finite,
connected graph with a transitive $G$-action, whose vertex
stabilizers are compact, open subgroups of $G$. This graph is called
a {\em rough Cayley graph} of $G$. As demonstrated in
\cite{KronMoller2008}, and summarized in this section,
a rough Cayley graph can be used to study compactly generated,
locally compact groups in a similar way as an ordinary Cayley
graph is used to study a finitely generated group.
In this article, we illustrate this approach, by using rough
Cayley graphs to generalize the concept of ends of groups and
to study compactly generated, locally compact groups of
polynomial growth.
Below, we explain how to construct a rough Cayley graph and
it is also shown that any two rough Cayley graphs for a given group
are quasi-isometric.
The applications of the rough Cayley graph to the theory of ends
of groups and to groups of polynomial growth are only sketched;
details can be found in \cite{KronMoller2008},
where tools from \cite{Dunwoody1982} and \cite{DicksDunwoody1989} are
used extensively.
\mbox{\bf s}ubsection{Definition of a rough Cayley graph}
\label{SDefRough}
\begin{definition}{\rm (\cite[Definition~2.1]{KronMoller2008})}
Let $G$ be a topological group. A connected graph $\Gamma$ is said to be a
{\em rough Cayley graph} of $G$ if $G$ acts transitively on $\Gamma$ and
the stabilizers of vertices are compact, open subgroups of $G$.
\end{definition}
In this section we show that if $G$ is a compactly
generated locally compact group that contains a compact open subgroup
then $G$ has a locally finite rough Cayley graph and any two rough
Cayley graphs are
quasi-isometric to each other. The approach here is different from
the approach in \cite{KronMoller2008}.
Let $G$ be a compactly generated topological group. For a compact
generating set $S$ we form the Cayley graph $\Gamma=\mbox{${\rm Cay}$}(G, S)$ of $G$ with
respect to $S$. The vertex set of $\Gamma$ is equal to $G$ and thus
carries a topology. The compactness of $S$ and the continuity of
multiplication in $G$ implies that if $A$ is a relatively compact set
of vertices in $\Gamma$ then the neighbourhood of $A$ in $\Gamma$ is
contained in the set $A\cdot S$ and is thus relatively compact also.
This can be used
to prove that a set $A$ of vertices in $\Gamma$ is relatively compact
if and only if it has finite diameter in the graph metric on $\Gamma$,
see \cite[2.3 Heine-Borel-Eigenschaft]{Abels1974}.
\begin{definition} \label{Dorbit}
Let $G$ be a group acting transitively on a connected
graph $\Gamma$. Suppose $U$ is a subgroup of $G$ that contains the
stabilizer of some vertex $\alpha$. The orbit $U\alpha$ is a block
of imprimitivity. Let $\Gamma_U$ denote the quotient graph
with respect to the $G$-congruence whose classes are the translates
under $G$ of the set $U\alpha$.
\end{definition}
\begin{lemma} \label{Llocallyfinite}
Let $G$ be a compactly generated topological group. Suppose $S$
is a compact generating set and
$U$ is a compact open subgroup of $G$.
Then the graph
$\mbox{${\rm Cay}$}(G,S)_U$ is locally finite.
\end{lemma}
\noindent
\begin{proof} The neighbourhood of a coset $gU$ in $\Gamma$ is compact
and can thus be covered by finitely many right cosets of $U$. Whence
$\Gamma_U$ is locally finite. \end{proof}
Lemma~\ref{Llocallyfinite} shows that every compactly generated group $G$
that has a compact, open subgroup $U$, has a locally finite rough Cayley
graph, namely, $\mbox{${\rm Cay}$}(G,S)_U$.
The proof of a result of Sabidussi \cite[Theorem 2]{Sabidussi1964}, restated
below in our own terminology, can be used to show that every locally finite
rough Cayley graph for a compactly generated group with a compact, open
subgroup can be obtained in this fashion.
\begin{theorem}{\rm (Cf.~\cite[Theorem 2]{Sabidussi1964})}
Let $\Gamma$ be a connected graph and $G$ a transitive subgroup of
$\mbox{${\rm Aut}$}(\Gamma)$. Then there is a set $S$ of generators of $G$ such
that $\Gamma\cong \mbox{${\rm Cay}$}(G, S)_U$ where $U$ is the stabilizer in $G$
of some vertex in $\Gamma$.
\end{theorem}
In our setting we are thinking of a topological group $G$ acting on a
locally finite rough Cayley graph $\Gamma$ such that stabilizers of vertices are compact open subgroups but the action is not necessarily
faithfully. Take a
vertex $\alpha$ in $\Gamma$ and let $U$ denote the stabilizer in $G$
of $\alpha$. Looking at Sabidussi's proof of his theorem we deduce that if
$S$ is the union of $U$ and all elements $h$ in $G$ such that
$\alpha$ and $h\alpha$ are adjacent in $\Gamma$
then $\Gamma\cong \mbox{${\rm Cay}$}(G, S)_U$.
Note also that the orbit $S\alpha$ consists precisely of
$\alpha$ and all
its neighbours. The graph $\Gamma$ is by assumption locally finite
so $S$ is a finite
union of cosets of $U$ and thus compact.
The influential concept of
quasi-isometry was introduced by Gromov \cite{Gromov1986} and has been
widely used since.
\begin{definition}
Two metric spaces $(X, d_X)$ and $(Y, d_Y)$ are said to be {\em
quasi-isometric} if there is a map $\varphi:X\rightarrow Y$ and
constants $a\geq 1$ and $b\geq 0$ such that for all points
$x_1$ and $x_2$ in $X$
\[a^{-1}d_X(x_1, x_2)-a^{-1}b\leq d_Y(\varphi(x_1), \varphi(x_2))
\leq ad_X(x_1, x_2)+ab,\]
and for all points $y\in Y$ we have
\[d_Y(y, \varphi(X))\leq b.\]
A map $\varphi$ between two metric spaces satisfying the above
conditions is called a {\em quasi-isometry}.
\end{definition}
Two connected graphs $X$ and $Y$ are called quasi-isometric if $(VX,
d_X)$ and $(VY, d_Y)$ are quasi-isometric.
Being quasi-isometric is an equivalence relation on the class of
metric spaces.
\begin{theorem}\label{Tquasi}
Let $G$ be a compactly generated group. Assume that $G$ admits a rough Cayley graph.
All rough Cayley graphs for $G$ are quasi-isometric.
\end{theorem}
The first step in the proof of Theorem \ref{Tquasi} is the following
Lemma. The proof
of the Lemma depends on the {\em Heine-Borel Eigenschaft},
\cite[2.3 Heine-Borel-Eigenschaft]{Abels1974}, mentioned in the
paragraph preceding Definition~\ref{Dorbit}.
\begin{lemma} \label{Lquasi}
(i) Let $G$ be a compactly generated topological group. Suppose $S_1$ and
$S_2$ are compact generating sets for $G$. Then the Cayley-graphs
$\mbox{${\rm Cay}$}(G, S_1)$ and $\mbox{${\rm Cay}$}(G, S_2)$ are quasi-isometric.
(ii) Suppose $S$ is a
compact generating set and
$U$ is a compact subgroup of $G$.
Then the graphs
$\mbox{${\rm Cay}$}(G, S)$ and $\mbox{${\rm Cay}$}(G, S)_U$ are quasi-isometric.
\end{lemma}
\noindent
\begin{proof} (i) There is a constant $C$, such that the elements in $S_1$ can be
expressed as words of length $\leq C$ in the elements in $S_2$
and {\em vice versa}. For each element of $S_1$ respectively $S_2$
fix a word in $S_2$ respectively $S_1$ with this property.
Using these correspondences, words in $S_1$ and $S_2$ may be
expressed as words in the other set, that are at most $C$ times longer.
This shows that the identity map on $G$ extends to quasi-isometries
$\operatorname{Cay}(G, S_1)\to \operatorname{Cay}(G, S_2)$ and
$\operatorname{Cay}(G, S_2)\to \operatorname{Cay}(G, S_1)$
that are inverse to each other.
(ii) By part (i) we know that $\mbox{${\rm Cay}$}(G, S)$ and
$\mbox{${\rm Cay}$}(G, S\cup U)$ are quasi-isometric. Thus we may assume that $S$
contains $U$. Under this assumption, each of the right cosets of $U$ has therefore diameter 1.
The quotient graph $\mbox{${\rm Cay}$}(G, S)_U$ is obtained by contracting each of these
cosets. Clearly this operation preserves quasi-isometry and the claim follows. \end{proof}
\begin{proof} (Theorem~\ref{Tquasi}) Suppose $\Gamma_1$ and
$\Gamma_2$ are rough Cayley graphs of $G$. Then we can find compact
generating sets $S_1$ and $S_2$ and compact, open subgroups $U_1$ and
$U_2$ such that $\Gamma_1=\mbox{${\rm Cay}$}(G, S_1)_{U_1}$ and
$\Gamma_2=\mbox{${\rm Cay}$}(G, S_2)_{U_2}$. By part~(ii) of Lemma~\ref{Lquasi}, $\Gamma_1$ is quasi-isometric to
$\mbox{${\rm Cay}$}(G, S_1)$, which in turn is quasi-isometric to $\mbox{${\rm Cay}$}(G,
S_2)$ by part~(i) of Lemma~\ref{Lquasi}, which is again
quasi-isometric to $\Gamma_2=\mbox{${\rm Cay}$}(G, S_2)_{U_2}$
by part~(ii) of the same result.
\end{proof}
Suppose $G$ is a locally compact group with a compact, open subgroup.
If the group is compactly generated, Lemma \ref{Llocallyfinite} gives
the existence of a locally finite rough Cayley graph. Conversely,
the existence of a rough Cayley graph implies that $G$ is compactly
generated.
\begin{proposition} {\rm (\cite[Corollary~1]{Moller2003})}
\label{Pcompactgenerated}
Suppose that $G$ is a locally compact topological group and that
$G$ acts transitively on a connected, locally finite graph such that
the stabilizers of vertices in $G$ are compact, open subgroups of $G$.
Then $G$ is compactly generated.
\end{proposition}
The proof of the proposition is based on the Lemma below.
\begin{lemma} \label{Ltransitive}
Let $G$ be a group acting transitively on a connected, locally finite
graph $\Gamma$. Then $G$ has a finitely generated, transitive subgroup.
\end{lemma}
\noindent
\begin{proof} Fix a reference vertex $\alpha$. Denote the neighbours
of $\alpha$ by $\beta_1, \ldots, \beta_n$. Choose elements $h_1,
\ldots, h_n$ such that $h_i\alpha=\beta_i$. We claim that
$H=\langle h_1, \ldots, h_n\rangle$ is transitive on the vertices of
$\Gamma$. Note that all the vertices in $\Gamma$ that are adjacent
to the vertex $\alpha$ are in the $H$-orbit of $\alpha$. Suppose that
$\beta=h\alpha$ for some $h\in H$. Then $hh_1h^{-1}\beta, \ldots,
hh_nh^{-1}\beta$ is an enumeration of all the neighbours of $\beta$.
Whence the neighbours of $\beta$ are also contained in the $H$-orbit
of $\alpha$. Since our graph is assumed to be connected, we conclude
that $H$ acts transitively on the vertices. \end{proof}
Proposition~\ref{Pcompactgenerated}
follows from Lemma \ref{Ltransitive}, because the union of the
stabilizer of a vertex with a finite generating set for a transitive subgroup
forms a compact generating set
for $G$.
The following theorem concludes our basic considerations of rough Cayley graphs.
Its first part is well known.
\begin{theorem} \label{TCocompact}
{\rm (\cite[Corollary~2.11]{KronMoller2008})}
Let $G$ be a compactly generated topological group that
has a compact open subgroup. Assume that $H$ is a cocompact closed
subgroup of $G$. Then $H$ is compactly generated and if $\Gamma_G$ is a
rough Cayley graph for $G$ and $\Gamma_H$ is a rough Cayley graph for $H$
then $\Gamma_G$ and $\Gamma_H$ are quasi-isometric.
\end{theorem}
\begin{proof} Let $X$ be a locally finite rough Cayley graph for $G$. By
Lemma~\ref{Lcocompact} we know $H$ acts with finitely many orbits on $X$.
Choose a vertex $\alpha$ in $\Gamma$.
Then there is a number $k$ such that every vertex in $\Gamma$ is in
distance at most $k$ from the orbit $H\alpha$. Now form the graph
$\Gamma'$ which has the same vertex set as $\Gamma$ but two
vertices being
adjacent if and only if their distance in $\Gamma$ is at most $2k+1$.
Note that $\Gamma'$ is also
locally finite. Consider the subgraph
$\Delta$ of $\Gamma'$
spanned by the vertices in $H\alpha$. Suppose $\alpha$ and $\beta$
are vertices in $\Delta$ and that $\alpha=\alpha_0,
\alpha_1, \ldots, \alpha_n=\beta$ is a path in
$\Gamma$ (and thus also a path in $\Gamma'$) from $\alpha$
to $\beta$. For each $\alpha_i$
choose a vertex $\beta_i$ in $H\alpha$ such that
$d_\Gamma(\alpha_i, \beta_i)\leq k$.
Then $d(\beta_i, \beta_{i+1})\leq 2k+1$ so either
$\beta_i=\beta_{i+1}$
or $\beta_i$ and
$\beta_{i+1}$ are adjacent in $\Gamma'$. Whereupon we conclude that
$\Delta$ is
connected. The action of $H$ on the connected, locally finite graph $\Delta$
is transitive with compact, open vertex-stabilizers. Hence $\Delta$
is a rough Cayley graph for $H$ and $H$ is compactly generated
by Proposition~\ref{Pcompactgenerated}.
From the above it is clear that $\Delta$ is
quasi-isometric to $\Gamma$. The second part of the theorem
follows by Theorem~\ref{Tquasi}. \end{proof}
{\em Remark.} For the rest of Section~\ref{SRough} we will focus on compactly generated, totally disconnected, locally compact groups.
Corresponding results hold for compactly generated, locally compact groups that contain a compact, open subgroup.
\mbox{\bf s}ubsection{Application to FC$^-$-groups}
As mentioned in Section~\ref{STrofimov}, an element $g$ of a topological
group $G$ is called a FC$^-$-element if its conjugacy class in $G$
has compact closure. It is an easy exercise to show that the
FC$^-$-elements of $G$ form a normal subgroup $B(G)$ of $G$. If $G=B(G)$
then $G$ itself is called a FC$^-$-group. These concepts have been
extensively studied, see for example the paper
\cite{GrosserMoskowitz1971} by Grosser and Moskowitz and various
papers by Wu and his collaborators, e.g. \cite{WuYu1972} and
\cite{Wu1991}.
A basic question about the subgroup $B(G)$ is whether it is closed or
not. This question was discussed by Tits in \cite{Tits1964}, where it
proved that $B(G)$ is closed if $G$ is a connected locally compact
group. But, Tits also gives an example of a locally compact, totally
disconnected group where $B(G)$ is not closed. Below is another
example of such a group described by using graphs.
{\em Example.} Let $\Gamma$ denote a directed tree such that each
vertex has in-valency 1 and out-valency 2. Choose a directed line
$\ldots, \alpha_{-1}, \alpha_0, \alpha_1, \alpha_2, \ldots$ in $\Gamma$.
We say that vertices $\alpha$
and $\beta$ are in the same {\em horocycle} if there is a number $n$
such that the unique path in $\Gamma$ from $\alpha$ to $\beta$
starts by going backwards along
$n$ arcs and then going forward along $n$ arcs.
(This concept is
also discussed in Section~\ref{Shat}.) Membership in the same
horocycle is an equivalence
relation on vertices. We denote by $C_i$ the equivalence
class of $\alpha_i$.
Define $H$ as the subgroup of $\mbox{${\rm Aut}$}ga$ that stabilizes $C_0$. Let $G$
denote the permutation group that $H$ induces on $C_0$ and endow $G$
with the permutation topology arising from that action. We think of
$C_0$ as a metric space, the metric being the restriction of the graph
metric on $\Gamma$. If $g$ is an
element of $G$ that fixes all but finitely many vertices in $C_0$ then
by Lemma~\ref{LWoessBounded} the element $g$ is an FC$^-$-element of
$G$. It is also clear that if $g$ and $h$ are elements of $G$, both
with finite support, then there is a conjugate of $h$ whose support is
disjoint from the support of $g$. Let $g_i$ be an element if $G$ with
finite support such that there is a vertex $\alpha\in C_0$ such that
$d(\alpha, g_i \alpha)=2i$. (We could define $g_i$ explicitly by
defining $\beta_{i+1}$ as the vertex such that $(\alpha_{-i-2},
\beta_{i+1})$ is an arc in $\Gamma$ and $\beta_{i+1}\neq \alpha_{-i-1}$
and then let $g$ transpose the two outward directed arcs from
$\beta_{i+1}$.) We may assume that $\mbox{\bf s}upp g_i\cap \mbox{\bf s}upp g_j=\emptyset$ if
$i\neq j$. All the $g_i$'s are contained in $B(G)$ and $h_i=g_1\ldots
g_1$ is also in $B(G)$. Then
we define $g$ as the limit of the sequence of the $h_i$'s ($g\alpha=h_i\alpha$
if $\alpha$ is in $\mbox{\bf s}upp h_i$ and $g\alpha=\alpha$ if $\alpha$ is not
in the support of any of the $h_i$'s). Clearly $g$ is not a bounded
isometry of $C_0$ and thus $g$ is not in $B(G)$. Hence $B(G)$ is not closed.
Indeed, one can easily show that $B(G)$ is dense in $G$.
The construction of a rough Cayley graph can be used in the study
of FC$^-$-elements in groups. Trofimov in \cite{Trofimov1985a} proved
the following.
\begin{theorem}{\rm (\cite{Trofimov1985a}, cf. \cite[Theorem~3]{Woess1992})}
Let $\Gamma$ be a vertex transitive, connected, locally finite graph. The
subgroup of bounded automorphisms in $\mbox{${\rm Aut}$}ga$ is closed in $\mbox{${\rm Aut}$}ga$.
\end{theorem}
Using a rough Cayley graph for the group and this theorem allows us
to prove the following result.
\begin{theorem}{\rm (\cite[Theorem~2]{Moller2003})} Let $G$ be a
compactly generated totally disconnected locally compact group.
Then the subgroup of FC$^-$-elements is closed in $G$.
\end{theorem}
The proof is simple. One constructs a locally finite rough Cayley
graph for $G$. The action of $G$ on $\Gamma$ gives a continuous
homomorphism $G\rightarrow\mbox{${\rm Aut}$}ga$. It is easy to show that the kernel
of this homomorphism is compact and the image is closed in $\mbox{${\rm Aut}$}ga$ (see
\cite[Corollary~1]{Moller2003}). The subgroup of bounded automorphism
in $\mbox{${\rm Aut}$}ga$ is
closed by the theorem of Trofimov above,
and the subgroup of FC$^-$-elements is closed in $G$, because it
is the preimage of the subgroup of bounded automorphisms in
$\mbox{${\rm Aut}$}ga$.
\mbox{\bf s}ubsection{Rough ends}
The space of ends of a connected, locally finite graph $\Gamma$ is the
boundary of a certain compactification of $V\Gamma$, where we think of
$V\Gamma$ as having the discrete topology. One can
think of different ends as representing the "different ways of
going to infinity" in $\Gamma$. Ends of graphs can both be defined by
using topological concepts and by purely graph theoretical means.
The graph theoretical method extends to graphs that are not locally
finite, but then the ends do not give a compactification of the vertex
set like in the locally finite case.
Recall that a {\em ray} in a graph is a sequence $\alpha_0, \alpha_1,
\ldots$ of
distinct vertices such that $\alpha_i$ is adjacent to $\alpha_{i+1}$
for all $i$.
The graph theoretic approach is to define the
ends of a graph $\Gamma$ as equivalence classes
of rays. Two rays are equivalent
(i.e.~are in the same {\em end}) if
there is the third ray that intersects both infinitely often. This
definition goes back to the paper \cite{Halin1964} by Halin.
Ends can also be defined with reference to connected components when
finite sets of edges are removed from the graph. For a finite set
$\Phi$ of edges define ${\cal C}_\Phi$ as the set of connected
components of $\Gamma\mbox{\bf s}etminus \Phi$.
Suppose $\Phi_1$ and
$\Phi_2$ are two finite sets of edges such that
$\Phi_1\mbox{\bf s}ubseteq\Phi_2$.
There is a natural map ${\cal C}_{\Phi_2}\rightarrow {\cal
C}_{\Phi_1}$ that takes a component $c$ of $\Gamma\mbox{\bf s}etminus \Phi_2$ to
the component of $\Gamma\mbox{\bf s}etminus \Phi_1$ that contains $c$. The collection of all the sets ${\cal C}_\Phi$ where
$\Phi$ ranges over all finite sets of edges in $\Gamma$, together
with the connecting natural maps forms an
inverse system. The space of ends $\Omega\Gamma$ of $\Gamma$
(with the natural topology of the
inverse limit) is then defined as the inverse
limit of this system. One could also look at the components of
$\Gamma\mbox{\bf s}etminus \Phi$ where $\Phi$ is a finite set of vertices, but
when the graph is locally finite the inverse limits are homeomorphic.
This approach goes back to the thesis of Freudenthal in 1931. In later works
Freudenthal and Hopf built up a theory of ends of spaces, see
\cite{Freudenthal1942}, \cite{Freudenthal1945} and \cite{Hopf1944}.
The graph theoretic approach to ends also leads to a natural definition of a topology. The {\em co-boundary} of a set $c\mbox{\bf s}ubseteq V\Gamma$ is the set of all edges in
$\Gamma$ such that one of its end vertices is in $c$ and the other is in
$V\Gamma\mbox{\bf s}etminus c$. Denote the co-boundary of $c$ with $\mbox{${\rm deg}$}lta c$.
Suppose $|\mbox{${\rm deg}$}lta c|<\infty$.
One sees that if $c$ contains all but finitely many vertices
from a ray $R$ then $c$ also contains also all but
finitely many vertices from
any ray in the same end as $R$. Thus we can say that the end
belongs to $c$. Define $\Omega c$ as the set of ends that belong to $c$.
The topology on the
space of ends has as a basis the sets $\Omega c$ where $c\mbox{\bf s}ubset
V\Gamma$ and $|\mbox{${\rm deg}$}lta c|<\infty$.
The graph $\Gamma\mbox{\bf s}etminus\mbox{${\rm deg}$}lta c$ is not connected. If
$\omega_1$ and $\omega_2$ are two ends of $\Gamma$ and $\omega_1$ belongs
to $c$ and $\omega_2$ belongs to $V\Gamma \mbox{\bf s}etminus c$ then we can say that
$\mbox{${\rm deg}$}lta c$ separates $\omega_1$ and $\omega_2$.
It is easy to show that a quasi-isometry between two locally finite
connected graphs induces a homeomorphism between the end spaces
of the graphs. In particular the number of ends is a quasi-isometry
invariant.
For a detailed introduction to ends of graphs the reader can
consult \cite{Moller1995} or \cite{DiestelKuhn2003}.
\mbox{\bf s}ubsubsection{Stallings' Theorem}
For a finitely generated group the number of ends is defined as the
number of ends of a Cayley graph of the group with respect to a finite
generating set. This is well defined, because the number of ends of a space is
invariant under quasi-isometry, as noted above. It can be shown
that a finitely generated group has $0$, $1$, $2$ or $\infty$ many ends.
Let $G$ be a compactly generated, totally disconnected, locally
compact group. Since $G$ admits a rough Cayley graph $\Gamma$,
which is unique up to quasi-isometry, we may define the
\emph{space of rough ends} of $G$ as the space of ends of $\Gamma$.
The cardinality of the space of rough ends of $G$ will be called the
\emph{number of rough ends} of $G$. These definitions obviously
extend the traditional concepts for finitely generated groups.
Because of Lemma~\ref{Ltransitive}, every compactly generated, totally
disconnected, locally compact group has $0$, $1$, $2$ or
$\infty$~many ends.
A compactly generated, totally disconnected, locally compact group
has $0$~rough ends, if and only if it is compact (in particular, a finitely
generated group has $0$~ends if and only if it is finite).
Finitely generated groups with precisely two ends are characterized by
the following
result, which is a conjunction of results of Hopf and C.~T.~C.~Wall.
\begin{theorem}\label{TTwoEnds}
{\rm (\cite[Satz 5]{Hopf1944} and \cite[Lemma~4.1]{Wall1967})}
Let $G$ be a finitely generated group. Then the following
are equivalent:
(i) $G$ has precisely two ends;
(ii) $G$ has an infinite cyclic subgroup of finite index;
(iii) $G$ has a finite normal subgroup $N$ such that $G/N$ is
either isomorphic to the infinite cyclic group or to the infinite
dihedral group.
\end{theorem}
For compactly generated, totally disconnected, locally compact group we
have the following analogue.
\begin{theorem}\label{TStructertree_two} {\rm
(Cf. \cite{MollerSeifter1998})}
Let $G$ be a compactly generated, totally disconnected, locally compact
group. Suppose that the space of rough ends has precisely two
points. Then $G$ has a compact, open, normal subgroup $N$ such that
$G/N$ is either isomorphic to the infinite cyclic group or to the infinite
dihedral group.
\end{theorem}
Finitely generated groups with more than one end are described in a
famous result of Stallings from 1968.
\begin{theorem}{\rm \cite{Stallings1968}}
Suppose $G$ is a finitely generated group with more than one end.
Then $G$ can be written as a non-trivial free product with amalgamation
$A*_CB$ (with $A\neq C\neq
B$) where $C$ is finite, or $G$ can be written as a non-trivial
HNN-extension $A*_Cx$ where $C$ is finite.
\end{theorem}
The converse also holds, if $A*_CB$ (with $A\neq C\neq
B$) where $C$ is finite, or $G$ can be written as a non-trivial
HNN-extension $A*_Cx$ where $C$ is finite, then $G$
has more than one end.
In 1974 Abels \cite{Abels1974} proved an analogue of Stallings' Theorem for
topological groups. Abels uses similar ideas in his proof as used by
Stallings. Essentially the same result as in \cite{Abels1974} is proved in
\cite{KronMoller2008}, using Dunwoody's theory of structure trees (see
\cite{Dunwoody1982} and \cite{DicksDunwoody1989}) and the Bass-Serre
theory of group actions on trees.
\begin{theorem}\label{TStructertree_infinite}
{\rm (Cf.~\cite{Abels1974})}
Let $G$ be a compactly generated, totally disconnected, locally compact
group. Suppose that some (equivalently, any) rough Cayley graph of $G$
has infinitely many ends. Then $G=A*_C B$ (with $A\neq C\neq
B$) or $G=A*_C x$, where
$A$ and $B$ are open, compactly generated subgroups of $G$, and
$C$ is a compact, open subgroup.
\end{theorem}
That it is possible to write $G$ as either a free product with
amalgamation $G=A*_C B$ or an HNN-extension $G=A*_C x$ is often expressed by
saying that $G$ splits over $C$. Using this expression, Stallings
theorem becomes the statement that a finitely generated group with
infinitely many ends splits over a finite subgroup, and
Theorem~\ref{TStructertree_infinite} above becomes the statement that
a compactly generated, totally disconnected, locally compact group with
more than one end splits over a compact, open subgroup.
If $G$ is totally disconnected, locally compact group with closed,
cocompact subgroup $H$, then a rough Cayley graphs for $G$
and a rough Cayley graph for
$H$ are quasi-isometric by Theorem~\ref{TCocompact}. In particular,
$G$ and $H$ have the same number of rough ends. Hence,
Theorem~\ref{TStructertree_infinite} has the following corollary.
\begin{corollary}{\rm (\cite[Corollary~3.22]{KronMoller2008})}
Let $G$ be a totally disconnected, locally compact group and $H$ a
closed, cocompact subgroup. Then $G$ splits over a compact, open
subgroup if and only if $H$ splits over a compact, open subgroup.
\end{corollary}
\mbox{\bf s}ubsubsection{Free subgroups}
A well known theorem of Gromov (also proved by Woess
\cite{Woess1989a}) says that a finitely generated group is
quasi-isometric to a tree if and only if it has a finitely generated
free subgroup of finite index.
The next theorem provides an analogue of this result for
compactly generated, totally disconnected, locally compact groups.
\begin{theorem}{\rm (\cite[Theorem~3.28]{KronMoller2008})}
\label{TQuasiTree}
Let $G$ be a compactly generated, totally disconnected,
locally compact group.
(i)
Some (hence, every) rough Cayley graph of $G$ is
quasi-isometric to a tree if and only if $G$ has an expression as a
fundamental group of a finite graph of groups such that all the vertex and edge
groups are compact open subgroups of $G$.
(ii)
Assume also that the group $G$ is unimodular. Then
some (hence, every) rough Cayley graph of $G$ is
quasi-isometric to some tree if and only if $G$ has a finitely
generated free subgroup that is cocompact and discrete.
\end{theorem}
The proof of Theorem \ref{TQuasiTree} uses information about ends,
quasi-isometries and structure trees from \cite{Woess1989a}, \cite{Kron2001a},
\cite{KronMoller2008a}, \cite{Moller1992a} and
\cite{ThomassenWoess1993}. The essential result used about graphs
quasi-isometric to trees is that a transitive, locally finite, graph is
quasi-isometric to a tree if and only if it has no {\em thick ends}
(cf.~\cite{Woess1989a} and \cite[Theorem~5.5]{KronMoller2008a}). This property can then be
used to show that for a
locally finite, transitive graph that is quasi-isometric to a tree
one can find a locally finite structure tree on which the automorphism
group of the original graph acts.
For the proof of existence of a cocompact, discrete, finitely
generated, free subgroup in part~(\romannumeral2) of
Theorem~\ref{TQuasiTree} the theory of tree lattices in~\cite{BassKulkarni1990} is used.
\begin{corollary} {\rm (\cite[Corollary~3.29]{KronMoller2008})}
Let $G$ be a totally disconnected,
locally compact group. If $G$ has a cocompact, finitely generated,
free, discrete subgroup, then $G$ splits over some compact, open
subgroup and $G$ can be written as $G=A*_C B$ (with $A\neq C\neq
B$) or $G=A*_C x$ where
$A, B$ and $C$ are compact, open subgroups of $G$.
\end{corollary}
The above Corollary implies a special case of a result
of Mosher, Sageev and Whyte
\cite[Theorem~9]{MosherSageevWhyte2003}.
\begin{corollary} {\rm (\cite[Corollary~3.30]{KronMoller2008})}
Let $G$ be a totally disconnected,
locally compact group. If $G$ has a cocompact, finitely generated,
free, discrete subgroup, then $G$ has an action on a locally finite
tree, such that $G$ fixes neither an edge nor a vertex.
\end{corollary}
Consider a finitely generated group $H$ and a Cayley graph $\Gamma$ of
$H$. The action of $H$ on the Cayley graph gives an embedding of $H$
as a closed, cocompact subgroup into the totally disconnected, locally
compact group $G=\mbox{${\rm Aut}$}ga$.
Willis asks in \cite[Section 6]{Willis2004} whether various invariants
of $G$ can be bounded in terms of $H$. For example, he asks if
it is possible to deduce that there are only finitely many
prime numbers that occur as factors in ${\bf s}(x)$ for $x\in G$
(the scale function ${\bf s}$ is discussed in Section~\ref{STidyScale}).
This question is motivated by the following result.
\begin{theorem}{\rm (\cite[Theorem 3.4]{Willis2001})}
Let $G$ be a compactly generated, totally disconnected, locally compact
group. Then there are finitely many primes $p_1, p_2, \ldots, p_t$ such that
the number ${\bf s}_G(x)$ for all elements $x\in G$ can be written in the
form ${\bf s}_G(x)=p_1^{s_1}p_2^{s_2}\cdots p_t^{s_t}$.
\end{theorem}
Baumgartner \cite{Baumgartner2007} has applied the above mentioned result of
Mosher, Sageev and Whyte to the program suggested by Willis.
Baumgartner has also extended the scope of the program, by
considering not only the special type of embedding $G\rightarrow
\mbox{${\rm Aut}$}ga$, where $\Gamma$ is a Cayley graph of $H$.
A topological group $G$ is called an {\em envelope} of
a group $H$, if $H$ embeds as a closed, cocompact subgroup of $G$.
\begin{theorem}{\rm \cite[Corollary 11]{Baumgartner2007})}
Let $H$ be a virtually free group of rank at least 2. Then there are
finitely many primes $p_1, p_2, \ldots, p_t$ such that
for all totally disconnected, locally
compact envelopes $G$ of $H$ all elements $x\in G$
the number ${\bf s}_G(x)$ can be written in the
form ${\bf s}_G(x)=p_1^{s_1}p_2^{s_2}\cdots p_t^{s_t}$.
\end{theorem}
The following result comes from a totally different direction, but it also
indicates the possible fruitfulness of the
study of envelopes and embeddings of a finitely
generated group into automorphism groups of its Cayley graphs.
\begin{theorem}{\rm \cite[Theorem~4.1]{MollerSeifter1998})}
Let $N$ be a finitely generated, torsion free, nilpotent group and
$\Gamma$ a Cayley graph of $N$ with respect to some finite generating
set of $N$. Put $G=\mbox{${\rm Aut}$}ga$. Then $G$ is discrete in the permutation
topology, and $N$ embeds into $G$ as normal subgroup.
\end{theorem}
This theorem is proved with the aid of Theorem~\ref{TTrofimovoauto}.
\mbox{\bf s}ubsubsection{Accessibility}
\begin{definition}\label{DAccessibility}
A finitely generated group is said to be {\em accessible} if it has an
action on a tree $T$ such that:
{(i)} the number of orbits of $G$ on the edges of $T$ is finite;
{(ii)} the stabilizers of edges in $T$ are finite subgroups of
$G$;
{(iii)} every stabilizer of a vertex in $T$ is a finitely
generated subgroup of $G$ with at most one end.
\end{definition}
The question by C.~T.~C.~Wall, \cite{Wall1971}, of whether or not every
finitely generated group is accessible motivated several important
developments in combinatorial group theory, among them
Dunwoody's theory of structure trees. Wall's question, after being
open for a long time, was settled by examples of finitely generated
groups that are not accessible, which were constructed by Dunwoody \cite{Dunwoody1991}.
\begin{definition}{\rm (\cite[p.~249]{ThomassenWoess1993})}
\label{DGraphAccessibility}
Let $\Gamma$ be a connected, locally finite graph. If there is a
number $k$ such that any two distinct ends can be separated by
removing $k$ or fewer vertices from $\Gamma$ then the graph $\Gamma$
is said to be {\em accessible}.
\end{definition}
Thomassen and Woess
\cite[Theorem~1.1]{ThomassenWoess1993} show that a finitely generated
group is accessible if and only if every Cayley graph with respect
to a finite generating set is accessible.
The notion of accessibility can be generalized to compactly generated,
totally disconnected, locally compact groups as follows.
\begin{definition}\label{DTopoAccessibility}
A compactly generated, totally disconnected, locally compact
group is said to be {\em accessible} if it has an
action on a tree $T$ such that:
(i) the number of orbits of $G$ on the edges of $T$ is finite;
(ii) the stabilizers of edges in $T$ are compact, open subgroups of
$G$;
(iii) every stabilizer of a vertex in $T$ is a compactly
generated, open subgroup of $G$ with at most one rough end.
\end{definition}
Then one can link accessibility
of compactly generated totally disconnected locally
compact groups to rough Cayley graphs in analogues
way as Thomassen and Woess link accessibility of finitely generated
groups to Cayley graphs.
\begin{theorem}{\rm (\cite[Theorem~3.27]{KronMoller2008})}
\label{TAccessible}
Let $G$ be a compactly generated, totally disconnected, locally compact
group. Then $G$ is accessible if and only if every rough Cayley graph
of $G$ is accessible.
\end{theorem}
In fact, the group in Theorem \ref{TAccessible} is accessible if and only if
any of its rough Cayley graphs is accessible, because the
property of a graph being accessible is invariant under
quasi-isometries by~\cite[Theorem~0.4]{PapasogluWhyte2002}. By the same result,
a compactly generated, totally disconnected, locally compact group
with a closed, cocompact, accessible subgroup is itself accessible.
Since every finitely presentable group is accessible by a result
of Dunwoody \cite{Dunwoody1985}, we deduce the following theorem as a corollary.
\begin{theorem}
Let $G$ be a compactly generated, totally disconnected, locally compact
group. If $G$ has a cocompact, finitely presented subgroup then $G$ is
accessible.
\end{theorem}
\mbox{\bf s}ubsubsection{Ends of pairs of groups}
By Stallings' Theorem the ends of a finitely generated group can be
used to detect if the
group splits over a finite subgroup. The concept of ends of pairs of
groups is an attempt to define a geometric invariant that can be used
to detect splittings of $G$ over subgroups that are not
finite. This concept was first introduced in the papers \cite{Houghton1974}
by Houghton and \cite{Scott1977} by Scott.
For a survey of these
and related concepts see \cite{Wall2003}.
\begin{definition}{\rm (Cf.~\cite[Lemma~1.1]{Scott1977})} Let $G$ be a finitely
generated group and $C$ a subgroup of $G$. Let $\Gamma$ be a
Cayley graph of $G$ with respect to some finite generating set. The
{\em number of ends of the pair} $(G,C)$, denoted with $e(G,C)$,
is defined as the number of ends
of the quotient graph $C\backslash\Gamma$ (quotient with respect to the
natural $C$-action on $\Gamma$).
\end{definition}
It can be shown that the number of ends of a pair of groups does not
depend on the choice of generating set. While a transitive, locally finite
graph has $0$, $1$, $2$ or infinitely many ends, a pair of groups can
have any number of ends, see~\cite[Example~2.1]{Scott1977}.
The following conjecture generalizing Stallings' Theorem
is due to Kropholler, see \cite{NibloSageev2006}.
\begin{conjecture}
Let $G$ be a finitely generated group and $C$ a subgroup of $G$. If
$G$ contains a subset $A$ such that
(i) $A=CA$;
(ii) for every element $g\in G$ the symmetric difference of $a$ and
$Ag$ is contained
in a finite union of right $C$ cosets;
(iii) neither $A$ nor $G\mbox{\bf s}etminus A$ is contained in any finite union
of right $C$ cosets;
(iv) $A=AC$
\noindent
then $G$ splits over a subgroup that is commensurable
with a subgroup of $C$. (Conditions (i)-(iii) above are equivalent to
$e(G,C)\geq 2$.)
\end{conjecture}
Here, two subgroups are are said to be \emph{commensurable},
if their intersection has finite index in both subgroups. Furthermore the
\emph{commensurator} of a subgroup $C$ of $G$ is the set
of elements $g\in G$ such that $C$ and $gCg^{-1}$ are commensurable.
The commensurator of a subgroup is itself a subgroup.
Kropholler's conjecture above has been verified under various
additional hypotheses. A sample result is the following theorem.
\begin{theorem}{\rm (\cite[p.~30]{DunwoodyRoller1993})}\label{Tcommensurator}
Let $G$ be a finitely generated group and $C$ a finitely generated
subgroup of $G$. If $e(G, C)>1$ and
the commensurator of $C$ is the whole group $G$ then $G$
splits over a subgroup commensurable with $C$.
\end{theorem}
This result has also been proved in papers
by Niblo \cite[cf. Theorem~B]{Niblo2002} and Scott and Swarup
\cite[Theorem~3.12]{ScottSwarup2000}.
In \cite[Section~3.7]{KronMoller2008} there is further discussion of how
the concepts of ends of pairs of groups, coends and rough ends relate
and how these concepts can be interpreted graph theoretically.
Amongst other things these considerations lead to a prove of
Theorem~\ref{Tcommensurator} above.
\mbox{\bf s}ubsection{Polynomial growth}\label{SPolynomial}
Recall from Section~\ref{STrofimov} that
a connected, locally finite graph is said to have {\em polynomial
growth} if for every vertex the number of vertices in distance less
than or equal to $n$ grows polynomially with $n$. A finitely
generated group is said to have polynomial growth if some (hence,
every) Cayley graph with respect to a finite generating set has
polynomial growth.
The concept of polynomial growth can be generalized to compactly
generated, locally compact groups.
\begin{definition}\label{DTopoPolynomial}
Let $G$ be a locally compact group generated by a compact
symmetric neighbourhood of the identity $V$. Set
$V^n=\{g_1 g_2\cdots g_n\mid g_i\in V\}$.
Let $\mu$ denote a Haar measure on
$G$. If there are constants $c$ and $d$ such that $\mu(V^n)\leq cn^d$
for all natural numbers $n$, $G$ is said to have {\em polynomial growth}.
\end{definition}
The following theorem characterizes compactly generated,
totally disconnected groups of polynomial growth in terms of
their rough Cayley graphs.
\begin{theorem}{\rm (\cite[Theorem~4.4]{KronMoller2008})}
\label{TopoPolynomial}
Let $G$ be a compactly generated, totally disconnected,
locally compact group and $\Gamma$
some rough Cayley graph for $G$. Then $\Gamma$ has polynomial growth
if and only if $G$ has polynomial growth (in the sense of
Definition~\ref{DTopoPolynomial}).
\end{theorem}
Viewed in this context, Trofimov's theorem about automorphism groups of
graphs with polynomial growth, Theorem~\ref{TTrofimovPoly}, can now be
seen as a version of Gromov's Theorem for compactly generated, totally
disconnected, locally compact groups.
As already mentioned, Losert proved a generalization
of Gromov's Theorem for topological groups in \cite{Losert1987} and
Woess deduced Trofimov's theorem from Lostert's result
in \cite{Woess1992}.
The following theorem \cite[Theorem~4.6]{KronMoller2008} is a
combination of Theorem~\ref{TopoPolynomial} and Trofimov's theorem, but
can also be seen as Corollary to Losert's results.
\begin{theorem} {\rm (Cf.~Trofimov \cite{Trofimov1985} and Losert
\cite{Losert1987})}
Let $G$ be a compactly generated, totally disconnected, locally compact
group. Then $G$ has polynomial growth if and only if $G$ has a normal,
compact, open
subgroup $K$ such that $G/K$ is a finitely generated almost nilpotent group.
\end{theorem}
\frenchspacing
\end{document} | math |
تِمہٕ گٔے تہٕ مٲرٕکھ سۄ ٹۄپَو سٟتۍ | kashmiri |
बड़ी छवि : धातुई स्वचालित स्क्रीन प्रिंटिंग मशीन एकल रंग प्रिंटिंग प्रेस
रंग और पृष्ठ: सिंगल-रंग प्रिंटिंग प्रेस आवेदन: दवा की बोतलें प्रिंटर
आयाम (एल * डब्ल्यू * एच): २८०० * २६०० * २९००म्म हालत: नई
मल्टी फंक्शन के साथ मशीन पूरी तरह से ऑटो सिंगल रंग स्क्रीन प्रिंटिंग मशीन है। विभिन्न प्रकार के गोल, अंडाकार, सपाट वस्तुओं को प्रिंट करना उपयुक्त है। यह न केवल एक रंगीन मशीन के रूप में काम करता है बल्कि मल्टीकोलर स्क्रीन प्रिंटिंग मशीन में एकजुट हो सकता है।
यह व्यापक रूप से कांच की बोतल, शराब, सौंदर्य प्रसाधन आदि के पैकेज पर लागू होता है और इसकी उच्च गुणवत्ता वाले मूल्य अनुपात के कारण छोटे और मध्यम आकार के उद्यमों द्वारा इसे व्यापक रूप से स्वीकार किया जाता है।
प्लास्टिक ट्यूब, कॉस्मेटिक बोतलें, दूध की बोतलें, चिकित्सा की बोतलें, पीने की बोतलें, तेल की बोतल चिकनाई, कांच सिलिकॉन, कलम आस्तीन और आदि के प्रिंट करने योग्य।
१. ऑपरेशन सरल और स्थिर सुनिश्चित करने के लिए जापान से पीएलसी और टच स्क्रीन।
२. ड्राइव को आसान बनाने के लिए स्वत: लोड हो रहा है और उतारने के लिए इष्टतम डिजाइन, उच्च गति पर स्थिर चलने वाली मशीन की गारंटी दें।
३. मशीन के गुणवत्ता और मुद्रण की स्थिरता सुनिश्चित करने के लिए सभी घटक जर्मनी, अमेरिका और जापान से हैं।
४. तीन भाषाओं जैसे चीनी, अंग्रेजी और स्पेनिश समर्थित हैं।
५. हम विशेष उत्पादों के अनुसार मशीन को अनुकूलित कर सकते हैं।
मुद्रण गति: ४०००-५०००पस / घंटा
प्रिंटिंग आकार: दीया १५-६० मिमी
लंबाई २०-२०0 मिमी
संपीड़ित हवा: ०.६-०.८म्पा
आकार: २८०० * २६०० * २९००म्म | hindi |
पटना में बारिश से तबाही के लिए नीतीश और सुशील जिम्मेवार : गिरिराज - रेजियोनल न्यूज - समय लाइव
पटना में बारिश से तबाही के लिए नीतीश और सुशील जिम्मेवार : गिरिराज
केंद्रीय मंत्री एवं भारतीय जनता पार्टी (भाजपा) के वरिष्ठ नेता गिरिराज सिंह
केंद्रीय मंत्री एवं भारतीय जनता पार्टी (भाजपा) के वरिष्ठ नेता गिरिराज सिंह ने पिछले दिनों भारी बारिश के कारण बिहार की राजधानी पटना में हुए जलजमाव के लिए आज मुख्यमंत्री नीतीश कुमार और उप मुख्यमंत्री सुशील कुमार मोदी सीधे तौर पर जिम्मेवार ठहराया है।
श्री सिंह ने माइक्रो ब्लॉगिंग साइट ट्विटर पर ट्वीट कर कहा, पटना में बारिश से तबाही के लिए मुख्यमंत्री नीतीश कुमार एवं उप मुख्यमंत्री सुशील कुमार मोदी जिम्मेवार हैं। उन्होंने कटाक्ष करते हुए कहा कि मुंह देख कर राहत सामग्री का वितरण न किया जाए।
प्रखंड कार्यालय में गिरिराज पदाधिकारियों के साथ कर रहे थे बाढ़ समीक्षा,बाहर पीड़ित राहत सामग्री के लिए कर रहे थे नारेबाजी।
पटना में बारिश से तबाही के लिए मुख्यमंत्री व उपमुख्यमंत्री जिम्मेवार।
केंद्रीय मंत्री ने इससे पूर्व बुधवार को भी पटना में जलजमाव के लिए बिहार की राष्ट्रीय जनतांत्रिक गठबंधन (राजग) सरकार को जिम्मेवार ठहराते हुए कहा था कि इस बुरे हालात के लिए केवल प्रकृति को जिम्मेवार नहीं ठहराया जा सकता। यह तो व्यवस्था और प्रशासन की घोर लापरवाही का नतीजा है।
राज्य सरकार ने सिर्फ लोगों को अलर्ट किया लेकिन जल निकासी के लिए स्वयं अपडेट नहीं हुई। उन्होंने इसे अपनी आंतरिक पीड़ा भी बताया था। | hindi |
वो आग जो धुंधली पड़ गयी
आज के दिन वो सर्द हवा ना तो वो रूमानी थी, ना ही और सर्द रातों की तरह वो रात और काली थी क्योंकि एक बेसहारा लड़की की इज्ज़त दिल्ली की सङकों पर सरेआम नीलाम हो रही थी| हवस में मदमस्त दरिंदे उस बेबस के शरीर के चीथड़े कर रहे थे और देश की सुरक्षा का जिम्मा लिए लोग चैन की नींद में खर्राटें मार रहे थे| कभी किसी के सामने ना झुका सिर अपनी इज्जत के लिए राक्षसों के सामनें हाथ जोड़ कर दुहाई माँग रहा था| उसके होंठों पर बस एक ही शब्द मुझे छोड़ दो और आँखों में आंसू बनकर निकलते लहू की हर एक बूँद भगवान की बजाए शैतान से आस लगा बैठी थी कि शायद उन्हें दया आ जाये और उसकी इज्जत पर लग रही आग ठण्डी पड़ जाये| पर दरिंदों को ना तो दया की तनिक भी छुअन लगी| ना ही उनके दिल में सरकार, समाज और प्रशासन का डर|
हो भी क्यूँ ना? सरकार वादें करती है, लाशों पर राजनीति तो इनका पुराना पेशा है| समाज- समाज तो एक मजाक बन गया है| प्रशासन हमेशा घटना की इंक्वायरी ही करती रह जाती है|
जंतर -मंतर पर प्रदर्शनकारियों को ज़बरदस्ती हटाती हुई तात्कालिक मनमोहन सरकार की पुलिस
शायद यहाँ स्वतंत्रता ही गुनाह है| अपनी स्वतंत्रता को भुनाने में उसे यह कीमत चुकानी पड़ेगी, शायद यही सवाल उसके मन में गूँज रहा था|शायद वह खुद से पूछ रही थी कि क्या अपने ही देश में उसका वजूद नहीं है? सिर्फ इसलिए क्योंकि वह एक महिला है, सुरक्षित नहीं रह सहती? खुलेआम स्वतंत्र सड़कों पर चल नहीं सकती? दस लोगों के बीच बैठ नहीं सकती, जिसे हम समाज कहते है?
स्वतंत्र भारत, अतुलनीय भारत के सीने पर तलवार से वार पर वार किए जा रहे थे, और वाह रे भारत! तूने आह तक नहीं भरी! आज फिर एक द्रोपदी इज्ज़त की दहलीज़ पर बिलख रही थी पर कृष्ण नदारद थे| उसकी चीख एक चार पहिये की बस तक ही सिमट कर रह गयी| आखिर पूरा शहर बहरा हो गया था या अंधा?
खैर सवाल तो सवाल है पर हकीक़त को सवाल में बदलने से क्या मिलेगा? सवाल से जवाब मिलते है, घाव के मरहम नहीं; और फिर बिना गलती के मिले घाव के लिए मरहम का क्या वजूद? आज ना कोई सांप्रदायिक दल था जो महिला सुरक्षा के लिए ताल ठोक कर सड़कों पर नंगा नाच करते है और ना ही कोई समूह था जो खुद को महिलाओं का हितैषी बताकर सत्ता की रोटी सेंकने की फिराक में रहता है| आज थे तो सिर्फ बेबस लड़की के आंसू, तार-तार होती इज्ज़त और बेख़ौफ़ दरिंदे| अपने पुरूष मित्र के साथ घूमना उसका गुनाह था या फिर बदकिस्मती?
सवाल बहुत छोटा है लेकिन बहुत अहम|
घंटों तक उसकी रूह काँपती रहीं, बदन थरथराता रहा, खून से लथपथ वो अपनी इज्जत के लिए लड़ती रही और पाँच हैवान अपनी प्यास बुझाते रहे| देश की राजधानी के सबसे अधिक चहलकदमी वाले इलाके मुनरिका से शुरू हुई घटना सड़कों पर यूँ ही कौधती रही पर अपनी धुन में अंधे समाज को कुछ ना दिखाई दिया, ना सुनाई दिया!
लाईफ आफ पाई देखते वक्त उसने कभी नहीं सोचा होगा कि उसकी ज़िन्दगी खुद मौत से जूझ पड़ेगी| एक फिजियोथिरेपी इंटर्न, २३ वर्षीय लड़की के हालात पर पूरा देश क्या पूरा विश्व रो पड़ा लेकिन देश चलाने वाले, राजनीति और सत्ता की रोटी सेंकने में ही लगे थे! मैं पूछता हूँ कि क्या इनका ज़मीर मर गया है या फिर ये इतना बेख़ौफ़ है कि हमें मूर्ख समझ बैठे है?
रात के ९.३० से ११ बजे तक वो लड़ती रहीपर अकेलीलाचारबेबस (माफ करना शब्द नहीं है अभिव्यक्ति के लिए)
अर्धनग्न वो दिल्ली की सड़कों पर बेसुध फेंक दिए गये थे लेकिन प्रशासन अभी भी सपनों में लीन था और हद तो तब हुई जब वो आपस में ही सीमा विवाद को लेकर लड़ने लगें और एक निर्लज्ज महिला अधिकारी को घर जाने की जल्दी थी क्योंकि उसके घर में सब्जी नहीं थी! वाह रे सुरक्षा के सिपाहियों! वाह!
सफदरजंग अस्पताल में वो जिन्दगी मौत से लड़ती रही और बाहर युवा पीढ़ी ने घटिया प्रशासन और निर्लज्ज सरकार के खिलाफ़ जंग का ऐलान कर दिया| ये वो युवा थे जो ना तो कभी उस बेसहारा लड़की को देखे थे, ना ही उससे कभी मिले थे, पर उनकी आँखों में आंसू और दिल में हजारों सवाल थे कि क्या यही है स्वतंत्र भारत? क्या यही है वो समाज, संस्कृति जिस पर हम थोथा गर्व करते है? क्या हालात सुधरेंगे या फिर बद्तर होंगे?
वो लड़ते रहे, इंडिया गेट से राजघाट के सीने पर चढ़ बैठे और पुलिस की लाठियों आंसू के गोलों के बावजूद एक कदम भी पीछे नहीं हटे|
उन युवाओं को मेरा सलाम! लेकिन फलस्वरूप हमें क्या मिला एक और झूठा वादा!
आज के हालात सबके सामने है| तब से लेकर अब तक हजारों दामिनी अपना सर्वस्व लुटा चुकी है और आगधुँधली पड़ गयी हैआखिर क्यों? | hindi |
NewTek Discussions > LightWave 3D User Community > LW - Community > Partial morphs based on incidence angle??
View Full Version : Partial morphs based on incidence angle??
I was wondering if it was possible to have vertices of an object morph based on the incidence angle of a light or the camera.
Just to make this clear, the light or camera will not trigger a whole object morph but a morph of the individual vertices based on the incidence angle to said light or camera.
Should be possible. You'd have to do it through nodes.
Take the Forward direction of the light/camera and dot that with the normal of the object, then scale that using a Scalar multiply node and feed it into a morph node which is plugged into the displacement input.
Sorry, could you show me a snapshot of the node setup?
I still use LW9.6. Would this be a problem?
Would that be a PointInfo or a SpotInfo kinda deal?
This is the node setup. Because you want the inflation to occur at the out edge, you need to find the arccos of the result of the dot (the dot function returns the cosine of the angle between the two vectors, in this case the forward vector and the normal of the surface) and then sine that (as sine goes from 0 to 1 instead of 1 to 0 which is what cosine does). The multiply scales the displacement strength. The normal give the displacement direction so feeding that into the displacement makes a surface bulge outwards.
Alternatively use a gradient to control how much it flares out. The gradient will need to go from -1 (the reverse of the forward direction) to 0 (the sideways direction) to 1 (the forward direction). I use the Alpha since that's a percentage and easy to use. Connect the Alpha output to the Vector Scale scalar input.
Thanks, I'll give that a try. The reason I wanted to see if this worked is to possibly come up with a way of having variable line widths to create more ink like edges for toon renders. This would work in conjunction with creating a copy of the mesh, flipping the normals and setting this copy to 100% black luminosity 0% diffuse.
If it does work like I imagine then it might open up interesting options with the technique.
Ah, you know you can taper the render edges in a node editor now right? You can even do this same trick in there, so rotating the light affects the width of edges.
Ah, I did not know that. Anyway, I tried the node setup and it didn't work. I don't know why it didn't work but I don't understand the actual setup itself so anything could have been the issue.
Anyway, no big deal. I just thought of the technique when I was half asleep one day.
FWIW, IIRC Jen Hachigian (Celshader) did the article/tutorial on using nodes to adjust edge widths. No idea where it might be, but she does a lot of presentations on the LALWUG channel on YouTube.
Also, I believe Proton's article on achieving Dr. Suess style edge weighting and cross hatching addresses this concept. Again, good luck finding it-- should be doable. | english |
تم کٔمن ہنٛز نمایندگی چھ کران | kashmiri |
इत्र लास्ट दते एक्स्टेंशन: क्या इनकम टैक्स रिटर्न की आखिरी तारीख ३१ अगस्त से आगे बढ़ेगी?
इत्र लास्ट दते एक्स्टेंशन: कबट ने इनकम टैक्स रिटर्न भरने की आखिरी तारीख ३१ जुलाई से १ महीना बढ़ाकर ३१ अगस्त कर दी थी। आज इत्र फाइल करने का आखिरी दिन है। क्या कबट आईटीआर भरने का आखिरी तारीख बढ़ाएगा?
इनकम टैक्स रिटर्न की आखिरी तारीख पहले ही १ महीना बढ़ चुकी है। | तस्वीर साभार: तिंक्स्टॉक
नई दिल्ली: इनकम टैक्स रिटर्न भरने की आखिरी तारीख ३१ अगस्त है। पहले ३१ जुलाई तक रिटर्न भरना था लेकिन सरकार ने लोगों की मांग को देखते हुए इसकी तारीख १ महीना बढ़ा दी थी। टैक्स रिटर्न भरने की तारीख बढ़ने के कारण लोगों को रिटर्न भरने का और समय मिल गया। इस कारण से रिटर्न भरने वालों की संख्या में बढ़ोतरी हो रही है।
३१ जुलाई तक इस साल रिटर्न भरने वालों की संख्या में ५३ फीसदी की बढ़ोतरी हुई है। ३१ जुलाई तक देश में करीब ३.4३ करोड़ लोगों ने टैक्स रिटर्न भर दिया था। पिछले वित्तवर्ष में रिटर्न भरने वालों की संख्या २.२4 करोड़ ही थी। आयकर रिटर्न भरने में बढ़ोतरी के कारण सरकार को इस साल ३0 हजार करोड़ की अतिरिक्त आय का अनुमान है।
इनकम टैक्स रिटर्न की इस तरह घर बैठे ऑनलाइन ए-फिलिंग कर लें
जानकारों के मुताबिक टैक्स रिटर्न भरने के लिए १ महीने की बढ़ोतरी पर्याप्त है। इसमें छूटे हुए अधिकतर लोगों ने अपने रिटर्न भर दिए हैं। इस तारीख में अभी और बढ़ोतरी की कोई उम्मीद नहीं है। इसका मतलब है सरकार अब 3१ अगस्त से आगे रिटर्न भरने की तारीख नहीं बढ़ाएगी। इस बार वित्तवर्ष 20१7-१8 (आई 20१8-१9) के लिए आयकर रिटर्न भरे जा रहे हैं। अगर आपने अब तक रिटर्न नहीं भरा तो आज आपके पास रिटर्न भरने का समय है। आप खुद भी रिटर्न भर सकते हैं या किसी सीए की मदद भी ले सकते हैं।
क्या ३१ अगस्त की डेडलाइन के बाद इनकम टैक्स रिटर्न फाइल कर सकते हैं?
वित्त मंत्रालय ने केरल के लोगों के लिए इनकम टैक्स रिटर्न भरने की तारीख १५ दिन बढ़ा कर १५ सितंबर कर दी है। केरल में बाढ़ के कारण तबाही मच गई है। वित्त मंत्रालय बाढ़ से प्रभावित क्षेत्रों के लिए इनकम टैक्स रिटर्न की डेडलाइन बढ़ाने पर विचार कर रहा था। इससे अब ये भी स्पष्ट हो गया है कि बाकि लोगों के लिए टैक्स रिटर्न भरने की तारीख नहीं बढ़ेगी। इसका मतलब है आपको अगले २ दिन में अपना रिटर्न फाइल करना है।
अगर इनकम टैक्स रिटर्न नहीं भरा तो क्या होगा?
अगर आपने इनकम टैक्स रिटर्न नहीं भरा तो आपको १० हजार रुपए तक की पेनल्टी देनी पड़ सकती है। अगर किसी की आय ५ लाख रुपए तो उस पर १ हजार रुपए की पेनल्टी लगाई जाएगी। अगर कोई १ सितंबर से 3१ दिसंबर के बीच रिटर्न भरता है तो उस पर ५ हजार रुपए की पेनल्टी लगाई जाएगी। 3१ दिसंबर 20१8 के बाद रिटर्न भरने पर १० हजार रुपए की पेनल्टी लगाई जाएगी। अगर टैक्स की देनदारी बनती है तो उस पर ब्याज भी वसूला जाएगी।
इत्र लास्ट दते एक्स्टेंशन: क्या इनकम टैक्स रिटर्न की आखिरी तारीख ३१ अगस्त से आगे बढ़ेगी? डेस्क्रिप्शन: इत्र लास्ट दते एक्स्टेंशन: कबट ने इनकम टैक्स रिटर्न भरने की आखिरी तारीख ३१ जुलाई से १ महीना बढ़ाकर ३१ अगस्त कर दी थी। आज इत्र फाइल करने का आखिरी दिन है। क्या कबट आईटीआर भरने का आखिरी तारीख बढ़ाएगा? टाइम्स नो | hindi |
CNBC-TV18 Polls had predicted a profit of Rs 167 crore for the quarter under review.
The company had posted a net profit of Rs 154.3 crore in the October-December quarter of 2017-18.
South India's leading two wheeler maker, TVS Motor Company, on Tuesday reported 16 percent rise in standalone net profit at Rs 178.4 crore for the third quarter ended December 31, 2018.
The company's revenue increased to Rs 4,664.6 crore in the October-December period, over Rs 3,703.1 crore in the same quarter a year ago.
EBITDA for the quarter ended December 2018 is Rs 375.7 crore compared to Rs 300.5 crore for the quarter ended December 2017, a growth of 25.0 percent.
During the quarter ended December 2018, the overall two-wheeler sales of the company including exports grew by 18.9 percent to 9.50 lakh units from 7.99 lakh units in the quarter ended December 2017.
Motorcycle sales grew by 20.3 percent to 3.78 lakh units in the quarter ended December 2018 from 3.14 lakh units registered in the quarter ended December 2017. Scooter sales of the company grew by 31.7 percent to 3.54 lakh units in the quarter ended December 2018 from 2.69 lakh units in the quarter ended December 2017.
The total export of the Company grew by 25.8 percent to 1.77 lakh units during the quarter ended December 2018 from 1.40 lakh units in the quarter ended December 2017. Total three wheelers sales grew by 47.0 percent to 0.40 lakh units in the quarter ended December 2018 from 0.27 lakh units in the quarter ended December 2017, the company said. | english |
Affects all viewers for many years.
Line them all up and (optionally) with your LOD factor set to 1 to make the LOD shift happen sooner pull the camera back. The meshes with 3 and 4 faces should crumple immediately followed by the 5,6,7 and 8 face meshes that will crumple slightly earlier than the remaining 1 and 2 face meshes.
There are 3 different results for the LODBias depending on the number of material faces, as this is clearly arbitrary and has nothing to do with geometry it must be considered a bug. The table of results by material count is as follows.
First reported in 2011, a bug exists that results in different (and generally incorrect) LOD radius being used for Meshes depending upon the number of material faces. I have linked a number of the associated reports some of which are no longer accessible to non-Lindens.
As the LOD is dictated by the bounding box of the object, and not any other arbirtary detail, all mesh objects of the same scale should LOD identically.
Arguably the correct bias is that shown for 5,6,7,8 as this would give half of the long diagonal of the object bounding box.
I have a fix available but I cannot yet submit until https://jira.secondlife.com/browse/BUG-40625 has been resolved and I have completed the form.
In the fix I force all mesh to have a bias equivalent to the existing behaviour for 1 and 2 material meshes as this ensures that no Mesh is adversely affected. However I suspect that strictly correct behaviour is to move everything to the 0.5 all axis bias seen on 5,6,7 & 8 material face meshes. | english |
The Nikila Emitter Small Water Spout is an example of fine detail crafted into a smaller fountain emitter. With a floral motif, the Nikila is the perfect emitter spout for smaller water features or pool or spa areas without sacrificing intricate detailing of larger pieces. Also available in a larger version. | english |
हॉय रे जमाना: जायदाद के लिए मां-बाप को मार डाला, दिल्ली की तंग गलियों का सच - भारतखबर.कॉम
एजेंसी, दिल्ली। कहते हैं जर, जोरू और जमीन किसी से भी बैर करवाने सक्षम होती हैं इसी का ताजा उदाहरण दिल्ली में देखने को मिला जब एक महिला ने अपने ही मां-बाप का बेरहमी से कत्ल कर दिया। देश की राजधानी के पश्चिम विहार इलाके में डबल मर्डर का बेहद हैरतअंगेज मामला सामने आया है। पुलिस के मुताबिक, एक बेटी ने बॉयफ्रेंड की मदद से अपने मां-बाप की न सिर्फ हत्या कर दी, बल्कि उनकी डेडबॉडी को सुनियोजित तरीके से सूटकेस में पैक करके बहते हुए नाले में फेंक आए। लेकिन ड्रेन में सूटकेस के फंस जाने से इस दोहरे हत्याकांड की साजिश से पर्दा उठ गया। पुलिस ने बेटी और उसके बॉयफ्रेंड को गिरफ्तार कर लिया है।
बेटी ने पुलिस के सामने कबूल किया कि मां-बाप की लाखों की प्रॉपर्टी पाने के लिए उसने मां-बाप की हत्या की थी, जिसमें बॉयफ्रेंड प्रिंस ने उसका साथ दिया था। पुलिस डबल मर्डर में इनके साथ रहे कुछ अन्य लोगों की तलाश में दबिश दे रही है।
प्रॉपर्टी पाने के लिए अपनों का किया कत्ल
हॉय रे जमाना: जायदाद के लिए मां-बाप को मार डाला, दिल्ली की तंग गलियों का सच अद्देड बाय भारतखबरमें ऑन मार्च १२, २०१९ ९:३४ आम | hindi |
I consider myself lucky to happen upon such a talented band. When one enjoys what they do, they do it as well as the Cosmic Sausages. My 6 kids always want me to pop in the Great Hits CD when we get into our van.
I never saw the band in person, but they musically paint a technicolor picture for one's mind.
I am looking forward to getting A Fistful of Sausages! If I ever get over to London again... Yes, I'm a Bloody Yank. | english |
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