text
stringlengths 1
7.94M
| lang
stringclasses 5
values |
|---|---|
سیاسی متحرک گژنچ مخصوص ضروریات کس سلسلس منٛز حد کہ لحاظہٕ تنگ یا وسیع ہیٛکہٕ کرنہٕ یتھ۔ | kashmiri |
function NamespacesMosaicsConfig($stateProvider) {
'ngInject';
$stateProvider
.state('app.explorerNamespacesMosaics', {
url: '/explorer/namespaces-and-mosaics',
controller: 'ExplorerNamespacesMosaicsCtrl',
controllerAs: '$ctrl',
templateUrl: 'modules/explorer/namespaces-mosaics/namespaces-mosaics.html',
title: 'Explorer - Namespaces & Mosaics'
});
};
export default NamespacesMosaicsConfig; | code |
HOSTS Zimbabwe shrugged off a strong second half challenge by Kenya to burst into the final of the Under-19 Rugby Africa Cup Group 1A tournament at St George’s College yesterday.
After putting themselves in the driving seat in the first half, the Young Sables had to endure some nervous moments in the second period but still hung on to set up a date in the final against bitter rivals Namibia at the same venue on Saturday.
Namibia thumped Uganda 49-12 in an earlier match at the same venue.
The focus was however, always on the Young Sables and the hosts soon took the lead with Brian Matoramusha touching down after just five minutes.
Matoramusha was outstanding throughout the match, scoring two first half tries that put his team in the lead.
After Matoramusha’s two tries, the Kenyans had their first touchdown through Tysos Okoth.
After the resumption scrumhalf Ernest Mudzengere also registered his name on the Young Sables score sheet when he scored a try while Mathew Ushewokunze also chipped in with a 56th minute try as Zimbabwe lived up to expectations and won the game. Zimbabwe coach Brendon Brider said they still have some areas they need to address before their final match against Namibia who have dominated both the Sables and Young Sables.
“We are happy with the result and reaching the finals is good for us and we now look forward to the Saturday game.
“We need to sort our defence after watching the Namibia game they are strong. I was disappointed by some tactics but we will look at it,” Brider said.
Kenya assistant coach Mike Aung conceded defeat.
“I am happy with the youngsters they have a passion of the game and they went down fighting we are now focusing on the third place playoff.
“It will be an East African contest and I look forward to a tough match. We have managed to beat Zimbabwe but they were tight in the opening match and they managed to maintain.
“They were under pressure to perform at home and the support was there,” said Aung. In the earlier match Namibia proved too strong for Uganda grabbing four tries in the opening half before they added three more in the second period.
The winner of the tournament will qualify for the next edition of the Junior World Rugby Under-20 Trophy competition. | english |
\begin{document}
\title[Strong Maximum Principles]{The Strong Elliptic Maximum Principle For Vector Bundles \\
and Applications to Minimal Maps
}
\author[Andreas Savas-Halilaj]{\textsc{Andreas Savas-Halilaj}}
\author[Knut Smoczyk]{\textsc{Knut Smoczyk}}
\address{Andreas Savas-Halilaj\newline
Leibniz Universit\"at Hannover\newline
Institut f\"ur Differentialgeometrie\newline
Welfengarten 1\newline
30167 Hannover\newline
Germany\newline
{\sl E-mail address:} {\bf savasha@math.uni-hannover.de}
}
\address{Knut Smoczyk\newline
Leibniz Universit\"at Hannover\newline
Institut f\"ur Differentialgeometrie\newline
Welfengarten 1\newline
30167 Hannover\newline
Germany\newline
{\sl E-mail address:} {\bf smoczyk@math.uni-hannover.de}
}
\date{}
\begin{abstract}
Based on works by Hopf, Weinberger, Hamilton and Evans, we state and prove the strong
elliptic maximum principle for smooth sections in vector bundles over Riemannian manifolds
and give some applications in Differential Geometry.
Moreover, we use this maximum principle to obtain various rigidity theorems and Bernstein
type theorems in higher codimension for minimal maps between Riemannian manifolds.
\end{abstract}
\maketitle
\section{Introduction}
The maximum principle is one of the most powerful tools used in the theory
of PDEs and Geometric Analysis. In general, maximum principles
for solutions of second order elliptic differential equations, that are defined in the closure of a
bounded domain of the euclidean space, appear in two forms.
The \textit{weak maximum principle} states that
the maximum of the solution is attained at the boundary of the domain, but in principle it
might occur in the interior as well. On the other hand, the \textit{strong maximum principle} asserts that the
solution achieves its maximum only at boundary points, unless it is constant. For instance,
H. Hopf \cite{hopf} established such strong maximum principles for a wide class of general
second order differential equations. For example, he proved that if a solution $u$ of
the uniformly elliptic differential equation
\begin{gather}\label{elliptic}
\mathscr{L}u=0,\quad\mathscr{L}=\sum_{i,j=1}^{m}a^{ij}\partial^2_{ij}+\sum_{j=1}^{m}b^j\partial_j, \tag{$\ast$}
\end{gather}
attains its supremum or infimum at an interior point
of its domain $D$ of definition, then it must be constant.
Equivalently, the above strong elliptic maximum principle of Hopf can be interpreted as follows: If
a solution $u$ of $\mathscr{L}u=0$ maps
an interior point of $D$ to the boundary of the set $K=(\inf_{D}u,\sup_{D}u)$, then $u$ maps
any point of $D$ to the boundary of $K$ and hence it must be constant. For the proof of
this strong maximum principle Hopf used the \textit{Hopf Lemma}, which implies that the subset
$B\subset D$ consisting of points where $u$ attains a value in $\partial K$ is open. Since by continuity
$B$ is also closed, one has $B=D$, if $D$ is connected and $B$ is non-empty.
The generalization of Hopf's maximum principle to elliptic and semi-linear parabolic systems has
been first considered by H. Weinberger \cite{weinberger}. Let us recall briefly here the elliptic
version of this strong maximum principle: Assume that the vector valued map $$u:D\subset\real{m}\to\real{n},\quad u:=(u_1,\dots,u_n),$$
is a solution of the differential system
$$\mathscr{L}u+\operatorname{P}si(u)=0,$$
such that $u(D)$ is contained in a closed convex set $K\subset\real{n}$.
Here $\mathscr{L}$ is a second order uniformly elliptic
differential operator of the form given in (\ref{elliptic}), $D$ is an open domain of $\real{m}$ and $\operatorname{P}si:\real{n}\to\real{n}$ is a Lipschitz
continuous map. Suppose further that for any boundary point $y_0\in\partial K$ the vector $\operatorname{P}si(y_0)$
belongs to the tangent cone of $K$ at $y_0$ (for the exact definition see Section $2.1$). Under various additional
assumptions on the regularity of the boundary of the convex set $K$, Weinberger proved that, if an interior point of $D$ is
mapped via $u$ to a boundary point of $K$, then every point of $D$ is mapped to the boundary of $K$. Recently,
L.C. Evans \cite{evans1} gave a proof of Weinberger's maximum principle without imposing any regularity assumption on the boundary of the convex set $K$.
In his seminal papers, R. Hamilton \cite{hamilton2,hamilton1} derived parabolic maximum principles for sections in Riemannian vector bundles. There one compares the solution of a parabolic differential equation with a solution of an associated ODE. The weak parabolic maximum principle of Weinberger
can be seen as a special case of Hamilton's more general maximum principle in \cite{hamilton1}
since Weinberger's result follows from the application of Hamilton's maximum principle in the
case of a trivial bundle.
Hamilton's maximum principle appears in many different forms and became an important tool
in the study of geometric evolution equations (cf. \cite{ecker,ni1,brendle,andrews}).
Here we state and prove the strong elliptic maximum principle for sections in Riemannian vector bundles. This maximum principle is in the most general form and contains all the previous results by Hopf, Weinberger, Evans and it also
contains the elliptic version of Hamilton's parabolic
maximum principle. It turns out that it is extremely powerful and
we apply it to derive optimal Bernstein type results for minimal maps between Riemannian manifolds.
In order to state the elliptic version of the strong maximum principle for sections in vector bundles,
we must introduce an appropriate notion of convexity for subsets of Riemannian vector bundles. In
\cite{hamilton2} Hamilton gave the following definition:
\begin{definition}{\bf(Hamilton).}
Let $(E,\pi,M)$ be a vector bundle over the manifold $M$ and let $K$ be a closed subset of $E$.
\begin{enumerate}[(i)]
\item
The set $K$ is said to be fiber-convex or convex in the fiber, if for each point $x$ of $M$, the
set $K_x:=K\cap E_x$ is a convex subset of the fiber $E_x=\pi^{-1}(x)$.
\item
The set $K$ is said to be invariant under parallel transport, if for every smooth
curve $\gamma:[0,b]\to M$ and any vector $v\in K_{\gamma(0)}$, the unique parallel section
$v(t)\in E_{\gamma(t)}$, $t\in [0,b]$, along $\gamma(t)$ with $v(0)=v$, is contained
in $K$.
\item
A fiberwise map $\operatorname{P}si:E\to E$ is a map such that $\pi\circ\operatorname{P}si=\pi$, where $\pi$ denotes the bundle projection. We say a fiberwise map $\operatorname{P}si$ points
into $K$ (or is inward pointing), if for any $x\in M$ and any
$\vartheta\in \partial K_x$ the vector $\operatorname{P}si(\vartheta)$ belongs to
the tangent cone $C_{\vartheta}K_x$ of $K_x$ at $\vartheta$.
\end{enumerate}
\end{definition}
Next we state the strong elliptic maximum principle for sections in Riemannian vector bundles.
Throughout the paper all manifolds will be smooth and connected.
Let $(E,\pi,M)$ be a vector bundle of rank $k$ over a smooth manifold $M$. Suppose $\operatorname{g}_E$ is a
bundle metric on $E$ and that $\nabla$ is a metric connection on $E$.
In this paper
we consider uniformly elliptic operators $\mathscr{L}$ on $\Gamma(E)$ of second order
that are given locally by
\begin{equation}\label{dast}
\mathscr{L}=\sum_{i,j=1}^ma^{ij}\nabla^{2}_{e_{i},e{_{j}}}+\sum_{j=1}^mb^{j}\nabla_{e_{j}}, \tag{$\ast\ast$}
\end{equation}
where $a\in\Gamma(TM\otimes TM)$ is a symmetric, uniformly positive definite tensor and
$b\in\Gamma(TM)$ is a smooth vector field such that
$$a=\sum_{i,j=1}^m\operatorname{u}u aij e_i\otimes e_j\quad\text{and}\quad b=\sum_{j=1}^mb^je_j$$
in a local frame field $\{e_1,\dots,e_k\}$ of $TM$.
\begin{mythm}{\bf (Strong Elliptic Maximum Principle).}\label{mp1}\\
Let $(M,{\operatorname{g}_M})$ be a Riemannian manifold
and $(E,\pi,M)$ a vector bundle
over $M$ equipped with a Riemannian metric $\operatorname{g}_{E}$ and a metric connection $\nabla$.
Let $K$ be a closed fiber-convex subset of the bundle $E$ that is invariant under parallel
transport and let $\phi\in K$ be a smooth section such that
\begin{equation*}
\mathscr{L}\phi+\operatorname{P}si(\phi)=0,
\end{equation*}
where here $\mathscr{L}$ is a uniformly elliptic operator of second order of the form given in (\ref{dast}) and $\operatorname{P}si$ is a smooth
fiberwise map that points into $K$. If there exists a point $x_0$ in the interior of $M$ such that $\phi(x_0)\in\partial K_{x_0}$, then $\phi(x)\in\partial K_x$ for any point $x\in M$. If, additionally, $K_{x_0}$ is strictly convex at $\phi(x_0)$, then $\phi$ is a parallel section.
\end{mythm}
For the classification of minimal maps between Riemannian manifolds we will later use
a special case of Theorem \ref{mp1} for
smooth, symmetric tensors $\phi\in\operatorname{Sym}(E^*\otimes E^*)$. Before stating the result let us
recall the following definition due to Hamilton \cite[Section 9]{hamilton2}.
\begin{definition}{\bf (Hamilton).}\label{def null}
A fiberwise map $\operatorname{P}si:\operatorname{Sym}(E^*\otimes E^*)\to\operatorname{Sym}(E^*\otimes E^*)$ is said to satisfy the null-eigenvector
condition, if whenever $\vartheta$ is a non-negative symmetric $2$-tensor at a point $x\in M$ and
if $v\in T_{x}M$ is a null-eigenvector of $\vartheta$, then $\operatorname{P}si(\vartheta)(v,v)\ge 0$.
\end{definition}
The next theorem is the elliptic analogue of the
maximum principle of Hamilton \cite[Lemma 8.2, p. 174]{hamilton1}.
More precisely:
\begin{mythm}\label{mp2}
Let $(M,{\operatorname{g}_M})$ be a Riemannian manifold and $(E,\pi,M)$ a Riemannian vector bundle
over $M$ equipped with a metric connection.
Suppose that $\phi\in\operatorname{Sym}(E^*\otimes E^*)$ is non-negative definite
and satisfies
$$\mathscr{L}\phi+\operatorname{P}si(\phi)=0,$$
where here $\operatorname{P}si$ is a smooth fiberwise map satisfying the null-eigenvector condition.
If there is an interior point of $M$ where $\phi$ has a zero-eigenvalue, then
$\phi$ must have a zero-eigenvalue everywhere. Additionally, if $\phi$ vanishes identically at an
interior point of $M$, then $\phi$ vanishes everywhere.
\end{mythm}
Since the maximum principle for scalar functions has uncountable many applications in Geometric
Analysis we expect that the strong maximum principle for sections in vector bundles will have plenty of applications as well. In Section \ref{sec3} we will apply this strong maximum principle to derive a classification of minimal maps between Riemannian manifolds.
Before stating our results in this direction, let us introduce some new definitions. Let $(M,{\operatorname{g}_M})$ and
$(N,{\operatorname{g}_N})$ be two Riemannian manifolds of dimensions $m$ and $n$ respectively.
For any smooth map $f:M\to N$
its differential $\operatorname{d}\hspace{-3pt}f$ induces a map $\Lambda^{k}\operatorname{d}\hspace{-3pt}f:\Lambda^{k}T^*M\to\Lambda^{k}T^*N$ given by
\begin{equation*}
\left(\Lambda^{k}\operatorname{d}\hspace{-3pt}f\right)(v_1\,,\cdots,v_k):=\operatorname{d}\hspace{-3pt}f(v_1)\wedge\cdots\wedge\operatorname{d}\hspace{-3pt}f(v_k),
\end{equation*}
for any smooth vector fields $v_1,\dots,v_k\in TM$.
The map $\Lambda^{k}\operatorname{d}\hspace{-3pt}f$ is called the $k$-\textit{Jacobian} of $f$. The \textit{supremum norm} or
the $k$-\textit{dilation} $\|\Lambda^{k}\operatorname{d}\hspace{-3pt}f\|(x)$ of the map $f$ at a point $x\in M$
is defined as the supremum of
$$\sqrt{\det\big([f^{\ast}{\operatorname{g}_N}(v_i,v_j)]_{1\le i,j\le k}\bigr)}$$
when $\{v_1,\dots,v_m\}$ runs over all orthonormal bases of $T_xM$.
The $k$-dilation measures how much the map stretches $k$-dimensional volumes. The map
$f:M\to N$ is called \textit{weakly $k$-volume decreasing} if $\|\Lambda^k\operatorname{d}\hspace{-3pt}f\|\le 1$, \textit{strictly
$k$-volume decreasing} if $\|\Lambda^k\operatorname{d}\hspace{-3pt}f\|<1$ and \textit{$k$-volume preserving} if $\|\Lambda^k\operatorname{d}\hspace{-3pt}f\|=1$.
As usual for $k=1$ we use the term \textit{length} instead of $1$-volume and if $k=2$ we use the term
\textit{area} instead of $2$-volume. The map $f:M\to N$ is called an \textit{isometric immersion}, if $f^*{\operatorname{g}_N}={\operatorname{g}_M}$.
A smooth map $f:M\to N$ is called \textit{minimal}, if its graph
$$\Gamma(f):=\{(x,f(x))\in M\times N:x\in M\}$$
is a minimal submanifold of $(M\times N,{\operatorname{g}_{M\times N}}={\operatorname{g}_M}\times{\operatorname{g}_N})$.
One of the main objectives in the present article is to prove the following results:
\begin{mythm}\label{thmD}
Let $(M,{\operatorname{g}_M})$ and $(N,{\operatorname{g}_N})$ be two Riemannian manifolds. Suppose $M$ is compact, $m=\dim M\ge 2$
and that there exists a constant $\sigma>0$
such that the sectional curvatures $\sigma_M$ of $M$ and $\sigma_N$ of $N$ and
the Ricci curvature $\operatorname{Ric}_M$ of $M$ satisfy
$$\sigma_M\,\,>\,\, -\sigma,\quad\,\,\frac{1}{m-1}\operatorname{Ric}_M\,\,\ge\,\,\sigma\,\,\ge\,\,\sigma_N.$$
If $f:M\to N$ is a minimal map that is weakly length decreasing, then one of the following
holds:
\begin{enumerate}[(i)]
\item
$f$ is constant.
\item
$f$ is an isometric immersion, $M$ is Einstein with $\operatorname{Ric}_M=(m-1)\sigma$ and the
restriction of $\sigma_N$ to $\operatorname{d}\hspace{-3pt}f(TM)$ is equal to $\sigma$.
\end{enumerate}
In particular,
any strictly length decreasing minimal map is constant.
\end{mythm}
A similar statement holds in the case of weakly area decreasing maps.
\begin{mythm}\label{thmC}
Let $(M,{\operatorname{g}_M})$ and $(N,{\operatorname{g}_N})$ be two Riemannian manifolds. Suppose $M$ is compact, $m=\dim M\ge 2$
and that there exists a constant $\sigma>0$
such that the sectional curvatures $\sigma_M$ of $M$ and $\sigma_N$ of $N$ and
the Ricci curvature $\operatorname{Ric}_M$ of $M$ satisfy
$$\sigma_M\,\,>\,\, -\sigma,\quad\,\,\frac{1}{m-1}\operatorname{Ric}_M\,\,\ge\,\,\sigma\,\,\ge\,\,\sigma_N.$$
If $f:M\to N$ is a smooth minimal map that is weakly area decreasing, then one of the following
holds:
\begin{enumerate}[(i)]
\item
$f$ is constant.
\item
There exists
a non-empty closed set $D$ such that $f$ is an isometric immersion on $D$ and $f$ is strictly
area decreasing on the complement
of $D$. Moreover, $M$ is Einstein on $D$ with $\operatorname{Ric}_M=(m-1)\sigma$ and the
restriction of $\sigma_N$ to $\operatorname{d}\hspace{-3pt}f(TD)$ is equal to $\sigma$.
\end{enumerate}
In particular, any strictly area decreasing minimal map is constant
and any area preserving minimal map is an isometric immersion.
\end{mythm}
In the special case where the manifold $N$ is one-dimensional we have the following stronger theorem:
\begin{mythm}\label{thmE}
Let $(M,{\operatorname{g}_M})$ and $(N,{\operatorname{g}_N})$ be two Riemannian manifolds. Suppose that $M$ is compact, $\dim M\ge 2$,
$\Ric_{M}>0$ and that $\dim N=1$. Then any smooth minimal map $f:M\to N$ is constant.
\end{mythm}
As pointed out in the final remarks of Section $3.6$, Theorems \ref{thmD}, \ref{thmC} and \ref{thmE} are optimal
in various ways. We include some examples and remarks concerning the imposed
assumptions at the end of the paper.
The paper is organized as follows. In Section \ref{sec2}
we recall the strong maximum principle for uniformly elliptic
systems of second order by Weinberger-Evans and give the proofs of
Theorems \ref{mp1} and \ref{mp2}. The geometry of
graphs will be treated in Section \ref{sec3}, where we also derive the crucial formula needed
in the proof of Theorems \ref{thmD}, \ref{thmC} and \ref{thmE}.
\section{Strong elliptic maximum principles for sections in vector bundles}\label{sec2}
In this section we shall derive strong elliptic maximum principles for smooth sections in
Riemannian vector bundles. The original idea goes back to
the fundamental work of Hamilton \cite{hamilton2,hamilton1} on the Ricci flow, where a strong
parabolic maximum principle for symmetric tensors and weak
parabolic maximum principles for sections in vector bundles were proven.
\subsection{Convex sets}
In this subsection we review the basic definitions about the geometry of convex sets
in euclidean space such as supporting half-spaces, tangent cones and normal vectors.
A brief exposition can be found in the book by Andrews and Hopper \cite[Appendix B]{andrews}.
Recall that a subset $K$ of $\real{n}$ is called \textit{convex} if for any pair of points $z$, $w\in K$,
the segment
$$\mathcal{E}_{z,w}:=\{tz+(1-t)w\in\real{n}:t\in(0,1)\}$$
is contained in $K$. The set $K$ is said to be \textit{strictly convex}, if for
any pair $z,w\in K$ the segment $\mathcal{E}_{z,w}$ belongs to the interior of $K$.
A convex set $K\subset\real{n}$ may have non-smooth boundary. Hence, there is no well-defined tangent
or normal space of $K$ in the classical sense. However, there is a way to generalize these important
notions for closed convex subsets of $\real{n}$. This difficulty can be overcome by using the property
that points lying outside of the given set can be separated from the set itself by half-spaces. This
property, leads to the notion of generalized tangency.
Let $K$ be a closed convex subset of the euclidean space $\real{n}$. A supporting half-space of the set $K$ is
a closed half-space of $\real{n}$ which contains $K$ and has points of $K$ on its boundary. A supporting
hyperplane of $K$ is a hyperplane which is the boundary of a supporting half-space of $K$.
The \textit{tangent cone} $C_{y_0}K$ of $K$
at $y_0\in \partial K$ is defined as the intersection of all supporting half-spaces of $K$ that contain $y_0$.
We may also introduce the notion of normal vectors
to the boundary of a closed convex set. Let $K\subset\real{n}$ be a closed convex subset and $y_0\in\partial K$. Then
\begin{enumerate}[(i)]
\item
A non-zero vector $\xi$ is called \textit{normal vector} of $\partial K$ at $y_0$, if $\xi$ is normal to a
supporting hyperplane of $K$ passing through $y_0$. This normal vector is called \textit{inward pointing}, if it points into the half-space containing the set $K$.
\item
A vector $\eta$ is called \textit{inward pointing} at $y_0\in\partial K$, if
$$\langle\xi,\eta\rangle\ge 0$$
for any inward pointing normal vector $\xi$ at $y_0$. Here, $\langle\cdot,\cdot\rangle$ denotes the usual inner product in $\real{n}$.
\end{enumerate}
\subsection{Maximum principles for systems}
In \cite{weinberger}, H. Weinberger established a strong maximum principle for vector valued
maps with values in a convex set $K\subset\real{n}$ whose boundary $\partial K$ satisfies regularity
conditions that he called \textquotedblleft{\textit{slab conditions}}\textquotedblright. Inspired by
the ideas of Weinberger, X. Wang in \cite{wang4} gave a geometric proof of the strong maximum principle,
in the case where the boundary of $K$ is of class $C^2$. The idea of Wang was to apply
the classical maximum principle of Hopf to the function $d(u):D\to\real{}$, whose value at $x$ is
equal to the distance of $u(x)$ from the boundary $\partial K$ of $K$. Very recently, L.C. Evans \cite{evans1}
was able to remove all additional regularity requirements on the boundary of the convex set $K$ by showing that even
if $d(u)$ is not twice differentiable, it is still a viscosity super-solution of an appropriate partial
differential equation. The argument of Evans is completed by applying a strong maximum principle
due to F. Da Lio \cite{dalio} for viscosity super-solutions of partial differential equations.
\begin{theorem}{\bf (Weinberger-Evans).}\label{weinberger}
Let $K$ be a closed, convex set of $\real{n}$ and $u:D\subset\real{m}\to K\subset\real{n}$ a
solution of the uniformly elliptic system of partial differential equations
$$(\mathscr{L}u)(x)+\operatorname{P}si(x,u(x))=0,\quad x\in D,$$
where $D$ is a domain of $\real{m}$, $\operatorname{P}si:D\times\real{n}\to\real{n}$ is a continuous map that is locally Lipschitz continuous in the second variable and $\mathscr{L}$ is a uniformly elliptic operator given in (\ref{elliptic}). Suppose that
\begin{enumerate}[(i)]
\item
there is a point $x_0$ in the interior of $D$ such that
$u(x_0)\in\partial K$,
\item
for any $(x,y)\in D\times\partial K$, the vector $\operatorname{P}si(x,y)$ points into $K$ at the point $y\in\partial K$.
\end{enumerate}
Then $u(x)\in\partial K$ for any $x\in D$. If $\partial K$ is strictly convex at $u(x_0)$, then $u$ is constant.
\end{theorem}
\begin{remark}
The above maximum principle is not valid without the convexity assumption. We illustrate this
by an example. Let
$$D=\{(x,y)\in\real{2}:x^2+y^2<1\}$$
be the unit open disc in $\real{2}$ and let $h:\partial D\to\Gamma\subset\real{2}$ be a
continuous function that maps $\partial D$ onto the upper semicircle
$$\Gamma=\{(x,y)\in\real{2}:x^2+y^2=1\,\text{and}\,y\ge 0\}.$$
Denote now by $u:D\to\real{2}$ the solution of the Dirichlet problem with boundary data
given by the function $h$. Let us examine the image of the harmonic map $u$.
We claim at first that the image of $u$ is contained in the convex hull
$\mathcal{C}(\Gamma)$ of the upper semicircle. That is,
$$K:=u\left(\overline{D}\right)\subset\,\mathcal{C}(\Gamma)=\{(x,y)\in\real{2}:x^2+y^2\le 1\,\text{and }y\ge 0\}.$$
Arguing indirectly, let us assume that this is not true. The convex hull $\mathcal{C}(K)$
of $K$ contains $\mathcal{C}(\Gamma)$. Since $K$ is compact, the set
$\mathcal{C}(K)$ is also compact. Consequently, there exist a point $(x_0,y_0)$ in $D$ such that
$u(x_0,y_0)\in\partial\,\mathcal{C}(K)$ and $u(x_0,y_0)\not\in\partial\,\mathcal{C}(\Gamma)$.
Then, from the maximum principle of Weinberger-Evans we deduce that $u(x,y)\in\partial\,\mathcal{C}(K)$ for any
$(x,y)\in D$. This contradicts with the boundary data imposed by the Dirichlet condition.
Therefore, $K$ is contained in $\mathcal{C}(\Gamma)$. From Theorem \ref{weinberger},
we conclude that there is no common point of $K$ with the $x$-axes. Hence, $K$ is not convex.
The same argument yields that there is no point of $D$ which is mapped to $\Gamma$ via $u$. On
the other hand, because $K$ is compact, there are infinitely many points of $D$ which are mapped
to the boundary of $K$. Furthermore, we claim that the set $K$ has non-empty interior.
In order to show this, suppose to the contrary that $K\setminus\partial K=\emptyset$. Then,
$$\rank(\operatorname{d}\hspace{-2pt}u)\le 1$$
which implies that the closure of the set $u(D)$ is a continuous curve $L$ joining the points
$(-1,0)$ and $(1,0)$. But then, the continuity of $u$ leads to a contradiction. Indeed, for any
sequence $\{p_{k}\}_{k\in\mathbb{N}}$ of points of $D$ converging to a point $p\in u^{-1}(0,1)$, we have
$\lim u(p_k)\neq (0,1)$.
\end{remark}
\subsection{Maximum principles for sections in vector bundles}
Here we give
the analogue version of the Weinberger-Evans strong maximum principle for sections
in Riemannian vector bundles.
Our approach is inspired by ideas developed by Weinberger \cite{weinberger} and Hamilton \cite{hamilton2,hamilton1}.
For the proof of the strong maximum principle we will use a beautiful result
due to C. B\"{o}hm and B. Wilking \cite[Lemma 1.2, p. 670]{bohm}.
\begin{lemma}{\bf (B\"{o}hm-Wilking).}\label{wilking}
Suppose that $M$ is a Riemannian manifold and that $(E,\pi,M)$ is a Riemannian vector bundle over
$M$ equipped with a metric connection. Let $K$ be a closed and fiber-convex subset
of the bundle $E$ that is invariant under parallel transport. If $\phi$ is a smooth section with
values in $K$ then, for any $x\in M$ and $v\in T_xM$, the Hessian
$$\nabla^{2}_{v,v}\phi=\nabla_{v}\nabla_v\phi-\nabla_{\nabla_{v}v}\phi$$
belongs to the tangent cone of $K_x$ at the point $\phi(x)$.
\end{lemma}
The following result is an immediate consequence of the above lemma.
\begin{lemma}\label{wilking2}
Suppose that $M$ is a Riemannian manifold and that $(E,\pi,M)$ is a Riemannian vector bundle over
$M$ equipped with a metric connection. Let $K$ be a closed and fiber-convex subset
of $E$ that is invariant under parallel transport. If $\phi$ is a smooth section with
values in $K$ then, for any $x\in M$, the vector $(\mathscr{L}\phi)(x)$ belongs to the tangent cone
$C_{\phi(x)}K_x$, for any operator $\mathscr{L}$ of the form given in (\ref{dast}).
\end{lemma}
Now we derive the proof of the strong elliptic maximum principle formulated in Theorem \ref{mp1}.
{\bf Proof of Theorem \ref{mp1}.}
Let $\{\phi_1,\dots,\phi_k\}$ be a geodesic
orthonormal frame field of smooth sections in $E$, defined in a sufficiently small neighborhood
$U$ of a local trivialization around $x_0\in M$. Hence,
$$\phi=\sum_{i=1}^{k}u_{i}\phi_{i}$$
where $u_{i}:U\to\real{}$, $i\in\{1,\dots,k\}$, are smooth functions.
With respect to this frame we have
\begin{eqnarray*}
\mathscr{L}\phi&=&\sum^k_{i=1}\Big\{\mathscr{L} u_{i}+\big(\text{gradient terms of }u_i\big)
+\sum\limits^{k}_{j=1}u_{j}\,{\operatorname{g}_{E}}(\mathscr{L}\phi_j,\phi_i)\Big\}\phi_i \\
&=&-\sum^{k}_{i=1}{\operatorname{g}_{E}}(\operatorname{P}si(\phi),\phi_i)\phi_i
\end{eqnarray*}
Therefore, the map $u:U\to\real{k}$, $u=(u_1,\dots,u_k)$, satisfies a uniformly elliptic system of second order of the form
\begin{equation}\label{pde}
\mathscr{\tilde L}u+\operatorname{P}hi(u)=0,
\end{equation}
where here $\operatorname{P}hi:\real{k}\to\real{k}$,
$$\operatorname{P}hi:=(\operatorname{P}hi_1,\dots,\operatorname{P}hi_k),$$
is given by
\begin{equation}\label{newphi}
\operatorname{P}hi_{i}(u)={\operatorname{g}_{E}}\left(\operatorname{P}si\left(\textstyle\sum\limits_{j=1}^k u_j\phi_j\right)+\textstyle\sum\limits_{j=1}^{k}u_j\mathscr{L}\phi_j\,,\phi_i\right),
\end{equation}
for any $i\in\{1,\dots,k\}$.
Consider the convex set
$$\mathcal{K}:=\{(y_1,\dots,y_k)\in\real{k}:\textstyle\sum\limits_{i=1}^{k}y_{i}\phi_{i}(x_0)\in K_{x_0}\}.$$
\textit{{\bf Claim 1:} For any point $x\in U$ we have $u(x)\in\mathcal{K}$}.
Indeed, fix a point $x\in U$ and
let $\gamma:[0,1]\to U$ be the geodesic curve joining the points $x$ and $x_0$. Denote by $\theta$ the
parallel section which is obtained by the parallel transport of $\phi(x)$ along the geodesic
$\gamma$. Then,
$$\theta\circ\gamma=\sum_{i=1}^{k}y_{i}\,\phi_{i}\circ\gamma,$$
where $y_{i}:[0,1]\to\real{}$, $i\in\{1,\dots,k\}$, are smooth functions. Because,
$\theta$ and $\phi_{i}$, $i\in\{1,\dots,k\}$ are parallel along $\gamma$, it follows
that
\begin{eqnarray*}
0=\nabla_{\partial_{t}}(\theta\circ\gamma)=\sum_{i=1}^{k}y'_{i}(t)\phi_{i}(\gamma(t)).
\end{eqnarray*}
Hence, $y_{i}(t)=y_{i}(0)=u_{i}(x),$ for any $t\in [0,1]$ and $i\in\{1,\dots,k\}$. Therefore,
$$\theta(\gamma(1))=\theta(x_0)=\sum_{i=1}^{k}u_{i}(x)\phi_{i}(x_0).$$
Since by our assumptions $K$ is invariant under parallel transport, it follows that
$\theta(x_0)\in K_{x_0}$. Hence, $u(U)\subset\mathcal{K}$ and this proves Claim 1.
\textit{{\bf Claim 2:} For any $y\in\partial \mathcal{K}$ the vector $\operatorname{P}hi(y)$ as
defined in $(\ref{newphi})$ points
into $\mathcal{K}$ at $y$.}
First note that the boundary of each slice $K_x$ is invariant under parallel transport.
From (\ref{newphi}) we deduce that it suffices to prove that both terms appearing on
the right hand side of (\ref{newphi}) point into $\mathcal{K}$. The first term points into
$\mathcal{K}$ by assumption on $\operatorname{P}si$. The second term is inward pointing due to
Lemma \ref{wilking2} by B\"ohm and Wilking.
This completes the proof of Claim 2.
The solution of the uniformly second order elliptic partial differential system $(\ref{pde})$ satisfies
all the assumptions of Theorem \ref{weinberger}. Therefore, because $u(x_0)\in\partial\mathcal{K}$
it follows that $u(U)$ is contained in the boundary $\partial\mathcal{K}$ of $\mathcal{K}$. Consequently,
$\phi(x)\in\partial K$ for any $x\in U$. Since $M$ is connected, we deduce that $\phi(M)\subset\partial K$.
Note, that if $\mathcal{K}$ is additionally strictly convex at
$u(x_0)$, then the map $u$ is constant. This
implies that
$$\phi(x)=\sum_{i=1}^{k}u_i(x_0)\phi_i(x)$$
for any $x\in U$. Consequently, $\phi$ is a parallel section taking all its values in $\partial K$.
This completes the proof of Theorem \ref{mp1}.
{$\square$}
\begin{remark}
Theorem \ref{mp1} implies the following: Suppose the fibers of $K$ are cones with vertices at $0$
and that they are strictly convex at $0$. If $\phi(x)=0$ in a point $x\in M$, then $\phi$ vanishes everywhere.
\end{remark}
We can now prove Theorem \ref{mp2}.
{\bf Proof of Theorem \ref{mp2}.}
Let $K$ be the set of all non-negative definite symmetric $2$-tensors on $M$, i.e.
$$K:=\{\vartheta\in\operatorname{Sym}(E^*\otimes E^*):\vartheta\ge 0\}.$$
Each fiber $K_{x}$ is a closed convex cone that is strictly convex at $0$. Moreover, $K$ is invariant under parallel transport.
The set of all boundary points of $K_x$ is given by
$$\partial K_{x}=\{\vartheta\in K_{x}:\exists\text{ a non-zero }v\in T_{x}M\text{ such that }
\vartheta(v,v)=0\}.$$
It is a classical fact in Convex Analysis (see for example the book \cite[Appendix B]{andrews}), that the tangent cone of
$K_{x}$ at a point $\vartheta$ of its boundary is given by
$$C_{\vartheta}K_{x}=\{\psi\in\operatorname{Sym}(E_{x}^*\otimes E_{x}^*):
\psi(v,v)\ge 0,\forall\, v\in E_x \text{ with }\vartheta(v,v)=0\}.$$
Thus $\psi$ is in the tangent cone of $K_x$ at $\vartheta$, if and only if it satisfies the null-eigenvector
condition of Hamilton given in Definition \ref{def null}.
By Theorem \ref{mp1} we immediately get the proof of Theorem \ref{mp2}.
$\square$
\subsection{A second derivative criterion for symmetric $2$-tensors}
For $\phi\in\operatorname{Sym}(E^*\otimes E^*)$ a real number $\lambda$ is called \textit{eigenvalue} of $\phi$ with
respect to ${\operatorname{g}_{E}}$ at the point $x\in M$, if there exists a non-zero vector $v\in E_{x}$, such that
\begin{equation*}
\phi(v,w)=\lambda{\operatorname{g}_{E}}(v,w),
\end{equation*}
for any $w\in E_{x}$. The linear subspace $\operatorname{Eig}_{\lambda,\phi}(x)$ of $E_x$ given by
\begin{equation*}
\operatorname{Eig}_{\lambda,\phi}(x):=\{v\in E_x:\phi(v,w)=\lambda{\operatorname{g}_{E}}(v,w),\, \text{for any}\,w\in E_x\},
\end{equation*}
is called the \textit{eigenspace} of $\lambda$ at $x$. Since $\phi$ is symmetric it admits $k$ real
eigenvalues $\lambda_1(x),\dots,\lambda_k(x)$ at each point $x\in M$. We will always arrange the
eigenvalues such that $\lambda_{1}(x)\leq\cdots\leq\lambda_{k}(x)$.
\begin{theorem}{\bf(Second Derivative Criterion)}\label{test}
Let $(M,{\operatorname{g}_M})$ be a Riemannian manifold and $(E,\pi,M)$ a Riemannian vector bundle
of rank $k$ over the manifold $M$ equipped with a metric connection $\nabla$.
Suppose that $\phi\in\operatorname{Sym}(E^*\otimes E^*)$ is a smooth symmetric $2$-tensor. If the
biggest eigenvalue $\lambda_k$ of $\phi$ admits a local maximum $\lambda$ at an interior
point $x_0\in M$, then
$$(\nabla\phi)(v,v)=0\quad\text{and}\quad(\mathscr{L}\phi)(v,v)\le 0,$$
for all vectors $v$ in the eigenspace of $\lambda$ at $x_0$ and for all uniformly elliptic
second order operators $\mathscr{L}$.
\end{theorem}
{\bf Remark.} Replacing $\phi$ by $-\phi$ in Theorem \ref{test} gives a similar result for
the smallest eigenvalue $\lambda_1$ of $\phi$.
\begin{proof}
Let $v\in \operatorname{Eig}_{\lambda,\phi}(x_0)$ be a unit vector and $V\in\Gamma(E)$ a smooth section such that
\begin{equation*}
V(x_0)=v\quad \text{and}\quad(\nabla V)(x_0)=0.
\end{equation*}
Define the symmetric $2$-tensor $\operatorname{S}$ given by $\operatorname{S}:=\phi-\lambda{\operatorname{g}_{E}}$.
From our assumptions, the symmetric $2$-tensor $\operatorname{S}$ is non-positive definite in a small neighborhood
of $x_0$. Moreover, the biggest eigenvalue of $\operatorname{S}$ at $x_0$ equals $0$. Consider the smooth function
$f:M\to\real{}$, given by
\begin{equation*}
f(x):=\operatorname{S}(V(x),V(x)).
\end{equation*}
The function $f$ is non-positive in the same neighborhood around $x_0$
and attains a local maximum in an interior point $x_0$. In particular,
\begin{equation*}
f(x_0)=0,\quad \operatorname{d}\hspace{-3pt}f(x_0)=0\quad\text{and}\quad(\mathscr{L}f)(x_0)\le 0.
\end{equation*}
Consider a local orthonormal frame field $\{e_1,\dots,e_m\}$ with respect to ${\operatorname{g}_M}$ defined in a neighborhood
of the point $x_0$ and assume that the expression of $\mathscr{L}$ with respect to this frame is
\begin{equation*}
\mathscr{L}=\sum_{i,j=1}^ma^{ij}\nabla^{2}_{e_{i},e{_{j}}}+\sum_{j=1}^mb^{j}\nabla_{e_{j}}.
\end{equation*}
A simple calculation yields
\begin{equation*}
\nabla_{e_{i}}f=\operatorname{d}\hspace{-3pt}f(e_i)=\left(\nabla_{e_{i}}\operatorname{S}\right)(V,V)+2\operatorname{S}\left(\nabla_{e_{i}}V,V\right).
\end{equation*}
Taking into account that ${\operatorname{g}_{E}}$ is parallel, we deduce that
$$0=(\nabla f)(x_0)=(\nabla\operatorname{S})(v,v)=(\nabla\phi)(v,v).$$
Furthermore,
\begin{eqnarray*}
\nabla^{2}_{e_{i},e_{j}}f&=&(\nabla^{2}_{e_{i},e_{j}}\operatorname{S})(V,V)
+2\operatorname{S}(V,\nabla^{2}_{e_{i},e_{j}}V)\\
&&+2\left(\nabla_{e_i}\operatorname{S}\right)(\nabla_{e_j}V,V)+2\left(\nabla_{e_j}\operatorname{S}\right) (\nabla_{e_i}V,V)\\
&&+2\operatorname{S}(\nabla_{e_i}V,\nabla_{e_j}V).
\end{eqnarray*}
Bearing in mind the definition of $S$ and using the fact that ${\operatorname{g}_{E}}$ is parallel with respect to $\nabla$,
we obtain
\begin{eqnarray*}
\mathscr{L}f&=&(\mathscr{L}\phi)(V,V)+2\operatorname{S}(V,\mathscr{L}V)\\
&&+\sum_{i,j=1}^{m}2a^{ij}\left\{\operatorname{S}(\nabla_{e_i}V,\nabla_{e_j}V)
+2(\nabla_{e_i}\operatorname{S})(\nabla_{e_j}V,V)\right\}\\
&=&(\mathscr{L}\phi)(V,V)+2\operatorname{S}(V,\mathscr{L}V)\\
&&+\sum_{i,j=1}^{m}2a^{ij}\left\{\operatorname{S}(\nabla_{e_i}V,\nabla_{e_j}V)
+2(\nabla_{e_i}\operatorname{S})(\nabla_{e_j}V,V)\right\}.
\end{eqnarray*}
Estimating at $x_0$ and taking into account that $V(x_0)=v$ is a null eigenvector
of $\operatorname{S}$ at $x_0$, we get
$$0\ge(\mathscr{L}f)(x_0)=(\mathscr{L}\phi)(v,v).$$
This completes the proof.
\end{proof}
\subsection{An application}
In order to demonstrate how to apply the strong elliptic maximum principle and the second
derivative criterion, we shall give here an example in the case of hypersurfaces in euclidean space.
Let $M$ be an oriented $m$-dimensional hypersurface of $\real{m+1}$. Denote by
$\xi$ a unit normal vector field along the hypersurface. The most natural symmetric $2$-tensor on
$M$ is the \textit{scalar second fundamental form} $h$ of the hypersurface with respect
to the unit normal direction $\xi$, that is
$$h(v,w):=-\langle \operatorname{d}\hspace{-2pt}\xi(v),w\rangle,$$
for any $v,w\in TM$.
The eigenvalues
$$\lambda_1\le\cdots\le\lambda_m$$
of $h$ with respect to the induced metric $\operatorname{g}$ are called the \textit{principal curvatures} of the hypersurface.
The quantity $H$ given by
$$H:=\lambda_1+\cdots+\lambda_m$$
is called the \textit{scalar mean curvature} of the hypersurface and the function $\|h\|$
given by
$$\|h\|^2:=\lambda^2_1+\cdots+\lambda^{2}_m$$
is called the
\textit{norm of the second fundamental form} with respect to the metric $\operatorname{g}$. It is well
known that if $h$ is non-negative definite, then $M$ is locally the boundary
of a convex subset of $\real{m+1}$. For this reason, the hypersurface $M$ is called \textit{convex} whenever
its scalar second fundamental form is non-negative definite.
In the sequel we will give an alternative short proof of a well-known theorem,
first proven by W. S\"uss \cite{suss}.
\begin{theorem}{\bf{(S\"uss)}}\label {thmalex}
Any closed and convex hypersurface $M$ in $\real{m+1}$ with constant mean curvature is a round
sphere.
\end{theorem}
\begin{proof}
The Laplacian of the second fundamental form $h$ with respect to the induced Riemannian metric $\operatorname{g}$,
is given by Simons' identity \cite{simons}
\begin{equation}\label{simons}
\Delta h+\|h\|^2h-Hh^{(2)}=0\,,
\end{equation}
where
$$h^{(2)}(v,w):=\operatorname{trace}\bigl(h(v,\cdot\,)\otimes h(w,\cdot\,)\bigr).$$
Since the manifold $M$ is closed, there exists an interior point $x_0\in M$, where the smallest
principal curvature $\lambda_1$ attains its global minimum $\lambda_{\min}$. Recall that by convexity we
have that $\lambda_{\min}\ge0$.
The fiberwise map $\operatorname{P}si$ given by
$$\operatorname{P}si(\vartheta)=\|\vartheta\|^2\vartheta-H\vartheta^{(2)},$$
obviously satisfies the null-eigenvector condition.
If $\lambda_{\min}=0$, then due to Theorem B, the smallest principal curvature
of $M$ vanishes everywhere. Hence, $\rank h<m$. It is a well known fact in
Differential Geometry that the set
$$M_{0}:=\{x\in M:\rank h_{x}={\max}_{z\in M}\rank h_{z}\},$$
is open and dense in $M$ (a standard reference is \cite{ferus}).
From the Codazzi equation, it follows that the nullity distribution
$$\mathcal{D}:=\{v\in TM_0:h(v,w)=0,\,\,\text{for all }w\in TM_0\},$$
is integrable and its integrals are totally geodesic submanifolds of $M$. On
the other hand, the Gau{\ss} formula says that these submanifolds are totally geodesic in
$\real{m+1}$. Moreover, because $M$ is complete it follows that these submanifolds must be
also complete. This contradicts with the assumption of compactness. Consequently,
the minimum $\lambda_{\min}$ of the smallest principal curvature must be strictly positive.
Let $v$ be a unit eigenvector of $h$ corresponding to $\lambda_{\min}$ at $x_0$. Applying
Theorem \ref{test}, we obtain
\begin{eqnarray*}
0&\ge&\|h\|^2(x_0)\lambda_{\min}-H\lambda_{\min}^2 \\
&=&\lambda_{\min}\left(\|h\|^2(x_0)-H\lambda_{\min}\right),
\end{eqnarray*}
Because $\|h\|^2\ge H^2/m$, we deduce that
$$\|h\|^2(x_0)-H\lambda_{\min}\ge H\left(H/m-\lambda_{\min}\right)\ge 0.$$
Consequently,
$$0\ge \lambda_{\min}H\left(H/m-\lambda_{\min}\right)\ge 0,$$
and so $H/m=\lambda_{\min}$. On the other hand $\lambda_{\min}$ is the global minimum of all
principal curvatures on $M$ and $H$ is constant. This shows that
the smallest principal curvature $\lambda_1(x)$ at an arbitrary point $x\in M$ satisfies
$$\lambda_{\min}\le\lambda_1(x)\le H/m=\lambda_{\min}.$$
Therefore $M$ is everywhere umbilic. It is well-known that the only closed and totally
umbilic hypersurfaces are the round spheres.
\end{proof}
\section{Bernstein Type Theorems for Minimal Maps}\label{sec3}
In this section we shall develop the relevant geometric identities for graphs induced by
smooth maps $f:M\to N$. Moreover, we will derive estimates that
will be crucial in the proofs of Theorems \ref{thmD}, \ref{thmC} and \ref{thmE}.
According to the Bernstein theorem \cite{bernstein}, all complete minimal graphs in the three
dimensional euclidean space are generated by affine maps. This result cannot be extended to complete minimal
graphs in any euclidean space without imposing further assumptions. There is a very rich and long literature
concerning complete minimal graphs which are generated by maps between euclidean
spaces, marked by works of W. Fleming \cite{fleming}, S.S. Chern and R. Osserman \cite{chern1}, J. Simons
\cite{simons}, E. Bombieri, E. de Giorgi and E. Giusti \cite{bombieri}, R. Schoen, L. Simon and S.T. Yau
\cite{schoen1}, S. Hildebrandt, J. Jost and K.-O. Widmann \cite{hildebrandt} and many others.
In the last decade there have been obtained several Bernstein type theorems in higher codimension,
see for instance \cite{swx}, \cite{li}, \cite{h-s2, h-s1} and \cite{jxy}.
The generalized Bernstein type problem that we are investigating here is to determine under which
additional geometric conditions minimal graphs generated by maps $f:M\to N$ are totally
geodesic. There are several recent results involving mean curvature flow in the case where both $M$ and $N$
are compact. For instance, we mention \cites{wang2,wang3,wang1}, \cite{sw}, \cite{tsui1}
and \cite{lee}. In these papers the authors prove that the mean curvature flow of graphs,
generated by smooth maps $f:M\to N$ satisfying suitable conditions,
evolves $f$ to a constant map or an isometric immersion as time tends to infinity. This implies in particular
Bernstein results for minimal graphs satisfying the same conditions as the initial map.
\subsection{Geometry of graphs}
Let $(M,{\operatorname{g}_M})$ and $(N,{\operatorname{g}_N})$ be Riemannian manifolds of dimension $m$ and $n$, respectively.
The induced metric on the product manifold will be denoted by
$${\operatorname{g}_{M\times N}}={\operatorname{g}_M}\times {\operatorname{g}_N}.$$
A smooth map
$f:M\to N$ defines an embedding $F:M\to M\times N$, by
$$F(x)=\bigl(x,f(x)\bigr),\quad x\in M.$$
The graph of $f$ is defined to be the submanifold $\Gamma(f):=F(M)$. Since $F$ is an embedding,
it induces another Riemannian metric
$\operatorname{g}:=F^*{\operatorname{g}_{M\times N}}$
on $M$. The two natural projections
$$\pi_{M}:M\times N\to M\,,\quad \pi_{N}:M\times N\to N$$
are submersions,
that is they are smooth and have maximal rank. Note that the tangent bundle of the product manifold
$M\times N$, splits as a direct sum
\begin{equation*}
T(M\times N)=TM\oplus TN.
\end{equation*}
The four metrics ${\operatorname{g}_M},{\operatorname{g}_N},{\operatorname{g}_{M\times N}}$ and $\operatorname{g}$ are related by
\begin{eqnarray}
{\operatorname{g}_{M\times N}}&=&\pi_M^*{\operatorname{g}_M}+\pi_N^*{\operatorname{g}_N}\,,\label{met1}\\
\operatorname{g}&=&F^*{\operatorname{g}_{M\times N}}={\operatorname{g}_M}+f^*{\operatorname{g}_N}\,.\label{met2}
\end{eqnarray}
Additionally, let us define the symmetric $2$-tensors
\begin{eqnarray}
{\operatorname{s}_{M\times N}}&:=&\pi_M^*{\operatorname{g}_M}-\pi_{N}^*{\operatorname{g}_N}\,,\label{met3}\\
\operatorname{s}&:=&F^*{\operatorname{s}_{M\times N}}={\operatorname{g}_M}-f^*{\operatorname{g}_N}\,.\label{met4}
\end{eqnarray}
Note that ${\operatorname{s}_{M\times N}}$ is a semi-Riemannian metric of signature $(m,k)$ on the manifold $M\times N$.
The Levi-Civita connection $\nabla^{{\operatorname{g}_{M\times N}}}$ associated to the Riemannian metric ${\operatorname{g}_{M\times N}}$ on $M \times N$ is
related to the Levi-Civita connections $\nabla^{{\operatorname{g}_M}}$ on $(M,{\operatorname{g}_M})$ and $\nabla^{{\operatorname{g}_N}}$ on
$(N,{\operatorname{g}_N})$ by
$$\nabla^{{\operatorname{g}_{M\times N}}}=\pi_M^*\nabla^{{\operatorname{g}_M}}\oplus\pi_N^*\nabla^{{\operatorname{g}_N}}\,.$$
The corresponding curvature operator ${\operatorname{R}_{M\times N}}$ on $M\times N$ with respect to the metric
${\operatorname{g}_{M\times N}}$ is related to the curvature
operators ${\operatorname{R}_M}$ on $(M,{\operatorname{g}_M})$ and ${\operatorname{R}_N}$ on $(N,{\operatorname{g}_N})$ by
\begin{equation*}
{\operatorname{R}_{M\times N}}=\pi^{*}_{M}{\operatorname{R}_M}\oplus\pi^{*}_{N}{\operatorname{R}_N}.
\end{equation*}
Denote the Levi-Civita connection on $M$ with respect to the induced metric
$\operatorname{g}=F^*{\operatorname{g}_{M\times N}}$ simply by $\nabla$ and the curvature tensor by $\operatorname{R}$.
On the manifold $M$ there are many interesting bundles. The most important
one is the \textit{pull-back bundle} $F^{\ast}T(M\times N)$.
The differential $\operatorname{d}\hspace{-3pt}F$ of the map $F$ is a section in $F^{\ast}T(M\times N)\otimes T^*M$.
The covariant derivative of it is called the \textit{second fundamental tensor}
$A$ of the graph. That is,
\begin{equation*}
A(v,w):=(\widetilde\nabla\hspace{-2pt}\operatorname{d}\hspace{-3pt}F)(v,w)=\nabla^{{\operatorname{g}_{M\times N}}}_{\operatorname{d}\hspace{-3pt}F(v)}\operatorname{d}\hspace{-3pt}F(w)-\operatorname{d}\hspace{-3pt}F(\nabla_vw)
\end{equation*}
where $v,w\in TM$, $\widetilde\nabla$ is the induced connection on $F^{\ast}T(M\times N)\otimes T^*M$
and $\nabla$ is the Levi-Civita connection associated to the Riemannian metric
$$\operatorname{g}:=F^*{\operatorname{g}_{M\times N}}.$$
The trace of $A$ with respect to the metric $\operatorname{g}$ is called the
{\it mean curvature vector field} of $\Gamma(f)$ and it
will be denoted by
$$\vec{H}:=\operatorname{trace}A.$$
Note that $\vec{H}$ is a section in the normal bundle of the
graph. If $\vec{H}$ vanishes identically the graph is said to be
minimal. Following Schoen's \cite{schoen} terminology, a map $f:M\to N$
between Riemannian manifolds is called \textit{minimal} if its graph
$\Gamma(f)$ is a minimal submanifold of the product space $(M\times
N,{\operatorname{g}_{M\times N}})$.
By \textit{Gau\ss' equation} the curvature tensors $\operatorname{R}$ and ${\operatorname{R}_{M\times N}}$
are related by the formula
\begin{eqnarray}
\operatorname{R}(v_1,w_1,v_2,w_2)&=&(F^*{\operatorname{R}_{M\times N}})(v_1,w_1,v_2,w_2)\nonumber\\
&&+{\operatorname{g}_{M\times N}}\bigl( A(v_1,v_2),A(w_1,w_2)\bigr)\nonumber\\
&&-{\operatorname{g}_{M\times N}}\bigl( A(v_1,w_2),A(w_1,v_2)\bigr),\label{gauss}
\end{eqnarray}
for any $v_1,v_2,w_1,w_2\in TM$. Moreover, the second fundamental form satisfies the
\textit{Codazzi equation}
\begin{eqnarray}
(\nabla_uA)(v,w)-(\nabla_vA)(u,w)&=&{\operatorname{R}_{M\times N}}\bigl(\operatorname{d}\hspace{-3pt}F(u),\operatorname{d}\hspace{-3pt}F(v)\bigr)\operatorname{d}\hspace{-3pt}F(w)\nonumber\\
&&-\operatorname{d}\hspace{-3pt}F\bigl(\operatorname{R}(u,v)w\bigr),\label{codazzi}
\end{eqnarray}
for any $u,v,w$ on $TM$.
\subsection{Singular decomposition}
In this subsection we closely follow the notations used in \cite{tsui1}.
For a fixed point $x\in M$, let
$$\lambda^2_{1}(x)\le\cdots\le\lambda^2_{m}(x)$$
be the eigenvalues of $f^{*}{\operatorname{g}_N}$ with respect to ${\operatorname{g}_M}$. The corresponding values $\lambda_i\ge 0$,
$i\in\{1,\dots,m\}$, are usually called
\textit{singular values} of the differential $\operatorname{d}\hspace{-3pt}f$ of $f$ and give rise to continuous functions on $M$. Let
$$r=r(x)=\rank\operatorname{d}\hspace{-3pt}f(x).$$
Obviously, $r\le\min\{m,n\}$ and $\lambda_{1}(x)=\cdots=\lambda_{m-r}(x)=0.$
At the point $x$ consider an orthonormal basis $\{\alpha_{1},\dots,\alpha_{m-r};\alpha_{m-r+1},
\dots,\alpha_{m}\}$ with respect to ${\operatorname{g}_M}$ which diagonalizes $f^*{\operatorname{g}_N}$. Moreover, at
$f(x)$ consider an orthonormal basis $\{\beta_{1},\dots,\beta_{n-r};\beta_{n-r+1},\dots,\beta_{n}\}$
with respect to ${\operatorname{g}_N}$ such that
$$\operatorname{d}\hspace{-3pt}f(\alpha_{i})=\lambda_{i}(x)\beta_{n-m+i},$$
for any $i\in\{m-r+1,\dots,m\}$. The above procedure is called the \textit{singular decomposition}
of the differential $\operatorname{d}\hspace{-3pt}f$.
Now we are going to define a special basis for the tangent and the normal space of the graph
in terms of the singular values. The vectors
\begin{equation}
e_{i}:=\left\{
\begin{array}{ll}
\alpha _{i}, & 1\le i\le m-r,\\
& \\
\frac{1}{\sqrt{1+\lambda _{i}^{2}\left( x\right) }}\left( \alpha
_{i}\oplus \lambda _{i}\left(x\right) \beta _{n-m+i}\right) , & m-r+1\leq
i\leq m,
\end{array}
\right.\label{tangent}
\end{equation}
form an orthonormal basis with respect to the metric ${\operatorname{g}_{M\times N}}$ of the tangent space $\operatorname{d}\hspace{-3pt}F\left(T_{x}M\right)$ of the graph $\Gamma(f)$ at
$x$. Moreover, the vectors
\begin{equation}
\xi_{i}:=\left\{
\begin{array}{ll}
\beta _{i}, & 1\leq i\leq n-r,\\
& \\
\frac{1}{\sqrt{1+\lambda _{i+m-n}^{2}\left( x\right) }}\left( -\lambda
_{i+m-n}(x)\alpha _{i+m-n}\oplus \beta _{i}\right) , & n-r+1\leq i\leq n, \\
\end{array}
\right.\label{normal}
\end{equation}
give an orthonormal basis with respect to ${\operatorname{g}_{M\times N}}$ of the normal space $\mathcal{N}_{x}M$ of the
graph $\Gamma(f)$ at the point $f(x)$. From the formulas above, we deduce that
\begin{equation}
{\operatorname{s}_{M\times N}}(e_{i},e_{j})=\frac{1-\lambda^{2}_{i}(x)}{1+\lambda^{2}_{i}(x)}\delta_{ij},\quad 1\le i,j\le m.
\end{equation}
Consequently, the eigenvalues of the $2$-tensor $\operatorname{s}$ with respect to $\operatorname{g}$, are
$$\frac{1-\lambda^{2}_{1}(x)}{1+\lambda^{2}_{1}(x)}\ge\cdots\ge\frac{1-\lambda^{2}_{m-1}(x)}{1+\lambda^{2}_{m-1}(x)}
\ge\frac{1-\lambda^{2}_{m}(x)}{1+\lambda^{2}_{m}(x)}.$$
Moreover,
\begin{eqnarray}
\hspace{-.5cm}
{\operatorname{s}_{M\times N}}(\xi_{i},\xi_{j})
&=&\begin{cases}
\displaystyle
-\delta_{ij},&\,1\le i\le n-r\\[4pt]\displaystyle
-\frac{1-\lambda^{2}_{i+m-n}(x)}{1+\lambda^{2}_{i+m-n}(x)}\delta_{ij},&\, n-r+1\le i\le n.
\end{cases}\label{normal}
\end{eqnarray}
and
\begin{equation}
{\operatorname{s}_{M\times N}}(e_{m-r+i},\xi_{n-r+j})=-\frac{2\lambda_{m-r+i}(x)}{1+\lambda^{2}_{m-r+i}(x)}\delta_{ij},\quad 1\le i,j\le r.\label{mixed}
\end{equation}
\subsection{Area decreasing maps}
Recall that a map $f:M\to N$ is \textit{weakly area decreasing} if
$\|\Lambda^{2}\operatorname{d}\hspace{-3pt}f\|\le 1$ and \textit{strictly area decreasing} if $\|\Lambda^{2}\operatorname{d}\hspace{-3pt}f\|<1$.
The above notions are expressed in terms of the singular values by the inequalities
$$\lambda_{i}^2(x)\lambda_{j}^2(x)\le1\quad\text{and}\quad\lambda_{i}^2(x)\lambda_{j}^2(x)<1,$$
for any $1\le i<j\le m$ and $x\in M$, respectively. On the other hand, the sum of two eigenvalues of
the tensor $\operatorname{s}$ with respect to $\operatorname{g}$ equals
$$\frac{1-\lambda^{2}_{i}}{1+\lambda^{2}_{i}}+\frac{1-\lambda^{2}_{j}}{1+\lambda^{2}_{j}}=
\frac{2(1-\lambda^{2}_{i}\lambda^{2}_{j})}{(1+\lambda^{2}_{i})(1+\lambda^{2}_{j})}.$$
Hence, the strictly area-decreasing property of the map $f$ is equivalent to the $2$-\textit{positivity} of
the symmetric tensor $\operatorname{s}$.
From the algebraic point of view, the $2$-positivity of a symmetric tensor $\operatorname{T}\in\operatorname{Sym}(T^*M\otimes T^*M)$ can
be expressed as the convexity of another symmetric tensor $\operatorname{T}^{[2]}\in\operatorname{Sym}(\Lambda^{2}T^*M\otimes\Lambda^{2}T^*M)$.
Indeed, let $\operatorname{P}$ and $\operatorname{Q}$ be two symmetric $2$-tensors. Then, the map $\operatorname{P} \varowedge\operatorname{Q}$ given
by
\begin{eqnarray*}
(P\varowedge\operatorname{Q})(v_1\wedge w_1,v_2\wedge w_2)&=&\operatorname{P}(v_1,v_2)\operatorname{Q}(w_1,w_2)+\operatorname{P}(w_1,w_2)\operatorname{Q}(v_1,v_2) \\
&-&\operatorname{P}(w_1,v_2)\operatorname{Q}(v_1,w_2)-P(v_1,w_2)\operatorname{Q}(w_1,v_2)
\end{eqnarray*}
gives rise to an element of $\operatorname{Sym}(\Lambda^{2}T^*M\otimes\Lambda^{2}T^*M)$. The operator $\varowedge$ is
called the \textit{Kulkarni-Nomizu product}.
Now we assign to each symmetric $2$-tensor $\operatorname{T}\in\operatorname{Sym}(T^*M\otimes T^*M)$ an element $\operatorname{T}^{[2]}$ of the bundle
$\operatorname{Sym}(\Lambda^{2}T^*M\otimes\Lambda^{2}T^*M)$, by setting
$$\operatorname{T}^{[2]}:=\operatorname{T}\varowedge\operatorname{g}.$$
The Riemannian metric $\operatorname{G}$ of the bundle $\Lambda^{2}TM$ is related
to the Riemannian metric $\operatorname{g}$ on the manifold $M$ by the formula
$$\operatorname{G}=\tfrac{1}{2}\operatorname{g}\varowedge\operatorname{g}=\tfrac{1}{2}\operatorname{g}^{[2]}.$$
The relation between the eigenvalues of $\operatorname{T}$ and the eigenvalues of
$\operatorname{T}^{[2]}$ is explained in the following lemma:
\begin{lemma}
Suppose that $\operatorname{T}$ is a symmetric $2$-tensor with eigenvalues $\mu_{1}\le\cdots\le\mu_{m}$
and corresponding eigenvectors $\{v_1,\dots,v_{m}\}$ with
respect to
$\operatorname{g}$. Then the eigenvalues of the symmetric $2$-tensor $\operatorname{T}^{[2]}$ with respect to $\operatorname{G}$
are
$$\mu_{i}+\mu_{j},\quad 1\le i<j\le m,$$
with corresponding eigenvectors
$$v_{i}\wedge v_{j},\quad 1\le i<j\le m.$$
\end{lemma}
\subsection{A Bochner-Weitzenb\"{o}ck formula}
Our next goal is to compute the Laplacians of the tensors $\operatorname{s}$ and $\operatorname{s}^{[2]}$. The next computations
closely follow those for a similarly defined tensor in \cite{sw}.
In order to control the smallest eigenvalue of $\operatorname{s}$, let us define the symmetric $2$-tensor $$\operatorname{P}hi_c:=\operatorname{s}-\frac{1-c}{1+c}\operatorname{g},$$
where $c$ is a non-negative constant.
At first let us compute the covariant derivative of the tensor $\operatorname{P}hi_{c}$. Since $\nabla\hspace{-3pt}\operatorname{g}=0$ and $\nabla^{{\operatorname{g}_{M\times N}}}{\operatorname{s}_{M\times N}}=0$, we have
\begin{eqnarray*}
(\nabla_v\operatorname{P}hi_c)(u,w)&=&(\nabla_v\operatorname{s})(u,w) \\
&=&(\nabla^{{\operatorname{g}_{M\times N}}}_{\operatorname{d}\hspace{-3pt}F(v)}{\operatorname{s}_{M\times N}})\bigr(\operatorname{d}\hspace{-3pt}F(u),\operatorname{d}\hspace{-3pt}F(w)\bigl) \\
&&+{\operatorname{s}_{M\times N}}\bigl(A(v,u),\operatorname{d}\hspace{-3pt}F(w)\bigr)+{\operatorname{s}_{M\times N}}\bigl(\operatorname{d}\hspace{-3pt}F(u),A(v,w)\bigr) \\
&=&{\operatorname{s}_{M\times N}}\bigl(A(v,u),\operatorname{d}\hspace{-3pt}F(w)\bigr)+{\operatorname{s}_{M\times N}}\bigl(\operatorname{d}\hspace{-3pt}F(u),A(v,w)\bigr),
\end{eqnarray*}
for any tangent vectors $u,v,w\in TM$.
Now let us compute the Hessian of $\operatorname{P}hi_c$.
Differentiating once more gives
\begin{eqnarray}
&&\left(\nabla^2_{v_1,v_2}\operatorname{P}hi_c\right)\hspace{-4pt}(u,w)\nonumber\\
&&\quad={\operatorname{s}_{M\times N}}\bigl((\nabla_{v_1}A)(v_2,u),\operatorname{d}\hspace{-3pt}F(w)\bigr)
+{\operatorname{s}_{M\times N}}\bigl(A(v_2,u),A(v_1,w)\bigr)\nonumber\\
&&\quad\quad+{\operatorname{s}_{M\times N}}\bigl(A(v_1,u),A(v_2,w)\bigr)+{\operatorname{s}_{M\times N}}\bigl(\operatorname{d}\hspace{-3pt}F(u),(\nabla_{v_1}A)(v_2,w)\bigr),
\nonumber
\end{eqnarray}
for any tangent vectors $v_1,v_2,u,w\in TM$. Applying Codazzi's equation (\ref{codazzi}) and exploiting the
symmetry of $A$ and ${\operatorname{s}_{M\times N}}$, we derive
\begin{eqnarray}
&&\left(\nabla^2_{v_1,v_2}\operatorname{P}hi_c\right)\hspace{-4pt}(u,w)\nonumber\\
&&\quad={\operatorname{s}_{M\times N}}\bigl((\nabla_{u}A)(v_1,v_2)+{\operatorname{R}_{M\times N}}\bigl(\operatorname{d}\hspace{-3pt}F(v_1),\operatorname{d}\hspace{-3pt}F(u)\bigr)\operatorname{d}\hspace{-3pt}F(v_2),\operatorname{d}\hspace{-3pt}F(w)\bigr)\nonumber\\
&&\quad+{\operatorname{s}_{M\times N}}\bigl((\nabla_{w}A)(v_1,v_2)+{\operatorname{R}_{M\times N}}\bigl(\operatorname{d}\hspace{-3pt}F(v_1),\operatorname{d}\hspace{-3pt}F(w)\bigr)\operatorname{d}\hspace{-3pt}F(v_2),\operatorname{d}\hspace{-3pt}F(u)\bigr)\nonumber\\
&&\quad+{\operatorname{s}_{M\times N}}\bigl(A(v_1,u),A(v_2,w)\bigr)+{\operatorname{s}_{M\times N}}\bigl(A(v_1,w),A(v_2,u)\bigr)\nonumber\\
&&\quad-\operatorname{s}\bigl(\operatorname{R}(v_1,u)v_2,w\bigr)-\operatorname{s}\bigl(\operatorname{R}(v_1,w)v_2,u\bigr).\nonumber
\end{eqnarray}
The decomposition of the tensors ${\operatorname{s}_{M\times N}}$ and ${\operatorname{R}_{M\times N}}$, implies
\begin{eqnarray}
&&{\operatorname{s}_{M\times N}}\Bigl({\operatorname{R}_{M\times N}}\bigl(\operatorname{d}\hspace{-3pt}F(v_1),\operatorname{d}\hspace{-3pt}F(u)\bigr)\operatorname{d}\hspace{-3pt}F(v_2),\operatorname{d}\hspace{-3pt}F(w)\Bigr)\nonumber\\
&&\quad=(\pi_M^*{\operatorname{g}_M})\bigl({\operatorname{R}_{M\times N}}\bigl(\operatorname{d}\hspace{-3pt}F(v_1),\operatorname{d}\hspace{-3pt}F(u)\bigr)\operatorname{d}\hspace{-3pt}F(v_2),\operatorname{d}\hspace{-3pt}F(w)\bigr)\nonumber\\
&&\quad\quad-(\pi_N^*{\operatorname{g}_N})\bigl({\operatorname{R}_{M\times N}}\bigl(\operatorname{d}\hspace{-3pt}F(v_1),\operatorname{d}\hspace{-3pt}F(u)\bigr)\operatorname{d}\hspace{-3pt}F(v_2),\operatorname{d}\hspace{-3pt}F(w)\bigr)\nonumber\\
&&\quad={\operatorname{g}_M}\bigl({\operatorname{R}_M}\bigl(v_1,u\bigr)v_2,w\bigr)-{\operatorname{g}_N}\bigl({\operatorname{R}_N}\bigl(\operatorname{d}\hspace{-3pt}f(v_1),\operatorname{d}\hspace{-3pt}f(u)\bigr)\operatorname{d}\hspace{-3pt}f(v_2),\operatorname{d}\hspace{-3pt}f(w)\bigr)\nonumber\\
&&\quad={\operatorname{R}_N}\bigl(\operatorname{d}\hspace{-3pt}f(v_1),\operatorname{d}\hspace{-3pt}f(u),\operatorname{d}\hspace{-3pt}f(v_2),\operatorname{d}\hspace{-3pt}f(w)\bigr)-{\operatorname{R}_M}\bigl(v_1,u,v_2,w\bigr)\,.\nonumber
\end{eqnarray}
Gau\ss' equation (\ref{gauss}) gives
\begin{eqnarray}
&-&\operatorname{s}\bigl(\operatorname{R}(v_1,u)v_2,w\bigr)\nonumber\\
&&=-\operatorname{P}hi_c\bigl(\operatorname{R}(v_1,u)v_2,w\bigr)-\frac{1-c}{1+c}\operatorname{g}\bigl(\operatorname{R}(v_1,u)v_2,w\bigr)\nonumber\\
&&=-\operatorname{P}hi_c\bigl(\operatorname{R}(v_1,u)v_2,w\bigr)+\frac{1-c}{1+c}\operatorname{R}\bigl(v_1,u,v_2,w\bigr)\nonumber\\
&&=-\operatorname{P}hi_c\bigl(\operatorname{R}(v_1,u)v_2,w\bigr)\nonumber\\
&&\,\,+\frac{1-c}{1+c}\left\{{\operatorname{g}_{M\times N}}(A(v_1,v_2),A(u,w))-{\operatorname{g}_{M\times N}}(A(v_1,w),A(v_2,u))\right\}\nonumber\\
&&\,\,+\frac{1-c}{1+c}{\operatorname{R}_M}(v_1,u,v_2,w)+\frac{1-c}{1+c}{\operatorname{R}_N}(\operatorname{d}\hspace{-3pt}f(v_1),\operatorname{d}\hspace{-3pt}f(u),\operatorname{d}\hspace{-3pt}f(v_2),\operatorname{d}\hspace{-3pt}f(w)).\nonumber
\end{eqnarray}
In the sequel consider any local orthonormal frame field $\{e_1,\dots,e_m\}$ with respect to the induced
metric $\operatorname{g}$ on $M$. Then, taking a trace, we derive the Laplacian of the tensor $\operatorname{P}hi_{c}$.
Let us now summarize the previous computations in the next lemma:
\begin{lemma}\label{laplacephi}
For any smooth map $f:M\to N$, the symmetric tensor $\operatorname{P}hi_c$ satisfies the identity
\begin{eqnarray*}
\bigl(\Delta\operatorname{P}hi_c\bigr)(v,w)&=&{\operatorname{s}_{M\times N}}\bigl(\nabla_v\vec{H},\operatorname{d}\hspace{-3pt}F(w)\bigr)+{\operatorname{s}_{M\times N}}(\nabla_w\vec{H},\operatorname{d}\hspace{-3pt}F(v)\bigr)\\
&&+2\frac{1-c}{1+c}{\operatorname{g}_{M\times N}}\bigl(\vec{H},A(v,w)\bigr) \\
&&+\operatorname{P}hi_c\bigl(\operatorname{Ric}v,w\bigr)+\operatorname{P}hi_c\bigl(\operatorname{Ric}w,v\bigr) \\
&&+2\sum_{k=1}^m\bigl({\operatorname{s}_{M\times N}}-\frac{1-c}{1+c}{\operatorname{g}_{M\times N}}\bigr)\bigl(A(e_k,v),A(e_k,w)\bigr) \\
&&+\frac{4}{1+c}\sum_{k=1}^m\Bigl(f^*{\operatorname{R}_N}(e_k,v,e_k,w)-c{\operatorname{R}_M}(e_k,v,e_k,w)\Bigr),
\end{eqnarray*}
where
$$\operatorname{Ric}v:=-\sum_{k=1}^m\operatorname{R}(e_k,v)e_k$$
is the Ricci operator on $(M,\operatorname{g})$ and $\{e_1,\dots,e_m\}$ is any orthonormal
frame with respect to the induced metric $\operatorname{g}$.
\end{lemma}
The expressions of the covariant derivative and the Laplacian of a symmetric
$2$-tensor $\operatorname{T}^{[2]}$ are given in the following Lemma. The proof follows by a straightforward
computation and for that reason will be omitted.
\begin{lemma}\label{laplacesind}
Any symmetric $2$-tensor $\operatorname{T}$ satisfies the identities,
\begin{enumerate}[(i)]
\item
$\nabla_{v}\operatorname{T}^{[2]}=\left(\nabla_{v}\operatorname{T}\right)^{[2]},$
\item
$\nabla^2_{v,v}\operatorname{T}^{[2]}=\left(\nabla^2_{v,v}\operatorname{T}\right)^{[2]},$
\item
$\Delta\operatorname{T}^{[2]}=\left(\Delta\operatorname{T}\right)^{[2]},$
\end{enumerate}
for any vector $v$ on the manifold $M$.
\end{lemma}
\subsection{Proofs of the Theorems \ref{thmD}, \ref{thmC} and \ref{thmE}}
We will first show the following lemma.
\begin{lemma}\label{lem cd}
Let $f:M\to N$ be weakly length decreasing. Suppose $\{e_1,\dots,e_m\}$ is orthonormal with
respect to $\operatorname{g}$ such that it diagonalizes the tensor $\operatorname{s}$. Then for any $e_l$ we have
\begin{eqnarray*}
&&2\sum_{k=1}^m\Bigl({\operatorname{R}_M}(e_k,e_l,e_k,e_l)-f^*{\operatorname{R}_N}(e_k,e_l,e_k,e_l)\Bigr) \\
&=&
2\sum_{k\neq l}\frac{\lambda_k^2}{1+\lambda_k^2}\Bigl\{\bigl(\sigma-\sigma_N(\operatorname{d}\hspace{-3pt}f(e_k)\wedge\operatorname{d}\hspace{-3pt}f(e_l))\bigr)
f^*{\operatorname{g}_N}(e_l,e_l)\\
&&\hspace{4cm}+\sigma\bigl({\operatorname{g}_M}(e_l,e_l)-f^*{\operatorname{g}_N}(e_l,e_l)\bigr)\Bigr\}\\
&&+\operatorname{Ric}_M(e_l,e_l)-(m-1)\sigma{\operatorname{g}_M}(e_l,e_l)\\
&&+\sum_{k\neq l}\frac{1-\lambda_k^2}{1+\lambda_k^2}\Bigl(\sigma_M(e_k\wedge e_l)+\sigma\Bigr)
{\operatorname{g}_M}(e_l,e_l),
\end{eqnarray*}
where $\operatorname{Ric}_M$ denotes the Ricci curvature with respect to ${\operatorname{g}_M}$, $\sigma_M(e_k\wedge e_l)$
and $\sigma_N(\operatorname{d}\hspace{-3pt}f(e_k)\wedge \operatorname{d}\hspace{-3pt}f(e_l))$ are the sectional
curvatures of the planes $e_k\wedge e_l$ on $(M,{\operatorname{g}_M})$ and $\operatorname{d}\hspace{-3pt}f(e_k)\wedge\operatorname{d}\hspace{-3pt}f (e_l)$ on $(N,{\operatorname{g}_N})$ respectively.
\end{lemma}
\begin{proof}
In terms of the singular
values we get
$$s(e_k,e_k)={\operatorname{g}_M}(e_k,e_k)-f^*{\operatorname{g}_N}(e_k,e_k)=\frac{1-\lambda_k^2}{1+\lambda_k^2}.$$
Since
$$1=\operatorname{g}(e_k,e_k)={\operatorname{g}_M}(e_k,e_k)+f^*{\operatorname{g}_N}(e_k,e_k)$$
we derive
$${\operatorname{g}_M}(e_k,e_k)=\frac{1}{1+\lambda_k^2},\quad f^*{\operatorname{g}_N}(e_k,e_k)=\frac{\lambda_k^2}{1+\lambda_k^2}$$
and
$$2{\operatorname{g}_M}(e_k,e_k)=\frac{1-\lambda_k^2}{1+\lambda_k^2}+1,\quad -2f^*{\operatorname{g}_N}(e_k,e_k)=\frac{1-\lambda_k^2}{1+\lambda_k^2}-1.$$
Note also that for any $k \neq l$ we have
$${\operatorname{g}_M}(e_k,e_l)=f^*{\operatorname{g}_N}(e_k,e_l)=\operatorname{g}(e_k,e_l)=0.$$
We compute
\begin{eqnarray*}
&&2\sum_{k=1}^m\Bigl({\operatorname{R}_M}(e_k,e_l,e_k,e_l)-f^*{\operatorname{R}_N}(e_k,e_l,e_k,e_l)\Bigr) \\
&=&2\sum_{k\neq l}\sigma_M(e_k\wedge e_l){\operatorname{g}_M}(e_k,e_k){\operatorname{g}_M}(e_l,e_l)\\
&&-2\sum_{k\neq l}\sigma_N(\operatorname{d}\hspace{-3pt}f(e_k)\wedge \operatorname{d}\hspace{-3pt}f(e_l))f^*{\operatorname{g}_N}(e_k,e_k)f^*{\operatorname{g}_N}(e_l,e_l).
\end{eqnarray*}
Hence the formula for ${\operatorname{g}_M}(e_k,e_k)$ implies
\begin{eqnarray*}
&&2\sum_{k=1}^m\Bigl({\operatorname{R}_M}(e_k,e_l,e_k,e_l)-f^*{\operatorname{R}_N}(e_k,e_l,e_k,e_l)\Bigr) \\
&=&\sum_{k\neq l}\left(1+\frac{1-\lambda_k^2}{1+\lambda_k^2}\right)\sigma_M(e_k\wedge e_l){\operatorname{g}_M}(e_l,e_l)\\
&&+2\sum_{k\neq l}f^*{\operatorname{g}_N}(e_k,e_k)\Bigl\{\bigl(\sigma-\sigma_N(\operatorname{d}\hspace{-3pt}f(e_k)\wedge\operatorname{d}\hspace{-3pt}f(e_l))\bigr)
f^*{\operatorname{g}_N}(e_l,e_l)\\
&&\hspace{4cm}+\sigma\bigl({\operatorname{g}_M}(e_l,e_l)-f^*{\operatorname{g}_N}(e_l,e_l)\bigr)\Bigr\}\\
&&-2\sigma\sum_{k\neq l}f^*{\operatorname{g}_N}(e_k,e_k){\operatorname{g}_M}(e_l,e_l)\\
&=&\sum_{k\neq l}\left(1+\frac{1-\lambda_k^2}{1+\lambda_k^2}\right)\sigma_M(e_k\wedge e_l){\operatorname{g}_M}(e_l,e_l)\\
&&+2\sum_{k\neq l}f^*{\operatorname{g}_N}(e_k,e_k)\Bigl\{\bigl(\sigma-\sigma_N(\operatorname{d}\hspace{-3pt}f(e_k)\wedge\operatorname{d}\hspace{-3pt}f(e_l))\bigr)
f^*{\operatorname{g}_N}(e_l,e_l)\\
&&\hspace{4cm}+\sigma\bigl({\operatorname{g}_M}(e_l,e_l)-f^*{\operatorname{g}_N}(e_l,e_l)\bigr)\Bigr\}\\
&&+\sigma\sum_{k\neq l}\left(\frac{1-\lambda_k^2}{1+\lambda_k^2}-1\right){\operatorname{g}_M}(e_l,e_l).
\end{eqnarray*}
We may then continue to get
\begin{eqnarray*}
&&2\sum_{k=1}^m\Bigl({\operatorname{R}_M}(e_k,e_l,e_k,e_l)-f^*{\operatorname{R}_N}(e_k,e_l,e_k,e_l)\Bigr) \\
&=&
2\sum_{k\neq l}f^*{\operatorname{g}_N}(e_k,e_k)\Bigl\{\bigl(\sigma-\sigma_N(\operatorname{d}\hspace{-3pt}f(e_k)\wedge\operatorname{d}\hspace{-3pt}f(e_l))\bigr)
f^*{\operatorname{g}_N}(e_l,e_l)\\
&&\hspace{4cm}+\sigma\bigl({\operatorname{g}_M}(e_l,e_l)-f^*{\operatorname{g}_N}(e_l,e_l)\bigr)\Bigr\}\\
&&+\operatorname{Ric}_M(e_l,e_l)-(m-1)\sigma\,{\operatorname{g}_M}(e_l,e_l)\\
&&+\sum_{k\neq l}\frac{1-\lambda_k^2}{1+\lambda_k^2}\Bigl(\sigma_M(e_k\wedge e_l)+\sigma\Bigr)
{\operatorname{g}_M}(e_l,e_l).
\end{eqnarray*}
This completes the proof.
\end{proof}
{\bf Proof of Theorem \ref{thmD}.}
Suppose that $f:M\to N$ is weakly length decreasing.
Then the tensor $\operatorname{s}$ satisfies
$$\operatorname{s}={\operatorname{g}_M}-f^*{\operatorname{g}_N}\ge 0.$$
In case $\operatorname{s} >0$ the map $f$ is also
strictly area decreasing. Thus in such a case the statement follows from
Theorem \ref{thmC} which we will prove further below. It remains to show that $\operatorname{s}$
vanishes identically, if $\operatorname{s}$ admits a null-eigenvalue somewhere.
{\bf Claim 1.} {\it The tensor $\operatorname{s}$ has a null-eigenvalue everywhere on $M$, if this is
the case in at least one point $x\in M$.}
Since $\operatorname{s}=\operatorname{P}hi_1$, from Lemma \ref{laplacephi} we get
$$\Delta \operatorname{s}+\operatorname{P}si(\operatorname{s})=0,$$
where
\begin{eqnarray*}
\bigl(\operatorname{P}si(\vartheta)\bigr)(v,w)&=&-\vartheta\bigl(\operatorname{Ric}v,w\bigr)-\vartheta\bigl(\operatorname{Ric}w,v\bigr) \\
&&-2\sum_{k=1}^m{\operatorname{s}_{M\times N}}\bigl(A(e_k,v),A(e_k,w)\bigr) \\
&&+2\sum_{k=1}^{m}\Bigl({\operatorname{R}_M}(e_k,v,e_k,w)-f^*{\operatorname{R}_N}(e_k,v,e_k,w)\Bigr).
\end{eqnarray*}
Let $v$ be a null-eigenvector of the symmetric, positive semi-definite tensor $\vartheta$.
Since $f$ is weakly length decreasing, equation (\ref{normal}) shows that ${\operatorname{s}_{M\times N}}$ is non-positive
definite on the normal bundle of the graph. Hence
\begin{eqnarray*}
\bigl(\operatorname{P}si(\vartheta)\bigr)(v,v)\ge
2\sum_{k=1}^{m}\Bigl({\operatorname{R}_M}(e_k,v,e_k,v)-f^*{\operatorname{R}_N}(e_k,v,e_k,v)\Bigr)\ge 0,
\end{eqnarray*}
where we have used Lemma \ref{lem cd} and the curvature assumptions on $(M,{\operatorname{g}_M})$, $(N,{\operatorname{g}_N})$
respectively. This shows that $\operatorname{P}si$ satisfies
the null-eigenvector condition and Claim 1 follows from the strong maximum principle
in Theorem \ref{mp2}.
{\bf Claim 2.} {\it If $\operatorname{s}$ admits a null-eigenvalue at some point $x\in M$, then $\operatorname{s}$ vanishes
at $x$.}
We already know that the tensor $\operatorname{s}$ admits a null-eigenvalue everywhere on $M$.
Since $\operatorname{s}\ge 0$ we may then apply the test criterion Theorem \ref{test} to the tensor $\operatorname{s}$
at an arbitrary point $x\in M$.
At $x$ consider a basis $\{e_1,\dots,e_m\}$, orthonormal with respect to $\operatorname{g}$ consisting of eigenvectors of $\operatorname{s}$,
such that $v:=e_m$ is a null-eigenvector of $\operatorname{s}$ and $\lambda_m^2=1$.
From Lemma \ref{lem cd}, we conclude
\begin{eqnarray}
0&\ge&\bigl(\operatorname{P}si(\operatorname{s})\bigr)(e_m,e_m)\nonumber\\
&\ge&2\sum_{k\neq m}\frac{\lambda_k^2}{1+\lambda_k^2}\Bigl\{\bigl(\operatorname{u}nderbrace{\sigma-\sigma_N(\operatorname{d}\hspace{-3pt}f(e_k)\wedge\operatorname{d}\hspace{-3pt}f(e_m))}_{\ge 0}\bigr)
\operatorname{u}nderbrace{f^*{\operatorname{g}_N}(e_l,e_l)}_{\ge 0}\nonumber\\
&&\hspace{4cm}+\sigma\bigl(\operatorname{u}nderbrace{{\operatorname{g}_M}(e_m,e_m)-f^*{\operatorname{g}_N}(e_m,e_m)}_{= 0}\bigr)\Bigr\}\nonumber\\
&&+\operatorname{u}nderbrace{\operatorname{Ric}_M(e_m,e_m)-(m-1)\sigma\,{\operatorname{g}_M}(e_m,e_m)}_{\ge 0}\nonumber\\
&&+\sum_{k\neq m}\operatorname{u}nderbrace{\frac{1-\lambda_k^2}{1+\lambda_k^2}}_{\ge 0}\Bigl(\operatorname{u}nderbrace{\sigma_M(e_k\wedge e_m)+\sigma}_{>0}\Bigr)\operatorname{u}nderbrace{{\operatorname{g}_M}(e_m,e_m)}_{=\frac{1}{2}}=0,\label{eq vanish}
\end{eqnarray}
because the curvature assumptions on $(M,{\operatorname{g}_M})$ and $(N,{\operatorname{g}_N})$ imply that the
right hand side is a sum of non-negative terms and thus we conclude that all of them must vanish.
In particular,
$$\sigma_M(e_k\wedge e_m)+\sigma>0$$
implies $\lambda_k^2=1$ for all $k$.
Now we can finish the proof of Theorem \ref{thmD}. Claim 1 and 2 imply that a weakly
length decreasing map $f$ which is not strictly length decreasing must be an isometric immersion.
Once we know that all tangent vectors at $x$ are
null-eigenvectors of $\operatorname{s}$, we may choose $e_m$ in (\ref{eq vanish}) arbitrarily. Then
$$\operatorname{Ric}_M(v,v)=(m-1)\sigma{\operatorname{g}_M}(v,v)$$
and
$$\sigma=\sigma_N\bigl(\operatorname{d}\hspace{-3pt}f(v),\operatorname{d}\hspace{-3pt}f (w)\bigr)$$
for all linearly independent vectors $v,w\in T_xM$. This completes the proof of Theorem \ref{thmD}.
$\square$
{\bf Proof of Theorem \ref{thmC}.}
Since the manifold $M$ is compact, there exists a point $x_{0}$ where the smallest
eigenvalue of $\operatorname{s}^{[2]}$ with respect to the metric $\operatorname{G}$ attains its minimum.
Let us denote this value by $\rho_{0}$. Note that in terms of the singular values
$$\lambda_1^2\le\cdots\le\lambda_m^2$$
we must have
$$\rho_{0}=\frac{1-\lambda^{2}_{m}(x_0)}{1+\lambda^{2}_{m}(x_0)}
+\frac{1-\lambda^{2}_{m-1}(x_0)}{1+\lambda^{2}_{m-1}(x_0)}\ge 0.$$
For simplicity we set
$$\kappa:=\lambda^{2}_{m-1}(x_0)\quad\text{and}\quad\mu:=\lambda^2_{m}(x_0).$$
Hence,
$$\rho_0=2\frac{1-\kappa\mu}{(1+\kappa)(1+\mu)}\,.$$
{\bf Claim 3.}\textit{ If $\mu=0$, then the map $f$ is constant.}
In this case we have $\rho_0=2$. Because $\rho_0$ is the minimum of the smallest eigenvalue
of the symmetric tensor $\operatorname{s}^{[2]}$, we obtain
\begin{equation*}
1\le \frac{1-\lambda^2_{i}(x)\lambda^2_{j}(x)}{(1+\lambda^2_i(x))(1+\lambda^2_j(x))},
\end{equation*}
for any $x\in M$ and $1\le i<j\le m$. From the above inequality one can readily see that
all the singular values of $f$ vanish everywhere. Thus, in this case $f$ is constant.
This completes the proof of Claim $3$.
Since we are assuming that $f$ is weakly area decreasing, we deduce that $\kappa\mu\le1$.
Consider now the symmetric $2$-tensor
$$\operatorname{P}hi:=\operatorname{P}hi_{\frac{2-\rho_{0}}{2+\rho_0}}=\operatorname{s}-\frac{\rho_0}{2}\operatorname{g}.$$
According to Lemma \ref{laplacesind},
$$\Delta\operatorname{P}hi^{[2]}=\left(\Delta\operatorname{P}hi\right)^{[2]}.$$
At $x_0$ consider an orthonormal bases $\{e_1,\dots,e_m\}$
with respect to $\operatorname{g}$ such that
$\operatorname{s}$ becomes diagonal and
$$\operatorname{s}(e_k,e_k)=\frac{1-\lambda_k^2}{1+\lambda_k^2}.$$
According to Theorem \ref{test},
we obtain
\begin{eqnarray*}
0&\le&\left(\Delta\operatorname{P}hi\right)^{[2]}\left( e_{m-1}\wedge e_m,
e_{m-1}\wedge e_m\right) \\
&=&\left(\Delta\operatorname{P}hi\right)\left( e_{m-1}, e_{m-1}\right)
+\left(\Delta\operatorname{P}hi\right)\left( e_m, e_m\right).
\end{eqnarray*}
In view of Lemma \ref{laplacephi}, we deduce that
\begin{eqnarray}
0&\le&2\operatorname{P}hi(\Ric e_{m-1}, e_{m-1})+2\operatorname{P}hi(\Ric e_m, e_m) \nonumber\\
&&+2\sum^{m}_{k=1}\Bigl({\operatorname{s}_{M\times N}}-\frac{\rho_0}{2}{\operatorname{g}_{M\times N}}\Bigr)(A( e_k, e_{m-1}),A( e_k, e_{m-1}))\nonumber \\
&&+2\sum^{m}_{k=1}\Bigl({\operatorname{s}_{M\times N}}-\frac{\rho_0}{2}{\operatorname{g}_{M\times N}}\Bigr)(A( e_k, e_m),A( e_k, e_m))\nonumber \\
&&+(2+\rho_0)\sum^{m}_{k=1}f^*{\operatorname{R}_N}( e_k,e_{m-1}, e_k, e_{m-1}) \nonumber \\
&&-(2-\rho_0)\sum^{m}_{k=1}{\operatorname{R}_M}( e_k, e_{m-1}, e_k, e_{m-1}) \nonumber \\
&&+(2+\rho_0)\sum^{m}_{k=1}f^*{\operatorname{R}_N}(e_k, e_m, e_k, e_m) \nonumber \\
&&-(2-\rho_0)\sum^{m}_{k=1}{\operatorname{R}_M}( e_k, e_m, e_k, e_m).\label{inequality5}
\end{eqnarray}
Because $ e_{m}$ is an eigenvector of $\operatorname{s}$ with respect to $\operatorname{g}$, we have
\begin{eqnarray*}
\operatorname{P}hi(\Ric e_{m}, e_{m})=\frac{\kappa-\mu}{(1+\kappa)(1+\mu)}\operatorname{g}(\Ric e_{m}, e_{m}).
\end{eqnarray*}
From the Gau{\ss} equation (\ref{gauss}) and the minimality of the graph, we obtain that
\begin{eqnarray*}
\operatorname{g}(\Ric e_{m}, e_{m})&=&\sum^{m}_{k=1}{\operatorname{R}_M}( e_k, e_{m}, e_k, e_{m}) \\
&&+\sum^{m}_{k=1}f^*{\operatorname{R}_N}(e_k,e_m,e_k,e_m) \\
&&-\sum^{m}_{k=1}{\operatorname{g}_{M\times N}}(A( e_k, e_{m}),A( e_k, e_{m})).
\end{eqnarray*}
Hence,
\begin{eqnarray}
\operatorname{P}hi(\Ric e_m, e_m)&=&\tfrac{\kappa-\mu}{(1+\kappa)(1+\mu)}\textstyle\sum\limits^{m}_{k=1}
{\operatorname{R}_M}( e_k, e_m, e_k, e_m) \label{phi1}\\
&+&\tfrac{\kappa-\mu}{(1+\kappa)(1+\mu)}\textstyle\sum\limits^{m}_{k=1}
f^*{\operatorname{R}_N}( e_k, e_m, e_k, e_m) \nonumber \\
&-&\tfrac{\kappa-\mu}{(1+\kappa)(1+\mu)}\textstyle\sum\limits^{m}_{k=1}{\operatorname{g}_{M\times N}}(A( e_k, e_m),A( e_k, e_m))\nonumber.
\end{eqnarray}
Similarly,
\begin{eqnarray}
\operatorname{P}hi(\Ric e_{m-1}, e_{m-1})&=&\tfrac{\mu-\kappa}{(1+\kappa)(1+\mu)}\textstyle\sum\limits^{m}_{k=1}{\operatorname{R}_M}
( e_k, e_{m-1}, e_k, e_{m-1})\label{phi2} \\
&+&\tfrac{\mu-\kappa}{(1+\kappa)(1+\mu)}\textstyle\sum\limits^{m}_{k=1}f^*{\operatorname{R}_N}
( e_k, e_{m-1}, e_k, e_{m-1})\nonumber \\
&-&\tfrac{\mu-\kappa}{(1+\kappa)(1+\mu)}\textstyle\sum\limits^{m}_{k=1}{\operatorname{g}_{M\times N}}(A(e_k, e_{m-1}),A( e_k, e_{m-1})).\nonumber
\end{eqnarray}
In view of (\ref{phi1}) and (\ref{phi2}), the inequality (\ref{inequality5}) can be now written equivalently in
the form
\begin{eqnarray}
0&\le&\sum^{m}_{k=1}({\operatorname{s}_{M\times N}}-\frac{1-\mu}{1+\mu}{\operatorname{g}_{M\times N}})(A( e_k, e_m),A( e_k, e_m))\nonumber \\
&&+\sum^{m}_{k=1}({\operatorname{s}_{M\times N}}-\frac{1-\kappa}{1+\kappa}{\operatorname{g}_{M\times N}})(A( e_k, e_{m-1}),A( e_k, e_{m-1}))\nonumber \\
&&+\frac{2}{1+\mu}\sum^{m}_{k=1}\left(f^*{\operatorname{R}_N}-\mu{\operatorname{R}_M}\right)( e_k, e_m, e_k, e_m)\nonumber \\
&&+\frac{2}{1+\kappa}\sum^{m}_{k=1}\left(f^*{\operatorname{R}_N}-\kappa{\operatorname{R}_M}\right)( e_k, e_{m-1}, e_k, e_{m-1}).\label{inequality6}
\end{eqnarray}
\textbf{Claim 4.}\textit{ The sum $\mathcal{A}$ of the first two terms on the right hand side of inequality $(\ref{inequality6})$
is non-positive.}
Indeed, if $\mu=0$, then $f$ is constant by Claim $3$ and thus $\mathcal{A}=0$. So, let us consider the case where $\mu>0$.
From Theorem \ref{test} again, we have
\begin{eqnarray}
0&=&(\nabla_{e_k}(\operatorname{s}^{[2]}-\rho_0\operatorname{G}))( e_m\wedge e_{m-1}, e_m\wedge e_{m-1}) \nonumber\\
&=&2(\nabla_{ e_k}\operatorname{s})( e_m, e_m)+2(\nabla_{ e_k}\operatorname{s})( e_{m-1}, e_{m-1})\nonumber\\
&=&4{\operatorname{s}_{M\times N}}(A( e_k, e_m), e_m)+4{\operatorname{s}_{M\times N}}(A( e_k, e_{m-1}), e_{m-1})\label{eq zero}
\end{eqnarray}
for any $k$. Since $\dim(N)=1$ implies that $\operatorname{rank}(\operatorname{d}\hspace{-3pt}f)\le 1$, from (\ref{mixed}) we obtain
\begin{eqnarray*}
0&=&A_{\xi_n}(e_k,e_m){\operatorname{s}_{M\times N}}(\xi_n,e_m)+A_{\xi_n}(e_k,e_{m-1})\operatorname{u}nderbrace{{\operatorname{s}_{M\times N}}(\xi_n,e_{m-1})}_{=0}\\
&=&-2A_{\xi_n}(e_k,e_m)\operatorname{u}nderbrace{\frac{\sqrt{\mu}}{1+\mu}}_{>0},
\end{eqnarray*}
where here
$$A_{\xi}(v,w):={\operatorname{g}_N}(A(v,w),\xi),\quad v,w\in T_xM,$$
stands for the second fundamental form of the graph $\Gamma(f)$ in the normal direction $\xi$
and the normal basis $\{\xi_1,\dots,\xi_n\}$ is chosen as in (\ref{normal}).
Hence, by the weakly area decreasing property of $f$, we get
$$\mathcal{A}=-\sum_{k=1}^m\left(\frac{1-\mu}{1+\mu}+\frac{1-\kappa}{1+\kappa}\right)A_{\xi_n}^2(e_k,e_{m-1})\le 0.$$
In case $\dim(N)\ge 2$, from (\ref{normal}), (\ref{eq zero}) and the weakly area decreasing condition we obtain
\begin{eqnarray*}
\mathcal{A}&=&\sum^{m}_{k=1}({\operatorname{s}_{M\times N}}-\frac{1-\mu}{1+\mu}{\operatorname{g}_{M\times N}})(A( e_k, e_m),A( e_k, e_m)) \\
&&+\sum^{m}_{k=1}({\operatorname{s}_{M\times N}}-\frac{1-\kappa}{1+\kappa}{\operatorname{g}_{M\times N}})(A( e_k, e_{m-1}),A( e_k, e_{m-1})) \\
&\le&-2\frac{1-\mu}{1+\mu}\sum^{m}_{k=1}A^2_{\xi_n}( e_k, e_m)
-2\frac{1-\kappa}{1+\kappa}\sum^{m}_{k=1}A^2_{\xi_{n-1}}( e_k, e_{m-1}).
\end{eqnarray*}
In view of equations (\ref{eq zero}) and (\ref{mixed}), we have
\begin{eqnarray*}
0&=&{\operatorname{s}_{M\times N}}(A( e_k, e_m), e_m)+{\operatorname{s}_{M\times N}}(A( e_k, e_{m-1}), e_{m-1}) \\
&=&-2\frac{\sqrt{\mu}}{1+\mu}A_{\xi_n}( e_k, e_m)-2\frac{\sqrt{\kappa}}{1+\kappa}A_{\xi_{n-1}}( e_k, e_{m-1}).
\end{eqnarray*}
Hence,
$$A^2_{\xi_n}( e_k, e_m)=\frac{\kappa(1+\mu)^2}{\mu(1+\kappa)^2}A^2_{\xi_{n-1}}( e_k, e_{m-1}).$$
Because, $\kappa\le\mu$ and $\kappa\mu\le1$, we deduce that
$$\frac{\kappa(1+\mu)^2}{\mu(1+\kappa)^2}\le 1.$$
This proves our assertion. Now it is clear that the quantity $\mathcal{A}$ is always non-positive which
proves Claim $4$.
{\bf Claim 5.} \textit{The sum $\mathcal{B}$ of the last two terms on the right hand side
of inequality (\ref{inequality6}) is non-positive}.
We have,
\begin{eqnarray*}
\mathcal{B}&=&\frac{1}{1+\mu}\operatorname{u}nderbrace{\sum^{m}_{k=1}2\left(f^*{\operatorname{R}_N}-\mu{\operatorname{R}_M}\right)
( e_k, e_m, e_k, e_m)}_{=:\mathcal{B}_1} \nonumber \\
&&+\frac{1}{1+\kappa}\operatorname{u}nderbrace{\sum^{m}_{k=1}2\left(f^*{\operatorname{R}_N}-\kappa{\operatorname{R}_M}\right)
( e_k, e_{m-1}, e_k, e_{m-1})}_{=:\mathcal{B}_2}\nonumber
\end{eqnarray*}
From the identities (\ref{met2}) and (\ref{met4}), we deduce that
$${\operatorname{g}_M}=\frac{1}{2}(\operatorname{g}+\operatorname{s})\quad\text{and}\quad f^*{\operatorname{g}_N}=\frac{1}{2}(\operatorname{g}-\operatorname{s}).$$
Since $\{ e_1,\dots, e_{m}\}$ diagonalizes $\operatorname{g}$ and $\operatorname{s}$, it follows that it
diagonalizes ${\operatorname{g}_M}$ and $f^*{\operatorname{g}_N}$ as well. In fact, for any $i\in\{1,\dots,m\}$, we have
$$f^*{\operatorname{g}_N}( e_{i}, e_{i})=\lambda^2_{i}{\operatorname{g}_M}( e_{i}, e_{i}).$$
Proceeding exactly as in the proof of Lemma \ref{lem cd}, but using $\mu\sigma_{M}$ instead of $\sigma_M$ and
$\mu\sigma$ instead of $\sigma$, we obtain that
\begin{eqnarray*}
\mathcal{B}_1&=&2\sum_{k\neq m}\Bigl(f^*{\operatorname{R}_N}(e_k,e_m,e_k,e_m)-\mu{\operatorname{R}_M}(e_k,e_m,e_k,e_m)\Bigr) \\
&=&-\frac{2\mu}{1+\mu}\sum_{k\neq m}f^*{\operatorname{g}_N}(e_k,e_k)\Bigl(\sigma-\sigma_N(\operatorname{d}\hspace{-3pt}f(e_k)\wedge\operatorname{d}\hspace{-3pt}f(e_m))\Bigr)\\
&&-\mu\Bigl(\operatorname{Ric}_M(e_m,e_m)-(m-1)\sigma\,{\operatorname{g}_M}(e_m,e_m)\Bigr)\\
&&-\frac{\mu}{1+\mu}\sum_{k\neq m}\frac{1-\lambda_k^2}{1+\lambda_k^2}\Bigl(\sigma_M(e_k\wedge e_{m})+\sigma\Bigr).
\end{eqnarray*}
Similarly,
\begin{eqnarray*}
\mathcal{B}_2&=&2\sum\limits_{k\neq m-1}\Bigl(f^*{\operatorname{R}_N}(e_k,e_{m-1},e_k,e_{m-1})-\kappa{\operatorname{R}_M}(e_k,e_{m-1},e_k,e_{m-1})\Bigr) \\
&=&-\frac{2\kappa}{1+\kappa}\sum_{k\neq m-1}f^*{\operatorname{g}_N}(e_k,e_k)\Bigl(\sigma-\sigma_N(\operatorname{d}\hspace{-3pt}f(e_k)\wedge\operatorname{d}\hspace{-3pt}f(e_{m-1}))\Bigr)\\
&&-\kappa\Bigl(\operatorname{Ric}_M(e_{m-1},e_{m-1})-(m-1)\sigma\,{\operatorname{g}_M}(e_{m-1},e_{m-1})\Bigr)\\
&&-\frac{\kappa}{1+\kappa}\sum_{k\neq m-1}\frac{1-\lambda_k^2}{1+\lambda_k^2}\Bigl(\sigma_M(e_k\wedge e_{m-1})+\sigma\Bigr).
\end{eqnarray*}
Taking into account that $\lambda^{2}_{1}\le\cdots\le\lambda^{2}_{m-2}\le 1$, we deduce that
\begin{eqnarray}
\mathcal{B}&=&-\frac{2\mu}{(1+\mu)^2}\sum_{k\neq m}f^*{\operatorname{g}_N}(e_k,e_k)\Bigl(\sigma-\sigma_N(\operatorname{d}\hspace{-3pt}f(e_k)\wedge\operatorname{d}\hspace{-3pt}f(e_m))\Bigr) \nonumber\\
&&-\frac{2\kappa}{(1+\kappa)^2}\sum_{k\neq m-1}f^*{\operatorname{g}_N}(e_k,e_k)\Bigl(\sigma-\sigma_N(\operatorname{d}\hspace{-3pt}f(e_k)\wedge\operatorname{d}\hspace{-3pt}f(e_{m-1}))\Bigr)\nonumber\\
&&-\frac{\mu}{1+\mu}\Bigl(\operatorname{Ric}_M(e_{m},e_{m})-(m-1)\sigma\,{\operatorname{g}_M}(e_{m},e_{m})\Bigr)\nonumber\\
&&-\frac{\kappa}{1+\kappa}\Bigl(\operatorname{Ric}_M(e_{m-1},e_{m-1})-(m-1)\sigma\,{\operatorname{g}_M}(e_{m-1},e_{m-1})\Bigr)\nonumber\\
&&-\frac{\mu}{(1+\mu)^2}\sum_{k=1}^{m-2}\frac{1-\lambda_k^2}{1+\lambda_k^2}\Bigl(\sigma_M(e_k\wedge e_{m})+\sigma\Bigr)\nonumber\\
&&-\frac{\kappa}{(1+\kappa)^2}\sum_{k=1}^{m-2}\frac{1-\lambda_k^2}{1+\lambda_k^2}\Bigl(\sigma_M(e_k\wedge e_{m})+\sigma\Bigr)\nonumber \\
&&-\frac{(\kappa+\mu)(1-\kappa\mu)}{(1+\kappa)(1+\mu)}\Bigl(\sigma_M(e_{m-1}\wedge e_{m})+\sigma\Bigr)\nonumber\\
&&\le 0\label{inequality7}.
\end{eqnarray}
This completes the proof of Claim $5$.
Now we shall distinguish two cases.
{\bf Case 1.} Assume at first that $f$ is strictly area decreasing. Hence, $\kappa\mu<1$. In view of our
curvature assumptions, Claim $4$, Claim $5$, (\ref{inequality7}) and from inequality (\ref{inequality6}) we deduce
that $\kappa=\mu=0$. Hence, from Claim $3$ the map $f$ must be constant.
{\bf Case 2.} Suppose now that there exists a point $x_0\in M$ such that
$\|\Lambda^2\operatorname{d}\hspace{-3pt}f\|(x_0)=1$.
In this case we have that $\kappa\mu=1$. From Claim $1$, (\ref{inequality7}) and
inequality (\ref{inequality6}) we deduce that at $x_0$ we must have
$$1=\lambda^{2}_{1}(x_0)=\cdots=\lambda^{2}_{m-2}(x_0)\le\kappa\le 1.$$
Hence, $\kappa=1$ and so $\mu=1$. Therefore, at each point $x$ where $\operatorname{P}hi^{[2]}$
has a zero eigenvalue, all the singular values of $f$ are equal to $1$. Thus, the set
$$D:=\{x\in M:\|\Lambda^2\operatorname{d}\hspace{-3pt}f\|=1\},$$
is closed, non-empty and moreover $D=\{x\in M:f^{\ast}{\operatorname{g}_N}={\operatorname{g}_M}\}.$
Obviously, the map $f$ is strictly area decreasing on the complement of $D$. Moreover,
by (\ref{inequality7}), $\Ric_{M}=(m-1)\sigma$ at any point of $D$ and the restriction of $\sigma_{N}$
to $\operatorname{d}\hspace{-3pt}f(TD)$ is equal to $\sigma$.
This completes the proof of Theorem \ref{thmC}.
$\square$
{\bf Proof of Theorem \ref{thmE}.}
Note that in this case the singular values of the map $f$ are
$$0=\lambda^{2}_{1}=\cdots=\lambda^{2}_{m-1}=\kappa\le\mu.$$
Hence, automatically, $f$ is strictly area decreasing. From Claim $4$, Claim $5$,
inequality (\ref{inequality5}) and (\ref{inequality7}), we deduce that
$$0\le-2\mu{\Ric}_{M}(e_m,e_m)\le 0.$$
Thus $\mu=0$ and $f$ is a constant map. This completes the proof of Theorem \ref{thmE}.
$\square$
\subsection{Final remarks}
We end this paper with examples and remarks concerning the imposed assumptions
in Theorems \ref{thmD}, \ref{thmC} and \ref{thmE}.
\begin{remark}
In several cases, graphical submanifolds over $(M,{\operatorname{g}_M})$ with \textit{parallel mean curvature}, i.e.,
$$\nabla^{\perp}H=0,$$
where $\nabla^{\perp}$ stands for the connection of the normal bundle, must be minimal.
This problem was first considered by Chern in \cite{c}. So, whenever graphs with parallel mean curvature
vector are minimal we can immediately apply Theorems \ref{thmD}, \ref{thmC} and \ref{thmE}.
For example this can be done for graphs considered in the paper by G. Li and I.M.C. Salavessa \cite{li}.
\end{remark}
\begin{remark}
The reason that the result of Theorem \ref{thmC} is weaker than that of Theorem \ref{thmD} is
due to the fact that in Theorem \ref{thmC} we cannot apply the strong elliptic maximum principle
stated in Theorem \ref{mp2}. In fact, the null-eigenvector condition of the corresponding
tensor $\operatorname{P}si(\vartheta^{[2]})$ in the equation of $\Delta \operatorname{s}^{[2]}$ seems to hold only for some
weakly $2$-positive definite tensors $\vartheta$, including $\operatorname{s}$.
\end{remark}
\begin{remark}
In some situations, a minimal map $f:M\to N$ satisfying the assumptions in Theorem \ref{thmC}
can only be constant. For instance, if $\dim M>\dim N$ the map $f$ cannot be an isometric immersion
since $\operatorname{rank}(\operatorname{d}\hspace{-3pt}f)<\dim M$. Moreover, if $M$ is not Einstein or the sectional curvature
of $N$ is strictly less than $\sigma$, then any such map must be constant.
\end{remark}
\begin{remark}
In this remark we show that the assumptions on the curvatures of $M$ and $N$ in Theorems
\ref{thmD} and \ref{thmC} are sharp.
\begin{enumerate}[i)]
\item{\bf Scaling.}
Suppose that $f:M\to N$ is a smooth map between two Riemannian manifolds
$(M,{\operatorname{g}_M})$ and $(N,{\operatorname{g}_N})$, and assume that there exists a constant $c>0$ such that
$f^*{\operatorname{g}_N}<c\,{\operatorname{g}_M}.$
Clearly such a constant exists, if $M$ is compact. Define the rescaled metrics
$$\widetilde{\operatorname{g}}_M:=c{\operatorname{g}_M}\,,\quad\widetilde{\operatorname{g}}_N:=c^{-1}{\operatorname{g}_N}\,.$$
One can verify that $f$ is a length (and obviously area) decreasing map with respect to the Riemannian metrics
$\widetilde{\operatorname{g}}_M$ and ${\operatorname{g}_N}$, as well as with respect to the metrics ${\operatorname{g}_M}$ and $\widetilde{\operatorname{g}}_N$.
Thus, any smooth map can be made a length decreasing map, if either the domain or the target is
scaled appropriately.
\item
{\bf Totally geodesic maps.}
There are plenty of non constant length decreasing minimal maps.
For instance, assume that $(M,{\operatorname{g}_M})$ is a Riemannian manifold and $c\in(0,1)$ a real constant. The
identity map $\Id:(M,{\operatorname{g}_M})\to(M,c^{-1}{\operatorname{g}_M})$ gives a length decreasing minimal map whose graph
$\Gamma(\Id)$ is even totally geodesic. If $\sigma_{M}$ and $\sigma_{N}$ are the sectional
curvatures of $(M,{\operatorname{g}_M})$ and $(N,c^{-1}{\operatorname{g}_M})$, respectively, then
$$\sigma_{N}=c^{-1}\sigma_{M}>\sigma_{M}.$$
Consequently, Theorems \ref{thmD} and \ref{thmC} are not valid if we assume $\sigma_{N}>\sigma_{M}$. Moreover, the assumption
$\sigma>0$ is essential in these theorems and cannot be removed. Indeed, consider the flat $2$-dimensional
torus $(\mathbb{T}^2,\operatorname{g}_{\mathbb{T}})$. By scaling properly the metric
$\operatorname{g}_{\mathbb{T}}$, the
identity map $\Id:\mathbb{T}^2\to\mathbb{T}^2$ produces a length decreasing map. On the other hand, the
scaled metric is again flat and $\Id$ is certainly neither constant nor an isometry.
\end{enumerate}
\end{remark}
\begin{example}
This example shows that there exists an abundance of length decreasing minimal maps that
are not totally geodesic.
\begin{enumerate}[i)]
\item{\bf Holomorphic maps.}
According to the
Schwarz-Pick {Lem\-ma}, any non-linear holomorphic map of the unit disc $D$ in the complex plane
$\mathbb{C}$ to itself is strictly length decreasing with respect to the Poincar\'e metric.
The holomorphicity implies that $f$ is a minimal map (cf., \cite{eells}). On the
other hand, L. Ahlfors \cite{ahlfors} exposed in his generalization of the Schwarz-Pick Lemma the
essential role played by the curvature. He proved that if $f:M\to N$ is a holomorphic map, where
$N$ is a Riemann surface with a metric ${\operatorname{g}_N}$ whose Gaussian curvature is bounded from above by a
negative constant $-b$ and $M:=D$ is the unit disc in $\mathbb{C}$ endowed with
an invariant metric ${\operatorname{g}_M}$ whose Gaussian curvature is a negative constant $-a$, then
$$f^{*}{\operatorname{g}_N}\leq \frac{a}{b}{\operatorname{g}_M}.$$
Ahlfors' result was extended by S.T. Yau \cite{yau1} for holomorphic maps between
complete K\"{a}hler manifolds. More precisely, Yau showed that any holomorphic map $f:M\to N$,
where here $M$ is a complete K\"{a}hler manifold with Ricci curvature bounded from below by a negative
constant $-a$ and $N$ is a Hermitian manifold with holomorphic bisectional curvature bounded
from above by a negative constant $-b$, then
$f^{*}{\operatorname{g}_N}\leq\frac{a}{b}{\operatorname{g}_M}.$
\item{\bf Biholomorphic maps.}
Let $M$ be a K\"{a}hler manifold and $\operatorname{Aut}(M)$ its \textit{automorphism group}, that is
the group of all biholomorphic maps of $M$. When $m\ge 4$, the group $\operatorname{Aut}(M)$ can be
arbitrary large (cf. \cite{akhiezer}). This indicates that the results of Theorem \ref{thmC},
cannot be extended for the $m$-Jacobian $\Lambda^{m}\operatorname{d}\hspace{-3pt}f$. For example, let $M$ be
compact, $y_0$ a fixed point on $M$, and $f\in\operatorname{Aut}(M)$. Then, the map $\tilde{f}:M\times M\to M\times M$,
$\tilde{f}(x,y)=(f(x),y_0),$
is minimal, as holomorphic, and has identically zero $m$-Jacobian.
In the flat case we can give even explicit
examples. For instance, consider the map $f:\mathbb{C}^2=\real{4}\to\mathbb{C}^2=\real{4}$, given by
$$f(z,w):=(\beta z+h(w),w),\quad z,w\in\mathbb{C},$$
where $h:\mathbb{C}\to\mathbb{C}$ is a non-affine holomorphic map and $\beta\le 1$ a positive
real number. Note that the graph $\Gamma(f)$ is minimal in $\real{8}$, $\|\Lambda^{4}\operatorname{d}\hspace{-3pt}f\|=\beta\le 1$
and $f$ is certainly not an isometry.
\end{enumerate}
\end{example}
\begin{remark}
Let $M$ and $N$ be two Riemannian manifolds satisfying the curvature assumptions in Theorem \ref{thmC}.
Following essentially the same computations as in the proof of Theorem \ref{thmD}, we can prove that the
strictly area decreasing property of a map $f:M\to N$ is preserved under mean
curvature flow. The convergence shall be explored in
another article where we shall also derive a parabolic analogue of Theorem \ref{mp1}.
\end{remark}
{\small{\bf{Acknowledgments:}}{ The first author would like to express his gratitude to the Max-Planck Institut
f\"{u}r Mathematik in den Nauturwissenschaften Leipzig and especially to Professor J. Jost for the scientific
support and everything that he benefited during the stay at the Institute. Moreover, he would like
to thank Dr. B. Hua for many stimulating conversations.}}
\begin{bibdiv}
\begin{biblist}
\bib{ahlfors}{article}{
author={Ahlfors, L.V.},
title={An extension of Schwarz's lemma},
journal={Trans. Amer. Math. Soc.},
volume={43},
date={1938},
pages={359--364},
}
\bib{akhiezer}{book}{
author={Akhiezer, D.N.},
title={Lie group actions in complex analysis},
series={Aspects of Mathematics, E27},
publisher={Friedr. Vieweg \& Sohn},
place={Braunschweig},
date={1995},
}
\bib{andrews}{book}{
author={Andrews, B.},
author={Hopper, C.},
title={The Ricci flow in Riemannian geometry},
series={Lecture Notes in Mathematics},
volume={2011},
note={A complete proof of the differentiable 1/4-pinching sphere
theorem},
publisher={Springer},
place={Heidelberg},
date={2011},
}
\bib{bernstein}{article}{
author={Bernstein, S.},
title={\"Uber ein geometrisches Theorem und seine Anwendung auf die
partiellen Differentialgleichungen vom elliptischen Typus},
journal={Math. Z.},
volume={26},
date={1927},
pages={551--558},
}
\bib{bohm}{article}{
author={B{\"o}hm, C.},
author={Wilking, B.},
title={Nonnegatively curved manifolds with finite fundamental groups
admit metrics with positive Ricci curvature},
journal={Geom. Funct. Anal.},
volume={17},
date={2007},
pages={665--681},
}
\bib{bombieri}{article}{
author={Bombieri, E.},
author={Giorgi, E. de},
author={Giusti, E.},
title={Minimal cones and the Bernstein theorem},
journal={Invent. Math.},
volume={7},
date={1969},
pages={243-269},
}
\bib{brendle}{book}{
author={Brendle, S.},
title={Ricci flow and the sphere theorem},
series={Graduate Studies in Mathematics},
volume={111},
publisher={American Mathematical Society},
place={Providence, RI},
date={2010},
}
\bib{chern1}{article}{
author={Chern, S.-S.},
author={Osserman, R.},
title={Complete minimal surfaces in euclidean $n$-space},
journal={J. d'Analyse Math.},
volume={19},
date={1967},
pages={15-34},
}
\bib{c}{article}{
author={Chern, S.-S},
title={On the curvatures of a piece of hypersurface in euclidean space},
journal={Abh. Math. Sem. Univ. Hamburg},
volume={29},
date={1965},
pages={77--91},
}
\bib{ni1}{book}{
author={Chow, B.},
author={Chu, S.-C.},
author={Glickenstein, D.},
author={Guenther, C.},
author={Isenberg, J.},
author={Ivey, T.},
author={Knopf, D.},
author={Lu, P.},
author={Luo, F.},
author={Ni, L.},
title={The Ricci flow: techniques and applications. Part II},
series={Mathematical Surveys and Monographs},
volume={144},
note={Analytic aspects},
publisher={American Mathematical Society},
place={Providence, RI},
date={2008},
}
\bib{dalio}{article}{
author={Da Lio, F.},
title={Remarks on the strong maximum principle for viscosity solutions to
fully nonlinear parabolic equations},
journal={Commun. Pure Appl. Anal.},
volume={3},
date={2004},
pages={395--415},
}
\bib{ecker}{book}{
author={Ecker, K.},
title={Regularity theory for mean curvature flow},
series={Progress in Nonlinear Differential Equations and their
Applications, 57},
publisher={Birkh\"auser Boston Inc.},
place={Boston, MA},
date={2004},
}
\bib{eells}{article}{
author={Eells, J.},
title={Minimal graphs},
journal={Manuscripta Math.},
volume={28},
date={1979},
pages={101--108},
}
\bib{evans1}{article}{
author={Evans, L.C.},
title={A strong maximum principle for parabolic systems in a convex set
with arbitrary boundary},
journal={Proc. Amer. Math. Soc.},
volume={138},
date={2010},
pages={3179--3185},
}
\bib{ferus}{article}{
author={Ferus, D.},
title={On the completeness of nullity foliations},
journal={Michigan Math. J.},
volume={18},
date={1971},
pages={61--64},
}
\bib{fleming}{article}{
author={Fleming, W.},
title={On the oriented Plateau problem},
journal={Rend. Circ. Mat. Palermo},
volume={11},
date={1962},
pages={69--90},
}
\bib{hamilton1}{article}{
author={Hamilton, R.},
title={Four-manifolds with positive curvature operator},
journal={J. Differential Geom.},
volume={24},
date={1986},
pages={153--179},
}
\bib{hamilton2}{article}{
author={Hamilton, R.},
title={Three-manifolds with positive Ricci curvature},
journal={J. Differential Geom.},
volume={17},
date={1982},
pages={255--306},
}
\bib{h-s1}{article}{
author={Hasanis, Th.},
author={Savas-Halilaj, A.},
author={Vlachos, Th.},
title={On the Jacobian of minimal graphs in $\Bbb R^4$},
journal={Bull. Lond. Math. Soc.},
volume={43},
date={2011},
pages={321--327},
}
\bib{h-s2}{article}{
author={Hasanis, Th.},
author={Savas-Halilaj, A.},
author={Vlachos, Th.},
title={Minimal graphs in $\Bbb R^4$ with bounded Jacobians},
journal={Proc. Amer. Math. Soc.},
volume={137},
date={2009},
pages={3463--3471},
}
\bib{hildebrandt}{article}{
author={Hildebrandt, S.},
author={Jost, J.},
author={Widman, K.-O.},
title={Harmonic mappings and minimal submanifolds},
journal={Invent. Math.},
volume={62},
date={1980/81},
pages={269--298},
}
\bib{hopf}{article}{
author={Hopf, E.},
title={Elementare Bemerkungen \"{u}ber die L\"{o}sungen partieller
Differentialgleichungen zweiter Ordnung vom elliptischen Typus},
journal={Sitzungsberichte Akad. Berlin.},
volume={19},
date={1927}
pages={147--152},
}
\bib{jxy}{article}{
author={Jost, J.},
author={Xin, Y.-L.},
author={Yang, L.},
title={The geometry of Grassmannian manifolds and Bernstein type theorems for higher codimension},
journal={arXiv:1109.6394},
date={2011},
pages={1--36},
}
\bib{lee}{article}{
author={Lee, K.-W.},
author={Lee, Y.-I.},
title={Mean curvature flow of the graphs of maps between compact
manifolds},
journal={Trans. Amer. Math. Soc.},
volume={363},
date={2011},
pages={5745--5759},
}
\bib{li}{article}{
author={Li, G.},
author={Salavessa, I.M.C.},
title={Bernstein-Heinz-Chern results in calibrated manifolds},
journal={Rev. Mat. Iberoam.},
volume={26},
date={2010},
pages={651--692},
}
\bib{schoen}{article}{
author={Schoen, R.},
title={The role of harmonic mappings in rigidity and deformation
problems},
conference={
title={Complex geometry},
address={Osaka},
date={1990},
},
book={
series={Lecture Notes in Pure and Appl. Math.},
volume={143},
publisher={Dekker},
place={New York},
},
date={1993},
pages={179--200},
}
\bib{schoen1}{article}{
author={Schoen, R.},
author={Simon, L.},
author={Yau, S.-T.},
title={Curvature estimates for minimal hypersurfaces},
journal={Acta Math.},
volume={134},
date={1975},
pages={275--288},
}
\bib{simons}{article}{
author={Simons, J.},
title={Minimal varieties in Riemannian manifolds},
journal={Ann. of Math.},
volume={88},
date={1968},
pages={62--105},
}
\bib{swx}{article}{
author={Smoczyk, K.},
author={Wang, G.},
author={Xin, Y.-L.},
title={Bernstein type theorems with flat normal bundle},
journal={Calc. Var. Partial Differential Equations},
volume={26},
date={2006},
pages={57--67},
}
\bib{sw}{article}{
author={Smoczyk, K.},
author={Wang, M.-T.},
title={Mean curvature flows of Lagrangians submanifolds with convex
potentials},
journal={J. Differential Geom.},
volume={62},
date={2002},
pages={243--257},
}
\bib{suss}{article}{
author={S\"uss, W.},
title={\"Uber Kennzeichnungen der Kugeln und Affinsph\"aren durch Herrn K.-P. Grotemeyer},
journal={Arch. Math. (Basel)},
volume={3},
date={1952},
pages={311-313},
}
\bib{tsui1}{article}{
author={Tsui, M.-P.},
author={Wang, M.-T.},
title={Mean curvature flows and isotopy of maps between spheres},
journal={Comm. Pure Appl. Math.},
volume={57},
date={2004},
pages={1110--1126},
}
\bib{yau1}{article}{
author={Yau, S.-T.},
title={A general Schwarz lemma for K\"ahler manifolds},
journal={Amer. J. Math.},
volume={100},
date={1978},
pages={197--203},
}
\bib{wang1}{article}{
author={Wang, M.-T.},
title={Long-time existence and convergence of graphic mean curvature flow
in arbitrary codimension},
journal={Invent. Math.},
volume={148},
date={2002},
pages={525--543},
}
\bib{wang2}{article}{
author={Wang, M.-T.},
title={Mean curvature flow of surfaces in Einstein four-manifolds},
journal={J. Differential Geom.},
volume={57},
date={2001},
pages={301--338},
}
\bib{wang3}{article}{
author={Wang, M.-T.},
title={Deforming area preserving diffeomorphism of surfaces by mean
curvature flow},
journal={Math. Res. Lett.},
volume={8},
date={2001},
pages={651--661},
}
\bib{wang4}{article}{
author={Wang, X.},
title={A remark on strong maximum principle for parabolic and elliptic
systems},
journal={Proc. Amer. Math. Soc.},
volume={109},
date={1990},
pages={343--348},
}
\bib{weinberger}{article}{
author={Weinberger, H.F.},
title={Invariant sets for weakly coupled parabolic and elliptic systems},
journal={Rend. Mat. (6)},
volume={8},
date={1975},
pages={295--310},
}
\end{biblist}
\end{bibdiv}
\end{document} | math |
var _ = require('./utils/mindash');
function createClass(properties, defaultOptions, BaseType) {
function Class(options) {
classCallCheck(this, Class);
this.id = properties.id;
this.displayName = properties.displayName;
var base = get(Object.getPrototypeOf(Class.prototype), 'constructor', this);
var baseOptions = _.extend({}, defaultOptions, options, properties);
base.call(this, baseOptions);
}
if (BaseType) {
inherits(Class, BaseType);
}
_.extend(Class.prototype, properties);
Class.id = properties.id;
Class.displayName = properties.displayName;
return Class;
}
function get(object, property, receiver) {
var desc = Object.getOwnPropertyDescriptor(object, property);
if (desc === undefined) {
var parent = Object.getPrototypeOf(object);
if (parent === null) {
return undefined;
} else {
return get(parent, property, receiver);
}
} else if ('value' in desc && desc.writable) {
return desc.value;
} else {
var getter = desc.get;
if (getter === undefined) {
return undefined;
}
return getter.call(receiver);
}
}
function inherits(subClass, superClass) {
if (typeof superClass !== 'function' && superClass !== null) {
throw new TypeError(`Super expression must either be null or a function, not ${typeof superClass}`);
}
subClass.prototype = Object.create(superClass && superClass.prototype, {
constructor: {
value: subClass,
enumerable: false,
writable: true,
configurable: true
}
});
if (superClass) {
subClass.__proto__ = superClass;
}
}
function classCallCheck(instance, Constructor) {
if (!(instance instanceof Constructor)) {
throw new TypeError('Cannot call a class as a function');
}
}
module.exports = createClass; | code |
Shame On UKIP Harbouring Zionist Filth & Anti Judaism!
The anti Jewish UKIP Group in The EU & The zionist parasites are feeding off of Judaism & Decent UKIP Members!
it would seem that the zionist behaviour includes removing any statements of fact or opinion that does not support their evil behaviour exploiting Judaism as a profit scam for the new concept of ‘Children of The Holocaust‘ – where they adopt the fascist genocidal methods of the holocaust to abuse and massacre Palastinians and their neighbours.
What vile creatures the zionists are.
This entry was posted on 27/05/2011 at 17:11 and is filed under Holocaust Denial, Jews, Palestine, ZIONISM. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site. | english |
\begin{example}gin{document}
\title{Existence and multiplicity results for a class of Kirchhoff-Choquard equations with a generalized sign-changing potential
}
\author{
Eduardo de S. Böer$^{1}$ \footnote{ E-mail address: eduardoboer04@gmail.com, Tel. +55.51.993673377}, Ol\'{\i}mpio H. Miyagaki$^{1}$ \thanks{E-mail address: ohmiyagaki@gmail.com, Tel.: +55.16.33519178 (UFSCar)} \,\, and \,\, Patrizia Pucci$^{2}$ \thanks{Corresponding author} \footnote{E-mail address: patrizia.pucci@unipg.it} \\
{\footnotesize $^{1}$ Department of Mathematics, Federal University of S\~ao Carlos,}\\ {\footnotesize 13565-905 S\~ao Carlos, SP - Brazil}\\
{\footnotesize $^{2}$ Dipartimento di Matematica e Informatica,
Universit\`{a} degli Studi di Perugia,}\\
{\footnotesize 06123
Perugia, Italy}
}
\noindent
\maketitle
\begin{example}gin{center}
{\emph{Dedicated to the memory of Professor Antonio Ambrosetti,
with high feelings of admiration\\for his notable contributions in Mathematics and great
affection}}
\end{center}
\noindent \textbf{Abstract:} In the present work we are concerned with the following Kirchhoff-Choquard-type equation $$-M(||\nabla u||_{2}^{2})\Delta u +Q(x)u + \mu(V(|\cdot|)\ast u^2)u = f(u) \mbox{ in \ } \mathbb{R}dois , $$ for $M: \mathbb{R} \mathbb{R}A \mathbb{R} \mbox{ \ given by \ } M(t)=a+bt$, $ \mu >0 $, $ V $ a sign-changing and possible unbounded potential, $ Q $ a continuous external potential and a nonlinearity $f$ with exponential critical growth. We prove existence and multiplicity of solutions in the \textit{nondegenerate} case and guarantee the existence of solutions in the \textit{degenerate} case.
\noindent
{\it \small Mathematics Subject Classification:} {\small 35J60, 35J15, 35Q55, 35B25. }\\
{\it \small Key words}. {\small Kirchhoff-Choquard equations, sign-changing potentials, exponential growth,
variational techniques, ground state solution.}
\section{Introduction}
The present work is devoted to study existence and multiplicity of solutions to the following class of Kirchhoff-Choquard equations
\begin{example}gin{equation}\label{P}
-M(||\nabla u||_{2}^{2})\Delta u +Q(x)u + \mu(V(|\cdot|)\ast u^2)u = f(u)\quad \mbox{ in \ } \mathbb{R}dois ,
\end{equation}
where $ \mu > 0 $, $M: \mathbb{R} \mathbb{R}A \mathbb{R}$ is a Kirchhoff type function, $ Q: \mathbb{R}dois \mathbb{R}A \mathbb{R} $ is a nonnegative potential, $V:\mathbb{R} \mathbb{R}A \mathbb{R}$ is a continuous sign-changing and possible unbounded potential and $ f:\mathbb{R} \mathbb{R}A \mathbb{R} $ is a continuous function with primitive $ F(t)=\int\limits_{0}^{t} f(s)ds $.
This paper was motivated by recent works dealing with Choquard equations with logarithmic kernel, such as \cite{[6], [10], [boer], [boer2]}, and some works of Kirchhoff-type equations, as for example \cite{[pucci], [olimpio]}. In the following, we make a quick literature overview.
On one hand, the following Choquard or nonlinear Schrödinger-Newton equation
\begin{example}q\label{i2}
-\Delta u + V(x) u + \gamma ( \mathcal{G}amma_N \ast |u|^2) u = b|u|^{p-2}u,\ p> 2, \ b>0, \textrm{ \ \ in \ } \mathbb{R}N,
\end{example}q
where
$ \mathcal{G}amma_N $ is the well-known fundamental solution of the Laplacian
$$
\mathcal{G}amma_N (x) = \begin{example}gin{cases}
\dfrac{1}{N(2-N)\sigma_N}|x|^{2-N} &\textrm{ if \ } N\geq 3 ,\\
\dfrac{1}{2\pi}\ln |x|& \textrm{ if \ } N= 2 ,
\end{cases}
$$
has been extensively studied in the case $ N=3 $, due to its relevance in physics. Although what the equation name suggests, it was first studied by Fröhlich and Pekar in \cite{[12] , [11] , [22]}, to describe the quantum mechanics of a polaron at rest, in the particular case $ V(x)\equiv Constant > 0 $ and $ \gamma > 0 $. Then, in 1976, Choquard introduced the same equation in the study of an electron trapped in its hole. Moreover, Penrose has derived equation (\ref{i2}) while discussing about the self gravitational collapse of a quantum-mechanical system in \cite{[18]}. See also \cite{[16]}.
In the case $ N=2 $, in \cite{[6]}, the authors have proved the existence of a ground state solution, using the Nehari manifold and the existence of infinitely many geometrically distinct solutions, when $ Q: \mathbb{R}dois \mathbb{R}A (0, \infty) $ is continuous and $ \mathbb{Z}^2 $-periodic, $ \mu > 0 $ and a nonlinearity of the form $ f(u)=b|u|^{p-2}u $, with $ b\geq 0 $ and $ p\geq 4 $. Then, intending to fill the gap, i.e., the situation $ 2<p<4 $, the paper \cite{[10]} deals with equation (\ref{P}) when $ Q(x)\equiv Constant > 0 $ and $ \mu >0 $, and $ f(u)=|u|^{p-2}u $, with $ 2<p<4 $, and provides existence of a mountain pass solution as well as of a ground state solution. Finally, in \cite{[boer]}, the authors prove existence and multiplicity results for the $p-$fractional Laplacian operator and in \cite{[boer2]} existence and multiplicity results are derived for $ (p, N)$-Laplacian equations. Moreover, in \cite{[boer2]} the authors prove for the first time that, up to subsequence, Cerami sequences are bounded in the solution space.
Let us recall that, from a physical point of view, the local nonlinear terms on the right side of equation (\ref{i2}), such as $ b|u|^{p-2}u $, for $ b\in \mathbb{R} $ and $ p>2 $, usually appears in the Schrödinger equations as a way of modelling the interaction among particles. We refer the reader to \cite{[boer]} for a complete overview in this topic.
On the other hand, the literature of Kirchhoff-type equations and its related elliptic problems is very interesting and quite large. As an example, we cite \cite{[jin]} where the authors consider the following equation
$$
\left\{\begin{example}gin{array}{ll}
-\left(a+ b \displaystyle\int\limits_{\mathbb{R}^N}|\nabla u|^2 dx \right)\Delta u + u = f(x, u) \mbox{ in } \mathbb{R}^N ,\\[2ex]
u\in H^1(\mathbb{R}^N),
\end{array}
\right.
$$
and prove the existence of a sequence of radial solutions $(u_k)\subset H^1(\mathbb{R}^N)$ satisfying $ I(u_k)\mathbb{R}A \infty $, as $ k \mathbb{R}A \infty $. For a more detailed overview in the numerous results involving Kirchhoff equations, we refer the reader to \cite{[pucci], [olimpio], [pucci2], [liang]} and the references therein.
To finish, we emphasize that nonlinearities with exponential behaviour appear frequently in applied problems, from physics to biology, which show us the importance of the studies on this topic. In this sense, we cite some works that deal with nonlinearities of Moser-Trudinger type \cite{[5], [Lam], [17], [olimpio]} and the references therein.
We intend to extend or complement the above mentioned works, considering a generalized sign-changing convolution potential, the Kirchhoff operator and a nonlinearity with critical exponential growth.
In the sequel we present the features of equation \eqref{P}. Throughout this paper, $ \mathbb{R}^+ = \{t\in \mathbb{R} \ ; \ t > 0\} $. In our work we are going to consider the following Kirchhoff function
\noindent $ (M) \
M: \mathbb{R} \mathbb{R}A \mathbb{R} \mbox{ \ given by \ } M(t)=a+bt \mbox{ , for all \ } t \in \mathbb{R} \mbox{ , with \ }a> 0 \mbox{ \ and \ } b \geq 0 \mbox{ or } a = 0 \mbox{ \ and \ } b >0.
$
The case where $ a>0 $ is called \textit{nondegenerate} while the situation in which $ a=0 $ is said to be \textit{degenerate}. We are going to consider both cases here.
Since our intention is to provide a way to solve problems with sign-changing potentials that can be unbounded from below, we require that $ V $ has a nontrivial negative part, $ V^- = \max\{-V, 0\} $. But some of the arguments can be modified in order to apply these techniques to positive potentials as well. The positive part of $V$ is defined as $\max\{ V, 0\}$. Thus, we assume that $V: \mathbb{R}^+ \mathbb{R}A \mathbb{R}$ is a real function verifying the following properties
\noindent $
(V_1) \ \mbox{There are real functions } a_1, a_2: \mathbb{R}^+ \mathbb{R}A \mathbb{R} \mbox{ such that } a_2 \in L^{\infty}(\mathbb{R}^+), a_{1, 0}= \inf\limits_{t\geq 2} a_1(t) > 0, a_{2, 0}=\inf\limits_{t \in \mathbb{R}^+}a_2(t) > 0 \mbox{ and }
$
$$
a_1(t)\ln(1+ t) \leq V^{+}(t) \leq a_2(t) \ln(1+t) , \forall \ t> 0.
$$
\noindent $(V_2)$ There exists a real function $ a_3: \mathbb{R}^+ \mathbb{R}A \mathbb{R} $ such that $a_3(t) > 0$ in a subset of $\mathbb{R}^+$ with positive measure,
$$V^{-}(t) \leq \dfrac{a_3(t)}{t}\quad \forall \ t> 0\quad\mbox{and}\quad
\begin{example}gin{cases}
a_3 \in L^{\infty}(\mathbb{R}),\\
\mbox{or}\\
a_3(t)= t^{-\lambda}, \mbox{ for some } \lambda \in [1, 3) \mbox{ and for all } t > 0,
\end{cases}
$$
$$
\mbox{ \ There exists an open subset } \mathcal{I} \subset \mathbb{R}^+ \mbox{ such that } V(t) < 0 \mbox{ for all } t \in \mathcal{I}.\leqno{(V_3)}
$$
Natural examples of potentials $ V $, satisfying conditions $ (V_1)-(V_3) $, have the following geometry: $ V(|x|)\mathbb{R}A \infty $ as $ |x|\mathbb{R}A \infty $ and, either its negative part is bounded or $ V(|x|)\mathbb{R}A - \infty $ as $ |x|\mathbb{R}A 0 $. Such behaviour is studied in Lemma \ref{l7}. Below, we present some examples of possible $ V $. Note that, in example \textbf{(b)}, the negative part is bounded.
\begin{example}\label{ex1}
\textbf{(a)} {\rm The most important example as a potential $ V $ is the logarithmic kernel, $ V(|x|)=\ln|x| $, for all $ x \in \mathbb{R}dois $. Observe that condition $ (V_1) $ is satisfied with $ a_2(t)\equiv 1 $ and
$$
a_1(t) = \left\{ \begin{example}gin{array}{lll}
\dfrac{\ln t}{2\ln(1+t)} \ \ , \mbox{ if } t \geq 2 , \\[2ex]
\dfrac{\ln 2}{2 \ln 3}(t-1) , \mbox{ if } 1 \leq t \leq 2 ,\\
0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \mbox{ if } 0 \leq t \leq 1.
\end{array}
\right.
$$
Conditions $(V_2)$ and $ (V_3) $ are verified with functions $ a_3, a_4, a_5 \equiv 1 $.
\noindent \textbf{(b)} Define $ V(|x|)=|x|^\alpha - |x|^\begin{example}ta $, for $ 0< \begin{example}ta < \alpha < 1 $ suitably chosen.
\noindent \textbf{(c)} One can also consider more exotic potentials, such as $ V:\mathbb{R}^+ \mathbb{R}A \mathbb{R} $ given by
$$
V(|x|)=
\begin{example}gin{cases}
-\dfrac{1}{|x|},&\mbox{if \ } 0<|x|\leq 1\\
2|x| - 3 ,&\mbox{if \ } 1\leq |x| \leq \dfrac{3}{2}\\
\ln\left(|x|-\dfrac{1}{2}\right),&\mbox{ if \ } \dfrac{3}{2}\leq |x|.
\end{cases}
$$}
\end{example}
\noindent
As for the external potential $ Q:\mathbb{R}dois \mathbb{R}A \mathbb{R} $, we ask
condition
\noindent $ (Q) \
Q \in C(\mathbb{R}dois , \mathbb{R}), \inf\limits_{x\in \mathbb{R}dois}Q(x) = Q_0 > 0 \mbox{ and there exists } p \in (1, \infty] \mbox{ such that } Q \in L^{p}(\mathbb{R}dois) .
$
Finally, we recall that a function $ h $ has \textit{subcritical} exponential growth at $ +\infty $, if
$$
\lim\limits_{s\mathbb{R}A + \infty}\dfrac{h(s)}{e^{\alpha s^2}-1} = 0 \textrm{ \ , for all \ } \alpha >0 ,
$$
and we say that $ h $ has $ \alpha_0 $-\textit{critical} exponential growth at $ \infty $, if
$$
\lim\limits_{s\mathbb{R}A + \infty}\dfrac{h(s)}{e^{\alpha s^2}-1} = \left\{ \begin{example}gin{array}{ll}
0, \ \ \ \forall \ \alpha > \alpha_{0} \\
\infty , \ \ \ \forall \ \alpha < \alpha_0
\end{array} \right. .
$$
Thus, inspired by works such as \cite{[ruf],[olimpio], [boer]}, we consider the following conditions over $ f $.
$$f\in C(\mathbb{R} , \mathbb{R}), f(0)=0 \mbox{ and has critical exponential growth with } \alpha_0 = 4\pi . \leqno{(f_1)}$$
$$ \lim\limits_{|t|\mathbb{R}A 0} \dfrac{|f(t)|}{|t|^\tau }=0 \mbox{, for some } \tau > 1. \leqno{(f_2)}$$
$$ \mbox{ There exists }\ \theta \geq 4 \ \mbox{ such that}\ f(t)t \geq \theta F(t) > 0, \ \mbox{for all }\ t\in \mathbb{R} \setminus \{0\}. \leqno{(f_3)}$$
$$ \mbox{There exist}\ q>4 \ \mbox{ and} \ C_q > 0 \ \mbox{ such that}\ F(t) \geq C_q |t|^q , \ \mbox{for all} \ t \in \mathbb{R} . \leqno{(f_4)}$$
\noindent
From conditions $ (f_1) $ and $(f_2)$, given $ \varepsilon >0 $, $\alpha > 4\pi,$ fixed, for all $ p>2 $, we can find two constants $ K_1=K_1(p, \alpha , \varepsilon) > 0 $ and $ K_2=K_2(p, \alpha , \varepsilon) > 0$ such that
\begin{example}gin{equation}\label{eq2}
f(t)\leq \varepsilon |t|^\tau +K_1 |t|^{p-1}(e^{\alpha t^2}-1) \ , \ \ \ \forall \ t \in \mathbb{R},
\end{equation}
and
\begin{example}gin{equation}\label{eq3}
F(t) \leq \varepsilon|t|^{\tau + 1} + K_2 |t|^p(e^{\alpha t^2}-1) \ , \ \ \ \forall t \in \mathbb{R} .
\end{equation}
\begin{example}\label{ex2}
{\rm As a prototype for nonlinearity $ f $ satisfying conditions $ (f_1)-(f_4) $, we can consider $ f: \mathbb{R} \mathbb{R}A \mathbb{R} $ given by
$$
f(t) = C_q \left\{ \begin{example}gin{array}{ll}
t^q \ \ \ \ \ \ \ \ \ \ \ \ \ , \mbox{ if } 0 \leq t \leq 1 \\
t^q e^{4\pi (t^2 -1)} , \mbox{ if } t > 1
\end{array} \right. ,
$$
for $ C_q > 0 $ sufficiently large and $ q> 3 $ and consider the its odd extension.}
\end{example}
\begin{example}\label{ex4}
{\rm From Example \ref{ex1}, one can see that problem \eqref{P} includes, as a very important particular case, the planar Schrödinger-Poisson system
$$
-\Delta u + Q(x) u + \mu (\ln|\cdot|\ast |u|^{2})u = f(u) \textrm{ \ in \ } \mathbb{R}^2.
$$}
\end{example}
We are now ready to enunciate our first main result.
\begin{example}gin{theorem}\label{t1}
Suppose $ (V_1)-(V_3) $, $ (Q) $, $ (f_1)-(f_4) $, $ a>0 $, $ b\geq 0 $, $ \mu > 0 $, $ q>4 $ and $ C_q>0 $ sufficiently large. Then,
\begin{example}gin{itemize}
\item[(a)] problem \eqref{P} has a nontrivial solution at the mountain pass level, that is, there exists $ u\in X\setminus\{0\} $ such that $ u $ is a critical point for $ I $ and $ I(u)=c_{mp} $, where
\begin{example}gin{equation}\label{eq16}
c_{mp} = \inf\limits_{\gamma \in \mathcal{G}amma}\max\limits_{t \in [0, 1]}I(\gamma(t)),
\end{equation}
with $ \mathcal{G}amma = \{\gamma\in C([0, 1], X) \ ; \ \gamma(0)=0 \mbox{ and } I(\gamma(1))< 0 \} $.
\item[(b)] Problem \eqref{P} has a nontrivial ground state solution, in the sense that, there is $ u\in X\setminus \{0\} $ that is a critical point to $ I $ and satisfies
$$
I(u)=c_g=\inf \{ I(v) \ ; \ v \in \mathcal{K} \}, \mbox{ \ where \ } \mathcal{K} = \{ v \in X \setminus\{0\} \ ; \ I'(v)=0\}.
$$
\end{itemize}
\end{theorem}
Then, in order to get multiple solutions for \eqref{P}, we are going to apply a symmetric version of mountain pass theorem. To do so, we need to change condition $ (f_1) $ by the following.
$$f\in C(\mathbb{R} , \mathbb{R}), f(0)=0, f \mbox{ is odd and has critical exponential growth with } \alpha_0 = 4\pi . \leqno{(f_1')}$$
As a prototype example for this case, one can consider the odd extension of $ f $ given in Example \ref{ex2}.
\begin{example}gin{theorem}\label{t2}
Suppose $ (V_1)-(V_3) $, $ (Q) $, $ (f_1') $, $ (f_2)-(f_4) $, $ a>0 $, $ b\geq 0 $, $ \mu > 0 $, $ q>4 $ and $ C_q>0 $ sufficiently large. Then, problem \eqref{P} has infinitely many solutions.
\end{theorem}
In the \textit{degenerate} case we also need some changes in the hypotheses for $ f $. First of all, in order to get the mountain pass geometry, we ask
$$ \lim\limits_{|t|\mathbb{R}A 0} \dfrac{|f(t)|}{|t|^\tau }=0 \mbox{, for some } \tau > 3. \leqno{(f_2')}$$
Moreover, to obtain boundedness for Cerami sequences in $ H^{1}(\mathbb{R}^2) $, we need
$$ \mbox{ There exists }\ \theta \geq 8 \ \mbox{ such that}\ f(t)t \geq \theta F(t) > 0, \ \mbox{for all }\ t\in \mathbb{R} \setminus \{0\}. \leqno{(f_3')}$$
\begin{example}gin{theorem}\label{t3}
Suppose $ (V_1)-(V_3) $, $ (Q) $, $ (f_1), (f_2'), (f_3'), (f_4) $, $ a=0 $, $ b> 0 $, $ q>4 $ and $ C_q>0 $ sufficiently large. Then,
\begin{example}gin{itemize}
\item[(a)] there exists a value $ \mu_\ast >0 $ such that, for all $ \mu \in (0, \mu_\ast) $, problem \eqref{P} has a nontrivial solution at the mountain pass level, i.e., there exits $ u\in X\setminus\{0\} $ a critical point for $ I $ satisfying $ I(u)=c_{mp} $, where
\begin{example}gin{equation*}
c_{mp} = \inf\limits_{\gamma \in \mathcal{G}amma}\max\limits_{t \in [0, 1]}I(\gamma(t)),
\end{equation*}
with $ \mathcal{G}amma = \{\gamma\in C([0, 1], X) \ ; \ \gamma(0)=0 \mbox{ and } I(\gamma(1))< 0 \} $.
\item[(b)] There exists a value $ \mu_{\ast \ast}\in (0, \mu_\ast] $ such that, for all $ \mu \in (0, \mu_{\ast \ast}) $, problem \eqref{P} has a nontrivial ground state solution, in the sense that, there is $ u\in X\setminus \{0\} $ that is a critical point to $ I $ and satisfies
$$
I(u)=c_g=\inf \{ I(v) \ ; \ v \in \mathcal{K} \}, \mbox{ \ where \ } \mathcal{K} = \{ v \in X \setminus\{0\} \ ; \ I'(v)=0\}.
$$
\end{itemize}
\end{theorem}
Throughout the paper, we consider the following notations: $ L^{s}(\mathbb{R}^2) $ denotes the usual Lebesgue space with norm $ ||\cdot ||_s $; \ $ X' $ stands as the dual space of $ X $; \ $ B_r(x) $ is the ball centred in $ x $ with radius $ r>0 $, simply $ B_r $ if $ x=0 $; \ $ r_1, r_2 $ will be real values verifying $ r_1, r_2 >1 $, $ r_1 \sim 1 $ and $ \frac{1}{r_1}+\frac{1}{r_2}=1 $; \ $ (y\ast u)(x) = u(x-y) $, for all $ x, y\in \mathbb{R}dois $; \ $ x_n \searrow x $ will mean that $ x_n \mathbb{R}A x $ and $ x_n \geq x $, for all $ n\in \mathbb{N} $; \ $ K_i $, $ i\in \mathbb{N} $, denote important constants present in the estimates; \ $ C_i $, $ i\in\mathbb{N} $, denote different positive constants whose exact values are not essential to the exposition of the arguments.
The paper is organized as follows: in Section 2 we present some technical results concerning the framework and boundedness of sequences in the solution space. Section 3 is devoted to analyse the geometry of the functional and the involved potentials. Moreover, we verify some boundedness and convergence properties. In Section 4 we consider the \textit{nondegenerate} case and present the proof of a key proposition and of Theorems \ref{t1} and \ref{t2}. Finally, in Section 5, we study the \textit{degenerate} case and prove the existence result related to it.
\section{Framework and Technical Results}
In this section, we are going to present the space where we are going to look for solutions to \eqref{P} and some technical results concerning sequences in such space. To begin with, since we are going to use a variational approach, we introduce the Euler-Lagrange functional $I:H^{1}(\mathbb{R}^2) \mathbb{R}A \mathbb{R} \cup \{\infty\}$ associated to \eqref{P}, given by
\begin{example}gin{equation}\label{eq4}
I(u) = \dfrac{a}{2}\displaystyle\int\limits_{\mathbb{R}^2} |\nabla u|^2 dx + \dfrac{b}{4}\left(\displaystyle\int\limits_{\mathbb{R}^2} |\nabla u|^2 dx \right)^2 + \dfrac{1}{2}\displaystyle\int\limits_{\mathbb{R}^2} Q(x) u^2(x) dx + \dfrac{\mu}{4}P(u) - \displaystyle\int\limits_{\mathbb{R}^2} F(u) dx ,
\end{equation}
where $ P: H^{1}(\mathbb{R}^2) \mathbb{R}A \mathbb{R} \cup \{\infty\} $ is defined as
\begin{example}gin{equation}\label{eq5}
P(u) = \displaystyle\int\limits_{\mathbb{R}^2} \displaystyle\int\limits_{\mathbb{R}^2} V(|x-y|)u^2(x)u^2(y) dx dy .
\end{equation}
We also consider two auxiliary bilinear symmetric and positive forms $ \overline{P}_1 , \overline{P}_2 : H^{1}(\mathbb{R}^2) \mathbb{R}A \mathbb{R} \cup \{\infty\} $ given by
\begin{example}gin{small}
\begin{example}gin{equation}\label{eq6}
\overline{P}_1(u) = \displaystyle\int\limits_{\mathbb{R}^2} \displaystyle\int\limits_{\mathbb{R}^2} V^+ (|x-y|)u(x)v(y) dx dy \mbox{ \ and \ } \overline{P}_2(u) = \displaystyle\int\limits_{\mathbb{R}^2} \displaystyle\int\limits_{\mathbb{R}^2} V^- (|x-y|)u(x)v(y) dx dy ,
\end{equation}
\end{small}
respectively, and the functionals $ P_1, P_2: H^{1}(\mathbb{R}^2) \mathbb{R}A \mathbb{R} \cup \{\infty\} $ defined as $ P_1 = \overline{P}_1(u^2, u^2)$ and $ P_2 = \overline{P}_2(u^2 , u^2) $. Observe that $ P(u)=P_1(u)-P_2(u) $.
Now, based on \cite{[6]}, we consider the slightly smaller Hilbert space
$$
X = \{ u\in H^{1}(\mathbb{R}^2) \ ; \ ||u||_\ast < \infty \},\quad \mbox{where \ \ \ } ||u||_{\ast}^{2}= \displaystyle\int\limits_{\mathbb{R}^2} \ln(1+|x|) u^2(x) dx ,
$$
endowed with the norm $ ||\cdot||_{X}^{2}=||\cdot||^2 + ||\cdot||_{\ast}^{2} $, where $ ||\cdot|| $ is the usual norm in $ H^{1}(\mathbb{R}^2) $, and $ ||\cdot||_\ast $ comes from the inner product
$$
\langle u, v \rangle_\ast = \displaystyle\int\limits_{\mathbb{R}^2} \ln(1+|x|) u(x) v(x)\ dx.
$$
Clearly $ ||\cdot|| \leq ||\cdot||_X $, so that $ X \mathcal{I}C H^{1}(\mathbb{R}^2) \mathcal{I}C L^{s}(\mathbb{R}^2) $, for all $ s \geq 2 $.
Our first aim in this section is to prove that $ I\in C^1(X, \mathbb{R}) $. Thus, we present some preliminary results to achieve this aim.
We start recalling the reader the well-known Moser-Trudinger inequality.
\begin{example}gin{lemma}\label{l4}
\cite{[5]} If $\alpha >0$ and $ u\in H^{1}(\mathbb{R}^2) $, then
$$
\displaystyle\int\limits_{\mathbb{R}^2} \left(e^{\alpha |u|^2} - 1 \right) \ dx < \infty .
$$
Moreover, if $ ||\nabla u||_{2}^{2}\leq 1 $, $ ||u||_{2}^{2}\leq M < \infty $ and $ \alpha < 4 \pi $, then there exists a constant $ K_{\alpha, M}=K(M, \alpha) $, such that
$$
\displaystyle\int\limits_{\mathbb{R}^2} \left(e^{\alpha |u|^2} - 1 \right) dx < K_{\alpha, M} .
$$
\end{lemma}
Then, combined with \eqref{eq3} and Hölder inequality, for $ r_1, r_2 > 1 $, $ r_1 \sim 1 $ and $ \frac{1}{r_1}+\frac{1}{r_2}=1 $, we have
\begin{example}gin{equation}\label{eq7}
\displaystyle\int\limits_{\mathbb{R}^2} F(u) \ dx \leq \varepsilon ||u||_{\tau + 1}^{\tau + 1} + K_2 ||u||_{pr_2}^{p}\left(\displaystyle\int\limits_{\mathbb{R}^2} (e^{r_1 \alpha|u|^2}-1) dx \right)^{\frac{1}{r_1}} <
\infty , \forall \ u \in H^{1}(\mathbb{R}^2) .
\end{equation}
On the other side, from condition $ (V_1) $, there exists a constant $ K_3>0 $ such that
\begin{example}gin{equation}\label{eq9}
P_1(u) \leq ||a_2||_\infty \displaystyle\int\limits_{\mathbb{R}^2} \displaystyle\int\limits_{\mathbb{R}^2} \ln(1+|x - y|) u^2(x)u^2(y) dxdy \leq K_3 ||u||_{\ast}^{2}||u||_{2}^{2},
\end{equation}
and, from condition $ (V_2) $ and the Hardy-Littlewood-Sobolev inequality (HLS) (found in \cite{[15]}), there is $ K_4 > 0 $ satisfying
$$
P_2(u) \leq ||a_3||_\infty \displaystyle\int\limits_{\mathbb{R}^2} \displaystyle\int\limits_{\mathbb{R}^2} \dfrac{1}{|x-y|}u^2(x)u^2(y) dx dy \leq K_4 ||u||_{\frac{8}{3}}^{4} ,
$$
if $ a_3 \in L^\infty (\mathbb{R}) $, and
\begin{example}gin{equation}\label{eq10}
P_2(u) \leq \displaystyle\int\limits_{\mathbb{R}^2} \displaystyle\int\limits_{\mathbb{R}^2} \dfrac{1}{|x-y|^{\lambda+1}}u^2(x)u^2(y) dx dy \leq K_{HLS} ||u||_{\frac{8}{3-\lambda}}^{4},
\end{equation}
if $ a_3(t)=t^{-\lambda} $, for all $ t >0 $ and $ \lambda \in [-1, 3) $. One should observe that the constant $ K_4 $ also depend on the best Hardy-Littlewood-Sobolev constant, denoted by $ K_{HLS} $. From now on we consider only the second case, since $a_3 \in L^{\infty}(\mathbb{R}) $ can be treated similarly. Moreover, from condition $ (Q) $, we have, for $ p>1 $ given in $ (Q) $, with $ \frac{1}{p}+\frac{1}{p'}=1 $, that
\begin{example}gin{equation}\label{eq13}
\displaystyle\int\limits_{\mathbb{R}^2} Q(x) u^2 \ dx \leq ||Q||_p ||u||_{2p'}^{2} \mbox{ \ \ \ and \ \ \ } \displaystyle\int\limits_{\mathbb{R}^2} Q(x)uv \ dx \leq ||Q||_p ||u||_{2p'}||v||_{2p'} .
\end{equation}
Furthermore, from \cite[Lemma 2.2]{[6]}, we have the following compact embedding.
\begin{example}gin{lemma}\label{l1}
The space $ X $ is compactly embedded in $ L^{s}(\mathbb{R}^2) $ for all $ s \geq 2 $.
\end{lemma}
Consequently, one can easily verify by standard arguments that $ I $ is well-defined on $ X $, $ I\in C^1(X, \mathbb{R}) $ and
$$
P'(u)(v) = 4 \displaystyle\int\limits_{\mathbb{R}^2} \displaystyle\int\limits_{\mathbb{R}^2} V(|x-y|)u^2(x)u(y)v(y) \ dx dy , \forall \ v \in X .
$$
The derivative of $ P $ can be treated in a similar way as that in \cite[Lemma 2.2]{[6]}. We recall the reader that a nontrivial \textit{weak solution} for \eqref{P} is a function $ u \in X \setminus \{0\} $ satisfying
\begin{example}gin{align*}
& a \displaystyle\int\limits_{\mathbb{R}^2} \nabla u \nabla v dx + b\left(\displaystyle\int\limits_{\mathbb{R}^2} |\nabla u|^2 dx \right)\displaystyle\int\limits_{\mathbb{R}^2} \nabla u \nabla v dx + \displaystyle\int\limits_{\mathbb{R}^2} Q(x) u v dx \\
& + \mu \displaystyle\int\limits_{\mathbb{R}^2} \displaystyle\int\limits_{\mathbb{R}^2} V(|x-y|)u^2(x)u(y)v(y) \ dx dy = \displaystyle\int\limits_{\mathbb{R}^2} f(u) v dx , \forall \ v \in X.
\end{align*}
Hence, critical points for $ I $ will be weak solutions for \eqref{P}.
In the sequence we provide two crucial technical lemmas. The first one states when we have boundedness or convergence in $ X $ and the second one shows an important integral convergence. We will prove a general version of theses results, in order to provide a version that can be possible used in some other problems. In this sense, consider a continuous function $ g: \mathbb{R}dois \mathbb{R}A \mathbb{R} $ satisfying the following condition:
\noindent $
(g) \ g \in C(\mathbb{R} , \mathbb{R}), g(0)=0 \mbox{ and there are constants } K_5\in \mathbb{R} \setminus\{0\}, K_6 > 0 \mbox{ such that } $
$$
K_5 |t| \leq |g(t)| \leq K_6 |t| , \forall \ t \in \mathbb{R}
$$
From condition $ (g) $, we have
\begin{example}gin{equation}\label{eq1}
\frac{K_5}{2} |t|^2 \leq G(t) \leq \frac{K_6}{2} |t|^2 , \forall \ t \in \mathbb{R} .
\end{equation}
\begin{example}\label{ex3}
\textbf{(1)} Clearly, the prototype for $ g $ is given by $ g(t)=t $, for all $ t \in \mathbb{R} $.
\textbf{(2)} We also have more general examples for $g$, such as $ g: \mathbb{R} \mathbb{R}A \mathbb{R} $ given by
$$
g(t) = \left\{ \begin{example}gin{array}{lll}
t \ , \mbox{ if } t \in [0, 1] \\
t^3 , \mbox{ if } t \in (1, 2] \\
4t , \mbox{ if } t \in (2, \infty)
\end{array} \right. .
$$
One can easily verify that $ g $ satisfies the desired condition.
\end{example}
\begin{example}gin{lemma}\label{l2}
Let $ u\in L^{2}(\mathbb{R}^2) \setminus\{0\} $. Suppose that $ (u_{n}), (v_{n}) \subset X $ are two sequences satisfying $ u_{n}(x) \mathbb{R}A u(x) $ a.e. in $ \mathbb{R}dois $ and $ (v_{n}) $ is bounded in $ L^{2}(\mathbb{R}^2) $. Set
$$
\omega_n = \displaystyle\int\limits_{\mathbb{R}^2} \displaystyle\int\limits_{\mathbb{R}^2} V^+ (|x-y|) G(u_{n}(x))G(v_{n}(y)) \ dx dy .
$$
Then, if $ \sup\limits_{n \in \mathbb{N}} \omega_n < \infty $, $ (||v_{n}||_\ast) \subset \mathbb{R} $ is bounded. Moreover, if $ \omega_n \mathbb{R}A 0 $ and $ v_{n} \mathbb{R}A 0 $ in $ L^{2}(\mathbb{R}^2) $, then $ ||v_{n}||_\ast \mathbb{R}A 0 $.
\end{lemma}
\begin{example}gin{proof}
From Egorov's Theorem, there are $ R \in \mathbb{N} $, $ \delta > 0 $, $ n_0 \in \mathbb{N} $ and $ A \subset B_R $ such that $ A $ is a measurable set with $ |A| > 0 $ and $ u_{n}(x) > \delta $, for all $ n \geq n_0 $. Without loss of generality we can consider $ R > 2 $. Thus, if $ x\in B_R $ and $ y\in B_{2R}^{c} $, we have $ 1+|x-y| \geq \sqrt{1+|y|} $ and $ |x-y|> 2 $. Therefore, for each $ n \geq n_0 $, we have
\begin{example}gin{align*}
\omega_n & \geq \dfrac{K_{5}^{2}}{4}\displaystyle\int\limits_{B_{2R}^{c}} \displaystyle\int\limits_{A} a_1(|x-y|)\ln(1+|x-y|)u_{n}^2(x)v_{n}^2(y) \ dxdy \\
& \geq \dfrac{K_{5}^{2}a_{1, 0}\delta^2 |A|}{8} \displaystyle\int\limits_{B_{2R}^{c}} \ln(1+|y|)v_{n}^2 (y) \ dy \\
& = \dfrac{K_{5}^{2}a_{1, 0}\delta^2 |A|}{8} (||v_{n}||_{\ast}^{2}-\ln(1+2R)||v_{n}||_{2}^{2}),
\end{align*}
and the result follows.
\end{proof}
\begin{example}gin{lemma}\label{l3}
Let $ (u_{n})\subset X $ such that $ u_{n} \rightharpoonup u $ in $ X $. Then,
$$
\lim\limits_{n \mathbb{R}A \infty} \displaystyle\int\limits_{\mathbb{R}^2} \displaystyle\int\limits_{\mathbb{R}^2} V^+ (|x-y|) G(u_{n}(x))g(u(y))(u_{n}(y)-u(y)) \ dxdy = 0.
$$
\end{lemma}
\begin{example}gin{proof}
For simplicity, for each $ n \in \mathbb{N} $, set
$$
A_n = \displaystyle\int\limits_{\mathbb{R}^2} \displaystyle\int\limits_{\mathbb{R}^2} V^+ (|x-y|) |G(u_{n}(x))| |g(u(y))| |(u_{n}(y)-u(y))| \ dxdy .
$$
Since $ u_{n} \rightharpoonup u $ in $ X $, from Lemma \ref{l1}, $ u_{n} \mathbb{R}A u $ in $ L^{s}(\mathbb{R}^2) $ for all $ s \geq 2 $. From this, condition $ (g) $ and \eqref{eq1}, we have
\begin{example}gin{align*}
A_n & \leq ||a_2||_\infty \dfrac{K_{6}^{2}}{2} u^2(x) |u(y)||u_{n}(y) -u(y)| \ dx dy \\
& \leq \dfrac{K_{6}^{2}||a_2||_\infty}{2} \left[ ||u_{n}||_{\ast}^{2}||u||_{2}^{2}||u_{n} - u ||_{2}^{2} + ||u_{n}||_{2}^{2}\left(\displaystyle\int\limits_{\mathbb{R}^2} \ln(1+|y|) |u(y)||u_{n}(y)-u(y)| \ dy \right)\right].
\end{align*}
To finish the proof one can agree similarly as in \cite[Lemma 2.6]{[6]}.
\end{proof}
\section{Geometry Properties and Convergence Results}
In the present section, we verify that $ I $ has the mountain pass geometry and prove that, up to a subsequence, Cerami sequences are bounded in $ X $. Moreover, we analyse the geometry of the potential $ V $ and the functional $ P $. For the next computations, we need the following useful inequality,
\begin{example}gin{equation}\label{eq8}
\displaystyle\int\limits_{\mathbb{R}^2} f(u)u \ dx \leq \varepsilon ||u||_{\tau + 1}^{\tau + 1} + K_2||u||_{qr_2}^{q}\left(\displaystyle\int\limits_{\mathbb{R}^2} (e^{r_1 \alpha |u|^2}-1) dx \right)^{\frac{1}{r_1}}.
\end{equation}
\begin{example}gin{lemma}\label{l5}
There exists a value $ \rho > 0 $, sufficiently small, such that
$$
m_\begin{example}ta = \inf \{ I(u) \ ; \ u\in X , ||u|| = \begin{example}ta \} > 0 \ , \ \forall \begin{example}ta \in (0, \rho]
$$
and
$$
l_\begin{example}ta = \inf \{ I'(u)(u) \ ; \ u\in X , ||u|| = \begin{example}ta \} > 0 \ , \ \forall \begin{example}ta \in (0, \rho].
$$
\end{lemma}
\begin{example}gin{proof}
Let $ \alpha > 4\pi $ and $ u\in X $ such that $ r_1 \alpha ||u||^2 < 4\pi $.
\noindent \textit{Case $a>0$}: From $ (Q) $, equations \eqref{eq7}, \eqref{eq8} and \eqref{eq10}, Lemma \ref{l4} and the embeddings, we have
\begin{example}gin{align*}
I(u) & \geq \dfrac{a}{2}||\nabla u||_{2}^{2} + \dfrac{b}{4}||\nabla u||_{2}^{4}+\dfrac{Q_0}{2}||u||_{2}^{2}-\dfrac{\mu}{4} K_4||u||_{\frac{8}{3-\lambda}}^{4}-\varepsilon ||u||_{\tau + 1}^{\tau + 1}-K_2 K_\alpha ||u||_{qr_2}^{q} \\
& \geq C_1 ||u||^2 [1-\mu C_2||u||^2 - \varepsilon C_3 ||u||^{\tau-1} - C_4||u||^{q-2}] + \dfrac{b}{4}||\nabla u||_{2}^{4},
\end{align*}
where $ C_1 = \min \left\{ \frac{a}{2}, \frac{Q_0}{2}\right\} >0$ and, similarly,
$$
I'(u)(u) \geq C_5 ||u||^2 [1-\mu C_6||u||^2 - \varepsilon C_7 ||u||^{\tau-1} - C_8||u||^{q-2}] + b||\nabla u||_{2}^{4},
$$
where $ C_5 = \min\{a , Q_0\} $. Hence, for any $ \mu >0 $ and for $ \rho, \varepsilon > 0 $ sufficiently small, the result is valid for this case.
\noindent \textit{Case $a=0$}: Since we are going to take $ \rho >0 $ small, we can assume that $ ||u||^2 < 1 $. Then, $ ||u||_{2}^{2}\geq ||u||_{2}^{4} $. Moreover,
$$
\dfrac{b}{4}||\nabla u||_{2}^{4}+\dfrac{Q_0}{2}||u||_{2}^{4}\geq C_9(||\nabla u||_{2}^{4}+||u||_{2}^{4}) \geq \dfrac{C_9}{4}||u||^4,
$$
where $ C_9=\min\left\{\frac{b}{4} , \frac{Q_0}{2}\right\} $. Consequently,
\begin{example}gin{equation}\label{eq11}
I(u) \geq ||u||^{4}\left(\dfrac{C_9}{4}-C_{10}\mu -\varepsilon C_{11} ||u||^{\tau - 3}- C_{12}||u||^{q-4}\right) .
\end{equation}
Therefore, for $ \rho, \varepsilon, \mu > 0 $ sufficiently small, we also have the result in this case.
\end{proof}
\begin{example}gin{observation}\label{obs1}
In inequality \eqref{eq11}, we can write the right hand side as
\begin{example}gin{small}
\begin{example}gin{equation}\label{eq12}\begin{example}gin{aligned}
\dfrac{C_9}{4}-C_{10}\mu &-\varepsilon C_{11} ||u||^{\tau - 3}- C_{12}||u||^{q-4}\\
& = \left( \dfrac{C_9}{8}-C_{10}\mu \right) + \left(\dfrac{C_9}{8} -\varepsilon C_{11} ||u||^{\tau - 3}- C_{12}||u||^{q-4}\right).\end{aligned}
\end{equation}
\end{small}
For the first term to be positive we need
$$
\dfrac{C_9}{8C_{10}}> \mu .
$$
To be more precise, we can explicit the constants $ C_9 $ and $ C_{10} $. First of all, from the Gagliardo-Nirenberg inequality, we have
$$
||u||_{\frac{8}{3-\lambda}}^{4}\leq K_{GN}^{\frac{3-\lambda}{2}}||u||^4,
$$
where $ K_{GN}>0 $ is the best constant. Hence, taking into account that
$$
C_{9}= \min\left\{\frac{b}{4} , \frac{Q_0}{2}\right\} \mbox{ \ \ \ and \ \ \ } C_{10}= K_{HLS} K_{GN}^{\frac{3-\lambda}{2}},
$$
we get
$$
\dfrac{\min\left\{\frac{b}{4} , \frac{Q_0}{2}\right\}}{8 K_{HLS}K_{GN}^{\frac{3-\lambda}{2}}} > \mu.
$$
Therefore, considering a sufficiently small $ \mu_0 >0 $ such that the above equation is satisfied and the second term in \eqref{eq12} is positive we have the result of Lemma \ref{l5}, in the \textit{degenerate} case, for all $ \mu \in (0, \mu_0) $.
Finally, we highlight that, in the case which $ a_3 \in L^{\infty}(\mathbb{R}) $, we have
$$
\dfrac{\min\left\{\frac{b}{4} , \frac{Q_0}{2}\right\}}{8||a_3||_\infty K_{HLS}K_{GN}^{\frac{3}{2}}} > \mu.
$$
\end{observation}
\begin{example}gin{lemma}\label{l6}
Let $ u\in X \setminus\{0\} $ and $ q>4 $. Then,
$$
I(tu)\searrow 0 \mbox{, as \ } t\mathbb{R}A 0 \ , \ \sup\limits_{t>0}I(tu) < \infty \mbox{ \ and \ } I(tu)\mathbb{R}A - \infty \mbox{, as \ } t\mathbb{R}A \infty .
$$
\end{lemma}
\begin{example}gin{proof}
Let $ u\in X\setminus\{0\} $, $ q>4 $ and $ t> 0 $. From $ (f_4) $ \eqref{eq13} and \eqref{eq9}, we have
$$
I(tu)\leq \dfrac{a}{2}t^2||\nabla u||_{2}^{2} + \dfrac{b}{4}t^4 ||\nabla u||_{2}^{4}+\dfrac{t^2}{2}||Q||_p ||u||_{2p'}^{2}+\dfrac{\mu K_3}{4}t^4||u||_{X}^{4}-C_q t^q ||u||_{q}^{q} \mathbb{R}A - \infty ,
$$
as $ t\mathbb{R}A \infty $. Now, let $ t>0 $ sufficiently small such that $ r_1 \alpha t^2 ||u||^2 < 4 \pi $. Then, from Lemma \ref{l4} and \eqref{eq7},
$$
\displaystyle\int\limits_{\mathbb{R}^2} F(tu) \ dx \leq \varepsilon t^{\tau + 1}||u||_{\tau + 1}^{\tau + 1} + C_1 t^q ||u||_{qr_2}^{q}\mathbb{R}A 0 \mbox{, as \ } t\mathbb{R}A 0.
$$
Thus, we conclude that $ I(tu)\mathbb{R}A 0 $ as $ t\mathbb{R}A 0 $ and, since $ I\in C^1(X, \mathbb{R}) $, $ \sup\limits_{t>0} I(tu) < \infty $.
\end{proof}
From Lemmas \ref{l5} and \ref{l6}, the value $ c_{mp} $ stated in \eqref{eq16} is well-defined and satisfies $ 0< m_\rho \leq c_{mp} < \infty $. Moreover, since $ I $ has the mountain pass geometry, there exists a Cerami sequence for $ I $ at the level $ c_{mp} $, that is, there exists $ (u_{n})\subset X $ such that
\begin{example}gin{equation}\label{eq14}
I(u_{n})\mathbb{R}A c_{mp} \mbox{ \ \ \ and \ \ \ } ||I'(u_{n})||_{X'}(1+||u_{n}||_X) \mathbb{R}A 0 \mbox{, as \ } n \mathbb{R}A \infty .
\end{equation}
Before we investigate boundedness and convergence for such sequences, we will study the geometry of $ P $ and $ V $.
\begin{example}gin{lemma}\label{l7}
For the potential $ V $ and the functional $ P $, we have the following properties:
\begin{example}gin{itemize}
\item[(i)] $ V^+ (t)\mathbb{R}A \infty $, as $ t\mathbb{R}A \infty $, and $ V^+ (t)\mathbb{R}A 0 $, as $ t\mathbb{R}A 0 $;
\item[(ii)] $ V^- (t)\mathbb{R}A 0 $, as $ t\mathbb{R}A \infty $;
\item[(iii)] $ V(t)\mathbb{R}A \infty $, as $ t\mathbb{R}A \infty $.
\item[(iv)] There exists a function $ u_0 \in X \setminus\{0\} $ such that $ P(u_0)<0 $.
\end{itemize}
\end{lemma}
\begin{example}gin{proof}
\textbf{(i)} From condition $ (V_1) $, for $ t\geq 2 $, we have
$$
0<a_{1, 0}\ln(1+t)<a_1(t)\ln(1+t)\geq V^+(t) \mathbb{R}A 0 \mbox{, as \ } t\mathbb{R}A \infty .
$$
On the other side, also from condition $ (V_1) $,
$$
0 \leq V^+(t) \leq ||a_2||_\infty \ln(1+t)\mathbb{R}A 0, \mbox{ as \ } t\mathbb{R}A 0.
$$
\textbf{(ii)} From condition $ (V_2) $, follows that
$$
0\leq V^-(t)\leq \dfrac{a_3(t)}{t} \leq \left\{\begin{example}gin{array}{ll}
\dfrac{||a_3||_\infty}{t}, \mbox{ if \ } a_3 \in L^{\infty}(\mathbb{R}) \\
\dfrac{1}{t^{1+\lambda}} \ \ \ \ , \mbox{ if \ } a_3(t)=t^{-\lambda}
\end{array}
\right. \mathbb{R}A 0, \mbox{ as \ } t\mathbb{R}A \infty .
$$
\textbf{(iii)} It follows immediately from items (i) and (ii).
\noindent \textbf{(iv)} From condition $ (V_3) $ we can consider an open interval $ (c, d)\subset \mathcal{I} $ in which $ V(t) < 0 $. Take $ x_0 \in (B_{c}^{c}\cap B_{d}) $. Let $ \psi $ be a function such that $ \psi \in C^{\infty}(\mathbb{R}dois,\mathbb{R}) $ and $ supp \ \psi \subset B_{\frac{|c-d|}{4}}(x_0) $. Then, $ \psi \in X \setminus\{0\} $ and $ P(\psi) < 0 $.
\end{proof}
As an immediately consequence, we have the following corollary.
\begin{example}gin{corollary}\label{c1}
The set $ \mathcal{A} = \{ u \in X \ ; \ u \neq 0 , P(u) \leq 0 \} \neq \emptyset $.
\end{corollary}
Hence, we are able to find an useful upper bound to the mountain pass level which will make possible to obtain our main results.
\begin{example}gin{lemma}\label{l8}
There exists a constant $K_{7}=K_7(a, b, q, Q, p)>0 $ such that $ c_{mp}\leq \dfrac{K_7}{C_{q}^{\frac{2}{q-2}}} $.
\end{lemma}
\begin{example}gin{proof}
From the continuous Sobolev embeddings, for $ q>4 $, there exists a constant $ C>0 $ such that $ ||u||\geq C||u||_q $, for all $ u\in H^{1}(\mathbb{R}^2)\setminus\{0\} $. Thus, by Corollary \ref{c1}, it makes sense to define
$$
S_q(v)=\dfrac{||v||}{||v||_q} \mbox{ \ \ \ and \ \ \ } S_q = \inf\limits_{v\in \mathcal{A}} S_q(v) \geq \inf\limits_{v\neq 0}S_q(v) > 0 .
$$
Now, from Lemma \ref{l6}, for $ v\in \mathcal{A} $ and $ T>0 $ sufficiently large, $ I(Tv)<0 $. So, we can define a path $ \gamma\in \mathcal{G}amma $ by $ \gamma(t)=tTv $, for $ t \in [0, 1] $, such that
$$
c_{mp} \leq \max\limits_{0 \leq t \leq 1}I(\gamma(t)) = \max\limits_{0 \leq t \leq 1}I(tTv) \leq \max\limits_{t> 0}I(tv) .
$$
Consequently, from $ (Q) $, $ (f_4) $ and the Gagliardo-Nirenberg inequality, for $ \psi \in \mathcal{A} $, we have
\begin{example}gin{align*}
c_{mp} & \leq \max\limits_{t>0} \left\{ \left(\dfrac{a+||Q||_p K_{GN}^{\frac{p-1}{p}}}{2}\right)S_q(\psi)^2 t^2 ||\psi||_{q}^{2} - \dfrac{C_q}{2}t^q ||\psi||_{q}^{q}\right\} \\
& + \max\limits_{t>0} \left\{ \dfrac{b}{4}S_q(\psi)^4 t^4 ||\psi||_{q}^{4} - \dfrac{C_q}{2}t^q ||\psi||_{q}^{q} \right\}.
\end{align*}
Considering the auxiliary functions $ h_1 , h_2: \mathbb{R} \mathbb{R}A \mathbb{R} $ given, respectively, by $ h_1(t)= \mathfrak{a}t^2 - \mathfrak{b}t^q $ and $ h_2(t)=\mathfrak{c}t^4 + \mathfrak{d}t^q $, for $ \mathfrak{a}, \mathfrak{b}, \mathfrak{c}, \mathfrak{d} >0 $, we obtain that
$$
c_{mp} \leq \left(2^{\frac{4-q}{q-2}}-\dfrac{2^{\frac{2}{q-2}}}{q}\right)(a+||Q||_p K_{GN}^{\frac{p-1}{p}})^{\frac{q}{q-2}}S_{q}(\psi)^{\frac{2q}{q-2}}\left(\dfrac{1}{qC_q}\right)^{\frac{2}{q-2}}.
$$
Therefore, taking the infimum over all $ \psi \in \mathcal{A} $, we get the desired result.
\end{proof}
Finally, in the last results of this section we verify when Cerami sequences are, up to subsequences, bounded in $ X $. Consider $ (u_{n})\subset X $ satisfying
\begin{example}gin{equation}\label{eq15}
\exists \ d >0 \mbox{ \ s.t. \ } I(u_{n}) \leq d , \mbox{ for all } n \in \mathbb{N} \mbox{ and } ||I'(u_{n})||_{X'}(1+||u_{n}||_X) \mathbb{R}A 0 , \mbox{ as } n \mathbb{R}A \infty .
\end{equation}
\begin{example}gin{lemma}\label{l9}
Let $ (u_{n})\subset X $ be bounded in $ H^{1}(\mathbb{R}^2) $ such that
\begin{example}gin{equation*}
\liminf\limits_{n\mathbb{R}A \infty} \sup\limits_{y\in \mathbb{Z}^2} \displaystyle\int\limits_{B_2(x)}u_{n}^2(x) dx > 0 .
\end{equation*}
Then, there exists $ u\in H^{1}(\mathbb{R}^2) \setminus\{0\} $ and $ (y_{n})\subset \mathbb{Z}^2 $ such that, up to a subsequence, $ y_{n} \ast u_{n} = u_{n}til \rightharpoonup u\in H^{1}(\mathbb{R}^2) $. Particularly, $ u\neq 0 $ in $ L^{2}(\mathbb{R}^2) $.
\end{lemma}
\begin{example}gin{lemma}\label{l11}
Let $ (u_{n})\subset X $ be a sequence satisfying (\ref{eq15}), bounded in $ H^{1}(\mathbb{R}^2) $ and such that $ ||\nabla u_{n} ||_2 < 2\sqrt{\frac{\pi}{r_1 \alpha}} $, for all $ n \in \mathbb{N} $, and
$$
\liminf_{n\mathbb{R}A \infty} \sup\limits_{y\in \mathbb{Z}^2} \displaystyle\int\limits_{B_2(y)}u_{n}^2(x) dx > 0 .
$$
Then, up to a subsequence, $ (u_{n}til) $ is bounded in $ X $.
\end{lemma}
\begin{example}gin{proof}
The proof follows from Lemmas \ref{l1}, \ref{l2} and \ref{l9}, equations \eqref{eq7} and \eqref{eq10} and the facts that, $ P_1 $ is invariant under $ \mathbb{Z}^2$-translations and that, for all $ n \in \mathbb{N} $,
$$
\dfrac{\mu}{4}P_1(u_{n}) = I(u_{n}) - \dfrac{a}{2}||\nabla u_{n}||_{2}^{2} - \dfrac{b}{4}||\nabla u_{n} ||_{2}^{4} - \dfrac{1}{2}\displaystyle\int\limits_{\mathbb{R}^2} Q(x) u_{n}^2 (x) dx + \dfrac{\mu}{4}P_2(u_{n}) + \int F(u_{n}) dx,
$$
as desired.
\end{proof}
We highlight that the next technical lemma is the key in obtaining multiplicity of solutions for problem \eqref{P}, since it makes possible to verify the validity of $(PS)$ condition at some suitable levels.
\begin{example}gin{corollary}\label{l12}
Let $ (u_{n})\subset X $ under the hypotheses given in Lemma \ref{l11}. Then, up to a subsequence, $ (u_{n}) $ is bounded in $ X $.
\end{corollary}
\begin{example}gin{proof}
To begin with, from Lemma \ref{l11}, passing to a subsequence if necessary, there exists $ (y_{n})\subset \mathbb{Z}^2 $ such that $ u_{n}til \rightharpoonup u $ in $ X $, with $ u \neq 0 $ in $ L^{2}(\mathbb{R}^2) $, $ u_{n}til(x) \mathbb{R}A u(x) $ pointwise a.e. in $ \mathbb{R}dois $ and, from Lemma \ref{l1}, $ u_{n}til \mathbb{R}A u $ in $ L^{s}(\mathbb{R}^2) $, for all $ s \geq 2 $.
Moreover, one can see that there are $ R_1, C_1 >0 $ and $ n_1 \in \mathbb{N} $ such that $ ||u_{n}||_{p, B_{R_1}}^{p} \geq C_1 > 0 $, for all $ n \geq n_1 $. From this we can conclude that $ (y_{n}) $ is bounded in $ \mathbb{Z}^2 $ and, using that
$$
||u_{n}||_{\ast}^{2} = \displaystyle\int\limits_{\mathbb{R}^2} \ln(1+|x- y_{n}|) u_{n}til^2(x) dx \leq ||u_{n}til||_{\ast}^{2}+\ln(1+|y_{n}|)||u_{n}til||_{2}^{2} , \forall \ n \in \mathbb{N},
$$
and that $ (u_{n}) $ is already bounded in $ H^{1}(\mathbb{R}^2) $, the result follows.
\end{proof}
\begin{example}gin{lemma}\label{l10}
Assume $ q>4 $ and $ \alpha > 4\pi $, fixed. Let $ (u_{n})\subset X $ be a sequence satisfying (\ref{eq15}), $ ||\nabla u_{n} ||_2 < 2\sqrt{\frac{\pi}{r_1 \alpha}} $ and that does not verify $ ||u_{n}||\mathbb{R}A 0 $ and $ I(u_{n}) \mathbb{R}A 0 $. Then,
$$
\liminf_{n\mathbb{R}A \infty} \sup\limits_{y\in \mathbb{Z}^2} \displaystyle\int\limits_{B_2(y)}u_{n}^2(x) dx > 0 .
$$
\end{lemma}
\begin{example}gin{proof}
The proof is done by contradiction, applying the Lion's Lemma, and using \eqref{eq10}, \eqref{eq8}, Moser-Trudinger inequality and that $ I'(u_{n})(u_{n})\mathbb{R}A 0 $, as $ n \mathbb{R}A \infty $.
\end{proof}
\section{The \textit{nondegenerate} case ($a>0$)}
This section is devoted to prove Theorems \ref{t1} and \ref{t2}. Since we are going to handle the \textit{nondegerate} case, throughout this section we will assume $ a>0 $ and $ b \geq 0 $. Our strategy consists in proving boundedness of Cerami sequences in $H^{1}(\mathbb{R}^2)$, guaranteeing that it is possible to apply Moser-Trudinger inequality for such sequences and, under what conditions, $ I $ has nontrivial critical points in $ X $. To finish this section, we verify that \eqref{P} has infinitely many solutions.
\begin{example}gin{lemma}\label{l13}
Suppose that $ a>0 $ and $ b\geq 0 $. Let $ (u_{n})\subset X $ a sequence satisfying \eqref{eq15}. Then, $ (u_{n}) $ is bounded in $ H^{1}(\mathbb{R}^2) $.
\end{lemma}
\begin{example}gin{proof}
From condition $ (f_3) $ and \eqref{eq15}, we have
\begin{example}gin{align*}
d+ o(1) & \geq I(u_{n}) - \dfrac{1}{4}I'(u_{n})(u_{n}) = \dfrac{a}{4}||\nabla u_{n}||_{2}^{2}+\dfrac{1}{4}\displaystyle\int\limits_{\mathbb{R}^2} Q(x)u_{n}^2 dx +\displaystyle\int\limits_{\mathbb{R}^2} \left[\dfrac{f(u_{n})u_{n}}{4}-F(u_{n}) \right] dx \\
& \geq \dfrac{\min\{a, Q_0\}}{4}||u_{n}||^2 , \forall \ n \in \mathbb{N}.
\end{align*}
Hence, for all $ n \in \mathbb{N} $,
$$
\left(\dfrac{4d}{\min\{a, Q_0\}}\right)^{\frac{1}{2}}+o(1) \geq ||u_{n}|| , \forall \ n\in \mathbb{N},
$$
and the result follows.
\end{proof}
\begin{example}gin{corollary}\label{c2}
Let $ (u_{n})\subset X $ be a sequence satisfying \eqref{eq15}, with $ d\in (0, c_{mp}] $, or being a Cerami sequence in level $ c_{mp} $. Then, up to a subsequence, there exists a constant $ K_8=K_8(a, b, q, Q, p)>0 $ such that $ ||u_{n}||\leq \dfrac{K_8}{C_{q}^{\frac{1}{q-2}}} $, for all $ n \in \mathbb{N} $.
\end{corollary}
\begin{example}gin{proof}
The proof follows directly from Lemmas \ref{l8} and \ref{l13} and $ \limsup $ properties.
\end{proof}
\begin{example}gin{proposition}\label{p1}
Suppose $ q>4 $ and $ C_q > 0 $ sufficiently large. Let $ (u_{n})\subset X $ a sequence satisfying \eqref{eq15}, with $ d\in (0, c_{mp}] $, or being a Cerami sequence in level $ c_{mp} $. Then, passing to a subsequence, if necessary, only one between the following alternatives hold:
\begin{example}gin{itemize}
\item[(a)] $ ||u_{n}||\mathbb{R}A 0 $ and $ I(u_{n})\mathbb{R}A 0 $.
\item[(b)] There exists a function $ u\in X\setminus\{0\} $ such that $ u_{n} \mathbb{R}A u $ in $ X $ and $ u $ is a critical point to $ I $ in $ X $.
\end{itemize}
\end{proposition}
\begin{example}gin{proof}
Let us suppose that item (a) does not hold. Then, from Lemmas \ref{l13}, \ref{l9}, \ref{l10}, \ref{l11} and Corollary \ref{l12}, passing to a subsequence if necessary, $ u_{n} \rightharpoonup u $ in $ X $, for $ u\in X\setminus\{0\} $. Moreover, from Lemma \ref{l1}, $ u_{n} \mathbb{R}A u $ in $ L^{s}(\mathbb{R}^2) $, for all $ s \geq 2 $.
Now, from Corollary \ref{c2}, up to a subsequence, we can assume that $ r_1 \alpha ||u_{n}||^2 < 4\pi $, for all $ n \in \mathbb{N} $ and $ C_q > 0 $ sufficiently large. Thus, from \eqref{eq15}, \eqref{eq13}, Lemma \ref{l4} and (HLS), we have the following
main properties
\noindent \textbf{(i)} $ |I'(u_{n})(u_{n} -u)|\leq ||I'(u_{n})||_{X'}||u_{n} - u||_X \mathbb{R}A 0 $, as $ n\mathbb{R}A \infty $;
\noindent \textbf{(ii)} $ P_2'(u_{n})(u_{n} - u) \mathbb{R}A 0 $, $ \displaystyle\int\limits_{\mathbb{R}^2} Q(x) u_{n}^2 dx \mathbb{R}A 0 $ and $ \displaystyle\int\limits_{\mathbb{R}^2} f(u_{n})u_{n} dx \mathbb{R}A 0 $, as $ n \mathbb{R}A \infty $.
Moreover, from $ u_{n} \rightharpoonup u $ in $ H^{1}(\mathbb{R}^2) $, the weakly sequentially lower semicontinuity of $ ||\cdot||_2 $ and $ \liminf $ properties, passing to a subsequence if necessary, we have
$$
\langle \nabla u_{n} , \nabla (u_{n} -u) \rangle = ||\nabla u_{n}||_{2}^{2}-||\nabla u||_{2}^{2}+o(1).
$$
Consequently, from Lemma \ref{l3} with $ g(t)=t $, (i) and (ii), follows that
\begin{example}gin{align*}
o(1) & = I'(u_{n})(u_{n} -u) \\
& \geq a (||\nabla u_{n}||_{2}^{2}-||\nabla u||_{2}^{2}) + b||\nabla u_{n}||_{2}^{2}(||\nabla u_{n}||_{2}^{2}-||\nabla u||_{2}^{2}+o(1)) P_{1}'(u_{n})(u_{n}-u) +o(1) \\
& = a (||\nabla u_{n}||_{2}^{2}-||\nabla u||_{2}^{2}) + \displaystyle\int\limits_{\mathbb{R}^2} \displaystyle\int\limits_{\mathbb{R}^2} V^+(|x-y|)u_{n}^2(x)(u_{n}-u)^2(y) dx dy \\
& + \displaystyle\int\limits_{\mathbb{R}^2} \displaystyle\int\limits_{\mathbb{R}^2} V^+(|x-y|)u_{n}^2(x)u(y)(u_{n}(y)-u(y)) dxdy + o(1) \\
& \geq a (||\nabla u_{n}||_{2}^{2}-||\nabla u||_{2}^{2}) + o(1) \geq o(1).
\end{align*}
Hence, we obtain that $ ||\nabla u_{n}||_{2}^{2} - ||\nabla u||_{2}^{2} \mathbb{R}A 0 $ and, since $ u_{n} \mathbb{R}A u $ in $ L^{2}(\mathbb{R}^2) $, $ u_{n} \mathbb{R}A u $ in $ H^{1}(\mathbb{R}^2) $. Moreover, returning to the above inequality we conclude that
$$
\displaystyle\int\limits_{\mathbb{R}^2} \displaystyle\int\limits_{\mathbb{R}^2} V^+(|x-y|)u_{n}^2(x)(u_{n}-u)^2(y) dx dy \mathbb{R}A 0
$$
and, from Lemma \ref{l2}, $ ||u_{n} - u||_\ast \mathbb{R}A 0 $, which implies that $ u_{n} \mathbb{R}A u $ in $ X $. Finally, for $ v \in X $,
$$
|I'(u)(v)| \leq |I'(u)(v)-I'(u_{n})(v)|+||I'(u_{n})||_{X'}||v||\mathbb{R}A 0, \mbox{ as } n \mathbb{R}A \infty .
$$
Therefore, $ u $ is a nontrivial critical point for $ I $ in $ X $.
\end{proof}
\begin{example}gin{proof}[Proof of Theorem \ref{t1}]
Item (a) follows immediately from \eqref{eq14}, Lemma \ref{l5} and Proposition \ref{p1}. Let us prove item (b). From item (a), $ \mathcal{K} \neq \emptyset $. Let $ (u_{n})\subset \mathcal{K} $ such that $ I(u_{n})\mathbb{R}A c_g $.
Observe that $ c_g \in [-\infty, c_{mp}] $. If $ c_g=c_{mp} $ nothing remains to be proved. Assume that $ c_g < c_{mp} $. Thus, combined with the definition of $ \mathcal{K} $, we have that $ (u_{n}) $ satisfies \eqref{eq15} with $ d=c_{mp} $. Hence, from Lemma \ref{l5} and Proposition \ref{p1}, there exists $ u\in X\setminus\{0\} $ such that $ u_{n} \mathbb{R}A u $ in $ X $ and $ u $ is a critical point for $ I $. Moreover, we have $ I(u)=c_g $ which implies, particularly, that $ c_g>-\infty $.
\end{proof}
In order to prove our second main result, let $ k\in \mathbb{N} $, arbitrary but fixed, and $ Z\subset X $ a subspace with $ \dim Z =k $ and norm $ ||\cdot||_Z $. Our goal is to apply a symmetric version of the mountain pass theorem, due to Ambrosetti and Rabinowitz \cite{[ambrosseti]}(see also \cite{[bartolo], [silva]}).
\begin{example}gin{theorem}\label{t4}
(\cite[Theorem 4.1]{[albuquerque]}) Let $ E= E_1 \oplus E_2 $, where $ E $ is a real Banach space and $ E_1 $ is finite dimensional. Suppose that $ J \in C^{1}(E, \mathbb{R}) $ is even, $ J(0)=0 $, and that it verifies
\begin{example}gin{itemize}
\item[$ (J_1) $] there exists $ \tau , r > 0 $ such that $ J(u) \geq \tau $ if $ ||u||_E = r $, $ u\in E_2 $,
\item[$ (J_2) $] there exists a finite-dimensional subspace $ \mathcal{F} \subset E $, with $ \dim E_1 < \dim \mathcal{F} $, and a constant $ \mathcal{B} > 0 $ such that $ \max\limits_{u\in \mathcal{F}} J(u) \leq \mathcal{B} $,
\item[$ (J_3) $] $ J $ satisfies the $ (PS)_c $ condition for all $ c\in (0, \mathcal{B}) $.
\end{itemize}
Then, $ J $ possess at least $ \dim \mathcal{F} - \dim E_1 $ pairs of nontrivial critical points.
\end{theorem}
In the sequence we need to verify the conditions of Theorem \ref{t4}. First of all, one should observe that, under conditions $ (f_1 ')-(f_4) $, $ (Q) $, $ (M) $ and $ (V_1)-(V_3) $ we already have that $ I \in C^{1}(X, \mathbb{R}) $, is even, $ I(0)=0 $ and, from Lemma \ref{l5}, $ I $ verifies $ (J_1) $. So, it remains to prove that $ I $ also verifies $ (J_2) $ and $ (J_3) $.
\begin{example}gin{lemma}\label{l14}
Let $ q>4 $. Then, there exists $ R>0 $ such that $ I(u) \leq 0 $ for all $ u\in X $ verifying $ ||u||_Z \geq R $.
\end{lemma}
\begin{example}gin{proof}
Since $ \dim Z < \infty $, all norms are equivalent. Thus, from condition $ (f_4) $ and \eqref{eq9}, we have
$$
I(u) \leq C_1 ||u||_{Z}^{2}+C_2 ||u||_{Z}^{4} - C_3 ||u||_{Z}^{q} \mathbb{R}A - \infty , \mbox{ as } ||u||_Z \mathbb{R}A \infty .
$$
\end{proof}
\begin{example}gin{lemma}\label{l15}
Let $ q>4 $. Then, there exists $ \end{theorem}a > 0 $, sufficiently small, such that $ \max\limits_{u\in Z}I(u) \leq \end{theorem}a $ and $ r_1 \alpha \frac{\end{theorem}a}{\min\{a, Q_0\}} < \pi $.
\end{lemma}
\begin{example}gin{proof}
Let $ u\in Z \setminus \{0\} $. Thus, from $ \dim Z < \infty $, condition $ (f_4) $ and \eqref{eq9}, there are constants constants $ C_1, C_2, C_3 >0 $, depending on $ a, b, q $ and $ Q $ such that
$$
I(u) \leq C_1 ||u||_{Z}^{2}+ C_2 ||u||_{Z}^{4} - C_q C_3 ||u||_{Z}^{q}.
$$
Arguing in a similar way as in Lemma \ref{l8}, one can find a constant $ C_4>0 $ satisfying
$$
I(u)\leq \dfrac{C_4}{C_{q}^{\begin{example}ta}}, \mbox{ for some exponent } \begin{example}ta = \begin{example}ta(q)>1.
$$
Consequently,
$$
\max\limits_{u\in Z} I(u) \leq \dfrac{C_4}{C_{q}^{\begin{example}ta}}
$$
and, taking $ C_q > 0 $ sufficiently large we find a value $ \end{theorem}a >0 $ sufficiently small as desired.
\end{proof}
In the next proposition we guarantee that $ I $ satisfies the $ (PS)_d $ condition for all $ d \in (0, \end{theorem}a) $. One can observe that the proof can be done in a very similar way as that of Proposition \ref{p1}, so we will omit it here. We highlight that the validity of following lemma is possible only in virtue of Lemma \ref{l12}.
\begin{example}gin{lemma}\label{l16}
The functional $ I $ satisfies condition $ (PSC)_d $ for all $ d\in (0, \end{theorem}a) $.
\end{lemma}
\begin{example}gin{proof}[Proof of Theorem \ref{t2}]
From Lemmas \ref{l5}, \ref{l15} and \ref{l16} and an immediate application of Theorem \ref{t4}, with $ E=X $, $ E_1 = \{0\} $, $ \mathcal{F} = Z $, $ J=I $, $ \tau = m_\rho $, $ r= \rho $ and $ \mathcal{B} = \end{theorem}a $, we get that $ I $ possess at least $ k $ nontrivial critical points. Therefore, as we can make $ k $ as large as we want, we conclude that (\ref{P}) has infinitely many solutions.
\end{proof}
\section{The \textit{degenerate} case ($a=0$)}
In this section we investigate the existence of solutions for \eqref{P} in the \textit{degenerate} case. So, we assume $ a=0 $ and $ b>0 $ throughout it. Since we have the same multiplying constant $ \frac{1}{4} $ in both terms, that one depending on $ ||\nabla \cdot||_2 $ and that on with $ V $, we need a different approach than was used in Section 4. The technique is based in Lemmas \ref{l17} and \ref{l19}, which are inspired in similar results of \cite{[6]}.
\begin{example}gin{lemma}\label{l17}
Let $ (u_{n})\subset X $ a sequence satisfying \eqref{eq15} and $ (t_n)\subset \left. \left(0, \left(\frac{\theta - 4}{\theta}\right)^{\frac{1}{4}}\right]\right.$. Then, $ I(t_n u_{n}) \leq I(u_{n}) $, for all $ n \in \mathbb{N} $.
\end{lemma}
\begin{example}gin{proof}
Observe that
\begin{example}gin{equation}\label{eq17}
I(t_n u_{n}) = \dfrac{b}{4}t_{n}^{4}||\nabla u_{n} ||_{2}^{4}+\dfrac{t_{n}^{2}}{2}\displaystyle\int\limits_{\mathbb{R}^2} Q(x)u_{n}^2 dx + \dfrac{\mu}{4}t_{n}^{4}P(u_{n})-\displaystyle\int\limits_{\mathbb{R}^2} F(t_n u_{n}) dx
\end{equation}
and
\begin{example}gin{equation}\label{eq18}
\mu P'(u_{n})(u_{n}) = I'(u_{n})(u_{n})-b||\nabla u_{n}||_{2}^{4}-\displaystyle\int\limits_{\mathbb{R}^2} Q(x) u_{n}^2 dx + \displaystyle\int\limits_{\mathbb{R}^2} f(u_{n}) u_{n} dx .
\end{equation}
Thus, combining \eqref{eq17} and \eqref{eq18},
\begin{example}gin{small}
\begin{example}gin{align*}
I(t_n u_{n}) - I(u_{n}) & = \dfrac{b}{4}(t_{n}^{4}-1)||\nabla u_{n} ||_{2}^{4}+\dfrac{t_{n}^{2}-1}{2}\displaystyle\int\limits_{\mathbb{R}^2} Q(x)u_{n}^2 dx + \dfrac{\mu}{4}(t_{n}^{4}-1)P(u_{n})\\
&\hspace{3cm}+\displaystyle\int\limits_{\mathbb{R}^2} [F(u_{n})- F(t_n u_{n})] dx \\
& = \dfrac{1}{2}\left(t_{n}^{2}-\dfrac{t_{n}^{4}}{2}-\dfrac{1}{2}\right)\displaystyle\int\limits_{\mathbb{R}^2} Q(x) u_{n}^2 dx + \displaystyle\int\limits_{\mathbb{R}^2} \left[F(u_{n})-F(t_n u_{n}) + \dfrac{t_{n}^{4}-1}{4}f(u_{n})u_{n} \right] dx \\
& \leq 0,
\end{align*}
\end{small}
since $ t_{n}^{2}-\frac{t_{n}^{4}}{2}-\frac{1}{2}\leq 0 $, for all $ n \in \mathbb{N} $, and
$$
F(u_{n})+\dfrac{t_{n}^{4}-1}{4}f(u_{n})u_{n} \leq \left(\dfrac{1}{\theta}+\dfrac{t_{n}^{4}-1}{4}\right)f(u_{n})u_{n} \leq 0, \forall \ n \in \mathbb{N}.
$$
Therefore, $ I(t_n u_{n}) \leq I(u_{n}) $, for all $ n \in \mathbb{N} $.
\end{proof}
\begin{example}gin{lemma}\label{l19}
Let $ (u_{n})\subset X $ satisfying \eqref{eq15}. Then, $ (u_{n}) $ is bounded in $ H^{1}(\mathbb{R}^2) $.
\end{lemma}
\begin{example}gin{proof}
Suppose by contradiction that $ ||u_{n}||\mathbb{R}A \infty $. For a fixed $ \alpha > 4\pi $, define $ v_{n} = \sqrt{\frac{\pi}{r_1 \alpha}}\frac{u_{n}}{||u_{n}||} $, for each $ n \in \mathbb{N} $. Thus, $ ||v_{n}||=\sqrt{\frac{\pi}{r_1 \alpha}} $, for all $ n \in \mathbb{N} $. Consequently, $ (v_{n}) $ is bounded in $ H^{1}(\mathbb{R}^2) $.
\noindent \textbf{Claim:} $ \liminf\limits_{n \mathbb{R}A \infty}\sup\limits_{y \in \mathbb{Z}^2}\displaystyle\int\limits_{B_2(y)}v_{n}^2(x) dx >0 $.
Otherwise, by Lion's Lemma, $ v_{n} \mathbb{R}A 0 $ in $ L^{s}(\mathbb{R}^2) $, for all $ s \in (2, \infty) $. Thus, from \eqref{eq10}, \eqref{eq8}, \eqref{eq15} and Lemma \ref{l4}, we have
\begin{example}gin{align*}
0 \leq \mu P_1(v_{n})+b||\nabla v_{n}||_{2}^{4}+\dfrac{1}{2}Q(x)v_{n}^2 dx & = I'(v_{n})(v_{n})+\mu P_2(v_{n})+\displaystyle\int\limits_{\mathbb{R}^2} f(v_{n})v_{n} dx \\
& \leq \mu K_5 ||v_{n}||_{\frac{8}{3-\lambda}}^{4}+\varepsilon ||v_{n}||_{\tau + 1}^{\tau + 1} + C_1 ||v_{n}||_{qr_2}^{q} \mathbb{R}A 0,
\end{align*}
as $ n \mathbb{R}A \infty $. Consequently, $ P_1(v_{n}) \mathbb{R}A 0 $, $ ||\nabla v_{n}||_2 \mathbb{R}A 0 $ and, by $ (Q) $, $ ||v_{n}||_2 \mathbb{R}A 0 $. Hence, $ ||v_{n}||\mathbb{R}A 0 $, which is a contradiction.
Therefore, from Lemma \ref{l12}, up to a subsequence, $ v_{n} \rightharpoonup v $ in $ X $, for $ v \in X \setminus\{0\} $. We can assume, without loss of generality, that $ v_{n}(x)\mathbb{R}A v(x) $ a.e. in $ \mathbb{R}dois $. Moreover, from the continuous Sobolev embeddings, $ v_{n} \rightharpoonup v $ in $ L^q(\mathbb{R}dois) $. Thus, from weakly sequentially lower semicontinuity, boundedness in $ X $, \eqref{eq13}, the Gagliardo-Nirenberg inequality and condition $ (f_4) $, there exist $ n_0\in \mathbb{N} $ such that, for $ t>0 $,
$$
I(tv_{n}) \leq t^4 \dfrac{b}{4}\left(\dfrac{\pi}{r_1 \alpha}\right)^2 + \dfrac{t^2}{2}\dfrac{\pi}{r_1 \alpha}||Q||_p K_{GN}^{\frac{1}{p'}}+ t^4 C_2 -C_q t^q ||v||_{q}^{q}, \forall \ n \geq n_0.
$$
So, if we choose $ t_0 > 0 $ sufficiently large, $ I(t_0 v_{n}) \leq -1 $, for all $ n\geq n_0 $. But, by other hand,
$$
t_0 \sqrt{\dfrac{\pi}{r_1 \alpha}} \dfrac{1}{||u_{n}||}\mathbb{R}A 0, \mbox{ as } n \mathbb{R}A \infty,
$$
which contradicts Lemma \ref{l6}. Therefore, $ (u_{n}) $ is bounded in $ H^{1}(\mathbb{R}^2) $.
\end{proof}
Let $ (u_{n})\subset X $ be the sequence given in equation \eqref{eq14}. From Lemma \ref{l19}, there exists a constant $ K_{mp}>0 $ such that, passing to a subsequence ie necessary, $ ||u_{n}||\leq K_{mp} $, for all $ n \in \mathbb{N} $. Although we already have a bound for this Cerami sequence, we still need a sufficiently small bound for $ ||\nabla u_{n}||_2 $ in order to apply Lemma \ref{l4}.
\begin{example}gin{lemma}\label{l20}
Let $ (u_{n})\subset X $ a sequence satisfying \eqref{eq14}. Then, there are $ \mu_{mp} > 0 $ sufficiently small and a constant $ K_9 > 0 $ such that, up to a subsequence, $ ||\nabla u_{n}||_2 \leq K_9 c_{mp}^{\frac{1}{4}} $, for all $ n \in \mathbb{N} $ and $ \mu \in (0, \mu_{mp}) $.
\end{lemma}
\begin{example}gin{proof}
From \eqref{eq10} and the Gagliardo-Nirenberg inequality, we have
$$
P_2(u_{n}) \leq K_5 ||u_{n}||_{\frac{8}{3-\lambda}}^{4}\leq K_5 K_{GN}^{\frac{3-\lambda}{2}}||u_{n}||^4 \leq K_5 K_{GN}^{\frac{3-\lambda}{2}} K_{mp}^{4}, \forall \ n \in \mathbb{N}.
$$
Consequently,
\begin{example}gin{align*}
c_{mp}+o(1) & \geq I(u_{n}) - \dfrac{1}{8}I'(u_{n})(u_{n}) \\
& \geq \dfrac{b}{8}||\nabla u_{n}||_{2}^{4}+ \dfrac{3}{8}Q_0 ||u_{n}||_{2}^{2}-\mu \dfrac{K_5 K_{GN}^{\frac{3-\lambda}{2}} K_{mp}^{4}}{4}+ \displaystyle\int\limits_{\mathbb{R}^2} \left(\dfrac{1}{8}f(u_{n})u_{n} - F(u_{n}) \right) dx \\
& \geq \dfrac{b}{8}||\nabla u_{n}||_{2}^{4}-\mu \dfrac{K_5 K_{GN}^{\frac{3-\lambda}{2}} K_{mp}^{4}}{4}.
\end{align*}
Hence, considering $ \mu_{mp}>0 $ sufficiently small, we have
$$
c_{mp}+ o(1)\geq \mathfrak{e} b ||\nabla u_{n}||_{2}^{4},
$$
for all $ n \in \mathbb{N} $, $ \mu \in (0, \mu_{mp}) $ and a value $ \mathfrak{e}\in \left(0, \frac{1}{8}\right) $, depending on $ \mu_{mp} $.
Therefore, the result follows for $ K_9 = \left(\frac{1}{\mathfrak{e} b}\right)^{\frac{1}{4}} > 0 $.
\end{proof}
\begin{example}gin{proof}[Proof of Theorem \ref{t3} - (a)]
The proof follows from Lemmas \ref{l3}, \ref{l4}, \ref{l5}, \ref{l9}, \ref{l10}, \ref{l11}, \ref{l19} and \ref{l20} and Corollary \ref{l12}, arguing in a very similar way as Proposition \ref{p1} and Theorem \ref{t1}.
\end{proof}
Since we already have item (a), we can consider the set $ \mathcal{K} = \{ v \in X \setminus\{0\} \ ; \ I'(v) = 0\} $ that is not empty. So, $ c_{g}\in [-\infty, c_{mp}] $ and there exists a sequence $ (u_{n})\subset \mathcal{K} $ such that
$
I(u_{n}) \mathbb{R}A c_g
$. We will assume $ c_g < c_{mp} $ once if the equality holds, nothing remains to be proved. Also, by the definition of $ \mathcal{K} $, one can see that $ (u_{n}) $ satisfies
$$
||I'(u_{n})||_{X'}(1+||u_{n}||_X)\mathbb{R}A 0, \mbox{ as } n \mathbb{R}A \infty.
$$
Thus, from Lemma \ref{l19}, there exists a constant $ K_g > 0 $ such that $ ||u_{n}||\leq K_g $, for all $ n \in \mathbb{N} $.
\begin{example}gin{lemma}\label{l21}
Let $ (u_{n})\subset X $ be the minimizing sequence for $ c_g $. Then, there are $ \mu_{g} > 0 $ sufficiently small and a constant $ K_{10} > 0 $ such that, up to a subsequence, $ ||\nabla u_{n}||_2 \leq K_{10} c_{mp}^{\frac{1}{4}} $, for all $ n \in \mathbb{N} $ and $ \mu \in (0, \mu_{g}) $.
\end{lemma}
\begin{example}gin{proof}
Similarly as done in Lemma \ref{l20}, since $ c_g < c_{mp} $, up to a subsequence, we get
$$
c_{mp}+o(1) \geq \dfrac{b}{8}||\nabla u_{n}||_{2}^{4}-\mu \dfrac{K_5 K_{GN}^{\frac{3-\lambda}{2}} K_{g}^{4}}{4}.
$$
Thus, considering $ \mu_{g}>0 $ sufficiently small, we have
$$
c_{mp}+ o(1)\geq \mathfrak{r} b ||\nabla u_{n}||_{2}^{4},
$$
for all $ n \in \mathbb{N} $, $ \mu \in (0, \mu_{g}) $ and a value $ \mathfrak{r}\in \left(0, \frac{1}{8}\right) $, depending on $ \mu_{g} $.
Therefore, the result follows for $ K_{10} = \left(\frac{1}{\mathfrak{r} b}\right)^{\frac{1}{4}} > 0 $.
\end{proof}
\begin{example}gin{observation}\label{obs2}
One should observe that, since the nonemptiness of $ \mathcal{K} $ depends on the existence of a solution at the mountain pass level, the value $ \mu_g $ must also satisfy $ \mu_g \leq \mu_{mp} $.
\end{observation}
\begin{example}gin{proof}[Proof of Theorem \ref{t3}-(b)]
The proof follows from Lemmas \ref{l3}, \ref{l4}, \ref{l5}, \ref{l9}, \ref{l10}, \ref{l11}, \ref{l19} and \ref{l21} and Corollary \ref{l12}, arguing in a very similar way as Proposition \ref{p1} and Theorem \ref{t1}.
\end{proof}
\noindent \textbf{Acknowledgements:} The first author was supported by Coordination of Superior Level Staff Improvement-(CAPES)-Finance Code 001 and S\~ao Paulo Research Foundation-(FAPESP), grant $\sharp $ 2019/22531-4, while the second author was supported by National Council for Scientific and Technological Development-(CNPq), grant $\sharp $ 307061/2018-3 and FAPESP grant $\sharp $ 2019/24901-3. The third author
is a member of the {\em Gruppo Nazionale per
l'Analisi Ma\-te\-ma\-ti\-ca, la Probabilit\`a e le loro Applicazioni} (GNAMPA)
of the {\em Istituto Nazionale di Alta Matematica} (INdAM) and was partly supported by
the {\em Fondo Ricerca di Base di Ateneo -- Eser\-ci\-zio 2017--2019} of the University of Perugia, named {\em PDEs and Nonlinear Analysis}.
\begin{example}gin{thebibliography}{2}
\bibitem{[albuquerque]} Albuquerque, F. S. B. (2014) Nonlinear Schrodinger elliptic systems involving exponential critical growth in $\mathbb{R}^2$, \textit{Electronic Journal of Differential Equations}. Vol. 2014, n. 59, pp. 1-12.
\bibitem{[ambrosseti]} Ambrosetti, A. and Rabinowitz P.H. (1973) Dual variational methods in critical point theory and applications, \textit{Journal of Functional Analysis}. 14, 349–381.
\bibitem{[bartolo]} Bartolo, P., Benci, V., Fortunato, D. (1983) Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity, \textit{Nonlinear Analysis: Theory, Methods \& Applications}. 7, 981–1012.
\bibitem{[boer]} Böer, E. de S. and Miyagaki, O. H. (2021) Existence and multiplicity of solutions for the fractional $p$-Laplacian Choquard logarithmic equation involving a nonlinearity with exponential critical and subcritical growth, \textit{J. Math. Phys.} 62, 051507.
\bibitem{[boer2]} Böer, E. de S. and Miyagaki, O.H. (2021) $ (p, N)$-Choquard logarithmic equation involving a nonlinearity with exponential critical growth: existence and multiplicity, \textit{ArXiv:2105.11442 [Math]}, submitted.
\bibitem{[5]} Cao, D. M. (1992) Nontrivial solution of semilinear elliptic equations with critical exponent in $\mathbb{R}^2$. \textit{Communications in Partial Differential Equations}, 17, 407–435.
\bibitem{[6]} Cingolani, S. and Weth, T. (2016) On the planar Schrödinger–Poisson system. \textit{Annales de l’Institut Henri Poincare (C) Non Linear Analysis}, 33, 169–197.
\bibitem{[10]} Du, M. and Weth, T. (2017) Ground states and high energy solutions of the planar Schrödinger–Poisson system. \textit{Nonlinearity}, 30, 3492–3515.
\bibitem{[11]} Fröhlich, H. (1937) Theory of electrical breakdown in Ionic crystals. \textit{Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences}, 160, 230–241.
\bibitem{[12]} Fröhlich, H. (1954) Electrons in lattice fields. \textit{Advances in Physics}, 3, 325–361.
35
\bibitem{[jin]} Jin, J. and Wu, X. (2010) Infinitely many radial solutions for Kirchhoff-type problems in RN, \textit{Journal of Mathematical Analysis and Applications}. 369, 564–574.
\bibitem{[Lam]} Lam, N. and Lu, G. (2014) Elliptic equations and systems with subcritical and critical exponential growth without the Ambrosetti-Rabinowitz condition, \textit{J. Geom. Anal.}, 24,118-143.
\bibitem{[pucci]} Liang, S., Pucci, P. and Zhang, B. (2020) Multiple solutions for critical Choquard-Kirchhoff type equations, \textit{Advances in Nonlinear Analysis}. 10, 400–419.
\bibitem{[liang]} Liang, S. and Zhang, J. (2015) Existence of solutions for Kirchhoff type problems with critical nonlinearity in $\mathbb{R}^N$, \textit{Z. Angew. Math. Phys.} 66, 47–562.
\bibitem{[15]} Lieb, E. H. (1983) Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. \textit{The Annals of Mathematics}, 118, 349.
\bibitem{[16]} Lions, P.-L. (1987) Solutions of Hartree-Fock equations for Coulomb systems. \textit{Communications in Mathematical Physics}, 109, 33–97.
\bibitem{[olimpio]} Miyagaki, O.H. and Pucci, P. (2019) Nonlocal Kirchhoff problems with Trudinger–Moser critical nonlinearities, \textit{Nonlinear Differ. Equ. Appl.}, 26:27, 1-26.
\bibitem{[17]} Moser, J. (1971) A Sharp form of an inequality by N. Trudinger. \textit{Indiana University
Mathematics Journal}, 20, 1077–1092.
\bibitem{[18]} Penrose, R. (1996) On gravity’s role in quantum state reduction. \textit{General Relativity and Gravitation}, 28, 581–600.
\bibitem{[pucci2]} Pucci, P., Xiang, M. and Zhang, B. (2019) Existence results for Schrödinger–Choquard–Kirchhoff equations involving the fractional p-Laplacian, \textit{Advances in Calculus of Variations}, 12, 253–275.
\bibitem{[ruf]} Ruf, B. and Sani, F. (2013) Ground states for elliptic equations in $\mathbb{R}^{2}$ with exponential critical growth. Magnanini, R., Sakaguchi, S., and Alvino, A. (eds.), \textit{Geometric properties for parabolic and elliptic PDE’s}, vol. 2, pp. 251–267, Springer, Milan.
\bibitem{[silva]} Silva, E. A. B. (1988) \textit{Critical point theorems and applications to differential equations}, PhD. Thesis, University of Wisconsin-Madison.
\bibitem{[22]} Wilson, A. J. C. (1955) Untersuchungen über die Elektronentheorie der Kristalle by S. I.
Pekar. \textit{Acta Crystallographica}, 8, 70–70.
\end{thebibliography}
\end{document} | math |
۔ زبان تہٕ ادبٕچ دستاویزکٲری ڈِجٹل فارمیٹس منٛز | kashmiri |
BC Tent & Awning Co., Inc. is an event rental and photo booth business based in Avon, Massachusetts. This company, owned by Bob Costa, has been providing elegant and personal tent services since 1980. Over the years, Bob Costa has greatly expanded his selection of tents that he provides to couples. This premier business is happy to serve couples all throughout not only Massachusetts, but also Rhode Island and Southern New Hampshire.
We rented a tent for our reception along with tables, chairs, dance floor and lanterns. It was beautiful. The staff at BC Tent were very accommodating with all of our changes.
When my daughter became engaged we searched for the perfect venue and fell in love with the Bradley Estate in Canton Ma. The drawback was we could only have up to 150 people for an in house wedding; if we were to use the Estate we would have to have a tent off the side of the Estate. I was nervous about this since I had never been to an outdoor wedding before but Valerie and Bob from BC Tent were very helpful (Bob is the the owner of BC Tent). Anytime I would think of something we might need they would email me pictures and make suggestions; sometimes they would say.... you really don't need that... or I suggest you do it this way. In the end the tent was spectacular the tables and chairs we rented for the ceremony were perfect as well as the seating for dinner was just right. The Crew was at the site early and everything was flawless from set up to the breakdown. I didn't need to worry about anything!
Thank you Valerie and Bob for all your help. It really was a night to remember!
Always a pleasure to work with, customer service is one of the highlights of this company. Always friendly and ready to accommodate. Tents are beautiful and well-maintained. We'd recommend B.C. Tent & Awning Co. for your outdoor rehearsal or wedding! We personally look forward to working with them soon! | english |
The mission of FusionCare Pharmacy is to see our business through the eyes of our customer, to exceed their expectations, and to provide the greatest selection and value in their pharmacy needs.
© 2016 FusionCare Pharmacy. All rights reserved. | english |
#include <stdio.h>
void p(unsigned char);
int main() {
//p:1 assign int from char
unsigned char t;
t = 5;
printf("c: %c\n", t);
printf("d: %d\n", t);
printf("u: %u\n", t);
//p:2 difference between unsigned char and char
t = 0x80;
p(t);
t = 0x7F;
p(t);
return 0;
}
void p(unsigned char v) {
char c = v;
unsigned uc = v;
int i = v, j = uc;
unsigned int ui = v, uj = uc;
printf("c: %c %c\n", c, uc);
printf("d: %d %d\n", i, j);
printf("u: %u %u\n", ui, uj);
} | code |
بۄدٕ برٲرۍ ووٚنُس یورٕ آ یِم چھِ سؠٹھہٕے عجیب جاناوار | kashmiri |
پَنٛجاب چھُ شُمٲلؠ ہِندوستانَس مَنٛز اَکھ رِیاسَتھ۔
== حوالہ == | kashmiri |
\begin{document}
\title{Bessel polynomials, double factorials and context-free grammars ootnote{This work is supported by~NSFC (11126217) and the Fundamental Research Funds for the Central Universities (N100323013).}
\begin{abstract}
The purpose of this paper is to show that Bessel polynomials, factorials and Catalan triangle can be generated by using
context-free grammars.
\\
{\sl Keywords:}\quad Bessel polynomials; Double factorials; Catalan triangle; Context-free grammars
\end{abstract}
\section{Introduction}\label{sec:intro}
The grammatical method was introduced by Chen~\cite{Chen93} in the study of exponential structures in combinatorics. Let $A$ be an alphabet whose letters are regarded as independent commutative indeterminates. Following Chen~\cite{Chen93}, a {\it context-free grammar} $G$ over $A$ is defined as a set of substitution rules replacing a letter in $A$ by a formal function over $A$.
The formal derivative $D$ is a linear operator defined with respect to a context-free grammar $G$.
For example, if $G=\{x\rightarrow xy, y\rightarrow y\}$, then $$D(x)=xy,D(y)=y,D^2(x)=x(y+y^2),D^3(x)=x(y+3y^2+y^3).$$
For any formal functions $u$ and $v$, we have
$$D(u+v)=D(u)+D(v),\quad D(uv)=D(u)v+uD(v) \quad and\quad D(f(u))=\frac{\partial f(u)}{\partial u}D(u),$$
where $f(x)$ is a analytic function.
Using Leibniz's formula, we obtain
\begin{equation}\label{Dnab-Leib}
D^n(uv)=\sum_{k=0}^n\binom{n}{k}D^k(u)D^{n-k}(v).
\end{equation}
Let $[n]=\{1,2,\ldots,n\}$. {\it The Stirling number of the second kind}
$\Stirling{n}{k}$ is the number of ways to partition $[n]$ into $k$ blocks.
Let ${\mathcal S}_n$ denote the symmetric group of all permutations of $[n]$.
The {\it Eulerian number} $\Eulerian{n}{k}$ enumerates the number of permutations in ${\mathcal S}_n$
with $k$ descents (i.e., $i<n,\pi(i)>\pi(i+1)$). The numbers $\Eulerian{n}{k}$
satisfy the recurrence relation
$$\Eulerian{n}{k}=(k+1)\Eulerian{n-1}{k}+(n-k)\Eulerian{n-1}{k-1},$$
with initial condition $\Eulerian{0}{0}=1$ and boundary conditions $\Eulerian{0}{k}=0$ for $k\geq 1$.
There is a close relationship between context-free grammars and combinatorics.
The reader is referred to~\cite{Chen121,Ma12} for recent results on this subject.
Let us now recall two classical results.
\begin{proposition}[{\cite[Eq. 4.8]{Chen93}}]
If $G=\{x\rightarrow xy, y\rightarrow y\}$,
then
\begin{equation*}
D^n(x)=x\sum_{k=1}^n\Stirling{n}{k}y^k.
\end{equation*}
\end{proposition}
\begin{proposition}[{\cite[Section 2.1]{Dumont96}}]
If $G=\{x\rightarrow xy, y\rightarrow xy\}$,
then
\begin{equation*}
D^n(x)=x\sum_{k=0}^{n-1}\Eulerian{n}{k}x^{k}y^{n-k}.
\end{equation*}
\end{proposition}
The purpose of this paper is to show that Bessel polynomials, factorials and Catalan triangle can be generated by using context-free grammars.
\section{Bessel polynomials}\label{sec:intro}
The well known {\it Bessel polynomials} $y_n(x)$ were introduced by Krall and Frink~\cite{Krall45}
as the polynomial solutions of the second-order differential equation
$$x^2\frac{d^2y_n(x)}{dx^2}+(2x+2)\frac{dy_n(x)}{dx}=n(n+1)y_n(x).$$
The Bessel polynomials $y_n(x)$ are a family of orthogonal polynomials and they have been extensively studied and applied (see~\cite[\textsf{A001498}]{Sloane}).
The polynomials $y_n(x)$ can be generated by using the Rodrigues formula
$$y_n(x)=\frac{1}{2^n}e^{\frac{2}{x}}\frac{d^n}{dx^n}\left(x^{2n}e^{-\frac{2}{x}}\right).$$
Explicitly, we have
$$y_n(x)=\sum_{k=0}^n\frac{(n+k)!}{(n-k)!k!}\left(\frac{x}{2}\right)^k.$$
These polynomials satisfy the recurrence relation
\begin{equation*}\label{ynx-recu}
y_{n+1}(x)=(2n+1)xy_n(x)+y_{n-1}(x)\quad {\text for}\quad n\geq 0,
\end{equation*}
with initial conditions $y_{-1}(x)=y_{0}(x)=1$.
The first few of the polynomials $y_n(x)$ are
\begin{align*}
y_1(x)&=1+x, \\
y_2(x)&=1+3x+3x^2, \\
y_3(x)&=1+6x+15x^2+15x^3.
\end{align*}
We present here a grammatical characterization of the Bessel polynomials $y_n(x)$.
\begin{theorem}\label{Thm11}
If $G=\{a\rightarrow ab, b\rightarrow b^2c, c\rightarrow bc^2\}$, then
\begin{equation*}\label{Dnab}
D^n(ab)=ab^{n+1}y_n(c)\quad {\text for}\quad n\geq 0.
\end{equation*}
\end{theorem}
\begin{proof}
Let $$a(n,k)=\frac{(n+k)!}{2^k(n-k)!k!}.$$
Then $y_n(c)=\sum_{k=0}^na(n,k)c^k$.
It is easy to verify that
\begin{equation}\label{ank-recurrence}
a(n+1,k)=a(n,k)+(n+k)a(n,k-1).
\end{equation}
For $n\geq 0$,
we define
\begin{equation}\label{Dnab-def}
D^n(ab)=ab^{n+1}\sum_{k=0}^{n}E(n,k)c^k.
\end{equation}
Note that $D(ab)=ab^2(1+c)$. Hence $E(1,0)=a(1,0),E(1,1)=a(1,1)$.
It follows from~\eqref{Dnab-def} that
$$D^{n+1}(ab)=D(D^n(ab))=ab^{n+2}\sum_{k=0}^nE(n,k)c^k+ab^{n+2}\sum_{k=0}^n(n+k+1)E(n,k)c^{k+1}.$$
Therefore,
$$E(n+1,k)=E(n,k)+(n+k)E(n,k-1).$$
Comparing with~\eqref{ank-recurrence}, we see that
the coefficients $E(n,k)$ satisfy the same recurrence relation and initial conditions as $a(n,k)$, so they agree.
\end{proof}
For the context-free grammar $$G=\{a\rightarrow ab, b\rightarrow b^2c, c\rightarrow bc^2\},$$
in the same way as above we find that
$$D^n(a^2b)=2^na^2b^{n+1}y_n\left(\frac{c}{2}\right)\quad {\text for}\quad n\geq 0.$$
By Theorem~\ref{Thm11}, we obtain $D^k(a)=ab^ky_{k-1}(c)$ for $k\geq 0$. The {\it double factorial} of odd numbers are defined by
\begin{equation*}
(2n-1)!!=1 \cdot 3 \cdot 5 \cdot \dots \cdot (2n-1),
\end{equation*}
and for even numbers
\begin{equation*}
(2n)!!=2 \cdot 4 \cdot 6 \cdot \dots \cdot (2n).
\end{equation*}
As usual, set $(-1)!!=0!!=1$. It is clear that
\begin{equation*}\label{Dnb}
D^{n}(b)=(2n-1)!!b^{n+1}c^n\quad {\text for}\quad n\geq 0.
\end{equation*}
By~\eqref{Dnab-Leib}, the following corollary is immediate.
\begin{corollary}
For $n\geq 0$, we have
$$y_n(x)=\sum_{k=0}^n(2n-2k-1)!!\binom{n}{k}y_{k-1}(x)x^{n-k}.$$
\end{corollary}
\section{Polynomials associated with diagonal Pad\'e approximation to the exponential function}\label{approximation}
The Pad\'e approximations arise naturally in many branches of mathematics and have been extensively investigated (see~\cite{Prevost10} for instance).
The {\it diagonal Pad\'e approximation} to the exponential function $e^x$ is the unique rational function $$R_n(x)=\frac{P_n(x)}{P_n(-x)},$$
where $$P_n(x)=\sum_{k=0}^nM(n,k)x^{n-k} \quad {\text and}\quad M(n,k)=\frac{(n+k)!}{(n-k)!k!}.$$
Clearly, $P_n(1)=y_n(2)$, where $y_n(x)$ is the Bessel polynomials.
It is easy to verify that the numbers $M(n,k)$ satisfy the recurrence relation
\begin{equation}\label{Mnk-recu-1}
M(n+1,k)=M(n,k)+(2n+2k)M(n,k-1).
\end{equation}
The first few of the polynomials $P_n(x)$ are given as follows (see~\cite[A113025]{Sloane}):
\begin{align*}
P_0(x)& =1, \\
P_1(x)& =x+2, \\
P_2(x)& =x^2+6x+12, \\
P_3(x)& =x^3+12x^2+60x+120.
\end{align*}
We present here a grammatical characterization of the polynomials $P_n(x)$.
\begin{theorem}\label{thm8}
If $G=\{a\rightarrow ab^2, b\rightarrow b^3c^2, c\rightarrow b^2c^3\}$,
then
\begin{equation*}
D^n(ab^2)=ab^{2n+2}c^{2n}P_n\left(\frac{1}{c^2}\right).
\end{equation*}
\end{theorem}
\begin{proof}
For $n\geq0$, we define
\begin{equation}\label{Nnk-recu-2}
D^n(ab^2)=ab^{2n+2}\sum_{k=0}^nN(n,k)c^{2k}.
\end{equation}
Note that $D(ab^2)=ab^4(1+2c^2)$.
Hence $N(1,0)=M(1,0),N(1,1)=M(1,1)$.
It follows from~\eqref{Nnk-recu-2} that
$$D^{n+1}(ab^2)=ab^{2n+4}\sum_{k=0}^nN(n,k)c^{2k}+ab^{2n+4}\sum_{k=0}^n(2n+2k+2)N(n,k)c^{2k+2}.$$
Therefore,
$$N(n+1,k)=N(n,k)+(2n+2k)N(n,k-1).$$
Comparing with~\eqref{Mnk-recu-1}, we see that
the coefficients $N(n,k)$ satisfy the same recurrence relation and initial conditions as $M(n,k)$, so they agree.
\end{proof}
Along the same lines, we immediately deduce the following corollary.
\begin{corollary}
Let $y_n(x)$ be the Bessel polynomials.
If $G=\{a\rightarrow ab^2, b\rightarrow b^3c^2, c\rightarrow b^2c^3\}$,
then
\begin{equation*}
D^n(a^2b^2)=2^na^2b^{2n+2}y_n(c^2).
\end{equation*}
\end{corollary}
\section{Double factorials}\label{sec:factorial}
The following identity was studied systematically by
Callan~\cite[Section 4.8]{Callan}:
\begin{equation}\label{Callan}
\sum_{k=1}^nk!\binom{2n-k-1}{k-1}(2n-2k-1)!!=(2n-1)!!.
\end{equation}
As pointed out by Callan~\cite{Callan}, the identity~\eqref{Callan} counts different combinatorial structures, such as {\it increasing ordered trees} of $n$ edges by outdegree $k$ of the root and the sum of the weights of all vertices labeled $k$ at depth $n-1$ in the {\it Catalan tree} (see~\cite[\textsf{A102625}]{Sloane}).
Let $$R(n,k)=k!\binom{2n-k-1}{k-1}(2n-2k-1)!!.$$
Thus, $\sum_{k=1}^nR(n,k)=(2n-1)!!$. It is easy to verify that
\begin{equation}\label{Tnk-recu}
R(n+1,k)=(2n-k)R(n,k)+kR(n,k-1),
\end{equation}
with initial conditions $R(0,0)=1$ and $R(0,k)=0$ for $k\geq 1$ or $k<0$. For $n\geq 1$,
let $R_n(x)=\sum_{k=1}^nR(n,k)x^k$.
The first few of the polynomials $R_n(x)$ are
\begin{align*}
R_1(x)& =x, \\
R_2(x)& =x+2x^2, \\
R_3(x)& =3x+6x^2+6x^3,\\
R_4(x)& =15x+30x^2+36x^3+24x^4.
\end{align*}
\begin{theorem}\label{Thm5}
If $G=\{a\rightarrow a^2b, b\rightarrow b^2c, c\rightarrow bc^2\}$, then
\begin{equation}\label{Dnab-3}
D^n(a)=ab^{n}\sum_{k=1}^{n}R(n,k)a^{k}c^{n-k}\quad {\text for}\quad n\geq 0.
\end{equation}
\end{theorem}
\begin{proof}
Note that $D(a)=a^2b$ and $D^2(a)=ab^2(ac+2a^2)$.
For $n\geq 1$, we define
\begin{equation}\label{Dna-def}
D^n(a)=ab^{n}\sum_{k=1}^{n}r(n,k)a^{k}c^{n-k}.
\end{equation}
Hence $r(1,1)=R(1,1),r(2,1)=R(2,1)$ and $r(2,2)=R(2,2)$.
It follows from~\eqref{Dna-def} that
$$D(D^n(a))=ab^{n+1}\sum_{k=1}^n(2n-k)r(n,k)a^kc^{n-k+1}+ab^{n+1}\sum_{k=0}^n(k+1)r(n,k)a^{k+1}c^{n-k}.$$
Therefore,
$$r(n+1,k)=(2n-k)r(n,k)+kr(n,k-1).$$
Comparing with~\eqref{Tnk-recu}, we see that
the coefficients $r(n,k)$ satisfy the same recurrence relation and initial conditions as $R(n,k)$, so they agree.
\end{proof}
In the following discussion, we also consider the context-free grammar
\begin{equation*}\label{grammar}
G=\{a\rightarrow a^2b, b\rightarrow b^2c, c\rightarrow bc^2\}.
\end{equation*}
Note that $$D(ab)=a^2b^2+ab^2c,D^2(ab)=ab^3(3c^2+3ac+2a^2).$$
For $n\geq 0$, we define
\begin{equation}\label{Hnk-def}
D^n(ab)=ab^{n+1}\sum_{k=0}^nH(n,k)a^kc^{n-k}.
\end{equation}
It follows that
$$D(D^n(ab))=ab^{n+2}\sum_{k=0}^n(2n-k+1)H(n,k)a^kc^{n-k+1}+ab^{n+2}\sum_{k=0}^n(k+1)H(n,k)a^{k+1}c^{n-k}.$$
Hence, the numbers $H(n,k)$ satisfy the recurrence relation
\begin{equation}\label{Gnk-recu}
H(n+1,k)=(2n-k+1)H(n,k)+kH(n,k-1),
\end{equation}
with initial conditions $H(1,0)=H(1,1)=1$ and $H(1,k)=0$ for $k\geq 2$ or $k<0$.
Using~\eqref{Gnk-recu}, it is easy to verify that
$$H(n,k)=\frac{(2n-k)!}{2^{n-k}(n-k)!}.$$
It should be noted that the numbers $H(n,k)$ are entries in a {\it double factorial triangle} (see~\cite[\textsf{A193229}]{Sloane}).
In particular, we have $H(n,0)=(2n-1)!!, H(n,n)=n!$ and $\sum_{k=0}^nH(n,k)=(2n)!!$.
Moreover, combining~\eqref{Dnab-Leib}, \eqref{Dnab-3} and~\eqref{Hnk-def}, we obtain
$$H(n,k)=\sum_{m=k}^n\binom{n}{m}(2n-2m-1)!!R(m,k)$$
for $n\geq 1$ and $0\leq k\leq n$.
For $n\geq 1$, we define
$$x(x+2)(x+4)\cdots (x+2n-2)=\sum_{k=1}^np(n,k)x^k$$
and
$$(x+1)(x+3)\cdots (x+2n-1)=\sum_{k=0}^nq(n,k)x^k.$$
The the triangular arrays $\{p({n,k})\}_{n\geq 1,1\leq k\leq n}$ and $\{q({n,k})\}_{n\geq 1,0\leq k\leq n}$ are both {\it double Pochhammer triangles} (see~\cite[\textsf{A039683,A028338}]{Sloane}).
The following theorem is in a sense ``dual" to Theorem~\ref{Thm5},
and we omit the proof for brevity.
\begin{theorem}\label{thmpq}
If $G=\{a\rightarrow ab^2, b\rightarrow b^2c, c\rightarrow bc^2\}$,
then we have
$$D^n(a)=ab^n\sum_{k=1}^np(n,k)b^kc^{n-k}$$
and
$$D^n(ab)=ab^{n+1}\sum_{k=0}^nq(n,k)b^kc^{n-k}.$$
\end{theorem}
Set $p(0,0)=q(0,0)=1$. By~\eqref{Dnab-Leib},
we immediately obtain
$$q(n,k)=\sum_{m=k}^n\binom{n}{m}(2n-2m-1)!!p(m,k)$$
for $n\geq 0$ and $0\leq k\leq n$.
\section{Catalan triangle}\label{sec:Catalan's triangle}
The classical {\it Catalan triangle}
is defined by the recurrence relation
$$T(n,k)=T(n-1,k)+T(n,k-1),$$
with initial conditions $T(0,0)=1$ and $T(0,k)=0$ for $k>0$ or $k<0$ (see~\cite[\textsf{A009766}]{Sloane}).
The numbers $T(n,k)$ are often called {\it ballot numbers}. Explicitly,
\begin{equation}\label{Catalan}
T(n,k)=\binom{n+k}{k}\frac{n-k+1}{n+1}\quad {\text for}\quad 0\leq k\leq n.
\end{equation}
Moreover, $\sum_{k=0}^{n}T(n,k)=T(n+1,n+1)=C(n+1)$, where $C(n)$ is the well known {\it Catalan number}.
Catalan numbers appear in a wide range of problems (see~\cite{Sagan12} for instance).
It follows from~\eqref{Catalan} that
\begin{equation}\label{Catalan-recu}
(n+2)T(n+1,k)=(n-k+2)T(n,k)+(2n+2k)T(n,k-1).
\end{equation}
This recurrence relation gives rise to the following result.
\begin{theorem}
If $G=\{a\rightarrow a^2b^2, b\rightarrow b^3c^2, c\rightarrow b^2c^3\}$,
then we have
\begin{equation*}\label{Catalan-Dnab}
D^n(a^2b^2)=(n+1)!a^2b^{2n+2}\sum_{k=0}^nT(n,k)a^{n-k}c^{2k}.
\end{equation*}
\end{theorem}
\begin{proof}
It is easy to verify that $D(a^2b^2)=2a^2b^4(a+c^2)$ and $D^2(a^2b^2)=3!a^2b^6(a^2+2ac^2+2c^4)$.
For $n\geq 0$, we define
\begin{equation*}\label{tnk}
D^n(a^2b^2)=(n+1)!a^2b^{2n+2}\sum_{k=0}^nt(n,k)a^{n-k}c^{2k}.
\end{equation*}
Note that
$$\frac{D^{n+1}(a^2b^2)}{(n+1)!a^2b^{2n+4}}=\sum_{k=0}^n(n-k+2)t(n,k)a^{n-k+1}c^{2k}+\sum_{k=0}^n(2n+2k+2)t(n,k)a^{n-k}c^{2k+2}.$$
Thus, we get
$$(n+2)t(n+1,k)=(n-k+2)t(n,k)+(2n+2k)t(n,k-1).$$
Comparing with~\eqref{Catalan-recu}, we see that
the coefficients $t(n,k)$ satisfy the same recurrence relation and initial conditions as $T(n,k)$, so they agree.
\end{proof}
In the same way as above we find that if $G=\{a\rightarrow a^2b^2, b\rightarrow b^3c^2, c\rightarrow b^2c^3\}$, then
\begin{equation*}\label{Catalan-Dnab}
D^n(ab^2)=n!ab^{2n+2}\sum_{k=0}^n\binom{n+k}{k}a^{n-k}c^{2k}
\end{equation*}
and
\begin{equation*}\label{Catalan-Dnab}
D^n(b)=\prod_{k=0}^{n-1}(4k+1)b^{2n+1}c^{2n}.
\end{equation*}
It should be noted that $\binom{n+k}{k}$ is the number of lattice paths from $(0,0)$ to $(n,k)$ using steps $(1,0)$ and $(0,1)$ (see~\cite[\textsf{A046899}]{Sloane}) and $\prod_{k=0}^{n-1}(4k+1)$ is the {\it quartic factorial number} (see~\cite[A007696]{Sloane}).
\end{document} | math |
सउदी ने अमेरिका के ९/११ कानून के विनाशकारी नतीजे होने की चेतावनी दी
सउदी अरब ने चेतावनी दी है कि अमेरिका के ९/११ कानून के परिणाम विध्वसंकारी होंगे। इसके बाद से लंबे समय से सहयोगी रहे दोनों देशों के बीच तनाव बढ़ गया है।
रियाद: सउदी अरब ने चेतावनी दी है कि अमेरिका के ९/११ कानून के परिणाम विध्वसंकारी होंगे। इसके बाद से लंबे समय से सहयोगी रहे दोनों देशों के बीच तनाव बढ़ गया है। दरअसल इस कानून के तहत हमले के पीडि़तों को सउदी के खिलाफ मुकदमा करने की इजाजत होगी। सउदी की ओर से यह चेतावनी इसलिए दी गई क्योंकि बुधवार को अमेरिकी कांग्रेस ने देशों के बीच संबंधों से जुड़े कानून आतंकी गतिविधियों के प्रायोजकों के खिलाफ न्याय :जेएएसटीए: के पक्ष में राष्ट्रपति बराक ओबामा के वीटो की अवहेलना करते हुए बड़ी संख्या में मतदान किया।
जेएएसटीए हमले के पीडि़तों और आतंकवाद पीडि़तों के संबंधियों को अमेरिकी संघीय अदालत में विदेशी सरकारों के खिलाफ मामले दायर की अनुमति देता है और अमेरिकी धरती पर हुए हमलों में इन देशों की सरकारों की जवाबदेही पाए जाने पर पीडि़तों के लिए मुआवजे की वकालत भी करता है। सउदी विदेश मंत्रालय के एक सूत्र ने कल अमेरिकी कांग्रेस से कहा है कि वह कानून के विध्वंसक और खतरनाक परिणामों से निपटने के लिए जरूरी कदम उठाए। सउदी की आधिकारिक प्रेस एजेंसी ने एक प्रवक्ता के हवाले से कहा है कि यह कानून चिंता का बड़ा कारण है। रियाद और वाशिंगटन के बीच संबंध दशकों पुराना है। इसमें सउदी अमेरिका को तेल देता है जबकि अमेरिका बदले में उसे सुरक्षा मुहैया करवाता है।
अमेरिका में ११ सितंबर, २००१ को हमला करने वाले अल कायदा के १९ विमान अपहर्ताओं में से १५ सउदी से हैं। उस हमले में ३,००० लोगों की मौत हो गई थी। हालांकि रियाद की ओर से हमलवारों से किसी भी तरह के संबंधों से इनकार किया जाता रहा है।
प्रेवियस स्टोरीभारत ने सावधानीपूर्वक आकलन करने के बाद किया लक्षित हमला: उस थिंक-टैंक नेक्स्ट स्टोरीन्यूजर्सी ट्रेन दुर्घटना में एक की मौत, ११४ घायल | hindi |
There was much excitement at our house on Tuesday morning; we had got our first two blobs of frog spawn in the pond, we now have at least eight blobs. The frogs started to appear in the pond on Saturday 8th February and have been croaking on damp, mild days and especially nights ever since, I did a rough head count today from the window with my binoculars and counted 22 but I am sure that there are more under the water too. Last year the frogs appeared on 16th February but didn’t lay any spawn until 9th March but we had that long spell of cold, frosty weather making spring late, in 2013 the first blob appeared on 28th February.
Frogs spend much of their adult lives on land but return to water (usually the same pond) in spring to mate and lay their spawn, each blob of spawn can contain 1,000-2,000 eggs this may seem a lot but only a small percentage will survive the journey to adulthood, frogspawn can freeze or dry out and many tadpoles and young frogs are eaten by predators. Frogs lay their spawn in a single clump near the water’s edge often in the sunniest and warmest part of the pond; fresh frog spawn is firm and solid and will sink to the bottom, where it will absorb water and then float. Each single ball is made up of two distinct parts, in the centre is the dark ovum (egg), this will develop and grow into a tadpole, surrounding the ovum is a sphere of clear jelly, this jelly provides nourishment for the developing tadpole before it hatches as well as insulation from frosts and protection from diseases, predators and the sun. If you handle frog spawn you will notice that the top of the ovum is black/dark brown and the underside is pale grey, this pale underside is the developing tadpole’s yolk reserve, always return the spawn to the water the right way up.
Frogs can lay their eggs in the most inhospitable places – damp ditches, tractor ruts, temporary ponds in flooded fields and even puddles all of which can dry up very quickly. If you come across some wild stranded frog spawn or tadpoles rescue them and find them a new home in a nearby wild pond, to prevent the risk of spreading diseases it is not advisable to put them into a garden pond or likewise transfer garden frog spawn/tadpoles/frogs into wild ponds.
Both myself, Kim and Sylvia have ponds in our gardens all of which are home to many frogs, this year I am the first to have frog spawn – not that I am competitive!
Why not have a look for some frog spawn this weekend and take some photographs of it, make a note on your calendar or in a Wildlife Diary when you see your first frog spawn of the year. | english |
اکھ وجہ چِھ یہٕ زِ دوٛن کمپیوٹیبل نمبرن ہنز مساوات کہ جانچ خاطرٕ چُھنہ کنہہ الگورتھم۔ | kashmiri |
Philosophy News | What is Skepticism?
The roots of skepticism are almost as deep as the roots of philosophy itself. The word "skepticism" is derived from the ancient Greek word "skeptikos" (σκεπτικός), an adjective meaning "inquiring" or "doubting". Today, the word has come to mean a sort of extreme or corrosive doubt that denies the existence of the subject matter with which the skeptic is concerned.
If I'm a skeptic about UFOs, for example, I doubt the existence of UFOs and question any evidence that someone proposes in favor of their existence. Used in this way -- "a skeptic about X" -- skepticism is local or confined to a particular subject matter.
In contrast, there is also the possibility of being a global skeptic. A global skeptic does not just doubt whether a particular subject area makes sense – like the study of UFOs, say, or astrology -- but doubts the existence of knowledge or even evidence more generally. The most common form of global skepticism is skepticism about knowledge of—or evidence about—the external world, the world beyond our senses. However, it is also possible to be a skeptic about other minds, or morality. In each case, the global skeptic is doubting that there can be any good evidence capable of supporting knowledge about the external world, or other minds, or moral judgments.
As we’ll see, philosophers have tended to see global skepticism as a challenge to be defeated. However, even if global skeptical arguments overplay the doubting, questioning impulse that characterizes skepticism, there is still a place for the sorts of challenges to dogmatism that skepticism represents. However, the lessons of skepticism must be applied carefully.
For example, the rise of fake news and demagoguery might suggest that the skeptical message of "Question Authority" is one that should be applied without reservation. However, the widespread public inability to understand scientific topics like global warming, genetically modified organisms, or nuclear power—not to mention the shape of the earth—suggests that it's not blind skepticism that is called for in today's networked society, but rather appropriate skepticism.
To explore skepticism more deeply, we'll begin with a brief look at ancient skepticism before discussing a new form of skepticism and the challenges it presents. After considering some contemporary answers to the challenges posed by what is called “Cartesian skepticism”, we'll look at a way of thinking of the role of skepticism that will help to appreciate when skeptical stances may be of value.
Suppose I tell you that it's going to be sunny this afternoon. You're trying to decide how to spend your day so you double-check with me: how do I know what the weather is going to be like? What evidence do I have for my claim? At this point, I might respond by saying, "Well, I looked outside, and the weather is nice and sunny now." Alternatively, I might say, "I just checked the weather app on my phone; it's forecasting sun for the afternoon."
You might follow up with a counter-claim: how the weather looks now isn't a guarantee of how it will look later, or that different weather apps are more trustworthy than others. And perhaps I'll respond with further information, or perhaps, at some point, I might not be able or willing to give you any more evidence and say that that's all the evidence I need to support my original claim about the weather. Most of us engage in conversations like this every day and think nothing of it. But this is the basis of a process for searching for truth that was prized by the ancients. It is a process that, when understood correctly, we moderns should prize as well.
The ancient Greek Mediterranean and the Hindu and Buddhist schools of the Indian Subcontinent were two centers of culture that put a high premium on debate and argument. It's little wonder, then, that thinkers in both cultural contexts spent a lot of time thinking about features of the general structure of argument.
In other words, there are five ways that a request for additional evidence can ultimately end: (1) Dissent, or suspension of judgment; (2) Infinite regress, where the giving of evidence goes on forever; (3) Relativism, where each person arrives at something that is "his (or her) truth"; (4) Dogmatism, where the giving of evidence stops at an assumption that isn't questioned, but also isn't supported by further evidence; and (5) Circular reasoning, where the evidence loops back on itself.
When searching for truth, none of these options is appealing. According to Agrippa, it seems that the very nature of giving reasons means that we are doomed to base our arguments ultimately on very shaky foundations! This is the crux of classical skepticism: the general structure of argument makes it impossible to present an argument that is immune to criticism.
Whereas ancient skepticism motivated doubt on the basis of general features of argumentation and the giving of evidence, French philosopher Rene Descartes (1596-1650) derived his skeptical doubt from a different source. Descartes believed that the mind was the seat of thought and radically separate from physical matter. For Descartes, then, the only evidence about which I can be immediately certain is evidence involving my current mental states: what I am thinking, feeling, and experiencing right now, in the moment in which I am thinking, feeling, and experiencing it.
Contemporary versions of Descartes's skeptical argument generally involve a consideration of the plight of the brain-in-a-vat. Imagine that an evil scientist has kidnapped you, removed your brain, keeping it alive in a vat of nutrient solution, and hooked your brain to a sophisticated computer. The computer feeds your brain information so that you believe you are still a person, with a body, moving around in the world -- all of this despite the fact that you are now a disembodied brain, floating in a vat of nutrient solution in a lab somewhere! This may remind you of the popular movie The Matrix. It’s the same idea.
Since the claim that I have hands was just a representative claim and could instead have been any other claim about the world outside of my immediate experience, the skepticism prompted by the brain-in-a-vat scenario is very far-reaching indeed!
The brain-in-a-vat argument can help us consider the various responses to skepticism that have been proposed in contemporary discussions. The argument contains three assumptions—premises 1, 2, and 3 above—and the various responses each involve rejections of one or more of those assumptions. While there are contemporary responses that focus on each of the premises, we’ll concentrate on premises 1 and 3, since it’s those two that will help us appreciate where those of us who are unable to take comfort in Cartesian certainty might still find value in skepticism. Let's take them in order.
One of the most radical solutions to the skeptical argument has been suggested, in different forms, by philosophers such as Fred Dretske and Robert Nozick. According to them, the problem with the skeptical argument is the first premise, and the principle of deductive closure that underlies it.
What is deductive closure? Suppose you have good evidence for the claim that p. Suppose furthermore that you have good evidence for the claim that, if p, then q. The principle of deductive closure allows you to conclude that you therefore have good evidence for the claim that q.
Applied to the brain-in-a-vat argument, it's deductive closure that provides support for the first premise. I have good evidence that, if I have hands, then I'm not handless—including that I'm not a handless brain-in-a-vat. So I have good evidence that, if I have hands, I'm not a (handless) brain-in-a-vat.
Dretske and Nozick argue, however, that deductive closure only holds within a certain context of evidence. If I'm considering evidence about whether I have hands, then that evidence will include what I see while I'm typing on the computer, say, or what I feel when my fingers touch the keyboard, or what I hear when my fingertips hit the keys. All of which, in circumstances when I'm trying to think about normal, everyday physical objects in my vicinity, counts as excellent evidence that I have hands.
But there’s a problem. Dretske and Nozick accept premise two: I would have all of the same experiential evidence about the normal, everyday physical objects in my vicinity if I were a brain-in-a-vat that I do right now. For them, as soon as realize this I must accept that the consideration of the brain-in-a-vat scenario means that none of the evidence about normal, everyday physical objects in my vicinity counts in the same way.
What Nozick and Dretske argue is that deductive closure no longer holds because the standards for the evaluation of evidence have changed between the normal scenario in which I'm considering whether I have hands—the scenario in which hands are everyday, physical objects interacting with other everyday, physical objects—and the extraordinary scenario in which I'm considering whether I am a brain-in-a-vat—a scenario in which experiential evidence no longer has the same evidentiary weight.
A more everyday analogy might motivate the rejection of deductive closure. When I arrived at work this morning, I parked my car on a parking deck some distance from my office—far enough away that I cannot now see my car, or hear my car alarm, etc. I have good reason for thinking that my car is where I parked it, on the top floor of the parking deck.
Unbeknownst to me, the parking deck might have been built with a structural flaw, so that while I was at work this morning, the parking deck collapsed, along with my car. (Luckily, nobody was injured when the deck collapsed.) Of course, if I have good reason for thinking that my car is where I parked it, on the top floor of the parking deck, then I also have good reason for thinking that the parking deck has not collapsed, along with my car. But I'm no civil engineer and I've not done an evaluation of the structural soundness of the parking structure at my work! I have no good reason for thinking that the parking deck has not collapsed—I have no evidence about the structural soundness of the parking structure at all.
I said that the denial of deductive closure was a radical solution, and it is. Without deductive closure, then there is no guarantee, even if we have a deductively valid argument, and even if we have strong evidence for each of the premises, that we are evidentially supported in drawing the conclusion of that argument. And this would be a very radical claim indeed!
One fairly recent attempt to draw on the lesson of Dretske and Nozick without having to give up on deductive closure is contextualism. According to the contextualist solution to the skeptical argument, Dretske and Nozick were right to draw our attention to the extraordinary nature of skeptical scenarios. Their mistake, according to the contextualist, is to interpret the consequences of the extraordinariness of those scenarios as affecting deductive closure. Rather, the contextualists suggest that we take the consequences of the extraordinariness of skeptical scenarios to affect the functioning of the skeptical argument as a whole.
If I have good evidence that I have hands, then I also have good evidence that I am not a (handless) brain-in-a-vat.
This premise can be the basis of an anti-skeptical argument, as follows: AS1. If I have good evidence that I have hands, then I also have good evidence that I am not a (handless) brain-in-a-vat. AS2. I have good evidence that I have hands. AS3. Thus, I have good evidence that I am not a (handless) brain-in-a-vat.
Or it can be the basis of a pro-skeptical argument, as follows: PS1. If I have good evidence that I have hands, then I also have good evidence that I am not a (handless) brain-in-a-vat. PS2. I don't have good evidence that I am not a (handless) brain-in-a-vat. PS3. Thus, I don't have good evidence that I have hands.
Now, the contextualist says that in the ordinary, everyday context in which you are sitting at your computer, watching your fingers dance across the keyboard, feeling the keys under your fingertips and hearing the clicking as you type, your evidence is such that the argument from AS1 - AS3 is true of you. It is important to note, though, that in those contexts you are not even considering the possibility that you might be a brain-in-a-vat!
As soon as you even consider the possibility that you might be a brain-in-a-vat, the contextualist suggests, you now change the context of evaluation of your evidence. And, according to the new context of evaluation, the argument from PS1 - PS3 is now true of you. So, as soon as you consider a skeptical scenario, the skeptical argument wins. To paraphrase a line from the eighties movie War Games, the only way to win against the brain-in-a-vat skeptic is not to play.
[everyday activity] cures me of this philosophical melancholy and delirium [of skepticism], either by relaxing this bent of mind, or by some avocation and lively impression of my senses, which obliterates all these chimeras. I dine, I play a game of back-gammon, and I am merry with my friends; and when after three or four hours’ amusement, I wou’d return to these speculations, they appear so cold, and strain’d, and ridiculous, that I cannot find it in my heart to enter into them any farther.
So here’s where we are. We have one solution–Nozick and Dretske’s–that requires us to deny deductive closure. This led us to a second solution, the contextualist one. The problem with this one, however, is that it is not a rejection of skepticism, but an abject surrender to skepticism. As soon as the skeptic enters into debate with us, it’s the skeptic who wins.
There is a further way of dealing with the skeptical argument that actually rejects skepticism. This way is to block the brain-in-a-vat argument by rejecting premise three and denying that the only evidence I can have is the sort of experiential evidence that is available to the brain-in-a-vat. For example, one of the contemporary theories of knowledge is externalism, according to which at least some of the evidence that supports our beliefs is external to our minds. There are different forms of externalism, but they all share the idea that people could differ with respect to the quality of the evidence that they have for their beliefs, in ways that are unknown to them.
Here's one way that this could work that is relevant to the brain-in-a-vat case. Consider people who suffer from Capgras Delusion -- the delusional belief that familiar, significant others, like family, friends, and loved ones, have been replaced by cleverly disguised robots or aliens. A standard explanation for the cause of Capgras Delusion is that, though the facial recognition system of those who suffer from the delusion is undamaged, their emotional system is damaged so that they don't have the expected emotional feedback when they see the familiar faces of their loved ones. It is this that causes them to believe that the faces that they see are imposters.
An externalist would attempt to solve this by claiming that those suffering from the delusion have all of the same evidence as a person not suffering from the delusion—the visual evidence of seeing the familiar face—despite the fact that they are not reliably forming the appropriate beliefs on the basis of that evidence; namely, that the familiar face that they see is the face of their loved one. What the externalist could say in such a case is that the experiential evidence alone is not sufficient to account for what constitutes good evidence: your face recognition system also has to be reliably integrated with your emotional system in order to form the appropriate beliefs on the basis of that experiential evidence.
Whether your face recognition system is reliably integrated in the appropriate way, however, is not something of which you can be aware. Those who suffer from Capgras Delusion are not aware that anything is wrong with them. Rather, they think there is something wrong with their loved ones: they have all been replaced by robots or aliens.
Note that the externalist provides us with a solution to ancient skepticism by establishing that there can be forms either of infinite regress or of dogmatism that are in fact virtuous because they depend on processes that reliably form true beliefs on the basis of evidence despite the fact that the parties to the dispute are unaware of the reliability of those processes.
For example, suppose that my spouse, knowing that I use whatever weather app is on the first screen that I see when I unlock my phone, has researched which weather app is most reliable, and, without telling me, has put that one on the first screen that I see when I unlock my phone. When I tell you that it’s going to be sunny this afternoon, and I do so on the basis of the weather app on my phone, I am in fact relaying reliable information to you–indeed, information that my spouse has curated on the basis of its reliability–although I am unaware of any of those facts myself. I’m giving you reliable and trustworthy information about the weather but I don’t know that I am.
In a case like this, if I am dogmatic in relying on the weather app on my phone, that strikes me as a different sort of case than that of someone who dogmatically relies on unreliable evidence. It’s different not because of the psychology of the dogmatists themselves, but because of the quality of the evidence that they’re relying on.
Although externalism provides a genuine rejection of skepticism, it doesn't do a very good job of explaining why skepticism exerts any pull on our imagination at all. But, as Hume’s description of skepticism as “delirium” suggests, skepticism does exert such a pull—at least at times. This is where the lesson from contextualism comes in. What the contextualist gets right is that the skeptic attempts to raise the standards of evidence so high that no normal response will be satisfactory. What contextualism gets wrong, however, is in abandoning the field to the skeptic.
On the contextualist’s account, the skeptic doesn’t just have the last word in every debate, but deserves to have the last word in every debate. This, however, doesn’t seem right either. Although the contextualist explains why the skeptic’s challenge seems compelling, she doesn’t explain why arguing with the skeptic is so frustrating. If the skeptic deservedly wins every debate, then presumably we shouldn’t be frustrated when she does.
The problem with the contextualist is that she treats the raising of standards of evidence by the skeptic as a legitimate move, and one that the skeptic’s interlocutor has to accept. In other words, the contextualist treats skepticism as a serious contribution to discussion–even if it’s to a different discussion than the one that we might have thought we were having.
What we need, however, is a way to determine what forms of skepticism are appropriate and in what contexts. It's at this point that we can apply what we've learned from the discussion of Nozick and Dretske. My evidence that my car is parked on the top floor of the parking structure at my work is based on the reliability of my memory and not on me being a civil engineer with information about the structural integrity of the parking structure. What we should take from this is not that everyone who parks in parking structures should study civil engineering and gather data on the structural integrity of the places they park. Rather, it's that even our everyday knowledge depends on the world around us being reliable in ways that we normally don't even consider.
Furthermore, this suggests that the value of skepticism doesn't always consist in each person's being skeptical all of the time. The result of that would be paralysis, not more knowledge.
Rather, we should seek to try to ensure that each, more circumscribed community of knowledge is one in which skepticism has a place. I can trust the parking structure because if the civil engineers who serve as inspectors in my town notice structural deficiencies in the parking garage, they'll have the power to shut it down until they're confident it's been fixed.
Here’s a great way to think about which types of considerations it pays to be skeptical about. If you’re faced with a claim, ask yourself how much money you would bet on the truth of that claim. In effect, I’m already doing that when I park my car on the top floor of the parking structure at work: I’m betting at least the replacement value of the car that the parking structure is of good structural integrity. Then, to see when skepticism is warranted, ask yourself what sort of information it would take for you to change your bet. For example, if I learned that code inspection in my town was very shoddy and the inspectors are poorly trained, would I still be willing to bet the value of my car in parking there, or would I choose to park somewhere else.
It can also help to see how people who actually have money on the line with respect to a certain claim behave–not how they talk, but what they actually do. To take one example, I don’t know what sorts of opinions Exxon investors express about the reality of global climate change, but I do know that they are pushing Exxon to do more to plan for the effects of climate change.
The more precise you can be in formulating your claims, and the more honest you can be with yourself about the actual cash value you would wager on outcomes, the better this strategy can be. For more concrete tips, a great resource is the book Superforecasting by University of Pennsylvania professor Philip E. Tetlock and Dan Gardner.
Joseph H. Shieber is Associate Professor of Philosophy at Lafayette College. Prior to teaching at Lafayette, he taught at Brown University, the Massachusetts Institute of Technology, and Connecticut College. He is the author of numerous articles on epistemology, the philosophy of language, and the history of philosophy, and of the book Testimony: A Philosophical Introduction.
Testimony: A Philosophical Introduction. The epistemology of testimony has experienced a growth in interest over the last twenty-five years that has been matched by few, if any, other areas of philosophy. Testimony: A Philosophical Introduction provides an epistemology of testimony that surveys this rapidly growing research area while incorporating a discussion of relevant empirical work from social and developmental psychology, as well as from the interdisciplinary study of knowledge-creation in groups. The past decade has seen a number of scholarly monographs on the epistemology of testimony, but there is a dearth of books that survey the current field. This book fills that gap, assessing the strengths and weaknesses of all major competing theories. All chapters conclude with Suggestions for Further Reading and Discussion Questions (from the publisher’s website). | english |
/*
* max77828.c - mfd core driver for the Maxim 77828
*
* Copyright (C) 2011 Samsung Electronics
* SeoYoung Jeong <seo0.jeong@samsung.com>
*
* 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., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
*
* This driver is based on max8997.c
*/
#include <linux/module.h>
#include <linux/slab.h>
#include <linux/i2c.h>
#include <linux/irq.h>
#include <linux/interrupt.h>
#include <linux/mutex.h>
#include <linux/mfd/core.h>
#include <linux/mfd/max77828.h>
#include <linux/mfd/max77828-private.h>
#include <linux/muic/max77828-muic.h>
#if defined (CONFIG_OF)
#include <linux/of_device.h>
#include <linux/of_gpio.h>
#endif /* CONFIG_OF */
#define I2C_ADDR_PMIC (0x92 >> 1) /* Top sys, Haptic */
#define I2C_ADDR_MUIC (0x4A >> 1)
#define I2C_ADDR_LED (0x94 >> 1)
static struct mfd_cell max77828_devs[] = {
#if defined(CONFIG_MUIC_MAX77828)
{ .name = MUIC_DEV_NAME, },
#endif /* CONFIG_MUIC_MAX77828 */
#if defined(CONFIG_MOTOR_DRV_MAX77828)
{ .name = "max77828-haptic", },
#endif /* CONFIG_MAX77828_HAPTIC */
#if defined(CONFIG_LEDS_MAX77828_RGB)
{ .name = "leds-max77828-rgb", },
#endif /* CONFIG_LEDS_MAX77828_RGB */
};
int max77828_read_reg(struct i2c_client *i2c, u8 reg, u8 *dest)
{
struct max77828_dev *max77828 = i2c_get_clientdata(i2c);
int ret;
mutex_lock(&max77828->i2c_lock);
ret = i2c_smbus_read_byte_data(i2c, reg);
mutex_unlock(&max77828->i2c_lock);
if (ret < 0) {
pr_info("%s:%s reg(0x%x), ret(%d)\n", MFD_DEV_NAME, __func__, reg, ret);
return ret;
}
ret &= 0xff;
*dest = ret;
return 0;
}
EXPORT_SYMBOL_GPL(max77828_read_reg);
int max77828_bulk_read(struct i2c_client *i2c, u8 reg, int count, u8 *buf)
{
struct max77828_dev *max77828 = i2c_get_clientdata(i2c);
int ret;
mutex_lock(&max77828->i2c_lock);
ret = i2c_smbus_read_i2c_block_data(i2c, reg, count, buf);
mutex_unlock(&max77828->i2c_lock);
if (ret < 0)
return ret;
return 0;
}
EXPORT_SYMBOL_GPL(max77828_bulk_read);
int max77828_write_reg(struct i2c_client *i2c, u8 reg, u8 value)
{
struct max77828_dev *max77828 = i2c_get_clientdata(i2c);
int ret;
mutex_lock(&max77828->i2c_lock);
ret = i2c_smbus_write_byte_data(i2c, reg, value);
mutex_unlock(&max77828->i2c_lock);
if (ret < 0)
pr_info("%s:%s reg(0x%x), ret(%d)\n",
MFD_DEV_NAME, __func__, reg, ret);
return ret;
}
EXPORT_SYMBOL_GPL(max77828_write_reg);
int max77828_bulk_write(struct i2c_client *i2c, u8 reg, int count, u8 *buf)
{
struct max77828_dev *max77828 = i2c_get_clientdata(i2c);
int ret;
mutex_lock(&max77828->i2c_lock);
ret = i2c_smbus_write_i2c_block_data(i2c, reg, count, buf);
mutex_unlock(&max77828->i2c_lock);
if (ret < 0)
return ret;
return 0;
}
EXPORT_SYMBOL_GPL(max77828_bulk_write);
int max77828_update_reg(struct i2c_client *i2c, u8 reg, u8 val, u8 mask)
{
struct max77828_dev *max77828 = i2c_get_clientdata(i2c);
int ret;
mutex_lock(&max77828->i2c_lock);
ret = i2c_smbus_read_byte_data(i2c, reg);
if (ret >= 0) {
u8 old_val = ret & 0xff;
u8 new_val = (val & mask) | (old_val & (~mask));
ret = i2c_smbus_write_byte_data(i2c, reg, new_val);
}
mutex_unlock(&max77828->i2c_lock);
return ret;
}
EXPORT_SYMBOL_GPL(max77828_update_reg);
#if defined(CONFIG_OF)
static int of_max77828_dt(struct device *dev, struct max77828_platform_data *pdata)
{
struct device_node *np_max77828 = dev->of_node;
if(!np_max77828)
return -EINVAL;
pdata->irq_gpio = of_get_named_gpio(np_max77828, "max77828,irq-gpio", 0);
pdata->wakeup = of_property_read_bool(np_max77828, "max77828,wakeup");
pr_info("%s: irq-gpio: %u \n", __func__, pdata->irq_gpio);
return 0;
}
#endif /* CONFIG_OF */
static int max77828_i2c_probe(struct i2c_client *i2c,
const struct i2c_device_id *dev_id)
{
struct max77828_dev *max77828;
struct max77828_platform_data *pdata = i2c->dev.platform_data;
u8 reg_data;
int ret = 0;
pr_info("%s:%s\n", MFD_DEV_NAME, __func__);
max77828 = kzalloc(sizeof(struct max77828_dev), GFP_KERNEL);
if (!max77828) {
dev_err(&i2c->dev, "%s: Failed to alloc mem for max77828\n", __func__);
return -ENOMEM;
}
if (i2c->dev.of_node) {
pdata = devm_kzalloc(&i2c->dev, sizeof(struct max77828_platform_data),
GFP_KERNEL);
if (!pdata) {
dev_err(&i2c->dev, "Failed to allocate memory \n");
ret = -ENOMEM;
goto err;
}
ret = of_max77828_dt(&i2c->dev, pdata);
if (ret < 0){
dev_err(&i2c->dev, "Failed to get device of_node \n");
return ret;
}
i2c->dev.platform_data = pdata;
} else
pdata = i2c->dev.platform_data;
max77828->dev = &i2c->dev;
max77828->i2c = i2c;
max77828->irq = i2c->irq;
if (pdata) {
max77828->pdata = pdata;
pdata->irq_base = irq_alloc_descs(-1, 0, MAX77828_IRQ_NR, -1);
if (pdata->irq_base < 0) {
pr_err("%s:%s irq_alloc_descs Fail! ret(%d)\n",
MFD_DEV_NAME, __func__, pdata->irq_base);
ret = -EINVAL;
goto err;
} else
max77828->irq_base = pdata->irq_base;
max77828->irq_gpio = pdata->irq_gpio;
max77828->wakeup = pdata->wakeup;
} else {
ret = -EINVAL;
goto err;
}
mutex_init(&max77828->i2c_lock);
i2c_set_clientdata(i2c, max77828);
if (max77828_read_reg(i2c, MAX77828_PMIC_REG_PMICREV, ®_data) < 0) {
dev_err(max77828->dev,
"device not found on this channel (this is not an error)\n");
ret = -ENODEV;
goto err;
} else {
/* print rev */
max77828->pmic_rev = (reg_data & 0x7);
max77828->pmic_ver = ((reg_data & 0xF8) >> 0x3);
pr_info("%s:%s device found: rev.0x%x, ver.0x%x\n",
MFD_DEV_NAME, __func__,
max77828->pmic_rev, max77828->pmic_ver);
}
max77828->muic = i2c_new_dummy(i2c->adapter, I2C_ADDR_MUIC);
i2c_set_clientdata(max77828->muic, max77828);
max77828->led = i2c_new_dummy(i2c->adapter, I2C_ADDR_LED);
i2c_set_clientdata(max77828->led, max77828);
ret = max77828_irq_init(max77828);
if (ret < 0)
goto err_irq_init;
ret = mfd_add_devices(max77828->dev, -1, max77828_devs,
ARRAY_SIZE(max77828_devs), NULL, 0, NULL);
if (ret < 0)
goto err_mfd;
device_init_wakeup(max77828->dev, pdata->wakeup);
return ret;
err_mfd:
mfd_remove_devices(max77828->dev);
err_irq_init:
i2c_unregister_device(max77828->muic);
i2c_unregister_device(max77828->led);
err:
kfree(max77828);
return ret;
}
static int max77828_i2c_remove(struct i2c_client *i2c)
{
struct max77828_dev *max77828 = i2c_get_clientdata(i2c);
mfd_remove_devices(max77828->dev);
i2c_unregister_device(max77828->muic);
i2c_unregister_device(max77828->led);
kfree(max77828);
return 0;
}
static const struct i2c_device_id max77828_i2c_id[] = {
{ MFD_DEV_NAME, TYPE_MAX77828 },
{ }
};
MODULE_DEVICE_TABLE(i2c, max77828_i2c_id);
#if defined(CONFIG_OF)
static struct of_device_id max77828_i2c_dt_ids[] = {
{ .compatible = "maxim,max77828" },
{ },
};
MODULE_DEVICE_TABLE(of, max77828_i2c_dt_ids);
#endif /* CONFIG_OF */
#if defined(CONFIG_PM)
static int max77828_suspend(struct device *dev)
{
struct i2c_client *i2c = container_of(dev, struct i2c_client, dev);
struct max77828_dev *max77828 = i2c_get_clientdata(i2c);
if (device_may_wakeup(dev))
enable_irq_wake(max77828->irq);
disable_irq(max77828->irq);
return 0;
}
static int max77828_resume(struct device *dev)
{
struct i2c_client *i2c = container_of(dev, struct i2c_client, dev);
struct max77828_dev *max77828 = i2c_get_clientdata(i2c);
if (device_may_wakeup(dev))
disable_irq_wake(max77828->irq);
enable_irq(max77828->irq);
return 0;
}
#else
#define max77828_suspend NULL
#define max77828_resume NULL
#endif /* CONFIG_PM */
#ifdef CONFIG_HIBERNATION
/*
u8 max77828_dumpaddr_pmic[] = {
#if 0
MAX77828_LED_REG_IFLASH,
MAX77828_LED_REG_IFLASH1,
MAX77828_LED_REG_IFLASH2,
MAX77828_LED_REG_ITORCH,
MAX77828_LED_REG_ITORCHTORCHTIMER,
MAX77828_LED_REG_FLASH_TIMER,
MAX77828_LED_REG_FLASH_EN,
MAX77828_LED_REG_MAX_FLASH1,
MAX77828_LED_REG_MAX_FLASH2,
MAX77828_LED_REG_VOUT_CNTL,
MAX77828_LED_REG_VOUT_FLASH,
MAX77828_LED_REG_VOUT_FLASH1,
MAX77828_LED_REG_FLASH_INT_STATUS,
#endif
MAX77828_PMIC_REG_PMICID1,
MAX77828_PMIC_REG_PMICREV,
MAX77828_PMIC_REG_MAINCTRL1,
MAX77828_PMIC_REG_MCONFIG,
};
*/
u8 max77828_dumpaddr_muic[] = {
MAX77828_MUIC_REG_INTMASK1,
MAX77828_MUIC_REG_INTMASK2,
MAX77828_MUIC_REG_INTMASK3,
MAX77828_MUIC_REG_CDETCTRL1,
MAX77828_MUIC_REG_CDETCTRL2,
MAX77828_MUIC_REG_CTRL1,
MAX77828_MUIC_REG_CTRL2,
MAX77828_MUIC_REG_CTRL3,
};
/*
u8 max77828_dumpaddr_haptic[] = {
MAX77828_HAPTIC_REG_CONFIG1,
MAX77828_HAPTIC_REG_CONFIG2,
MAX77828_HAPTIC_REG_CONFIG_CHNL,
MAX77828_HAPTIC_REG_CONFG_CYC1,
MAX77828_HAPTIC_REG_CONFG_CYC2,
MAX77828_HAPTIC_REG_CONFIG_PER1,
MAX77828_HAPTIC_REG_CONFIG_PER2,
MAX77828_HAPTIC_REG_CONFIG_PER3,
MAX77828_HAPTIC_REG_CONFIG_PER4,
MAX77828_HAPTIC_REG_CONFIG_DUTY1,
MAX77828_HAPTIC_REG_CONFIG_DUTY2,
MAX77828_HAPTIC_REG_CONFIG_PWM1,
MAX77828_HAPTIC_REG_CONFIG_PWM2,
MAX77828_HAPTIC_REG_CONFIG_PWM3,
MAX77828_HAPTIC_REG_CONFIG_PWM4,
};
*/
u8 max77828_dumpaddr_led[] = {
MAX77828_RGBLED_REG_LEDEN,
MAX77828_RGBLED_REG_LED0BRT,
MAX77828_RGBLED_REG_LED1BRT,
MAX77828_RGBLED_REG_LED2BRT,
MAX77828_RGBLED_REG_LED3BRT,
MAX77828_RGBLED_REG_LEDBLNK,
MAX77828_RGBLED_REG_LEDRMP,
};
static int max77828_freeze(struct device *dev)
{
struct i2c_client *i2c = container_of(dev, struct i2c_client, dev);
struct max77828_dev *max77828 = i2c_get_clientdata(i2c);
int i;
for (i = 0; i < ARRAY_SIZE(max77828_dumpaddr_pmic); i++)
max77828_read_reg(i2c, max77828_dumpaddr_pmic[i],
&max77828->reg_pmic_dump[i]);
for (i = 0; i < ARRAY_SIZE(max77828_dumpaddr_muic); i++)
max77828_read_reg(i2c, max77828_dumpaddr_muic[i],
&max77828->reg_muic_dump[i]);
for (i = 0; i < ARRAY_SIZE(max77828_dumpaddr_led); i++)
max77828_read_reg(i2c, max77828_dumpaddr_led[i],
&max77828->reg_led_dump[i]);
disable_irq(max77828->irq);
return 0;
}
static int max77828_restore(struct device *dev)
{
struct i2c_client *i2c = container_of(dev, struct i2c_client, dev);
struct max77828_dev *max77828 = i2c_get_clientdata(i2c);
int i;
enable_irq(max77828->irq);
for (i = 0; i < ARRAY_SIZE(max77828_dumpaddr_pmic); i++)
max77828_write_reg(i2c, max77828_dumpaddr_pmic[i],
max77828->reg_pmic_dump[i]);
for (i = 0; i < ARRAY_SIZE(max77828_dumpaddr_muic); i++)
max77828_write_reg(i2c, max77828_dumpaddr_muic[i],
max77828->reg_muic_dump[i]);
for (i = 0; i < ARRAY_SIZE(max77828_dumpaddr_led); i++)
max77828_write_reg(i2c, max77828_dumpaddr_led[i],
max77828->reg_led_dump[i]);
return 0;
}
#endif
const struct dev_pm_ops max77828_pm = {
.suspend = max77828_suspend,
.resume = max77828_resume,
#ifdef CONFIG_HIBERNATION
.freeze = max77828_freeze,
.thaw = max77828_restore,
.restore = max77828_restore,
#endif
};
static struct i2c_driver max77828_i2c_driver = {
.driver = {
.name = MFD_DEV_NAME,
.owner = THIS_MODULE,
#if defined(CONFIG_PM)
.pm = &max77828_pm,
#endif /* CONFIG_PM */
#if defined(CONFIG_OF)
.of_match_table = max77828_i2c_dt_ids,
#endif /* CONFIG_OF */
},
.probe = max77828_i2c_probe,
.remove = max77828_i2c_remove,
.id_table = max77828_i2c_id,
};
static int __init max77828_i2c_init(void)
{
pr_info("%s:%s\n", MFD_DEV_NAME, __func__);
return i2c_add_driver(&max77828_i2c_driver);
}
/* init early so consumer devices can complete system boot */
subsys_initcall(max77828_i2c_init);
static void __exit max77828_i2c_exit(void)
{
i2c_del_driver(&max77828_i2c_driver);
}
module_exit(max77828_i2c_exit);
MODULE_DESCRIPTION("MAXIM 77828 multi-function core driver");
MODULE_AUTHOR("SeoYoung Jeong <seo0.jeong@samsung.com>");
MODULE_LICENSE("GPL");
| code |
argparse
numpy
#matplotlib
networkx==1.11
| code |
Being diagnosed with cancer will come as a terrible shock for most people. You may look back on the experience and remember it as all being 'a bit of a blur'. Often you will be given so much information that you are only able to take in part of what you were told.
It is likely that you will also experience a huge range of emotions during the period immediately after diagnosis, and you will probably have many questions to ask about your condition and its treatment.
The following webpages contain comprehensive, yet easy to understand information about bowel cancer and what may be ahead of you in the coming months. We hope this information will assist you at this difficult time and will be useful now, and in the weeks ahead, as you want to know more about your diagnosis and treatment options.
As the news sinks in and you are ready to talk about what's going to happen, the most important thing to remember is that it's your body you are discussing. Don't be rushed into making decisions and don't be frightened to ask the doctor or nurse to explain things again if you don't understand. You may also wish to request a second opinion.
Your cancer has a whole new language and it is going to be a while before you understand it all. The experts treating you can sometimes forget this – don't let them!
You will need to have more tests to find out the extent of the cancer and whether it has spread to other parts of your body. This process is known as 'staging'.
The doctor will choose from a range of investigations which will usually include blood tests, x-rays and a CT scan. You may also have an MRI scan, PET scan and/or an ultrasound. At each stage, your specialist team should take time to tell you what the results are and they will explain why you need to have these and any other tests done.
Once your doctor has received your test results, they will be able to discuss your diagnosis more fully with you. This will help you to understand your options as you start to get involved in putting together your individual treatment plan. This will be unique to your own personal circumstances, and will depend on a number of factors including the type, size and location of the cancer and your general health.
All decision making will be done jointly between you and your multi-disciplinary team (MDT). Your doctors will help you to understand the advantages and disadvantages of what is being proposed so that you can be confident in the decisions taken, and satisfied that your individual needs and wishes have been fully considered.
Your specialist nurse will also make an assessment of your general health and fitness – known as Holistic Needs Assessment – and consider any underlying health problems. It is also important that they understand what home and family issues need to be considered as well as any practical concerns, as these might also have an impact on your health and treatment choices. This process will also help you to manage your own care much more effectively so that you will know when and how to ask for help.
There are three main types of treatment for bowel cancer, based on – surgery, chemotherapy and radiotherapy techniques. Depending on the stage and location of your cancer, you will usually receive one or a combination of these treatments. Bowel cancer patients may also receive targeted therapies if the bowel cancer has spread to other parts of the body.
Patients sometimes choose to seek a second opinion from another specialist or hospital. This may be at the suggestion of family members keen to ensure all possible treatment options are being explored. Or it may be that patients are unhappy regarding their treatment pathway or decisions that have been made by their current team.
We would recommend that you discuss your concerns with your GP or specialist first. It may be that talking things through can address some of your worries, allowing you to continue on your current pathway uninterrupted.
It is possible to get a second opinion by asking your GP or your current specialists to refer you on. Asking for a second opinion can feel uncomfortable, but it shouldn't be an issue. Most doctors would prefer that you are confident in your team and the treatment being planned.
A second opinion will require all your scans and reports to be sent over to the other specialist, and your case to be discussed at their multi-disciplinary team meeting. This will quite often mean a wait of a week or two to allow all of this to happen, which can be quite stressful for the patient and their family.
Generally speaking, we wouldn't recommend delaying treatment in order to have a second opinion. However, some patients feel that it is worth taking the time to look at other options before starting treatment. | english |
Google Pixel 3 XL vs. Google Pixel 2 XL: Should you upgrade?
There's no shortage of great Android phones out there, and Google makes some of the best available. We test each and every one to see just how they compare year-over-year, and against the latest competition, so we can give you the best recommendations on which to buy and when to upgrade.
Google refreshed its hardware and jumped to a 6.3-inch, 18.5:9 display — but it's not all about size, the display is dramatically better as well. The rest of the phone builds on all of Google's strengths: simple hardware, capable specs, smooth software, and an amazing camera experience front and back. It also added wireless charging and kept stereo speakers.
This was Google's top device a year ago, and that means for the most part it still feels modern today. The specs, hardware, and software are all up to speed and the camera is still fantastic. But its screen was bad then and looks even worse now.
The Pixel 3 XL isn't a massive upgrade from the Pixel 2 XL, but there are a few changes that could make it worth it if you're particularly upset with a few aspects of the phone.
Is the Google Pixel 3 XL a worthy upgrade from the Pixel 2 XL?
This comparison effectively hinges on two points: how much do you dislike the Pixel 2 XL's display, and how big of a deal is it to spend $900 on a phone?
Google went to great lengths to make sure the Pixel 3 XL's screen didn't repeat any of the problems of the Pixel 2 XL — and if you have the Pixel 2 XL, you know exactly what I'm talking about. It's dim, lacking that "punch" of color so many other phones have, and has proven to be more susceptible to burn-in and image retention than the competition. The Pixel 3 XL addresses all of these points: it's brighter, more accurate, more colorful, and generally exhibits all of the qualities you expect for a phone of this price. If you're underwhelmed by your Pixel 2 XL's screen, the Pixel 3 XL may be appealing just from that point alone.
The screen upgrade is so important because elsewhere, the Pixel 3 XL isn't necessarily a huge upgrade from the 2 XL. There's a new processor inside, but the rest of the specs remain unchanged. The external hardware didn't change in size or shape, which is a good thing in most people's eyes, but the new glass back does offer a better feel and wireless charging capabilities.
The display is unequivocally better, and there are a few hardware upgrades — is that worth $900?
If the display upgrade alone wasn't enough to change your mind, perhaps you'll be interested in the new camera features. The front-facing camera has been upgraded with auto focus and a new sensor, and it's now assisted by a secondary wide-angle camera for better portrait mode and group selfie shots. The rear camera takes fundamentally better photos, and better leverages the Pixel Visual Core — but chances are that many of its new features can actually make their way back to the Pixel 2 XL (in due time). The Pixel 3 XL takes great photos, but the Pixel 2 XL is no slouch — it spent a year at the top of the charts in image quality.
So let's bring this all back together and talk about price: the Pixel 3 XL costs $899. If that seems like a lot of money to you, and you own a Pixel 2 XL already, this upgrade probably doesn't make a whole lot of sense. The screen is unequivocally better, the camera experience is an upgrade, and a couple of extra hardware features are nice — and in the end, you don't lose anything. But are those relatively small changes worth scrapping your phone and paying another $899 for a Pixel 3 XL?
Still a modern and capable phone.
The Pixel 2 XL has been updated to Android 9 Pie and is still a modern phone in so many respects. If it weren't for the poor display quality, it'd be easier to enjoy the performance, software, and quality camera — but on the whole, it's still a good phone if you find it at a discount. | english |
Being on the wrong end of bad reputation issues comes with a price. How big a price? Just ask United Airlines. The price tag after one particular incident hit $180 million. In this age of 24/7 instantaneous digital communications, companies and corporations of every size and shape can’t afford to allow these types of reputation problems get out of hand because the damage to their bottom line is huge.
In the case of United Airlines, it was all the result of damaged guitar. The company declined to pay for the damage to the instrument and musician David Carroll turned the episode into a song that became a runaway YouTube with more than 15 million views. United Airlines was left crying the blues. What was good for Carroll’s music career was disastrous for the airline when United's stock fell by 10% in the weeks following the video's release.
Companies with bad or even tarnished reputations find that with social media, businesses suffer almost instantaneously with costs going up and revenues falling. It’s harder to hold onto customers whose reaction is to stay away from businesses being bombarded by negative press. There are usually many other options for customers to choose from, so it’s easy to simply do business with someone else that isn’t getting clobbered on Twitter.
Stocks fall when reputations get hammered.
Stock markets are so sensitive to bad news, as in the case of Kellogg’s that shares fall hard and fast when that company got caught up in a fight with Breitbart News. When the cereal giant pulled its ads off Breitbart, the news organization called for a boycott and hundreds of thousands responded to it immediately. That sent shares of the company tumbling just as fast. According to one estimate, the annual cost of bad reputation episodes is more than $537 billion. That amount factors in missed revenues accruing to companies whose customers have switched due to poor experiences.
Mind you one company’s loss may be another’s gain, as competitors reap rewards from being the beneficiary of another firm’s reputation woes. This corporate phenomenon means the benefits of having a pristine or at least stellar reputation is worth the billions companies spend to try and keep their customers happy. This is actually a positive thing for consumers who, armed with a Facebook account, can have a lot of leverage when it comes to getting satisfaction from companies wary of making enemies of social network-savvy people.
Samsung and Apple rank at or near the top of companies with stellar reputations according to a study by Yahoo! Finance. It’s no accident the companies are the two leading sellers of smartphones in the world. The good reputation list also includes Microsoft and Sony, makers of the world's two most popular video game consoles.
Profitable, competitive companies also have great reputations.
Is your company suffering from a bad restaurant review, public production snafu or service glitch? You’d better get on it quickly. A bad movie review, for instance, can doom that picture to terrible box-office returns costing a studio tens of millions of dollars. In the same way, small businesses can also suffer.
In another study, researchers found that a four out of five consumers will change their mind about making a purchase due to negative information online.
Given the high stakes, doesn’t it make sense to hire an expert firm whose job it is to protect a reputation. It’s almost always easier to try and prevent a disastrous episode through best practices than it is to salvage a tattered one after something has happened.
Think of it as an insurance policy or a preventative maintenance. Better safe than sorry…right United Airlines? | english |
मध्य प्रदेश के नवनियुक्त मुख्यमंत्री कमलनाथ अपने विवादित बयान का बचाव करने के बजाय उस पर कायम नजर आए.
कमलनाथ, मध्य प्रदेश के मुख्यमंत्री
मध्य प्रदेश के नवनियुक्त मुख्यमंत्री कमलनाथ अपने विवादित बयान पर कायम नजर आए. मध्य प्रदेश के लोगों को रोजगार देने के बयान पर कमलनाथ ने कहा, ''ये सब जगह है, मैंने कौन सी नई बात कही?'' बुधवार को बिहार की दो अदालतों में कमलनाथ के खिलाफ अलग-अलग परिवाद पत्र दाखिल किए गए. यह परिवाद पत्र कमलनाथ के उस बयान के विरोध में दायर किया गया है, जिसमें उन्होंने कहा था कि बिहार और उत्तर प्रदेश के लोग नौकरियां पा लेते हैं और मध्य प्रदेश के नौजवान रोजगार से वंचित रह जाते हैं. बिहार के बेतिया में कमलनाथ के खिलाफ अधिवक्ता मुराद अली ने एक परिवाद पत्र दायर किया है.
प्रदेश के मुख्यमंत्री कमलनाथ के खिलाफ बिहार की दो अदालतों में परिवाद पत्र दायर, जानें पूरा मामला...
कांग्रेस लगातार केंद्र सरकार पर वादे के मुताबिक रोजगार नहीं दे पाने को लेकर निशाना साध रही है. मध्य प्रदेश में शपथ लेने के बाद से ही कमलनाथ रोजगार पैदा करने की योजना की बात कर रहे हैं. इसी कड़ी में उन्होंने कहा था कि हम देखते आ रहे हैं कि कई इंडस्ट्रीज में बिहार और उत्तर प्रदेश के युवक नौकरी करने लगते हैं जबकि मध्य प्रदेश के नौजवान बेरोजगार रह जाते हैं. मैं यूपी और बिहार से आए लोगों की आलोचना नहीं कर रहा हूं लेकिन कंपनियों को प्रदेश के लोगों के रोजगार के बारे में सोचना होगा.
कमलनाथ ने कहा ''मध्य प्रदेश में ऐसे उद्योगों को ही सरकार की तरफ से वित्तीय और अन्य सुविधाओं का लाभ मिलेगा, जिसमें ७० प्रतिशत रोजगार मध्य प्रदेश के लोगों को दिया जाएगा. मैंने इससे संबंधित फाइल पर हस्ताक्षर कर दिए हैं.वहीं इस मामले पर उत्तर प्रदेश के पूर्व मुख्यमंत्री अखिलेश यादव ने नाराजगी जाहिर की थी.
बिहार में परिवारवाद पत्र दाखिल करने वाले अधिवक्ता मुराद अली ने बताया कि कमलनाथ के खिलाफ परिवाद पत्र में भादवि की धारा १२४-ए, १५३ ए, १५३ बी, १८१ और ५०४ के तहत आरोप लगाए गए हैं. परिवाद में कहा गया है कि मुख्यमंत्री कमलनाथ ने १७ दिसंबर को शपथ ग्रहण के बाद से ऐसा बयान देकर उस शपथ का उल्लंघन किया है, जिसमें उन्होंने 'देश की एकता, अखंडता' का वचन लिया था.
इसके अलावा दूसरा परिवार पत्र दाखिल करने वाले हाशमी ने आरोप लगाया है कि मध्य प्रदेश के मुख्यमंत्री कमलनाथ के बयान से बिहार और उत्तर प्रदेश के लोगों का जहां अपमान हुआ है, वहीं यह बयान देश को तोड़ने वाला भी है. परिवाद पत्र में कमलनाथ के बयान को बिहार की प्रतिभाओं को अपमानित करने वाला बताते हुए अदालत से मुख्यमंत्री पर कार्रवाई करने का आग्रह किया गया है. | hindi |
صفوان بن بیضا ٲسؠ اَکھ صُحابی.
زٲتی زِندگی
== حَوالہٕ == | kashmiri |
६२ वर्ष की उम्र में भी ज़बरदस्त फिट सनी देओल की क्या है डाइट, जानें यहाँ - बऐकटोबॉलिवुड
६२ वर्ष की उम्र में भी ज़बरदस्त फिट सनी देओल की क्या है डाइट, जानें यहाँ
सुननी देओल फिटनेस रूटीन- ६२ वर्ष की उम्र में भी ज़बरदस्त फिट सनी देओल की क्या है डाइट, जानें यहाँ
सनी देओल अपने ढाई किलो के हाँथ के लिए तो जाने ही जाते हैं साथ ही अपने बलिष्ठ शरीर के लिए भी प्रसिद्ध हैं। शायद इसलिए लोग उन्हें एक्शन रोल्स में देखना चाहते हैं। सनी देओल स्क्रीन पर जब हैंड पम्प उखाड़ते हैं तो लोगों को भी यकीन होता है कि ये इंसान कर सकता है क्योंकि उनका स्क्रीन प्रेजेंस इतना दमदार है। ३६ वर्षों से सनी देओल एक दमदार एक्शन हीरो के रूप में विख्यात हैं और इस बात का क्रेडिट उनके खान पान और वर्जिश को भी जाता है,जो उन्हें ६२ वर्ष की उम्र में भी फिट बनाए हुए आइये जानते हैं क्या है सनी पाजी की
सनी पाजी का दिन सुबह ५:३०- ६:०० बजे शुरू होता है, वे रात में जल्दी सोते हैं और जल्दी उठते हैं। बॉलीवुड हंगामा में दिए एक साक्षात्कार के अनुसार सनी देओल का ब्रेकफ़ास्ट होता है स्क्रेम्ब्लड एग और टोस्ट। साथ में वे कॉफ़ी पीना पसंद करते हैं। सुबह का समय ही उनके एकसरसाइज का समय होता है, वे जिम में कसरत करने को महत्व देते हैं और १ घंटे तगड़ी वर्जिश करते हैं।
सनी देओल लंच में प्रोटीन ज़्यादा लेना पसंद करते हैं और लंच उनका हेवी मील होता है। जिसमें वे चिकन, बॉयल्ड या ग्रिल्ड, करके खाना पसंद करते हैं। साथ ही पास्ता लेते हैं। हालांकि लंच उनका मौसम के हिसाब से परिवर्तित होता जाता है। खासकर ठंड के दिनों में सनी मूली के परांठे और दही खाना पसंद करते हैं। देसी डाइट और देसी बॉडी।
शाम का समय सनी के स्पोर्ट्स का टाइम होता है। जिसमें सनी प्रतिदिन एक घंटा कोई न कोई स्पोर्ट्स खेलते हैं। इन दिनों सनी देओल टेबल टेनिस खेलना एन्जॉय कर रहे हैं। शाम के नाश्ते में सनी पाजी फलों का सेवन करते हैं खास कर ऐप्पल को योगर्ट के साथ मिलाकर।
सनी देओल रात में जल्दी सोते हैं इसलिए उनका डिनर भी जल्दी होता है। रात में वे मिस्सी रोटी और घर का बना हुआ मक्खन खाते हैं। साथ ही कोई सब्ज़ी। उनकी पसंदीदा सब्ज़ी है आलू गोभी। हालांकि 'मेंस एक्स्पी' में छपी खबर के अनुसार सनी मेथी का परांठा और लौकी की सब्ज़ी भी बेहद पसंद करते हैं। | hindi |
बंगा(चमन, राकेश): बंगा-फगवाड़ा मुख्य मार्ग पर पड़ते गांव मजारी के नजदीक हुए सड़क हादसे में एक लड़की की मौत हो गई और २ महिलाएं गंभीर रूप से घायल हो गईं।
जानकारी के अनुसार तरलोक सिंह पुत्र गुलजारी लाल निवासी बहराम टवेरा गाड़ी में पारिवारिक मैंबरों पिता गुलजारी लाल, माता प्रेम प्यारी, प्रियंशा, सुखजिन्द्र कौर, नवजोत और तमन्ना के साथ गढ़शंकर में किसी पारिवारिक खुशी समागम में से वापस अपने घर बहराम जा रहा था कि रास्ते में फगवाड़ा साइड से आ रही गाड़ियों की तेज लाइट पड़ने से सामने कुछ भी नजर न आने के कारण उसकी गाड़ी सड़क के बीच बने डिवाइडर पर लगे लोहे के जंगले के साथ टकराई, जिसके उसके बाद वह मजारी में एक पैट्रोल पंप के बाहर लगे खम्भे से टकरा गई।
इसके फलस्वरूप गाड़ी का काफी नुक्सान हुआ, वहीं गाड़ी में सवार तमन्ना बेटी कुलवंत राय निवासी नूरमहल रोड फिल्लौर जालंधर, प्रेम प्यारी पत्नी गुलजारी लाल, प्रियंशा बेटी तरलोक सिंह निवासी बहराम गंभीर रूप से जख्मी हो गईं। उन्हें मौके पर मौजूद लोगों की मदद से स्थानीय गुरु नानक मिशन अस्पताल पहुंचाया गया, जहां डाक्टरों ने तमन्ना को मृत घोषित कर दिया, जबकि बाकी दोनों महिलाओं की हालत नाजुक बताई जा रही है। हादसे की सूचना मिलते ही थाना सदर के ए.एस.आई. रछपाल सिंह सहित पुलिस पार्टी मौके पर पहुंची और मृतका के शव व हादसाग्रस्त कार को कब्जे में लेकर आगामी कार्रवाई शुरू कर दी। वहीं शव को पोस्टमार्टम करने के उपरांत वारिसों को सौंप दिया गया है। | hindi |
हेल्लो दोस्तों! आज हम ऐसे विषय पर बात करने वाले है जिसकी जरूरत सबको पड़ती है| वह विषय है "एक्सम टिप्स इन हिन्दी | कैसे करें परीक्षा की तैयारी | हॉ तो प्रेपरे फॉर एक्सम इन हिन्दी" जब एक्सम का नाम लिया जाता है तब सबके पसीने छुट जाते है| खास करके १०त और १२त के विद्यार्थी को एक्सम का बहुत ज्यादा तनाव रहता है, इसीलिए आज हम जानेंगे कि परीक्षा की तैयारी कैसे करें या कम समय में बोर्ड एग्जाम की तैयारी कैसे करें|
सबसे पहले आपको ये बता दे कि आपको एग्जाम से डरना नही चाहिए, डरने से आपके दिमाग पर मानसिक दबाव पड़ता है लेकिन ज्यादा पढाई तभी हो पाती है जब की दिमाग बिलकुल हल्का हो, इसीलिए आप सबसे पहले अपने मन और दिमाग से एग्जाम का डर निकल फेंके|
अब हम कुछ चीजो के बारे में जानेंगे जिनका उपयोग करके आप एग्जाम की तैयारी जल्दी से जल्दी और ज्यादा से ज्यादा कर सकेंगे|
जब आपक बुक से पढाई कर रहे हो उस समय अपने साथ एक नोट बुक जरुर रखे और जो महत्वपूर्ण बाते हो उसे नोट बुक में नोट कर ले| नोटस बना लेने से आपको आसानी होगी, आप महत्वपूर्ण चीजो को आसानी से दोहरा सकोगे, इससे आपका समय भी बचेगा और बचे हुए समय में आप अन्य विषयों पर ध्यान दे सकोगे|
टिप्स संख्या २. प्राणायाम और ध्यान करें
जैसे स्वस्थ रहने के लिए कसरत करना जरूरी होता है वैसे ही दिमाग को स्वस्थ रखने के लिए प्राणायाम और ध्यान करना चाहिए| ध्यान करने से आपकी एकाग्रता(कन्सेंट्रेशन) की शक्ति बढ़ेगी और आपको चीजे ज्यादा जल्दी याद होंगी |
एग्जाम की तैयारी करने से पहले आप अपने विषयों को अलग अलग भागों में बाँट लीजिये, जैसे कि आपको कौन से विषय का कौन सा भाग आसन लगता है और कौन सा भाग कठिन लगता है| इससे आपको अच्छे से पता लग जायेगा की आपको कौन सा विषय ज्यादा पढ़ना है और कौन सा कम|
कहा जाता है कि "सफल लोगो को भी उतना ही समय मिलता है जितना कि असफल लोगो को मिलता है " कहने का अर्थ ये है की सफल व्यक्ति अपने समाय को सही से उपयोग करता है जबकि असफल समय को सही से उपयोग नही कर पाता| इसिलए आप अपने पढाई का एक उचित समय सारणी बनाये, समय सारणी बनाते समाय ध्यान रखे कि आपको अपने कमजोर विषयों पर ज्यादा समय देना है और एक ही विषय को लगातार न पढ़ कर बदल बदल कर पढ़े, इससे आपकी रूचि विषय में बनी रहेगी|
टिप्स संख्या ५. पढाई को मजेदार बनाये(ल्र्न वित फन)
मजे-मजे में सीखी हुई बाते ज्यादा देर तक याद रहती है, इसिलए आप भी कोशिश करे कि पढाई को मजेदार तरीके से करें|आप पढाई को मजेदार बनाने के लिए पुस्तकालय में जाकर नए नए किताबो से जानकारी इकठा करके पढ़ सकते हो या फिर इन्टरनेट से अपनी विषय से सम्बंधित जानकारी प्राप्त कर सकते हो|
टिप्स संख्या ६. आत्म मूल्याङ्कन
आपको जब कोई पाठ या विषय याद हो जाये तो खुद से अपना परीक्षा ले और अपना आत्म मूल्याङ्कन करे,इससे आपको अपनी गलतियों का पता चल जायेगा और आप उसको सुधार भी पाएंगे| हो सके तो सप्ताह में एक बार दोस्तों के साथ ग्रुप स्टडी भी करें, इससे कुछ नयी चीजे आपको समझ में आ जाएँगी|
अंत में मै आपको कहना चाहूँगा कि परीक्षा में ज्यादा तनाव न लें और बताई गयी टिप्स को फॉलो करे|हमने तो आपको बेस्ट टिप्स बता दिया लेकिन इसको फॉलो तो आपको ही करना होगा| हमारी शुभकामनाएं आपके साथ हैं|
अगर आपको कोई प्रश्न पूछना हो तो निचे कमेंट करके पूछ ले| | hindi |
>भारत में ब्रिटिश शासन की शुरूआत किस लड़ाई द्वारा हुई ? >एंगस्ट्रोम से किसका मापन किया जाता है ? >पृथ्वी के सबसे निकट कौन सा ग्रह है ? >भारतीय संविधान के अनुच्छेद ४५ का सम्बन्ध किससे है ? >गरीबी हटाने के लिए पहली बार किस पंचवर्षीय योजना में जोर दिया गया था ? >गीतगोविन्द के लेखक कौन थे ? >पृथ्वी को १ अक्षांश घूमने में कितना समय लगता है ? >लोकसभा का विघटन कौन कर सकता है ? >अद्वैत वाद मत का प्रवर्तन किसने किया था ? >महमूद गजवनी के आक्रमणों में सबसे महत्त्वपूर्ण आक्रमण कौन-सा था ? >बैंकिंग परिचालनों में हम बहुत बार क्ब्स शब्द पढ़ते हैं। क्ब्स शब्द में च से क्या शब्द बनता है ? >भारतीय रिजर्व बैंक का राष्ट्रीयकरण किस वर्ष हुआ था ? >अंग्रेजी ईस्ट इंडिया कंपनी का प्रतिनिधि कैप्टन हॉकिंस किसके राजदरबार में राजकीय अनुग्रह प्राप्त करने के लिए उपस्थित हुआ था ? >महान लॉन टेनिस खिलाड़ी बोर्न बॉर्ग किस देश का है ? >विलास वस्तुओं में क्रय के लिए बैंको द्वारा किस प्रकार का ऋण दिया जाता है ? >रॉकीज, एण्डीज, एटलस, आल्पस, हिमालय आदि किस प्रकार के पर्वत हैं ? >जवाहरलाल नेहरू राष्ट्रीय सौर मिशन कब शुरू किया गया था ? >केन्द्रीय मन्त्रिपरिषद् ने किस तिथि को राष्ट्रीय शहरी स्वास्थ्य मिशन को मंजूरी दी ? >द्वितीय पंचवर्षीय योजना में किस क्षेत्र के विकास पर सर्वाधिक बल दिया गया था ? >आनुपातिक प्रतिनिधित्व प्रणाली किसके चुनाव के लिए प्रयोग की जाती है ? >किसने सबसे पहले यह पता लगाया कि शुक्र ग्रह पूरी तरह से बादलों से घिरा हुआ है ? >डाइनामाइट बनाने में किस द्रव का प्रयोग किया जाता है ? >कौन सा सागर सबसे लवणीय सागर है ? >सौरमंडल का केंद्र कौन-सा है ? >द वे ऑफ नाइफः द सी आईए, ए सीक्रेट आर्मी एण्ड ए वार एट द एण्ड्स ऑफ द अर्थ पुस्तक के लेखक कौन हैं ? >मध्य प्रदेश की राजधानी निम्नलिखित में से कौन-सी है ? >वेबसाइट में प्रयुक्त ।ककतमेे को क्या कहा जाता है ? >भूमध्य रेखा पर सूर्य वर्ष में कितनी बार सीधा चमकता है ? >प्रथम जैन तीर्थंकर कौन थे ? >मानव-रूधिर का फ क्या है ? >यदि सूरत में बनी वस्तुएं मुंबई या दिल्ली में बेची जाएं, तो यह कौन सा व्यापार हुआ ? >उत्तर प्रदेश में नॉलेज पार्क की स्थापना कहाँ की जा रही है ? >उत्तर प्रदेश शैक्षणिक अनुसन्धान एवं विकास परिषद् की स्थापना कब की गई ? >भारत सरकार द्वारा दिया जाने वाला सर्वोच्च पुरस्कार कौन सा है ? >हड़प्पाई स्थलों में कांस्य नर्तकी की मूर्ति कहाँ से प्राप्त हुई है ? >मानस पशुविहार किस राज्य में स्थित है ? >संसार की अधिकतम वर्षा किस रूप में होती है ? >१2वीं पंचवर्षीय योजना में उत्तर प्रदेश के किन दो शहरों में एम्स जैसे अस्तपाल खोलने की घोषणा केन्द्रीय स्वास्थ्य तथा परिवार कल्याण मंत्रालय ने की है ? >गाँधी सागर, जवाहर सागर तथा राणा प्रताप सागर बाँध किस नदी पर निर्मित हैं ? >माई बेस्ट गेम ऑफ चेस किस शतरंज खिलाड़ी की प्रसिध्द पुस्तक है ? >सीपीयू का सबसे महत्त्वपूर्ण भाग कौन सा होता है ? >चित्रकला की बंगाल शैली का अग्रदूत कौन था ? >अशोक के किस शिलालेख में उसकी कंलिंग विजय का वर्णन है ? >चंगेज खाँ शब्द का क्या अर्थ होता है ? >किस ग्रह को सान्ध्य तारा कहा जाता है ? >फुटबॉल में ब्लैक पर्ल के उपनाम से किसे जाना जाता है ? >खाद्य ऊर्जा को हम किस इकाई में माप सकते हैं ? >वर्तमान में सम्पत्ति का अधिकार किस प्रकार का अधिकार है ? >संविधान सभा ने राष्ट्रीय गान को कब स्वीकार किया ? >भारत का राष्ट्रीय खेल क्या है ? >भारत में साइमन कमीशन के बहिष्कार का मुख्य कारण क्या था ? >भारत का दक्षिणतम स्थान इन्दिरा प्वाॅइण्ट कहाँ स्थित है ? >रामड्डष्ण मिशन की स्थापना किसने की ? >काले वन किस देश में पाए जाते हैं ? >राष्ट्रपति द्वारा राज्यसभा के सदस्यों के नामांकन का नियम किस देश के संविधान से लिया गया था ? >किशन महाराज किस वाद्य यंत्र से सम्बन्धित हैं ? >विश्व में सबसे विशाल मरुभूमि कौनसी है ? >भारतीय राष्ट्रीय कांग्रेस की स्थापना कहाँ हुई ? >विश्व का सबसे ऊँचा ज्वालामुखी कौन सा है ? >हल्दीघाटी की लड़ाई में अकबर ने किसको हराया था ? >ई.मेल संदेशों के लिए स्टोरेज क्षेत्र को क्या कहते हैं ? >भारत का राष्ट्रीय खेल क्या है ? >साँची के स्तूप का निर्माण किसने कराया था ? >सम्पत्ति के बँटवारे (निपटारे) के सम्बन्ध में न्यायालय के आदेशों का यथोचित पालन हुआ है कि नहीं, यह देखने का दायित्व किसका है ? >टैगोर पुरस्कार किस क्षेत्र में शुरू किया गया है ? >भारत का सबसे पुराना हॉकी टूर्नामेन्ट कौन-सा है ? >नॉक आउट किस खेल में सम्बन्धित है ? >भगत सिंह, सुखदेव एवं राजगुरू को फाँसी पर कब चढ़ाया गया ? >उत्तर प्रदेश में १920 ई. में भारतीय कला परिषद् की स्थापना कहाँ की गई ? >संसार का विशालतम स्तनधारी कौन-सा है ? >ड्डष्ण.भक्त मीरा का ब्याह किस राजवंश में हुआ था ? >मुहर्रम के अवसर पर प्रयोग में लिया जाने वाला प्रसिध्द वाद्ययंत्र कौनसा है ? >केन्द्रीय सरकार ने गाँव की ओर एक कदम की नीति स्वीकार करते हुए एक योजना जिसका नाम भारत निर्माण योजना था, प्रारम्भ की। यह योजना कब शुरू हुई ? >महात्मा गाँधी द्वारा सविनय अवज्ञा आंदोलन किस वर्ष शुरू किया गया था ? >कठपुतली किस राज्य का प्रमुख लोक नृत्य है ? >फिरोजशाह कोटला ग्राउण्ड कहाँ स्थित है ? >अकल पै के नाम से प्रसिध्द अनंत पै ने किस कॉमिक श्रृंखला की रचना की थी ? >पृथ्वी की ऊपरी परत को क्या कहा जाता हैं ? >जल की कठोरता जल में क्या डालकर दूर की जाती है ? >सुन्दरवन का डेल्टा कौनसी नदी बनाती है ? >सुरक्षा परिषद में स्थायी सदस्यों की संख्या कितनी है ? >मेसोपोटामिया की सभ्यता कहाँ विकसित हुई थी ? >विश्व की सबसे बड़ी वित्तीय संस्थाओं में से एक - बैंक ऑफ अमेरिका के बोर्ड में निदेशक के रूप में नियुक्त पहला गैर.अमेरिकी व्यक्ति कौन है ? >तेन्दुलकर समिति के द्वारा भारत में गरीबी का कितना प्रतिशत अनुमानित किया गया है ? >भारत के नियन्त्रक एवं महालेखा परीक्षक की नियुक्ति कौन करता है ? >भारतीय संविधान ने किस देश के संविधान से राज्य के नीर्ति.निर्देशन सिध्दांत ग्रहण किए हैं ? >किस राजा के शासनकाल में ईसाई धर्म प्रचारक सेण्ट थॉमस भारत आया ? >भारत में एकल नागरिकता की अवधारणा किस देश के संविधान से अपनाई गई है ? >हड्डी खाद के रूप में प्रयुक्त की जाती है, इसमें कौन सा पौध तत्व उपलब्ध होता है ? >भारतीय संविधान के किस अनुच्छेद में किसी भी प्रकार छुआछूत को असंवैधानिक घोषित किया गया है ? >विंग्स ऑफ फायर पुस्तक किसके द्वारा लिखी गई है ? >व्यायाम के दौरान मानव शरीर में पसीना आना किस प्रक्रिया का होना इंगित करता है ? >संविधान के कौन-से अनुच्छेद के अधीन भारत के राष्ट्रपति पर महाभियोग चलाया जा सकता है ? >मानव द्वारा सर्वप्रथम किस धातु का प्रयोग किया गया ? >सर्वाधिक सोने के सिक्के किस काल में चलाए गए ? >डबल रोटी बनाने में प्रयुक्त किए जाने वाला बेकिंग पाउडर क्या होता है ? >कावेरी नदी किस खाड़ी में गिरती है ? >भारतीय संविधान में मूल कर्तव्यों का वर्णन किस अनुच्छेद में है ? >भारत एक धर्मनिरपेक्ष राज्य है इसका उल्लेख किसमें किया गया है ? >मौर्य शासक अशोक ने कलिंग पर कब आक्रमण किया था ? >भारतीय उपमहाद्वीप की पहली सभ्यता का विकास कहाँ हुआ ? >४ अक्टूबर किस दिवस के रूप में मनाया जाता है ? >भारत में सबसे बड़ा जनजातीय समुदाय कौन सा है ? >पेट में भोजन को पचाने के लिए किसकी खास आवश्यकता होती है ? >नीला ग्रह के नाम से कौन जाना जाता है ? >धोने के सोडे का रासायनिक सूत्र क्या है ? >चण्डीगढ़ का वास्तुविद् ले कोर्बुजिया किस देश का नागरिक था ? >किस जंतु में तंत्रिका तंत्र नहीं होता ? >कौन-से विटामिन को डेंगू जैसी खतरनाक बीमारी के इलाज के लिए मददगार बताया गया है ? >संयुक्त राज्य अमेरिका में किसने थियोसोफिकल सोसायटी की स्थापना की थी ? >हेमेराइट किसका अयस्क है ? >शरीर में सबसे बड़ी अंतःस्रावी ग्रंथि कौनसी है ? >सर्वोच्च न्यायालय के न्यायाधीश की सेवानिवृत्ति की आयु कितनी होती है ? >उत्तर प्रदेश में किस वर्ष पंचायती राज प्रणाली का शुभारम्भ हुआ था ? >उत्तर प्रदेश में लोकायुक्त संगठन कब बना ? >संयुक्त राष्ट्र संघ का महासचिव बनने वाला पहला अफ्रीकी राष्ट्रीय व्यक्ति कौन था ? >चक्रवर्ती राजगोपालचारी से ठीक पहले कौन भारत का गवर्नर जनरल था ? >किस बीमारी में रक्त में शर्करा का स्तर बढ़ जाता है ? >दिल्ली में लाल किला किसके द्वारा बनवाया गया ? >अति लोकप्रिय धार्मिक पत्रिका कल्याण कहाँ से प्रकाशित होती है ? >ईकोमार्क का प्रतीक र्चिी क्या है ? >कौनण्से राज्य ने सर्वप्रथम ईण्कोर्ट फी सिस्टम लागू किया? >अहिंसा का चरम स्वरूप किस धर्म में सर्वाधिक पालन किया जाता है ? >किस सुल्तान ने बाजार नियन्त्रण व्यवस्था लागू की ? >उत्तर प्रदेश को पूर्व में किस नाम से जाना जाता था ? >मथने के पश्चात दूध से क्रीम किस कारण से पृथक् हो जाती है ? >अन्नपूर्णा योजना किस वर्ष कार्यन्वित की गई थी ? >किस ग्रह द्वारा सूर्य की परिक्रमा की गति के अवलोकन के आधार पर जोहानेस केप्लर ने अपने तीन सिध्दांतो की रचना की ? >महात्मा बुध्द को ज्ञान की प्राप्ति कहाँ हुई ? >भारत में पहला सफल यड्डत प्रत्यारोपण किसने किया ? >किस मुगल शासक ने अंग्रेजो को भारत में व्यापार करने की अनुमति दी थी ? >भारत का पहला कंप्यूटर कहां स्थापित किया गया था ? >डी.आर.डी.ओ. द्वारा विकसित अग्नि.५ मिसाइल की मारक क्षमता कितने कि.मी. तक है ? >यूनाइटेड किंगडम किसका एक उत्तम उदाहरण है ? >पृथ्वी का ऊपरी वायुमंडल सूर्य की ऊर्जा का कुल कितना भाग अंतरिक्ष में प्रतिबिंबित करता है ? >मुगलकाल में न्यायालयों में किस भाषा का प्रयोग किया जाता था ? >लोकसभा और राज्यसभा की संयुक्त बैठक की अध्यक्षता कौन करता है ? >भारतीय संविधान के किस अनुच्छेद में राज्य के नीति.निदेशक तत्त्वों का उल्लेख है ? >सल्तनतकालीन किस सुल्तान ने सर्वप्रथम किसानों पर सिंचाई कर तथा ब्राह्मणों पर जजिया कर लगाया ? >जनगणना 20११ के अनुसार देश में प्रति हजार पुरुषों पर महिलाओं की संख्या कितनी है ? >हीमोग्लोबिन का क्या कार्य है ? >मनुष्य एक सामाजिक प्राणी है. यह कथन किसका है ? >वर्तमान में संविधान में मूल कर्तव्यों की कुल संख्या कितनी है ? >राज्यालय/हड़प नीति/गोद निषेध सिध्दान्त किसके द्वारा लागू किया गया था ? >भारतीय अंतरिक्ष कार्यक्रम का पिता किसे कहा गया है ? >हिन्दू.मुस्लिम एकता का प्रतीक सुलहकुल उत्सव उत्तर प्रदेश के किस शहर में आयोजित होता है ? >कंप्यूटर के घटक उचित रूप से जोड़े गए हैं तथा कार्यरत है, इसे सुनिश्चित करने वाली कौनसी जांच-प्रक्रिया है ? >सुल्ताना रजिया बेगम किसकी बेटी थी ? >चेतना ऊतक के नाम से जाने वाला ऊतक कौन है ? >राजभाषा विभाग किस मंत्रालय के अधीन आता है ? मैनेज सर्विसेस बाय: समीक्षा सोफ्त्वार् | hindi |
I-5 Design relied on Elemental LED’s products and service as an essential solution to their clients’ custom design needs.
Goal: Design and oversee a complete renovation of the casino floor, gaming floor and customer service areas, including custom signage and integrated LED accent lighting within an aggressive timeline.
Challenge: I-5 Design required a lighting product with a flexible form factor, low long-term maintenance, and low energy consumption. They partnered with Elemental LED to help achieve the look they wanted while satisfying specification requirements. I-5 Design had previously worked with other lighting manufacturers who were often unable to fulfill quick orders. Elemental LED was able to meet their 12-week implementation deadline with extensive inventory held in their Emeryville, CA distribution center.
Since this tape light is field-cuttable and versatile, it was easily installed and featured throughout the casino, most prominently lighting the under counter areas at the bar, and overhead circular ceiling designs.
Elemental LED Value: One of the reasons I-5 Design partnered with Elemental LED was the responsive customer support. Faced with a 15,000 square foot remodel within the time frame of 12 weeks, I-5 Design relied on Elemental LED help advise on the best lighting product to meet their needs and deliver it within the timeline. Elemental LED manufactures and stocks inventory, including the FLUID VIEW 12V LED Tape Light, in the United States, which allow for same-day shipping directly to the project site for a quick installation that fits within a project timeline.
Installation was easy and completed on schedule. I-5 Design has received a lot of positive feedback from patrons and employees of the casino. The client was so impressed with the improvement, they asked I-5 Design to return and remodel the remaining spaces including the hotel lobby, gift shop, buffet, boardroom, and conference hall. I-5 Design continues to work with Elemental LED as its exclusive LED lighting provider for 4 Bears Casino. | english |
Additionally, Beech outfitted the 200 with more powerful Pratt & Whitney Canada PT6-42 engines. In essence, the King Air 200, originally dubbed the ‘Super King Air,’ was a response to the demand and need for increased capacity in the King Air 90. The result was a twin-engine airplane that went on to dominate the turboprop charter market (it still does to this day). Our clients like to charter King Air 200 turboprops because they offer a blend of speed and range. They have the ability to hit speeds of 333 mph and can cruise about 1,800 nautical miles. Furthermore, this aircraft can take off from runways as short as 2,579 feet. This is virtually impossible for most light and midsize jets, which gives it a major advantage when it comes to flight planning.
There’s also 55 cubic feet of cargo area inside the cabin, perfect for storing luggage and business equipment. The aircraft also features a private potty in the aft. The seats in a charter King Air 200 swivel and recline, allowing for easy communication during your flight. The square-oval cabin design maximizes head and shoulder room for added passenger comfort. To enhance the cabin experience further, Beechcraft installed vibration dampening tuning forks to mitigate propeller noise. The result is a remarkably quiet cabin. The King Air 200 is a favorite among pilots. Nimble and easy to fly, it comes equipped with a fully integrated Collins Pro Line 21 avionics system. | english |
سہ چھ ہیٚچھناوان | kashmiri |
I have wide feet, they IC ones fit ok. I paid £50 each for them. I am very happy with them and love the colours.
09/10/11 What a Fandabulous day!
A pair of shoes can change your life. Just ask Cinderella!
Absolutely love your shoes, Alana. Fabulous!
i like the pleats in the side as well and a little bling but not overstated... just the right amount of bling.
Thanks stressed to the max - was worried about ordering online and then wanting to return them.
Wohoooooooo h2b has just informed me I can order the JC's I wanted. Excited!!
Yeah stressed they are, perfect for my colour scheme although i defo wouldnt say they are teal as they describe them on the website. Were they the wrong colour for you then ?
Aw thats fab melody what ones?
OOH! Lovely, Melody! You lucky, lucky girl!
mine are in my album but i might change them yet i've seen some gorgeous irregular choice ones in purple. I also think i might go for a mary jane type court shoe rather than strappy. still lots of time to change my mind!!!!!
following my moan on this thread about shoes, is anybody wearing white dress but have bought ivory shoes. i have just been through all the sites you girls have recommend and most shoes i have kinda earmarked as a 'like' are mostly in ivory and just wondering if someone has done the same, what they look like together.
Why not go with a totally different colour if you can't find white shoes you like?
Thanks Becca! yes it is a spurs garter! h2b is a huge fan and we went and did the stadium tour in feb and he saw it in the shop and asked me to buy it and i said no way! then bought it online as a surprise!
Hi girls, in relation to my spec on shoes. today i received two pairs of shoes that i ordered both of which i love, but now i am worried that they may be too high in relation to the dress.
ashlil - one pair i ordered were detailed as winter white, but they are pristine white!!!! so fingers crossed.
Here are a few pics of my lovely shoes. I can not at the moment get them on as I have really really bad water retention. The before picture is in my photos. I have added millions of Swarovski crystals. | english |
"There is no governor present anywhere"
From Only Maybe, where you can see another piece of art from Bobby.
Great image, Bobby! Thanks for sharing this, Tom. | english |
Visit our beautiful cemetery "cimitero monumentale delle porte sante" and stroll along the cypress alleys leading to many graveyard chapels and enjoy the peace and quiet.
Treat yourself to a bus excursion with the local bus company and go to the "Infiorata di Spello" - a traditional, religious Corpus Christi festival - several kilometers of the main street of Spello are decorated with petals during the night of Saturday and Sunday. You should not miss this event (155km).
Be our guest - we are looking forward to you! | english |
Call today, 830-299-4076, Frigidaire Washer Repair in Pipe Creek for a same day or next day appointment for a Washer Repair. If you are located in Pipe Creek or around in the Bandera County area and need Frigidaire Washer Repair, please call Pipe Creek Appliance Repair Men. If you need an experienced Frigidaire Washer Repair technician in Pipe Creek, we can send out a service technician to diagnose your Washer. All Frigidaire Washer Repair technicians have extensive experience servicing all types of models and type of Washers including, Frigidaire Washer, Frigidaire Top Load Washer, Frigidaire Washer Installation, Frigidaire Front Load Washer and Frigidaire Washer.
Do not try troubleshooting your Frigidaire Washer at home by yourself as you can damage or ruin your appliance. The technician will not be able to work on your Frigidaire Washer if it has been tampered with or taken apart by another technician. Pipe Creek Frigidaire Washer repair technicians are available most of the time for same day appointments especially when it comes to Washers as we know how important it is to service quickly.
Call today, 830-299-4076, for an Frigidaire Washer Repair Service to schedule a same day or next day repair service for a small diagnostic fee cheaper than the industry average.
Call today, 830-299-4076, for a Frigidaire Washer repair and schedule a same day or next day appointment for a small diagnostic fee. You want a local technician that is located in Pipe Creek that services the entire Bandera County especially when dealing with a Washer repair. | english |
पॉक्सो कानून में बदलाव को मंजूरी, मृत्युदंड का प्रावधान शामिल - लेगेंड न्यूज
पॉक्सो कानून में बदलाव को मंजूरी, मृत्युदंड का प्रावधान शामिल
नई दिल्ली। केंद्रीय कैबिनेट ने यौन अपराधों से बच्चों का संरक्षण विधेयक पॉक्सो २०१२ के संशोधन में जेंडर न्यूट्रल और सख्त सजा के प्रावधानों को मंजूरी दे दी है। इसके अलावा जुर्माना चाइल्ड पोर्नोग्राफी के मामलों में भी सख्त कार्रवाई के साथ सजा के प्रस्ताव को भी मंजूरी दे दी है। पॉक्सो विधेयक को इस साल के शुरू में दोनों सदनों में पेश किया गया था लेकिन इन्हें पारित नहीं किया गया था।
विधेयक में विकल्प प्रदान करने के लिए अधिनियम की धारा ४,५,६ और ९ में संशोधन की मांग की गई थी। इस पर कैबिनेट ने बुधवार को मासूमों से दुष्कर्म और प्रयास को कड़ी सजा और मृत्युदंड की परिधि में लाया गया है। इससे बच्चों के साथ होने वाले यौन उत्पीड़न पर सजा के वर्तमान प्रावधानों को कठोर कर दिया गया है।
अब इस प्रस्ताव के पास होने के बाद १८ साल से कम उम्र के बच्चे वह चाहे किसी भी लिंग के हो यदि उनके विरुद्ध अपराध होता है तो मृत्युदंड का प्रावधान है। साथ ही अन्य मामलों में भी कड़ी सजा की सिफारिश की गई है। ताकि समाज के समाने कानून का डर बने और बच्चों के प्रति अपराधों में कमी आए। अब इस बिल को इन संशोधनों के साथ दोनों सदनों के पटल पर लाया जाएगा। ताकि इसे पास कराके कड़े कानूनों के तुरंत प्रभाव से लागू कराया जा सके।
बच्चों के साथ होने वाले यौन अपराधों की घटनाएं समाज को शर्मसार करती हैं। इस तरह के मामलों की बढ़ती संख्या देखकर सरकार ने वर्ष २०१२ में एक विशेष कानून बनाया था। पॉक्सो कानून यानी की प्रोटेक्शन ऑफ चिल्ड्रन फ्र्म सेक्सुअल ऑफेंसेस एक्ट २०१२ जिसको हिंदी में लैंगिक उत्पीड़न से बच्चों के संरक्षण का अधिनियम २०१२ कहा जाता है।
इस कानून के तहत अलग-अलग अपराध में अलग-अलग सजा का प्रावधान है और यह भी ध्यान दिया जाता है कि इसका पालन कड़ाई से किया जा रहा है या नहीं। इस कानून की धारा चारा में वो मामले आते हैं जिसमें बच्चे के साथ कुकर्म या फिर दुष्कर्म किया गया हो। इस अधिनियम में सात साल की सजा से लेकर उम्रकैद तक का प्रावधान है साथ ही साथ जुर्माना भी लगाया जा सकता है।
इस अधिनियम की धारा छह के अंतर्गत वो मामले आते हैं जिनमें बच्चों के साथ कुकर्म, दुष्कर्म के बाद उनको चोट पहुंचाई गई हो। इस धारा के तहत १० साल से लेकर उम्रकैद तक की सजा का प्रावधान है साथ ही जुर्माना भी लगाया जा सकता है। अगर धारा सात और आठ की बात की जाए तो उसमें ऐसे मामले आते हैं जिनमें बच्चों के गुप्तांग में चोट पहुंचाई जाती है। इसमें दोषियों को पांच से सात साल की सजा के साथ जुर्माना का भी प्रावधान है।
१८ साल से कम किसी भी मासूम के साथ अगर दुराचार होता है तो वह पॉक्सो एक्ट के तहत आता है। इस कानून के लगने पर तुरंत गिरफ्तारी का प्रावधान है। इसके अतिरिक्त अधिनियम की धारा ११ के साथ यौन शोषण को भी परिभाषित किया जाता है। जिसका मतलब है कि यदि कोई भी व्यक्ति अगर किसी बच्चे को गलत नीयत से छूता है या फिर उसके साथ गलत हरकतें करने का प्रयास करता है या उसे पॉर्नोग्राफी दिखाता है तो उसे धारा ११ के तहत दोषी माना जाएगा। इस धारा के लगने पर दोषी को तीन साल तक की सजा हो सकती है।
बिजनोर: मदरसे में मिला अवैध हथियारों का जखीरा, ६ गिरफ्तार
विदेशी फंडिंग केस में सुप्रीम कोर्ट की मशहूर वकील के यहां क्बी का छापा
लगातार ११वें दिन भी बढ़े पेट्रोल-डीजल के दाम
शोपियां में मुठभेड़: दो आतंकवादी मार गिराए, एनकाउंटर जारी | hindi |
For many, their day doesn’t start till they don’t sip their favourite hot tea. Be it tea bags, loose tea leaves or green tea, Spencer’s has it all. It is your one-stop shop destination for all things grocery. Make your selection and order it online.
Shop from Top Brands like Brooke Bond, Lipton, Tetley & Other prolific Spencer’s Brands. | english |
In Texas, we embrace hard work, the belief in opportunity for all, and treating others like we want to be treated. Discrimination toward anyone, including gay and transgender people—is out of line with values Texans hold dear.
All Texans should have the rights to provide for themselves and their families, to live in a safe place, and to access public spaces, safely and without fear of losing these rights because of who they are or whom they love. Though we believe that everyone should be treated equally, our laws do not always reflect that belief.
Most Texans believe that gay and transgender people should be protected in the workplace and many believe that it’s already illegal to fire someone solely for being gay or transgender. But in Texas, it is completely legal to do just that—terminate an employee, even though that person is performing satisfactorily, because of their sexual orientation or gender identity or expression.
The lack of safety for LGBT Texans in Texas classrooms, public accommodations, health care facilities, and other spaces fosters an un-Texan culture of bias and prejudice.
Hardworking Texans who do their job and are working to provide for themselves and their families have no protection from being fired because they identify as LGBT. And this practice puts our state at a disadvantage when attracting and keeping a talented workforce that believes in diversity and equality.
Texas will be at its strongest when discrimination is prohibited in employment, housing, and public accommodations and all Texans are treated fairly and equally.
Status: 4/24//17 Committee report sent to Calendars. | english |
ब्डी स्प्क्ल: अमृता सिंह की इस हरकत की वजह से परेशान सैफ अली खान ने तोड़ दी थी शादी
नई दिल्ली/टीम डिजिटल। बॉलीवुड के छोटे नवाब के नाम से मशहूर सैफ अली खान (साफ अली खान) आज अपना ४९वां जन्मदिन मना रहे हैं। वहीं पटौदी खानदान के बेटे सैफ अली खान अपनी फिल्मों से ज्यादा अपनी पर्सनल लाइफ को लेकर खूब चर्चा में रहे हैं। हालांकि अभी वो अपने जिंदगी के बेस्ट पल को जी रहे हैं लेकिन एक समय ऐसा भी था जब उनके लाइफ में कुछ भी अच्छा नहीं था। आइए आज हम आपको उनके जीवन से जुड़ी कुछ अनसुनी बातें बताएंगे।
एक्स्स्लसिव विडियो: मिशन मंगल की टीम ने खोले फिल्म के कई राज, देखें वीडियो
साल १९९१ में सैफ ने खुद से १२ साल बड़ी लड़की से शादी कर सभी को हैरान कर दिया था। ये लड़की कोई और नहीं बल्कि उस दौर की मशहूर एक्ट्रेस अमृता सिंह (अमृता सिंह) थीं। जी हां, ये बात उस समय की है जब सैफ ने इंडस्ट्री में कुछ खास मुकाम हासिल नहीं किया था। वो उस वक्त इंडस्ट्री में स्ट्रगल ही कर रहे थे कि जब उनकी मुलाकात से हुई और वो उनके प्यार में पागल हो गएं। बात इतनी ज्यादा बढ़ गई कि दोनों ने बिना किसी को बताए छुप छुपाकर शादी रचा ली।
आयुष्मान खुराना की फिल्म 'ड्रीम गर्ल' का पहला गाना 'राधे' आज होगा रिलीज
वहीं ये शादी पटौदी खानदान के लिए भी चौंका देने वाली थी। शादी के बाद अमृता ने फिल्मों में काम ना करने का फैसला किया। बता दें कि उस वक्त अमृता अपने करियर की उचाइयों पर थीं। वहीं सैफ भी अमृता के इस फैसले से हैरान हो गए थे लेकिन वो उस समय चुप रहें। लेकिन शादि के कुछ सालों बाद सैफ की भी फिल्में लगातार फ्लॉप होने लगीं और इस वजह से सैफ को भी कम फिल्में मिलने लगी।
ऐसे में अमृता फिल्मों में सैफ की असफलता से हताश हो गईं और आए दिन उन्हें तानें मारने लगीं। धीरे-धीरे दोनों में प्यार खत्म होने लगा और लड़ाईयां बढ़नी शुरू हो गईं।
बात इतनी ज्यादा बढ़ गई की अमृता सैफ को निकम्मा और नाकारा कह कर ताने मारने लगीं। वहीं अमृता सैफ की मां और उनकी बहन सोहा अली खान को से भी गाली-गलौच करने लगीं थी। ऐसे में सैफ अपनी शादी शुदा जिंदगी को लेकर काफी परेशान रहने लगे और जब पानी सर से ऊपर चला गया तो उन्होंने अमृता से अलग रहने का फैसला ले लिया।
हालांकि ये सब करना इतना आसान नहीं रहा लेकिन सैफ ने हिम्मत जुटाई और अमृता के साथ अपने इस शादी को तोड़ दिया। वहीं कहा जाता है कि अमृता अपने बच्चों को सैफ से मिलने तक नहीं देती थीं।
चंद्रमा की कक्षा में पहुंचा चंद्रयान-२, कम केजरीवाल ने गर्व से कही ये...
सेना के खिलाफ शेहला रशीद के आरोप को गृह राज्य मंत्री ने किया खारिज,... | hindi |
Q: Pat: I have followed Avalon Advanced Materials for some time. What are your thoughts on this company and its prospects?
A: Avalon Advanced Materials, $0.12, symbol AVL on Toronto (Shares outstanding: 179.5 million; Market cap: $21.5 million; www.avalonadvancedmaterials.com), is a Canadian junior mineral exploration and development company. It is largely focused on lithium and rare earth elements.
IAMGOLD $2.90 (Toronto symbol IMG; TSINetwork Rating: Speculative) (1-888-464-9999; www.iamgold.com; Shares outstanding: 376.9 million; Market cap: $1.0 billion; No dividends paid) now holds over $800 million U.S. of cash and gold bullion after the sale of its Niobec niobium mine in Quebec’s Saguenay-Lac-Saint-Jean region.
DOREL INDUSTRIES, $40.09, symbol DII.B on Toronto, is our Stock of the Year for 2015.
The company makes a range of items, including ready-to-assemble home and office furniture; juvenile products, such as car seats, strollers, high chairs, toddler beds and cribs; and recreational goods, mainly bicycles.
Pat: What do you think about Arafura Resources?
Arafura Resources, $0.05, symbol ARU on the Australian exchange (Shares outstanding: 396.0 million; Market cap: $19.8 million; www.arultd.com), is an Australian firm with a rare earth property in the country’s Northern Territory. | english |
Poison’s Kiss was the story of a Visha Kanya, a poison girl, who was raised to kiss and kill. Living her life essential as a free prisoner, Marinda has to kiss at her handlers bidding per the request of the Raja. As much as she wants to escape, she can’t because of her brother and the man she just might be in love with, and who happens to be her latest assignment.
This was a really quick, easy read. The world building was simple, but adequate as you are introduced into a Indian land with four different entities that are worshiped. The Hindu inspirations were intriguing as well as the words that were incorporated into the book: visha kanya – poison girl, rajakumari-princess, janu-sweetheart, vish bimari- poison disease. Having foreign words to me always adds in a little more flavor and culture into a book. I loved how mithridatism was incorporated into the book that someone could become immune to poisons by slowly introducing it into their systems.
The characters were alright. Marinda was a little too trusting and was a little annoying how she relied so much on her handler for medicines. She was naive to think that the people who would hurt her or her friend would be telling her everything honestly. Deven was all fine and dandy like a good antagonist would be, but their romance wasn’t quite believable. I felt that it could have been built up a little bit more, but their relationship was sort of shallow at best. Iyla was an interesting character and I liked her relationship with Marinda, how her character was used, and how understandable her feelings would be to those that were involved with her life.
Although predictable at times, it was still a fun read full of culture, loyalty, and friendship. I would read the next book just to see where the story is going. | english |
یہِ چُھ تِمن لُکن ہِنٛدِ خٲطرٕ اَصٕل حَل یِم کانٛٹیکٹ لیٚنٛس ژٲلِتھ ہیٚکن | kashmiri |
पिक - विराट ने शेयर की अपनी शादी की अनदेखी तस्वीरें, कहा अनुष्का मेरी बेस्टफ्रेंड है ! | ए२४बॉलिवुड
पिक - विराट ने शेयर की अपनी शादी की अनदेखी तस्वीरें, कहा अनुष्का मेरी बेस्टफ्रेंड है !
होम पिक - विराट ने शेयर की अपनी शादी की अनद...
नीतू कुमार - विराट अनुष्का की शादी को एक साल हो चुके हैं । दोनों शादी की सालगिरह ऑस्ट्रेलिया में मना रहे हैं। दोनों ने पिछले साल दिसंबर में इटली के फ्लोंरेस में डेस्टिनेशन वेडिंग की । पहली सालगिरह पर विराट कोहली ने शादी की अनदेखी तस्वीरें शेयर की हैं। साथ ही विराट ने लिखा है कि पता नहीं चला कि कैसे एक साल बीत गया। साथ ही अनुष्का को उन्होंने अपना बेस्ट फ्रेंड और सोलमेट कहा है। | hindi |
मार्केट एक्सपर्ट उदयन मुखर्जी ने कहा कि यूएस-चीन ट्रेड वॉर खत्म हुआ तो बाजार दौड़ेंगे।
बाजार की आगे की चाल और दिशा पर बात करते हुए मार्केट एक्सपर्ट उदयन मुखर्जी ने कहा कि बाजार के मूड में पिछले १० दिन में हेरफेर हुआ है। लंबी अवधि में भारत-पाक तनाव का बाजार में कोई बड़ा असर देखने को नहीं मिलेगा। भारत-पाक तनाव को दौरान सरकार ने जो कदम उठाए हैं उससे मोदी सरकार की संभावनाएं मजबूत हुई हैं। इसी के चलते सेंटीमेंट अच्छा हुआ है और मिड और स्मॉल कैप शेयरों में रैली देखने को मिल रही है। बाजार को लग रहा है कि मौजूदा सरकार की वापसी होगी। ऐसे में चुनाव से पहले मिड और स्मॉल कैप शेयरों में पैसे बनाने के अच्छे मौके दिख रहे हैं।
अमेरिका-भारत ट्रेड इस समय बाजार के लिए बड़ी चिंता नहीं है। बाजार पर इस समय सिर्फ इलेक्शन का भूत सवार है। बाजार की नजर सिर्फ भारत-पाक तनाव और चुनावों पर है। यूएस-चीन ट्रेड वॉर खत्म हुआ तो बाजार दौड़ेंगे। मिड और स्मॉलकैप शेयरों में सीमेंट, इंफ्रा, मेटल और कुछ कंज्यूमर स्टॉक्स काफी अच्छे लग रहे हैं। इनमें नियर टर्म में अच्छे पैसे बन सकते हैं। | hindi |
کیا ژٕ ہیٛککھا ۱۸۳۸۱۳۹۱۲۳۰۴ پران نمبر والہِ اے پی واٮٔی اکاؤنٹس مَنٛز میٲنۍ کل ہولڈنگ چیک کٔرتھ | kashmiri |
Pocketknife by Boker Plus Elegance. This traditional and versatile pocket knife offers an extraordinarily smooth blade action and perfectly fitted Bubinga wood scales, crowned by brushed stainless steel bolsters. The classic nail nick provides comfortable opening of the 440C stainless steel blade, which locks securely with the sturdy lockback mechanism. The Bubinga scales and the polished blade make this knife special, in showcase or pocket. | english |
شیتیا ونیسم یا نِراول یا نیراول چھِ أکہِ گؠونٕچ اکھ یا زٕ سطرن ہُنٛد بار بار گؠوُن یتھ منٛز پرؠٹھ دۄہراونس منٛز اصلٲحی نمٲئش چھِ گژھان | kashmiri |
सेल्फ स्टडी के लिए पहले तो हमें अपना अपनी लोकेशन को अच्छी तरह से देख लेना चाहिए शांत वातावरण में हमें अभ्यास की प्रक्रिया शुरू करनी चाहिए एकांत और एकाग्रता को सामने रखकर हमें अपना कंसंट्रेशन पूरी तरह ...
सेल्फ स्टडी के लिए पहले तो हमें अपना अपनी लोकेशन को अच्छी तरह से देख लेना चाहिए शांत वातावरण में हमें अभ्यास की प्रक्रिया शुरू करनी चाहिए एकांत और एकाग्रता को सामने रखकर हमें अपना कंसंट्रेशन पूरी तरह से किताब और अपनी पढ़ाई पर रखना चाहिए तभी हम सेल्फ स्टडी कैसे कर पाएंगे सेल्फ स्टडी मतलब जो हमने पढ़ा उसका चिंतन हमको करना होगा और अच्छी तरह से हम यूपीएससी में अंक ले सकते हैं यदि हम अपना फुल कंसंट्रेट पढ़ाई पर ध्यान रखें तो हम जल्दी यूपीसी कोएजू कर पाएंगे धन्यवाद
अगर आप यह जानना चाहते हैं कि कोई भी इंसान का सरल तरीका समय क्या...जवाब पढ़िये
इंग्लिश आती है आपको जो है इंग्लिश को समझना होगा ओके तो उसके बाद जो...जवाब पढ़िये
आरआरबी जे एक्जाम की सेल्फी कैसे करें...जवाब पढ़िये
स्टडी करने के लिए मैं ज्यादा कुछ तो बोल कर बोलूंगा नहीं सिंपल सी बात...जवाब पढ़िये
आपकी आर्थिक स्थिति अच्छी नहीं है तो क्या आप पीएसी का एग्जाम की तैयारी कर...जवाब पढ़िये
दिखने में पांच से छह सूची लखनऊ हर साल आते जिसमें से तीन चार चारों...जवाब पढ़िये
दीपिका रेटिंग कैसे रखा जाए आप देखेंगे क्या है आप पढ़ सकते हैं टिकट बना...जवाब पढ़िये
अगर आप सेल्फ स्टडी करना चाहते थे उसको बहुत सारे तरीके हैं पहले तो आप जो भी सभी का संस्कारी करना चाहते हैं ताकि स्कूल कॉलेज के एग्जाम पैटर्न देख लीजिए एग्जाम का सिलेबस ले लीजिए हमने जो है आप दिन में आप... | hindi |
مرا دم ہے شکستہ سپر جاناں | kashmiri |
<!DOCTYPE html>
<!--[if IE 8]> <html lang="en" class="ie8"> <![endif]-->
<!--[if IE 9]> <html lang="en" class="ie9"> <![endif]-->
<!--[if !IE]><!--> <html lang="en"> <!--<![endif]-->
<!-- BEGIN HEAD -->
<head>
<meta charset="utf-8" />
<title>Metronic | UI Features - Buttons</title>
<meta content="width=device-width, initial-scale=1.0" name="viewport" />
<meta content="" name="description" />
<meta content="" name="author" />
<link href="assets/bootstrap/css/bootstrap.min.css" rel="stylesheet" />
<link href="assets/css/metro.css" rel="stylesheet" />
<link href="assets/bootstrap/css/bootstrap-responsive.min.css" rel="stylesheet" />
<link href="assets/font-awesome/css/font-awesome.css" rel="stylesheet" />
<link href="assets/glyphicons/css/glyphicons.css" rel="stylesheet" />
<link href="assets/glyphicons_halflings/css/halflings.css" rel="stylesheet" />
<link href="assets/css/style.css" rel="stylesheet" />
<link href="assets/css/style_responsive.css" rel="stylesheet" />
<link href="assets/css/style_default.css" rel="stylesheet" id="style_color" />
<link rel="stylesheet" type="text/css" href="assets/uniform/css/uniform.default.css" />
<link rel="shortcut icon" href="favicon.ico" />
</head>
<!-- END HEAD -->
<!-- BEGIN BODY -->
<body class="fixed-top">
<!-- BEGIN HEADER -->
<div class="header navbar navbar-inverse navbar-fixed-top">
<!-- BEGIN TOP NAVIGATION BAR -->
<div class="navbar-inner">
<div class="container-fluid">
<!-- BEGIN LOGO -->
<a class="brand" href="index.html">
<img src="assets/img/logo.png" alt="logo" />
</a>
<!-- END LOGO -->
<!-- BEGIN RESPONSIVE MENU TOGGLER -->
<a href="javascript:;" class="btn-navbar collapsed" data-toggle="collapse" data-target=".nav-collapse">
<img src="assets/img/menu-toggler.png" alt="" />
</a>
<!-- END RESPONSIVE MENU TOGGLER -->
<!-- BEGIN TOP NAVIGATION MENU -->
<ul class="nav pull-right">
<!-- BEGIN NOTIFICATION DROPDOWN -->
<li class="dropdown" id="header_notification_bar">
<a href="#" class="dropdown-toggle" data-toggle="dropdown">
<i class="icon-warning-sign"></i>
<span class="badge">6</span>
</a>
<ul class="dropdown-menu extended notification">
<li>
<p>You have 14 new notifications</p>
</li>
<li>
<a href="#">
<span class="label label-success"><i class="icon-plus"></i></span>
New user registered.
<span class="time">Just now</span>
</a>
</li>
<li>
<a href="#">
<span class="label label-important"><i class="icon-bolt"></i></span>
Server #12 overloaded.
<span class="time">15 mins</span>
</a>
</li>
<li>
<a href="#">
<span class="label label-warning"><i class="icon-bell"></i></span>
Server #2 not respoding.
<span class="time">22 mins</span>
</a>
</li>
<li>
<a href="#">
<span class="label label-info"><i class="icon-bullhorn"></i></span>
Application error.
<span class="time">40 mins</span>
</a>
</li>
<li>
<a href="#">
<span class="label label-important"><i class="icon-bolt"></i></span>
Database overloaded 68%.
<span class="time">2 hrs</span>
</a>
</li>
<li>
<a href="#">
<span class="label label-important"><i class="icon-bolt"></i></span>
2 user IP blocked.
<span class="time">5 hrs</span>
</a>
</li>
<li class="external">
<a href="#">See all notifications <i class="m-icon-swapright"></i></a>
</li>
</ul>
</li>
<!-- END NOTIFICATION DROPDOWN -->
<!-- BEGIN INBOX DROPDOWN -->
<li class="dropdown" id="header_inbox_bar">
<a href="#" class="dropdown-toggle" data-toggle="dropdown">
<i class="icon-envelope-alt"></i>
<span class="badge">5</span>
</a>
<ul class="dropdown-menu extended inbox">
<li>
<p>You have 12 new messages</p>
</li>
<li>
<a href="#">
<span class="photo"><img src="./assets/img/avatar2.jpg" alt="" /></span>
<span class="subject">
<span class="from">Lisa Wong</span>
<span class="time">Just Now</span>
</span>
<span class="message">
Vivamus sed auctor nibh congue nibh. auctor nibh
auctor nibh...
</span>
</a>
</li>
<li>
<a href="#">
<span class="photo"><img src="./assets/img/avatar3.jpg" alt="" /></span>
<span class="subject">
<span class="from">Richard Doe</span>
<span class="time">16 mins</span>
</span>
<span class="message">
Vivamus sed congue nibh auctor nibh congue nibh. auctor nibh
auctor nibh...
</span>
</a>
</li>
<li>
<a href="#">
<span class="photo"><img src="./assets/img/avatar1.jpg" alt="" /></span>
<span class="subject">
<span class="from">Bob Nilson</span>
<span class="time">2 hrs</span>
</span>
<span class="message">
Vivamus sed nibh auctor nibh congue nibh. auctor nibh
auctor nibh...
</span>
</a>
</li>
<li class="external">
<a href="#">See all messages <i class="m-icon-swapright"></i></a>
</li>
</ul>
</li>
<!-- END INBOX DROPDOWN -->
<!-- BEGIN TODO DROPDOWN -->
<li class="dropdown" id="header_task_bar">
<a href="#" class="dropdown-toggle" data-toggle="dropdown">
<i class="icon-tasks"></i>
<span class="badge">5</span>
</a>
<ul class="dropdown-menu extended tasks">
<li>
<p>You have 12 pending tasks</p>
</li>
<li>
<a href="#">
<span class="task">
<span class="desc">New release v1.2</span>
<span class="percent">30%</span>
</span>
<span class="progress progress-success ">
<span style="width: 30%;" class="bar"></span>
</span>
</a>
</li>
<li>
<a href="#">
<span class="task">
<span class="desc">Application deployment</span>
<span class="percent">65%</span>
</span>
<span class="progress progress-danger progress-striped active">
<span style="width: 65%;" class="bar"></span>
</span>
</a>
</li>
<li>
<a href="#">
<span class="task">
<span class="desc">Mobile app release</span>
<span class="percent">98%</span>
</span>
<span class="progress progress-success">
<span style="width: 98%;" class="bar"></span>
</span>
</a>
</li>
<li>
<a href="#">
<span class="task">
<span class="desc">Database migration</span>
<span class="percent">10%</span>
</span>
<span class="progress progress-warning progress-striped">
<span style="width: 10%;" class="bar"></span>
</span>
</a>
</li>
<li>
<a href="#">
<span class="task">
<span class="desc">Web server upgrade</span>
<span class="percent">58%</span>
</span>
<span class="progress progress-info">
<span style="width: 58%;" class="bar"></span>
</span>
</a>
</li>
<li>
<a href="#">
<span class="task">
<span class="desc">Mobile development</span>
<span class="percent">85%</span>
</span>
<span class="progress progress-success">
<span style="width: 85%;" class="bar"></span>
</span>
</a>
</li>
<li class="external">
<a href="#">See all tasks <i class="m-icon-swapright"></i></a>
</li>
</ul>
</li>
<!-- END TODO DROPDOWN -->
<!-- BEGIN USER LOGIN DROPDOWN -->
<li class="dropdown user">
<a href="#" class="dropdown-toggle" data-toggle="dropdown">
<img alt="" src="assets/img/avatar1_small.jpg" />
<span class="username">Bob Nilson</span>
<i class="icon-angle-down"></i>
</a>
<ul class="dropdown-menu">
<li><a href="extra_profile.html"><i class="icon-user"></i> My Profile</a></li>
<li><a href="calendar.html"><i class="icon-calendar"></i> My Calendar</a></li>
<li><a href="#"><i class="icon-tasks"></i> My Tasks</a></li>
<li class="divider"></li>
<li><a href="login.html"><i class="icon-key"></i> Log Out</a></li>
</ul>
</li>
<!-- END USER LOGIN DROPDOWN -->
</ul>
<!-- END TOP NAVIGATION MENU -->
</div>
</div>
<!-- END TOP NAVIGATION BAR -->
</div>
<!-- END HEADER -->
<!-- BEGIN CONTAINER -->
<div class="page-container row-fluid">
<!-- BEGIN SIDEBAR -->
<div class="page-sidebar nav-collapse collapse">
<!-- BEGIN SIDEBAR MENU -->
<ul>
<li>
<!-- BEGIN SIDEBAR TOGGLER BUTTON -->
<div class="sidebar-toggler hidden-phone"></div>
<!-- BEGIN SIDEBAR TOGGLER BUTTON -->
</li>
<li>
<!-- BEGIN RESPONSIVE QUICK SEARCH FORM -->
<form class="sidebar-search">
<div class="input-box">
<a href="javascript:;" class="remove"></a>
<input type="text" placeholder="Search..." />
<input type="button" class="submit" value=" " />
</div>
</form>
<!-- END RESPONSIVE QUICK SEARCH FORM -->
</li>
<li class="start ">
<a href="index.html">
<i class="icon-home"></i>
<span class="title">Dashboard</span>
</a>
</li>
<li class="active has-sub ">
<a href="javascript:;">
<i class="icon-bookmark-empty"></i>
<span class="title">UI Features</span>
<span class="selected"></span>
<span class="arrow open"></span>
</a>
<ul class="sub">
<li ><a href="ui_general.html">General</a></li>
<li class="active"><a href="ui_buttons.html">Buttons</a></li>
<li ><a href="ui_tabs_accordions.html">Tabs & Accordions</a></li>
<li ><a href="ui_sliders.html">Sliders</a></li>
<li ><a href="ui_tiles.html">Tiles</a></li>
<li ><a href="ui_typography.html">Typography</a></li>
<li ><a href="ui_tree.html">Tree View</a></li>
<li ><a href="ui_nestable.html">Nestable List</a></li>
</ul>
</li>
<li class="has-sub ">
<a href="javascript:;">
<i class="icon-table"></i>
<span class="title">Form Stuff</span>
<span class="arrow "></span>
</a>
<ul class="sub">
<li ><a href="form_layout.html">Form Layouts</a></li>
<li ><a href="form_samples.html">Advance Form Samples</a></li>
<li ><a href="form_component.html">Form Components</a></li>
<li ><a href="form_wizard.html">Form Wizard</a></li>
<li ><a href="form_validation.html">Form Validation</a></li>
<li ><a href="form_fileupload.html">Multiple File Upload</a></li>
<li ><a href="form_dropzone.html">Dropzone File Upload</a></li>
</ul>
</li>
<li class="has-sub ">
<a href="javascript:;">
<i class="icon-th-list"></i>
<span class="title">Data Tables</span>
<span class="arrow "></span>
</a>
<ul class="sub">
<li ><a href="table_basic.html">Basic Tables</a></li>
<li ><a href="table_managed.html">Managed Tables</a></li>
<li ><a href="table_editable.html">Editable Tables</a></li>
</ul>
</li>
<li class="has-sub ">
<a href="javascript:;">
<i class="icon-th-list"></i>
<span class="title">Portlets</span>
<span class="arrow "></span>
</a>
<ul class="sub">
<li ><a href="portlet_general.html">General Portlets</a></li>
<li ><a href="portlet_draggable.html">Draggable Portlets</a></li>
</ul>
</li>
<li class="has-sub ">
<a href="javascript:;">
<i class="icon-map-marker"></i>
<span class="title">Maps</span>
<span class="arrow "></span>
</a>
<ul class="sub">
<li ><a href="maps_google.html">Google Maps</a></li>
<li ><a href="maps_vector.html">Vector Maps</a></li>
</ul>
</li>
<li class="">
<a href="charts.html">
<i class="icon-bar-chart"></i>
<span class="title">Visual Charts</span>
</a>
</li>
<li class="">
<a href="calendar.html">
<i class="icon-calendar"></i>
<span class="title">Calendar</span>
</a>
</li>
<li class="">
<a href="gallery.html">
<i class="icon-camera"></i>
<span class="title">Gallery</span>
</a>
</li>
<li class="has-sub ">
<a href="javascript:;">
<i class="icon-briefcase"></i>
<span class="title">Extra</span>
<span class="arrow "></span>
</a>
<ul class="sub">
<li ><a href="extra_profile.html">User Profile</a></li>
<li ><a href="extra_faq.html">FAQ</a></li>
<li ><a href="extra_search.html">Search Results</a></li>
<li ><a href="extra_invoice.html">Invoice</a></li>
<li ><a href="extra_pricing_table.html">Pricing Tables</a></li>
<li ><a href="extra_404.html">404 Page</a></li>
<li ><a href="extra_500.html">500 Page</a></li>
<li ><a href="extra_blank.html">Blank Page</a></li>
<li ><a href="extra_full_width.html">Full Width Page</a></li>
</ul>
</li>
<li class="">
<a href="login.html">
<i class="icon-user"></i>
<span class="title">Login Page</span>
</a>
</li>
</ul>
<!-- END SIDEBAR MENU -->
</div>
<!-- END SIDEBAR -->
<!-- BEGIN PAGE -->
<div class="page-content">
<!-- BEGIN SAMPLE PORTLET CONFIGURATION MODAL FORM-->
<div id="portlet-config" class="modal hide">
<div class="modal-header">
<button data-dismiss="modal" class="close" type="button"></button>
<h3>Widget Settings</h3>
</div>
<div class="modal-body">
<p>Here will be a configuration form</p>
</div>
</div>
<!-- END SAMPLE PORTLET CONFIGURATION MODAL FORM-->
<!-- BEGIN PAGE CONTAINER-->
<div class="container-fluid">
<!-- BEGIN PAGE HEADER-->
<div class="row-fluid">
<div class="span12">
<!-- BEGIN STYLE CUSTOMIZER -->
<div class="color-panel hidden-phone">
<div class="color-mode-icons icon-color"></div>
<div class="color-mode-icons icon-color-close"></div>
<div class="color-mode">
<p>THEME COLOR</p>
<ul class="inline">
<li class="color-black current color-default" data-style="default"></li>
<li class="color-blue" data-style="blue"></li>
<li class="color-brown" data-style="brown"></li>
<li class="color-purple" data-style="purple"></li>
<li class="color-white color-light" data-style="light"></li>
</ul>
<label class="hidden-phone">
<input type="checkbox" class="header" checked value="" />
<span class="color-mode-label">Fixed Header</span>
</label>
</div>
</div>
<!-- END BEGIN STYLE CUSTOMIZER -->
<!-- BEGIN PAGE TITLE & BREADCRUMB-->
<h3 class="page-title">
Buttons <small>buttons, icons, dropdowns and more</small>
</h3>
<ul class="breadcrumb">
<li>
<i class="icon-home"></i>
<a href="index.html">Home</a>
<i class="icon-angle-right"></i>
</li>
<li>
<a href="#">UI Features</a>
<i class="icon-angle-right"></i>
</li>
<li><a href="#">Buttons</a></li>
</ul>
<!-- END PAGE TITLE & BREADCRUMB-->
</div>
</div>
<!-- END PAGE HEADER-->
<!-- BEGIN PAGE CONTENT-->
<div class="row-fluid">
<div class="span6">
<!-- BEGIN BUTTONS PORTLET-->
<div class="portlet box green">
<div class="portlet-title">
<h4><i class="icon-reorder"></i>Buttons</h4>
<div class="tools">
<a href="javascript:;" class="collapse"></a>
<a href="#portlet-config" data-toggle="modal" class="config"></a>
<a href="javascript:;" class="reload"></a>
<a href="javascript:;" class="remove"></a>
</div>
</div>
<div class="portlet-body">
<p>
<button type="button" class="btn">Default</button>
<button type="button" class="btn red">Primary</button>
<button type="button" class="btn blue">Info</button>
<button type="button" class="btn green">Success</button>
</p>
<div class="btn-group">
<button class="btn">Left</button>
<button class="btn">Middle</button>
<button class="btn">Right</button>
</div>
<p>
<a href="#" class="btn red-stripe">Red Stripe</a>
<a href="#" class="btn purple-stripe">Purple stripe</a>
</p>
<p>
<a href="#" class="btn disabled">Disabled</a>
<a href="#" class="btn blue disabled">Disabled</a>
<a href="#" class="btn red disabled">Disabled</a>
<a href="#" class="btn green disabled">Disabled</a>
</p>
<p>
<a href="#" class="btn red mini">Mini size</a>
<a href="#" class="btn blue">Default size</a>
<a href="#" class="btn green big">Large size</a>
</p>
<p>
<button class="btn yellow btn-block" type="button">Block button</button>
</p>
</div>
</div>
<!-- END BUTTONS PORTLET-->
<!-- BEGIN BUTTONS WITH ICONS PORTLET-->
<div class="portlet box red">
<div class="portlet-title">
<h4><i class="icon-reorder"></i>Icon Buttons</h4>
<div class="tools">
<a href="javascript:;" class="collapse"></a>
<a href="#portlet-config" data-toggle="modal" class="config"></a>
<a href="javascript:;" class="reload"></a>
<a href="javascript:;" class="remove"></a>
</div>
</div>
<div class="portlet-body">
<p>Examples to use buttons with font awesome icons.</p>
<p>
<a href="#" class="btn icn-only"><i class="icon-share"></i></a>
<a href="#" class="btn red icn-only"><i class="icon-remove icon-white"></i></a>
<a href="#" class="btn blue icn-only"><i class="m-icon-swapright m-icon-white"></i></a>
<a href="#" class="btn green icn-only"><i class="icon-user icon-white"></i></a>
</p>
<p>Examples to use buttons with glyphicon halflings icons.</p>
<p>
<a href="#" class="btn icn-only"><i class="halflings-icon share"></i></a>
<a href="#" class="btn red icn-only"><i class="halflings-icon remove white"></i></a>
<a href="#" class="btn blue icn-only"><i class="halflings-icon user white"></i></a>
<a href="#" class="btn green icn-only"><i class="halflings-icon white"></i></a>
</p>
<p>Buttons with both text and icon.</p>
<p>
<a href="#" class="btn mini red"><i class="icon-trash"></i> Delete Item</a>
<a href="#" class="btn"><i class="icon-plus"></i> Add Item</a>
<a class="btn purple-stripe">Listen <i class="icon-headphones"></i></a>
</p>
<p>
<a href="#" class="btn blue"><i class="icon-plus"></i> Submit Entry</a>
<a class="btn purple big">pricing options <i class="m-icon-big-swapright m-icon-white"></i></a>
</p>
<p>Navigation icons.</p>
<p>
<a href="#" class="btn bigicn-only"><i class="m-icon-big-swapleft"></i></a>
<a href="#" class="btn bigicn-only green"><i class="m-icon-big-swapright m-icon-white"></i></a>
<a href="#" class="btn bigicn-only blue"><i class="m-icon-big-swapdown m-icon-white"></i></a>
<a href="#" class="btn bigicn-only black"><i class="m-icon-big-swapup m-icon-white"></i></a>
</p>
<p>
<a href="#" class="btn icn-only"><i class="m-icon-swapleft"></i></a>
<a href="#" class="btn icn-only green"><i class="m-icon-swapright m-icon-white"></i></a>
<a href="#" class="btn icn-only blue"><i class="m-icon-swapdown m-icon-white"></i></a>
<a href="#" class="btn icn-only black"><i class="m-icon-swapup m-icon-white"></i></a>
</p>
<p>Toolbar icon example</p>
<div class="btn-group hidden-phone">
<a href="javascript:;" class="btn">Tools</a>
<a href="javascript:;" class="btn">Settings</a>
<a href="javascript:;" class="btn active">About</a>
<a href="javascript:;" class="btn">Help</a>
<a href="javascript:;" class="btn">Contact</a>
</div>
<div class="btn-group visible-phone">
<a href="javascript:;" class="btn">Tools</a>
<a href="javascript:;" class="btn">Settings</a>
<a href="javascript:;" class="btn active">About</a>
</div>
<div class="btn-group visible-phone">
<a href="javascript:;" class="btn">Help</a>
<a href="javascript:;" class="btn">Contact</a>
</div>
<div class="btn-group hidden-phone">
<button class="btn"><i class="icon-step-backward"></i></button>
<button class="btn"><i class="icon-fast-backward"></i></button>
<button class="btn"><i class="icon-backward"></i></button>
<button class="btn"><i class="icon-play"></i></button>
<button class="btn"><i class="icon-stop"></i></button>
<button class="btn"><i class="icon-forward"></i></button>
<button class="btn"><i class="icon-fast-forward"></i></button>
<button class="btn"><i class="icon-step-forward"></i></button>
</div>
<div class="btn-group visible-phone">
<button class="btn"><i class="icon-step-backward"></i></button>
<button class="btn"><i class="icon-fast-backward"></i></button>
<button class="btn"><i class="icon-backward"></i></button>
<button class="btn"><i class="icon-play"></i></button>
</div>
<div class="btn-group visible-phone">
<button class="btn"><i class="icon-stop"></i></button>
<button class="btn"><i class="icon-forward"></i></button>
<button class="btn"><i class="icon-fast-forward"></i></button>
<button class="btn"><i class="icon-step-forward"></i></button>
</div>
<p>Star Rating Example</p>
<div>
<span class="rating">
<span class="star"></span>
<span class="star"></span>
<span class="star"></span>
<span class="star"></span>
<span class="star"></span>
</span>
</div>
</div>
</div>
<!-- END BUTTONS WITH ICONS PORTLET-->
</div>
<div class="span6">
<!-- BEGIN DROPDOWNS PORTLET-->
<div class="portlet box purple">
<div class="portlet-title">
<h4><i class="icon-reorder"></i>Dropdowns</h4>
<div class="tools">
<a href="javascript:;" class="collapse"></a>
<a href="#portlet-config" data-toggle="modal" class="config"></a>
<a href="javascript:;" class="reload"></a>
<a href="javascript:;" class="remove"></a>
</div>
</div>
<div class="portlet-body">
<p>Dropdown buttons</p>
<div class="btn-group">
<a class="btn dropdown-toggle" data-toggle="dropdown" href="#">
Tools <i class="icon-angle-down"></i>
</a>
<ul class="dropdown-menu">
<li><a href="#">Settings</a></li>
<li><a href="#">Preferences</a></li>
<li><a href="#">Window Options</a></li>
<li><a href="#">Help</a></li>
</ul>
</div>
<div class="btn-group">
<button class="btn red dropdown-toggle" data-toggle="dropdown">Primary <i class="icon-angle-down"></i></button>
<ul class="dropdown-menu">
<li><a href="#">Action</a></li>
<li><a href="#">Another action</a></li>
<li><a href="#">Something else here</a></li>
<li class="divider"></li>
<li><a href="#">Separated link</a></li>
</ul>
</div>
<div class="btn-group">
<button class="btn purple dropdown-toggle" data-toggle="dropdown">Success <i class="icon-angle-down"></i>
</button>
<ul class="dropdown-menu">
<li><a href="#">Action</a></li>
<li><a href="#">Another action</a></li>
<li><a href="#">Something else here</a></li>
<li class="divider"></li>
<li><a href="#">Separated link</a></li>
</ul>
</div>
<div class="btn-toolbar hide">
<div class="btn-group">
<button class="btn green dropdown-toggle" data-toggle="dropdown">Success <i class="icon-angle-down"></i>
</button>
<ul class="dropdown-menu">
<li><a href="#">Action</a></li>
<li><a href="#">Another action</a></li>
<li><a href="#">Something else here</a></li>
<li class="divider"></li>
<li><a href="#">Separated link</a></li>
</ul>
</div>
<!-- /btn-group -->
<div class="btn-group">
<button class="btn blue dropdown-toggle" data-toggle="dropdown">Info <i class="icon-angle-down"></i>
</button>
<ul class="dropdown-menu">
<li><a href="#">Action</a></li>
<li><a href="#">Another action</a></li>
<li><a href="#">Something else here</a></li>
<li class="divider"></li>
<li><a href="#">Separated link</a></li>
</ul>
</div>
<!-- /btn-group -->
<div class="btn-group">
<button class="btn black dropdown-toggle" data-toggle="dropdown">Inverse <i class="icon-angle-down"></i>
</button>
<ul class="dropdown-menu opens-left">
<li><a href="#">Action</a></li>
<li><a href="#">Another action</a></li>
<li><a href="#">Something else here</a></li>
<li class="divider"></li>
<li><a href="#">Separated link</a></li>
</ul>
</div>
<!-- /btn-group -->
</div>
<p>Dropdown button with icons</p>
<div class="btn-toolbar">
<div class="btn-group">
<a class="btn green" href="#" data-toggle="dropdown">
<i class="icon-user"></i> User
<i class="icon-angle-down"></i>
</a>
<ul class="dropdown-menu">
<li><a href="#"><i class="icon-pencil"></i> Edit</a></li>
<li><a href="#"><i class="icon-trash"></i> Delete</a></li>
<li><a href="#"><i class="icon-ban-circle"></i> Ban</a></li>
<li class="divider"></li>
<li><a href="#"><i class="i"></i> Make admin</a></li>
</ul>
</div>
<div class="btn-group">
<a class="btn purple" href="#" data-toggle="dropdown">
<i class="icon-user"></i> Settings
<i class="icon-angle-down"></i>
</a>
<ul class="dropdown-menu">
<li><a href="#"><i class="icon-plus"></i> Add</a></li>
<li><a href="#"><i class="icon-trash"></i> Edit</a></li>
<li><a href="#"><i class="icon-remove"></i> Delete</a></li>
<li class="divider"></li>
<li><a href="#"><i class="i"></i> Full Settings</a></li>
</ul>
</div>
</div>
<p>Form inputs with icons</p>
<form>
<div class="control-group">
<div class="controls">
<div class="input-prepend">
<span class="add-on"><i class="icon-envelope"></i></span><input class="m-wrap" id="inputIcon" type="text" placeholder="Email address" />
</div>
</div>
</div>
<div class="control-group">
<div class="controls">
<div class="input-prepend">
<span class="add-on"><i class="icon-key"></i></span><input class="m-wrap" id="inputIcon2" type="text" placeholder="Password" />
</div>
</div>
</div>
</form>
</div>
</div>
<!-- END DROPDOWNS PORTLET-->
<!-- BEGIN CUSTOM BUTTONS WITH ICONS PORTLET-->
<div class="portlet box blue">
<div class="portlet-title">
<h4><i class="icon-reorder"></i>Font Awesome Buttons</h4>
</div>
<div class="portlet-body">
<div class="row-fluid">
<a href="#" class="icon-btn span3">
<i class="icon-group"></i>
<div>Users</div>
<span class="badge badge-important">2</span>
</a>
<a href="#" class="icon-btn span3">
<i class="icon-barcode"></i>
<div>Products</div>
<span class="badge badge-success">4</span>
</a>
<a href="#" class="icon-btn span3">
<i class="icon-bar-chart"></i>
<div>Reports</div>
</a>
<a href="#" class="icon-btn span3">
<i class="icon-sitemap"></i>
<div>Categories</div>
</a>
</div>
<div class="row-fluid">
<a href="#" class="icon-btn span3">
<i class="icon-calendar"></i>
<div>Calendar</div>
<span class="badge badge-success">4</span>
</a>
<a href="#" class="icon-btn span3">
<i class="icon-envelope"></i>
<div>Inbox</div>
<span class="badge badge-info">12</span>
</a>
<a href="#" class="icon-btn span3">
<i class="icon-bullhorn"></i>
<div>Notification</div>
<span class="badge badge-important">3</span>
</a>
<a href="#" class="icon-btn span3">
<i class="icon-map-marker"></i>
<div>Locations</div>
</a>
</div>
<div class="row-fluid">
<a href="#" class="icon-btn span3">
<i class="icon-money"><i></i></i>
<div>Finance</div>
</a>
<a href="#" class="icon-btn span3">
<i class="icon-plane"></i>
<div>Projects</div>
<span class="badge badge-info">21</span>
</a>
<a href="#" class="icon-btn span3">
<i class="icon-thumbs-up"></i>
<div>Feedback</div>
<span class="badge badge-info">2</span>
</a>
<a href="#" class="icon-btn span3">
<i class="icon-cloud"></i>
<div>Servers</div>
<span class="badge badge-important">2</span>
</a>
</div>
<div class="row-fluid">
<a href="#" class="icon-btn span3">
<i class="icon-globe"></i>
<div>Regions</div>
</a>
<a href="#" class="icon-btn span3">
<i class="icon-heart-empty"></i>
<div>Popularity</div>
<span class="badge badge-info">221</span>
</a>
<a href="#" class="icon-btn span3">
<i class="icon-wrench"></i>
<div>Settings</div>
</a>
<a href="#" class="icon-btn span3">
<i class="icon-search"></i>
<div>Search</div>
</a>
</div>
</div>
</div>
<!-- END CUSTOM BUTTONS WITH ICONS PORTLET-->
</div>
</div>
<div class="row-fluid">
<div class="span12">
<div class="tabbable tabbable-custom">
<ul class="nav nav-tabs">
<li class="active"><a href="#tab_1_1" data-toggle="tab">Font Awesome Icons</a></li>
<li><a href="#tab_1_2" data-toggle="tab">Glyphicons Pro Icons</a></li>
<li><a href="#tab_1_3" data-toggle="tab">Glyphicons Pro Halfing Icons</a></li>
</ul>
<div class="tab-content">
<div class="tab-pane active" id="tab_1_1">
<div class="row-fluid">
<h3>Web Application Icons</h3>
</div>
<div class="row-fluid">
<div class="span3">
<ul class="unstyled">
<li><i class="icon-adjust"></i> icon-adjust</li>
<li><i class="icon-asterisk"></i> icon-asterisk</li>
<li><i class="icon-ban-circle"></i> icon-ban-circle</li>
<li><i class="icon-bar-chart"></i> icon-bar-chart</li>
<li><i class="icon-barcode"></i> icon-barcode</li>
<li><i class="icon-beaker"></i> icon-beaker</li>
<li><i class="icon-bell"></i> icon-bell</li>
<li><i class="icon-bolt"></i> icon-bolt</li>
<li><i class="icon-book"></i> icon-book</li>
<li><i class="icon-bookmark"></i> icon-bookmark</li>
<li><i class="icon-bookmark-empty"></i> icon-bookmark-empty</li>
<li><i class="icon-briefcase"></i> icon-briefcase</li>
<li><i class="icon-bullhorn"></i> icon-bullhorn</li>
<li><i class="icon-calendar"></i> icon-calendar</li>
<li><i class="icon-camera"></i> icon-camera</li>
<li><i class="icon-camera-retro"></i> icon-camera-retro</li>
<li><i class="icon-certificate"></i> icon-certificate</li>
<li><i class="icon-check"></i> icon-check</li>
<li><i class="icon-check-empty"></i> icon-check-empty</li>
<li><i class="icon-cloud"></i> icon-cloud</li>
<li><i class="icon-cog"></i> icon-cog</li>
<li><i class="icon-cogs"></i> icon-cogs</li>
<li><i class="icon-comment"></i> icon-comment</li>
<li><i class="icon-comment-alt"></i> icon-comment-alt</li>
<li><i class="icon-comments"></i> icon-comments</li>
<li><i class="icon-comments-alt"></i> icon-comments-alt</li>
<li><i class="icon-credit-card"></i> icon-credit-card</li>
<li><i class="icon-dashboard"></i> icon-dashboard</li>
<li><i class="icon-download"></i> icon-download</li>
<li><i class="icon-download-alt"></i> icon-download-alt</li>
<li><i class="icon-edit"></i> icon-edit</li>
<li><i class="icon-envelope"></i> icon-envelope</li>
<li><i class="icon-envelope-alt"></i> icon-envelope-alt</li>
</ul>
</div>
<div class="span3">
<ul class="unstyled">
<li><i class="icon-exclamation-sign"></i> icon-exclamation-sign</li>
<li><i class="icon-external-link"></i> icon-external-link</li>
<li><i class="icon-eye-close"></i> icon-eye-close</li>
<li><i class="icon-eye-open"></i> icon-eye-open</li>
<li><i class="icon-facetime-video"></i> icon-facetime-video</li>
<li><i class="icon-film"></i> icon-film</li>
<li><i class="icon-filter"></i> icon-filter</li>
<li><i class="icon-fire"></i> icon-fire</li>
<li><i class="icon-flag"></i> icon-flag</li>
<li><i class="icon-folder-close"></i> icon-folder-close</li>
<li><i class="icon-folder-open"></i> icon-folder-open</li>
<li><i class="icon-gift"></i> icon-gift</li>
<li><i class="icon-glass"></i> icon-glass</li>
<li><i class="icon-globe"></i> icon-globe</li>
<li><i class="icon-group"></i> icon-group</li>
<li><i class="icon-hdd"></i> icon-hdd</li>
<li><i class="icon-headphones"></i> icon-headphones</li>
<li><i class="icon-heart"></i> icon-heart</li>
<li><i class="icon-heart-empty"></i> icon-heart-empty</li>
<li><i class="icon-home"></i> icon-home</li>
<li><i class="icon-inbox"></i> icon-inbox</li>
<li><i class="icon-info-sign"></i> icon-info-sign</li>
<li><i class="icon-key"></i> icon-key</li>
<li><i class="icon-leaf"></i> icon-leaf</li>
<li><i class="icon-legal"></i> icon-legal</li>
<li><i class="icon-lemon"></i> icon-lemon</li>
<li><i class="icon-lock"></i> icon-lock</li>
<li><i class="icon-unlock"></i> icon-unlock</li>
<li><i class="icon-magic"></i> icon-magic</li>
<li><i class="icon-magnet"></i> icon-magnet</li>
<li><i class="icon-map-marker"></i> icon-map-marker</li>
<li><i class="icon-minus"></i> icon-minus</li>
<li><i class="icon-minus-sign"></i> icon-minus-sign</li>
</ul>
</div>
<div class="span3">
<ul class="unstyled">
<li><i class="icon-money"></i> icon-money</li>
<li><i class="icon-move"></i> icon-move</li>
<li><i class="icon-music"></i> icon-music</li>
<li><i class="icon-off"></i> icon-off</li>
<li><i class="icon-ok"></i> icon-ok</li>
<li><i class="icon-ok-circle"></i> icon-ok-circle</li>
<li><i class="icon-ok-sign"></i> icon-ok-sign</li>
<li><i class="icon-pencil"></i> icon-pencil</li>
<li><i class="icon-picture"></i> icon-picture</li>
<li><i class="icon-plane"></i> icon-plane</li>
<li><i class="icon-plus"></i> icon-plus</li>
<li><i class="icon-plus-sign"></i> icon-plus-sign</li>
<li><i class="icon-print"></i> icon-print</li>
<li><i class="icon-pushpin"></i> icon-pushpin</li>
<li><i class="icon-qrcode"></i> icon-qrcode</li>
<li><i class="icon-question-sign"></i> icon-question-sign</li>
<li><i class="icon-random"></i> icon-random</li>
<li><i class="icon-refresh"></i> icon-refresh</li>
<li><i class="icon-remove"></i> icon-remove</li>
<li><i class="icon-remove-circle"></i> icon-remove-circle</li>
<li><i class="icon-remove-sign"></i> icon-remove-sign</li>
<li><i class="icon-reorder"></i> icon-reorder</li>
<li><i class="icon-resize-horizontal"></i> icon-resize-horizontal</li>
<li><i class="icon-resize-vertical"></i> icon-resize-vertical</li>
<li><i class="icon-retweet"></i> icon-retweet</li>
<li><i class="icon-road"></i> icon-road</li>
<li><i class="icon-rss"></i> icon-rss</li>
<li><i class="icon-screenshot"></i> icon-screenshot</li>
<li><i class="icon-search"></i> icon-search</li>
<li><i class="icon-share"></i> icon-share</li>
<li><i class="icon-share-alt"></i> icon-share-alt</li>
<li><i class="icon-shopping-cart"></i> icon-shopping-cart</li>
</ul>
</div>
<div class="span3">
<ul class="unstyled">
<li><i class="icon-signal"></i> icon-signal</li>
<li><i class="icon-signin"></i> icon-signin</li>
<li><i class="icon-signout"></i> icon-signout</li>
<li><i class="icon-sitemap"></i> icon-sitemap</li>
<li><i class="icon-sort"></i> icon-sort</li>
<li><i class="icon-sort-down"></i> icon-sort-down</li>
<li><i class="icon-sort-up"></i> icon-sort-up</li>
<li><i class="icon-star"></i> icon-star</li>
<li><i class="icon-star-empty"></i> icon-star-empty</li>
<li><i class="icon-star-half"></i> icon-star-half</li>
<li><i class="icon-tag"></i> icon-tag</li>
<li><i class="icon-tags"></i> icon-tags</li>
<li><i class="icon-tasks"></i> icon-tasks</li>
<li><i class="icon-thumbs-down"></i> icon-thumbs-down</li>
<li><i class="icon-thumbs-up"></i> icon-thumbs-up</li>
<li><i class="icon-time"></i> icon-time</li>
<li><i class="icon-tint"></i> icon-tint</li>
<li><i class="icon-trash"></i> icon-trash</li>
<li><i class="icon-trophy"></i> icon-trophy</li>
<li><i class="icon-truck"></i> icon-truck</li>
<li><i class="icon-umbrella"></i> icon-umbrella</li>
<li><i class="icon-upload"></i> icon-upload</li>
<li><i class="icon-upload-alt"></i> icon-upload-alt</li>
<li><i class="icon-user"></i> icon-user</li>
<li><i class="icon-user-md"></i> icon-user-md</li>
<li><i class="icon-volume-off"></i> icon-volume-off</li>
<li><i class="icon-volume-down"></i> icon-volume-down</li>
<li><i class="icon-volume-up"></i> icon-volume-up</li>
<li><i class="icon-warning-sign"></i> icon-warning-sign</li>
<li><i class="icon-wrench"></i> icon-wrench</li>
<li><i class="icon-zoom-in"></i> icon-zoom-in</li>
<li><i class="icon-zoom-out"></i> icon-zoom-out</li>
</ul>
</div>
</div>
<div class="row-fluid">
<div class="span12">
<h3>Text Editor Icons</h3>
</div>
</div>
<div class="row-fluid">
<div class="span3">
<ul class="unstyled">
<li><i class="icon-file"></i> icon-file</li>
<li><i class="icon-cut"></i> icon-cut</li>
<li><i class="icon-copy"></i> icon-copy</li>
<li><i class="icon-paste"></i> icon-paste</li>
<li><i class="icon-save"></i> icon-save</li>
<li><i class="icon-undo"></i> icon-undo</li>
<li><i class="icon-repeat"></i> icon-repeat</li>
<li><i class="icon-paper-clip"></i> icon-paper-clip</li>
</ul>
</div>
<div class="span3">
<ul class="unstyled">
<li><i class="icon-text-height"></i> icon-text-height</li>
<li><i class="icon-text-width"></i> icon-text-width</li>
<li><i class="icon-align-left"></i> icon-align-left</li>
<li><i class="icon-align-center"></i> icon-align-center</li>
<li><i class="icon-align-right"></i> icon-align-right</li>
<li><i class="icon-align-justify"></i> icon-align-justify</li>
<li><i class="icon-indent-left"></i> icon-indent-left</li>
<li><i class="icon-indent-right"></i> icon-indent-right</li>
</ul>
</div>
<div class="span3">
<ul class="unstyled">
<li><i class="icon-font"></i> icon-font</li>
<li><i class="icon-bold"></i> icon-bold</li>
<li><i class="icon-italic"></i> icon-italic</li>
<li><i class="icon-strikethrough"></i> icon-strikethrough</li>
<li><i class="icon-underline"></i> icon-underline</li>
<li><i class="icon-link"></i> icon-link</li>
<li><i class="icon-columns"></i> icon-columns</li>
<li><i class="icon-table"></i> icon-table</li>
</ul>
</div>
<div class="span3">
<ul class="unstyled">
<li><i class="icon-th-large"></i> icon-th-large</li>
<li><i class="icon-th"></i> icon-th</li>
<li><i class="icon-th-list"></i> icon-th-list</li>
<li><i class="icon-list"></i> icon-list</li>
<li><i class="icon-list-ol"></i> icon-list-ol</li>
<li><i class="icon-list-ul"></i> icon-list-ul</li>
<li><i class="icon-list-alt"></i> icon-list-alt</li>
</ul>
</div>
</div>
<div class="row-fluid">
<div class="span12">
<h3>Directional Icons</h3>
</div>
</div>
<div class="row-fluid">
<div class="span3">
<ul class="unstyled">
<li><i class="icon-arrow-down"></i> icon-arrow-down</li>
<li><i class="icon-arrow-left"></i> icon-arrow-left</li>
<li><i class="icon-arrow-right"></i> icon-arrow-right</li>
<li><i class="icon-arrow-up"></i> icon-arrow-up</li>
<li><i class="icon-chevron-down"></i> icon-chevron-down</li>
</ul>
</div>
<div class="span3">
<ul class="unstyled">
<li><i class="icon-circle-arrow-down"></i> icon-circle-arrow-down</li>
<li><i class="icon-circle-arrow-left"></i> icon-circle-arrow-left</li>
<li><i class="icon-circle-arrow-right"></i> icon-circle-arrow-right</li>
<li><i class="icon-circle-arrow-up"></i> icon-circle-arrow-up</li>
<li><i class="icon-chevron-left"></i> icon-chevron-left</li>
</ul>
</div>
<div class="span3">
<ul class="unstyled">
<li><i class="icon-caret-down"></i> icon-caret-down</li>
<li><i class="icon-caret-left"></i> icon-caret-left</li>
<li><i class="icon-caret-right"></i> icon-caret-right</li>
<li><i class="icon-caret-up"></i> icon-caret-up</li>
<li><i class="icon-chevron-right"></i> icon-chevron-right</li>
</ul>
</div>
<div class="span3">
<ul class="unstyled">
<li><i class="icon-hand-down"></i> icon-hand-down</li>
<li><i class="icon-hand-left"></i> icon-hand-left</li>
<li><i class="icon-hand-right"></i> icon-hand-right</li>
<li><i class="icon-hand-up"></i> icon-hand-up</li>
<li><i class="icon-chevron-up"></i> icon-chevron-up</li>
</ul>
</div>
</div>
<div class="row-fluid">
<div class="span12">
<h3>Video Player Icons</h3>
</div>
</div>
<div class="row-fluid">
<div class="span3">
<ul class="unstyled">
<li><i class="icon-play-circle"></i> icon-play-circle</li>
<li><i class="icon-play"></i> icon-play</li>
<li><i class="icon-pause"></i> icon-pause</li>
<li><i class="icon-stop"></i> icon-stop</li>
</ul>
</div>
<div class="span3">
<ul class="unstyled">
<li><i class="icon-step-backward"></i> icon-step-backward</li>
<li><i class="icon-fast-backward"></i> icon-fast-backward</li>
<li><i class="icon-backward"></i> icon-backward</li>
<li><i class="icon-forward"></i> icon-forward</li>
</ul>
</div>
<div class="span3">
<ul class="unstyled">
<li><i class="icon-fast-forward"></i> icon-fast-forward</li>
<li><i class="icon-step-forward"></i> icon-step-forward</li>
<li><i class="icon-eject"></i> icon-eject</li>
</ul>
</div>
<div class="span3">
<ul class="unstyled">
<li><i class="icon-fullscreen"></i> icon-fullscreen</li>
<li><i class="icon-resize-full"></i> icon-resize-full</li>
<li><i class="icon-resize-small"></i> icon-resize-small</li>
</ul>
</div>
</div>
<div class="row-fluid">
<div class="span12">
<h3>Social Icons</h3>
</div>
</div>
<div class="row-fluid">
<div class="span3">
<ul class="unstyled">
<li><i class="icon-phone"></i> icon-phone</li>
<li><i class="icon-phone-sign"></i> icon-phone-sign</li>
<li><i class="icon-facebook"></i> icon-facebook</li>
<li><i class="icon-facebook-sign"></i> icon-facebook-sign</li>
</ul>
</div>
<div class="span3">
<ul class="unstyled">
<li><i class="icon-twitter"></i> icon-twitter</li>
<li><i class="icon-twitter-sign"></i> icon-twitter-sign</li>
<li><i class="icon-github"></i> icon-github</li>
<li><i class="icon-github-sign"></i> icon-github-sign</li>
</ul>
</div>
<div class="span3">
<ul class="unstyled">
<li><i class="icon-linkedin"></i> icon-linkedin</li>
<li><i class="icon-linkedin-sign"></i> icon-linkedin-sign</li>
<li><i class="icon-pinterest"></i> icon-pinterest</li>
<li><i class="icon-pinterest-sign"></i> icon-pinterest-sign</li>
</ul>
</div>
<div class="span3">
<ul class="unstyled">
<li><i class="icon-google-plus"></i> icon-google-plus</li>
<li><i class="icon-google-plus-sign"></i> icon-google-plus-sign</li>
<li><i class="icon-sign-blank"></i> icon-sign-blank</li>
</ul>
</div>
</div>
</div>
<div class="tab-pane" id="tab_1_2">
<div class="glyphicons-demo">
<a href="#" class="glyphicons no-js glass"><i></i>glass</a>
<a href="#" class="glyphicons no-js leaf"><i></i>leaf</a>
<a href="#" class="glyphicons no-js dog"><i></i>dog</a>
<a href="#" class="glyphicons no-js user"><i></i>user</a>
<a href="#" class="glyphicons no-js girl"><i></i>girl</a>
<a href="#" class="glyphicons no-js car"><i></i>car</a>
<a href="#" class="glyphicons no-js user_add"><i></i>user_add</a>
<a href="#" class="glyphicons no-js user_remove"><i></i>user_remove</a>
<a href="#" class="glyphicons no-js film"><i></i>film</a>
<a href="#" class="glyphicons no-js magic"><i></i>magic</a>
<a href="#" class="glyphicons no-js envelope"><i></i>envelope</a>
<a href="#" class="glyphicons no-js camera"><i></i>camera</a>
<a href="#" class="glyphicons no-js heart"><i></i>heart</a>
<a href="#" class="glyphicons no-js beach_umbrella"><i></i>beach_umbrella</a>
<a href="#" class="glyphicons no-js train"><i></i>train</a>
<a href="#" class="glyphicons no-js print"><i></i>print</a>
<a href="#" class="glyphicons no-js bin"><i></i>bin</a>
<a href="#" class="glyphicons no-js music"><i></i>music</a>
<a href="#" class="glyphicons no-js note"><i></i>note</a>
<a href="#" class="glyphicons no-js heart_empty"><i></i>heart_empty</a>
<a href="#" class="glyphicons no-js home"><i></i>home</a>
<a href="#" class="glyphicons no-js snowflake"><i></i>snowflake</a>
<a href="#" class="glyphicons no-js fire"><i></i>fire</a>
<a href="#" class="glyphicons no-js magnet"><i></i>magnet</a>
<a href="#" class="glyphicons no-js parents"><i></i>parents</a>
<a href="#" class="glyphicons no-js binoculars"><i></i>binoculars</a>
<a href="#" class="glyphicons no-js road"><i></i>road</a>
<a href="#" class="glyphicons no-js search"><i></i>search</a>
<a href="#" class="glyphicons no-js cars"><i></i>cars</a>
<a href="#" class="glyphicons no-js notes_2"><i></i>notes_2</a>
<a href="#" class="glyphicons no-js pencil"><i></i>pencil</a>
<a href="#" class="glyphicons no-js bus"><i></i>bus</a>
<a href="#" class="glyphicons no-js wifi_alt"><i></i>wifi_alt</a>
<a href="#" class="glyphicons no-js luggage"><i></i>luggage</a>
<a href="#" class="glyphicons no-js old_man"><i></i>old_man</a>
<a href="#" class="glyphicons no-js woman"><i></i>woman</a>
<a href="#" class="glyphicons no-js file"><i></i>file</a>
<a href="#" class="glyphicons no-js coins"><i></i>coins</a>
<a href="#" class="glyphicons no-js airplane"><i></i>airplane</a>
<a href="#" class="glyphicons no-js notes"><i></i>notes</a>
<a href="#" class="glyphicons no-js stats"><i></i>stats</a>
<a href="#" class="glyphicons no-js charts"><i></i>charts</a>
<a href="#" class="glyphicons no-js pie_chart"><i></i>pie_chart</a>
<a href="#" class="glyphicons no-js group"><i></i>group</a>
<a href="#" class="glyphicons no-js keys"><i></i>keys</a>
<a href="#" class="glyphicons no-js calendar"><i></i>calendar</a>
<a href="#" class="glyphicons no-js router"><i></i>router</a>
<a href="#" class="glyphicons no-js camera_small"><i></i>camera_small</a>
<a href="#" class="glyphicons no-js dislikes"><i></i>dislikes</a>
<a href="#" class="glyphicons no-js star"><i></i>star</a>
<a href="#" class="glyphicons no-js link"><i></i>link</a>
<a href="#" class="glyphicons no-js eye_open"><i></i>eye_open</a>
<a href="#" class="glyphicons no-js eye_close"><i></i>eye_close</a>
<a href="#" class="glyphicons no-js alarm"><i></i>alarm</a>
<a href="#" class="glyphicons no-js clock"><i></i>clock</a>
<a href="#" class="glyphicons no-js stopwatch"><i></i>stopwatch</a>
<a href="#" class="glyphicons no-js projector"><i></i>projector</a>
<a href="#" class="glyphicons no-js history"><i></i>history</a>
<a href="#" class="glyphicons no-js truck"><i></i>truck</a>
<a href="#" class="glyphicons no-js cargo"><i></i>cargo</a>
<a href="#" class="glyphicons no-js compass"><i></i>compass</a>
<a href="#" class="glyphicons no-js keynote"><i></i>keynote</a>
<a href="#" class="glyphicons no-js paperclip"><i></i>paperclip</a>
<a href="#" class="glyphicons no-js power"><i></i>power</a>
<a href="#" class="glyphicons no-js lightbulb"><i></i>lightbulb</a>
<a href="#" class="glyphicons no-js tag"><i></i>tag</a>
<a href="#" class="glyphicons no-js tags"><i></i>tags</a>
<a href="#" class="glyphicons no-js cleaning"><i></i>cleaning</a>
<a href="#" class="glyphicons no-js ruller"><i></i>ruller</a>
<a href="#" class="glyphicons no-js gift"><i></i>gift</a>
<a href="#" class="glyphicons no-js umbrella"><i></i>umbrella</a>
<a href="#" class="glyphicons no-js book"><i></i>book</a>
<a href="#" class="glyphicons no-js bookmark"><i></i>bookmark</a>
<a href="#" class="glyphicons no-js wifi"><i></i>wifi</a>
<a href="#" class="glyphicons no-js cup"><i></i>cup</a>
<a href="#" class="glyphicons no-js stroller"><i></i>stroller</a>
<a href="#" class="glyphicons no-js headphones"><i></i>headphones</a>
<a href="#" class="glyphicons no-js headset"><i></i>headset</a>
<a href="#" class="glyphicons no-js warning_sign"><i></i>warning_sign</a>
<a href="#" class="glyphicons no-js signal"><i></i>signal</a>
<a href="#" class="glyphicons no-js retweet"><i></i>retweet</a>
<a href="#" class="glyphicons no-js refresh"><i></i>refresh</a>
<a href="#" class="glyphicons no-js roundabout"><i></i>roundabout</a>
<a href="#" class="glyphicons no-js random"><i></i>random</a>
<a href="#" class="glyphicons no-js heat"><i></i>heat</a>
<a href="#" class="glyphicons no-js repeat"><i></i>repeat</a>
<a href="#" class="glyphicons no-js display"><i></i>display</a>
<a href="#" class="glyphicons no-js log_book"><i></i>log_book</a>
<a href="#" class="glyphicons no-js adress_book"><i></i>adress_book</a>
<a href="#" class="glyphicons no-js building"><i></i>building</a>
<a href="#" class="glyphicons no-js eyedropper"><i></i>eyedropper</a>
<a href="#" class="glyphicons no-js adjust"><i></i>adjust</a>
<a href="#" class="glyphicons no-js tint"><i></i>tint</a>
<a href="#" class="glyphicons no-js crop"><i></i>crop</a>
<a href="#" class="glyphicons no-js vector_path_square"><i></i>vector_path_square</a>
<a href="#" class="glyphicons no-js vector_path_circle"><i></i>vector_path_circle</a>
<a href="#" class="glyphicons no-js vector_path_polygon"><i></i>vector_path_polygon</a>
<a href="#" class="glyphicons no-js vector_path_line"><i></i>vector_path_line</a>
<a href="#" class="glyphicons no-js vector_path_curve"><i></i>vector_path_curve</a>
<a href="#" class="glyphicons no-js vector_path_all"><i></i>vector_path_all</a>
<a href="#" class="glyphicons no-js font"><i></i>font</a>
<a href="#" class="glyphicons no-js italic"><i></i>italic</a>
<a href="#" class="glyphicons no-js bold"><i></i>bold</a>
<a href="#" class="glyphicons no-js text_underline"><i></i>text_underline</a>
<a href="#" class="glyphicons no-js text_strike"><i></i>text_strike</a>
<a href="#" class="glyphicons no-js text_height"><i></i>text_height</a>
<a href="#" class="glyphicons no-js text_width"><i></i>text_width</a>
<a href="#" class="glyphicons no-js text_resize"><i></i>text_resize</a>
<a href="#" class="glyphicons no-js left_indent"><i></i>left_indent</a>
<a href="#" class="glyphicons no-js right_indent"><i></i>right_indent</a>
<a href="#" class="glyphicons no-js align_left"><i></i>align_left</a>
<a href="#" class="glyphicons no-js align_center"><i></i>align_center</a>
<a href="#" class="glyphicons no-js align_right"><i></i>align_right</a>
<a href="#" class="glyphicons no-js justify"><i></i>justify</a>
<a href="#" class="glyphicons no-js list"><i></i>list</a>
<a href="#" class="glyphicons no-js text_smaller"><i></i>text_smaller</a>
<a href="#" class="glyphicons no-js text_bigger"><i></i>text_bigger</a>
<a href="#" class="glyphicons no-js embed"><i></i>embed</a>
<a href="#" class="glyphicons no-js embed_close"><i></i>embed_close</a>
<a href="#" class="glyphicons no-js table"><i></i>table</a>
<a href="#" class="glyphicons no-js message_full"><i></i>message_full</a>
<a href="#" class="glyphicons no-js message_empty"><i></i>message_empty</a>
<a href="#" class="glyphicons no-js message_in"><i></i>message_in</a>
<a href="#" class="glyphicons no-js message_out"><i></i>message_out</a>
<a href="#" class="glyphicons no-js message_plus"><i></i>message_plus</a>
<a href="#" class="glyphicons no-js message_minus"><i></i>message_minus</a>
<a href="#" class="glyphicons no-js message_ban"><i></i>message_ban</a>
<a href="#" class="glyphicons no-js message_flag"><i></i>message_flag</a>
<a href="#" class="glyphicons no-js message_lock"><i></i>message_lock</a>
<a href="#" class="glyphicons no-js message_new"><i></i>message_new</a>
<a href="#" class="glyphicons no-js inbox"><i></i>inbox</a>
<a href="#" class="glyphicons no-js inbox_plus"><i></i>inbox_plus</a>
<a href="#" class="glyphicons no-js inbox_minus"><i></i>inbox_minus</a>
<a href="#" class="glyphicons no-js inbox_lock"><i></i>inbox_lock</a>
<a href="#" class="glyphicons no-js inbox_in"><i></i>inbox_in</a>
<a href="#" class="glyphicons no-js inbox_out"><i></i>inbox_out</a>
<a href="#" class="glyphicons no-js cogwheel"><i></i>cogwheel</a>
<a href="#" class="glyphicons no-js cogwheels"><i></i>cogwheels</a>
<a href="#" class="glyphicons no-js picture"><i></i>picture</a>
<a href="#" class="glyphicons no-js adjust_alt"><i></i>adjust_alt</a>
<a href="#" class="glyphicons no-js database_lock"><i></i>database_lock</a>
<a href="#" class="glyphicons no-js database_plus"><i></i>database_plus</a>
<a href="#" class="glyphicons no-js database_minus"><i></i>database_minus</a>
<a href="#" class="glyphicons no-js database_ban"><i></i>database_ban</a>
<a href="#" class="glyphicons no-js folder_open"><i></i>folder_open</a>
<a href="#" class="glyphicons no-js folder_plus"><i></i>folder_plus</a>
<a href="#" class="glyphicons no-js folder_minus"><i></i>folder_minus</a>
<a href="#" class="glyphicons no-js folder_lock"><i></i>folder_lock</a>
<a href="#" class="glyphicons no-js folder_flag"><i></i>folder_flag</a>
<a href="#" class="glyphicons no-js folder_new"><i></i>folder_new</a>
<a href="#" class="glyphicons no-js edit"><i></i>edit</a>
<a href="#" class="glyphicons no-js new_window"><i></i>new_window</a>
<a href="#" class="glyphicons no-js check"><i></i>check</a>
<a href="#" class="glyphicons no-js unchecked"><i></i>unchecked</a>
<a href="#" class="glyphicons no-js more_windows"><i></i>more_windows</a>
<a href="#" class="glyphicons no-js show_big_thumbnails"><i></i>show_big_thumbnails</a>
<a href="#" class="glyphicons no-js show_thumbnails"><i></i>show_thumbnails</a>
<a href="#" class="glyphicons no-js show_thumbnails_with_lines"><i></i>show_thumbnails_with_lines</a>
<a href="#" class="glyphicons no-js show_lines"><i></i>show_lines</a>
<a href="#" class="glyphicons no-js playlist"><i></i>playlist</a>
<a href="#" class="glyphicons no-js imac"><i></i>imac</a>
<a href="#" class="glyphicons no-js macbook"><i></i>macbook</a>
<a href="#" class="glyphicons no-js ipad"><i></i>ipad</a>
<a href="#" class="glyphicons no-js iphone"><i></i>iphone</a>
<a href="#" class="glyphicons no-js iphone_transfer"><i></i>iphone_transfer</a>
<a href="#" class="glyphicons no-js iphone_exchange"><i></i>iphone_exchange</a>
<a href="#" class="glyphicons no-js ipod"><i></i>ipod</a>
<a href="#" class="glyphicons no-js ipod_shuffle"><i></i>ipod_shuffle</a>
<a href="#" class="glyphicons no-js ear_plugs"><i></i>ear_plugs</a>
<a href="#" class="glyphicons no-js phone"><i></i>phone</a>
<a href="#" class="glyphicons no-js step_backward"><i></i>step_backward</a>
<a href="#" class="glyphicons no-js fast_backward"><i></i>fast_backward</a>
<a href="#" class="glyphicons no-js rewind"><i></i>rewind</a>
<a href="#" class="glyphicons no-js play"><i></i>play</a>
<a href="#" class="glyphicons no-js pause"><i></i>pause</a>
<a href="#" class="glyphicons no-js stop"><i></i>stop</a>
<a href="#" class="glyphicons no-js forward"><i></i>forward</a>
<a href="#" class="glyphicons no-js fast_forward"><i></i>fast_forward</a>
<a href="#" class="glyphicons no-js step_forward"><i></i>step_forward</a>
<a href="#" class="glyphicons no-js eject"><i></i>eject</a>
<a href="#" class="glyphicons no-js facetime_video"><i></i>facetime_video</a>
<a href="#" class="glyphicons no-js download_alt"><i></i>download_alt</a>
<a href="#" class="glyphicons no-js mute"><i></i>mute</a>
<a href="#" class="glyphicons no-js volume_down"><i></i>volume_down</a>
<a href="#" class="glyphicons no-js volume_up"><i></i>volume_up</a>
<a href="#" class="glyphicons no-js screenshot"><i></i>screenshot</a>
<a href="#" class="glyphicons no-js move"><i></i>move</a>
<a href="#" class="glyphicons no-js more"><i></i>more</a>
<a href="#" class="glyphicons no-js brightness_reduce"><i></i>brightness_reduce</a>
<a href="#" class="glyphicons no-js brightness_increase"><i></i>brightness_increase</a>
<a href="#" class="glyphicons no-js circle_plus"><i></i>circle_plus</a>
<a href="#" class="glyphicons no-js circle_minus"><i></i>circle_minus</a>
<a href="#" class="glyphicons no-js circle_remove"><i></i>circle_remove</a>
<a href="#" class="glyphicons no-js circle_ok"><i></i>circle_ok</a>
<a href="#" class="glyphicons no-js circle_question_mark"><i></i>circle_question_mark</a>
<a href="#" class="glyphicons no-js circle_info"><i></i>circle_info</a>
<a href="#" class="glyphicons no-js circle_exclamation_mark"><i></i>circle_exclamation_mark</a>
<a href="#" class="glyphicons no-js remove"><i></i>remove</a>
<a href="#" class="glyphicons no-js ok"><i></i>ok</a>
<a href="#" class="glyphicons no-js ban"><i></i>ban</a>
<a href="#" class="glyphicons no-js download"><i></i>download</a>
<a href="#" class="glyphicons no-js upload"><i></i>upload</a>
<a href="#" class="glyphicons no-js shopping_cart"><i></i>shopping_cart</a>
<a href="#" class="glyphicons no-js lock"><i></i>lock</a>
<a href="#" class="glyphicons no-js unlock"><i></i>unlock</a>
<a href="#" class="glyphicons no-js electricity"><i></i>electricity</a>
<a href="#" class="glyphicons no-js ok_2"><i></i>ok_2</a>
<a href="#" class="glyphicons no-js remove_2"><i></i>remove_2</a>
<a href="#" class="glyphicons no-js cart_out"><i></i>cart_out</a>
<a href="#" class="glyphicons no-js cart_in"><i></i>cart_in</a>
<a href="#" class="glyphicons no-js left_arrow"><i></i>left_arrow</a>
<a href="#" class="glyphicons no-js right_arrow"><i></i>right_arrow</a>
<a href="#" class="glyphicons no-js down_arrow"><i></i>down_arrow</a>
<a href="#" class="glyphicons no-js up_arrow"><i></i>up_arrow</a>
<a href="#" class="glyphicons no-js resize_small"><i></i>resize_small</a>
<a href="#" class="glyphicons no-js resize_full"><i></i>resize_full</a>
<a href="#" class="glyphicons no-js circle_arrow_left"><i></i>circle_arrow_left</a>
<a href="#" class="glyphicons no-js circle_arrow_right"><i></i>circle_arrow_right</a>
<a href="#" class="glyphicons no-js circle_arrow_top"><i></i>circle_arrow_top</a>
<a href="#" class="glyphicons no-js circle_arrow_down"><i></i>circle_arrow_down</a>
<a href="#" class="glyphicons no-js play_button"><i></i>play_button</a>
<a href="#" class="glyphicons no-js unshare"><i></i>unshare</a>
<a href="#" class="glyphicons no-js share"><i></i>share</a>
<a href="#" class="glyphicons no-js chevron-right"><i></i>chevron-right</a>
<a href="#" class="glyphicons no-js chevron-left"><i></i>chevron-left</a>
<a href="#" class="glyphicons no-js bluetooth"><i></i>bluetooth</a>
<a href="#" class="glyphicons no-js euro"><i></i>euro</a>
<a href="#" class="glyphicons no-js usd"><i></i>usd</a>
<a href="#" class="glyphicons no-js gbp"><i></i>gbp</a>
<a href="#" class="glyphicons no-js retweet_2"><i></i>retweet_2</a>
<a href="#" class="glyphicons no-js moon"><i></i>moon</a>
<a href="#" class="glyphicons no-js sun"><i></i>sun</a>
<a href="#" class="glyphicons no-js cloud"><i></i>cloud</a>
<a href="#" class="glyphicons no-js direction"><i></i>direction</a>
<a href="#" class="glyphicons no-js brush"><i></i>brush</a>
<a href="#" class="glyphicons no-js pen"><i></i>pen</a>
<a href="#" class="glyphicons no-js zoom_in"><i></i>zoom_in</a>
<a href="#" class="glyphicons no-js zoom_out"><i></i>zoom_out</a>
<a href="#" class="glyphicons no-js pin"><i></i>pin</a>
<a href="#" class="glyphicons no-js albums"><i></i>albums</a>
<a href="#" class="glyphicons no-js rotation_lock"><i></i>rotation_lock</a>
<a href="#" class="glyphicons no-js flash"><i></i>flash</a>
<a href="#" class="glyphicons no-js google_maps"><i></i>google_maps</a>
<a href="#" class="glyphicons no-js anchor"><i></i>anchor</a>
<a href="#" class="glyphicons no-js conversation"><i></i>conversation</a>
<a href="#" class="glyphicons no-js chat"><i></i>chat</a>
<a href="#" class="glyphicons no-js male"><i></i>male</a>
<a href="#" class="glyphicons no-js female"><i></i>female</a>
<a href="#" class="glyphicons no-js asterisk"><i></i>asterisk</a>
<a href="#" class="glyphicons no-js divide"><i></i>divide</a>
<a href="#" class="glyphicons no-js snorkel_diving"><i></i>snorkel_diving</a>
<a href="#" class="glyphicons no-js scuba_diving"><i></i>scuba_diving</a>
<a href="#" class="glyphicons no-js oxygen_bottle"><i></i>oxygen_bottle</a>
<a href="#" class="glyphicons no-js fins"><i></i>fins</a>
<a href="#" class="glyphicons no-js fishes"><i></i>fishes</a>
<a href="#" class="glyphicons no-js boat"><i></i>boat</a>
<a href="#" class="glyphicons no-js delete"><i></i>delete</a>
<a href="#" class="glyphicons no-js sheriffs_star"><i></i>sheriffs_star</a>
<a href="#" class="glyphicons no-js qrcode"><i></i>qrcode</a>
<a href="#" class="glyphicons no-js barcode"><i></i>barcode</a>
<a href="#" class="glyphicons no-js pool"><i></i>pool</a>
<a href="#" class="glyphicons no-js buoy"><i></i>buoy</a>
<a href="#" class="glyphicons no-js spade"><i></i>spade</a>
<a href="#" class="glyphicons no-js bank"><i></i>bank</a>
<a href="#" class="glyphicons no-js vcard"><i></i>vcard</a>
<a href="#" class="glyphicons no-js electrical_plug"><i></i>electrical_plug</a>
<a href="#" class="glyphicons no-js flag"><i></i>flag</a>
<a href="#" class="glyphicons no-js credit_card"><i></i>credit_card</a>
<a href="#" class="glyphicons no-js keyboard-wireless"><i></i>keyboard-wireless</a>
<a href="#" class="glyphicons no-js keyboard-wired"><i></i>keyboard-wired</a>
<a href="#" class="glyphicons no-js shield"><i></i>shield</a>
<a href="#" class="glyphicons no-js ring"><i></i>ring</a>
<a href="#" class="glyphicons no-js cake"><i></i>cake</a>
<a href="#" class="glyphicons no-js drink"><i></i>drink</a>
<a href="#" class="glyphicons no-js beer"><i></i>beer</a>
<a href="#" class="glyphicons no-js fast_food"><i></i>fast_food</a>
<a href="#" class="glyphicons no-js cutlery"><i></i>cutlery</a>
<a href="#" class="glyphicons no-js pizza"><i></i>pizza</a>
<a href="#" class="glyphicons no-js birthday_cake"><i></i>birthday_cake</a>
<a href="#" class="glyphicons no-js tablet"><i></i>tablet</a>
<a href="#" class="glyphicons no-js settings"><i></i>settings</a>
<a href="#" class="glyphicons no-js bullets"><i></i>bullets</a>
<a href="#" class="glyphicons no-js cardio"><i></i>cardio</a>
<a href="#" class="glyphicons no-js t-shirt"><i></i>t-shirt</a>
<a href="#" class="glyphicons no-js pants"><i></i>pants</a>
<a href="#" class="glyphicons no-js sweater"><i></i>sweater</a>
<a href="#" class="glyphicons no-js fabric"><i></i>fabric</a>
<a href="#" class="glyphicons no-js leather"><i></i>leather</a>
<a href="#" class="glyphicons no-js scissors"><i></i>scissors</a>
<a href="#" class="glyphicons no-js bomb"><i></i>bomb</a>
<a href="#" class="glyphicons no-js skull"><i></i>skull</a>
<a href="#" class="glyphicons no-js celebration"><i></i>celebration</a>
<a href="#" class="glyphicons no-js tea_kettle"><i></i>tea_kettle</a>
<a href="#" class="glyphicons no-js french_press"><i></i>french_press</a>
<a href="#" class="glyphicons no-js coffe_cup"><i></i>coffe_cup</a>
<a href="#" class="glyphicons no-js pot"><i></i>pot</a>
<a href="#" class="glyphicons no-js grater"><i></i>grater</a>
<a href="#" class="glyphicons no-js kettle"><i></i>kettle</a>
<a href="#" class="glyphicons no-js hospital"><i></i>hospital</a>
<a href="#" class="glyphicons no-js hospital_h"><i></i>hospital_h</a>
<a href="#" class="glyphicons no-js microphone"><i></i>microphone</a>
<a href="#" class="glyphicons no-js webcam"><i></i>webcam</a>
<a href="#" class="glyphicons no-js temple_christianity_church"><i></i>temple_christianity_church</a>
<a href="#" class="glyphicons no-js temple_islam"><i></i>temple_islam</a>
<a href="#" class="glyphicons no-js temple_hindu"><i></i>temple_hindu</a>
<a href="#" class="glyphicons no-js temple_buddhist"><i></i>temple_buddhist</a>
<a href="#" class="glyphicons no-js bicycle"><i></i>bicycle</a>
<a href="#" class="glyphicons no-js life_preserver"><i></i>life_preserver</a>
<a href="#" class="glyphicons no-js share_alt"><i></i>share_alt</a>
<a href="#" class="glyphicons no-js comments"><i></i>comments</a>
<a href="#" class="glyphicons no-js flower"><i></i>flower</a>
<a href="#" class="glyphicons no-js baseball"><i></i>baseball</a>
<a href="#" class="glyphicons no-js rugby"><i></i>rugby</a>
<a href="#" class="glyphicons no-js ax"><i></i>ax</a>
<a href="#" class="glyphicons no-js table_tennis"><i></i>table_tennis</a>
<a href="#" class="glyphicons no-js bowling"><i></i>bowling</a>
<a href="#" class="glyphicons no-js tree_conifer"><i></i>tree_conifer</a>
<a href="#" class="glyphicons no-js tree_deciduous"><i></i>tree_deciduous</a>
<a href="#" class="glyphicons no-js more_items"><i></i>more_items</a>
<a href="#" class="glyphicons no-js sort"><i></i>sort</a>
<a href="#" class="glyphicons no-js filter"><i></i>filter</a>
<a href="#" class="glyphicons no-js gamepad"><i></i>gamepad</a>
<a href="#" class="glyphicons no-js playing_dices"><i></i>playing_dices</a>
<a href="#" class="glyphicons no-js calculator"><i></i>calculator</a>
<a href="#" class="glyphicons no-js tie"><i></i>tie</a>
<a href="#" class="glyphicons no-js wallet"><i></i>wallet</a>
<a href="#" class="glyphicons no-js piano"><i></i>piano</a>
<a href="#" class="glyphicons no-js sampler"><i></i>sampler</a>
<a href="#" class="glyphicons no-js podium"><i></i>podium</a>
<a href="#" class="glyphicons no-js soccer_ball"><i></i>soccer_ball</a>
<a href="#" class="glyphicons no-js blog"><i></i>blog</a>
<a href="#" class="glyphicons no-js dashboard"><i></i>dashboard</a>
<a href="#" class="glyphicons no-js certificate"><i></i>certificate</a>
<a href="#" class="glyphicons no-js bell"><i></i>bell</a>
<a href="#" class="glyphicons no-js candle"><i></i>candle</a>
<a href="#" class="glyphicons no-js pushpin"><i></i>pushpin</a>
<a href="#" class="glyphicons no-js iphone_shake"><i></i>iphone_shake</a>
<a href="#" class="glyphicons no-js pin_flag"><i></i>pin_flag</a>
<a href="#" class="glyphicons no-js turtle"><i></i>turtle</a>
<a href="#" class="glyphicons no-js rabbit"><i></i>rabbit</a>
<a href="#" class="glyphicons no-js globe"><i></i>globe</a>
<a href="#" class="glyphicons no-js briefcase"><i></i>briefcase</a>
<a href="#" class="glyphicons no-js hdd"><i></i>hdd</a>
<a href="#" class="glyphicons no-js thumbs_up"><i></i>thumbs_up</a>
<a href="#" class="glyphicons no-js thumbs_down"><i></i>thumbs_down</a>
<a href="#" class="glyphicons no-js hand_right"><i></i>hand_right</a>
<a href="#" class="glyphicons no-js hand_left"><i></i>hand_left</a>
<a href="#" class="glyphicons no-js hand_up"><i></i>hand_up</a>
<a href="#" class="glyphicons no-js hand_down"><i></i>hand_down</a>
<a href="#" class="glyphicons no-js fullscreen"><i></i>fullscreen</a>
<a href="#" class="glyphicons no-js shopping_bag"><i></i>shopping_bag</a>
<a href="#" class="glyphicons no-js book_open"><i></i>book_open</a>
<a href="#" class="glyphicons no-js nameplate"><i></i>nameplate</a>
<a href="#" class="glyphicons no-js nameplate_alt"><i></i>nameplate_alt</a>
<a href="#" class="glyphicons no-js vases"><i></i>vases</a>
<a href="#" class="glyphicons no-js bullhorn"><i></i>bullhorn</a>
<a href="#" class="glyphicons no-js dumbbell"><i></i>dumbbell</a>
<a href="#" class="glyphicons no-js suitcase"><i></i>suitcase</a>
<a href="#" class="glyphicons no-js file_import"><i></i>file_import</a>
<a href="#" class="glyphicons no-js file_export"><i></i>file_export</a>
<a href="#" class="glyphicons no-js bug"><i></i>bug</a>
<a href="#" class="glyphicons no-js crown"><i></i>crown</a>
<a href="#" class="glyphicons no-js smoking"><i></i>smoking</a>
<a href="#" class="glyphicons no-js cloud-upload"><i></i>cloud-upload</a>
<a href="#" class="glyphicons no-js cloud-download"><i></i>cloud-download</a>
<a href="#" class="glyphicons no-js restart"><i></i>restart</a>
<a href="#" class="glyphicons no-js security_camera"><i></i>security_camera</a>
<a href="#" class="glyphicons no-js expand"><i></i>expand</a>
<a href="#" class="glyphicons no-js collapse"><i></i>collapse</a>
<a href="#" class="glyphicons no-js collapse_top"><i></i>collapse_top</a>
<a href="#" class="glyphicons no-js globe_af"><i></i>globe_af</a>
<a href="#" class="glyphicons no-js global"><i></i>global</a>
<a href="#" class="glyphicons no-js spray"><i></i>spray</a>
<a href="#" class="glyphicons no-js nails"><i></i>nails</a>
<a href="#" class="glyphicons no-js claw_hammer"><i></i>claw_hammer</a>
<a href="#" class="glyphicons no-js classic_hammer"><i></i>classic_hammer</a>
<a href="#" class="glyphicons no-js hand_saw"><i></i>hand_saw</a>
<a href="#" class="glyphicons no-js riflescope"><i></i>riflescope</a>
<a href="#" class="glyphicons no-js electrical_socket_eu"><i></i>electrical_socket_eu</a>
<a href="#" class="glyphicons no-js electrical_socket_us"><i></i>electrical_socket_us</a>
<a href="#" class="glyphicons no-js pinterest"><i></i>pinterest</a>
<a href="#" class="glyphicons no-js dropbox"><i></i>dropbox</a>
<a href="#" class="glyphicons no-js google_plus"><i></i>google_plus</a>
<a href="#" class="glyphicons no-js jolicloud"><i></i>jolicloud</a>
<a href="#" class="glyphicons no-js yahoo"><i></i>yahoo</a>
<a href="#" class="glyphicons no-js blogger"><i></i>blogger</a>
<a href="#" class="glyphicons no-js picasa"><i></i>picasa</a>
<a href="#" class="glyphicons no-js amazon"><i></i>amazon</a>
<a href="#" class="glyphicons no-js tumblr"><i></i>tumblr</a>
<a href="#" class="glyphicons no-js wordpress"><i></i>wordpress</a>
<a href="#" class="glyphicons no-js instapaper"><i></i>instapaper</a>
<a href="#" class="glyphicons no-js evernote"><i></i>evernote</a>
<a href="#" class="glyphicons no-js xing"><i></i>xing</a>
<a href="#" class="glyphicons no-js zootool"><i></i>zootool</a>
<a href="#" class="glyphicons no-js dribbble"><i></i>dribbble</a>
<a href="#" class="glyphicons no-js deviantart"><i></i>deviantart</a>
<a href="#" class="glyphicons no-js read_it_later"><i></i>read_it_later</a>
<a href="#" class="glyphicons no-js linked_in"><i></i>linked_in</a>
<a href="#" class="glyphicons no-js forrst"><i></i>forrst</a>
<a href="#" class="glyphicons no-js pinboard"><i></i>pinboard</a>
<a href="#" class="glyphicons no-js behance"><i></i>behance</a>
<a href="#" class="glyphicons no-js github"><i></i>github</a>
<a href="#" class="glyphicons no-js youtube"><i></i>youtube</a>
<a href="#" class="glyphicons no-js skitch"><i></i>skitch</a>
<a href="#" class="glyphicons no-js foursquare"><i></i>foursquare</a>
<a href="#" class="glyphicons no-js quora"><i></i>quora</a>
<a href="#" class="glyphicons no-js badoo"><i></i>badoo</a>
<a href="#" class="glyphicons no-js spotify"><i></i>spotify</a>
<a href="#" class="glyphicons no-js stumbleupon"><i></i>stumbleupon</a>
<a href="#" class="glyphicons no-js readability"><i></i>readability</a>
<a href="#" class="glyphicons no-js facebook"><i></i>facebook</a>
<a href="#" class="glyphicons no-js twitter"><i></i>twitter</a>
<a href="#" class="glyphicons no-js instagram"><i></i>instagram</a>
<a href="#" class="glyphicons no-js posterous_spaces"><i></i>posterous_spaces</a>
<a href="#" class="glyphicons no-js vimeo"><i></i>vimeo</a>
<a href="#" class="glyphicons no-js flickr"><i></i>flickr</a>
<a href="#" class="glyphicons no-js last_fm"><i></i>last_fm</a>
<a href="#" class="glyphicons no-js rss"><i></i>rss</a>
<a href="#" class="glyphicons no-js skype"><i></i>skype</a>
<a href="#" class="glyphicons no-js e-mail"><i></i>e-mail</a>
</div>
</div>
<div class="tab-pane halfings-demo" id="tab_1_3">
<h3>Image</h3>
<p><i class="halflings-icon glass"></i>glass</p>
<p><i class="halflings-icon music"></i>music</p>
<p><i class="halflings-icon search"></i>search</p>
<p><i class="halflings-icon envelope"></i>envelope</p>
<p><i class="halflings-icon heart"></i>heart</p>
<p><i class="halflings-icon star"></i>star</p>
<p><i class="halflings-icon star-empty"></i>star-empty</p>
<p><i class="halflings-icon user"></i>user</p>
<p><i class="halflings-icon film"></i>film</p>
<p><i class="halflings-icon th-large"></i>th-large</p>
<p><i class="halflings-icon th"></i>th</p>
<p><i class="halflings-icon th-list"></i>th-list</p>
<p><i class="halflings-icon ok"></i>ok</p>
<p><i class="halflings-icon remove"></i>remove</p>
<p><i class="halflings-icon zoom-in"></i>zoom-in</p>
<p><i class="halflings-icon zoom-out"></i>zoom-out</p>
<p><i class="halflings-icon off"></i>off</p>
<p><i class="halflings-icon signal"></i>signal</p>
<p><i class="halflings-icon cog"></i>cog</p>
<p><i class="halflings-icon trash"></i>trash</p>
<p><i class="halflings-icon home"></i>home</p>
<p><i class="halflings-icon file"></i>file</p>
<p><i class="halflings-icon time"></i>time</p>
<p><i class="halflings-icon road"></i>road</p>
<p><i class="halflings-icon download-alt"></i>download-alt</p>
<p><i class="halflings-icon download"></i>download</p>
<p><i class="halflings-icon upload"></i>upload</p>
<p><i class="halflings-icon inbox"></i>inbox</p>
<p><i class="halflings-icon play-circle"></i>play-circle</p>
<p><i class="halflings-icon repeat"></i>repeat</p>
<p><i class="halflings-icon refresh"></i>refresh</p>
<p><i class="halflings-icon list-alt"></i>list-alt</p>
<p><i class="halflings-icon lock"></i>lock</p>
<p><i class="halflings-icon flag"></i>flag</p>
<p><i class="halflings-icon headphones"></i>headphones</p>
<p><i class="halflings-icon volume-off"></i>volume-off</p>
<p><i class="halflings-icon volume-down"></i>volume-down</p>
<p><i class="halflings-icon volume-up"></i>volume-up</p>
<p><i class="halflings-icon qrcode"></i>qrcode</p>
<p><i class="halflings-icon barcode"></i>barcode</p>
<p><i class="halflings-icon tag"></i>tag</p>
<p><i class="halflings-icon tags"></i>tags</p>
<p><i class="halflings-icon book"></i>book</p>
<p><i class="halflings-icon bookmark"></i>bookmark</p>
<p><i class="halflings-icon print"></i>print</p>
<p><i class="halflings-icon camera"></i>camera</p>
<p><i class="halflings-icon font"></i>font</p>
<p><i class="halflings-icon bold"></i>bold</p>
<p><i class="halflings-icon italic"></i>italic</p>
<p><i class="halflings-icon text-height"></i>text-height</p>
<p><i class="halflings-icon text-width"></i>text-width</p>
<p><i class="halflings-icon align-left"></i>align-left</p>
<p><i class="halflings-icon align-center"></i>align-center</p>
<p><i class="halflings-icon align-right"></i>align-right</p>
<p><i class="halflings-icon align-justify"></i>align-justify</p>
<p><i class="halflings-icon list"></i>list</p>
<p><i class="halflings-icon indent-left"></i>indent-left</p>
<p><i class="halflings-icon indent-right"></i>indent-right</p>
<p><i class="halflings-icon facetime-video"></i>facetime-video</p>
<p><i class="halflings-icon picture"></i>picture</p>
<p><i class="halflings-icon pencil"></i>pencil</p>
<p><i class="halflings-icon map-marker"></i>map-marker</p>
<p><i class="halflings-icon adjust"></i>adjust</p>
<p><i class="halflings-icon tint"></i>tint</p>
<p><i class="halflings-icon edit"></i>edit</p>
<p><i class="halflings-icon share"></i>share</p>
<p><i class="halflings-icon check"></i>check</p>
<p><i class="halflings-icon move"></i>move</p>
<p><i class="halflings-icon step-backward"></i>step-backward</p>
<p><i class="halflings-icon fast-backward"></i>fast-backward</p>
<p><i class="halflings-icon backward"></i>backward</p>
<p><i class="halflings-icon play"></i>play</p>
<p><i class="halflings-icon pause"></i>pause</p>
<p><i class="halflings-icon stop"></i>stop</p>
<p><i class="halflings-icon forward"></i>forward</p>
<p><i class="halflings-icon fast-forward"></i>fast-forward</p>
<p><i class="halflings-icon step-forward"></i>step-forward</p>
<p><i class="halflings-icon eject"></i>eject</p>
<p><i class="halflings-icon chevron-left"></i>chevron-left</p>
<p><i class="halflings-icon chevron-right"></i>chevron-right</p>
<p><i class="halflings-icon plus-sign"></i>plus-sign</p>
<p><i class="halflings-icon minus-sign"></i>minus-sign</p>
<p><i class="halflings-icon remove-sign"></i>remove-sign</p>
<p><i class="halflings-icon ok-sign"></i>ok-sign</p>
<p><i class="halflings-icon question-sign"></i>question-sign</p>
<p><i class="halflings-icon info-sign"></i>info-sign</p>
<p><i class="halflings-icon screenshot"></i>screenshot</p>
<p><i class="halflings-icon remove-circle"></i>remove-circle</p>
<p><i class="halflings-icon ok-circle"></i>ok-circle</p>
<p><i class="halflings-icon ban-circle"></i>ban-circle</p>
<p><i class="halflings-icon arrow-left"></i>arrow-left</p>
<p><i class="halflings-icon arrow-right"></i>arrow-right</p>
<p><i class="halflings-icon arrow-up"></i>arrow-up</p>
<p><i class="halflings-icon arrow-down"></i>arrow-down</p>
<p><i class="halflings-icon share-alt"></i>share-alt</p>
<p><i class="halflings-icon resize-full"></i>resize-full</p>
<p><i class="halflings-icon resize-small"></i>resize-small</p>
<p><i class="halflings-icon plus"></i>plus</p>
<p><i class="halflings-icon minus"></i>minus</p>
<p><i class="halflings-icon asterisk"></i>asterisk</p>
<p><i class="halflings-icon exclamation-sign"></i>exclamation-sign</p>
<p><i class="halflings-icon gift"></i>gift</p>
<p><i class="halflings-icon leaf"></i>leaf</p>
<p><i class="halflings-icon fire"></i>fire</p>
<p><i class="halflings-icon eye-open"></i>eye-open</p>
<p><i class="halflings-icon eye-close"></i>eye-close</p>
<p><i class="halflings-icon warning-sign"></i>warning-sign</p>
<p><i class="halflings-icon plane"></i>plane</p>
<p><i class="halflings-icon calendar"></i>calendar</p>
<p><i class="halflings-icon random"></i>random</p>
<p><i class="halflings-icon comments"></i>comments</p>
<p><i class="halflings-icon magnet"></i>magnet</p>
<p><i class="halflings-icon chevron-up"></i>chevron-up</p>
<p><i class="halflings-icon chevron-down"></i>chevron-down</p>
<p><i class="halflings-icon retweet"></i>retweet</p>
<p><i class="halflings-icon shopping-cart"></i>shopping-cart</p>
<p><i class="halflings-icon folder-close"></i>folder-close</p>
<p><i class="halflings-icon folder-open"></i>folder-open</p>
<p><i class="halflings-icon resize-vertical"></i>resize-vertical</p>
<p><i class="halflings-icon resize-horizontal"></i>resize-horizontal</p>
<p><i class="halflings-icon hdd"></i>hdd</p>
<p><i class="halflings-icon bullhorn"></i>bullhorn</p>
<p><i class="halflings-icon bell"></i>bell</p>
<p><i class="halflings-icon certificate"></i>certificate</p>
<p><i class="halflings-icon thumbs-up"></i>thumbs-up</p>
<p><i class="halflings-icon thumbs-down"></i>thumbs-down</p>
<p><i class="halflings-icon hand-right"></i>hand-right</p>
<p><i class="halflings-icon hand-left"></i>hand-left</p>
<p><i class="halflings-icon hand-top"></i>hand-top</p>
<p><i class="halflings-icon hand-down"></i>hand-down</p>
<p><i class="halflings-icon circle-arrow-right"></i>circle-arrow-right</p>
<p><i class="halflings-icon circle-arrow-left"></i>circle-arrow-left</p>
<p><i class="halflings-icon circle-arrow-top"></i>circle-arrow-top</p>
<p><i class="halflings-icon circle-arrow-down"></i>circle-arrow-down</p>
<p><i class="halflings-icon globe"></i>globe</p>
<p><i class="halflings-icon wrench"></i>wrench</p>
<p><i class="halflings-icon tasks"></i>tasks</p>
<p><i class="halflings-icon filter"></i>filter</p>
<p><i class="halflings-icon briefcase"></i>briefcase</p>
<p><i class="halflings-icon fullscreen"></i>fullscreen</p>
<p><i class="halflings-icon dashboard"></i>dashboard</p>
<p><i class="halflings-icon paperclip"></i>paperclip</p>
<p><i class="halflings-icon heart-empty"></i>heart-empty</p>
<p><i class="halflings-icon link"></i>link</p>
<p><i class="halflings-icon phone"></i>phone</p>
<p><i class="halflings-icon pushpin"></i>pushpin</p>
<p><i class="halflings-icon euro"></i>euro</p>
<p><i class="halflings-icon usd"></i>usd</p>
<p><i class="halflings-icon gbp"></i>gbp</p>
<p><i class="halflings-icon sort"></i>sort</p>
<p><i class="halflings-icon sort-by-alphabet"></i>sort-by-alphabet</p>
<p><i class="halflings-icon sort-by-alphabet-alt"></i>sort-by-alphabet-alt</p>
<p><i class="halflings-icon sort-by-order"></i>sort-by-order</p>
<p><i class="halflings-icon sort-by-order-alt"></i>sort-by-order-alt</p>
<p><i class="halflings-icon sort-by-attributes"></i>sort-by-attributes</p>
<p><i class="halflings-icon sort-by-attributes-alt"></i>sort-by-attributes-alt</p>
<p><i class="halflings-icon unchecked"></i>unchecked</p>
<p><i class="halflings-icon expand"></i>expand</p>
<p><i class="halflings-icon collapse"></i>collapse</p>
<p><i class="halflings-icon collapse-top"></i>collapse-top</p>
<br /><br /><br />
<div class="white-content">
<h3>Image - white</h3>
<p><i class="halflings-icon white glass"></i>glass</p>
<p><i class="halflings-icon white music"></i>music</p>
<p><i class="halflings-icon white search"></i>search</p>
<p><i class="halflings-icon white envelope"></i>envelope</p>
<p><i class="halflings-icon white heart"></i>heart</p>
<p><i class="halflings-icon white star"></i>star</p>
<p><i class="halflings-icon white star-empty"></i>star-empty</p>
<p><i class="halflings-icon white user"></i>user</p>
<p><i class="halflings-icon white film"></i>film</p>
<p><i class="halflings-icon white th-large"></i>th-large</p>
<p><i class="halflings-icon white th"></i>th</p>
<p><i class="halflings-icon white th-list"></i>th-list</p>
<p><i class="halflings-icon white ok"></i>ok</p>
<p><i class="halflings-icon white remove"></i>remove</p>
<p><i class="halflings-icon white zoom-in"></i>zoom-in</p>
<p><i class="halflings-icon white zoom-out"></i>zoom-out</p>
<p><i class="halflings-icon white off"></i>off</p>
<p><i class="halflings-icon white signal"></i>signal</p>
<p><i class="halflings-icon white cog"></i>cog</p>
<p><i class="halflings-icon white trash"></i>trash</p>
<p><i class="halflings-icon white home"></i>home</p>
<p><i class="halflings-icon white file"></i>file</p>
<p><i class="halflings-icon white time"></i>time</p>
<p><i class="halflings-icon white road"></i>road</p>
<p><i class="halflings-icon white download-alt"></i>download-alt</p>
<p><i class="halflings-icon white download"></i>download</p>
<p><i class="halflings-icon white upload"></i>upload</p>
<p><i class="halflings-icon white inbox"></i>inbox</p>
<p><i class="halflings-icon white play-circle"></i>play-circle</p>
<p><i class="halflings-icon white repeat"></i>repeat</p>
<p><i class="halflings-icon white refresh"></i>refresh</p>
<p><i class="halflings-icon white list-alt"></i>list-alt</p>
<p><i class="halflings-icon white lock"></i>lock</p>
<p><i class="halflings-icon white flag"></i>flag</p>
<p><i class="halflings-icon white headphones"></i>headphones</p>
<p><i class="halflings-icon white volume-off"></i>volume-off</p>
<p><i class="halflings-icon white volume-down"></i>volume-down</p>
<p><i class="halflings-icon white volume-up"></i>volume-up</p>
<p><i class="halflings-icon white qrcode"></i>qrcode</p>
<p><i class="halflings-icon white barcode"></i>barcode</p>
<p><i class="halflings-icon white tag"></i>tag</p>
<p><i class="halflings-icon white tags"></i>tags</p>
<p><i class="halflings-icon white book"></i>book</p>
<p><i class="halflings-icon white bookmark"></i>bookmark</p>
<p><i class="halflings-icon white print"></i>print</p>
<p><i class="halflings-icon white camera"></i>camera</p>
<p><i class="halflings-icon white font"></i>font</p>
<p><i class="halflings-icon white bold"></i>bold</p>
<p><i class="halflings-icon white italic"></i>italic</p>
<p><i class="halflings-icon white text-height"></i>text-height</p>
<p><i class="halflings-icon white text-width"></i>text-width</p>
<p><i class="halflings-icon white align-left"></i>align-left</p>
<p><i class="halflings-icon white align-center"></i>align-center</p>
<p><i class="halflings-icon white align-right"></i>align-right</p>
<p><i class="halflings-icon white align-justify"></i>align-justify</p>
<p><i class="halflings-icon white list"></i>list</p>
<p><i class="halflings-icon white indent-left"></i>indent-left</p>
<p><i class="halflings-icon white indent-right"></i>indent-right</p>
<p><i class="halflings-icon white facetime-video"></i>facetime-video</p>
<p><i class="halflings-icon white picture"></i>picture</p>
<p><i class="halflings-icon white pencil"></i>pencil</p>
<p><i class="halflings-icon white map-marker"></i>map-marker</p>
<p><i class="halflings-icon white adjust"></i>adjust</p>
<p><i class="halflings-icon white tint"></i>tint</p>
<p><i class="halflings-icon white edit"></i>edit</p>
<p><i class="halflings-icon white share"></i>share</p>
<p><i class="halflings-icon white check"></i>check</p>
<p><i class="halflings-icon white move"></i>move</p>
<p><i class="halflings-icon white step-backward"></i>step-backward</p>
<p><i class="halflings-icon white fast-backward"></i>fast-backward</p>
<p><i class="halflings-icon white backward"></i>backward</p>
<p><i class="halflings-icon white play"></i>play</p>
<p><i class="halflings-icon white pause"></i>pause</p>
<p><i class="halflings-icon white stop"></i>stop</p>
<p><i class="halflings-icon white forward"></i>forward</p>
<p><i class="halflings-icon white fast-forward"></i>fast-forward</p>
<p><i class="halflings-icon white step-forward"></i>step-forward</p>
<p><i class="halflings-icon white eject"></i>eject</p>
<p><i class="halflings-icon white chevron-left"></i>chevron-left</p>
<p><i class="halflings-icon white chevron-right"></i>chevron-right</p>
<p><i class="halflings-icon white plus-sign"></i>plus-sign</p>
<p><i class="halflings-icon white minus-sign"></i>minus-sign</p>
<p><i class="halflings-icon white remove-sign"></i>remove-sign</p>
<p><i class="halflings-icon white ok-sign"></i>ok-sign</p>
<p><i class="halflings-icon white question-sign"></i>question-sign</p>
<p><i class="halflings-icon white info-sign"></i>info-sign</p>
<p><i class="halflings-icon white screenshot"></i>screenshot</p>
<p><i class="halflings-icon white remove-circle"></i>remove-circle</p>
<p><i class="halflings-icon white ok-circle"></i>ok-circle</p>
<p><i class="halflings-icon white ban-circle"></i>ban-circle</p>
<p><i class="halflings-icon white arrow-left"></i>arrow-left</p>
<p><i class="halflings-icon white arrow-right"></i>arrow-right</p>
<p><i class="halflings-icon white arrow-up"></i>arrow-up</p>
<p><i class="halflings-icon white arrow-down"></i>arrow-down</p>
<p><i class="halflings-icon white share-alt"></i>share-alt</p>
<p><i class="halflings-icon white resize-full"></i>resize-full</p>
<p><i class="halflings-icon white resize-small"></i>resize-small</p>
<p><i class="halflings-icon white plus"></i>plus</p>
<p><i class="halflings-icon white minus"></i>minus</p>
<p><i class="halflings-icon white asterisk"></i>asterisk</p>
<p><i class="halflings-icon white exclamation-sign"></i>exclamation-sign</p>
<p><i class="halflings-icon white gift"></i>gift</p>
<p><i class="halflings-icon white leaf"></i>leaf</p>
<p><i class="halflings-icon white fire"></i>fire</p>
<p><i class="halflings-icon white eye-open"></i>eye-open</p>
<p><i class="halflings-icon white eye-close"></i>eye-close</p>
<p><i class="halflings-icon white warning-sign"></i>warning-sign</p>
<p><i class="halflings-icon white plane"></i>plane</p>
<p><i class="halflings-icon white calendar"></i>calendar</p>
<p><i class="halflings-icon white random"></i>random</p>
<p><i class="halflings-icon white comments"></i>comments</p>
<p><i class="halflings-icon white magnet"></i>magnet</p>
<p><i class="halflings-icon white chevron-up"></i>chevron-up</p>
<p><i class="halflings-icon white chevron-down"></i>chevron-down</p>
<p><i class="halflings-icon white retweet"></i>retweet</p>
<p><i class="halflings-icon white shopping-cart"></i>shopping-cart</p>
<p><i class="halflings-icon white folder-close"></i>folder-close</p>
<p><i class="halflings-icon white folder-open"></i>folder-open</p>
<p><i class="halflings-icon white resize-vertical"></i>resize-vertical</p>
<p><i class="halflings-icon white resize-horizontal"></i>resize-horizontal</p>
<p><i class="halflings-icon white hdd"></i>hdd</p>
<p><i class="halflings-icon white bullhorn"></i>bullhorn</p>
<p><i class="halflings-icon white bell"></i>bell</p>
<p><i class="halflings-icon white certificate"></i>certificate</p>
<p><i class="halflings-icon white thumbs-up"></i>thumbs-up</p>
<p><i class="halflings-icon white thumbs-down"></i>thumbs-down</p>
<p><i class="halflings-icon white hand-right"></i>hand-right</p>
<p><i class="halflings-icon white hand-left"></i>hand-left</p>
<p><i class="halflings-icon white hand-top"></i>hand-top</p>
<p><i class="halflings-icon white hand-down"></i>hand-down</p>
<p><i class="halflings-icon white circle-arrow-right"></i>circle-arrow-right</p>
<p><i class="halflings-icon white circle-arrow-left"></i>circle-arrow-left</p>
<p><i class="halflings-icon white circle-arrow-top"></i>circle-arrow-top</p>
<p><i class="halflings-icon white circle-arrow-down"></i>circle-arrow-down</p>
<p><i class="halflings-icon white globe"></i>globe</p>
<p><i class="halflings-icon white wrench"></i>wrench</p>
<p><i class="halflings-icon white tasks"></i>tasks</p>
<p><i class="halflings-icon white filter"></i>filter</p>
<p><i class="halflings-icon white briefcase"></i>briefcase</p>
<p><i class="halflings-icon white fullscreen"></i>fullscreen</p>
<p><i class="halflings-icon white dashboard"></i>dashboard</p>
<p><i class="halflings-icon white paperclip"></i>paperclip</p>
<p><i class="halflings-icon white heart-empty"></i>heart-empty</p>
<p><i class="halflings-icon white link"></i>link</p>
<p><i class="halflings-icon white phone"></i>phone</p>
<p><i class="halflings-icon white pushpin"></i>pushpin</p>
<p><i class="halflings-icon white euro"></i>euro</p>
<p><i class="halflings-icon white usd"></i>usd</p>
<p><i class="halflings-icon white gbp"></i>gbp</p>
<p><i class="halflings-icon white sort"></i>sort</p>
<p><i class="halflings-icon white sort-by-alphabet"></i>sort-by-alphabet</p>
<p><i class="halflings-icon white sort-by-alphabet-alt"></i>sort-by-alphabet-alt</p>
<p><i class="halflings-icon white sort-by-order"></i>sort-by-order</p>
<p><i class="halflings-icon white sort-by-order-alt"></i>sort-by-order-alt</p>
<p><i class="halflings-icon white sort-by-attributes"></i>sort-by-attributes</p>
<p><i class="halflings-icon white sort-by-attributes-alt"></i>sort-by-attributes-alt</p>
<p><i class="halflings-icon white unchecked"></i>unchecked</p>
<p><i class="halflings-icon white expand"></i>expand</p>
<p><i class="halflings-icon white collapse"></i>collapse</p>
<p><i class="halflings-icon white collapse-top"></i>collapse-top</p>
</div>
<h3>Fonts</h3>
<a href="" class="halflings glass"><i></i>glass</a>
<a href="" class="halflings music"><i></i>music</a>
<a href="" class="halflings search"><i></i>search</a>
<a href="" class="halflings envelope"><i></i>envelope</a>
<a href="" class="halflings heart"><i></i>heart</a>
<a href="" class="halflings star"><i></i>star</a>
<a href="" class="halflings star-empty"><i></i>star-empty</a>
<a href="" class="halflings user"><i></i>user</a>
<a href="" class="halflings film"><i></i>film</a>
<a href="" class="halflings th-large"><i></i>th-large</a>
<a href="" class="halflings th"><i></i>th</a>
<a href="" class="halflings th-list"><i></i>th-list</a>
<a href="" class="halflings ok"><i></i>ok</a>
<a href="" class="halflings remove"><i></i>remove</a>
<a href="" class="halflings zoom-in"><i></i>zoom-in</a>
<a href="" class="halflings zoom-out"><i></i>zoom-out</a>
<a href="" class="halflings off"><i></i>off</a>
<a href="" class="halflings signal"><i></i>signal</a>
<a href="" class="halflings cog"><i></i>cog</a>
<a href="" class="halflings trash"><i></i>trash</a>
<a href="" class="halflings home"><i></i>home</a>
<a href="" class="halflings file"><i></i>file</a>
<a href="" class="halflings time"><i></i>time</a>
<a href="" class="halflings road"><i></i>road</a>
<a href="" class="halflings download-alt"><i></i>download-alt</a>
<a href="" class="halflings download"><i></i>download</a>
<a href="" class="halflings upload"><i></i>upload</a>
<a href="" class="halflings inbox"><i></i>inbox</a>
<a href="" class="halflings play-circle"><i></i>play-circle</a>
<a href="" class="halflings repeat"><i></i>repeat</a>
<a href="" class="halflings refresh"><i></i>refresh</a>
<a href="" class="halflings list-alt"><i></i>list-alt</a>
<a href="" class="halflings lock"><i></i>lock</a>
<a href="" class="halflings flag"><i></i>flag</a>
<a href="" class="halflings headphones"><i></i>headphones</a>
<a href="" class="halflings volume-off"><i></i>volume-off</a>
<a href="" class="halflings volume-down"><i></i>volume-down</a>
<a href="" class="halflings volume-up"><i></i>volume-up</a>
<a href="" class="halflings qrcode"><i></i>qrcode</a>
<a href="" class="halflings barcode"><i></i>barcode</a>
<a href="" class="halflings tag"><i></i>tag</a>
<a href="" class="halflings tags"><i></i>tags</a>
<a href="" class="halflings book"><i></i>book</a>
<a href="" class="halflings bookmark"><i></i>bookmark</a>
<a href="" class="halflings print"><i></i>print</a>
<a href="" class="halflings camera"><i></i>camera</a>
<a href="" class="halflings font"><i></i>font</a>
<a href="" class="halflings bold"><i></i>bold</a>
<a href="" class="halflings italic"><i></i>italic</a>
<a href="" class="halflings text-height"><i></i>text-height</a>
<a href="" class="halflings text-width"><i></i>text-width</a>
<a href="" class="halflings align-left"><i></i>align-left</a>
<a href="" class="halflings align-center"><i></i>align-center</a>
<a href="" class="halflings align-right"><i></i>align-right</a>
<a href="" class="halflings align-justify"><i></i>align-justify</a>
<a href="" class="halflings list"><i></i>list</a>
<a href="" class="halflings indent-left"><i></i>indent-left</a>
<a href="" class="halflings indent-right"><i></i>indent-right</a>
<a href="" class="halflings facetime-video"><i></i>facetime-video</a>
<a href="" class="halflings picture"><i></i>picture</a>
<a href="" class="halflings pencil"><i></i>pencil</a>
<a href="" class="halflings map-marker"><i></i>map-marker</a>
<a href="" class="halflings adjust"><i></i>adjust</a>
<a href="" class="halflings tint"><i></i>tint</a>
<a href="" class="halflings edit"><i></i>edit</a>
<a href="" class="halflings share"><i></i>share</a>
<a href="" class="halflings check"><i></i>check</a>
<a href="" class="halflings move"><i></i>move</a>
<a href="" class="halflings step-backward"><i></i>step-backward</a>
<a href="" class="halflings fast-backward"><i></i>fast-backward</a>
<a href="" class="halflings backward"><i></i>backward</a>
<a href="" class="halflings play"><i></i>play</a>
<a href="" class="halflings pause"><i></i>pause</a>
<a href="" class="halflings stop"><i></i>stop</a>
<a href="" class="halflings forward"><i></i>forward</a>
<a href="" class="halflings fast-forward"><i></i>fast-forward</a>
<a href="" class="halflings step-forward"><i></i>step-forward</a>
<a href="" class="halflings eject"><i></i>eject</a>
<a href="" class="halflings chevron-left"><i></i>chevron-left</a>
<a href="" class="halflings chevron-right"><i></i>chevron-right</a>
<a href="" class="halflings plus-sign"><i></i>plus-sign</a>
<a href="" class="halflings minus-sign"><i></i>minus-sign</a>
<a href="" class="halflings remove-sign"><i></i>remove-sign</a>
<a href="" class="halflings ok-sign"><i></i>ok-sign</a>
<a href="" class="halflings question-sign"><i></i>question-sign</a>
<a href="" class="halflings info-sign"><i></i>info-sign</a>
<a href="" class="halflings screenshot"><i></i>screenshot</a>
<a href="" class="halflings remove-circle"><i></i>remove-circle</a>
<a href="" class="halflings ok-circle"><i></i>ok-circle</a>
<a href="" class="halflings ban-circle"><i></i>ban-circle</a>
<a href="" class="halflings arrow-left"><i></i>arrow-left</a>
<a href="" class="halflings arrow-right"><i></i>arrow-right</a>
<a href="" class="halflings arrow-up"><i></i>arrow-up</a>
<a href="" class="halflings arrow-down"><i></i>arrow-down</a>
<a href="" class="halflings share-alt"><i></i>share-alt</a>
<a href="" class="halflings resize-full"><i></i>resize-full</a>
<a href="" class="halflings resize-small"><i></i>resize-small</a>
<a href="" class="halflings plus"><i></i>plus</a>
<a href="" class="halflings minus"><i></i>minus</a>
<a href="" class="halflings asterisk"><i></i>asterisk</a>
<a href="" class="halflings exclamation-sign"><i></i>exclamation-sign</a>
<a href="" class="halflings gift"><i></i>gift</a>
<a href="" class="halflings leaf"><i></i>leaf</a>
<a href="" class="halflings fire"><i></i>fire</a>
<a href="" class="halflings eye-open"><i></i>eye-open</a>
<a href="" class="halflings eye-close"><i></i>eye-close</a>
<a href="" class="halflings warning-sign"><i></i>warning-sign</a>
<a href="" class="halflings plane"><i></i>plane</a>
<a href="" class="halflings calendar"><i></i>calendar</a>
<a href="" class="halflings random"><i></i>random</a>
<a href="" class="halflings comments"><i></i>comments</a>
<a href="" class="halflings magnet"><i></i>magnet</a>
<a href="" class="halflings chevron-up"><i></i>chevron-up</a>
<a href="" class="halflings chevron-down"><i></i>chevron-down</a>
<a href="" class="halflings retweet"><i></i>retweet</a>
<a href="" class="halflings shopping-cart"><i></i>shopping-cart</a>
<a href="" class="halflings folder-close"><i></i>folder-close</a>
<a href="" class="halflings folder-open"><i></i>folder-open</a>
<a href="" class="halflings resize-vertical"><i></i>resize-vertical</a>
<a href="" class="halflings resize-horizontal"><i></i>resize-horizontal</a>
<a href="" class="halflings hdd"><i></i>hdd</a>
<a href="" class="halflings bullhorn"><i></i>bullhorn</a>
<a href="" class="halflings bell"><i></i>bell</a>
<a href="" class="halflings certificate"><i></i>certificate</a>
<a href="" class="halflings thumbs-up"><i></i>thumbs-up</a>
<a href="" class="halflings thumbs-down"><i></i>thumbs-down</a>
<a href="" class="halflings hand-right"><i></i>hand-right</a>
<a href="" class="halflings hand-left"><i></i>hand-left</a>
<a href="" class="halflings hand-top"><i></i>hand-top</a>
<a href="" class="halflings hand-down"><i></i>hand-down</a>
<a href="" class="halflings circle-arrow-right"><i></i>circle-arrow-right</a>
<a href="" class="halflings circle-arrow-left"><i></i>circle-arrow-left</a>
<a href="" class="halflings circle-arrow-top"><i></i>circle-arrow-top</a>
<a href="" class="halflings circle-arrow-down"><i></i>circle-arrow-down</a>
<a href="" class="halflings globe"><i></i>globe</a>
<a href="" class="halflings wrench"><i></i>wrench</a>
<a href="" class="halflings tasks"><i></i>tasks</a>
<a href="" class="halflings filter"><i></i>filter</a>
<a href="" class="halflings briefcase"><i></i>briefcase</a>
<a href="" class="halflings fullscreen"><i></i>fullscreen</a>
<a href="" class="halflings dashboard"><i></i>dashboard</a>
<a href="" class="halflings paperclip"><i></i>paperclip</a>
<a href="" class="halflings heart-empty"><i></i>heart-empty</a>
<a href="" class="halflings link"><i></i>link</a>
<a href="" class="halflings phone"><i></i>phone</a>
<a href="" class="halflings pushpin"><i></i>pushpin</a>
<a href="" class="halflings euro"><i></i>euro</a>
<a href="" class="halflings usd"><i></i>usd</a>
<a href="" class="halflings gbp"><i></i>gbp</a>
<a href="" class="halflings sort"><i></i>sort</a>
<a href="" class="halflings sort-by-alphabet"><i></i>sort-by-alphabet</a>
<a href="" class="halflings sort-by-alphabet-alt"><i></i>sort-by-alphabet-alt</a>
<a href="" class="halflings sort-by-order"><i></i>sort-by-order</a>
<a href="" class="halflings sort-by-order-alt"><i></i>sort-by-order-alt</a>
<a href="" class="halflings sort-by-attributes"><i></i>sort-by-attributes</a>
<a href="" class="halflings sort-by-attributes-alt"><i></i>sort-by-attributes-alt</a>
<a href="" class="halflings unchecked"><i></i>unchecked</a>
<a href="" class="halflings expand"><i></i>expand</a>
<a href="" class="halflings collapse"><i></i>collapse</a>
<a href="" class="halflings collapse-top"><i></i>collapse-top</a>
<br /><br /><br />
<div class="white-content">
<h3>Fonts - white</h3>
<a href="" class="halflings white glass"><i></i>glass</a>
<a href="" class="halflings white music"><i></i>music</a>
<a href="" class="halflings white search"><i></i>search</a>
<a href="" class="halflings white envelope"><i></i>envelope</a>
<a href="" class="halflings white heart"><i></i>heart</a>
<a href="" class="halflings white star"><i></i>star</a>
<a href="" class="halflings white star-empty"><i></i>star-empty</a>
<a href="" class="halflings white user"><i></i>user</a>
<a href="" class="halflings white film"><i></i>film</a>
<a href="" class="halflings white th-large"><i></i>th-large</a>
<a href="" class="halflings white th"><i></i>th</a>
<a href="" class="halflings white th-list"><i></i>th-list</a>
<a href="" class="halflings white ok"><i></i>ok</a>
<a href="" class="halflings white remove"><i></i>remove</a>
<a href="" class="halflings white zoom-in"><i></i>zoom-in</a>
<a href="" class="halflings white zoom-out"><i></i>zoom-out</a>
<a href="" class="halflings white off"><i></i>off</a>
<a href="" class="halflings white signal"><i></i>signal</a>
<a href="" class="halflings white cog"><i></i>cog</a>
<a href="" class="halflings white trash"><i></i>trash</a>
<a href="" class="halflings white home"><i></i>home</a>
<a href="" class="halflings white file"><i></i>file</a>
<a href="" class="halflings white time"><i></i>time</a>
<a href="" class="halflings white road"><i></i>road</a>
<a href="" class="halflings white download-alt"><i></i>download-alt</a>
<a href="" class="halflings white download"><i></i>download</a>
<a href="" class="halflings white upload"><i></i>upload</a>
<a href="" class="halflings white inbox"><i></i>inbox</a>
<a href="" class="halflings white play-circle"><i></i>play-circle</a>
<a href="" class="halflings white repeat"><i></i>repeat</a>
<a href="" class="halflings white refresh"><i></i>refresh</a>
<a href="" class="halflings white list-alt"><i></i>list-alt</a>
<a href="" class="halflings white lock"><i></i>lock</a>
<a href="" class="halflings white flag"><i></i>flag</a>
<a href="" class="halflings white headphones"><i></i>headphones</a>
<a href="" class="halflings white volume-off"><i></i>volume-off</a>
<a href="" class="halflings white volume-down"><i></i>volume-down</a>
<a href="" class="halflings white volume-up"><i></i>volume-up</a>
<a href="" class="halflings white qrcode"><i></i>qrcode</a>
<a href="" class="halflings white barcode"><i></i>barcode</a>
<a href="" class="halflings white tag"><i></i>tag</a>
<a href="" class="halflings white tags"><i></i>tags</a>
<a href="" class="halflings white book"><i></i>book</a>
<a href="" class="halflings white bookmark"><i></i>bookmark</a>
<a href="" class="halflings white print"><i></i>print</a>
<a href="" class="halflings white camera"><i></i>camera</a>
<a href="" class="halflings white font"><i></i>font</a>
<a href="" class="halflings white bold"><i></i>bold</a>
<a href="" class="halflings white italic"><i></i>italic</a>
<a href="" class="halflings white text-height"><i></i>text-height</a>
<a href="" class="halflings white text-width"><i></i>text-width</a>
<a href="" class="halflings white align-left"><i></i>align-left</a>
<a href="" class="halflings white align-center"><i></i>align-center</a>
<a href="" class="halflings white align-right"><i></i>align-right</a>
<a href="" class="halflings white align-justify"><i></i>align-justify</a>
<a href="" class="halflings white list"><i></i>list</a>
<a href="" class="halflings white indent-left"><i></i>indent-left</a>
<a href="" class="halflings white indent-right"><i></i>indent-right</a>
<a href="" class="halflings white facetime-video"><i></i>facetime-video</a>
<a href="" class="halflings white picture"><i></i>picture</a>
<a href="" class="halflings white pencil"><i></i>pencil</a>
<a href="" class="halflings white map-marker"><i></i>map-marker</a>
<a href="" class="halflings white adjust"><i></i>adjust</a>
<a href="" class="halflings white tint"><i></i>tint</a>
<a href="" class="halflings white edit"><i></i>edit</a>
<a href="" class="halflings white share"><i></i>share</a>
<a href="" class="halflings white check"><i></i>check</a>
<a href="" class="halflings white move"><i></i>move</a>
<a href="" class="halflings white step-backward"><i></i>step-backward</a>
<a href="" class="halflings white fast-backward"><i></i>fast-backward</a>
<a href="" class="halflings white backward"><i></i>backward</a>
<a href="" class="halflings white play"><i></i>play</a>
<a href="" class="halflings white pause"><i></i>pause</a>
<a href="" class="halflings white stop"><i></i>stop</a>
<a href="" class="halflings white forward"><i></i>forward</a>
<a href="" class="halflings white fast-forward"><i></i>fast-forward</a>
<a href="" class="halflings white step-forward"><i></i>step-forward</a>
<a href="" class="halflings white eject"><i></i>eject</a>
<a href="" class="halflings white chevron-left"><i></i>chevron-left</a>
<a href="" class="halflings white chevron-right"><i></i>chevron-right</a>
<a href="" class="halflings white plus-sign"><i></i>plus-sign</a>
<a href="" class="halflings white minus-sign"><i></i>minus-sign</a>
<a href="" class="halflings white remove-sign"><i></i>remove-sign</a>
<a href="" class="halflings white ok-sign"><i></i>ok-sign</a>
<a href="" class="halflings white question-sign"><i></i>question-sign</a>
<a href="" class="halflings white info-sign"><i></i>info-sign</a>
<a href="" class="halflings white screenshot"><i></i>screenshot</a>
<a href="" class="halflings white remove-circle"><i></i>remove-circle</a>
<a href="" class="halflings white ok-circle"><i></i>ok-circle</a>
<a href="" class="halflings white ban-circle"><i></i>ban-circle</a>
<a href="" class="halflings white arrow-left"><i></i>arrow-left</a>
<a href="" class="halflings white arrow-right"><i></i>arrow-right</a>
<a href="" class="halflings white arrow-up"><i></i>arrow-up</a>
<a href="" class="halflings white arrow-down"><i></i>arrow-down</a>
<a href="" class="halflings white share-alt"><i></i>share-alt</a>
<a href="" class="halflings white resize-full"><i></i>resize-full</a>
<a href="" class="halflings white resize-small"><i></i>resize-small</a>
<a href="" class="halflings white plus"><i></i>plus</a>
<a href="" class="halflings white minus"><i></i>minus</a>
<a href="" class="halflings white asterisk"><i></i>asterisk</a>
<a href="" class="halflings white exclamation-sign"><i></i>exclamation-sign</a>
<a href="" class="halflings white gift"><i></i>gift</a>
<a href="" class="halflings white leaf"><i></i>leaf</a>
<a href="" class="halflings white fire"><i></i>fire</a>
<a href="" class="halflings white eye-open"><i></i>eye-open</a>
<a href="" class="halflings white eye-close"><i></i>eye-close</a>
<a href="" class="halflings white warning-sign"><i></i>warning-sign</a>
<a href="" class="halflings white plane"><i></i>plane</a>
<a href="" class="halflings white calendar"><i></i>calendar</a>
<a href="" class="halflings white random"><i></i>random</a>
<a href="" class="halflings white comments"><i></i>comments</a>
<a href="" class="halflings white magnet"><i></i>magnet</a>
<a href="" class="halflings white chevron-up"><i></i>chevron-up</a>
<a href="" class="halflings white chevron-down"><i></i>chevron-down</a>
<a href="" class="halflings white retweet"><i></i>retweet</a>
<a href="" class="halflings white shopping-cart"><i></i>shopping-cart</a>
<a href="" class="halflings white folder-close"><i></i>folder-close</a>
<a href="" class="halflings white folder-open"><i></i>folder-open</a>
<a href="" class="halflings white resize-vertical"><i></i>resize-vertical</a>
<a href="" class="halflings white resize-horizontal"><i></i>resize-horizontal</a>
<a href="" class="halflings white hdd"><i></i>hdd</a>
<a href="" class="halflings white bullhorn"><i></i>bullhorn</a>
<a href="" class="halflings white bell"><i></i>bell</a>
<a href="" class="halflings white certificate"><i></i>certificate</a>
<a href="" class="halflings white thumbs-up"><i></i>thumbs-up</a>
<a href="" class="halflings white thumbs-down"><i></i>thumbs-down</a>
<a href="" class="halflings white hand-right"><i></i>hand-right</a>
<a href="" class="halflings white hand-left"><i></i>hand-left</a>
<a href="" class="halflings white hand-top"><i></i>hand-top</a>
<a href="" class="halflings white hand-down"><i></i>hand-down</a>
<a href="" class="halflings white circle-arrow-right"><i></i>circle-arrow-right</a>
<a href="" class="halflings white circle-arrow-left"><i></i>circle-arrow-left</a>
<a href="" class="halflings white circle-arrow-top"><i></i>circle-arrow-top</a>
<a href="" class="halflings white circle-arrow-down"><i></i>circle-arrow-down</a>
<a href="" class="halflings white globe"><i></i>globe</a>
<a href="" class="halflings white wrench"><i></i>wrench</a>
<a href="" class="halflings white tasks"><i></i>tasks</a>
<a href="" class="halflings white filter"><i></i>filter</a>
<a href="" class="halflings white briefcase"><i></i>briefcase</a>
<a href="" class="halflings white fullscreen"><i></i>fullscreen</a>
<a href="" class="halflings white dashboard"><i></i>dashboard</a>
<a href="" class="halflings white paperclip"><i></i>paperclip</a>
<a href="" class="halflings white heart-empty"><i></i>heart-empty</a>
<a href="" class="halflings white link"><i></i>link</a>
<a href="" class="halflings white phone"><i></i>phone</a>
<a href="" class="halflings white pushpin"><i></i>pushpin</a>
<a href="" class="halflings white euro"><i></i>euro</a>
<a href="" class="halflings white usd"><i></i>usd</a>
<a href="" class="halflings white gbp"><i></i>gbp</a>
<a href="" class="halflings white sort"><i></i>sort</a>
<a href="" class="halflings white sort-by-alphabet"><i></i>sort-by-alphabet</a>
<a href="" class="halflings white sort-by-alphabet-alt"><i></i>sort-by-alphabet-alt</a>
<a href="" class="halflings white sort-by-order"><i></i>sort-by-order</a>
<a href="" class="halflings white sort-by-order-alt"><i></i>sort-by-order-alt</a>
<a href="" class="halflings white sort-by-attributes"><i></i>sort-by-attributes</a>
<a href="" class="halflings white sort-by-attributes-alt"><i></i>sort-by-attributes-alt</a>
<a href="" class="halflings white unchecked"><i></i>unchecked</a>
<a href="" class="halflings white expand"><i></i>expand</a>
<a href="" class="halflings white collapse"><i></i>collapse</a>
<a href="" class="halflings white collapse-top"><i></i>collapse-top</a>
</div>
</div>
</div>
</div>
</div>
</div>
<!-- END PAGE CONTENT-->
</div>
<!-- END PAGE CONTAINER-->
</div>
<!-- END PAGE -->
</div>
<!-- END CONTAINER -->
<!-- BEGIN FOOTER -->
<div class="footer">
2013 © Metronic by keenthemes.
<div class="span pull-right">
<span class="go-top"><i class="icon-angle-up"></i></span>
</div>
</div>
<!-- END FOOTER -->
<!-- BEGIN JAVASCRIPTS -->
<!-- Load javascripts at bottom, this will reduce page load time -->
<script src="assets/js/jquery-1.8.3.min.js"></script>
<script src="assets/breakpoints/breakpoints.js"></script>
<script src="assets/bootstrap/js/bootstrap.min.js"></script>
<script src="assets/glyphicons/scripts/modernizr.js"></script>
<script src="assets/js/jquery.blockui.js"></script>
<script src="assets/js/jquery.cookie.js"></script>
<!-- ie8 fixes -->
<!--[if lt IE 9]>
<script src="assets/js/excanvas.js"></script>
<script src="assets/js/respond.js"></script>
<![endif]-->
<script type="text/javascript" src="assets/uniform/jquery.uniform.min.js"></script>
<script src="assets/js/app.js"></script>
<script>
jQuery(document).ready(function() {
// initiate layout and plugins
App.init();
});
</script>
</body>
<!-- END BODY -->
</html> | code |
भ्रष्ट इंडिया : सेना प्रमुख ने प्रधानमंत्री राहत कोष में एक करोड़ रुपया कहां से दिया ?
सेना प्रमुख ने प्रधानमंत्री राहत कोष में एक करोड़ रुपया कहां से दिया ?
नई दिल्ली। सेना द्वारा एक आरटीआई के जवाब ने यह प्रश्न खड़ा कर दिया है कि सेना प्रमुख ने प्रधानमंत्री राहत कोष में एक करोड़ रुपया कहां से दिया ?
दरअसल, सेना का कहना है कि उसने सैनिकों के वेतन से कोई चंदा नहीं दिया। ऐसे में उस चेक पर सवाल उठ रहा है, जिस पर लिखा गया था कि रकम सेना के वेतन से ली गई है।
चार महीने पहले आर्मी चीफ दलबीर सिंह सुहाग ने प्रधानमंत्री राहत कोष के लिए १०० करोड़ रुपये का चेक दिया था। यह चेक उन्होंने खुद पीएम नरेंद्र मोदी को दिया। मगर अब सेना की तरफ से जारी बयान में कहा गया है कि सैनिकों की सैलरी से या अन्य तरीके से ऐसा कोई दान नहीं दिया गया है।
देहरादून के रहने वाले प्रभु डंडरियाल की तरफ से डाली गई आरटीआई के जवाब में आर्मी के सीपीआईओ लेफ्टिनेंट कर्नल राजीव गुलेरिया ने लिखा है संबंधित एजेंसी ने सूचित किया है कि सेना के जवानों के वेतन से प्रधानमंत्री राहत कोष में कोई अनुदान नहीं दिया गया है। यह मामला अभी विचाराधीन है।
गौरतलब है कि प्रधानमंत्री नरेद्र मोदी के ऑफिस की वेबसाइट पर डाले गए ६७वें सेना दिवस समारोह की तस्वीरों में एक चेक दिख रहा था। जनरल सुहाग इस चेक को पीएम मोदी को सौंप रहे हैं। इस चेक में लिखा है, भारतीय सेना के सभी रैंक्स का एक दिन का वेतन।
डंडरियाल ने अब प्रधानमंत्री कार्यालय से आरटीआई के जरिए इस चेक के बारे में जानकारी मांगी है। डंडरियाल ने कहा, जब २० मार्च तक दान के लिए वेतन से कुछ नहीं लिया गया था, तो आर्मी चीफ ने १०० करोड़ रुपये का चेक कैसे दे दिया? इसीलिए मैंने आरटीआई के जरिए पीएमओ से जवाब मांगा है।
सेना प्रमुख ने प्रधानमंत्री राहत कोष में एक करोड़ ...
जानियेनेपाल जैसे भूकंप से ब्रज वसुंधरा का क्या...
यूपी: आईपीएस अमिताभ ठाकुर के खिलाफ गवर्नर ने दिए ज... | hindi |
रनवे पर उतरते समय दुर्घटनाग्रस्त हुआ प्रशिक्षु विमान, पायलट सुरक्षित - ट्रेनी प्लान क्रैश इन अमेठी
अमेठीः इंदिरा गांधी राष्ट्रीय उड़ान अकादमी पर उतरते समय एक प्रशिक्षु विमान दुर्घटनाग्रस्त हो गया और उसमें आग लग गई। गनीमत रही कि ट्रेनी पायलट बाल-बाल बच गया।
फुरसतगंज स्थित अकादमी के मुख्य प्रशासनिक अधिकारी संदीप पुरी ने बताया कि एक विमान रनवे पर उतरते समय फिसलकर घास के मैदान में चला गया और उसमें आग लग गई। विमान का कुछ हिस्सा क्षतिग्रस्त हो गया। विमान पायलट ने कूदकर अपनी जान बचाई।
अधिकारियों के मुताबिक यह विमान चेकस्लोवाकिया का बना हुआ था और प्रशिक्षु पायलट की यह दूसरी अकेली उड़ान थी। | hindi |
پادَر سٕہہ ( کٲشُر : /paːdar sɨh/ ) یا شیرِ بَبَر ( کٲشُر : /ʃeːri babar/ ) چھُ اَکھ جَنٛگلی جانوَر۔ | kashmiri |
مگر ٲخٕر کر ہے تہٕ کَرِ کیٛاہ | kashmiri |
#!/bin/sh
set -e
[ -d "build" ] || mkdir "build"
cd "build"
find .. -type f -name "*.c" | grep -Fxv "../demo/demo.c" | sort -R | C_INCLUDE_PATH="/usr/include/libxml2" xargs -rd '\n' \
gcc -Wall -Wextra -pedantic -std=c99 -O2 -shared -fPIC -lcrypto -lcurl -lxml2 -o "libdecrypt.so"
gcc -Wall -Wextra -pedantic -std=c99 -O2 -L. -ldecrypt -o "demo" "../demo/demo.c"
cp -t . "../demo/test."{rsdf,ccf,dlc}
LD_LIBRARY_PATH="`readlink -f .`" ./demo
| code |
import { DocumentUnknown20 } from "../../";
export = DocumentUnknown20;
| code |
\begin{document}
\begin{frontmatter}[classification=text]
\author[rk]{Robert Kleinberg }
\author[ds]{David E Speyer}
\author[ws]{Will Sawin}
\begin{abstract}
Let $G$ be an abelian group. A tri-colored sum-free set in $G$ is a collection of triples $(\vctr{a}_i, \vctr{b}_i, \vctr{c}_i)$ in $G$ such that $\vctr{a}_i+\vctr{b}_j+\vctr{c}_k=0$ if and only if $i=j=k$. Fix a prime $q$ and let $C_q$ be the cyclic group of order $q$. Let $\theta = \min_{\rho>0} (1+\rho+\cdots + \rho^{q-1}) \rho^{-(q-1)/3}$. Blasiak, Church, Cohn, Grochow, Naslund, Sawin, and Umans (building on previous work of Croot, Lev and Pach, and of Ellenberg and Gijswijt) showed that a tri-colored sum-free set in $C_q^n$ has size at most $3 \theta^n$. Between this paper and a paper of Pebody, we will show that, for any $\delta > 0$, and $n$ sufficiently large, there are tri-colored sum-free sets in $C_q^n$ of size $(\theta-\delta)^n$. Our construction also works when $q$ is not prime.
\end{abstract}
\end{frontmatter}
\section{Introduction}
Let $G$ be an abelian group. Let $\vctr{t} \in G^n$. We make the following slightly nonstandard definition: a \newword{sum-free set in $G^n$ with target $\vctr{t}$} is a collection of triples $(\vctr{a}_i, \vctr{b}_i, \vctr{c}_i)$ in $G^n \times G^n \times G^n$ such that $\vctr{a}_i+\vctr{b}_j+\vctr{c}_k=\vctr{t}$ if and only if $i=j=k$.
We may always replace $(\vctr{a}_i, \vctr{b}_i, \vctr{c}_i)$ by $(\vctr{a}_i, \vctr{b}_i, \vctr{c}_i - \vctr{t})$ to make the target $\vctr{0}$ (as we did in the abstract, and as is more standard), but allowing an arbitrary target will simplify our notation.
The usual terminology is ``tri-colored sum-free set", but we omit the reference to the coloring as we never consider any other kind.
If $X \subset G^n$ is a set with no three-term arithmetic progressions, then $\{ (\vctr{x}, \vctr{x}, -2 \vctr{x}) : \vctr{x} \in X \}$ is sum-free with target $\vctr{0}$, so lower bounds on sets without three-term arithmetic progressions are also bounds on sum-free sets. The reverse does not hold: the largest known three-term arithmetic progression free subsets of $C_3^n$ (where $C_q$ is the cyclic group of order $q$) are of size $2.217^n$~\cite{Edel}. Before this paper, the largest known sum-free sets in $C_3^n$ were of size $2.519^n$~\cite{Alon-Shpilka-Umans}; this paper will raise the bound to $2.755^n$ and show that this bound is tight.
Letting $r_3(G^n)$ denote the largest subset of $G^n$ with no three-term arithmetic progressions, the question of
whether $\lim \sup_{n \to \infty} r_3(G^n)^{1/n} < |G|$ was open, until recently, for every abelian $G$ containing elements of order greater than two.
The breakthrough work of Croot, Lev, and Pach~\cite{CrootLP} introduced a polynomial method to
prove that strict inequality holds when $G$ is cyclic of order 4,
and Ellenberg and Gijswijt~\cite{EllenbergG} built upon their ideas to prove it for cyclic groups of odd prime order.
Blasiak et al.~\cite{BCCGNSU} applied the same method to prove upper bounds for sum-free sets in $G^n$ for any fixed finite abelian group $G$.
We recall here one case of their bound. Let $C_q$ be the cyclic group of order $q$. Let $\theta = \min_{\beta>0} (1+\beta+\cdots + \beta^{q-1}) \beta^{-(q-1)/3}$ and let $\rho$ be the value of $\beta$ at which the minimum is attained. We note that the minimum is attained at a unique point which belongs to $(0,1)$ because $(1+\beta+\cdots + \beta^{q-1}) \beta^{-(q-1)/3}$ approaches $\infty$ as $\beta$ goes to $0$ from above, is increasing on the interval $[1,\infty)$, and has increasing first derivative on the interval $(0,1)$.
The following result of~\cite{BCCGNSU} is closely related to the results of~\cite{EllenbergG} (for primes) and~\cite[Theorem 4]{Petrov} (for prime powers). (What we denote $\theta$ is called $q J(q)$ in~\cite{BCCGNSU}.)
\begin{theorem}[{\cite[Theorem~4.14]{BCCGNSU}}] \label{UpperBound}
If $q$ is a prime power, then sum-free sets in $C_q^n$ have size at most $3 \theta^n$.
\end{theorem}
Prior to this paper, it was not clear whether any of these applications of the polynomial method yielded tight
bounds. In fact, Theorem \ref{UpperBound} is tight to within a subexponential factor.
\begin{theorem} \label{LowerBound}
Fix an integer $q \geq 2$. Define $\theta$ as above. For $n$ sufficiently large, there are sum-free sets in $C_q^n$ with size $\geq \theta^n e^{- 2 \sqrt{(2 \log 2 \log \theta ) n} - O_q(\log n)}$.
\end{theorem}
In this paper, we show Theorem \ref{LowerBound} except for a hypothesis on the existence of a probability distribution satisfying certain properties (Theorem \ref{DistributionExists}). In~\cite{Pebody}, Pebody will verify Theorem \ref{DistributionExists}, completing the proof of Theorem \ref{LowerBound}.\footnote{\cite{Pebody} was written in response to a preprint version of the paper which stated Theorem \ref{DistributionExists} as a conjecture, and we have chosen to preserve the chronology here, though otherwise updating the paper to reflect his result.}
The question of whether Theorem \ref{UpperBound} also yields a tight bound for $\lim \sup r_3(G^n)^{1/n}$ remains open.
Sum-free sets have applications in theoretical
computer science, especially the circle of ideas surrounding fast matrix multiplication algorithms. The
$O(n^{2.41})$ algorithm of Coppersmith and Winograd~\cite{Coppersmith} rests on a combinatorial
construction that can, in hindsight, be interpreted\footnote{This interpretation was made explicit by Fu and
Kleinberg~\cite{Fu}.} as a large sum-free set in $\mathbb{F}_2^n$. In the same paper they presented a
conjecture in additive combinatorics that, if true, would imply that
the exponent of matrix multiplication is 2, i.e., that there exist
matrix multiplication algorithms with running time $O(n^{2+\epsilon})$
for any $\epsilon>0$. This conjecture, along with another conjecture by
Cohn et al.~\cite{CohnKSU} that also implies the exponent of matrix
multiplication is 2, was shown
by Alon, Shpilka, and Umans~\cite{Alon-Shpilka-Umans}
to necessitate the existence of sum-free
sets of size $3^{n-o(n)}$ in $\mathbb{F}_3^n$. The upper bound on sum-free
sets by Blasiak et al.~\cite{BCCGNSU} thus refutes both of these conjectures.
Furthermore, Blasiak et al.~\cite{BCCGNSU} show that a more general family
of proposed fast matrix multiplication algorithms based on the ``simultaneous
triple product property'' (STPP)~\cite{CohnKSU} in an abelian group $H$ necessitates
the existence of sum-free sets of size $|H|^{1-o(1)}$. Their upper bound
on sum-free sets in abelian groups of bounded exponent thus precludes
achieving matrix multiplication exponent 2 using STPP constructions in such
groups.
A second application of sum-free sets in theoretical computer science concerns
property testing, the study of randomized algorithms for distinguishing functions
$f$ having a specified property from those which have large Hamming
distance from every function that satisfies the property. A famous example
is the Blum-Luby-Rubinfeld (BLR) linearity tester~\cite{BlumLR}, which
queries the function value at only $O(1/\delta)$ points and
succeeds, with error probability less than $1/3$, in
distinguishing linear functions
on $\mathbb{F}_2^n$ from those that have distance $\delta \cdot 2^n$ from any linear function.
Testers which can distinguish low-degree polynomials on $\mathbb{F}_2^n$
from those that are far from any low-degree polynomial constitute an
important ingredient in the celebrated PCP Theorem~\cite{ALMSS}.
Bhattacharya and Xie~\cite{BX} demonstrated that constructions
of large sum-free sets in $\mathbb{F}_2^n$ could be used to derive lower
bounds on the complexity of testing certain linear-invariant properties
of Boolean functions.
Finally, sum-free sets have applications to removal lemmas in
additive combinatorics, a topic that is heavily intertwined with property testing.
In particular, Green~\cite{Green05} proved an ``arithmetic removal lemma'' for
abelian groups which implies that for every $\epsilon>0$, there
is a $\delta>0$ such that for any abelian group $G$ and three subsets
$A, B, C$, either there are at least $\delta |G|^2$ distinct triples
$(a,b,c) \in A \times B \times C$ satisfying $a+b+c=0$, or one can
eliminate all such triples by deleting at most $\epsilon |G|$ elements
from each of $A,B,$ and $C$. Green's argument yields an upper bound
for $\delta^{-1}$ which is a tower of twos of height polynomial in
$\epsilon^{-1}$. This bound can be improved using
combinatorial\footnote{See~\cite{FoxTR}, building upon the
combinatorial proof of Green's result in~\cite{KralSV}.}
or Fourier analytic\footnote{See~\cite{HatamiST}, which pertains to the
case $G=\mathbb{F}_2^n$ and adapts the
proof idea of~\cite{FoxTR} to the analytic setting.} techniques,
but for general abelian groups $G$ the value of $\delta$ is not bounded
below by any polynomial function of $\epsilon$. However,
when $G$ is the group $\mathbb{F}_q^n$, Fox and Lovasz~\cite{FoxL} have applied our
nearly-tight construction of sum-free sets in $G$ to obtain bounds of the form
$$
\epsilon^{-C_q + o(1)} \, < \, \delta^{-1} \, <(\epsilon/3)^{-C_q} ,
$$
where $C_q$ is a constant depending on $q$ but not $n$, and where $o(1)$ goes to $0$ as $\epsilon$ goes to $0$ for any fixed $q$.
\section{Notation}
Throughout this paper, we will use the following conventions: Lower case Roman letters denote integers, elements of cyclic groups (denoted $C_q$), of finite fields (denoted $\mathbb{F}_q$), or general finite sets. Lower case Roman letters
in boldface denote elements of $\mathbb{Z}_{\geq 0}^m$ (for any $m$), $C_q^m$ or $\mathbb{F}_q^m$. Capital Roman letters denote subsets of $\mathbb{Z}_{\geq 0}^m$, $C_q^m$ or $\mathbb{F}_q^m$. Lower case Greek letters denote real numbers; lower case Greek letters
in boldface denote elements of $\mathbb{R}^m$.
A notation such as $\alpha(x)$ or $\vctr{\alpha}(x)$ refers to a function of $x$ valued in real numbers, or real vectors.
For any sets $U$ and $V$, we write $U^V$ for the set of $U$-valued functions on $V$. All logarithms are to base $e$.
We fix a positive integer $q$. In section $4$, we will fix $n$ to be a positive integer divisible by $3$. The notation $O_q( \ )$ will always refer to bounds as $n \to \infty$ through integers divisible by $3$, with $q$ fixed. Let $\vctr{t} = (q-1, q-1, \ldots, q-1) \in \mathbb{Z}_{\geq 0}^n$.
We define the following sets of lattice points:
\[ \begin{array}{rcl}
I &= & \{ 0,1,\ldots, q-1 \} \subset \mathbb{Z}_{\geq 0} \\
T &=& \{ (a,b,c) \in I^3 : a+b+c = q-1 \} \\
\end{array} \]
\section{Entropy}
Let $A$ be a finite set and let $\vctr{e} = (e_1, e_2, \ldots, e_n) \in A^n$. We define the probability distribution $\vctr{\sigma}(\vctr{e})$ on $A$ by $\vctr{\sigma}_a(\vctr{e}) = \# \{ r : e_r = a \}/n$. In other words, $\vctr{\sigma}(\vctr{e})$ is the probability distribution of uniformly randomly selecting an element of $\vctr{e}$.
Let $A$ be a finite set and $\vctr{\lambda} \in \mathbb{R}_{\geq 0}^A$ a probability distribution on $A$. The \newword{entropy}, $\eta(\vctr{\lambda})$, is defined by
\[ \eta(\vctr{\lambda}) = - \sum_{a \in A} \vctr{\lambda}_a \log(\vctr{\lambda}_a) \]
where $0 \log 0$ is considered to be $0$.
The importance of the entropy function in our situation is the following:
\begin{lemma} \label{Histogram}
Let $A$ be a finite set, and let $\vctr{e}_0 \in A^n$.
Then
\[
n \eta(\vctr{\sigma}(\vctr{e}_0)) - O_{|A|}(\log n)\leq \log \left( \# \left\{ \vctr{e} \in A^n : \vctr{\sigma}(\vctr{e}) = \vctr{\sigma}(\vctr{e}_0) \right\} \right) \leq n \eta(\vctr{\sigma}(\vctr{e}_0)). \]
\end{lemma}
The implied constant in $O$ depends only on $|A|$ and not on $n$ or $\vctr{e}_0$.
\begin{proof}
For $a \in A$, let $n_a = n \vctr{\sigma}_a(\vctr{e}_0)$ be the number of times $a$ appears in $\vctr{e}_0$.
The number of $\vctr{e} \in A^n$ such that $\vctr{\sigma}(\vctr{e}) = \vctr{\sigma}(\vctr{e}_0)$ is
equal to the multinomial coefficient
\[
\binom{n}{(n_a)_{a \in A}} : = \frac{n!}{\prod_{a \in A} n_a!}.
\]
For the upper bound, we take one term from the multinomial formula
\[ n^n = \left( \sum_{a\in A} n_a\right)^n \geq \binom{n}{(n_a)_{a\in A}} \prod_{a \in A} n_a^{n_a}, \]
so
\[ \binom{n}{(n_a)_{a\in A}} \leq \prod_{a \in A} \left(\frac{n}{n_a}\right)^{n_a} = \exp(n \eta(\vctr{\sigma}(e_0))).\]
For the lower bound, we use the
following version of Stirling's formula.
(See, e.g.,~\cite{Robbins}.)
\[
(n + \tfrac12) \log(n) - n + \tfrac12 \log(2 \pi) \;<\; \log(n!) \;<\; (n + \tfrac12) \log(n) - n + \tfrac12 \log(2 \pi)+\tfrac{1}{12}
\]
Applying this estimate to each of the factorial terms, and using $\sum_{a \in A} n_a = n$
we find that
\[
\left|
\log \binom{n}{(n_a)_{a \in A}} -
\sum_{a \in A} n_a \log \left( \frac{n}{n_a} \right)
\right|
\leq
|A| \left[ \log(n) + \log(2 \pi) + \frac{1}{6} \right].
\]
Note that $\eta(\vctr{\sigma}(\vctr{e}_0)) = \sum_{a \in A} \frac{n_a}{n} \log \left( \frac{n}{n_a} \right)$, so this gives
\[
\left|
\log \binom{n}{(n_a)_{a \in A}} -
n \eta(\vctr{\sigma}(\vctr{e}_0))
\right|
\leq
|A| \left[ \log(n) + \log(2 \pi) + \frac{1}{6} \right]. \qedhere
\]
\end{proof}
If $A$ and $B$ are finite sets, $f: A \to B$ is a map and $\vctr{\lambda}$ is a probability distribution on $A$, then we define the probability distribution $f_{\ast} \vctr{\lambda}$ on $B$ by
\[ (f_{\ast} \vctr{\lambda})_b = \sum_{a \in f^{-1}(b)} \vctr{\lambda}_a . \]
It is well known that $\eta(f_{\ast} \vctr{\lambda}) \leq \eta(\vctr{\lambda})$, with strict inequality if there are distinct elements $a_1$ and $a_2 \in A$ with $f(a_1) = f(a_2)$ and $\vctr{\lambda}_{a_1}$, $\vctr{\lambda}_{a_2} > 0$.
With $\rho$ and $\theta$ as defined before,
define a probability distribution $\vctr{\psi}$ on $I$ by
\[ \vctr{\psi}_k = \frac{\rho^k}{1+\rho+\cdots + \rho^{q-1}}. \]
Let $f: T \to I$ be the map $f((i,j,k)) = k$.
The following is proved in ~\cite{Pebody}.\footnote{A proof was also claimed in a preprint \cite{Norin}, but we are unable to confirm all the steps in the argument.}
\begin{theorem}[{\cite[Theorem 4]{Pebody}}] \label{DistributionExists}
There is an $S_3$-symmetric probability distribution $\vctr{\pi}$ on $T$ with $f_{\ast} (\vctr{\pi}) = \vctr{\psi}$.
\end{theorem}
More precisely, \cite{Pebody} proves that $\vctr{\psi},\vctr{\psi},\vctr{\psi}$ are compatible in the sense that there are random variables $X_1,X_2,X_3$ whose distributions are each $\vctr{\psi}$ and such that $X_1+X_2+X_3$ is constant. As each variable has expectation $(p-1)/3$, that constant is certainly $p-1$, so $(X_1,X_2,X_3)$ is a random $T$-valued variable. Its probability distribution is a probability distribution on $T$ whose three projections are each $\vctr{\psi}$. Symmetrizing it, we obtain an $S_3$-symmetric probability distribution on $T$ whose projection under $f$ is $\vctr{\psi}$, as stated in Theorem \ref{DistributionExists}.
We will need to compute:
\begin{lemma} \label{EntropyOfPsi}
With notation as above, $\eta(\vctr{\psi}) = \log \theta$.
\end{lemma}
\begin{proof}
Note that
\[ \vctr{\psi}_k = \frac{\rho^{k-(q-1)/3}}{\theta}. \]
We have
\begin{equation} \label{eq:EntropyOfPsi.1}
\eta(\vctr{\psi}) =- \sum_{k \in I} \vctr{\psi}_k \log \frac{\rho^{k-(q-1)/3}}{\theta} = \left( \sum_{k \in I} \vctr{\psi}_k \right) \log \theta - \left( \sum_{k \in I} (k-(q-1)/3) \vctr{\psi}_k \right) \log \rho.
\end{equation}
The result follows by substituting
\begin{align*}
& \sum_{k \in I} \vctr{\psi}_k = 1 \\
& \sum_{k \in I} (k-(q-1)/3) \vctr{\psi}_k =
\frac{\rho}{\theta} \cdot
\frac{d}{d \beta} \left[
(1 + \beta + \cdots + \beta^{q-1}) \beta^{-(q-1)/3}
\right]_{\beta = \rho} = 0,
\end{align*}
into~\eqref{eq:EntropyOfPsi.1}.
\end{proof}
\begin{remark}
If $\vctr{\pi}$ is any $S_3$-symmetric probability distribution on $T$ then $f_{\ast} (\vctr{\pi})$ has expected value $\tfrac{q-1}{3}$. Of all probability distributions on $I$ with expected value $\tfrac{q-1}{3}$, the distribution $\vctr{\psi}$ has the greatest entropy.
\end{remark}
\section{The construction}
Let $\vctr{\pi}$ be the probability distribution on $T$ guaranteed by Theorem~\ref{DistributionExists}. Fix $n$ divisible by $3$, so that when $S_3$ acts on the lattice $\mathbb{Z}^T$ by permuting the coordinates according to the $S_3$ action on $T$, the fixed point set of the action includes lattice vectors whose coordinates sum up to $n$. We can approximate $\vctr{\pi}$ to within $O_q(1/n)$ by an $S_3$-symmetric distribution $\vctr{\pi'}$ where the probability of each element is an integer multiple of $1/n$; such a $\vctr{\pi}'$ can be found by scaling down $\mathbb{Z}^T$ by $1/n$, taking the set of $S_3$-fixed points that belong to the probability simplex, and selecting the closest such point to $\vctr{\pi}$. Then the marginal distribution $\vctr{\psi'}$ will be within $O_q(1/n)$ of $\vctr{\psi}$. The entropy function of a probability distribution, viewed as function of the vector of the probabilities of the elements, is a differentiable function on the open set of probability distributions assigning positive probability to every element. Thus, because $\vctr{\psi}$ assigns positive probability to each element, the entropy is Lipschitz in a neighborhood of $\vctr{\psi}$. For large enough $n$, $\vctr{\psi'}$ is in that neighborhood, so
\begin{equation} \label{EntropyOfPsiPrime}
\eta(\vctr{\psi'}) = \eta(\vctr{\psi}) - O_q(1/n) = \log \theta - O_q(1/n) .
\end{equation}
(The second equality is~\Cref{EntropyOfPsi}.)
Define the following sets:
\begin{align*}
W &= \{ \vctr{a} \in I^n : \vctr{\sigma}(\vctr{a}) = \vctr{\psi'} \} \\
V &= \{ (\vctr{a}, \vctr{b}, \vctr{c}) \in W^3 : \vctr{a}+\vctr{b}+\vctr{c} = \vctr{t} \} .
\end{align*}
We will show in Lemma~\ref{LowerBound0} that $|V|$ and $|W|$ grow exponentially in $n$, with $|V|$ having the faster growth rate.
Our sum-free set in $C_q^n$ will be a subset of $V$.
Let $p$ be a prime number between $4 |V|/|W|$ and $8 |V|/|W|$ (such a prime exists by Bertrand's postulate). Since $|V|$ grows faster than $|W|$, the prime $p$ goes to $\infty$ as $n$ does. Let $S$ be a subset of $\mathbb{F}_p$ having no three distinct elements in arithmetic
progression.
Behrend's construction~\cite{Behrend}, with Elkin's improvement~\cite{Elkin}, implies that, for $p$ sufficiently large one can
choose such a set whose cardinality is at least $p \cdot e^{-2\sqrt{2 \log 2 \log p}}$.
Let ${h}: \mathbb{Z}^{n+2} \to \mathbb{F}_p$ be a linear map, chosen uniformly at random from all such linear maps.
For any $(\vctr{a},\vctr{b},\vctr{c}) \in V$, the sequence
\[
{h}(0,1,\vctr{a}), \;\; \tfrac12 {h}(1,1,\vctr{t}- \vctr{b}), \;\; {h}(1,0,\vctr{c})
\]
constitutes a (possibly degenerate) arithmetic progression in $\mathbb{F}_p$. Thus,
this arithmetic progression is contained in $S$ if and only if its three terms are
all equal to one another and lie in $S$. Define $V'$ to be the subset of $V$ given by
\[
V' = {\Big \{} (\vctr{a},\vctr{b},\vctr{c}) \in W^3 : \begin{array}{l} \vctr{a}+\vctr{b}+\vctr{c} = \vctr{t} \\ {h}(0,1,\vctr{a}) = \tfrac{1}{2} {h}(1,1,\vctr{t}- \vctr{b}) =
{h}(1,0,\vctr{c}) \in S \end{array} {\Big \}} .
\]
Define
$V''$ to be the set of all $(\vctr{a},\vctr{b},\vctr{c}) \in V'$
such that every other $(\vctr{a}',\vctr{b}',\vctr{c}') \in V'$ obeys
$\vctr{a'} \neq \vctr{a}, \vctr{b'} \neq \vctr{b}, \vctr{c'} \neq \vctr{c}$.
\begin{remark}
For this remark, assume $q$ is odd. Define a tri-colored $3$-AP-free set in $C_q^n$ to be a set of triples $(\vctr{a}_i, \vctr{b}'_i, \vctr{c}_i)$ in $(C_q^n)^3$ such that $\vctr{a}_i + \vctr{c}_k = 2 \vctr{b}'_j$ if and only if $i=j=k$. Replacing $(\vctr{a}_i, \vctr{b}_i, \vctr{c}_i)$ with $(\vctr{a}_i, \tfrac{1}{2}(\vctr{t} - \vctr{b}) \bmod q, \vctr{c}_j)$ turns any tri-colored sum-free set into a tri-colored $3$-AP-free set.
In our set $V''$, each of $\vctr{a}$, $\vctr{b}$ and $\vctr{c}$ has entries distributed over $I$ with probability distribution $\vctr{\psi}$. Therefore in the tri-colored $3$-AP free set, the entries of $\vctr{a}$ and $\vctr{c}$ will be distributed with probability $\vctr{\psi}$, but the entries of $\vctr{b}$ will be distributed with the different distribution $g_{\ast} \vctr{\psi}$ where $g: I \to I$ is the map $g(b) = \tfrac{1}{2} (q-1-b) \bmod q$. By contrast, if $X \subset C_q^n$ is a $3$-AP-free set in the standard sense, then $\{ (\vctr{x}, \vctr{x}, \vctr{x}) : \vctr{x} \in X \}$ is a tri-colored $3$-AP-free set but, for this tri-colored $3$-AP-free set, each of the three components has the same distribution. This discrepancy suggests that it may be hard to lift our constructions out of the colored setting.
\end{remark}
The set $V''$ will be our sum-free set. We verify that it is sum-free in~\Cref{LowerBound1}.
\begin{lemma} \label{PsiExpectation}
For any $\vctr{a} = (a_1,a_2, \ldots, a_n) \in W$, we have $\sum a_i = n(q-1)/3$.
\end{lemma}
\begin{proof}
By definition, $\vctr{\sigma}(\vctr{a})=\vctr{\psi}'$, so we want to show the expected value of the distribution $\vctr{\psi}'$ is $(q-1)/3$. But $\vctr{\psi}'$ is the marginal of the $S_3$ symmetric distribution $\vctr{\pi}'$ on $T$. As $\vctr{\pi'}$ is a symmetric distribution for a triple of random variables summing to $q-1$, the expectation of each variable must be $(q-1)/3$.
\end{proof}
\begin{lemma} \label{LowerBound1}
For any choice of the map ${h}$, the set $V''$ is a sum-free set
with target $\vctr{t}$ in $C_q^n$.
\end{lemma}
\begin{proof}
Suppose that we have three (not necessarily distinct) triples
$(\vctr{a}_i,\vctr{b}_i,\vctr{c}_i) \, (i=0,1,2)$ in $V''$ such that
$\vctr{a}_0 + \vctr{b}_1 + \vctr{c}_2 = \vctr{t}$ in $C_q^n$.
We claim that we also have $\vctr{a}_0 + \vctr{b}_1 + \vctr{c}_2 = \vctr{t}$ in $\mathbb{Z}^n$.
By~\Cref{PsiExpectation}, the entries of $\vctr{a}_0$, $\vctr{b}_1$ and $\vctr{c}_2$ each sum to $n(q-1)/3$ (in $\mathbb{Z}$) so the sum of all the entries of $\vctr{a}_0 + \vctr{b}_1 + \vctr{c}_2$ (with the sum taken in $\mathbb{Z}$) must be $n(q-1)$.
Now the sum $\vctr{a}_0 + \vctr{b}_1 + \vctr{c}_2$ in $\mathbb{Z}^n$ has each entry congruent to $q-1$ mod $q$, by the assumption $\vctr{a}_0 + \vctr{b}_1 + \vctr{c}_2 = \vctr{t}$ in $C_q^n$, and each entry is nonnegative, because the entries of $\vctr{a}_0,\vctr{b}_1,$ and $\vctr{c}_2$ are nonnegative. So each entry is at least $q-1$. We just saw that the sum of all the entries is $n(q-1)$, so each entry is exactly $q-1$, as claimed.
Now that we know $\vctr{a}_0 + \vctr{b}_1 + \vctr{c}_2 = \vctr{t}$ in $\mathbb{Z}^n$, we deduce that
$\left( {h}(0,1,\vctr{a}_0), \tfrac{1}{2} {h}(1,1,\vctr{t}-\vctr{b}_1), {h}(1,0,\vctr{c}_2) \right)$ is an arithmetic progression in $\mathbb{F}_p$.
Since $(\vctr{a}_0,\vctr{b}_0,\vctr{c}_0) \in V'$, we have $\vctr{a}_0 \in W$ and ${h}(0,1,\vctr{a}_0) \in S$. Similarly,
$\vctr{b}_1, \, \vctr{c}_2 \in W$ and $\tfrac{1}{2} {h}(1,1,\vctr{t}-\vctr{b}_1), \, {h}(1,0,\vctr{c}_2) \in S$.
So $\left( {h}(0,1,\vctr{a}_0), \tfrac{1}{2} {h}(1,1,\vctr{t}-\vctr{b}_1), {h}(1,0,\vctr{c}_2) \right)$ is a (possibly degenerate) arithmetic progression in $S$.
As $S$ is arithmetic-progression-free, we must have $ {h}(0,1,\vctr{a}_0)= \tfrac{1}{2} {h}(1,1,\vctr{t}-\vctr{b}_1)= {h}(1,0,\vctr{c}_2) \in S$.
We have now checked that $(\vctr{a}_0, \vctr{b}_1, \vctr{c}_2)$ obeys all the conditions to be an element of $V'$.
Now, recalling the definition of $V''$ and the fact that $(\vctr{a}_i,\vctr{b}_i,\vctr{c}_i)\in V'$ for $i=0$, $1$, $2$,
we may conclude that $(\vctr{a}_i,\vctr{b}_i,\vctr{c}_i) = (\vctr{a}_0, \vctr{b}_1, \vctr{c}_2)$ for
$i=0$, $1$, $2$. In other words, the three triples $(\vctr{a}_0,\vctr{b}_0,\vctr{c}_0)$,
$(\vctr{a}_1,\vctr{b}_1,\vctr{c}_1)$ and $(\vctr{a}_2,\vctr{b}_2,\vctr{c}_2)$ are
all equal to one another.
\end{proof}
We will now begin to estimate the expected value of $|V''|$.
\begin{lemma} \label{LowerBound0}
We have
\[
|V| \; \geq \; \exp(\eta(\vctr{\pi}') n - O_q(\log n))
\]
and
\[
\exp(\eta(\vctr{\psi'}) n) \; \geq \; |W|
\; \geq \; \exp(\eta(\vctr{\psi}')n - O_q(\log n)) .
\]
\end{lemma}
Since $\vctr{\psi}' = f_{\ast} \vctr{\pi}'$, we have $\eta(\vctr{\pi}') \geq \eta(\vctr{\psi}')$. Moreover, if $n$ is large enough that the distribution $\vctr{\pi}'$ is not a point-mass on $(\frac{q-1}{3},\frac{q-1}{3},\frac{q-1}{3})$, then we have strict inequality since $\vctr{\pi}'$ is $S_3$-symmetric, so $\vctr{\pi}'_{ijk}>0$ implies $\vctr{\pi}'_{jik}>0$. This establishes the previous claim that $|V|$ and $|W|$ grow exponentially, with $|V|$ having the faster rate.
\begin{proof}
Since $W= \{ \vctr{e} \in I^n : \vctr{\sigma}(\vctr{e}) = \vctr{\psi'} \} $, the lower and upper bounds for $|W|$ follow from \Cref{Histogram}.
We now need to establish the lower bound for $V$.
Let $V_0 = \{ \vctr{f} \in T^n : \vctr{\sigma}(\vctr{f}) = \vctr{\pi'} \}$. An element of $T^n$ is an $n$-tuple of triples of integers $((a_1, b_1, c_1), (a_2, b_2, c_2), \ldots, (a_n, b_n, c_n))$ with $a_i+b_i+c_i = q-1$. Reorganizing these integers as $((a_1, a_2, \ldots, a_n), (b_1, b_2, \ldots, b_n), (c_1, c_2, \ldots, c_n))$, we obtain a triple of length $n$ vectors $\vctr{a}$, $\vctr{b}$ and $\vctr{c}$ with $\vctr{a}+\vctr{b}+\vctr{c}= \vctr{t}$. Let us apply this construction to some $\vctr{f}$ in $V_0$ to get some $\vctr{a}$, $\vctr{b}$ and $\vctr{c}$. Since $\vctr{\pi}'$ is $S_3$ symmetric, we have $\vctr{\sigma}(\vctr{a}) = \vctr{\sigma}(\vctr{b}) = \vctr{\sigma}(\vctr{c}) = \vctr{\psi}'$ so $\vctr{a}$, $\vctr{b}$ and $\vctr{c}$ lie in $W$ and $(\vctr{a}, \vctr{b}, \vctr{c}) \in V$. This construction gives an injection from $V_0$ into $V$, so $|V| \geq |V_0|$.
By~\Cref{Histogram}, $|V_0| = \exp(\eta(\vctr{\pi}') n - O_q(\log n))$, so $|V| \geq \exp(\eta(\vctr{\pi}') n - O_q(\log n))$ as desired.
\end{proof}
\begin{lemma} \label{LinIndep}
Suppose $p > q$.
For any two distinct elements $(\vctr{a},\vctr{b},\vctr{c})$,
$(\vctr{a}',\vctr{b}',\vctr{c}') \in V$, the $(n+2) \times 6$-matrix over
$\mathbb{F}_p$ given by
\[
M = \begin{pmatrix}
0 & 0 & 1/2 & 1/2 & 1 & 1 \\
1 & 1 & 1/2 & 1/2 & 0 & 0 \\
\vctr{a} & \vctr{a}' & (\vctr{t}-\vctr{b})/2 & (\vctr{t}- \vctr{b}')/2 &
\vctr{c} &\vctr{c}' \\
\end{pmatrix}
\]
has rank at least $3$.
\end{lemma}
\begin{proof}
The first two rows already have rank $2$, so we simply must show that the bottom $n$ rows are not all in the span of the first two.
If the bottom $n$ rows were in the span of the first two, then modulo $p$ the first column would equal the second, the third column equal the fourth, and the fifth column equal the sixth. Since the entries of the matrix are between $0$ and $q-1$, and $p > q$, equality of columns modulo $p$ implies outright equality. This gives $\vctr{a} = \vctr{a}'$, $\vctr{b} = \vctr{b}'$ and $\vctr{c}=\vctr{c}'$, contrary to our assumption that $(\vctr{a},\vctr{b},\vctr{c})$ and
$(\vctr{a}',\vctr{b}',\vctr{c}')$ are distinct.
\end{proof}
\begin{lemma} \label{LowerBound2}
When $p >q $ and ${h}$ is a uniformly random homomorphism of $\mathbb{Z}^{n+2}$ to $\mathbb{F}_p$,
the expected cardinality of $V''$ is at least $\frac{1}{32}e^{-2\sqrt{2 \log 2 \log p}} \cdot |W|$.
\end{lemma}
\begin{proof}
For any $(\vctr{a},\vctr{b},\vctr{c}) \in V$, we want to compute the probability that
there exists $s \in S$ such that
\begin{equation} \label{eq:LB2.1}
{h}(0,1,\vctr{a}) = \tfrac{1}{2} {h}(1,1, \vctr{t}-\vctr{b}) = {h}(1,0,\vctr{c}) = s.
\end{equation}
Furthermore, since ${h}(0,1,\vctr{a}), \, \frac12 {h}(1,1, \vctr{t}-\vctr{b}), \,
{h}(1,0,\vctr{c})$ always form a (possibly degenerate) arithmetic
progression, if any two of these values are equal to $s$ then the third one
equals $s$ as well. The vectors $(0,1,\vctr{a})$ and $(1,0,\vctr{c})$ are
linearly independent modulo $p$, so the pair
$({h}(0,1,\vctr{a}), {h}(1,0,\vctr{c}))$ is
uniformly distributed in $\mathbb{F}_p^2$ and the probability
that~\eqref{eq:LB2.1} is satisfied for a fixed $s \in S$ is $p^{-2}$. Summing
over all $(\vctr{a},\vctr{b},\vctr{c}) \in V$ and $s \in S$ we obtain
\begin{equation} \label{eq:LB2.2}
\mathbb{E} ( |V'| ) = \frac{|V| |S| }{ p^2}.
\end{equation}
An element $(\vctr{a},\vctr{b},\vctr{c}) \in V'$ belongs to $V''$ unless
there exists some other $(\vctr{a}', \vctr{b}', \vctr{c}') \in V'$ such that
one of the equations $\vctr{a}=\vctr{a}', \, \vctr{b}=\vctr{b}'$, or
$\vctr{c}=\vctr{c}'$ holds. In order for any such equation to hold, it
must be the case that there is a single element $s \in S$ such that
\begin{equation} \label{eq:LB2.3}
s = {h}(0,1,\vctr{a}) = {h}(0,1,\vctr{a}') =
\tfrac12 {h}(1,1,\vctr{t}-\vctr{b}) = \tfrac12 {h}(1,1,\vctr{t}-\vctr{b}') =
{h}(1,0,\vctr{c}) = {h}(1,0,\vctr{c}').
\end{equation}
By \Cref{LinIndep}, the six-tuple $({h}(0,1,\vctr{a}), \, {h}(0,1,\vctr{a}'), \,
\tfrac12 {h}(1,1,\vctr{t}-\vctr{b}) , \, \tfrac12 {h}(1,1,\vctr{t}-\vctr{b}') , \,
{h}(1,0,\vctr{c}) , \, {h}(1,0,\vctr{c}'))$
is uniformly distributed on a subspace of $\mathbb{F}_p^6$ of dimension
at least 3.
Hence, for any $(\vctr{a},\vctr{b},\vctr{c}),(\vctr{a}',\vctr{b}',\vctr{c}')\in V$ and for a fixed $s$, the probability that~\eqref{eq:LB2.3} holds is
at most $p^{-3}$.
The probability that there exists some $s$ for which~\eqref{eq:LB2.3} holds is thus bounded by $|S| p^{-3}$.
For any $(\vctr{a},\vctr{b},\vctr{c}) \in V$, the number of elements $(\vctr{a}',\vctr{b}',\vctr{c}') \in V$ such that $\vctr{a}' = \vctr{a}$ is equal to $|V|/|W|$.
(To see this, note that the group $S_n$ acts on $V$ and $W$ by
permuting the coordinates of vectors. These actions are compatible with
the projection map $V \to W$ defined by
$(\vctr{a},\vctr{b},\vctr{c}) \mapsto \vctr{a}$.
The fibers of this projection map must be equinumerous
because the action of $S_n$ on $W$ is transitive.)
Thus, for any $(\vctr{a},\vctr{b},\vctr{c}) \in V$
the probability that $(\vctr{a},\vctr{b},\vctr{c})$ belongs
to $V'$ but not $V''$
because it ``collides'' with another ordered triple of the form $(\vctr{a},\vctr{b}',\vctr{c}')$
in $V'$ is bounded above by $\frac{|V|}{|W|} |S| p^{-3}$.
The analogous counting argument applies to collisions with triples of the form
$(\vctr{a}',\vctr{b},\vctr{c}')$ and $(\vctr{a}',\vctr{b}',\vctr{c})$.
Summing over $|V|$ choices of $(\vctr{a},\vctr{b},\vctr{c})$,
we find that the expected cardinality
of $V' \setminus V''$ is bounded above by
\[
3 |V| \frac{|V|}{|W|} |S| p^{-3} =
\frac{3 |V|}{p|W|} \cdot \frac{|V| |S|}{p^2} <
\frac{3}{4} \cdot \mathbb{E} ( |V'| ).
\]
Thus,
\[
\mathbb{E}( |V''|) \geq \frac14 \mathbb{E} (|V'|) = \frac{|V| |S|}{4 p^2} = \frac14 \cdot \frac{|V|}{p} \cdot \frac{|S|}{p}
> \frac{e^{-2\sqrt{2 \log 2 \log p}}}{32} \cdot |W|.
\]
\end{proof}
We now prove our main theorem.
\begin{theorem}
If $n$ is sufficiently large then
there exists a sum-free set in $C_q^n$ with target $\vctr{t}$
whose size is greater than $\theta^n e^{ - 2 \sqrt{2\log 2 \log \theta \ n } - O_q(\log n)}$.
\end{theorem}
\begin{proof}
The random set $V''$ constructed above is a sum-free set in $C_q^n$
with target $\vctr{t}$ (\Cref{LowerBound1}) and its expected size
is greater than $\frac{1}{32} e^{-2\sqrt{2 \log 2 \log p}}\cdot |W|$ (\Cref{LowerBound2}), because we may take $n$ large enough that $p>q$.
Using \Cref{LowerBound0} we have
\[
|W| \geq \exp(\eta(\vctr{\psi}') \, n - O_q(\log n) ) \geq \exp((\log \theta - O_q(1/n)) \, n - O_q(\log n)) \geq \theta^n \exp( - O_q(\log n))
\]
for all sufficiently large $n$.
The inequality $|V| \leq |W|^2$ holds because
the projection map $V \to W^2$ defined by
$(\vctr{a},\vctr{b},\vctr{c}) \mapsto (\vctr{a},\vctr{b})$
is one-to-one. This justifies the second inequality in
\[
p < 8 \frac{|V|}{|W|} \leq 8 |W| < 8 \exp(\eta(\vctr{\psi}') \, n)
\leq 8 \exp ( (\log \theta + O_q(1/n) ) \, n ),
\]
while the third inequality follows from \Cref{LowerBound0}.
Taking logarithms of both sides, we deduce that
$\log p < n \log \theta + O_q(1)$, and hence
\[
e^{- 2 \sqrt{2 \log 2 \log p} }> e^{- 2 \sqrt{2 \log 2 (n \log \theta + O_q(1))} } > e^{- 2 \sqrt{2 \log 2 \log \theta \ n} -O_q(1/\sqrt{n})} .
\]
Hence,
\[
\mathbb{E} (|V''|) > \frac{1}{32} e^{- 2 \sqrt{2 \log 2 \log \theta \ n} -O_q(1/\sqrt{n})}
\cdot |W|
\geq \theta^n e^{ - 2 \sqrt{2 \log 2 \log \theta \ n} - O_q(\log n)}
\]
for sufficiently large $n$. The theorem follows because there must exist at least one choice of
${h}$ for which the cardinality of the random set $V''$ is at least as large as its expected value.
\end{proof}
It follows from Roth's theorem that our construction produces sum-free sets $V'' \subseteq V$ of size $\mathbb{E}(|V''|) \leq\mathbb{E}(V') = \frac{V |S|}{p^2}=o(|W|)$ regardless of how we choose $S$. We do not know if an arbitrary sum-free set contained in $V$ must have size $o(|W|)$, only the trivial bound $ |W|$. It would be interesting to improve this situation.
\begin{dajauthors}
\begin{authorinfo}[rk]
Robert Kleinberg \\
Department of Computer Science \\ Cornell University \\ Ithaca, NY 14853, USA \\
robert\imagedot{}kleinberg\imageat{}cornell\imagedot{}edu \\
\end{authorinfo}
\begin{authorinfo}[ws]
Will Sawin \\
ETH Institute for Theoretical Studies \\ ETH Zurich \\ 8092 Z\"{u}rich, Switzerland \\
william\imagedot{}sawin\imageat{}math\imagedot{}ethz\imagedot{ch} \\
\end{authorinfo}
\begin{authorinfo}[ds]
David E Speyer \\
Department of Mathematics \\ University of Michigan \\ Ann Arbor, MI 48109, USA \\
speyer\imageat{}umich\imagedot{}edu \\
\end{authorinfo}
\end{dajauthors}
\end{document} | math |
Sidney Carlow [Parents] was born in 1869 in Birmingham, Warwickshire, England. He died in 1900 in Birmingham, Warwickshire, England. He married Annie Hardle on 20 Dec 1891 in St Davids, Birmingham, Warwickshire, England.
Annie Hardle [Parents] was born in 1870 in Birmingham, Warwickshire, England. She died in 1944 in Birmingham, Warwickshire, England. She married Sidney Carlow on 20 Dec 1891 in St Davids, Birmingham, Warwickshire, England.
M ii Henry Carlow was born in 1895 in Birmingham, Warwickshire, England.
M iv Edward Carlow was born in 1899 in Birmingham, Warwickshire, England.
Henry Hardle was born in 1844 in Restbury, Gloucestershire, England.
M ii Thomas H. Hardle was born in 1874 in Birmingham, Warwickshire, England.
Sidney Thomas Carlow [Parents] was born in 1893 in Birmingham, Warwickshire, England. He died on 10 May 1951 in Birmingham, Warwickshire, England. He married Amy Pamela Foster in 1921 in Birmingham, Warwickshire, England.
Amy Pamela Foster was born on 13 Jun 1903 in Birmingham, Warwickshire, England. She died in 1975 in Birmingham, Warwickshire, England. She married Sidney Thomas Carlow in 1921 in Birmingham, Warwickshire, England.
Samuel Nathaniel Skipp [Parents] was born in 1871 in Birmingham, Warwickshire, England. He died in 1944 in Birmingham, Warwickshire, England. He married Annie Hardle on 5 Apr 1906 in Bishop Ryder, Birmingham, Warwickshire,England.
Annie Hardle [Parents] was born in 1870 in Birmingham, Warwickshire, England. She died in 1944 in Birmingham, Warwickshire, England. She married Samuel Nathaniel Skipp on 5 Apr 1906 in Bishop Ryder, Birmingham, Warwickshire,England.
M i Samuel Skipp was born in 1907 in Birmingham, Warwickshire, England.
M ii Nathaniel Skipp was born in 1908 in Birmingham, Warwickshire, England.
William Lester was born in 1807 in Wolstan, Warwickshire, England. He married Maria about 1832.
Maria was born in 1808 in Coventry, Warwickshire, England. She married William Lester about 1832.
M i William Lester was born in 1834 in Coventry, Warwickshire, England.
M ii John Lester was born in 1837 in Coventry, Warwickshire, England.
F iii Catherine Lester was born in 1838 in Coventry, Warwickshire, England.
F iv Phoebe Lester was born in 1841 in Coventry, Warwickshire, England.
F vi Maria Lester was born in 1845 in Birmingham, Warwickshire, England.
M vii George Lester was born in 1847 in Birmingham, Warwickshire, England.
F viii Mary Ann Lester was born in 1849 in Birmingham, Warwickshire, England.
M ix Samuel Lester was born in 1849 in Birmingham, Warwickshire, England.
Joseph William Wootton [Parents] was born in 1888 in Birmingham, Warwickshire, England. He married Amelia Maria Hardiker on 7 Mar 1909 in St Saviours, Birmingham, Warwickshire, England.
Amelia Maria Hardiker [Parents] was born in Birmingham, Warwickshire, England. She was christened on 4 May 1890 in Birmingham, Warwickshire, England. She married Joseph William Wootton on 7 Mar 1909 in St Saviours, Birmingham, Warwickshire, England.
M i William Wootton was born in 1909 in Birmingham, Warwickshire, England. | english |
अमरनाथ यात्रा की तैयारी से महबूबा मुफ्ती को परेशानी जताई नाराज़गी
अमरनाथ यात्रा सालों से होती आ रही है,लेकिन दुर्भाग्य से जो इंतजाम इस साल किए गए हैं, वह कश्मीर के लोगों के खिलाफ हैं-महबूबा मुफ्ती
पीडीपी नेता महबूबा मुफ्ती ने अमरनाथ यात्रा के लिए सरकार द्वारा किए गए इंतजामों पर नाराजगी जताई है और कहा है कि ये कश्मीर के लोगों के खिलाफ हैं। उन्होंने कहा, अमरनाथ यात्रा सालों से होती आ रही है। लेकिन दुर्भाग्य से जो इंतजाम इस साल किए गए हैं, वह कश्मीर के लोगों के खिलाफ हैं। इससे स्थानीय लोगों को अपनी रोजमर्रा की जिंदगी में परेशानी का सामना करना पड़ा रहा है। मैं राज्यपाल से इस मामले पर संज्ञान लेने के लिए अनुरोध करती हूं।
घाटी के पहलगाम और बालटाल आधार शिविरों में अमरनाथ यात्रियों की संख्या ज्यादा हो जाने के कारण जम्मू के भगवती नगर आधार शिविर से जाने वाला यात्रियों का जत्था सोमवार को यहां से नहीं भेजा जाएगा। हालांकि देर रात तक इस बारे में कोई आधिकारिक सूचना जारी नहीं की गई थी।
ये भी पढ़े:खेत में मिला युवक का शव,लाश के पास मिला तमंचा और पांच कारतूस
पुलिस और प्रशासनिक सूत्रों के अनुसार, भगवती नगर आधार शिविर से प्रतिदिन तीन से चार हजार अमरनाथ यात्री विभिन्न जत्थों में रवाना किए जाते हैं। इसके अलावा बड़ी संख्या में श्रद्धालु निजी तौर पर सीधे घाटी पहुंच जाते हैं। इससे घाटी के आधार शिविरों में श्रद्धालुओं की संख्या अचानक काफी बढ़ जाती है। ऐसी स्थिति में मौसम अथवा कानून व्यवस्था बिगड़ने पर यात्रियों की सुरक्षा एक बड़ी चुनौती हो जाती है। हालांकि आधिकारिक रूप से कोई खुलकर बोलने को तैयार नहीं है। परंतु यात्रा व्यवस्था से जुड़े सूत्र इस बात की पुष्टि कर रहे हैं। | hindi |
أمِس جاناوارَس وُچھِتھ سَمَے ؤلۍ ؤلی اَتِکۍ واریٚہہ بٔسکٟن اَتہِ أنٛدۍ أنٛدۍ تہٕ لٲگِکھ أمِس واریَہہ سوال پرٛژھٕنؠ | kashmiri |
#------------------------------------------------------------------------------
#
# Copyright (c) 2007, Enthought, Inc.
# All rights reserved.
#
# This software is provided without warranty under the terms of the BSD
# license included in /LICENSE.txt and may be redistributed only
# under the conditions described in the aforementioned license. The license
# is also available online at http://www.enthought.com/licenses/BSD.txt
#
# Thanks for using Enthought open source!
#
#------------------------------------------------------------------------------
""" Test whether HasTraits objects with cycles can be garbage collected.
"""
from __future__ import absolute_import
import gc
import time
from traits.testing.unittest_tools import unittest
# Enthought library imports
from ..api import HasTraits, Any, DelegatesTo, Instance, Int
class TestCase(unittest.TestCase):
def _simple_cycle_helper(self, foo_class):
""" Can the garbage collector clean up a cycle with traits objects?
"""
# Create two Foo objects that refer to each other.
first = foo_class()
second = foo_class(child=first)
first.child = second
# get their ids
foo_ids = [id(first), id(second)]
# delete the items so that they can be garbage collected
del first, second
# tell the garbage collector to pick up the litter.
gc.collect()
# Now grab all objects in the process and ask for their ids
all_ids = [id(obj) for obj in gc.get_objects()]
# Ensure that neither of the Foo object ids are in this list
for foo_id in foo_ids:
self.assertTrue(foo_id not in all_ids)
def test_simple_cycle_oldstyle_class(self):
""" Can the garbage collector clean up a cycle with old style class?
"""
class Foo:
def __init__(self, child=None):
self.child = child
self._simple_cycle_helper(Foo)
def test_simple_cycle_newstyle_class(self):
""" Can the garbage collector clean up a cycle with new style class?
"""
class Foo(object):
def __init__(self, child=None):
self.child = child
self._simple_cycle_helper(Foo)
def test_simple_cycle_hastraits(self):
""" Can the garbage collector clean up a cycle with traits objects?
"""
class Foo(HasTraits):
child = Any
self._simple_cycle_helper(Foo)
def test_reference_to_trait_dict(self):
""" Does a HasTraits object refer to its __dict__ object?
This test may point to why the previous one fails. Even if it
doesn't, the functionality is needed for detecting problems
with memory in debug.memory_tracker
"""
class Foo(HasTraits):
child = Any
foo = Foo()
# It seems like foo sometimes has not finished construction yet, so
# the frame found by referrers is not _exactly_ the same as Foo(). For
# more information, see the gc doc: http://docs.python.org/lib/module-
# gc.html
#
# The documentation says that this (get_referrers) should be used for
# no purpose other than debugging, so this is really not a good way to
# test the code.
time.sleep(0.1)
referrers = gc.get_referrers(foo.__dict__)
self.assertTrue(len(referrers) > 0)
self.assertTrue(foo in referrers)
def test_delegates_to(self):
""" Tests if an object that delegates to another is freed.
"""
class Base(HasTraits):
""" Object we are delegating to. """
i = Int
class Delegates(HasTraits):
""" Object that delegates. """
b = Instance(Base)
i = DelegatesTo('b')
# Make a pair of object
b = Base()
d = Delegates(b=b)
# Delete d and thoroughly collect garbage
del d
for i in range(3):
gc.collect(2)
# See if we still have a Delegates
ds = [obj for obj in gc.get_objects() if isinstance(obj, Delegates)]
self.assertEqual(ds, [])
if __name__ == '__main__':
unittest.main()
| code |
आकाश विजयवर्गीय पर बोले मोदी, 'ऐसे लोगों को पार्टी से निकाल देना चाहिए' - इना न्यूज
होम / नई दिल्ली / राजनीती / आकाश विजयवर्गीय पर बोले मोदी, 'ऐसे लोगों को पार्टी से निकाल देना चाहिए'
आकाश विजयवर्गीय पर बोले मोदी, 'ऐसे लोगों को पार्टी से निकाल देना चाहिए'
नई दिल्ली, २ जुलाई- प्रधानमंत्री नरेंद्र मोदी ने एक सरकारी कर्मचारी पर बल्ले से हमला करने को लेकर पार्टी नेता कैलाश विजयवर्गीय के बेटे आकाश विजयवर्गीय की निंदा करते हुए मंगलवार को कहा, "बेटा किसी का भी हो, ऐसे लोगों को पार्टी से निकाल देना चाहिए। मोदी ने यह टिप्पणी संसद में भाजपा संसदीय दल की बैठक के दौरान की।
मोदी ने कहा, "हम ऐसा कोई नेता नहीं चाहते जो पार्टी की छवि को खराब करे। बेटा किसी का भी हो, ऐसे नेताओं को पार्टी से निकाल देना चाहिए।मोदी इंदौर के एक भाजपा विधायक आकाश विजयवर्गीय का जिक्र कर रहे थे, जिन्होंने २६ जून को नगर निगम के एक अधिकारी पर मकान गिराने के मामले में हमला किया था।
मोदी ने जेल से छूटने के बाद आकाश विजयवर्गीय का जोरदार स्वागत करने को लेकर भी पार्टी नेताओं की आलोचना की और कहा, "जिन्होंने उनका स्वागत किया, ऐसे नेताओं को भी पार्टी से बर्खास्त किया जाना चाहिए।बल्ले से पीटने के मामले में आकाश विजयवर्गीय को गिरफ्तार कर लिया गया था और बाद में उन्हें जमानत दे दी गई।
नई दिल्ली राजनीती | hindi |
बिहार के बाद अब इस राज्य में होगी पूर्ण शराबबंदी! | डन न्यूज न्यूज़
होम न्यूज़ बिहार के बाद अब इस राज्य में होगी पूर्ण शराबबंदी!
बिहार के बाद अब इस राज्य में होगी पूर्ण शराबबंदी!
पटना.न्यूज़डेस्क. नीतीश सरकार द्वारा बिहार में पूर्ण शराबबंदी के बाद कई अन्य राज्यों में भी पूर्ण शराबबंदी की मांग उठने लगी है. उत्तर प्रदेश, राजस्थान, झारखंड आदि राज्यों में शराब बंद करने की मांग उठ रही है. मांग करने वालों में महिलाएं शामिल है.
नहीं आएगा बिहार में तूफान मगर मौसम विभाग ने कहा, बिहार में इस बार की गर्मी होगी
झारखंड की गुलबी गैंग की महिलाओं ने राजभवन के सामने मांग की कि झारखंड में भी पूर्ण शराबबंदी लागू होनी चाहिए. विशेष कर ग्रामीण क्षेत्र की महिलाएं इस पर काफी जोर दे रही है कि झारखंड में भी शराब बंद होनी चाहिए.
ऐसा नही है कि महिलाओं के इस मांग की आवाज सिर्फ सड़कों तक सीमित है. झारखंड की राजनीतिक गलियारों तक भी उनकी आवाज पहुंची है. इस मामने पर बीजेपी के प्रदेश अध्यक्ष ने कहा कि शराबंदी को लेकर उनकी सकारात्मक है. सरकार उचित समय आने पर इसके लिए सही कदम उठाया जाएगा.
बीजेपी के इस बयान के बाद यह संभावना जताई जा रही है कि बिहार के बाद अब झारखंड में भी पूर्ण शराबबंदी लागू होगी. सरकार ने भी साफ कर दिया है कि वह सही समय का इंतजार कर रही है.
लालू परिवार पर इट की कार्रवाई के बाद नित्यानंद राय का आया अजीबोगरीब बयान
बिहार में मचे सियासी घमासान के बीच लालू और नीतीश पर बरस पड़े जीतन राम मांझी
कांवड़ियों के नारों से भड़की धार्मिक हिंसा, १२ से ज्यादा घायल | hindi |
Usually the speaker is referring to an amount significant enough to affect a person’s decision to buy something or participate in an activity. This is similar to a texting abbreviation, and is very informal. The use of three dollar signs serves two purposes: to make sure it isn't a typo and there should be an amount written next to it, and to emphasize a large expense.
Etymology : From the dollar sign $ which we use to symbolize the dollar (at least in the United States).
This is similar to a texting abbreviation, and is very informal. The use of three dollar signs serves two purposes: to make sure it isn't a typo and there should be an amount written next to it, and to emphasize a large expense. | english |
मैरी कॉम ने जीता एशियाई मुक्केबाज़ी चैंपियनशिप में सोना | हल्लाबोल तोडे
होम खेल मैरी कॉम ने जीता एशियाई मुक्केबाज़ी चैंपियनशिप में सोना
मैरी कॉम ने जीता एशियाई मुक्केबाज़ी चैंपियनशिप में सोना
भारतीय मुक्केबाजी की वंडर गर्ल एम सी मैरी कॉम ( ४८ किलो ) ने एशियाई मुक्केबाजी में पांचवीं बार गोल्ड मेडल अपने नाम कर लिया। पांच बार की विश्व चैंपियन और ओलिंपिक ब्रांज मेडल विजेता मैरी कॉम ने उत्तर कोरिया की किम ह्यांग मि को ५-० से हराया।
यह २०१४ एशियाई खेलों के बाद मैरी कॉम का पहला अंतरराष्ट्रीय गोल्ड मेडल है और एक साल में उनका पहला मेडल है। ३५ बरस की मैरी कॉम का सामना किम ह्यांग मि के रूप में सबसे आक्रामक प्रतिद्वंद्वी से था, लेकिन वह इस चुनौती के लिए तैयार थीं। अब तक पहले तीन मिनट एक दूसरे को आंकने में जाते रहे थे, लेकिन इस मुकाबले में शुरूआती पलों से ही खेल आक्रामक रहा।
मैरी कॉम ने अपनी प्रतिद्वंद्वी के हर वार का माकूल जवाब दिया, दोनों ओर से तेज पंच लगाए गए। मैरी कॉम उनके किसी भी वार से विचलित नहीं हुईं और पूरे सब्र के साथ खेलते हुए जीत दर्ज की।
बॉक्सिंग फेडरेशन ऑफ इंडिया की और से जारी विज्ञाप्ति में अध्यक्ष अजय सिंह ने इस जीत पर मैरी कॉम की तारीफ करते हुए कहा, ३५ साल की उम्र में तीन बच्चों की मां होने के बावजूद मैरी कॉम की यह उपलब्धि शानदार है। उन्होंने खुद को साबित किया है। भारतीय टीम में १० बॉक्सरों में से ७ बॉक्सर मेडल के सात स्वदेश लौटेंगे, यह वाकई बड़ी कामयाबी है। मैं इसके लिए टीम कोच को और स्पोर्टिंग स्टॉफ को भी बधाई देता हूं।
एशियाई मुक्केबाज़ी चैंपियनशिप
प्रेवियस आर्टियलफिटनेस का बेस्ट फंडा चाहिए तो हर्षित गुप्ता से ज़रुर मिलिए
नेक्स्ट आर्टियलपुलिस आपकी दोस्त क्यों नहीं? | hindi |
\begin{document}
\title[]{
Iterative methods for $k$-Hessian equations
}
\author{Gerard Awanou}
\address{Department of Mathematics, Statistics, and Computer Science, M/C 249. University of Illinois at Chicago, Chicago, IL 60607-7045, USA}
\maketitle
\begin{abstract}
On a domain of the $n$-dimensional Euclidean space, and for an integer $k=1,\ldots,n$, the $k$-Hessian equations are fully nonlinear elliptic equations for $k >1$ and consist of the Poisson equation for $k=1$ and the Monge-Amp\`ere equation for $k=n$. We analyze for smooth non degenerate solutions a 9-point finite difference scheme. We prove that the discrete scheme has a locally unique solution with a quadratic convergence rate.
In addition we propose new iterative methods which are numerically shown to work for non smooth solutions. A connection of the latter with a popular Gauss-Seidel method for the Monge-Amp\`ere equation is established and new Gauss-Seidel type iterative methods for $2$-Hessian equations are introduced.
\end{abstract}
\section{Introduction}
Let $\Omega$ be a bounded, connected open subset of $\mathbb{R}^n, n\geq 2$ with boundary denoted $\partial \Omega$.
Let $u \in C^2(\Omega)$ and for $x \in \Omega$, let $D^2 u(x)=\bigg( (\partial^2 u(x))(\partial x_i \partial x_j)\bigg)_{i,j=1,\ldots, n} $
denote its Hessian. We denote the eigenvalues of $D^2 u(x)$ by $\lambda_i(x), i=1,\ldots, n$.
For $1 \leq k \leq n$, the $k$-Hessian operator is defined as
\begin{align*}
S_k(D^2 u) = \sum_{i_1 < \cdots < i_k} \lambda_{i_1} \cdots \lambda_{i_k}.
\end{align*}
We note that $S_1(D^2 u) = \Delta u$ is the Laplacian operator and $S_n(D^2 u) = \det D^2 u$ is the Monge-Amp\`ere operator. For $k \geq 2$, we are interested in the numerical approximation of
solutions of the Dirichlet problem for the $k$-Hessian equation
\begin{equation}
S_k(D^2 u) = f \, \text{in} \, \Omega, u=g \, \text{on} \, \partial \Omega, \label{k-H1}
\end{equation}
with $f$ and $g$ given and $f \geq 0$.
\subsection{Local existence, uniqueness and quadratic convergence rate for a finite difference discretization}
Let $u^0$ be a sufficiently close initial guess to the smooth solution $u$ of \eqref{k-H1}. Consider the iterative method
\begin{align} \label{broyden}
\begin{split}
\operatorname{div} \bigg( \{S_k^{ij}(D^2 u^0) \} D u^{m+1} \bigg)& = \operatorname{div} \bigg( \{S_k^{ij}(D^2 u^0) \} D u^{m} \bigg) +f-S_k (D^2 u^m) \, \text{in} \, \Omega \\
u^{m+1} & = g \, \text{on} \, \partial \Omega,
\end{split}
\end{align}
where $ \{S_k^{ij}(D^2 u^0) \}$ is a matrix which generalizes the cofactor matrix of $D^2 u^0$.
We prove the convergence of \eqref{broyden} at the continuous level in H$\ddot{\text{o}}$lder spaces. A discrete version of \eqref{broyden} is also shown to converge to a solution of a 9-point stencil discretization of \eqref{k-H1}. This establishes the local existence and uniqueness of a discrete solution. In addition the convergence rate of the discretization is shown to be quadratic.
It is reasonable to expect that the discrete version of the iterative method \eqref{broyden} will retrieve the correct solution when it is smooth and non degenerate. As with Newton's method it is not effective for non smooth and degenerate solutions. For these, we advocate iterative methods like the subharmonicity preserving iterations described below. The discrete version of \eqref{broyden} is used in this paper to prove the local solvability of the 9-point scheme when $u$ is smooth and non degenerate. These results form a building block of a theory which explains why standard discretizations work for non smooth solutions \cite{Awanou-Std-fd-jsc}. In addition results for smooth solutions are also needed for the analysis of hybrid schemes where the 9 point scheme is used in part of the region where the solution is smooth and a monotone scheme elsewhere \cite{AwanouHybrid}.
\subsection{Newton's method}
If one is only interested in smooth solutions, Newton's method is the most appropriate method.
We analyze the convergence of Newton's method for solving \eqref{k-H1} when it has a smooth solution.
\subsection{Numerical work for subharmonicity preserving iterations}
A smooth function $u$ is said to be $k$-convex if $S_l (D^2 u) \geq 0, 1 \leq l \leq k$. Convexity of a function can be shown to be equivalent to $n$-convexity, Lemma \ref{n-convexity}.
It is of interest in some applications to be able to handle \eqref{k-H1} when it has a non smooth $k$-convex solution. It has only been recently understood, c.f. \cite{Awanou-Std-fd-jsc} for the Monge-Amp\`ere equation, that what is needed is a numerical method provably convergent for smooth solutions and numerically robust to handle non smooth solutions. The approach in \cite{Awanou-Std-fd-jsc} is to regularize the data and use approximation by smooth functions. The key to numerically handle non smooth solutions of \eqref{k-H1} is to preserve $k$-convexity in the iterations. For discrete $k$-convexity we simply require discrete analogues of the condition $S_l (D^2 u) \geq 0$ with a natural discretization of $D^2 u$. We refer to \cite{Aguilera2008} where this approach was first used for the discretization of $n$-convexity.
Consider the iterative method
\begin{align}
\begin{split}
\Delta u^{m+1} & = \bigg( (\Delta u^{m})^k + \frac{1}{c(k,n)}(f-S_k (D^2 u^m)) \bigg)^{\frac{1}{k}} \, \text{in} \, \Omega,
u^{m+1} = g \, \text{on} \, \partial \Omega, \label{k-H-iterative}
\end{split}
\end{align}
with
$c(k,n) = \binom{n}{k}/n^k$.
If $D^2 u$ has positive eigenvalues, we have the inequality
\begin{equation} \label{G-am}
S_k(D^2 u) \leq c(k,n) (\Delta u)^{k},
\end{equation}
which follows from the Maclaurin inequalities,
\cite[Proposition 1.1 (v i)]{Gavitone2009}.
For $k=2$, \eqref{G-am} also holds with {\it no convexity assumption} on $u$, \cite[Lemma 15.11]{Lieberman96}. Explicitly
$c(2,3)=1/3$. Also, $c(n,n)=1/n^n$ which gives
$$
\det D^2 u \leq \frac{1}{n^n} (\Delta u)^n,
$$
a direct consequence of the arithmetic mean - geometric mean inequality.
If one starts with an initial guess $u^0$ such that $\Delta u^0 \geq 0$, \eqref{k-H-iterative} enforces $\Delta u^m \geq 0$ for all $m$. Indeed recall that $f \geq 0$ and assume that $\Delta u^m \geq 0$. Then by \eqref{G-am}
$1/c(k,n) S_k (D^2 u^m) \leq (\Delta u^{m})^k$, and using \eqref{k-H-iterative} it follows that $(\Delta u^{m+1})^k \geq 0$. In other words, starting with an initial guess $u^0$ with $\Delta u^0 \geq 0$, \eqref{k-H-iterative} enforces subharmonicity
in arbitrary dimension for smooth convex solutions and subharmonicity for 2-Hessian equations with no convexity assumption on $u$. In addition for 2-Hessian equations, the limit solution solves $S_2(D^2 u)=f \geq 0$. That is, the sequence $u^{m+1}$ defined by \eqref{k-H-iterative} has a formal limit which solves $\Delta u \geq 0$ and $S_2(D^2 u)\geq 0$. Thus \eqref{k-H-iterative} enforces 2-convexity in arbitrary dimension for 2-Hessian equations.
Another class of iterative methods we introduce
in this paper are Gauss-Seidel type iterative methods.
The Gauss-Seidel methods are more efficient than \eqref{k-H-iterative} for large scale problems.
The simplicity of the methods discussed in this paper and the facility with which they can be implemented, make them attractive to researchers interested in Monge-Amp\`ere equations. The other major motivation to study the subharmonicity preserving iterations is that they can be adapted to the finite element context and have been numerically shown in that context to be robust for non smooth solutions.
In two dimension, \eqref{k-H-iterative} appears to perform well in the degenerate case $f \geq 0$ as discrete $k$-convexity is enforced in the iterations. The situation is different in three dimension with $k=2$.
We were not able to reproduce the solution $u(x,y,z)=|x-1/2|$ by solving \eqref{k-H1} with $k=2$ and using \eqref{k-H-iterative}. Here, since $u$ does not depend on $z$, we have $f(x,y,z)=0$ as in the two dimensional case.
However, for $n=3$ and $k=3$, we can preserve convexity in the degenerate case by using the sequence of nonlinear $2$-Hessian equations
\begin{align} \label{sigma2k}
S_2(D^2 u^{m+1}) = 3 \bigg(\bigg(\frac{1}{3} S_2 (D^2 u^m)\bigg)^{\frac{3}{2}} + f - \det D^2 u^m
\bigg)^{\frac{2}{3}},
\end{align}
with $u^{m+1}=g$ on $\partial \Omega$. Each of these equations is solved iteratively by \eqref{k-H-iterative} with $k=2, n=3$.
We note that $\bigg(\frac{1}{3} S_2 (D^2 u^m)\bigg)^{\frac{3}{2}} - \det D^2 u^m \geq 0$ when $S_2 (D^2 u^m)>0$, \cite[Lemma 15.12]{Lieberman96}. Starting with an initial guess which satisfies $S_2 (D^2 u^0) >0$ and setting $\det D^2 u^m=0$ in \eqref{sigma2k} whenever $S_2 (D^2 u^m)=0$, we obtain a double sequence iterative method which at the limit enforce $\Delta u \geq 0, S_2 (D^2 u) \geq 0$, and $ \det D^2 u = f \geq 0$.
The reason for setting $\det D^2 u^m=0$ in \eqref{sigma2k} whenever $S_2 (D^2 u^m)=0$ is motivated by the observation that in the case $f=0$, if $S_2 (D^2 u^m)=0$, $S_2 (D^2 u^{m+1})$ is ill-defined or complex valued if $\det D^2 u^m>0$. While \eqref{k-H-iterative} may be inexact for degenerate 2-Hessian equations, its use inside a double iterative method appears effective. This is reminiscent of inexact Uzawa algorithms.
\subsection{Relation with other work}
The $k$-Hessian equations have mainly applications in conformal geometry and physics. The Monge-Amp\`ere operator has received recently a lot of interest from numerical analysts. For $n=3$ and $k=2$,
the numerical resolution of \eqref{k-H1} has been considered in \cite{Sorensen10}, where it was referred to as the $\sigma_2$ problem.
The iterative method \eqref{k-H-iterative} generalizes an iterative method introduced in \cite{Benamou2010} for the two dimensional Monge-Amp\`ere equation. The latter corresponds to the choice $ k=n=2$ and the constant $c(2,2)=1/4$ replaced by 1/2. The $2$-Hessian equation has also been considered recently in \cite{FroeseObermanSalvago} from the point of view of monotone schemes.
We will see that if the central finite difference discretization of \eqref{k-H-iterative} is solved by a Gauss-Seidel iterative method, one recovers a Gauss-Seidel iterative method which has been used by many authors to solve the two dimensional Monge-Amp\`ere equation. We will refer to the latter method as the 2D Gauss-Seidel method for Monge-Amp\`ere equation. It has been used in the numerical simulation of Ricci flow \cite{Headrick05}, as a smoother in multigrid methods for the balance vortex model in meteorology, \cite{Chen2010b,Chen2010c} and has been recently shown numerically to capture the viscosity solution of the 2D Monge-Amp\`ere equation \cite{Benamou2010}. The connection between \eqref{k-H-iterative} and the 2D Gauss-Seidel method for the Monge-Amp\`ere equation is what enables us to introduce new Gauss-Seidel type iterative methods for $k$-Hessian equations.
The ingredients of our proof of the convergence rate for the finite difference discretization are discrete Schauder estimates and a suitable generalization of the combined fixed point iterative method used in \cite{Feng2009}.
Schauder estimates were also used in the proof of convergence of Newton's method at the continuous level \cite{Loeper2005}.
\subsection{Organization of the paper}
The paper is organized as follows: In the next section, we
give some notations, recall the Schauder estimates and their discrete analogues. In section \ref{elliptic} we prove our main results on the quadratic convergence rate of a finite difference discretization of \eqref{k-H1} and in section \ref{newton-sec} we prove the convergence of Newton's method. In section \ref{convexity} we
introduce new Gauss-Seidel type iterative methods and their connections with the subharmonicity preserving iterations \eqref{k-H-iterative}.
Section \ref{num} is devoted to numerical results. We conclude with some remarks.
The reader interested only in the Monge-Amp\`ere equation, or for a first reading, may assume that $k=n$.
\section{Notation and preliminaries} \label{notation}
\subsection{H$\ddot{\text{o}}$lder spaces and Schauder estimates} \label{notation1}
For a nonnegative integer $r$ or for $r=\infty$, we denote by $C^r(\Omega)$ the set of all functions having all derivatives of order $\leq r$ continuous on $\Omega$
and by $C^r(\tir{\Omega})$, the set of all functions in $C^r(\Omega)$
whose derivatives of order $\leq r$ have continuous
extensions to $\tir{\Omega}$. For a multi-index $\beta=(\beta_1,\ldots,\beta_n) \in \mathbb{N}^n$, put $|\beta|=\beta_1+\ldots+\beta_n$. We use the notation $D^{\beta} u(x)$ for the partial derivative
$(\partial /\partial x_1)^{\beta_1} \ldots (\partial /\partial x_n)^{\beta_n} u(x)$.
The norm in $C^r(\Omega)$ is given by
$$
||u||_{r;\Omega} = \sum_{j=0}^r \, |u|_{j;\Omega}, \quad |u|_{j;\Omega} = \text{sup}_{|\beta|=j} \text{sup}_{\Omega} |D^{\beta}u(x)|.
$$
We denote by $|x|$ the Euclidean norm of $x \in \mathbb{R}^n$. A function $u$ is said to be uniformly H$\ddot{\text{o}}$lder continuous with exponent $\alpha, 0 <\alpha \leq 1$ in $\Omega$ if
the quantity
$$
\text{sup}_{x \neq y} \frac{|u(x)-u(y)|}{|x-y|^{\alpha}},
$$
is finite.
The space $C^{r,\alpha}(\tir{\Omega})$
consists of functions whose $r$-th order derivatives are uniformly H$\ddot{\text{o}}$lder
continuous with exponent $\alpha$ in $\Omega$. It is a Banach space with norm
$$
||u||_{r,\alpha;\Omega} = ||u||_{r;\Omega} + [u]_{r,\alpha;\Omega},
$$
where
$$
[u]_{r,\alpha;\Omega} = \text{sup}_{|\beta|=r} \text{sup}_{x \neq y} \frac{|D^{\beta}u(x)-
D^{\beta} u(y)|}{|x-y|^{\alpha}}.
$$
The norms $|| \, ||_{r;\Omega}$ and $|| \, ||_{r,\alpha;\Omega}$ are naturally extended to vector fields and matrix fields by taking the supremum over all components. We make the standard convention of using $C$ for a generic constant. For $A=(a_{ij})_{i,j=1,\ldots,n}$ and $B=(b_{ij})_{i,j=1,\ldots,n}$
we recall that
$A:B=\sum_{i,j=1}^n a_{i j} b_{ij}$.
We will often use the following property
\begin{equation} \label{alpha-prod1}
|| f g ||_{0,\alpha;\Omega} \leq C || f ||_{0,\alpha;\Omega} || g ||_{0,\alpha;\Omega}, \, \text{for} \, f,g \in C^{0,\alpha}(\tir{\Omega}),
\end{equation}
from which it follows that if $A, B$ are matrix fields
\begin{equation} \label{alpha-prod2}
||A:B||_{0,\alpha;\Omega} \leq C \sum_{i,j=1}^n || a_{ij} ||_{0,\alpha;\Omega} || b_{ij} ||_{0,\alpha;\Omega}.
\end{equation}
We first state a global regularity result for the solution of strictly elliptic equations, which follows from \cite[Theorems 6.14, 6.6 and Corollary 3.8 ]{Gilbarg2001}.
\begin{thm} \label{SchauderPoisson}
Assume $0< \alpha < 1$. Let $\Omega$ be a $C^{2,\alpha}$ domain in $\mathbb{R}^n$ and $f, a^{ij} \in C^{\alpha}(\tir{\Omega})$, $\phi \in C^{2,\alpha}(\tir{\Omega})$. We consider the strictly elliptic operator
\begin{equation} \label{st-elliptic}
L u = \sum_{i,j=1}^n a^{ij}(x) \frac{\partial^2}{\partial x_i \partial x_j} u(x),
\end{equation}
with coefficients satisfying for positive constants $\lambda, \Lambda$,
$$
\sum_{i,j=1}^na^{ij}(x) \zeta_i \zeta_j \geq \lambda \sum_{l=1}^n \zeta_l^2, \zeta_l \in \mathbb{R}, \, \text{and} \, |a^{i,j}|_{0,\alpha;\Omega} \leq \Lambda.
$$
Then the solution $u$
of the equation
$$
L u =f \, \text{in} \, \Omega, u = \phi \, \text{on} \, \partial \Omega,
$$
satisfies
$$
||u||_{2,\alpha;\Omega} \leq C(||\phi||_{2,\alpha;\Omega}+ ||f||_{0,\alpha;\Omega}),
$$
where $C$ depends on $n, \alpha, \lambda, \Lambda, \Omega, \sup_{\partial \Omega} |\phi|$, and $\sup_{\Omega} |f|/\lambda$.
\end{thm}
We will make the slight abuse of language of also denoting by $S_k(x), x=(x_1,\ldots,x_n)$ the $k$th elementary symmetric polynomial of the variable
$x$, i.e.
$$
S_k(\lambda) = \sum_{i_1 < \cdots < i_k} \lambda_{i_1} \cdots \lambda_{i_k}.
$$
A function $u \in C^2(\Omega) \cap C^0(\tir{\Omega})$ with Hessian $D^2 u$ having eigenvalues $\lambda_i, i=1,\ldots,n$ is said to be $k$-admissible if
$S_j(\lambda) > 0, j=1,\ldots,k$. Solutions of the $k$-Hessian equation will be required to be $k$-admissible, thus requiring $f>0$.
Moreover, let $\kappa=(\kappa_1,\ldots,\kappa_{n-1})$ denote the principal curvatures of $\partial \Omega$.
\begin{defn} \label{k-convexity-domain}
The domain $\Omega$ is said to be $(k-1)$-convex if there exists $c_0 >0$ such that
$$
S_{k-1}(\kappa) \geq c_0 >0 \, \text{on} \, \partial \Omega.
$$
\end{defn}
We then have, (\cite[Theorems 3.3 and 3.4 ]{WangXJ09})
\begin{thm} \label{k-Hessian}
Assume that $\Omega$ is $(k-1)$-convex, $\partial \Omega \in C^{3,1}$, $f \in C^{1,1}(\tir{\Omega})$, inf $f >0$, $g \in C^{3,1}(\tir{\Omega})$. Then there is a unique $k$-admissible solution $u \in C^{3,1}(\tir{\Omega})$ to the Dirichlet problem \eqref{k-H1}.
\end{thm}
We will need some identities for the $k$-Hessian operator $S_k(D^2 u)$ which are derived explicitly for example in \cite[p. 5--6]{Gavitone2009}. See also \cite{WangXJ09}.
For a symmetric matrix $A=(a_{ij})_{i,j}=1,\ldots,n$ with eigenvalues $\lambda_i, i=1,\ldots,n$, let us also denote by $S_k(A)$ the $k$-th elementary symmetric polynomial of $\lambda$. This is equivalent to say that $S_k(A)$ is the sum of all $k \times k$ principal minors of $A$. Using the permutation definition of the determinant, we have
\begin{align} \label{k-minor}
S_k(A) = \frac{1}{k!}\sum_{1 \leq i_1,\cdots,i_k\leq n} \delta^{j_1,\cdots,j_k}_{i_1,\cdots,i_k} a_{i_1 j_1} \cdots a_{i_k j_k},
\end{align}
where $\delta^{j_1,\cdots,j_k}_{i_1,\cdots,i_k}$ is the generalized Kronecker delta which takes the value +1 if $i_1,\cdots,i_k$ differs from $j_1,\cdots,j_k$ by an even permutation and the value -1 in the case of an odd permutation. In other words, for a choice of $i_1,\ldots,i_k$, $\delta^{j_1,\cdots,j_k}_{i_1,\cdots,i_k}$ is the signature of the permutation $\sigma$ defined by $\sigma(i_l)=j_l, l=1,\ldots,k$. This implies that we only consider the case where the sets $\{i_1,\ldots,i_k\}$ and $\{j_1,\ldots,j_k\}$ are identical. Moreover we define $\delta^{j_1,\cdots,j_k}_{i_1,\cdots,i_k}$ to be 0 if $\{i_1,\ldots,i_k\} \neq \{j_1,\ldots,j_k\}$. Note also that $\{i_1,\ldots,i_k\}$ is a subset of $k$ elements of $\{ 1, \ldots, n \}$.
We have
\begin{align*}
S_k^{ij}(A)\coloneqq \frac{\partial}{\partial a_{ij}} S_k(A) = \frac{1}{(k-1)!}
\sum_{1 \leq i, i_1,\cdots,i_{k-1}\leq n} \delta^{j,j_1,\cdots,j_{k-1}}_{i,i_1,\cdots,i_{k-1}} a_{i_1 j_1} \cdots a_{i_{k-1} j_{k-1}},
\end{align*}
and so
$
S_k(A) = \frac{1}{k} \sum_{i,j=1}^n S_k^{ij}(A) a_{i j}
$
by the $k$-homogeneity of $S_k$ and Euler's theorem for homogeneous functions. Here $\{j_1,\ldots,j_{k-1}\}$ is the image of the set of $k-1$ elements $\{i_1,\ldots,i_{k-1}\}$ not containing $i$ by a permutation.
Let us denote by $\{S_k^{ij}(A) \}$ the symmetric matrix with entries $S_k^{ij}(A)$. We can write $S_k(A)=1/k \, \{S_k^{ij}(A) \}: A $, that is
$S_k(D^2 v) = \frac{1}{k} \{S_k^{ij}(D^2 v) \} : D^2 v$.
Using \eqref{k-minor} and observing that the expression of $S_k(A) $ can be written in terms of a multilinear map, we obtain
\begin{align} \label{k-Hdiv0}
S_k'(D^2 v) D^2 w = \{S_k^{ij}(D^2 v) \}: D^2 w.
\end{align}
Let us denote by $ \{S_k^{ij}(A) \}' $ the Fr\'echet derivative of the mapping $A \to \{S_k^{ij}(A) \}$. Since $\{S_k^{ij}(A) \}' (B)$ is a sum of terms each of which is a product of $k-2$ terms from $A$ and is linear in $B$, we have
\begin{equation} \label{Sij-der}
|| \{S_k^{ij}(D^2 v) \}' D^2 w ||_{0;\Omega} \leq C |v|_{2;\Omega}^{k-2} |w|_{2;\Omega}.
\end{equation}
Using \eqref{alpha-prod2} and \eqref{Sij-der} we also have
\begin{equation} \label{cof-estimate}
|| \{S_k^{ij}(D^2 v) \}' D^2 w ||_{0,\alpha;\Omega} \leq C |v|_{2,\alpha;\Omega}^{k-2} |w|_{2,\alpha;\Omega}.
\end{equation}
Finally we note that
\begin{lemma} \label{close}
Let $v$ be a $C^2$ strictly convex function with Hessian having smallest eigenvalue uniformly bounded below by a constant $a >0$. Then for $\eta=a/(2 n)$, we have $w$ strictly convex, whenever $||w-v||_{C^2(\Omega)} < \eta$.
\end{lemma}
\begin{proof}
It follows from \cite[Theorem 1 and Remark 2 p. 39]{Hoffman53} that for two symmetric $n \times n$ matrices $A$ and $B$,
\begin{equation} \label{cont-eig}
|\lambda_l(A) - \lambda_l(B)| \leq n \max_{i,j} |A_{ij} - B_{ij}|, l=1, \ldots, n.
\end{equation}
It follows that for $u, v \in C^2(\Omega)$,
\begin{align}
|\lambda_1( D^2 u(x)) - \lambda_1( D^2 v(x))| & \leq n ||w-v||_{C^2(\Omega)} \label{lambda1}.
\end{align}
The result then follows.
\end{proof}
We conclude this section with the equivalence of $n$-convexity and convexity in the usual sense.
\begin{lemma} \label{n-convexity}
A $C^2$ function $u$ is convex if and only if it is $n$-convex.
\end{lemma}
\begin{proof}
If $u$ is $C^2$, $\lambda_i \geq 0$ on $\Omega$ for all $i$ and thus $S_l(D^2 u) \geq 0, l=1,\ldots,n$.
Conversely let us assume that $A$ is a symmetric matrix with
$S_l(A) \geq 0, l=1,\ldots,n$. We show that its eigenvalues $\lambda_i$ are all positive. Let
$$
p(\lambda)= \lambda^n + c_1 \lambda^{n-1} + \ldots + c_n,
$$
denote the characteristic polynomial of $A$. It can be shown \cite[Theorem 1.2.12]{Horn85} that
$$
c_l = (-1)^l S_l(A), l=1,\ldots,n.
$$
We show that if $\lambda_i <0$ then $p(\lambda_i) \neq 0$. We have
\begin{align*}
p(\lambda_i) & = \lambda_i^n + c_1 \lambda_i^{n-1} + \ldots + c_n \\
& = \lambda_i^n + \sum_{l=1}^n (-1)^{l} S_l(A) \lambda_i^{n-l} \\
& = (-1)^n \bigg( (-\lambda_i)^n + \sum_{l=1}^{n} (-1)^{l-n} S_l(A) \lambda_i^{n-l}\bigg) \\
& = (-1)^n \bigg( (-\lambda_i)^n + \sum_{l=1}^{n} S_l(A) (-\lambda_i)^{n-l}\bigg).
\end{align*}
Since $-\lambda_i >0$ and $S_l(A) \geq 0$ for all $l$, we have $(-1)^n p(\lambda_i) \geq 0$. Moreover since
$\sum_{l=1}^{n} S_l(A) (-\lambda_i)^{n-l} \geq 0$ and $-\lambda_i > 0$ we have $(-1)^n p(\lambda_i) \neq 0$.
We conclude that $\lambda_i \geq 0$ for all $i$. This completes the proof.
\end{proof}
\subsection{Discrete Schauder estimates and related tools} \label{disc-schauder}
We will study the numerical approximation of \eqref{k-H1}--\eqref{k-H-iterative} by standard finite difference discretizations.
For simplicity, we consider a cuboidal domain $\Omega = (0,1)^n \subset \mathbb{R}^n$. Let $0 < h < 1 \, \text{with} \, 1/h \in \mathbb{Z}$. Put
\begin{align*}
\mathbb{Z}_h & = \{x=(x_1,\ldots,x_n)^T \in \mathbb{R}^n: x_i/h \in \mathbb{Z} \}\\
\Omega^h_0 &= \Omega \cap \mathbb{Z}_h, \Omega^h = \tir{\Omega} \cap \mathbb{Z}_h, \partial \Omega^h = \partial \Omega \cap \mathbb{Z}_h= \Omega^h \setminus \Omega^h_0.
\end{align*}
Let $e^i, i=1,\ldots,n$ denote the $i$-th unit vector of $\mathbb{R}^n$. We define the following first order difference operators on the space $\mathcal{M}(\Omega^h)$ of grid functions $v^h(x), x \in \mathbb{Z}_h$,
\begin{align*}
\partial^i_{+} v^h(x) & \coloneqq \frac{v^h(x+he^i)-v^h(x)}{h}, \\
\partial^i_{-} v^h(x) & \coloneqq \frac{v^h(x)-v^h(x-he^i)} {h},\\
\partial^i_h v^h(x) & \coloneqq \frac{v^h(x+he^i)-v^h(x-he^i)}{2 h}.
\end{align*}
Higher order difference operators are obtained by combining the above difference operators. For a multi-index $\beta=(\beta_1,\ldots,\beta_n) \in \mathbb{N}^n$, we define
$$
\partial^{\beta}_{+} v^h \coloneqq \partial^{\beta_1}_{+} \cdots \partial^{\beta_n}_{+}v^h.
$$
The operators $\partial^{\beta}_{-}$ and $\partial^{\beta}_{h}$ are defined similarly. Note that
\begin{align}
\partial^i_{+} \partial^i_{-} v^h(x) & = \frac{v^h(x+he^i)-2v^h(x)+v^h(x-he^i)}{h^2}, \label{second-disc1}
\end{align}
\begin{align}
\begin{split}
\partial^i_h \partial^j_h v^h(x) & = \frac{1}{4 h^2} \bigg\{v^h(x+he^i+h e^j)+v^h(x-he^i-h e^j) \\
& \qquad \qquad \qquad -v^h(x+he^i-h e^j)-v^h(x-he^i+ he^j)\bigg\}, i \neq j. \label{second-disc2}
\end{split}
\end{align}
The second order derivatives $\partial^2 v/\partial x_i \partial x_j$ are discretized using \eqref{second-disc1} and \eqref{second-disc2} for $i \neq j$. This gives a discretization of the Hessian $D^2 u$ which we denote by $\mathcal{H}_d(u^h)$.
Thus the discrete version of \eqref{k-H1} takes the form
\begin{align} \label{k-H1h}
S_k (\mathcal{H}_d \, u^h(x)) = f(x), x \in \Omega^h_0, u^h(x) = g(x)\, \text{on} \, \partial \Omega^h.
\end{align}
The discrete Laplacian takes the form
\begin{align} \label{second-disc3}
\Delta_d (u^h) = \sum_{i=1}^n \partial^i_{+} \partial^i_{-} u^h.
\end{align}
We consider a discrete uniformly elliptic linear operator
with low order terms
\begin{align*}
L_d v^h(x) = \sum_{i,j=1}^n a^{ij}(x) \partial^i_- \partial^j_+ v^h(x) + \sum_{i=1}^n b^{i}(x) \partial^i_+ v^h(x), x \in \Omega_0^h,
\end{align*}
i.e. the matrix $(a^{ij}(x))_{i,j=1,\ldots,n}$ is uniformly positive definite.
We now define discrete analogues of the H$\ddot{\text{o}}$lder norms and semi-norms following \cite{Johnson74}. Let $[\xi,\eta]$ denote the set of points $\zeta \in \Omega^h$ such that $\xi_j \leq \zeta_j \leq \eta_j, j=1,\ldots,n$. Then for $v^h \in \mathcal{M}(\Omega^h), 0 < \alpha < 1$, we define
\begin{align*}
|v^h|_{j;\Omega_0^h} & = \, \text{max} \, \{\, |\partial^{\beta}_{+}v^h (\xi)|, |\beta|=j, [\xi,\xi+\beta] \subset \Omega^h \, \}, \\
[v^h]_{j,\alpha;\Omega_0^h} & = \, \text{max} \, \bigg\{\, \frac{|\partial^{\beta}_{+}v^h (\xi)- \partial^{\beta}_{+}v^h (\eta)|}{( |\xi-\eta|)^{\alpha}},
|\beta|=j, \xi \neq \eta, [\xi,\xi+\beta] \cup [\eta,\eta+\beta] \subset \Omega^h \, \bigg\}, \\
||v^h||_{p;\Omega_0^h} & = \, \text{max}_{j \leq p} \, |v^h|_{j;\Omega_0^h}, \\
||v^h||_{p,\alpha;\Omega_0^h} & = ||v^h||_{p;\Omega_0^h}+ [v^h]_{p,\alpha;\Omega_0^h}.
\end{align*}
The above norms are extended canonically to vector fields and matrix fields by taking the maximum over all components. For $j=0$, we have discrete analogues of the maximum and $C^{0,\alpha}$ norms.
For a domain $O \subset \mathbb{R}^n$, we denote by $\mathcal{D}_h(O)$ the set of mesh functions on $\mathbb{R}^n$ which vanish outside $O$. If $v^h=0$ on $\partial \Omega^h$, extending $v^h$ by 0 to $\mathbb{Z}_h$, we obtain $v^h \in \mathcal{D}_h(\Omega)$. The following theorem then follows from
\cite[Lemma 3.4]{Thomee1970}.
\begin{thm} \label{discShauderPoisson}
Assume $ 0<\alpha<1$ and $v^h=0$ on $\partial \Omega^h$. Then there are constants $C$ and $h_0$ such that for $v^h \in \mathcal{M}(\Omega^h), h \leq h_0$
\begin{align} \label{discShauderPoisson0}
||v^h||_{2,\alpha;\Omega_0^h} \leq C ||L_d \, v^h||_{0,\alpha;\Omega_0^h},
\end{align}
with the constant $C$ independent of $h$.
\end{thm}
Since
\begin{align*}
\partial^i_{+} \partial^i_{-} v^h(x) & =\partial^i_{+} \partial^i_{+} v^h (x-h e^i) \, \text{and} \, \\
\partial^j_{h} \partial^i_{h} v^h(x) & = \frac{1}{4}\bigg(\partial^j_{+} \partial^i_{+} v^h (x) + \partial^j_{+} \partial^i_{+} v^h (x- h e^i) + \partial^j_{+} \partial^i_{+} v^h (x- h e^j) \\
& \qquad \qquad \qquad +\partial^j_{+} \partial^i_{+} v^h (x- h e^i-h e^j) \bigg),
\end{align*}
we have max $\{||\partial^i_{+} \partial^i_{-} v^h ||_{0,\alpha;\Omega_0^h}, ||\partial^j_{h} \partial^i_{h} v^h||_{0,\alpha;\Omega_0^h}, i,j=1,\ldots,n \} \leq ||v^h||_{2,\alpha;\Omega_0^h}$ and hence the above theorem also applies when the second order derivatives \eqref{second-disc1}
and \eqref{second-disc2} are used in the definition of $|| . ||_{2,\alpha;\Omega_0^h}$.
By Taylor series expansions, it is not difficult to verify that for $v \in C^2(\Omega)$
\begin{align*}
|v |_{j;\Omega_0^h} \leq |v|_{2;\Omega}, j \leq 2.
\end{align*}
Moreover, for $v \in C^{4,\alpha}(\Omega)$,
\begin{align} \label{consistent}
||D^2 v - \mathcal{H}_d(v)||_{0;\Omega_0^h} \leq C h^2 |v|_{4;\Omega},
\end{align}
and
\begin{align*}
[D^2 v - \mathcal{H}_d( v) ]_{0,\alpha;\Omega_0^h} \leq C h^{2} [v]_{4,\alpha;\Omega}.
\end{align*}
To see that the last inequality holds, it is enough to consider a function of one variable $v \in C^{4,\alpha}(-1,1)$ and estimate
$[v''(x)-(v(x+h)-2v(x)+v(x-h))/h^2]_{0,\alpha}$. Now,
$$
v''(x)-\frac{v(x+h)-2v(x)+v(x-h)}{h^2} = -\frac{h^2}{24} (v^{(4)}(x+ t_1 h) + v^{(4)}(x- t_2 h)), t_1, t_2 \in [0,1].
$$
Next we note that, using the definition, the $C^{0,\alpha}$ norm of $v^{(4)}(x+ t_1 h)$ is bounded above by the $C^{0,\alpha}$ norm of $v^{(4)}$. The result then follows.
We have for $v \in C^{4,\alpha}(\Omega)$,
\begin{align} \label{consistent2}
|| D^2 v - \mathcal{H}_d( v)||_{0,\alpha;\Omega_0^h} \leq C h^{2} ||v||_{4,\alpha;\Omega}.
\end{align}
\begin{lemma} \label{est-d2}
We have for $u \in C^{4,\alpha}(\Omega)$
$$
||S_k (D^2 u)-S_k (\mathcal{H}_d( u)) ||_{0,\alpha;\Omega_0^h} \leq C h^{2} |u|_{2;\Omega}^{k-1} ||u||_{4,\alpha;\Omega}.
$$
\end{lemma}
\begin{proof}
By the mean value theorem, using \eqref{k-Hdiv0}, we have for some $t$ in $[0,1]$, and $x\in \Omega_0^h$,
\begin{align*}
S_k (D^2 u) (x) - S_k (\mathcal{H}_d( u))(x) & = S_k'(t D^2(u)(x) + (1-t) \mathcal{H}_d( u)(x)): (D^2 u (x) \\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad - \mathcal{H}_d( u)(x)) \\
& = \sum_{i,j=1}^n S_k^{ij} (t D^2(u)(x) + (1-t) \mathcal{H}_d( u)(x)) (D^2 u (x) \\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad - \mathcal{H}_d( u)(x))_{ij}.
\end{align*}
Using \eqref{alpha-prod2}, it follows that
\begin{align*}
||S_k (D^2 u)-S_k (\mathcal{H}_d( u) ) ||_{0,\alpha;\Omega_0^h} & \leq C (|u|_{2;\Omega} + | u|_{2;\Omega_0^h} )^{k-1} ||D^2 u - \mathcal{H}_d( u)||_{0,\alpha;\Omega_0^h} \\
& \leq C h^{2} |u|_{2;\Omega}^{k-1} ||u||_{4,\alpha;\Omega}.
\end{align*}
\end{proof}
\section{Approximations by linear elliptic problems} \label{elliptic}
In this section, we prove the convergence of the iterative method \eqref{broyden} and its discrete version. As indicated in the introduction, we also obtain the existence and uniqueness of the solution of the discrete version of \eqref{k-H1}, i.e. \eqref {k-H1h}, as well as error estimates.
\subsection{Convergence at the operator level}
We assume that
there is a unique $k$-admissible solution $u \in C^{2,\alpha}(\tir{\Omega})$ of \eqref{k-H1} for $0 < \alpha < 1$. Let $u^0 \in C^{2,\alpha}(\tir{\Omega})$ such that
\begin{equation} \label{is-delta}
||u-u^0||_{2,\alpha;\Omega} < \delta.
\end{equation}
For $k=n$, using an eigenvalue argument, it is not difficult to prove that the cofactor matrix is uniformly positive definite under the assumption $f \geq f_0 >0$ for a constant $f_0$. We assume that the matrix $\{S_k^{ij}(D^2 u) \}$ is uniformly positive definite. We claim that this holds if $u \in C^2(\tir{\Omega})$ and there is $c_3 >0$ such that
$$
c_3 \leq S_l(D^2 u), 1 < l \leq k.
$$
We then have
\begin{equation} \label{c4}
c_3 \leq S_l(D^2 u) \leq c_4, 1 < l \leq k,
\end{equation}
for a constant $c_4$. The proof is essentially given as \cite[Theorem 1.3 ]{Gavitone2009}. We define
$$S_k^i(\lambda):= \frac{\partial}{\partial \lambda_i} S_k(\lambda).$$
First we note from the proof of \cite[Theorem 1.3 ]{Gavitone2009} that the eigenvalues of $\{S_k^{ij}(D^2 u) \}$ are given by $S_k^i(\lambda(D^2 u)), 1 \leq i \leq n$.
On the other hand, since $S_l(D^2 u) \geq c_3 >0, 1 < l \leq k$, we have by \cite[Proposition 1.1]{Caffarelli1985}
$$
\frac{\partial}{\partial \lambda_i} S_k(\lambda)^{\frac{1}{k}} >0 \text{ for } \lambda=\lambda(D^2 u).
$$
Finally, as $S_k(D^2 u) \leq c_4$ and $u \in C^2(\tir{\Omega})$, the result follows.
By the continuity of the smallest eigenvalue of a matrix as a function of its entries, $\{S_k^{ij}(D^2 u^0) \}$ is also uniformly positive definite for $|u-u^0|_{2;\Omega}$ sufficiently small.
Next, $\{S_k^{ij}(D^2 u^0)\}$ is a symmetric matrix and divergence free by \cite[ Formula 1.10 ]{Gavitone2009}. Thus we obtain
\begin{equation} \label{div-prop}
\operatorname{div}\bigg( \{S_k^{ij}(D^2 u^0) \} D v) \bigg) = \{S_k^{ij}(D^2 u^0) \}: D^2 v.
\end{equation}
We have
\begin{thm} \label{contT-broyden}
Under the assumption that there is a unique $k$-admissible solution $u \in C^{2,\alpha}(\tir{\Omega})$ of \eqref{k-H1} for $0 < \alpha < 1$, the sequence defined by \eqref{broyden} converges to $u$
for $u^0$ sufficiently close to $u$.
\end{thm}
\begin{proof}
We define the operator $R: C^{2,\alpha}(\tir{\Omega}) \to C^{2,\alpha}(\tir{\Omega})$ by
\begin{align*}
-\operatorname{div}\bigg( \{S_k^{ij}(D^2 u^0) \} D (v-R v)\bigg) &= - S_k (D^2 v) + f \, \text{in} \, \Omega \\
R(v) & = g \, \text{on} \, \partial \Omega.
\end{align*}
By Theorem \ref{SchauderPoisson}, the operator $R$ is well defined. We show that for $\rho>0$ sufficiently small, $R$ is a strict contraction in the ball
$B_{\rho}(u) = \{v \in C^{2,\alpha}(\tir{\Omega}), ||u-v||_{2,\alpha;\Omega} < \rho \}$.
For $v, w \in B_{\rho}(u)$ we have using \eqref{div-prop}
\begin{multline*}
\operatorname{div}\bigg( \{S_k^{ij}(D^2 u^0) \} D (R v-R w) \bigg) =\operatorname{div}\bigg(\{S_k^{ij}(D^2 u^0) \} D ( v- w) \bigg) + S_k (D^2 w) - S_k (D^2 v)\\
= -\{S_k^{ij}(D^2 u^0) \} : (D^2 w - D^2 v) + S_k (D^2 w) - S_k (D^2 v).
\end{multline*}
Next, by the mean value theorem and using \eqref{k-Hdiv0}, we have for some $t$ in $[0,1]$,
\begin{multline*}
S_k(D^2 w)-S_k(D^2 v) = \{S_k^{ij}(t D^2 w + (1-t) D^2 v) \} : D^2 (w-v) \\
= \{S_k^{ij}(t (D^2 w -D^2 u^0)+ (1-t) (D^2 v-D^2 u^0) + D^2 u^0) \} : D^2 (w-v).
\end{multline*}
We use \eqref{cof-estimate} to estimate the $C^{0,\alpha}$ norm of
$$
A=\{S_k^{ij}(t (D^2 w -D^2 u^0)+ (1-t) (D^2 v-D^2 u^0) + D^2 u^0) \} - \{S_k^{ij}(D^2 u^0) \}.
$$
For $0 \leq s \leq 1$ to be specified below, put
$$
\alpha_{s t} = s t (D^2 w -D^2 u^0)+ s (1-t) (D^2 v-D^2 u^0) + D^2 u^0.
$$
We have
\begin{equation} \label{a-st}
|\alpha_{s t}|_{0,\alpha;\Omega} \leq ||u^0-v||_{2,\alpha;\Omega}+||u^0-w||_{2,\alpha;\Omega} +||u^0||_{2,\alpha;\Omega}.
\end{equation}
By the mean value theorem, for some $s \in [0,1]$ we have
$$
A= \{S_k^{ij}(\alpha_{s t} ) \}' (t (D^2 w -D^2 u^0)+ (1-t) (D^2 v-D^2 u^0)),
$$
and thus by \eqref{cof-estimate}
\begin{equation} \label{a-st2}
||A||_{0,\alpha;\Omega} \leq C |\alpha_{s t}|_{0,\alpha;\Omega}^{k-2} (||u^0-v||_{2,\alpha;\Omega}+||u^0-w||_{2,\alpha;\Omega} ).
\end{equation}
By Schauder estimates (Theorem \ref{SchauderPoisson}), \eqref{alpha-prod2}, \eqref{a-st} and \eqref{a-st2} we obtain
\begin{align} \label{contraction-cont-level}
\begin{split}
||R(v) - R(w)||_{2,\alpha;\Omega} & \leq C ||A||_{0,\alpha;\Omega} ||D^2(v-w)||_{0,\alpha;\Omega} \\
& \leq C (||u^0-v||_{2,\alpha;\Omega}+||u^0-w||_{2,\alpha;\Omega} +||u^0||_{2,\alpha;\Omega} )^{k-2} \\
& \qquad \qquad (||u^0-v||_{2,\alpha;\Omega}+||u^0-w||_{2,\alpha;\Omega} ) ||v-w||_{2,\alpha;\Omega} \\
& \leq C (\rho+\delta+||u^0||_{2,\alpha;\Omega})^{k-2} (\rho+\delta)||v-w||_{2,\alpha;\Omega},
\end{split}
\end{align}
where $\delta$ measures how close $u^0$ is to $u$ \eqref{is-delta}.
Thus, for $\rho$ and $\delta$ sufficiently small, $R$ is a strict contraction in $B_{\rho}(u)$.
It remains to show that $R$ maps $B_{\rho}(u)$ into itself. We note by the definition of $R$ and unicity of the solution of \eqref{k-H1}, a fixed point of $R$ solves \eqref{k-H1}. Let $v \in B_{\rho}(u)$,
\begin{align*}
||u-R v||_{2,\alpha;\Omega} & =||R u-R v||_{2,\alpha;\Omega} \leq ||u-v||_{2,\alpha;\Omega} \leq \rho,
\end{align*}
which shows that $R$ maps $B_{\rho}(u)$ into itself. The existence of a fixed point follows from the Banach fixed point theorem. Moreover, the sequence defined by $u^{m+1}=R(u^m)$, i.e. the sequence defined by
\eqref{broyden}, converges for $\rho$ and $\delta$ sufficiently small to $u$.
\end{proof}
\subsection{Finite difference discretization}
Next
we consider the following discrete version of \eqref{broyden}
\begin{align} \label{broyden-D}
\begin{split}
\{S_k^{ij}(\mathcal{H}_d \, u^{0,h}) \} : \mathcal{H}_d u^{m+1,h} & = \{S_k^{ij}(\mathcal{H}_d \, u^{0,h}) \} : \mathcal{H}_d u^{m,h} \\
& \qquad \qquad \qquad \qquad +f-S_k (\mathcal{H}_d \, u^{m,h}) \, \text{in} \, \Omega^h_0 \\
u^{m+1,h} & = g \, \text{on} \, \partial \Omega^h.
\end{split}
\end{align}
Under the assumptions of Theorem \ref{disc-thm} below, we show that \eqref{k-H1h} has a unique solution to which the above sequence converges. Moreover, the convergence rate is O($h^{2}$).
Define
\begin{equation} \label{ball-h}
B_{\rho} ( u) = \{v^h \in \mathcal{M}(\Omega^h), ||v^h- u||_{2,\alpha;\Omega_0^h} \leq \rho \}.
\end{equation}
\begin{lemma} \label{sum-lem}
Let $S^h: \mathcal{M}(\Omega^h) \to \mathcal{M}(\Omega^h)$ be a strict contraction with contraction factor less than 1/2, i.e. for $v^h, w^h \in \mathcal{M}(\Omega^h)$
$$
||S^h(v^h) - S^h(w^h)||_{2,\alpha;\Omega_0^h} \leq\frac{1}{2} ||v^h - w^h||_{2,\alpha;\Omega_0^h}.
$$
Let us also assume that $S^h$ does not move the center $u$ of the ball $B_{\rho} ( u)$ too far, i.e.
$$
||S^h( u) - u||_{2,\alpha;\Omega_0^h} \leq C_0 h^2.
$$
Then $S^h$ maps $B_{\rho} ( u)$ into itself for $\rho=2 C_0 h^{2}$. Moreover $S^h$ has a unique fixed point $u^h$ in $B_{\rho} ( u)$ with the error estimate
$$
|| u - u^h ||_{2,\alpha;\Omega_0^h} \leq 2 C_0 h^2.
$$
\end{lemma}
\begin{proof}
For $v^h \in B_{\rho} ( u)$,
\begin{align*}
||S^h(v^h)- u||_{2,\alpha;\Omega_0^h} & \leq ||S^h(v^h)- S^h( u)||_{2,\alpha;\Omega_0^h}+ ||S^h( u)- u||_{2,\alpha;\Omega_0^h}\\
& \leq \frac{1}{2} ||v^h- u||_{2,\alpha;\Omega_0^h} + C_0 h^2 \\
& \leq \frac{\rho}{2}+C_0 h^2 \leq \frac{\rho}{2} + \frac{\rho}{2}=\rho.
\end{align*}
This proves that $S^h$ maps $B_{\rho} ( u)$ into itself.
The existence of a fixed point follows from the Banach fixed point theorem.
The convergence rate follows from the observation that
\begin{align*}
|| u - u^h ||_{2,\alpha;\Omega_0^h} & \leq || u - S^h( u) ||_{2,\alpha;\Omega_0^h} + ||S^h( u) - S^h(u^h) ||_{2,\alpha;\Omega_0^h} \\
& \leq C_0 h^2 + \frac{1}{2} ||u^h- u||_{2,\alpha;\Omega_0^h}.
\end{align*}
\end{proof}
\begin{rem} \label{u0h-rem}
For $h$ sufficiently small, $\mathcal{H}_d (u)$ is sufficiently close to $D^2 u$ and hence $\{S_k^{ij}(\mathcal{H}_d u)\}$ is positive definite, a property which also holds for $\{S_k^{ij}(\mathcal{H}_d \,u^{0,h})\}$ for $u^{0,h}$ sufficiently close to $u$. The arguments are similar to the ones of Lemma \ref{close}. See also Lemma \ref{lboundDelta} below.
\end{rem}
\begin{lemma}\label{lboundDelta} Let $u$ be a $k$-admissible solution of \eqref{k-H1}. Assume that inf $f >0$ and $u \in C^4(\Omega)$. Then for $h$ sufficiently small, $\Delta_d (u) \geq c_0 >0$ where
$c_0= 1/2 ((\text{inf} \, f)/c(k,n))^{1/k}$. Moreover, if $u$ is a strictly convex function, then for $\rho= O(h^{2})$, $\mathcal{H}_d( u)$ is a positive matrix and $v^h$ is a discrete convex function, when $v^h \in B_{\rho} ( u)$.
\end{lemma}
\begin{proof}
Since the eigenvalues of a matrix are continuous functions of its entries (as roots of the characteristic polynomial), for a matrix $A=(a_{ij})$ with
$S_k (A) >0$, we have for $\operatorname{\epsilon}ilon >0$, the existence of $\gamma >0$ depending only on the space dimension $n$ such that $|S_k ( B) - S_k ( A)| < \operatorname{\epsilon}ilon$ when $\text{sup}_{ij} |b_{ij}-a_{ij}| <\gamma$. This implies $S_k ( B)> S_k ( A) - \operatorname{\epsilon}ilon$. Thus with $\operatorname{\epsilon}ilon=S_k( A)/2$, we have $S_k ( B) > S_k ( A)/2$.
For $h$ sufficiently small we have $C h^2 |u|_{4;\Omega} < \gamma$ and thus since $S_k(D^2 u) =f > \text{inf} \, f >0 $, by \eqref{consistent} $S_k(\mathcal{H}_d( u)) \geq 1/2 \, \text{inf} \, f $.
By \eqref{G-am}
$$
\Delta_d (u) \geq \frac{1}{2} ((\text{inf} \, f)/c(k,n))^{1/k}.
$$
Let $v^h \in B_{\rho} ( u)$. Then by definition of $ B_{\rho} ( u)$ and \eqref{consistent}
\begin{align*}
||\mathcal{H}_d(v^h) - \mathcal{H}_d( u)||_{0,\alpha;\Omega_0^h} & \leq ||\mathcal{H}_d(v^h) - D^2 u||_{0,\alpha;\Omega_0^h} + ||D^2 u- \mathcal{H}_d( u)||_{0,\alpha;\Omega_0^h} \\
& \leq \rho + C h^2 |u|_{4;\Omega},
\end{align*}
which can be made smaller than $\gamma$ for $h$ and $\rho$ sufficiently small. Thus given that $\mathcal{H}_d( u)$ is positive definite, the same holds for $\mathcal{H}_d(v^h)$.
\end{proof}
\begin{thm}\label{disc-thm}
Assume that $u \in C^{4,\alpha}(\tir{\Omega})$ is $k$-admissible.
Choose $u^{0,h}$ such that
$||u^{0,h}- u||_{2,\alpha;\Omega_0^h} = O(h^2)$.
For $h$ sufficiently small, \eqref{k-H1h} has a locally unique solution $u^h$ which satisfies $\Delta_d (u^h) \geq 0$ and
$u^h$ converges to the unique solution $u$ of \eqref{k-H1} as $h \to 0$ at the rate O$(h^{2})$.
\end{thm}
\begin{proof}
It follows from the assumptions that inf $f >0$. We define the operator
$R^h: \mathcal{M}(\Omega^h) \to \mathcal{M}(\Omega^h)$ by
\begin{align*}
- \{S_k^{ij}(\mathcal{H}_d \, u^{0,h}) \} : \mathcal{H}_d (v^h-R^h v^h) &= - S_k (\mathcal{H}_d \, v^h) + f \, \text{in} \, \Omega^h_0 \\
R^h(v^h) & = g \, \text{on} \, \partial \Omega^h,
\end{align*}
and show that $R^h$ has a unique fixed point in $B_{\rho}( u)$ for $\rho=O(h^{2})$. By Remark \ref{u0h-rem} the above problem is then well defined.
It follows from \eqref{div-prop} that the operator $R^h$ is a discrete version of the operator $R$ used in the proof of Theorem \ref{contT-broyden}. Thus, as in the proof of Theorem \ref{contT-broyden}
we obtain
\begin{align*}
\{S_k^{ij}(\mathcal{H}_d \, u^{0,h}) \} : \mathcal{H}_d (R^h v^h - R^h w^h) & =
S_k (\mathcal{H}_d \, w^h) - S_k (\mathcal{H}_d \, v^h) \\
& \qquad + \{S_k^{ij}(\mathcal{H}_d \, u^{0,h}) \}:\mathcal{H}_d \,(v^h-w^h).
\end{align*}
And thus by the mean value theorem and discrete Schauder estimates, as in the proof of Theorem \ref{contT-broyden}
\begin{multline} \label{strict-step-01}
||R^h(v^h) - R^h(w^h)||_{2,\alpha;\Omega_0^h} \leq \\ C (\rho+\delta_h+||u^{0,h}||_{2,\alpha;\Omega_0^h})^{k-2} (\rho+\delta_h)||v^h-w^h||_{2,\alpha;\Omega_0^h}.
\end{multline}
Next, note that with \eqref{consistent2} applied to $u$ one has $|u|_{2,\alpha;\Omega_0^h} \leq C ||u||_{4,\alpha;\Omega}$.
It follows that $||u^{0,h}||_{2,\alpha;\Omega_0^h}\leq ||u^{}||_{2,\alpha;\Omega_0^h} + \delta_h \leq C ||u||_{4,\alpha;\Omega} + \delta_h$. We recall that by assumption
$||u^{0,h}- u||_{2,\alpha;\Omega_0^h} = O(h^2)$.
Thus
$R^h$ is a strict contraction in $B_{\rho}( u)$ for $\rho=$O$(h^{2})$ and $h$ sufficiently small.
Moreover, the contraction factor can be made smaller than 1/2 by choosing $h$ sufficiently small.
Since $f=S_k(D^2 u)$, by the discrete Schauder estimates Theorem \ref{discShauderPoisson} and Lemma \ref{est-d2}
\begin{align*}
||R^h( u) - u||_{2,\alpha;\Omega_0^h} \leq C ||S_k (D^2 u)-S_k (\mathcal{H}_d( u)) ||_{0,\alpha;\Omega_0^h} \leq C h^2.
\end{align*}
By Lemma \ref{sum-lem} we conclude that $R^h$ has a fixed point $u^h$ in $B_{\rho}( u)$ with the claimed convergence rate.
The claimed property of $u^h$ follows from the fact that $u^h \in B_{\rho}( u)$ and Lemma \ref{lboundDelta}.
\end{proof}
\section{Newton's method} \label{newton-sec}
As in the previous section, we assume that $ \{S_k^{ij}(D^2 u) \}$ is uniformly positive definite. By Remark \ref{u0h-rem},
for $h$ sufficiently small, there exists $m' >0$ such that for $v^h \in B_{\rho}( u)$, $ \{S_k^{ij}(\mathcal{H}_d v^h) \}$ has smallest eigenvalue greater than $m'$. We consider for $u^{0,h} \in B_{\rho}( u)$ the sequence of iterates
\begin{align} \label{newton}
\begin{split}
\{S_k^{ij}(\mathcal{H}_d u^{m,h}) \}: (\mathcal{H}_d u^{m+1,h} - \mathcal{H}_d u^{m,h}) & = f- S_k(\mathcal{H}_d u^{m,h}) \ \text{in} \ \Omega_0^h \\
u^{m+1,h} & = g \ \text{in} \ \partial \Omega^h.
\end{split}
\end{align}
We note that \eqref{newton} defines $u^{m+1,h}$ as the solution of a discrete second order elliptic equation in non divergence form, which is uniformly elliptic for $u^{m,h} \in B_{\rho}( u)$ for $h$ sufficiently small.
\begin{thm} \label{newton-th}
The sequence defined by \eqref{newton} satisfies
\begin{equation} \label{newton-quad}
||u^{m+1,h} - u^h||_{2,\alpha;\Omega_0^h} \leq C ||u^{m,h} - u^h||_{2,\alpha;\Omega_0^h}^2,
\end{equation}
for $\rho$ and $h$ sufficiently small and where $u^h$ denotes the solution of \eqref{k-H1h} in $B_{\rho}( u), \rho=O(h^{2})$.
\end{thm}
\begin{proof}
Put
\begin{equation} \label{B0}
B = \{S_k^{ij}(\mathcal{H}_d u^{m,h}) \}: (\mathcal{H}_d u^{m+1,h} - \mathcal{H}_d u^{h}) .
\end{equation}
We have by \eqref{k-H1h}
\begin{align} \label{B}
\begin{split}
B & = \{S_k^{ij}(\mathcal{H}_d u^{m,h}) \}: (\mathcal{H}_d u^{m,h} - \mathcal{H}_d u^{h}) +S_k (\mathcal{H}_d u^{h} )- S_k(\mathcal{H}_d u^{m,h}) \\
& = \bigg( \{S_k^{ij}(\mathcal{H}_d u^{m,h}) \} - \{S_k^{ij}(\mathcal{H}_d u^{h} ) \} \bigg): (\mathcal{H}_d u^{m,h} - \mathcal{H}_d u^{h}) \\
& \quad \quad + \{S_k^{ij}(\mathcal{H}_d u^{h}) \}: (\mathcal{H}_d u^{m,h} - \mathcal{H}_d u^{h}) +S_k (\mathcal{H}_d u^{h} )- S_k(\mathcal{H}_d u^{m,h}).
\end{split}
\end{align}
Put
\begin{equation} \label{B1}
B_1 = \bigg( \{S_k^{ij}(\mathcal{H}_d u^{m,h}) \} - \{S_k^{ij}(\mathcal{H}_d u^{h} ) \} \bigg): (\mathcal{H}_d u^{m,h} - \mathcal{H}_d u^{h}),
\end{equation}
and
\begin{equation} \label{B2}
B_2 = \{S_k^{ij}(\mathcal{H}_d u^{h}) \}: (\mathcal{H}_d u^{m,h} - \mathcal{H}_d u^{h}) +S_k (\mathcal{H}_d u^{h} )- S_k(\mathcal{H}_d u^{m,h}).
\end{equation}
By the mean value theorem, \eqref{k-Hdiv0} and \eqref{cof-estimate}, we have
$$
B_1 =\big( \{ S_k^{ij}(t \mathcal{H}_d u^{m,h} + (1-t) \mathcal{H}_d u^{h} ) \}'(\mathcal{H}_d u^{m,h} - \mathcal{H}_d u^{h}) \big): (\mathcal{H}_d u^{m,h} - \mathcal{H}_d u^{h}),
$$
for $t \in [0,1]$ and thus
\begin{align} \label{B1-est}
\begin{split}
||B_1||_{0,\alpha;\Omega_0^h} & \leq C (||u^h||_{2,\alpha;\Omega_0^h} + ||u^{m,h}||_{2,\alpha;\Omega_0^h} )^{k-2} ||u^{m,h} - u^h||_{2,\alpha;\Omega_0^h}^2 \\
& \leq C (|| u||_{2,\alpha;\Omega_0^h} +\rho )^{k-2} ||u^{m,h} - u^h||_{2,\alpha;\Omega_0^h}^2 \\
& \leq C (|| u||_{2,\alpha;\Omega_0^h} +\rho )^{k-2} ||u^{m,h} - u^h||_{2,\alpha;\Omega_0^h}^2.
\end{split}
\end{align}
We also have by the mean value theorem
\begin{align} \label{B2-est}
\begin{split}
B_2 & = \{S_k^{ij}(\mathcal{H}_d u^{h}) \}: (\mathcal{H}_d u^{m,h} - \mathcal{H}_d u^{h}) \\
& \qquad + \{S_k^{ij}( t \mathcal{H}_d u^{h} + (1-t) \mathcal{H}_d u^{m,h}) \}: (\mathcal{H}_d u^{h} - \mathcal{H}_d u^{m,h})\\
& = \bigg(\{S_k^{ij}(\mathcal{H}_d u^{h}) \} - \{S_k^{ij}( t \mathcal{H}_d u^{h} + (1-t) \mathcal{H}_d u^{m,h}) \} \bigg) : (\mathcal{H}_d u^{m,h} - \mathcal{H}_d u^{h}) \\
& = \bigg( \{S_k^{ij}( (1-s) \mathcal{H}_d u^{h} +s t \mathcal{H}_d u^{h} + s(1-t) \mathcal{H}_d u^{m,h}) \}' \\
& \qquad \qquad \qquad \big((1-t)(\mathcal{H}_d u^{h} - \mathcal{H}_d u^{m,h})\big) \bigg) : (\mathcal{H}_d u^{m,h} - \mathcal{H}_d u^{h}),
\end{split}
\end{align}
for $s,t \in [0,1]$. As for $B_1$ we obtain
\begin{align}
\begin{split}
||B_2||_{0,\alpha;\Omega_0^h} & \leq C (|| u||_{2,\alpha;\Omega_0^h} +\rho )^{k-2} ||u^{m,h} - u^h||_{2,\alpha;\Omega_0^h}^2.
\end{split}
\end{align}
Combining \eqref{B0}--\eqref{B2-est}
and using Schauder estimates, we obtain \eqref{newton-quad}.
\end{proof}
Choosing $\rho=O(h^{2})$ we have $C \rho < 1$ for $h$ sufficiently small. We conclude that $u^{m+1,h} \in B_{\rho}( u)$ when $u^{m,h} \in B_{\rho}( u)$ and the quadratic convergence rate of Newton's method.
\begin{rem}
Having established that the discrete problem has a locally unique solution and that $v^h$ is a discrete convex function for $v^h$ sufficiently close to $u$, the convergence of Newton's method also follows from the verification of standard assumptions given in \cite[p. 68]{Kelley95}. See \cite{Oberman2010a} for an example of verification of the standard assumptions for a wide stencil discretization.
\end{rem}
\section{Gauss-Seidel iterative methods} \label{convexity}
It is a natural idea to solve \eqref{k-H1h} by a nonlinear Gauss-Seidel method, that is solve \eqref{k-H1h} for $u^h(x)$ and solve the resulting nonlinear equations by a Gauss-Seidel method. Although this seems a daunting task for arbitrary $k$, we show that for $k=2$, this takes a very elegant form. We then establish a connection between the resulting nonlinear Gauss-Seidel iterative method for $2$-Hessian equations and the discrete version of \eqref{k-H-iterative}, i.e.
\begin{align} \label{k-H1-iterativeD}
\begin{split}
\Delta_d \, u^{m+1,h} & = \bigg( (\Delta_d \, u^{m,h})^k + \frac{1}{c(k,n)}( f-S_k (\mathcal{H}_d \, u^{m,h} ) )\bigg)^{\frac{1}{k}} \ \text{in} \ \Omega_0^h \\
u^{m+1,h} & = g \, \text{on} \, \partial \Omega^h,
\end{split}
\end{align}
when the Gauss-Seidel method is used to solve the Poisson equations.
\subsection{Nonlinear Gauss-Seidel method for 2-Hessian equations}
We start with the identity
\begin{align} \label{identity}
\Delta_d \, u^{h} = \bigg( (\Delta_d \, u^{h})^2 + \frac{1}{c(2,n)}(f-S_2 (\mathcal{H}_d \, u^{h} ) ) \bigg)^{\frac{1}{2}},
\end{align}
and show that the right hand side is independent of $u^h(x)$. Note that by \eqref{second-disc2}, $\partial^i_h \partial^j_h u^h(x), i \neq j$ is independent of $u^h(x)$ and
by \eqref{second-disc3},
$$
\frac{\partial (\Delta_d \, u^{h}(x))}{\partial (u^h(x))} = \sum_{i=1}^n -\frac{2}{h^2} = -\frac{2 n}{h^2}.
$$
Since $\partial S_k(A)/\partial z = \sum_{i,j=1}^n( \partial S_k(A)/\partial a_{ij}) ( \partial a_{ij}/\partial z)$,
we conclude that
\begin{align*}
\frac{\partial }{\partial (u^h(x))} S_2 (\mathcal{H}_d \, u^{h}(x) ) & = \sum_{i,j=1 \atop i \neq j}^n S_2^{ij}(\mathcal{H}_d \, u^{h}(x)) \frac{\partial}{\partial (u^h(x))} \partial^i_h \partial^j_h u^h(x) \\
& \qquad + \sum_{i=1}^n S_2^{i i}(\mathcal{H}_d \, u^{h}(x)) \frac{\partial}{\partial (u^h(x))} \partial^i_+ \partial^i_- u^h(x) \\
& = -\frac{2}{h^2} \sum_{i=1}^n S_2^{ii}(\mathcal{H}_d \, u^{h}(x)) = -\frac{2}{h^2} \sum_{i=1}^n \sum_{1 \leq p \leq n \atop p \neq i} \delta_{i p}^{i p} \,
\partial^p_+ \partial^p_- u^h(x)\\
& = -\frac{2}{h^2} \sum_{i=1}^n \sum_{p \neq i} \partial^p_+ \partial^p_- u^h(x)= -\frac{2}{h^2} (n-1) \Delta_d \, u^{h}(x)\\
& = -\frac{2}{h^2} (2 n) \, c(2,n) \Delta_d \, u^{h}(x) = -\frac{4 n}{h^2} c(2,n) \Delta_d \, u^{h}(x),
\end{align*}
and we recall that the definition of $\delta_{i p}^{i p}$ was given in section \ref{notation1}.
This gives
$$
\frac{\partial }{\partial (u^h(x))} \bigg( (\Delta_d \, u^{h}(x))^2 + \frac{1}{c(2,n)}(f-S_2 (\mathcal{H}_d \, u^{h}(x) )) \bigg) = 0.
$$
We can therefore rewrite \eqref{identity} as
\begin{align} \label{identity2}
\begin{split}
u^h(x) & = \frac{h^2}{2 n} \bigg[\sum_{i=1}^n \frac{u^h(x+he^i) + u^h(x-h e^i)}{h^2} \\
& \qquad \qquad \qquad \quad - \bigg( (\Delta_d \, u^{h}(x))^2 + \frac{1}{c(2,n)}(f-S_2 (\mathcal{H}_d \, u^{h}(x) ) \bigg)^{\frac{1}{2}} \bigg],
\end{split}
\end{align}
where the solution with $\Delta_d \, u^{h} \geq 0$ has been selected. For $n=2$, this is the identity which was solved in \cite{Headrick05,Chen2010b,Chen2010c,Benamou2010} by a Gauss-Seidel iterative method, as indicated in the introduction. For $n \geq 3$, this provides new iterative methods for the $2$-Hessian equations.
Henceforth, we shall assume that a row ordering of the elements of $\Omega^h$ is chosen. Note that if we apply the Gauss-Seidel method to the problem \eqref{k-H1-iterativeD}, we obtain a double sequence $u^{m,p,h}$ defined by
\begin{align*}
\begin{split}
u^{m+1,p+1,h}(x) & = \frac{h^2}{2 n} \bigg[\sum_{i=1}^n \frac{u^{m+1,p,h}(x+he^i) + u^{m+1,p+1,h}(x-h e^i)}{h^2} \\
& \qquad \qquad \qquad \quad - \bigg( (\Delta_d \, u^{m,h}(x))^2 + \frac{1}{c(2,n)}(f-S_2 (\mathcal{H}_d \, u^{m,h}(x) ) \bigg)^{\frac{1}{2}} \bigg],
\end{split}
\end{align*}
This leads us to consider the double sequence $u^{m,h}_p$ defined by
\begin{align*}
\begin{split}
u^{m+1,h}_{p+1}(x) & = \frac{h^2}{2 n} \bigg[\sum_{i=1}^n \frac{u^{m+1,h}_p(x+he^i) + u^{m+1,h}_{p+1}(x-h e^i)}{h^2} \\
& \qquad \qquad \qquad \quad - \bigg( (\Delta_d \, u^{m,h}_{p*}(x))^2 + \frac{1}{c(2,n)}(f-S_2 (\mathcal{H}_d \, u^{m,h}_{p*}(x) ) \bigg)^{\frac{1}{2}} \bigg],
\end{split}
\end{align*}
where $\Delta_d \, u^{m,h}_{p*}(x)$ and $S_2 (\mathcal{H}_d \, u^{m,h}_{p*}(x) )$ are the actions of the discrete Laplace and $2$-Hessian operators on $u^{m,h}_p$ updated with the most recently computed values.
Formally, as $m \to \infty$, this gives the nonlinear Gauss-Seidel method
\begin{align} \label{Gauss}
\begin{split}
u^{h}_{p+1}(x) & = \frac{h^2}{2 n} \bigg[\sum_{i=1}^n \frac{u^{h}_p(x+he^i) + u^{h}_{p+1}(x-h e^i)}{h^2} \\
& \qquad \qquad \qquad \quad - \bigg( (\Delta_d \, u^{h}_{p*}(x))^2 + \frac{1}{c(2,n)}(f-S_2 (\mathcal{H}_d \, u^{h}_{p*}(x) ) \bigg)^{\frac{1}{2}} \bigg],
\end{split}
\end{align}
where as above $\Delta_d \, u^{h}_{p*}(x)$ and $S_2 (\mathcal{H}_d \, u^{h}_{p*}(x) )$ are the actions of the discrete Laplace and $2$-Hessian operators on $u^{h}_p$ updated with the most recently computed values of $u^{h}_{p+1}$.
In particular, the right hand side of \eqref{Gauss} does not depend on $u^h_{p+1}$ since as shown above, the right hand side of \eqref{identity2} does not depend on $u^h(x)$.
\section{Numerical results} \label{num}
We give numerical results for the $\sigma_2$ problem, i.e. for $k=2, n=3$ using the subharmonicity preserving iterations.
Although our theoretical results only cover smooth solutions, as indicated in the abstract and in the introduction, the subharmonicity preserving iterations appear able to handle non smooth solutions. The initial guess in all of our numerical experiments is taken as the finite difference approximation of the solution of the Poisson equation $\Delta u = 2 \sqrt{f}$ in $\Omega$ with $u=g$ on $\partial \Omega$.
We use the following test functions on the unit cube $[0,1]^3$:
Test 1: A smooth solution which is strictly convex, $u(x,y,z)=e^{x^2+y^2+z^2}$ so that
$f(x,y,z)=4(3+x^2+y^2+z^2)e^{2(x^2+y^2+z^2)}$ and $g(x,y,z)=e^{x^2+y^2+z^2}$ on $\partial \Omega$.
Test 2: A smooth solution which is $2$-convex but not convex. It is known that for a radial function $u(x)=\phi(r), r=|x|, x \in \mathbb{R}^n$ the eigenvalues of $D^2 u$ are given by
$\lambda_1=\phi''(r)$ with multiplicity 1 and $\lambda_2=\phi'(r)/r$ with multiplicity $n-1$. See for example \cite[Lemma 2.1]{Felmer2003}. It follows that with $u(x,y,z)=\ln(a+x^2+y^2+z^2)$, we have $\phi(r)= \ln(a+r^2)$ and we get
$\Delta u = \frac{6 a + 2 r^2}{ (a+r^2)^2} \geq 0, \, S_2(D^2 u)= 4\frac{3 a-r^2}{ (a+r^2)^3} \geq 0, \, \det D^2 u = 2\frac{a-r^2} {(a+r^2)^2},$
in $[0,1]^3$. With $a=2$, $\det D^2 u$ takes negative values in $[0,1]^3$.
Test 3: A solution not in $H^2(\Omega)$, $u(x,y,z)=-\sqrt{3-x^2-y^2-z^2}$
so that $f(x,y,z)=-(x^2+y^2+z^2-9)/(-3+x^2+y^2+z^2)^2$ and $g(x,y,z)=-\sqrt{3-x^2-y^2-z^2}$ on $\partial \Omega$.
Test 4: No exact solution is known. Here
$f(x,y,z)=1$ and $g(x,y,z)=0$.
Test 5: A degenerate three dimensional Monge-Amp\`ere equation. We take $f(x,y,z)=0$ and $g(x,y,z)=|x-1/2|$.
We use the double iterative method based on \eqref{sigma2k}.
Numerically, the solution computed may not satisfy $S^2 D^2 u^{m} \geq 0$. At those points we set both $S_2 (D^2 u^{m})$ and $\det D^2 u^m$ to 0 in
\eqref{sigma2k}. If the numerical value of $S_2(D^2 u^{m})$ is negative, then 0 is a better approximate value. Since $S_2( D^2 u^{m})$ is computed from $u^m$, the numerical value of $\det D^2 u^m$ would also be inaccurate. Since $u^m$ is expected to be an approximate solution of $u$ for which $\det D^2 u \geq 0$, a better approximation of $\det D^2 u^m$ at any stage where the latter is negative is also 0. It would be interesting to analyze the effect of these rounding off errors on the overall numerical convergence of the method. For example, one may analyze the convergence of the inexact double iteration. Similar situations appear with inexact Newton's methods and inexact Uzawa algorithms.
The right hand side $f(x,y,z)$ can be computed from the exact solution $u(x,y,z)$ using the definition of $S_2(D^2 u)$ as the sum of the $2 \times 2$ principal minors.
For all tests but Test 3, we used the direct solver \eqref{k-H1-iterativeD}. For $h=2^6$, we run out of memory with \eqref{k-H1-iterativeD}. For Test 3, the Gauss-Seidel method was used since there is no memory issue for the latter with $h=2^6$. As expected, we have quadratic convergence (as $h \to 0$) for the smooth solutions of Tests 1 and 2 while enough data is not available to give the convergence rate for the singular solution of Test 3.
\begin{table}
\begin{tabular}{c|ccccc}
\multicolumn{6}{c}{$h$}\\
& $1/2^1$ & $1/2^2$ & $1/2^3$& $1/2^4$ & $1/2^5$ \\%& $1/2^{10}$\\ \hline
Error & 6.2328 $10^{-2}$ & 2.6556 $10^{-2}$ & 7.7836 $10^{-3}$ & 2.0616 $10^{-3}$ & 5.2449 $10^{-4}$ \\
Rate & &1.23 & 1.77&1.92 &1.97
\end{tabular}
\caption{Maximum error with Test 1.}
\end{table}
\begin{table}
\begin{tabular}{c|ccccc}
\multicolumn{6}{c}{$h$}\\
& $1/2^1$ & $1/2^2$ & $1/2^3$& $1/2^4$ & $1/2^5$ \\%& $1/2^{10}$\\ \hline
Error & 6.5241 $10^{-4}$ & 5.0653 $10^{-4}$ & 1.3850 $10^{-4}$ & 3.5587 $10^{-5}$ & 9.1276 $10^{-6}$ \\
Rate & &0.36 & 1.87&1.96 &1.96
\end{tabular}
\caption{Maximum error with Test 2.}
\end{table}
\begin{table}
\begin{tabular}{c|ccc}
\multicolumn{4}{c}{$h$}\\
& $1/2^4$ & $1/2^5$ & $1/2^6$ \\%& $1/2^7$ & $1/2^8$ \\%& $1/2^{10}$\\ \hline
Error & 1.1084 $10^{-3}$ & 9.7971 $10^{-4}$ & 7.6618 $10^{-4}$ \\&& & \\
Rate & &0.18 &0.35 \\%& &
\end{tabular}
\caption{Maximum error with Test 3.}
\end{table}
\begin{figure}
\caption{Test 4, $h=1/2^5$. Graph and contour in plane $z=1/2$.}
\end{figure}
\begin{figure}
\caption{Test 5, $h=1/2^4$. Graph in the plane $z=1/2$.}
\end{figure}
In
\cite{Benamou2010}, it was argued based on numerical evidence that the Gauss-Seidel method \eqref{Gauss} is faster than a certain variant of the direct solver \eqref{k-H1-iterativeD} for singular solutions. In our implementation we saw evidence of the contrary, that is, the Gauss-Seidel method is less efficient. We note that the Gauss-Seidel method requires much more loops which are not efficient in MATLAB.
\section{Concluding Remarks}
\begin{rem}
Although the pseudo-transient and time marching methods introduced in \cite{AwanouPseudo10} work as well for $k$-Hessian equations, and apply to more general fully nonlinear equations, the subharmonicity preserving iterative methods introduced in this paper are parameter free.
All these type of methods can be accelerated with fast Poisson solvers and multigrid methods.
\end{rem}
\begin{rem}
When it comes to numerical methods for fully nonlinear equations, there are two types of convergence to study. Since the equations are nonlinear, they must be solved iteratively. One must then address the convergence to the discrete solution of the iterative methods used. The second type of convergence is the convergence of the numerical solution to the exact solution as the discretization parameter converges to 0.
We have addressed both types of convergence in this paper.
\end{rem}
\begin{rem}
Existence of a discrete solution and convergence (as the mesh size $h \to 0$), for finite difference discretization of smooth solutions of fully nonlinear equations, are not often discussed.
It is clear that convergence does not simply follow from the consistency of standard finite difference discretization of the second order derivatives. For viscosity solutions, convergence of monotone, stable and consistent schemes follows immediately from the theory of Barles and Souganidis.
\end{rem}
\begin{rem}
The iterative method \eqref{k-H-iterative} can be viewed as a linearization of the fully nonlinear equation \eqref{k-H1}. It is possible to linearize \eqref{k-H1} in ways different from \eqref{broyden} and \eqref{k-H-iterative}. See for example the methods described in \cite{AwanouPseudo10}. The iterative method \eqref{k-H-iterative} has been shown numerically to select discrete solutions which converge to non smooth solutions. Since \eqref{k-H-iterative} consists of a sequence of Poisson equations, the numerical solution of \eqref{k-H1} can now be tackled with any good numerical method.
\end{rem}
{\bf Acknowledgments.}
The author would like to thank the referees for a careful reading of the manuscript. The author is grateful to M. Neilan for many useful discussions. The author was supported in part by NSF grants DMS-0811052, DMS-1319640 and the Sloan Foundation.
\end{document} | math |
یہِ کیٛاہ چھُ | kashmiri |
Little Sonny is now approaching the eve of his 86th birthday. Being blessed with a longevity gene courtesy of his mother Pasqualina (she lived to almost 100), he is a force of nature and amazement to anyone who has been lucky enough to know him.
He loves to cook in the old world ways just as his mother and father taught him, and he loves to share that gift with all around him. A more generous spirit of a man would be hard to find.
Upon entering the Rossi kitchen, his son Giorgio would ask, “What’s cooking, Pops?” and the answer shot back was always “You eat what I cook you!”. Picky eaters with dietary disciplines were never an interest in Alfredo and Pasqualina’s house, and as the head chef and bottle washer in his own kitchen, Poppa Nick wasn’t about to change tradition.
Which was fine. Anything he cooks, he cooks with love and a sense of history, along with a sense of artistry as well. “Eating what He Cooks You” is a real authentic experience.
A couple of years ago, Giorgio approached both Nick and Linda on the idea of recording the family cuisine on video for family use; both as tutorials for the third and fourth generations of the Rossi family, but also as a way to archive and preserve those old world culinary customs and tricks to ensure that they wouldn’t be immediately lost in the march of time.
Giorgio posted a few of Poppa Nick’s kitchen hi-jinks on YouTube, and well... here we are.
What started as an insignificant family portrait project through food preparation has grown into a fairly large “revolving virtual dinner party” with the advent of social networking; where if the recipes are followed to specification, anybody can now eat what Nick cooks for them. And people have started cooking them a lot!
He gets to potentially throw this party for anybody that wants to come, and conveniently avoid having to clean a mountain of dirty dishes in the aftermath.
As a direct descendant of Neopolitan fishermen and pasta manufacturers, Nick knows a thing or two about Italian "Peasant Cuisine".
This is the cuisine that he has taught his children and grandchildren how to prepare, and that he still prepares, serving it up and sharing it with friends and family in his home for over 85 years on planet earth. Its the family voodoo stew and secret family alchemical formulas.
In the end, the food and the preparation of it in tutorial form is just a delivery system of the love and history of a family.
Assisted rather clumsily by his son Giorgio “Secondo” Rossi, Poppa “Primo” knows YOU EAT WHAT I COOK YOU is about love; giving and sharing unconditionally, spreading warmth and good cheer whenever you can recognize an opportunity to do so.
This is just a big internet version of what anyone who crosses over the family’s threshold would experience when they are arriving for dinner.
“Siddown... have a glass of Prosecco and some Peppers! | english |
Boulder traps hiker for nine hours in Northern Idaho's Selkirk Mountains, results in Air Force rescue via helicopter airlift.
Ammi Midstokke is airlifted out of harm’s way by a UH-1N Iroquois helicopter from the Fairchild Air Force Base in Washington.
Ammi Midstokke after the accident at Northern Idaho’s Selkirk Mountains.
Ammi Midstokke’s foot after the boulder collapsed on it.
The 2010 film 127 Hours depicts every hiker’s worst nightmare: getting hurt and being trapped for an extended amount of time with nobody around to help.
Luckily for 36-year-old Ammi Midstokke, she was only trapped for nine hours and had the help of her hiking partner after a 1.5 ton boulder came crashing down on her right foot in Northern Idaho’s Selkirk Mountains.
Midstokke and her hiking partner, Jason Luthy, had successfully reached the summit of Chimney Rock on Sept. 19. The duo was climbing out when a boulder moved on Midstokke, hitting her in the head and then pinning her, landing on her foot.
Luthy, a trained paramedic, tried to move the boulder with no luck.
“We called search and rescue and tried to stabilize me with heat and good stories,” Midstokke wrote on Facebook after the ordeal.
Luthy called search and rescue around 5:30 p.m. By 8 p.m., more than 20 rescuers from Priest Lake Search and Rescue and the Bonner County EMS Wilderness Response Team set out to look for Midstokke, according to the Spokesman-Review.
The team reached Midstokke and Luthy around 1 a.m., after navigating steep and narrow terrain in the dark. Once there, it took less than an hour for a web and pulley system to lift the boulder off of Midstokke’s “very deformed, very dead looking foot,” as she said on Facebook.
Hiking out of the area, however, wasn’t an option. At 7:10 a.m., a four-member crew from the 36th Rescue Flight at Fairchild Air Force Base in Washington was contacted. The team used a UH-1N Iroquois helicopter to fly in at 7:45 a.m. and take Midstokke out of the area.
After securing Midstokke to a harness, the team airlifted her off of the mountain at 8:35 a.m.
The result of her injury?
“Initial X rays show tarsal breakage, but a remarkably whole foot,” Midstokke said.
Donations can be made to Priest Lake Search and Rescue here. | english |
مٔلِکانہِ ییٚلہِ یہِ وُچھ تَس گۄو یہِ تیوٗت خۄش زِ تَمہِ تُج نہٕ بادشاہَس تھپھٕے زِ سُہ انناوِ ہے بییٚہِ تہِ یِہوے کَپُر | kashmiri |
\begin{document}
\title{ THE ASYMPTOTIC BEHAVIOR OF QUSI-HARMONIC FUNCTIONS AND EIGENFUNCTIONS OF DRIFT LAPLACIAN AT INFINITY}
\keywords{Quasi-Laplacian, singularity, asymptotic behavior.}
\thanks{\noindent \textbf{MR(2010)Subject Classification} 47F05 58C40}
\author{min chen}
\author{jiayu Li}
\author{yuchen Bi}
\address[Corresponding author] {University of Science and Technology of China, No.96, JinZhai Road Baohe District,Hefei,Anhui, 230026,P.R.China.}
\email{cmcm@mail.ustc.edu.cn}
\thanks{The research is supported by the National Nature Science Foudation of China No. 11721101 No. 11526212 }
\pagestyle{fancy}
\fancyhf{}
\renewcommand{0pt}{0pt}
\fancyhead[CE]{}
\fancyhead[CO]{\leftmark}
\fancyhead[LE,RO]{\thepage}
\begin{abstract}
Note that $\mathbb{R}^m$ with the metric $g=e^{-\frac{|x|^2}{2(m-2)}}ds_0^2$ is actually a Riemannian manifold with a singularity at $\infty.$ The metric is quite singular at infinity and it is not complete. Colding-Minicozzi \cite{11} pointed out that the Ricci curvature of this metric does not have a sign and goes to negative infinity at infinity and thus there is no way to smoothly extend the metric to a neighborhood of infinity. Chen-Li \cite{7} proved that any non-constant quasi-harmonic function or eigenfunction of drift Laplacian is discontinuous at infinity. In this paper, we show expansions of quasi-harmonic functions and of eigenfunctions of drift-Laplacian in terms of spherical harmonics. Using these expansions, we have a more precise description of the asymptotic behavior of quasi-harmonic functions and of eigenfunctions of drift-Laplacian at infinity. Moreover, we improve the Liouville theorem of quasi-harmonic functions and eigenfunctions of drift-Laplacian by reducing the requirement of the conditions.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\section{Introduction}
\setcounter{equation}{0}
Chen-Li \cite{7} studied eigenfunctions of Quasi-Laplacian $\Delta_g = e^{\frac{|x|^2}{2(m-2)}}(\Delta_{g_0} - \nabla_{g_0} h \cdot \nabla_{g_0}) = e^{\frac{|x|^2}{2(m-2)}} \Delta_h $ for $h = \frac{|x|^2}{4}$ and proved that any non-constant quasi-harmonic function is discontinuous at infinity in Corollary 3.3 and any non-constant eigenfunction of drift Laplacian $\Delta_h = \Delta_{g_0} - \nabla_{g_0} h \cdot \nabla_{g_0}$ is discontinuous at infinity in Theorem 1.4, which means that any quasi-harmonic function or eigenfunction of drift Laplacian could not converge to a constant at infinity.
Recently, Colding-Minicozzi defined a frequency function $U(r) = \frac{r}{2} (\log{I})'$ where $I(r) = r^{1-n} \int_{\partial B_r} u^2$ and used the mean value $\sqrt{I}$ of $u$ to measure the rate of the growth of eigenfunctions of drift Laplacian $\mathcal{L} = \Delta_{h}$ in Theorem 4.8 in \cite{5} and Theorem 1.1 in \cite{6}.
\begin{theorem}\label {thm:1.4}
\cite{6} Given $\epsilon > 0$ and $\delta > 0$, there exist $r_1 > 0$ such that if $\mathcal{L}u = -\lambda u$ and $U(\bar{r}_1) \ge \delta + 2 \sup\{0, \lambda\}$ for some $\bar{r}_1 \ge r_1$, then for all $r \ge R(\bar{r}_1)$
\begin{align*}
U(r) > \frac{r^2}{2} - n - 2\lambda - \epsilon.
\end{align*}
\end{theorem}
$U(r) > \frac{r^2}{2} - n - 2\lambda - \epsilon$ implies that $\sqrt{I} \ge Ce^{\frac{r^2}{4}} r^{-(n + 2\lambda - \epsilon)}$. Theorem 1.4 shows that there is a sharp dichotomy for the growth of eigenfunctions of $\mathcal{L}$: either $\sqrt{I} \le Cr^{2(\delta + 2\lambda)}$, $u$ grows at most polynomially; or $\sqrt{I} \ge Ce^{\frac{r^2}{4}} r^{-(n + 2\lambda - \epsilon)}$, $u$ grows at least like $Ce^{\frac{r^2}{4}} r^{-(n + 2\lambda - \epsilon)}$. They use $\sqrt{I}$ to describe the asymptotic behavior of $u$. In this paper, we give the expansion of $u$ in terms of spherical harmonics in Lemma 3.2 to see its asymptotic behavior directly.
To know the sharp dichotomy phenomenon of the growth rate, we compare the expansion of quasi-harmonic function in Lemma 2.3 with the expansion harmonic function.
Assume $\varphi_k(\theta)$ is an eigenfunction of $L^2(S^{m-1})$ corresponding to the eigenvalue $\lambda_k$ and $C(N)$ is an Euclidean Cone $(0,\infty)\times N^{m-1}.$
\begin{theorem}\label {thm:1.5}
\cite{10} If $u$ is a harmonic function on $C(N)$, then
\[ u(r,\theta) = \sum c_k r^{p_k} \varphi_k(\theta),\]
where the $c_k$ are constants and $p_k = \frac{-(m-2) + \sqrt{(m-2)^2 + 4\lambda_k}}{4}$ increases strictly from $0$ to $+\infty$ as $k\rightarrow +\infty$. Furthermore, $u$ has polynomial growth if and only if this is a finite sum.
\end{theorem}
We know that the property of quasi-harmonic and harmonic functions are quiet different.
Let us recall some basic results about the metric $g=e^{-\frac{|x|^2}{2(m-2)}}ds_0^2.$ Lin-Wang \cite{1} introduced the quasi-harmonic sphere, which is a harmonic map from $M=\big(\mathbb{R}^m, e^{-\frac{|x|^2}{2(m-2)}}ds_0^2\big)$ to $N$ with finite energy when they study the regularity of the heat flow of harmonic maps(c.f.\cite{12}). Here $ds_0^2$ is Euclidean metric in $\mathbb{R}^m$. Colding-Minicozzi \cite{11} also pointed out self-shrinkers in $\mathbb{R}^{m-1}$ are minimal hypersurfaces for the metric $g=e^{-\frac{|x|^2}{2(m-2)}}ds_0^2.$ Note that $\mathbb{R}^m$ with this metric is actually a Riemannian manifold with a singularity at $\infty.$ The compactification of $\mathbb{R}^m$ provided by this metric is a topological $m-$sphere. Colding-Minicozzi \cite{11} mentioned that the Ricci curvature of this metrics does not have a sign and goes to negative infinity at infinity and thus there is no way to smoothly extend the metric to a neighborhood of infinity. The metric $g=e^{-\frac{|x|^2}{2(m-2)}}ds_0^2$ is quite singular at infinity and it is not complete. Ding-Zhao \cite{2} showed that if the target $N$ is a sphere, any equivariant quasi-harmonic sphere is discontinuous at infinity and conjectured that any non-constant quasi-harmonic sphere is discontinuous at infinity.
In this paper, we will give a more precise description of the behavior of quasi-harmonic function and eigenfunctions of $\Delta_h$ near the infinity.
Assume $u_0 = \frac{1}{2} \int_{0}^{r} e^{\frac{r^2}{4}} r^{1-m} d r,$ which is an radially symmetric solution of $\Delta_g u = 0$, we will show that $\frac{u(r,\theta)}{u_0(r)}$ could be asymptotic to any given function $g(\theta) \in H^{[\frac{m}{2}]+2}(S^{m-1})$ from the following result.
\begin{theorem}\label {thm:1.6}
Assume $\bar{M} = (\mathbb{R}^m, g)$, for any given function $g(\theta) \in H^{[\frac{m}{2}]+2}(S^{m-1})$, there exists a quasi-harmonic function $u(r,\theta)$ (i.e., $\Delta_g u = 0$) on $\mathbb{R}^m\backslash \{0\}$ such that $\lim\limits_{r \to +\infty} \frac{u(r,\theta)}{u_0(r)} = g(\theta)$. Moreover, if $\frac{u(r,\theta)}{u_0(r)}$ and $\frac{\bar{u}(r,\theta)}{u_0(r)}$ are asymptotic to the same function $g(\theta)$, then $u(r,\theta) = \bar{u}(r, \theta) + c.$
\end{theorem}
Similarly, assume $u_0 = e^{\frac{r^2}{4}} r^{-(m+2\lambda)}$, we have
\begin{theorem}\label {thm:1.7}
If $ 2\lambda$ is not an integer, for any given function $g(\theta) \in H^{[\frac{m}{2}]+2}(S^{m-1})$, there exists an eigenfunction $u(r,\theta)$ of $\Delta_h$ on $\mathbb{R}^m\backslash \{0\}$ such that $\lim\limits_{r \to +\infty} \frac{u(r,\theta)}{u_0(r)} = g(\theta)$. Moreover, if $\frac{u(r,\theta)}{u_0(r)}$ and $\frac{\bar{u}(r,\theta)}{u_0(r)}$ are asymptotic to the same function $g(\theta)$, then $u(r,\theta) = \bar{u}(r, \theta)+p(r)$, where $p(r) \sim r^{2\lambda}$. If $\lambda$ is an integer, for any given function $g(\theta) \in H^{[\frac{m}{2}]+2}(S^{m-1})$ satisfying that $\langle g, \varphi_k \rangle = 0$ when $k \in \{0, m-2, m, m+2, \cdots, m+2\lambda\} = A$. Then there exists an eigenfunction $u(r, \theta)$ of $\Delta_h$ on $\mathbb{R}^m\backslash \{0\}$ such that $\lim\limits_{r \to +\infty} \frac{u(r,\theta)}{u_0(r)} = g(\theta)$. If $2\lambda$ is an integer and $\lambda$ is not an integer, for any given function $g(\theta) \in H^{[\frac{m}{2}]+2}(S^{m-1})$ satisfying that $\langle g, \varphi_k \rangle = 0 $ when $k \in \{m-2, m, m+2, \cdots, m+2[\lambda]\} = B$. Then there exists an eigenfunction $u(r, \theta)$ of $\Delta_h$ on $\mathbb{R}^m\backslash \{0\}$ such that $\lim\limits_{r \to +\infty} \frac{u(r,\theta)}{u_0(r)} = g(\theta)$.
\end{theorem}
Finally, we consider Liouville theorem of quasi-harmonic function with these expansions.One may wonder whether the quasi-harmonic functions still possess the basic properties of harmonic functions. Cheng-Yau \cite{3} proved that any harmonic function with sub-linear growth on manifolds with non-negative Ricci curvature must be constant. Li-Wang \cite{4} showed that there is no non-constant positive quasi-harmonic function on $\mathbb{R}^m$ with polynomial growth in Theorem 4.2. In this paper, we can improve this result by replacing the condition of polynomial growth with exponential growth.
\begin{theorem}\label {thm:1.1}
Let $u$ be a quasi-harmonic function (i.e, $\Delta_gu=0$) in $\mathbb{R}^m$. If there exists a sequence $r_i \rightarrow +\infty$ such that $\big(\int_{S^{m-1}} ( u(r_i,\theta))^2\big)^{\frac{1}{2}} \le Ce^{\frac{r_i^2}{4}}r_i^{-(m+\epsilon)}$ for some $\epsilon > 0$, then $u$ is a constant.
\end{theorem}
Colding-Minicozzi mentioned that if $\Delta_hu=0$ and $||u||_{L^2(\mathbb{R}^m)}^2=\int_{\mathbb{R}^m}u^2e^{-h} < \infty$ in \cite{6}, then $u$ must be constant. More generally, they showed the following result in \cite{5}.
\begin{lemma}
\cite{5} If $\Delta_{h}u=-\lambda u$ on $\mathbb{R}^m$ and $\int_{\mathbb{R}^m}u^2e^{-h} < \infty$, then $\lambda$ is a half-integer and $u$ is a polynomial of degree $2\lambda$.
\end{lemma}
\begin{theorem}\label {thm:1.3}
If $\Delta_{h}u=-\lambda u$ on $\mathbb{R}^m\backslash \{0\}.$ If there exists a sequence $r_i \rightarrow +\infty$ such that $\big(\int_{S^{m-1}} ( u(r_i,\theta))^2\big)^{\frac{1}{2}} \le C e^{\frac{r_i^2}{4}} r_i^{-(m+2\lambda + \epsilon)}$ for some $\epsilon > 0$, then $u$ is a polynomial of degree $2 \lambda$ when $2 \lambda$ is an integer or $u=p(r)$ satisfying $p(r) \sim r^{2\lambda}$ when $2 \lambda$ is not an integer.
\end{theorem}
Since $\int_{\mathbb{R}^m}u^2e^{-h} =\int^{+\infty}_0\int_{S^{m-1}}u^2e^{-\frac{r^2}{4}}< \infty$ implies that there exists a sequence $r_i \rightarrow +\infty$ such that $\big(\int_{S^{m-1}} ( u(r_i,\theta))^2\big)^{\frac{1}{2}} \le Ce^{\frac{r_i^2}{8}}$. Theorem 1.5 and Theorem 1.7 can also be see as a generalization of Lemma 1.6.
\hspace{0.4cm}
\section{The asymptotic behavior of quasi-harmonic functions at infinity}
\setcounter{equation}{0}
Assume that $u$ is quasi-harmonic function , i.e.,
\begin{equation}
\Delta_g u = 0.
\end{equation}
We rewrite it in the following form
\begin{equation}
\Delta u - (\nabla h, \Delta u) = 0,
\end{equation}
where $h = \frac{r^2}{4}$. We know that the Euclidean metric of $\mathbb{R}^m$ can be written in spherical coordinates $(r, \theta)$ as
\[ ds^2 = dr^2 + r^2d\theta^2,\]
where $d\theta^2$ is the standard metric on $S^{m-1}$. Then
\[\Delta = \Delta_r + \frac{1}{r^2}\Delta_\theta,\]
where $\Delta_\theta$ is the Laplacian on the standard $S^{m-1}$. It is clear that
\[ \nabla h \cdot \nabla = \frac{r}{2} \frac{\partial}{\partial r}.\]
It follows from (2.2) that
\[ u_{rr} + \frac{m-1}{r} u_r + \frac{1}{r^2} \Delta_\theta u - \frac{r}{2} \frac{\partial u}{\partial r} = 0.\]
Let $\varphi_k$ be the orthonormal basis on $L^2(S^{m-1})$ corresponding to the eigenvalues,
\[ 0 = \lambda_0 < \lambda_1 \le \lambda_2 \le \cdots \le \lambda_k \to \infty \]
we have
\[ \Delta_\theta \varphi_k = - \lambda_k \varphi_k. \]
Let $\langle \cdot, \cdot \rangle$ denote $L^2$ inner product of $L^2 (S^{m-1})$. Then we have
\begin{align*}
\langle \Delta_r u, \varphi_k \rangle &= \Delta_r \langle u, \varphi_k \rangle,\\
\langle \Delta_\theta u, \varphi_k \rangle &= \langle u, \Delta_\theta \varphi_k \rangle = -\lambda_k \langle u, \varphi_k \rangle,\\
\langle \frac{\partial u}{\partial r}, \varphi _k \rangle &= \frac{\partial \langle u, \varphi_k \rangle}{\partial r}.
\end{align*}
Let $f_k(r) = \langle u(r,\cdot), \varphi_k \rangle$ for $k \ge 0$. Then we see that $f_k$ satisfies
\begin{equation}
(f_k)_{rr} + (\frac{m-1}{r} - \frac{r}{2})(f_k)_r = \frac{\lambda_k f_k}{r^2}.
\end{equation}
Let
\begin{align*}
&f_k(r) = w(z)z^{l_k}, \\
&z = r^2, \\
&l_k = \frac{-(m-2) + \sqrt{(m-2)^2 + 4\lambda_k}}{4}.
\end{align*}
Then
\begin{align*}
(f_k)_r &= 2(w'(z)z^{l_k + \frac{1}{2}} + l_kz^{l_k - \frac{1}{2}}w(z)), \\
(f_k)_{rr} &= 4w''(z)z^{l_k+1} + 2(4l_k + 1)z^{l_k}w'(z) + l_k(4l_k - 2)z^{l_k -1}w(z).
\end{align*}
Then by (2.3), we have
\begin{equation}
zw''(z) + ((2l_k + \frac{m}{2}) - \frac{z}{4})w'(z) - \frac{l_k}{4} w(z) = 0.
\end{equation}
Assume
\[ w(z) = e^{\frac{z}{4}}y(-\frac{z}{4}).\]
Then
\begin{align*}
w_z &= \frac{1}{4} e^{\frac{z}{4}}(y(-\frac{z}{4}) - y'(-\frac{z}{4})), \\
w_{zz} &= \frac{1}{16} e^{\frac{z}{4}}(y(-\frac{z}{4}) - 2y'(-\frac{z}{4}) + y''(-\frac{z}{4})).
\end{align*}
By (2.4), we have
\begin{equation}
\frac{z}{4} y''(-\frac{z}{4}) + (-\frac{z}{4} - (2l_k + \frac{m}{2}))y'(-\frac{z}{4}) + (l_k + \frac{m}{2})y(-\frac{z}{4}) = 0.
\end{equation}
Let $x = -\frac{z}{4}$, then we have
\begin{equation}
xy''(x) + ((2l_k + \frac{m}{2}) -x)y'(x) - (l_k + \frac{m}{2})y(x) = 0.
\end{equation}
\begin{lemma}
For $k\ge 1,$ the general solution of (2.3) is
\[ f_k(r) = c_1 e^{\frac{r^2}{4}} r^{2l_k} F[l_k + \frac{m}{2}, 2l_k + \frac{m}{2}; -\frac{r^2}{4}],\]
where $F[a,b;x]$ is a Kummer's function in \cite{9} (see Page 2).
\end{lemma}
\begin{proof}
Set $b = 2l_k + \frac{m}{2}, a = l_k + \frac{m}{2}$. Assume that one solution of the Kummer's equation (2.6) is
\[ y = a_0 x^c + a_1 x^{c+1} + a_2 x^{c+2} + \cdots + a_n x^{c+n} + \cdots .\]
If we substitute this series and its first two derivatives in the differential equation, and then equate to zero the coefficients of powers of $x$, we find that
\begin{align*}
a_0 c(c + b - 1) &= 0, \\
a_1 (c + 1)(c + b) &= a_0(c+a), \\
a_2(c+2)(c + b + 1) &= a_1 (c + a + 1), \\
&\cdots
\end{align*}
Since $a_0\ne 0,$ it has two roots:\\
(i) $c=0$, we have
\[ a_n = \frac{a\cdots (a+n-1)}{b\cdots (b+n-1)n!},\]
which gives one solution in terms of Kummer's series
\[ y = a_0 F [a,b;x].\]
(ii) $c = 1 - b$, which leads to a second solution
\begin{align*}
a_1 (2 - b)& = a_0 (1 - b + a), \\
&\cdots \\
a_{n-1} (n - b)(n-1) &= a_{n-2} (1-b+a+n-2), \\
a_n (n + 1 - b)n & = a_{n-1} (1 - b + a + n -1), \\
a_{n+1}(n+2-b)(n + 1)& = a_n(1-b+a+n), \\
&\cdots
\end{align*}
If $b = n $ for some integer $n$, since $b > a > 1$, then $1 - b + a \le 0$ and $1 - b + a +n - 2 > 0$. It implies that the solution is a polynomial of some degree $i\le n$
\[ y = x^{1-b} (a_0 + a_1x + \cdots + a_i x^i).\]
If $b \neq n $ for any integer $n > 0$, then
\[a_k = \frac{(1-b+a)\cdots(1-b+a + k -2)}{((k+1)-b)\cdots (2-b)k!},\]
and
\[y = x^{1-b}(a_0 + a_1x + \cdots + a_k x^k+ \cdots).\]
Then the solution of (2.3) is
\begin{align*}
f_k(r) &= c_1 e^{\frac{r^2}{4}} r^{2l_k} y(-\frac{r^2}{4}) \\
&= c_1 e^{\frac{r^2}{4}} r^{2l_k} (-\frac{r^2}{4})^{1-(2l_k + \frac{m}{2})}(a_0 + a_1(-\frac{r^2}{4}) + \cdots) \\
&= (-4)^{2l_k + \frac{m}{2} - 1} c_1 e^{\frac{r^2}{4}} r^{2 - (2l_k + m)}(a_0 + a_1(-\frac{r^2}{4}) + \cdots),
\end{align*}
which implies that $\lim\limits_{r \to 0} f_k(r) \neq 0$. This contradicts with the initial condition that
\[ f_k(0) = \lim_{r \to 0} f_k(r) = \langle \lim_{r \to 0} u(r,\theta), \varphi_k \rangle = \langle u(0), \varphi_k \rangle = 0.\]
Hence the second solution should be ruled out and the solution of (2.3) has the
following form
\[ f_k(r) = c_k e^{\frac{r^2}{4}} r^{2l_k} F[l_k + \frac{m}{2}, 2l_k + \frac{m}{2}; -\frac{r^2}{4}].\]
\end{proof}
\begin{lemma}
For any fixed $\delta>0$ and $k\ge 1,$ we set $a=l_k + \frac{m}{2}, b=2l_k + \frac{m}{2}, c=-a+\delta$. Assume $L> \delta$ and $x$ is a positive real number, the following
asymptotic relation holds:
\[ F[a, b; -x] = x^{-a} \frac{\Gamma(b)}{\Gamma(b-a)}\Big(1 +\sum_{n=1}^{[L-\delta]} \frac{(a)_n(1+a-b)_n}{n!} x^{-n} + \frac{\Gamma(b-a)}{\Gamma(a)}J_k(x)x^a\Big), \]
where $(a)_n=a(a+1)\cdots(a+n-1)$ and $J_k(x)= \int_{c-i\infty}^{c+i\infty} \frac{\Gamma(L-s)\Gamma(a - L +s)}{2\pi i \Gamma(b-L+s)} x^{s-L}ds$. Moreover, $\left| J_k(x)\right|\le C(m,L,\delta)l_k^{2(L-\delta)+\frac{m}{2}} |x|^{c-L}$ for $k$ sufficient large.
Assume $g_k(x)=\sum_{n=1}^{[L-\delta]} \frac{(a)_n(1+a-b)_n}{n!} x^{-n} + \frac{\Gamma(b-a)}{\Gamma(a)}J_k(x)x^a$, in particular, if $k$ is sufficient large, we set $L=2\delta,$ then
\[ |g_k(x)| \le C(m,\delta)\frac{l_k^{2\delta}}{x^{\delta}},\]
and if $k\le K$ for some $K>0$, we set $L=\delta+1,$ then
\[|g_k(x)| \le C(m,K,\delta)\frac{1}{x}.\]
\end{lemma}
\begin{proof}
Using the results in \cite{9} (see Page 36), we have
\begin{equation}
|z^s| = |z|^{\text{Re}(s)} e^{-\text{Im}(s)\arg{z}},
\end{equation}
and
\begin{equation}
\frac{\Gamma (a)}{\Gamma (b)} F[a, b ; -x] = \frac{1}{2\pi i}\int_{c-i\infty}^{c + i\infty}\frac{\Gamma(-s)\Gamma(a+s)}{\Gamma(b+s)} x^s ds,
\end{equation}
provided that $|\arg{x}| < \frac{1}{2} \pi$ and $b \neq 0, -1, -2, \cdots$. Now we will deduce the asymptotic expansion in $x$ for Kummer's function. Let us consider the integral
\[ I_k = \frac{1}{2\pi i} \int_{ADEF} \frac{\Gamma(-s)\Gamma(a+s)}{\Gamma(b+s)} x^s ds\]
round the rectangular contour
\[ A(c - iN),\ D(c + iM), \ E(c - L + iM),\ F(c - L - iN)\]
in the $s-$ space.
\begin{align*}
I_k &= \int_{AD} + \int_{DE} + \int_{EF} + \int_{FA} \\
&=I_{M,N} + J_4 + J_5 + J_6.
\end{align*}
As $M$ and $N \rightarrow \infty,$
\begin{align*}
&J_4 \to 0, \\
&J_6 \to 0, \\
&-J_5 \to J_k = \int_{c-i\infty}^{c+i\infty} \frac{\Gamma(L-s)\Gamma(a - L +s)}{2\pi i \Gamma(b-L+s)} x^{s-L}ds, \\
&I_{M,N} \to I_1 = \int_{c-i\infty}^{c+i\infty} \frac{\Gamma(-s)\Gamma(a + s)}{2\pi i \Gamma(b+s)} x^s ds \\
&= \frac{\Gamma(a)}{\Gamma(b)}F[a,b;-x],
\end{align*}
and
\[ I_k = -\sum_{n=0}^{[L]} \frac{\Gamma(a+n)}{\Gamma(b-a-n)} \frac{(-1)^{n-1}}{n!} x^{-a-n},\]
from the residues at the poles $s=-a, -a-1, \cdots, -a - [L - \delta]$. Then we have
\begin{align*}
F[a,b;x] &= \frac{\Gamma (b)}{\Gamma (a)}(I_k + J_k) \\
&= x^{-a} \frac{\Gamma(b)}{\Gamma(b-a)}(\sum_{n=0}^{[L-\delta]} \frac{(a)_n(1+a-b)_n}{n!} x^{-n} + \frac{\Gamma(b-a)}{\Gamma(a)}J_k(x)x^a).
\end{align*}
Assume $s= it + c$, then
\[ J_k = \int_{-\infty}^{+\infty} \frac{\Gamma(L-it-c)\Gamma(a-L+it+c)}{2\pi i \Gamma(b-L+it+c)}x^{it+c-L} idt.\]
By (2.7), we have
\[|x^{it+c-L}| = |x|^{c-L} e^{-(\arg{x})t}, \]
if $x$ is a positive real number, we have
\[|x^{it+c-L}| = x^{c-L} . \]
We know that the classical stirling's approximate formula for the Gamma-Function
in the form
\[ \Gamma(z) = \sqrt{2\pi} e^{-z} z^{z-\frac{1}{2}}(1 + O(\frac{1}{|z|})),\]
for $|\arg{z}| < \pi$ as $|z| \to \infty$. In absolute value:
\[ |\Gamma(x + iy)| = \sqrt{2\pi}e^{-x}(x^2 + y^2)^{\frac{x-\frac{1}{2}}{2}}e^{-y\arg{(x+iy)}}, \ as \ \sqrt{x^2 + y^2} \to +\infty. \]
If we set $a = l_k + \frac{m}{2}$, $b = 2l_k + \frac{m}{2}$ and $ c = -a + \delta,$ then\begin{align*}
&L-c = l_k + \frac{m}{2} + L - \delta, \\
&a +c -L = \delta -L,\\
&b + c - L = l_k + \delta - L.
\end{align*}
Thus
\begin{align*}
|\Gamma(L - it -c)| &\sim ((L -c)^2 + t^2)^{\frac{L-c-\frac{1}{2}}{2}} \exp({t \arg{(L-it-c)}}) e^{c-L}, \\
|\Gamma(a -L + it + c)| &\sim ((a + c - L)^2 + t^2)^{\frac{a-L-c-\frac{1}{2}}{2}} \\
&\exp(-t\arg{(a - L + c+ it)})\exp(-(a-L+c)),\\
|\Gamma(b -L + it + c)| &\sim ((b - L + c)^2 + t^2)^{\frac{b-L+ c-\frac{1}{2}}{2}} \\
&\exp(-t\arg{(b - L + c+ it)}) \exp(-(b-L+c)).
\end{align*}
If $k$ is large enough, we have
\begin{align*}
&\frac{((L-c)^2+t^2)^{\frac{L-c-\frac{1}{2}}{2}}((a+c -L)^2 + t^2)^{\frac{a-L+c-\frac{1}{2}}{2}}}{((b - L +c)^2 + t^2 )^{\frac{b-L+c - \frac{1}{2}}{2}}} \\
&=\frac{((L-\delta + l_k + \frac{m}{2})^2 + t^2)^{\frac{l_k + \frac{m}{2} + L -\delta -\frac{1}{2}}{2}}((L-\delta)^2 + t^2)^{\frac{\delta - L - \frac{1}{2}}{2}}}{((l_k +\delta - L)^2 + t^2)^{\frac{l_k - L +\delta - \frac{1}{2}}{2}}} \\
&=(1 + \frac{(m + 4(L-\delta))(l_k - L + \delta) + (\frac{m}{2} + 2(L-\delta))^2}{t^2 + (l_k - L +\delta)^2})^{\frac{l_k - L + \delta - \frac{1}{2}}{2}} \\
&((l_k + \frac{m}{2} + L - \delta)^2 + t^2)^{\frac{2(L-\delta) + \frac{m}{2}}{2}}((L-\delta)^2 + t^2)^{\frac{\delta -L - \frac{1}{2}}{2}} \\
&\le \frac{\exp(\frac{m}{2} + 2(L - \delta))}{(L-\delta)^{L-\delta}}((l_k + \frac{m}{2} + L -\delta)^2 + t^2)^{\frac{2(L-\delta) + \frac{m}{2}}{2}}\\
&\le C(m,L,\delta)((l_k + \frac{m}{2} + L -\delta)^2 + t^2)^{\frac{2(L-\delta) + \frac{m}{2}}{2}}.
\end{align*}
For $t > 0$, we have
\begin{align*}
&\frac{\exp(t\arg(L-it-c))\exp(-t\arg(a-L+c+it))}{\exp(-t\arg(b-L+c+it)} \\
&=\exp\big(-t \arctan \frac{t}{L-c} - t(\pi + \arctan \frac{t}{a-L+c}) + t\arctan \frac{t}{b-L+c}\big) \\
\end{align*}
\begin{align*}
&=\exp(t \arctan \frac{\frac{t}{b-L+c} - \frac{t}{L-c} }{1+ \frac{t^2}{(b-L+c)(L-c)}})\exp(-t(\pi + \arctan \frac{t}{a-L+c}))\\
&\le \exp(t\arctan \frac{2L-2c -b}{t + \frac{(b-L+c)(L-c)}{t} })\exp(-\frac{\pi}{2}t) \\
&= \exp(t\arctan \frac{2L-2\delta +\frac{m}{2}}{t + \frac{(l_k + \delta -L)(l_k + \frac{m}{2} + L -\delta)}{t} })\exp(-\frac{\pi}{2}t) \\
&\le \exp(-\frac{\pi}{4}t).
\end{align*}
For $t < 0$, we similarly have
\begin{align*}
&\frac{\exp(t\arg(L-it-c)) \exp(-t \arg(a-L+c+it))}{\exp(-t \arg(b-L+c+it)} \\
&=\exp(-t \arctan \frac{t}{L-c} -t(\arctan \frac{t}{a-L+c} - \pi) + t\arctan \frac{t}{b-L+c} \\
&\le \exp(\frac{\pi}{4}t).
\end{align*}
And
\begin{align*}
&\exp (c-L) \exp(-(a-L+c)) \exp(b-L+c)\\
&=\exp(-a+b-L+c) \\
&=\exp(-\frac{m}{2} - L + \delta).
\end{align*}
Then we can get
\begin{align*}
&|J_k| \le \int_{-\infty}^{+\infty}\left |\frac{\Gamma(L-it-c)\Gamma(a-L+it+c)}{2\pi i\Gamma(b-L+it+c)} x^{c-L}\right |dt \\
&=C(m,L,\delta) \int_{0}^{+\infty} \big((l_k + \frac{m}{2} + L -\delta)^2 + t^2\big)^{\frac{2(L-\delta)+\frac{m}{2}}{2} } \exp(-\frac{\pi}{4}t)dtx^{c-L} \\
&\le C(m,L,\delta)l_k^{2(L-\delta)+\frac{m}{2}} x^{c-L}.
\end{align*}
Since
\[ \frac{\Gamma(b-a)}{\Gamma(a)} \sim \frac{\sqrt{2\pi l_k}(\frac{l_k}{e})^{l_k}}{\sqrt{2\pi(l_k + \frac{m}{2})}(\frac{l_k + \frac{m}{2}}{e})^{l_k + \frac{m}{2}}},\]
then
\[ \left | \frac{\Gamma(b-a)}{\Gamma(a)} \right| \le \frac{C(m)}{(l_k + \frac{m}{2})^{\frac{m}{2}}}.\]
We set $L=2\delta$, then
\begin{align*}
&\left | \frac{\Gamma(b-a)}{\Gamma (a)} J_k(x) x^a \right| \\
& \le C(m,\delta) l_k^{2\delta} x^{-\delta}.
\end{align*}
Thus
\begin{align*}
|g_k(x)| &= \left | \sum_{n=1}^{[L-\delta]} \frac{a_n(1+a-b)_n}{n!}x^{-n} + \frac{\Gamma(b-a)}{\Gamma(a)}J_k(x) x^a \right | \\
&\le C(m,\delta) l_k^{2\delta}x^{-\delta}.
\end{align*}
If $k\le K$ for some $K>0$, $a,b,c$ are all bounded. We set $L=\delta+1$, then
\[|g_k(x)| \le C(m,K,\delta)\frac{1}{x}.\]
\end{proof}
If $k\ge 1,$ we can use the asymptotic relation in Lemma 2.2, then
\begin{align*}
f_k(r)& = c_k e^{\frac{r^2}{4}} r^{2l_k} F[l_k + \frac{m}{2}, 2l_k + \frac{m}{2}; -\frac{r^2}{4}]\\
&=c_k\frac{\Gamma(2l_k + \frac{m}{2})}{\Gamma(l_k)}e^{\frac{r^2}{4}}r^{-m}\Big(1+g_k(\frac{r^2}{4})\Big) \\
&= C_ke^{\frac{r^2}{4}}r^{-m}\Big(1+g_k(\frac{r^2}{4})\Big),
\end{align*}
where $C_k$ are constants.
If $k=0,$ the asymptotic relation of Kummer's function does not hold, we solves the equation directly.
\begin{equation}
(f_0)_{rr} + (\frac{m-1}{r} - \frac{r}{2})(f_0)_r = 0.
\end{equation}
Then
\begin{equation}
f_0 = c + C_0\int_0^r e^{\frac{r^2}{4}} r^{-m} dr
\end{equation}
is the general solution of (2.4).
Then we can easily get that:
\begin{lemma}
The general solution of $\Delta_g u=0$ on $\mathbb{R}^{m}\backslash \{0\}$ has the following form:
\[u(r,\theta)=c + C_0\int_0^r e^{\frac{r^2}{4}} r^{-m} dr+\sum_{k=1}^{\infty}C_ke^{\frac{r^2}{4}}r^{-m}\Big(1+g_k(\frac{r^2}{4})\Big)\varphi_k.\]
\end{lemma}
\begin{lemma}
Assume $u\in H^{2n}(S^{m-1}),$ the fourier coefficients $|\langle u, \varphi_k\rangle|\le \frac{C}{\lambda_k^n},$ where $C$ is independent of $k.$
\end{lemma}
\begin{proof}
\begin{align*}
\langle u,\varphi_k \rangle &=\int_{S^{m-1}}u\varphi_k\\
=&-\frac{1}{\lambda_k}\int_{S^{m-1}}u\Delta\varphi_k\\
=&-\frac{1}{\lambda_k}\int_{S^{m-1}}\Delta u\varphi_k\\
=&\frac{1}{\lambda_k^2}\int_{S^{m-1}}\Delta u\Delta \varphi_k\\
=&\frac{1}{\lambda_k^2}\int_{S^{m-1}}\Delta^2 u \varphi_k\\
&\vdots \\
&=(-\frac{1}{\lambda_k})^n\int_{S^{m-1}}\Delta^n u \varphi_k.
\end{align*}
\[|\langle u, \varphi_k\rangle| \le \frac{1}{\lambda_k^n}\int_{S^{m-1}}(\Delta^n u)^2\le \frac{C}{\lambda_k^n}.\]
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:1.1}]
For any fixed $k\ge 1,$ note that
\begin{align*}
\frac{1}{2}|C_k|e^{\frac{r_i^2}{4}}r_i^{-m}& \le |C_k||1+g_k(\frac{r_i^2}{4})|e^{\frac{r_i^2}{4}}r_i^{-m}\\
&=|\int_{S^{m-1}}u(r_i,\theta)\varphi_k|\\
&\le \big(\int_{S^{m-1}} ( u(r_i,\theta))^2\big)^{\frac{1}{2}} (\int_{S^{m-1}} \varphi_k^2)^{\frac{1}{2}}\\
&\le C\sqrt{\omega_n}e^{\frac{r_i^2}{4}}r_i^{-(m+\epsilon)},
\end{align*}
which implies that
\[C_k<Cr_i^{-\epsilon}.\]
Let $r_i \rightarrow +\infty$, we have
\[C_k\equiv 0.\]
If $k=0,$
\[|c+\frac{1}{2}C_0\int_0^{r_i} e^{\frac{r^2}{4}}r^{1-m}dr|\le C\sqrt{\omega_n}e^{\frac{r_i^2}{4}}r_i^{-(m+\epsilon)},\]
which implies that
\[C_0\equiv 0.\]
It then follows that
\[u\equiv c.\]
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:1.6}]
For any $g(\theta)\in H^{[\frac{m}{2}]+2}(S^{m-1}),$ by Lemma 2.4, we have $g(\theta)=\sum_{k=0}^{\infty} \bar C_k\varphi_k$ and $|\bar C_k|\le \frac{C}{\lambda_k^{\frac{1}{2}[\frac{m}{2}]+1}}.$ By Lemma 2.1, for any $k\ge 1,$
\[\bar f_k= \bar C_ke^{\frac{r^2}{4}}r^{-m}\Big(1+g_k(\frac{r^2}{4})\Big) \]
is a solution of (2.3). \\
Similarly,
\[\bar f_0(r)=\frac{1}{2}\bar C_0\int_0^re^{\frac{r^2}{4}}r^{-m}dr\]
is a solution of (2.9). Assume $L=2\delta,$ by using the fact that $|\varphi_k|^2\le C(m)k^{m-2},$ we have
\[|\bar f_k \varphi_k|\le Ce^{\frac{r^2}{4}}r^{-m}\frac{1}{k^{2}}\Big(1+C(m,K,\delta)\frac{k^{2\delta}}{r^{2\delta}}\Big)\]
for $k$ bounded by some $K>0$, where $C$ is independent of $k$. Assume $L=1+\delta,$ similarly we have
\[|\bar f_k \varphi_k|\le Ce^{\frac{r^2}{4}}r^{-m}\frac{1}{k^{2}}\Big(1+C(m,\delta)\frac{1}{r^{2}}\Big)\]
for $k$ finite, where $C$ is independent of $k$.
If we set $0<\delta<\frac{1}{2},$ the series $\sum_{k=0}^{\infty} \bar f_k\varphi_k$ is uniformly convergent on $\overline {B_{A}(0)} \setminus B_{\epsilon}(0)$ for any $\epsilon,A>0$, where $B_{\epsilon}(0)$ is an open ball. Assume $\bar u=\sum_{k=0}^{\infty} \bar f_k\varphi_k,$ then we have
\begin{align*}
\Delta_g(\bar u)&=\bar u_{rr}+\frac{m-1}{r}\bar u_r+\frac{1}{r^2}\Delta_{\theta}\bar u-\frac{r}{2}\frac{\partial \bar u}{\partial r}\\
&=\sum_{k=1}^{\infty} \Big((\bar f_k)_{rr}+(\frac{m-1}{r}-\frac{r}{2})(\bar f_k)_r-\frac{\lambda_k\bar f_k}{r^2}\Big)\varphi_k+(\bar f_0)_{rr}+(\frac{m-1}{r}-\frac{r}{2})(\bar f_0)_r\\
&=0.
\end{align*}
Thus $\bar u$ is a solution of $\Delta_g u=0$ on $\mathbb{R}^m\backslash \{0\}.$\\
Since $\sum_{k=0}^{\infty} \frac{\bar f_k}{u_0}\varphi_k$ is uniformly convergent on $\mathbb{R}^m\setminus B_{\epsilon}(0)$, we can obtain that
\begin{align*}
\lim\limits_{r\to+\infty}\frac{\bar u(r,\theta)}{u_0(r)}=\lim\limits_{r\to+\infty}\sum_{k=0}^{\infty} \frac{\bar f_k}{u_0}\varphi_k=\sum_{k=0}^{\infty}\lim\limits_{r\to + \infty}\frac{\bar f_k}{u_0}\varphi_k=\sum_{k=0}^{\infty} \bar C_k\varphi_k=g(\theta).
\end{align*}
Hence $\bar u$ is the quasi-harmonic function which satisfies that $\lim\limits_{r\to+\infty}\frac{\bar u(r,\theta)}{u_0(r)}=g(\theta)$ for the given function $g(\theta)\in H^{[\frac{m}{2}]+2}(S^{m-1}).$
Moreover, assume $\frac{\bar u(r,\theta)}{u_0(r)}$ and $\frac{ u(r,\theta)}{u_0(r)}$ are asymptotic to the same function $g(\theta)$, for $k\ge 1,$ we have
\begin{align*}
&C_k=\lim\limits_{r\to+\infty}\frac{ f_k}{u_0}=\lim\limits_{r\to+\infty}\langle \frac{ u(r,\theta)}{u_0(r)}, \varphi_k \rangle \\
&=\langle \lim\limits_{r\to+\infty}\frac{\bar u(r,\theta)}{u_0(r)}, \varphi_k \rangle=\lim\limits_{r\to + \infty}\frac{\bar f_k}{u_0}=\bar C_k.
\end{align*}
By (2.5), we also have
\[C_0=\bar C_0.\]
Hence
\begin{align*}
&u(r,\theta)-\bar u(r, \theta)\\
&=c-\bar c+(C_0-\bar C_0)\int^r_0e^{\frac{r^2}{4}}r^{1-m}dr +\sum_{k=0}^{\infty} (C_k-\bar C_k)e^{\frac{r^2}{4}}r^{-m}\Big(1+g_k(\frac{r^2}{4})\Big)\\
&=c-\bar c.
\end{align*}
\end{proof}
\hspace{0.4cm}
\section{The asymptotic behavior of eigenfunctions of the drift laplacian at infinity}
\setcounter{equation}{0}
Assume that $u$ is an eigenfunction of $\Delta_h$ , i.e.,
\begin{equation}
\Delta_h u = -\lambda u.
\end{equation}
We rewrite it in the following form
\begin{equation}
\Delta u - (\nabla h, \Delta u) = -\lambda u,
\end{equation}
where $h = \frac{r^2}{4}$.
Let $f_k(r)=\langle u(r,\cdot),\varphi_k\rangle$ for $k\ge 0,$ we have
\begin{equation}
(f_k)_{rr} + (\frac{m-1}{r} - \frac{r}{2})(f_k)_r = (\frac{\lambda_k }{r^2}-\lambda)f_k.
\end{equation}
Let
\begin{align*}
&f_k(r) = w(z)z^{l_k} ,\\
&z = r^2 ,\\
&l_k = \frac{-(m-2) + \sqrt{(m-2)^2 + 4\lambda_k}}{4}.
\end{align*}
Then by (3.3), we have
\begin{equation}
zw''(z) + ((2l_k + \frac{m}{2}) - \frac{z}{4})w'(z) - \frac{l_k-\lambda}{4} w(z) = 0.
\end{equation}
Assume
\[ w(z) = e^{\frac{z}{4}}y(-\frac{z}{4}).\]
Then by (3.4), we have
\begin{equation}
\frac{z}{4} y''(-\frac{z}{4}) + (-\frac{z}{4} - (2l_k + \frac{m}{2}))y'(-\frac{z}{4}) + (l_k + \frac{m}{2}+\lambda)y(-\frac{z}{4}) = 0.
\end{equation}
Let $x = -\frac{z}{4}$, then we have
\begin{equation}
xy''(x) + ((2l_k + \frac{m}{2}) -x)y'(x) - (l_k + \frac{m}{2}+\lambda)y(x) = 0.
\end{equation}
Set $b = 2l_k + \frac{m}{2}, a = l_k + \frac{m}{2}+\lambda$. The first solution of (3.6) in terms of Kummer's series is
\[y=c_kF[a,b;x].\]
The first solution of (3.3) is
\[f_k(r)=c_ke^{\frac{r^2}{4}}r^{2l_k}F[ l_k + \frac{m}{2}+\lambda,2l_k + \frac{m}{2};-\frac{r^2}{4}].\]
If $k\ge 1,$ the second solution of (3.3) should be ruled out with the same reason as in the proof of Lemma 2.1.
We can use the same method as in the proof of Lemma 2.2 to get the asymptotic relation of the Kummer's functions.
\begin{lemma}
For any fixed $\delta>0$ and $k$ satisfying that $l_k-\lambda\neq-i$ for any nonnegative integer $i$, i.e., $\frac{1}{\Gamma(l_k-\lambda)}\neq 0,$ we set $a=l_k + \frac{m}{2}+\lambda, b=2l_k + \frac{m}{2}, c=-a+\delta$. Assume $L> \delta$ and $x$ is a positive real number, the following
asymptotic relation holds:
\[ F[a, b; -x] = x^{-a} \frac{\Gamma(b)}{\Gamma(b-a)}\Big(1 +\sum_{n=1}^{[L-\delta]} \frac{a_n(1+a-b)_n}{n!} x^{-n} + \frac{\Gamma(b-a)}{\Gamma(a)}J_k(x)x^a\Big), \]
where $(a)_n=a(a+1)\cdots(a+n-1)$ and $J_k(x)= \int_{c-i\infty}^{c+i\infty} \frac{\Gamma(L-s)\Gamma(a - L +s)}{2\pi i \Gamma(b-L+s)} x^{s-L}ds$. Moreover, $\left| J_k(x)\right|\le C(m,L,\delta,\lambda)l_k^{2(L-\delta)+\frac{m}{2}} |x|^{c-L}$ for $k$ sufficient large. Assume $g_k(x)=\sum_{n=1}^{[L-\delta]} \frac{(a)_n(1+a-b)_n}{n!} x^{-n} + \frac{\Gamma(b-a)}{\Gamma(a)}J_k(x)x^a$, in particular, if $k$ is sufficient large, we set $L=2\delta$, then
\[ |g_k(x)| \le C(m,\delta,\lambda)\frac{l_k^{2\delta}}{x^{\delta}}\]
and if $k\le K$ for some $K>0$, we set $L=\delta+1,$ then
\[ |g_k(x)| \le C(m,\delta,\lambda,K)\frac{1}{x}.\]
\end{lemma}
Note that the points of nonpositive integer are the zero points of $\frac{1}{\Gamma(z)}.$ When $l_k-\lambda=-i$ for any nonnegative integer $i,$ the Kummer's functions $F[l_k+\frac{m}{2}+\lambda,2l_k+\frac{m}{2};-x]$ do not have the same asymptotic relation as in Lemma 3.1. We will deal with the Kummer's series directly.
Assume $l_k-\lambda=-i$ for some nonnegative integer $0\le i\le \lambda.$ Then the first solution of (3.6) is
\begin{align*}
y(x)&=\sum_{n=0}^{\infty}\frac{(l_k+\frac{m}{2}+\lambda)\cdots(l_k+\frac{m}{2}+\lambda+n-1)}{(l_k+\frac{m}{2}+\lambda-i)\cdots(l_k+\frac{m}{2}+n+1+\lambda-i)n!}x^n\\
&=\sum_{n=0}^{\infty}\frac{(l_k+\frac{m}{2}+\lambda-i+1+n-1)\cdots(l_k+\frac{m}{2}+\lambda+n-1)}{(l_k+\frac{m}{2}+\lambda-i)\cdots(l_k+\frac{m}{2}+\lambda-1)n!}x^n
\end{align*}
Multiply $x^{l_k+\frac{m}{2}+\lambda-i-1}$ on both sides, we have
\begin{align*}
&y(x)x^{l_k+\frac{m}{2}+\lambda-i-1}=\\
&\sum_{n=0}^{\infty}\frac{(l_k+\frac{m}{2}+\lambda-i+1+n-1)\cdots(l_k+\frac{m}{2}+\lambda+n-1)}{(l_k+\frac{m}{2}+\lambda-i)\cdots(l_k+\frac{m}{2}+\lambda-1)n!}x^{n+l_k+\frac{m}{2}+\lambda-i-1}.
\end{align*}
Integrating from $0$ to $x$ for $i$ times, then we have
\[\int_0^x\cdots \int_0^x y(x)x^{l_k+\frac{m}{2}+\lambda-i-1}=\sum_{n=0}^{\infty}\frac{x^{n+l_k+\frac{m}{2}+\lambda-1}}{(l_k+\frac{m}{2}+\lambda-i)\cdots(l_k+\frac{m}{2}+\lambda-1)n!}.\]
Thus
\[\int_0^x\cdots \int_0^x y(x)x^{l_k+\frac{m}{2}+\lambda-i-1}=\frac{x^{l_k+\frac{m}{2}+\lambda-1}}{(l_k+\frac{m}{2}+\lambda-i)\cdots(l_k+\frac{m}{2}+\lambda-1)}\sum_{n=0}^{\infty}\frac{x^{n}}{n!}.\]
Differentiate the equation on both sides for $i$ times, we have
\[y(x)x^{l_k+\frac{m}{2}+\lambda-i-1}=e^x\Big(x^{l_k+\frac{m}{2}+\lambda-i-1}+\cdots+\frac{x^{l_k+\frac{m}{2}+\lambda-1}}{(l_k+\frac{m}{2}+\lambda-i)\cdots(l_k+\frac{m}{2}+\lambda-1)}\Big).\]
Then
\[y(x)=e^x\Big(1+\cdots+\frac{x^{i}}{(l_k+\frac{m}{2}+\lambda-i)\cdots(l_k+\frac{m}{2}+\lambda-1)}\Big).\]
Thus
\begin{align*}
(f_k)_1(r)&=e^{\frac{r^2}{4}}r^{2l_k}y(-\frac{r^2}{4})\\
&=c_k\Big(r^{2l_k}+\cdots+(\frac{1}{4})^{2i}\frac{r^{2\lambda}}{(l_k+\frac{m}{2}+\lambda-i)\cdots(l_k+\frac{m}{2}+\lambda-1)}\Big)\\
&=c_k q_k(r).
\end{align*}
Here $q_k(r)=r^{2l_k}+\cdots+(\frac{1}{4})^{2(l_k-\lambda)}\frac{r^{2\lambda}}{(2l_k+\frac{m}{2})\cdots(l_k+\frac{m}{2}+\lambda-1)}$ is a polynomial of degree $2\lambda.$
\begin{lemma}
Assume $A=\{0,m-2,m,m+2,\cdots,m+2\lambda\}$ and $B=\{m-2,m,m+2,\cdots,m+2[\lambda]\}.$ The general solution of $\Delta_h u=-\lambda u$ on $\mathbb{R}^m\backslash \{0\}$ has the following form:\\
If $2\lambda$ is not an integer, $u(r,\theta)=p(r)+\sum_{k=0}^{\infty}C_ke^{\frac{r^2}{4}}r^{-(2\lambda+m)}\big(1+g_k(\frac{r^2}{4})\big)\varphi_k$ with $p(r)=ce^{\frac{r^2}{4}}r^{-(m+2\lambda)}\int e^{-\frac{r^2}{4}}r^{m+4\lambda+1}(1+O(\frac{1}{r^2}))$ and $p(r)\sim r^{2\lambda}$;\\
If $\lambda$ is an integer, $u(r,\theta)=q(r,\theta)+C_0e^{\frac{r^2}{4}}r^{-(2\lambda+m)}\big(1+O(\frac{1}{r^2})\big)+\sum_{k\notin A}C_ke^{\frac{r^2}{4}}r^{-(2\lambda+m)}\big(1+g_k(\frac{r^2}{4})\big)\varphi_k$ with a polynomial $q(r,\theta)$ of degree $2\lambda$;\\
If $2\lambda$ is an integer but $\lambda$ is not an integer, $u(r,\theta)=t(r,\theta)+\sum_{k\notin B}C_k e^{\frac{r^2}{4}}r^{-(2\lambda+m)}\big(1+g_k(\frac{r^2}{4})\big)\varphi_k$ with a polynomial $t(r,\theta)$ of degree $2\lambda.$
\end{lemma}
\begin{proof}
Considering that $l_k = \frac{-(m-2) + \sqrt{(m-2)^2 + 4\lambda_k}}{4},$ we have
\begin{equation}
l_k-\lambda=-i \Longleftrightarrow k=-2(\lambda-i) \quad \text{or} \quad k=m-2+2(\lambda-i).
\end{equation}
So we divide $\lambda$ into three cases.
Case 1. $2\lambda$ is not an integer.
By (3.7), we have $\frac{1}{\Gamma(l_k-\lambda)}\neq 0$ for any $k\ge 0.$ Then we can use Lemma 3.1 to get the asymptotic behavior of $f_k$ at infinity.
The first solution of (3.3) is
\begin{align*}
f_k(r)&=c_ke^{\frac{r^2}{4}}r^{2l_k}F[ l_k + \frac{m}{2}+\lambda,2l_k + \frac{m}{2};-\frac{r^2}{4}]\\
&=c_k\frac{\Gamma(2l_k+\frac{m}{2})}{\Gamma(l_k-\lambda)}e^{\frac{r^2}{4}}r^{-(2\lambda+m)}\big(1+g_k(\frac{r^2}{4})\big)\\
&=C_ke^{\frac{r^2}{4}}r^{-(2\lambda+m)}\big(1+g_k(\frac{r^2}{4})\big).
\end{align*}
The second solution of $f_0$ can be found by using the method of reduction of order. Let $(f_0)_2(r)=v(r)(f_0)_1(r)$ be the second linearly independent solution of (3.3), then
\[(f_0)_1(r)v''(r)+(2(f_0)'_1(r)+(\frac{m-1}{r} - \frac{r}{2})(f_0)_1)v'(r)=0.\]
Thus
\begin{equation}
(f_0)_2(r)=c_0'(f_0)_1(r)\int\frac{e^{\frac{r^2}{4}}r^{1-m}}{((f_0)_1(r))^2}=ce^{\frac{r^2}{4}}r^{-(m+2\lambda)}\int e^{-\frac{r^2}{4}}r^{m+4\lambda+1}(1+O(\frac{1}{r^2}))=p(r)
\end{equation}
and $p(r)\sim r^{2\lambda}.$
Thus
\[u(r,\theta)=\sum_{k=0}^{\infty}f_k\varphi_k=p(r)+\sum_{k=0}^{\infty}C_ke^{\frac{r^2}{4}}r^{-(2\lambda+m)}\big(1+g_k(\frac{r^2}{4})\big)\varphi_k.\]
Case 2. $\lambda$ is an integer.
$l_k-\lambda=-i $ for some integer $0\le i\le \lambda \Longleftrightarrow k\in A,$ where $A=\{0,m-2,m,m+2,\cdots,m+2\lambda\}.$
When $k=0,$ which implies that $i=\lambda,$ we have
\[(f_0)_1(r)=c_0q_{2\lambda}(r).\]
The second linearly independent solution can be found by using the method of reduction of order.
\begin{equation}
(f_0)_2(r)=c_0'(f_0)_1(r)\int\frac{e^{\frac{r^2}{4}}r^{1-m}}{((f_0)_1(r))^2}=l(r),
\end{equation}
where $l(r)=C_0e^{\frac{r^2}{4}}r^{-(2\lambda+m)}\big(1+O(\frac{1}{r^2})\big).$
When $k\ge 1,$ $f_k(r)=(f_k)_1(r),$ where $(f_k)_1(r)$ is the first solution of $f_k.$
Thus we have
\begin{align*}
u(r,\theta)&=\sum_{k=0}^{\infty}f_k\varphi_k\\
&=\sum_{k\in A}f_k\varphi_k+\sum_{k\notin A}f_k\varphi_k\\
&=\sum_{k\in A}(f_k)_1\varphi_k+(f_0)_2+\sum_{k\notin A}(f_k)_1\varphi_k\\
&=\sum_{k\in A}c_ke^{\frac{r^2}{4}}r^{2l_k}F[ l_k + \frac{m}{2}+\lambda,2l_k + \frac{m}{2};-\frac{r^2}{4}]\varphi_k+l(r)\\
&+\sum_{k\notin A}c_ke^{\frac{r^2}{4}}r^{2l_k}F[ l_k + \frac{m}{2}+\lambda,2l_k + \frac{m}{2};-\frac{r^2}{4}]\varphi_k\\
&=\sum_{k\in A}c_{k}q_{k} \varphi_{k}+l(r)+\sum_{k\notin A}C_ke^{\frac{r^2}{4}}r^{-(2\lambda+m)}\big(1+g_k(\frac{r^2}{4})\big)
\varphi_k\\
&=q(r,\theta)+l(r)+\sum_{k\notin A}C_ke^{\frac{r^2}{4}}r^{-(2\lambda+m)}\big(1+g_k(\frac{r^2}{4})\big)
\varphi_k,
\end{align*}
where $q(r,\theta)=\sum_{k\in A}c_{k}q_{k}\varphi_{k}$ is a polynomial of degree $2\lambda.$
Case 3. $2\lambda$ is an integer but $\lambda$ is not an integer.
$l_k-\lambda=-i$ for some integer $0\le i\le [\lambda] \Longleftrightarrow k\in B,$ where $B=\{m-2,m,m+2,\cdots,m+2[\lambda]\}.$
Since $0\notin B, \frac{1}{\Gamma(l_k-\lambda)}\neq 0.$ By Lemma 3.1, we have
\begin{align*}
(f_0)_1(r)&=c_0e^{\frac{r^2}{4}}r^{2l_k}F[ l_k + \frac{m}{2}+\lambda,2l_k + \frac{m}{2};-\frac{r^2}{4}]\\
&=C_0e^{\frac{r^2}{4}}r^{-(2\lambda+m)}\big(1+g_0(\frac{r^2}{4})\big).
\end{align*}
The second linearly independent solution can be found by using the method of reduction of order.
\[(f_0)_2(r)=p(r),\]
where $p(r)\sim r^{2\lambda}.$\\
When $k\ge 1,$ $f_k(r)=(f_k)_1(r),$ where $(f_k)_1(r)$ is the first solution of $f_k.$
Thus we have
\begin{align*}
u(r,\theta)&=\sum_{k=0}^{\infty}f_k\varphi_k\\
&=\sum_{k\in B}f_k\varphi_k+\sum_{k\notin B}f_k\varphi_k\\
&=\sum_{k\in B}(f_k)_1\varphi_k+(f_0)_2+\sum_{k\notin B}(f_k)_1\varphi_k\\
&=\sum_{k\in B}c_ke^{\frac{r^2}{4}}r^{2l_k}F[ l_k + \frac{m}{2}+\lambda,2l_k + \frac{m}{2};-\frac{r^2}{4}]\varphi_k+p(r)\\
&+\sum_{k\notin B}c_ke^{\frac{r^2}{4}}r^{2l_k}F[ l_k + \frac{m}{2}+\lambda,2l_k + \frac{m}{2};-\frac{r^2}{4}]\varphi_k\\
&=\sum_{k\in B}c_{k}q_{k} \varphi_{k}+p(r)+\sum_{k\notin B}C_ke^{\frac{r^2}{4}}r^{2\lambda+m}\big(1+g_k(\frac{r^2}{4})\big)\varphi_k\\
&=\sum_{k\in B}c_{k}q_{k} \varphi_{k}+p(r)+\sum_{k\notin B}C_ke^{\frac{r^2}{4}}r^{2\lambda+m}\big(1+g_k(\frac{r^2}{4})\big)
\varphi_k\\
&=s(r,\theta)+p(r)+\sum_{k\notin B}C_ke^{\frac{r^2}{4}}r^{2\lambda+m}\big(1+g_k(\frac{r^2}{4})\big)
\varphi_k,
\end{align*}
where $s(r,\theta)=\sum_{k\in B}c_{k}q_{k} \varphi_{k}$ is a polynomial of degree $2\lambda.$
If we denote $t(r,\theta)=s(r,\theta)+p(r),$ by Lemma 1.2 in \cite{5}, we know that $ t(r,\theta)$ is an polynomial of degree $2\lambda.$ Then
\[u(r,\theta)=t(r,\theta)+\sum_{k\notin B}C_ke^{\frac{r^2}{4}}r^{2\lambda+m}\big(1+g_k(\frac{r^2}{4})\big)\varphi_k.\]
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:1.3}]
We divide $\lambda$ into three cases:
Case1. $2\lambda$ is not an integer.
For any fixed $k\ge 0,$ note that
\begin{align*}
\frac{1}{2}|C_k|e^{\frac{r_i^2}{4}}r_i^{-(m+2\lambda)}& \le |C_k||1+g_k(\frac{r_i^2}{4})|e^{\frac{r_i^2}{4}}r_i^{-(m+2\lambda)}\\
&=|\int_{S^{m-1}}u(r_i,\theta)\varphi_k|\\
&\le (\int_{S^{m-1}}( u(r_i,\theta))^2)^{\frac{1}{2}}(\int_{S^{m-1}} \varphi_k^2)^{\frac{1}{2}}\\
&\le C\sqrt{\omega_n}e^{\frac{r_i^2}{4}}r_i^{-(m+2\lambda+\epsilon)},
\end{align*}
which implies that
\[C_k<Cr_i^{-\epsilon}.\]
Let $r_i\rightarrow+\infty,$ then
\[C_k\equiv 0.\]
It is clear that
\[u(r,\theta)\equiv p(r).\]
Case 2. $\lambda$ is an integer.
Similarly, we can obtain that
\[u(r,\theta)\equiv q(r,\theta).\]
Case 3. $2\lambda$ is an integer but $\lambda$ is not an integer.
Similarly, we can obtain that
\[u(r,\theta)\equiv t(r,\theta).\]
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:1.7}]
We divide $\lambda$ into three cases:
Case 1. $2\lambda$ is not an integer.
For any $g(\theta)\in H^{[\frac{m}{2}]+2}(S^{m-1}),$ by Lemma 2.4, we have $g(\theta)=\sum_{k=0}^{\infty} \bar C_k\varphi_k$ and $|\bar C_k|\le \frac{C}{\lambda_k^{\frac{1}{2}[\frac{m}{2}]+1}}.$ By Lemma 3.2, for any $k\ge 0,$
\[\bar f_k= \bar C_ke^{\frac{r^2}{4}}r^{-(2\lambda+m)}\Big(1+g_k(\frac{r^2}{4})\Big) \]
is a solution of (3.3).
Assume $L=2\delta,$ by using the fact that $|\varphi_k|^2\le C(m)k^{m-2},$ we have
\[|\bar f_k \varphi_k|\le Ce^{\frac{r^2}{4}}r^{-(m+2\lambda)}\frac{1}{k^{2}}\Big(1+C(m,\delta,\lambda)\frac{k^{2\delta}}{r^{2\delta}}\Big)\]
for $k$ sufficient large, where $C$ is independent of $k$. Assume $L=1+\delta,$ similarly we have
\[|\bar f_k \varphi_k|\le Ce^{\frac{r^2}{4}}r^{-(m+2\lambda)}\frac{1}{k^{2}}\Big(1+C(m,\delta,K,\lambda)\frac{1}{r^2}\Big)\]
when $k\le K$ for some $K>0$, where $C$ is independent of $k$.
If we set $0<\delta<\frac{1}{2},$ the series $\sum_{k=0}^{\infty} \bar f_k\varphi_k$ is uniformly convergent on $\overline { B_A(0)}\setminus B_{\epsilon}(0)$ for any $\epsilon,A>0.$ Assume $\bar u=\sum_{k=0}^{\infty} \bar f_k\varphi_k,$ then we have
\begin{align*}
\Delta_g(\bar u)&=\bar u_{r r}+\frac{m-1}{r}\bar u_r+\frac{1}{r^2}\Delta_{\theta}\bar u-\frac{r}{2}\frac{\partial \bar u}{\partial r}\\
&=\sum_{k=1}^{\infty} \Big((\bar f_k)_{rr}+(\frac{m-1}{r}-\frac{r}{2})(\bar f_k)_r-\frac{\lambda_k\bar f_k}{r^2}\Big)\varphi_k+(\bar f_0)_{r r}+(\frac{m-1}{r}-\frac{r}{2})(\bar f_0)_r\\
&=-\lambda\sum_{k=0}^{\infty}\bar f_k\varphi_k\\
&=-\lambda \bar u
\end{align*}
Thus $\bar u$ is a solution of $\Delta_g u=-\lambda u$ on $\mathbb{R}^m\backslash \{0\}.$\\
Since $\sum_{k=0}^{\infty} \frac{\bar f_k}{u_0}\varphi_k$ is uniformly convergent on $\mathbb{R}^m\setminus B_{\epsilon}(0)$ for any $\epsilon>0$, then we can obtain that
\begin{align*}
\lim\limits_{r\to+\infty}\frac{\bar u(r,\theta)}{u_0(r)}=\lim\limits_{r\to+\infty}\sum_{k=0}^{\infty} \frac{\bar f_k}{u_0}\varphi_k=\sum_{k=0}^{\infty} \lim\limits_{r\to + \infty}\frac{\bar f_k}{u_0}\varphi_k=g(\theta).
\end{align*}
Hence $\bar u$ is the solution of $\Delta_h u=-\lambda u$ which satisfies that $\lim\limits_{r\to+\infty}\frac{\bar u(r,\theta)}{u_0(r)}=g(\theta)$ for any given function $g(\theta)\in H^{[\frac{m}{2}]+2}(S^{m-1}).$
Moreover, assume $\frac{\bar u(r,\theta)}{u_0(r)}$ and $\frac{ u(r,\theta)}{u_0(r)}$ are asymptotic to the same function $g(\theta)$, then
\begin{align*}
&C_k=\lim\limits_{r\to+\infty} \frac{ f_k}{u_0}=\langle\lim\limits_{r\to+\infty} \frac{ u(r,\theta)}{u_0(r)}, \varphi_k \rangle \\
&=\langle \lim\limits_{r\to+\infty}\frac{\bar u(r,\theta)}{u_0(r)}, \varphi_k \rangle=\lim\limits_{r\to + \infty}\frac{\bar f_k}{u_0}=\bar C_k.
\end{align*}
Hence
\begin{align*}
&u(r,\theta)-\bar u(r, \theta)\\
&=p(r)-\bar p(r)+\sum_{k=0}^{\infty} (C_k-\bar C_k)e^{\frac{r^2}{4}}r^{-(2\lambda+m)}\Big(1+g_k(\frac{r^2}{4})\Big)\varphi_k\\
&=p(r)-\bar p(r).
\end{align*}
Case 2. $\lambda$ is an integer.
For any $g(\theta)\in H^{[\frac{m}{2}]+2}(S^{m-1})$ satisfying that $\langle g,\varphi_k\rangle=0$ when $k\in \{0,m-2,m,m+2,\cdots,m+2\lambda\}=A,$ by Lemma 2.4, we have $g(\theta)=\sum_{k\notin A}\bar C_k\varphi_k$ and $|\bar C_k|\le \frac{C}{\lambda_k^{\frac{1}{2}[\frac{m}{2}]+1}}.$\\
Set \[u(r,\theta)=\sum_{k\notin A}\bar C_ke^{\frac{r^2}{4}}r^{-(2\lambda+m)}\Big(1+g_k(\frac{r^2}{4})\Big)\varphi_k+\bar C_0l(r).\]
Then $\bar u$ is a solution of $\Delta_h=-\lambda u$ which satisfies that $\lim\limits_{r\to+\infty}\frac{\bar u(r,\theta)}{u_0(r)}=g(\theta).$
Case 3. $2\lambda$ is an integer but $\lambda$ is not an integer.
For any $g(\theta)\in H^{[\frac{m}{2}]+2}(S^{m-1})$ satisfying that $\langle g,\varphi_k\rangle=0$ when $k\in \{m-2,m,m+2,\cdots,m+2[\lambda]\}=B,$ by Lemma 2.4, we have $g(\theta)=\sum_{k\notin B}\bar C_k\varphi_k$ and $|\bar C_k|\le \frac{C}{\lambda_k^{\frac{1}{2}[\frac{m}{2}]+1}}.$\\
Set \[u(r,\theta)=\sum_{k\notin B}\bar C_ke^{\frac{r^2}{4}}r^{-(2\lambda+m)}\Big(1+g_k(\frac{r^2}{4})\Big)\varphi_k.\]
Then $\bar u$ is a solution of $\Delta_h u =-\lambda u$ which satisfies that $\lim\limits_{r\to+\infty}\frac{\bar u(r,\theta)}{u_0(r)}=g(\theta).$
\end{proof}
\end{document} | math |
var indicators = [
{ key: "absoluteVolumeOscillatorIndicator", text: "絶対出来高オシレーター" },
{ key: "averageTrueRangeIndicator", text: "ATR (アベレージ トゥルー レンジ)" },
{ key: "accumulationDistributionIndicator", text: "蓄積/分配" },
{ key: "averageDirectionalIndexIndicator", text: "平均方向性指数" },
{ key: "bollingerBandsOverlay", text: "ボリンジャー バンド オーバーレイ" },
{ key: "bollingerBandWidthIndicator", text: "ボリンジャー帯域幅" },
{ key: "chaikinOscillatorIndicator", text: "チャイキン オシレーター" },
{ key: "chaikinVolatilityIndicator", text: "チャイキン ボラティリティ" },
{ key: "commodityChannelIndexIndicator", text: "商品チャネル指数 (CCI)" },
{ key: "detrendedPriceOscillatorIndicator", text: "トレンド除去価格オシレーター" },
{ key: "easeOfMovementIndicator", text: "イーズ オブ ムーブメント" },
{ key: "fastStochasticOscillatorIndicator", text: "ファスト スト キャスティクス オシレーター" },
{ key: "forceIndexIndicator", text: "フォース インデックス" },
{ key: "fullStochasticOscillatorIndicator", text: "フル ストキャスティクス オシレーター" },
{ key: "marketFacilitationIndexIndicator", text: "マーケット ファシリテーション インデックス" },
{ key: "massIndexIndicator", text: "マス インデックス" },
{ key: "medianPriceIndicator", text: "メディアンプライス" },
{ key: "moneyFlowIndexIndicator", text: "マネー フロー インデックス " },
{ key: "movingAverageConvergenceDivergenceIndicator", text: "移動平均収束拡散" },
{ key: "negativeVolumeIndexIndicator", text: "負出来高指数 (NVI)" },
{ key: "onBalanceVolumeIndicator", text: "OBV (オン バランス ボリューム) インジケーター" },
{ key: "percentagePriceOscillatorIndicator", text: "パーセンテージ価格オシレーター" },
{ key: "percentageVolumeOscillatorIndicator", text: "パーセンテージ出来高オシレーター" },
{ key: "positiveVolumeIndexIndicator", text: "PVI (ポジティブ ボリューム インデックス)" },
{ key: "priceChannelOverlay", text: "プライス チャネル オーバーレイ" },
{ key: "priceVolumeTrendIndicator", text: "価格出来高トレンド" },
{ key: "rateOfChangeAndMomentumIndicator", text: "変化率およびモメンタム" },
{ key: "relativeStrengthIndexIndicator", text: "相対力指数" },
{ key: "slowStochasticOscillatorIndicator", text: "スロー ストキャスティクス オシレーター" },
{ key: "standardDeviationIndicator", text: "標準偏差" },
{ key: "stochRSIIndicator", text: "ストキャスティクス RSI" },
{ key: "trixIndicator", text: "Trix" },
{ key: "typicalPriceIndicator", text: "代表価格" },
{ key: "ultimateOscillatorIndicator", text: "アルティメット オシレーター" },
{ key: "weightedCloseIndicator", text: "重み付きクローズ" },
{ key: "williamsPercentRIndicator", text: "ウィリアム パーセント レンジ" }
];
| code |
Information on the government’s policy on the assessed and supported year in employment programme. Explains the ASYE programme for newly qualified social workers, and how organisations can register to take part.
These files contain information for suppliers developing software and management information systems (MIS) for local authorities and schools.
This includes information on learners who are studying on a course at a further education college, learners studying courses within their local community, employees undertaking an apprenticeship, and employees undertaking other qualifications in the workplace.Please note: data on providers is only published annually.
It has been designed to complement the main statistical releases, and act as a ‘one stop shop’ for data and information on learners, learning programmes and learner achievement.
Use the ready reckoner to calculate level 3 value-added results.
The transition matrices spreadsheet shows data for the subjects included in the level 3 value-added measure.
Level 3 value-added is a progress measure for school sixth forms and colleges which is used in the 16 to 18 performance tables.
You can also read the 16 to 18 accountability technical guide to learn more about the level 3 value-added measure.
How to check the infant class size information in your spring 2019 school census return.
The Selective Schools Expansion Fund (SSEF) will provide funding of £49.3 million for 16 expansion projects. This will create over 3,000 more grammar school places.
This guide helps you prepare and submit your school’s 2019 school-level annual school census (SLASC) return.
SLASC census day for registered independent schools is Thursday 17 January 2019.
A list of COLLECT queries and explanatory notes to help schools, academies and local authorities complete the school census. | english |
سُہ کھوٚت ہیرِ تہٕ دِژٕن اَتھ مچہِ نظرا اتہِ اوس تِیُتھ پھَکھ یِوان زِ سُہ اوسنہٕ أکِس لحظَس تہِ اتِتھ روزُن یژھان چؠنٕچ چھَنہٕ کَتھٕے | kashmiri |
Scuba diver Jimmy Roseman came face to face with a 12-foot great white shark while surveying the coral reefs at a depth of 90 feet just off the Atlantic Coast near Vero Beach, Florida.
All the action was caught on his GoPro camera that was mounted to his diver helmet.
The shark was at a distance when he took the dive but came close to Roseman really quickly.
The interaction lasted two minutes when finally Roseman's air tank was hit by the shark's fin. It was then that he used the spear gun to fend off the potentially deadly shark.
In the video it looks like the great white shark may have a baby with it.
Never does Roseman shot or injure the shark and both are left unharmed.
In a video interview with Fox 35, Roseman said, "I felt something hit me on the back and I then I seen the shark come over my head and swim off. He circled me...I had to poke him a few times. He wouldn't go away and then I finally jabbed him hard enough to where he left me long enough i could get to the surface and get out of there."
The great white shark is typically found in cold and warm-temperate waters throughout the world, although occurrences in tropical waters have been documented, according to the National Oceanic and Atmospheric Administration.
Research suggests that great whites migrate to Florida in a seasonal pattern with distribution limited by water temperature, food resources, or other factors.
Jimmy Roseman of West Melbourne, FL has an encounter with a Great White Shark off of the coast of Florida at Bethel Shoals. Obviously, the beginning is computer generated, but the GoPro video is 100% raw. | english |
\begin{document}
\selectlanguage{english}
\title{Subvarieties of moduli spaces of sheaves via finite coverings}
\author{Markus Zowislok\footnote{Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, UK}}\date{}\maketitle\newtheorem{theorem}{Theorem}[section]\newtheorem{definition}[theorem]{Definition}\newtheorem{corollary}[theorem]{Corollary}\newtheorem{lemma}[theorem]{Lemma}\newtheorem{proposition}[theorem]{Proposition}\newtheorem{example}[theorem]{Example} \newcommand{\textit}{\textit}
\begin{abstract}
Given a finite unbranched covering of a nonsingular projective scheme we analyse the morphism between moduli spaces of sheaves induced by pullback.
We have a closer look at cyclic coverings and, in particular, at canonical coverings of surfaces.
Our main application is the construction of Lagrangian subvarieties of certain irreducible holomorphically symplectic manifolds that arise from moduli spaces of sheaves on K3 or abelian surfaces.
\end{abstract}
\maketitle
\newcommand{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{Z}}}
\newcommand{\ensuremath{\mathbb{N}}}{\ensuremath{\mathbb{N}}}
\newcommand{\ensuremath{\mathcal{H}}}{\ensuremath{\mathcal{H}}}
\newcommand{\ensuremath{\mathrm{H}}}{\ensuremath{\mathrm{H}}}
\newcommand{\ensuremath{\mathcal{E}xt}}{\ensuremath{\mathcal{E}xt}}
\renewcommand{\hom}{\ensuremath{\mathrm{hom}}}\newcommand{\ensuremath{\mathrm{Hom}}}{\ensuremath{\mathrm{Hom}}}
\newcommand{\ensuremath{\mathrm{codim}}}{\ensuremath{\mathrm{codim}}}
\newcommand{\ensuremath{\mathrm{Coh}}}{\ensuremath{\mathrm{Coh}}}
\newcommand{\ensuremath{\mathrm{Num}}}{\ensuremath{\mathrm{Num}}}
\newcommand{\ensuremath{\mathrm{NS}}}{\ensuremath{\mathrm{NS}}}
\newcommand{\ensuremath{\mathrm{Amp}}}{\ensuremath{\mathrm{Amp}}}
\newcommand{\ensuremath{\Lambda'(Y)}}{\ensuremath{\Lambda'(Y)}}
\newcommand{\ensuremath{\mathrm{Hilb}}}{\ensuremath{\mathrm{Hilb}}}
\newcommand{\ensuremath{\;(\le_0)\;}}{\ensuremath{\;(\le_0)\;}}
\newcommand{\ensuremath{\mathrm{gcd}}}{\ensuremath{\mathrm{gcd}}}
\newcommand{\ensuremath{\mathrm{ord}}}{\ensuremath{\mathrm{ord}}}
\newcommand{
\[\qedhere\]}{
\[\qedhere\]}
\renewcommand{(\alph{enumii})\ }{(\alph{enumii})\ }
\newtheorem{question}[theorem]{Question}\newtheorem{conjecture}[theorem]{Conjecture}
\section{Introduction}
Is Gieseker stability preserved under pullback by a finite covering? How are the corresponding moduli spaces related?
In this article, we exhibit the behaviour of stability under pullback by a finite unbranched covering of a nonsingular projective scheme.
The roots of this question go back to Kim's article \cite{Kim98a}, which is on the canonical covering of an Enriques surface and $\mu$-stable sheaves.
Our main results hold not only for the notion of Gieseker stability, but also for twisted stability and for $(H,A)$-stability. Therefore we will always write (semi)stable whenever the statement allows all three notions. These stability notions are recalled in Section \ref{StabNot}.
For the whole article let $X$ be a nonsingular projective irreducible variety over $\IC$, $G$ a finite group acting freely on $X$, $f\colon X\to Y$ the quotient of $X$ by $G$, and $H$ an ample divisor on $Y$.
In Section \ref{arbdim} we show that the pullback $f^*E$ of a semistable sheaf $E$ on $Y$ is again semistable (Proposition \ref{PbStab}).
After some more precise results on the behaviour of the stability property, we apply these results to moduli spaces:\\
\noindent \textbf{Theorem \ref{MHApb}}. \titar{Let $P$ be a polynomial, $M_Y$ a quasiprojective and nonempty subscheme of the moduli space $M_Y(P)$ of semistable sheaves on $Y$ with (twisted) Hilbert polynomial $P$, $M_X=M_X(\deg f\cdot P)$ the moduli space of semistable sheaves on $X$ with (twisted) Hilbert polynomial $\deg f\cdot P$, and $M_Y^s$ and $M_X^s$ the respective loci of stable sheaves.
The pullback by $f$ induces a morphism $f^*\colon M_Y\to M_X$ which maps closed points $[E]$ to $[f^*E]$.
The closed points of its image are represented by polystable $G$-sheaves, and $(f^*)^{-1}( M_X^s )\subseteq M_Y^s\,.$}\\
The restriction to a cyclic covering given by a line bundle $L$ of finite order in Section \ref{arbCC} allows a deeper analysis of the pullback morphism. The main tool is the group action on the moduli space of sheaves on $Y$ induced by tensoring with $L$. This allows us to give a precise description of the locus of stable sheaves becoming strictly semistable in Theorem \ref{CycStab}.\\
Section \ref{Surfaces} contains the application to canonical coverings of surfaces. We investigate the double covering $X\to Y$ of an Enriques surface $Y$ by a K3 surface $X$ as well as the canonical covering $X\to Y$ of a bielliptic surface $Y$ by an abelian surface $X$.
The main results are given in Theorems \ref{CC} and \ref{CCsst}.
An interesting question is in what cases there are Lagrangian subvarieties as images of $f^*$ inside the moduli spaces of semistable sheaves on $X$, and what kind of varieties these subvarieties are.
If $X$ is a K3 surface and $f^*u$ is primitive then the moduli space $M_X(f^*u)$ in general is an irreducible symplectic manifold, and
whenever there is a suitable stable sheaf on $Y$, one gets a Lagrangian subvariety as described in Proposition \ref{CanDouSt}.
If $X$ is an abelian surface, the situation is different: in order to produce higher dimensional irreducible symplectic manifolds out of moduli spaces of sheaves on $X$ one has to get rid of superfluous factors in the Bogomolov decomposition by taking a fibre of the Albanese map. The last part of Section \ref{LIHS} explains how and why this reduction reduces the Lagrangian subvarieties to (smaller) Lagrangian subvarieties in the case of a double covering, resulting in Proposition \ref{CanDouStK}.
The most prominent Lagrangian subvarieties of irreducible holomorphically symplectic manifolds occur as fibres or sections of Lagrangian fibrations. Recently, smooth Lagrangian tori have attracted a lot of interest, sparked by the following question of Beauville \cite{Beau10}:
If an irreducible holomorphically symplectic manifold $M$ contains a smooth Lagrangian torus, is this a fibre of a Lagrangian fibration on $M$?
This has been answered positively combining \cite{GLR11}, \cite{HW12}, and \cite{Mat12}.
It would be interesting to understand better, which kind of Lagrangian subvarieties can be obtained by Propositions \ref{CanDouSt} and \ref{CanDouStK}.
Some concrete results are contained in Section \ref{example}.
Unfortunately, our knowledge on moduli spaces of sheaves on Enriques and bielliptic surfaces is quite limited at the moment.
However, we expect that complex tori do not occur as Lagrangian subvarieties in the case of sheaves of odd rank. In particular, they do not occur as image under pullback of the moduli space of sheaves of rank one on Enriques surfaces, i.e.\ in the Hilbert scheme case, as explained in Section \ref{example}.
The case of even rank seems to be more promising, as an example of Hauzer can be used to produce an elliptic curve as Lagrangian subvariety in a K3 surface, as explained in Section \ref{example} as well. Hence there is hope that higher-dimensional examples of Lagrangian tori inside symplectic varieties may occur.
On the other side, it is interesting in its own to construct different Lagrangian subvarieties and, in particular, to study their intersection. In \cite{BF09} the authors construct a Gerstenhaber algebra structure and a compatible Batalin-Vilkovisky module structure
giving rise to a de Rham type cohomology theory for Lagrangian intersections.
Shortly after the first version of this article appeared on arXiv.org, an independent article by Sacc\`a \cite{Sac12} on the case of one-dimensional sheaves on Enriques surfaces and the induced pullback by the canonical double covering appeared there as well. Her results are related to my Question \ref{OSgen}.
Finally we give an outlook in Section \ref{Outlook} considering the case of surfaces of general type,
and an appendix briefly recalls the notion of a general ample divisor which occurs in Section \ref{Surfaces}.\\
\acks
{\small The author would like to express his gratitude to S\"onke Rollenske for fruitful discussions and helpful suggestions.
He thanks Arvid Perego for directing his attention to Kim's articles and Hauzer's article and Manfred Lehn for useful comments.
Moreover, he thanks the DFG for the SFB 701, which provides a vibrant research atmosphere at Bielefeld.
This article was completed during a stay at Imperial College London supported by a DFG research fellowship (Az.: ZO 324/1-1).}
\section{Pullback by finite \'etale coverings and stability}\label{arbdim}
In this section we recall the considered notions of stability of sheaves, analyse their behaviour under pullback by $f$ and apply the results to get our general Theorem \ref{MHApb} on the moduli spaces of sheaves.
We assume familiarity with the material presented in \cite{HL10} and use the notation therein.
\subsection{Three stability notions}\label{StabNot}
Our main results hold for the notions of Gieseker stability, twisted stability and $(H,A)$-stability.
Twisted stability and $(H,A)$-stability are two generalisations of Gieseker stability, with an overlap in the case of a surface, see \cite[Corollary 6.2.6]{Zow10}.
We briefly recall the definitions.
\begin{enumerate}
\item \textit{Gieseker stability.}
A detailed treatment of this notion can be found e.g.\ in \cite[Section 1.2]{HL10}.
Let $H$ be an ample divisor on $Y$ and $E$ a nontrivial coherent sheaf on $Y$.
The Hilbert polynomial of $E$ is $P_H(E)(n):=\chi(E\otimes H^{\otimes n})$. Its leading coefficient multiplied by $(\dim E)!$ is called multiplicity of $E$ and denoted here by $\alpha^H(E)$. It is always positive, and $p_H(E)(n):=\frac{\chi(E\otimes H^{\otimes n})}{\alpha^H(E)}$ is called reduced Hilbert polynomial of $E$.
With these at hand, one says that $E$ is $H$-(semi)stable if $E$ is pure and for all nontrivial proper subsheaves $F\subseteq E$ one has that $p_H(F) \lel p_H(E)$, i.e.\ $p_H(F)(n) \lel p_H(E)(n)$ for $n\gg 0$.
In order to avoid case differentiation for stable and semistable sheaves we here follow the Notation 1.2.5 in \cite{HL10} using bracketed inequality signs, e.g.\ an inequality with $\lel$ for (semi)stable sheaves means that one has $\le$ for semistable sheaves and $<$ for stable sheaves.
\item \textit{Twisted stability as defined in \cite[Definition 4.1]{Yos03b}.}
Let $H$ be an ample divisor on $Y$, $V$ a locally free sheaf on $Y$ and $E$ a nontrivial coherent sheaf on $Y$.
The $V$-twisted Hilbert polynomial is $P^V_H(E)(n):=\chi(E\otimes V\otimes H^{\otimes n})$, and
the reduced $V$-twisted Hilbert polynomial is $$p^V_H(E)(n):=\frac{\chi(E\otimes V\otimes H^{\otimes n})}{\alpha^H(E\otimes V)}\,.$$
One says that $E$ is $V$-twisted $H$-(semi)stable if $E$ is pure and for all nontrivial proper subsheaves $F\subseteq E$ one has that $p^V_H(F) \lel p^V_H(E)$.
\item \textit{$(H,A)$-stability as defined in \cite[Definition 7.1]{Zow12}.}
Let $H$ and $A$ be two ample divisors on $Y$ and $E$ a nontrivial coherent sheaf on $Y$.
We defined
$$
P_{H,A}(E)(m,n):=\chi(E\otimes H^{\otimes m} \otimes A^{\otimes n}) \quad\mathrm{and}\quad
p_{H,A}(E):=\frac{P_{H,A}(E)}{\alpha^H(E)} \,.
$$
These are polynomials in $m$ and $n$ with degree $d:=\dim E$ in $n$ and $m$ and total degree $d$, and one has $P_{H,A}(E)(\bullet,0)=P_{H}(E)$ and $p_{H,A}(E)(\bullet,0)=p_{H}(E)$.
There is a natural ordering of polynomials in one variable given by the lexicographic ordering of their coefficients.
This generalises to polynomials of two variables by the identification $\IQ[m,n]=(\IQ[m])[n]$, i.e.\ we consider the elements as polynomials in $n$ and use the ordering of $\IQ[m]$ for comparing coefficients.
We introduce another ordering on $\IQ[m,n]$ by defining
$$f\le_0 g\quad:\Leftrightarrow\quad (f(\bullet,0),-f)\le(g(\bullet,0),-g)$$ for $f,g\in\IQ[m,n]$, where on the right hand side we use lexicographic ordering on the product $\IQ[m]\times\IQ[m,n]$, i.e.\ $f\le_0 g$ if and only if $f(\bullet,0)<g(\bullet,0)$ or $f(\bullet,0)=g(\bullet,0)$ and $f\ge g$.
Clearly one has $f=_0g$ if and only if $f=g$.
We say that $E$ is $(H,A)$-(semi)stable if it is pure and if for any proper nontrivial subsheaf $F\subset E$ one has that $p_{H,A}(F)\lelo p_{H,A}(E)\,.$
\end{enumerate}
The case of Gieseker stability can be regained by $V=\cO_Y$ from twisted stability or by $H=A$ from $(H,A)$-stability.
We will always write (semi)stable whenever the statement allows all three notions.
For twisted stability, we will always tacitly assume to have chosen a locally free sheaf $V$ on $Y$, and for $(H,A)$-stability to have chosen an additional ample divisor $A$ on $Y$.
If $H$ and $A$ are two ample divisors on $Y$ and $V$ is a locally free sheaf on $Y$, then the divisors $f^*H$ and $f^*A$ are ample divisors on $X$, and $f^*V$ is a locally free sheaf on $X$.
Also for a sheaf on $X$, we will write (semi)stable instead of Gieseker $f^*H$-(semi)stable, $f^*V$-twisted $f^*H$-(semi)stable or $(f^*H,f^*A)$-(semi)stable.
Moreover, we will denote the usual reduced Hilbert polynomial $p_H(E)$, the reduced $V$-twisted Hilbert polynomial $p_H^V(E)$ and the polynomial $p_{H,A}(E)$ by $p(E)$, and analogously for a sheaf $E'\in\ensuremath{\mathrm{Coh}}(X)$ one might insert $p_{f^*H}(E')$, $p_{f^*H}^{f^*V}(E')$ or $p_{f^*H,f^*A}(E')$ according to the notion one is interested in.
If we compare the polynomials of $E$ and of a nontrivial subsheaf $F\subseteq E$, then e.g.\ $p(F) \le p(E)$ has to be understood as $p_H(F) \le p_H(E)$, $p_H^V(F) \le p_H^V(E)$, and $p_{H,A}(F) \le_0 p_{H,A}(E)$ (!), respectively. Whenever we need to be more precise, we use the more explicit notation.
\subsection{$f^*$ preserves pureness}
As pureness is part of the definition of stability, we need to check its preservation under pullback.
\begin{lemma}\label{PurePbInv}
Let $E$ be a coherent sheaf on $Y$. $E$ is pure if and only if $f^*E$ is pure.
\end{lemma}
\begin{proof}
For a coherent sheaf $E$ on $Y$ the authors of \cite{HL10} define in Definition 1.1.7 the dual sheaf of $E$ to be $E^D:=\cExt^c(E,K_Y)$, where $K_Y$ is the canonical bundle and $c:=\dim Y-\dim E$ is the codimension of the sheaf $E$. This dualisation commutes with pullback:
By \cite{EGA31} (13.3.5) there is a canonical isomorphism
$$f^*\cExt^c_Y(E,K_Y) \cong \cExt^c_X(f^*E,f^*K_Y)\;.$$
By \cite[Section I.16]{BHPV} one has $f^*K_Y=K_X$, thus $f^*(E^D) \cong (f^*E)^D$.
Now let $E$ be pure. Then by \cite[Lemma 1.1.10]{HL10} the natural homomorphism $\theta_E\colon E\to E^{DD}$ is injective.
As $f^*$ is exact, $f^*(\theta_E)\colon f^*E\to f^*(E^{DD})$ is injective as well.
Using the isomorphism $f^*(E^{DD}) \cong (f^*E)^{DD}$ from above one gets the injectivity of the natural morphism $\theta_{f^*E}\colon f^*E\to (f^*E)^{DD}$, hence $f^*E$ is pure.
Conversely, let $E$ be not pure. Then the natural homomorphism $\theta_E\colon E\to E^{DD}$ has a nontrivial kernel. The exactness of $f^*$ gives a nontrivial kernel of $\theta_{f^*E}\colon f^*E\to (f^*E)^{DD}$, hence $f^*E$ is not pure.
\end{proof}
\subsection{$f^*$ and (semi)stability}
Let $H$ be an ample divisor on $Y$ and $E$ a coherent sheaf on $Y$.
In order to exhibit stability under pullback, we need the behaviour of the reduced Hilbert polynomial under pullback.
\begin{lemma}\label{rHpPb}
$p(f^*E) = p(E)\,.$
\end{lemma}
\begin{proof}
By \cite[\S 12 Theorem 2]{Mum70} one has $\chi(f^*E)=\deg f\; \chi(E)$. Thus on the one hand one has $p_{f^*H}^{f^*V}(f^*E) = p_H^V(E)$,
and on the other hand, one has $p_{f^*H,f^*A}(f^*E) = p_{H,A}(E)$.
\end{proof}
\begin{proposition}\label{PbStab}
$f^*E$ is semistable if and only if $E$ is semistable. If $f^*E$ is stable, then $E$ is stable.
More precisely,
\begin{enumerate}
\item
$f^*E$ is $f^*H$-semistable if and only if $E$ is $H$-semistable.
If $f^*E$ is $f^*H$-stable, then $E$ is $H$-stable.
\item
$f^*E$ is $f^*V$-twisted $f^*H$-semistable if and only if $E$ is $V$-twisted $H$-semistable.
If $f^*E$ is $f^*V$-twisted $f^*H$-stable, then $E$ is $V$-twisted $H$-stable.
\item
$f^*E$ is $(f^*H,f^*A)$-semistable if and only if $E$ is $(H,A)$-semistable.
If $f^*E$ is $(f^*H,f^*A)$-stable, then $E$ is $(H,A)$-stable.
\end{enumerate}
\end{proposition}
\begin{proof}
The pureness condition is given by Lemma \ref{PurePbInv}.
Let $f^*E$ be (semi)stable, and let $F\subseteq E$ be a nontrivial proper subsheaf.
Then $f^*F\subseteq f^*E$ is a nontrivial proper subsheaf, hence by (semi)stability one has
$p(f^*F)\lel p(f^*E)\,.$
Lemma \ref{rHpPb} yields $p(F)\lel p(E)\,,$ thus $E$ is (semi)stable.
In order to prove the other direction, let $E$ be semistable and assume that $f^*E$ is not semistable.
Let $F\subseteq f^*E$ be a maximal destabilising subsheaf, i.e.\ $F\subseteq f^*E$ is the first part of the Harder-Narasimhan filtration with respect to the considered semistability notion.
$f^*E$ is a $G$-sheaf in the sense of \cite[\S 7]{Mum70}. As $F$ is maximal, it is also $G$-invariant and thus a $G$-subsheaf of $f^*E$.
By \cite[\S 7 Proposition 2]{Mum70} one has $F\cong f^*\left((f_*F)^G\right)$ and $\left(f_*f^*E\right)^G \cong E$. In particular, $(f_*F)^G$ is isomorphic to a subsheaf of $E$.
The semistability of $E$ yields
$p((f_*F)^G) \le p(E)\,,$ and Lemma \ref{rHpPb} then
$p(F) \le p(f^*E)\,,$
which contradicts the assumption. Thus $f^*E$ is semistable.
\end{proof}
\noindent Clearly, if $E$ is strictly semistable, i.e.\ semistable but not stable, then $f^*E$ is strictly semistable.
The converse does not hold for stability: If $E$ is stable then $f^*E$ need not be stable. The following lemma and proposition should be well-known at least for Gieseker stability.
\begin{lemma}\label{StabDestab}
Let $E$ be semistable, and $F_1$ and $F_2$ two destabilising subsheaves with $F_1\cap F_2\neq 0$. Then $p(F_1+F_2)=p(E)$.
Moreover, if $F_1$ is stable, then $F_1\cap F_2 = F_1$ and $F_1 + F_2= F_2$. If $F_2$ is stable as well, then $F_1=F_2$.
\end{lemma}
\begin{proof}
As $F_1$ and $F_2$ are destabilising subsheaves, one has $p(F_1)=p(E)=p(F_2)$.
In particular, $F_1$ and $F_2$ are both semistable, as well as $F_1\oplus F_2$.
Hence one gets the inequality chain $p(F_1\oplus F_2)\le p(F_1+F_2) \le p(E)$,
and together with $p(E)=p(F_1\oplus F_2)$ one has equality everywhere.
The exact sequence
$$0\longrightarrow F_1\cap F_2 \longrightarrow F_1\oplus F_2 \longrightarrow F_1 + F_2 \longrightarrow 0$$
yields
$p(F_1\cap F_2)=p(F_1\oplus F_2)=p(F_1)\,.$
If $F_1$ is stable, then this implies $F_1\cap F_2 = F_1$ and $F_1 + F_2= F_2$.
If $F_2$ is stable as well, then $F_2\cap F_1 = F_2$ as we just proved.
\end{proof}
\begin{proposition}\label{PStGshDe}
Let $E'$ be a semistable sheaf on $X$ and $(F_i)_{i\in I}$ a family of destabilising stable subsheaves of $E'$.
Then there is a subset $J\subseteq I$ such that $E'\supseteq\sum_{i\in I} F_i=\oplus_{i\in J} F_i$.
In particular, if $E'$ is a semistable $G$-sheaf and $F\subseteq E'$ is a destabilising stable subsheaf, then one has $E'\supseteq\sum_{g\in G} g^*F=\oplus_{g\in G'} g^*F$ for a suitable subset $G'\subseteq G$.
\end{proposition}
\begin{proof}
This follows from Lemma \ref{StabDestab} by induction. More precisely, let $F_{1,2}$ be two destabilising stable subsheaves of $E'$. Then either $F_1\cap F_2=0$, i.e.\ $F_1+F_2=F_1\oplus F_2$, or $F_1\cap F_2\neq 0$ and thus $F_1+F_2=F_2$ by Lemma \ref{StabDestab}. This also holds if $F_2$ is not stable, which proves the inductive step.
\end{proof}
\noindent In the preceding Proposition \ref{PStGshDe} the pullbacks $f^*H$, $f^*A$ and $f^*V$, which are hidden in the notation, can be replaced by ample divisors $H'$ and $A'$ on $X$ and a locally free sheaf $V'$ on $X$, respectively.
\begin{proposition}\label{PbStabPStab}
If $E$ is stable then $f^*E\cong \oplus_{g\in G'} g^*F$ for a destabilising stable subsheaf $F\subset f^*E$ and a suitable subset $G'\subseteq G$.
In particular, $f^*E$ is polystable.
One has
$$\Ext^k(f^*E,f^*E)\cong\left(\bigoplus_{h\in G'}\Ext^k(F,h^*F)\right)^{|G'|}$$
for all $k$, and
$\ensuremath{\mathrm{Hom}}(f^*E,f^*E)\cong\IC^{|G'|\cdot |G''|}\,,$
where $G'':= \{ g\in G' \;|\; g^*F\cong F \}$. In particular,
$f^*E$ is stable if and only if it is simple.
\end{proposition}
\begin{proof}
Let $E$ be stable. By Proposition \ref{PbStab} $f^*E$ is semistable.
Let $F\subseteq f^*E$ be a destabilising stable subsheaf.
Then by Proposition \ref{PStGshDe} $f^*E\supseteq F':=\sum_{g\in G} g^*F=\oplus_{g\in G'} g^*F$ with $G'\subseteq G$ as in Proposition \ref{PStGshDe}.
As $F'$ is a $G$-subsheaf, by \cite[\S 7 Proposition 2]{Mum70} the sheaf $(f_*F')^G$ is isomorphic to a subsheaf of $E$, and
one has $F'\cong f^*\left((f_*F')^G\right)$. Applying Lemma \ref{rHpPb} to $p(F')=p(F)=p(f^*E)$
yields $p((f_*F')^G)=p(E)\,.$
Due to the stability of $E$, one has $(f_*F')^G\cong E$, hence $F'=f^*E$.
One has
\begin{eqnarray*}
\Ext^k(f^*E,f^*E)&\cong&\bigoplus_{g,h\in G'}\Ext^k(g^*F,h^*F)\\
&\cong&\bigoplus_{g,h\in G'}\Ext^k(g^*F,h^*g^*F)\cong \left(\bigoplus_{h\in G'}\Ext^k(F,h^*F)\right)^{|G'|}
\end{eqnarray*}
for all $k$.
As $\ensuremath{\mathrm{Hom}}(F,h^*F)=0$ unless $F\cong h^*F$, one has
\begin{eqnarray*}
\ensuremath{\mathrm{Hom}}(f^*E,f^*E)\cong\left(\bigoplus_{h\in G'}\ensuremath{\mathrm{Hom}}(F,h^*F)\right)^{|G'|}\cong\left(\bigoplus_{h\in G''}\ensuremath{\mathrm{Hom}}(F,h^*F)\right)^{|G'|}\cong\IC^{|G'|\cdot |G''|}\,.
\end{eqnarray*}
In particular, if $f^*E$ is not stable, then $|G'|\neq 1$, i.e.\ $f^*E$ is not simple. Conversely, any stable sheaf is always simple.
\end{proof}
\begin{corollary}\label{PbPStabPStab}
If $E$ is polystable then $f^*E$ is a polystable $G$-sheaf.
\end{corollary}
\subsection{The pullback morphism}
We want to apply these results to moduli spaces of sheaves. We keep considering the three stability notions at once:
\begin{enumerate}
\item \textit{Gieseker stability.} The moduli space of Gieseker semistable sheaves with given Hilbert polynomial is standard by now. A detailed treatment can be found in \cite{HL10}.
\item \textit{Twisted stability.} The moduli space of twisted semistable sheaves with given twisted Hilbert polynomial is constructed in \cite[Section 4]{Yos03b}.
\item \textit{$(H,A)$-stability.} The moduli space of $(H,A)$-semistable sheaves with given Hilbert polynomial is constructed in \cite[Section 8]{Zow12}.
\end{enumerate}
Our main result in the general setting is the following:
\begin{theorem}\label{MHApb}
Let $P$ be a polynomial, $M_Y$ a quasiprojective and nonempty subscheme of the moduli space $M_Y(P)$ of semistable sheaves on $Y$ with (twisted) Hilbert polynomial $P$, $M_X=M_X(\deg f\cdot P)$ the moduli space of semistable sheaves on $X$ with (twisted) Hilbert polynomial $\deg f\cdot P$, and $M_Y^s$ and $M_X^s$ the respective loci of stable sheaves.
The pullback by $f$ induces a morphism $f^*\colon M_Y\to M_X$ which maps closed points $[E]$ to $[f^*E]$.
The closed points of its image are represented by polystable $G$-sheaves, and $(f^*)^{-1}( M_X^s )\subseteq M_Y^s\,.$
\end{theorem}
\begin{proof}
Let $F\in\ensuremath{\mathrm{Coh}}(Y\times S)$ be a flat family of semistable sheaves on $Y$ with (twisted) Hilbert polynomial $P$, which is parametrised by a scheme $S$.
Then $(f\times\mathrm{id}_S)^*F\in\ensuremath{\mathrm{Coh}}(X\times S)$ is a flat family of semistable sheaves on $X$ with (twisted) Hilbert polynomial $\deg f\cdot P$ by Proposition \ref{PbStab}.
Hence one has a natural transformation between the moduli functors, which induces a morphism $f^*\colon M_Y\to M_X$.
If $[E]$ is a closed point represented by a polystable sheaf $E$ on $Y$ then $f^*E$ is a polystable $G$-sheaf by Corollary \ref{PbPStabPStab}.
The statement on the stable locus follows also from Proposition \ref{PbStab}.
\end{proof}
For simplicity we restrict to simple cyclic coverings for the rest of this article.
\section{Cyclic coverings}\label{arbCC}
We keep all notations and assumptions as before. Additionally, for the whole section, let $f$ be a cyclic covering given by a line bundle $L$ on $Y$ of finite order $n$, and let $\nu$ be the order of $\bc_1(L)$.
Moreover, let $M_Y\subseteq M_Y(P)$ be a nonempty quasiprojective subscheme containing classes of sheaves of rank $r$ and $m\big|\frac n{\gcf(n,r)}$ such that the morphism $\varphi\colon M_Y\to M_Y$ induced by $\otimes L^{\otimes m}$ is well-defined. We denote the stable locus by $M_Y^s$.
We are interested in the following particular examples:
\begin{example}\label{Mexa}
\begin{enumerate}
\item $M_Y=M_Y(P)$ and $m=1$. As the Hilbert polynomial of a sheaf is invariant under $\otimes L$, the morphism $\varphi$ is well-defined.
\item $M_Y$ is the subscheme of $M_Y(P)$ containing classes of sheaves with fixed determinant and $m=\frac n{\gcf(n,r)}$.
\item $M_Y$ is the subscheme of $M_Y(P)$ containing classes of sheaves with fixed first Chern class and $m=\frac {\nu}{\gcf(\nu,r)}$.
\end{enumerate}
\end{example}
\noindent The following two Lemmas \ref{PreimDiv2} and \ref{PreimDiv3} ensure that $\varphi$ is well-defined in the cases 2 and 3.
\begin{lemma}\label{PreimDiv2}
Let $E$ be a coherent sheaf on $Y$. Then
$\det(E)\cong \det(E\otimes L^{j})$ if and only if $\frac n{\gcf(n,\rk E)} \big| j$.
In particular, the number of nonisomorphic sheaves of the form $E\otimes L^{j}$ with $\det(E)=\det(E\otimes L^{j})$ is equal to $\frac {\ensuremath{\mathrm{ord}}_E(L)\;\gcf(n,\rk E)}n$.
\end{lemma}
\begin{proof}
One has $\det(E\otimes L^{j})\cong \det(E)\otimes L^{\rk E \cdot j}$.
Hence $\det(E\otimes L^{j})\cong\det(E)$ if and only if $L^{\rk E \cdot j}\cong\cO_Y$.
This is equivalent to $n\big|\rk E\cdot j$, i.e.\ $\frac n{\gcf(n,\rk E)}\big|j$.
The nonisomorphic sheaves of the form $E\otimes L^{j}$ with $\det(E)\cong \det(E\otimes L^{j})$ are (up to isomorphisms) precisely those with $0\le j<\ensuremath{\mathrm{ord}}_E(L)$ and $j$ a multiple of $\frac n{\gcf(n,\rk E)}$, and there are $\frac {\ensuremath{\mathrm{ord}}_E(L)\,\gcf(n,\rk E)}n$ such choices.
\end{proof}
\noindent Analogously one proves the following:
\begin{lemma}\label{PreimDiv3}
Let $E$ be a coherent sheaf on $Y$. Then
$\bc_1(E)=\bc_1(E\otimes L^{j})$ if and only if $\frac {\nu}{\gcf(\nu,\rk E)} \big| j$.
In particular, the number of nonisomorphic sheaves of the form $E\otimes L^{j}$ with $\bc_1(E)=\bc_1(E\otimes L^{j})$ is equal to $\frac {\ensuremath{\mathrm{ord}}_E(L)\;\gcf(\nu,\rk E)}{\nu}$.
\end{lemma}
\noindent Moreover, these two Lemmas \ref{PreimDiv2} and \ref{PreimDiv3} ensure that $\frac {\nu}{\gcf(\nu,r)}\big|\frac n{\gcf(n,r)}$, as equal determinants imply equal first Chern classes.
\subsection{The pullback morphism factorises}
The morphism between the moduli spaces induced by the pullback by $f$ factorises as follows:
\begin{proposition}\label{pbFact}
There is a commutative diagram
\begin{center}
\ \xymatrix{
M_Y \ar[rr]^{f^*} \ar@{->>}[rd]^{\pi} && M_X(nP) \\
&M_Y/\langle\varphi\rangle \ar[ru]^{\vartheta}
}\
\end{center}
with $f^*(M_Y)=\vartheta(M_Y/\langle\varphi\rangle)$.
If we choose one of the Examples \ref{Mexa} then $\vartheta|_{M_Y^s/\langle \varphi \rangle}$ is injective.
\end{proposition}
\begin{proof}
As the morphism $f^*\colon M_Y\to M_X(nP)$ is $\varphi$-invariant, it factors through the quotient morphism $\pi$ and yields the morphism $\vartheta$ satisfying $f^*=\vartheta\circ\pi$.
$f^*(M_Y)=\vartheta(M_Y/\langle\varphi\rangle)$ follows from the surjectivity of $\pi$.
By Lemma \ref{PreimStab} below any preimage of $[f^*E]$ for $E\in M_Y^s$ is isomorphic to $E\otimes L^{j}$ for some $j$ with $0\le j<|G|$.
Thus $\vartheta|_{M_Y^s/\langle \varphi \rangle}$ is injective in the Examples \ref{Mexa}, using Lemmas \ref{PreimDiv2} and \ref{PreimDiv3} for the cases 2 and 3, respectively.
\end{proof}
\noindent If we choose Example \ref{Mexa}.2 and if $\gcf(n,r)=1$, then $m=n$, $\pi$ is an isomorphism, and $f^*|_{M_Y^s}$ is injective.
\begin{lemma}\label{PreimStab}
Let $E$ and $F$ be two stable sheaves on $Y$ with $f^*E\cong f^*F$.
Then there is a $j$ such that $F\cong E\otimes L^{j}$.
\end{lemma}
\begin{proof}
By Lemma \ref{CycShDec1} below one has $f_*f^*E\cong \oplus_{j=0}^{n-1}E\otimes L^{j}$ and $f_*f^*F\cong \oplus_{j=0}^{n-1}F\otimes L^{j}$.
As we assume $f^*E\cong f^*F$, one has $\oplus_{j=0}^{n-1}E\otimes L^{j}\cong f_*f^*E\cong f_*f^*F\cong\oplus_{j=0}^{n-1}F\otimes L^{j}$.
Therefore $\ensuremath{\mathrm{Hom}}(\oplus_{j=0}^{n-1}E\otimes L^{j},\oplus_{k=0}^{n-1}F\otimes L^k)\cong \oplus_{j,k} \ensuremath{\mathrm{Hom}}(E\otimes L^j, F\otimes L^k)$ is nonzero, i.e.\ for some $j,k$ there is a nontrivial homomorphism $E\otimes L^j\to F\otimes L^k$ which must be an isomorphism because $E\otimes L^j$ and $F\otimes L^k$ are stable and have the same (twisted) Hilbert polynomial. This yields the isomorphism $E\otimes L^{j-k}\to F$.
\end{proof}
\begin{lemma}\label{CycShDec1}
Let $E$ be a coherent sheaf on $Y$. Then
$f_*f^*E\cong E\otimes f_*\cO_X\cong \oplus_{j=0}^{n-1}E\otimes L^{j}\,.$
\end{lemma}
\begin{proof}
In \cite[\S 7]{Mum70}
there is the statement that $f_*f^*E\cong E\otimes f_*\cO_X$, but without any further comment. Hence we give a proof:
Let $E_\bullet\twoheadrightarrow E$ be a locally free resolution.
As the functors $f_*$, $f^*$ and $\otimes f_*\cO_X$
are exact, we get the commutative diagram
\begin{center}
\ \xymatrix{
...\ar[r]& f_*f^*E_{-2}\ar[r]\ar[d]^{\cong} & f_*f^*E_{-1}\ar[r]\ar[d]^{\cong}& f_*f^*E \ar[r] &0 \\
...\ar[r]& E_{-2}\otimes f_*\cO_X\ar[r] & E_{-1}\otimes f_*\cO_X\ar[r]& E\otimes f_*\cO_X \ar[r] &0
}\
\end{center}
with exact rows, where the vertical arrows are the natural isomorphisms given by the projection formula.
Thus there is also an isomorphism $f_*f^*E\cong E\otimes f_*\cO_X$.
Using \cite[Lemma I.17.2]{BHPV}, which states that $f_*\cO_X=\oplus_{j=0}^{n-1}L^{j}$, one gets the second isomorphism.
\end{proof}
\subsection{The morphism $\pi\colon M_Y\to M_Y/\langle\varphi\rangle$}
In this section we exhibit the morphism $\pi\colon M_Y\to M_Y/\langle\varphi\rangle$ from Proposition \ref{pbFact}.
We start with the following definition:
\begin{definition}\label{Eorder}
For a coherent sheaf $E$ on $Y$ the $E$-order of $L$ is the number $\ensuremath{\mathrm{ord}}_E(L):=\min \{ j \in \IN \;|\; E\cong E\otimes L^{j}\}$.
\end{definition}
\noindent Clearly $1\le\ensuremath{\mathrm{ord}}_E(L)\le n$.
\begin{lemma}\label{PreimDiv1}
Let $E$ be a coherent sheaf on $Y$. Then
$E\otimes L^{j}\cong E\otimes L^{k}$ if and only if $\ensuremath{\mathrm{ord}}_E(L)|(k-j)$.
In particular, the sheaves $E\otimes L^{j}$ are pairwise nonisomorphic for $0\le j<\ensuremath{\mathrm{ord}}_E(L)$, and the number of nonisomorphic sheaves of the form $E\otimes L^{j}$ with $j\in\IZ$ is equal to $\ensuremath{\mathrm{ord}}_E(L)$.
\end{lemma}
\begin{proof}
One has $k-j=a\,\ensuremath{\mathrm{ord}}_E(L)+r$ for some $a,r\in\IZ$ with $0\le r<\ensuremath{\mathrm{ord}}_E(L)$, and
$$E\otimes L^{k-j}\cong E\otimes L^{a\,\ensuremath{\mathrm{ord}}_E(L)+r}\cong E\otimes (L^{\ensuremath{\mathrm{ord}}_E(L)})^{a}\otimes L^{r}\cong E\otimes L^{r}\,.$$
Due to the minimality of $\ensuremath{\mathrm{ord}}_E(L)$ one has that $E\cong E\otimes L^{k-j}$ if and only if $r=0$.
Thus $E\otimes L^{k}\cong E\otimes L^{j}$ if and only if $r=0$, which is equivalent to $\ensuremath{\mathrm{ord}}_E(L)|(k-j)$.
\end{proof}
\begin{lemma}\label{PreimDiv4}
Let $E$ be a coherent sheaf on $Y$. Then
$\frac n{\gcf(n,\rk E)} \big| \ensuremath{\mathrm{ord}}_E(L) \big| n$. In particular, $\frac {\ensuremath{\mathrm{ord}}_E(L)\,\gcf(n,\rk E)} n \big| \gcf(n,\rk E)$.
\end{lemma}
\begin{proof}
This follows from Lemma \ref{PreimDiv1} with $k=n$ and $j=0$, and Lemma \ref{PreimDiv2} with $j=\ensuremath{\mathrm{ord}}_E(L)$.
\end{proof}
\noindent As $1\le\ensuremath{\mathrm{ord}}_E(L)\le n$ for a coherent sheaf $E$ on $Y$, the following definition makes sense:
\begin{definition}\label{Morder}
The $M_Y$-order of a line bundle $L$ on $Y$ of order $n$ is the number $\ensuremath{\mathrm{ord}}_{M_Y}(L):=\lcm \{ \ensuremath{\mathrm{ord}}_{E}(L) \in \IN \;|\; [E]\in M_Y\}$, i.e. the least common multiple of all $E$-orders of $L$, where $E$ runs through all sheaves with equivalence class in $M_Y$.
\end{definition}
\begin{lemma}\label{orderDiv} For all sheaves $E$ with equivalence class in $M_Y$ one has that
$$\frac n{\gcf(n,\rk E)}\Big| \ensuremath{\mathrm{ord}}_E(L) \Big| \ensuremath{\mathrm{ord}}_{M_Y}(L) \Big|n$$ and therefore
$$\frac {\ensuremath{\mathrm{ord}}_E(L)\,\gcf(n,\rk E)}n \Big| \frac {\ensuremath{\mathrm{ord}}_{M_Y}(L)\,\gcf(n,\rk E)}n \Big| \gcf(n,\rk E)\,.$$
\end{lemma}
\begin{proof}
By Lemma \ref{PreimDiv4}, $\frac n{\gcf(n,\rk E)} \big| \ensuremath{\mathrm{ord}}_E(L) \big| n$ for a coherent sheaf $E$ on $Y$.
The first row now follows from the definition of the least common multiple, and the second is an immediate consequence.
\end{proof}
\noindent We are now ready for the main result on the first factorising morphisms $\pi\colon M_Y\to M_Y/\langle\varphi\rangle$ of Proposition \ref{pbFact}:
\begin{proposition}\label{Mdecomp}
Let $M_k:=\{ [E] \in M_Y \;|\; \ensuremath{\mathrm{ord}}_E(L)| k \}$ for $k\in\IN$, and let $j\in\IN$ such that $M_j\neq\emptyset$.
\begin{enumerate}
\item $\varphi(M_k)=M_k$.
\item $\frac n{\gcf(n,r)}\Big|\ensuremath{\mathrm{ord}}_{M_j}(L)\Big|j$.
\item If $k|\ensuremath{\mathrm{ord}}_{M_j}(L)$, then $M_k\subseteq M_j$.
\item $M_j$ is the fixed point set of the morphism $\varphi^{\frac j m}$ (composition of morphisms).
In particular, it is closed in $M_Y$.
\item One has that $M_k=M_Y$ if and only if $\ensuremath{\mathrm{ord}}_{M_Y}(L)|k$.
In particular, $M_{\ensuremath{\mathrm{ord}}_{M_Y}(L)}=M_Y$, and $\ensuremath{\mathrm{ord}}_{M_Y}(L)\ge 1$ is minimal with this property.
\item $\displaystyle\Sigma_j:=\{ [E] \in M_j \;|\; \ensuremath{\mathrm{ord}}_E(L)\neq\ensuremath{\mathrm{ord}}_{M_j}(L)\}=\hspace{-0.5cm}\bigcup_{\frac n{\gcf(n,r)}|k|\ensuremath{\mathrm{ord}}_{M_j}(L),k\neq\ensuremath{\mathrm{ord}}_{M_j}(L)}\hspace{-0.5cm}M_k$, and it is closed in $M_j$ and ${M_Y}$.
\item
Assume that $M_Y=M_Y^s$.
Then the number of preimages under the quotient morphism $\pi\colon M_j\to M_j/\langle\varphi\rangle$ of $\pi([E])$ with $[E]\in M_j$ is $\frac {\ensuremath{\mathrm{ord}}_E(L)}{m}$.
Moreover, if $M_j\neq \Sigma_j$, then $\pi$ is an $\frac {\ensuremath{\mathrm{ord}}_{M_j}(L)}{m}:1$ covering branched along $\Sigma_j$.
\end{enumerate}
\end{proposition}
\begin{proof}
\begin{enumerate}
\item Let $[E]\in M_k$. Then $\ensuremath{\mathrm{ord}}_{E\otimes L^{\otimes m}}(L)=\ensuremath{\mathrm{ord}}_{E}(L)$. Thus $\varphi([E])\in M_k$.
\item For all $[E] \in M_j$ one has that $\ensuremath{\mathrm{ord}}_E(L)|j$. By definition of the least common multiple, $\ensuremath{\mathrm{ord}}_{M_j}(L)|j$. By Lemma \ref{orderDiv} $\frac n{\gcf(n,r)}\big|\ensuremath{\mathrm{ord}}_{M_j}(L)$.
\item Let $[E]\in M_Y$ such that $\ensuremath{\mathrm{ord}}_E(L)| k$, i.e.\ $[E]\in M_k$. By assumption, $k|\ensuremath{\mathrm{ord}}_{M_j}(L)$, and by item 2, $\ensuremath{\mathrm{ord}}_{M_j}(L)|j$. Thus $\ensuremath{\mathrm{ord}}_E(L)|j$, i.e.\ $[E]\in M_j$.
\item Let $[E] \in M_Y$. By Lemma \ref{PreimDiv1} $\ensuremath{\mathrm{ord}}_E(L)|j$ if and only if $E\cong E\otimes L^{\otimes j}$, and the morphism induced by $\otimes L^{\otimes j}$ is $\varphi^{\frac j m}\colon M_Y\to M_Y$. Here we use that $m\big|\frac n{\gcf(n,r)}$ by assumption, and $\frac n{\gcf(n,r)}\big|j$ by item 2. The fixed point set of an automorphism of a separated scheme is closed.
\item If $\ensuremath{\mathrm{ord}}_{M_Y}(L)|k$ then clearly $\ensuremath{\mathrm{ord}}_E(L)|k$ for any $[E] \in M_Y$ by definition, hence $[E]\in M_k$.
Conversely, if $M_k=M_Y$ then $\ensuremath{\mathrm{ord}}_M(L)|k$ by item 2.
\item
Let $[E]\in M_k$ with $k\neq\ensuremath{\mathrm{ord}}_{M_j}(L)$ and $\frac n{\gcf(n,r)}\big|k\big|\ensuremath{\mathrm{ord}}_{M_j}(L)$. By item 3 $[E]\in M_j$, and as $\ensuremath{\mathrm{ord}}_E(L)|k$ one has that $\ensuremath{\mathrm{ord}}_E(L)\le k<\ensuremath{\mathrm{ord}}_{M_j}(L)$.
Conversely, let $[E]\in M_j$ with $\ensuremath{\mathrm{ord}}_E(L)\neq\ensuremath{\mathrm{ord}}_{M_j}(L)$. Clearly $[E]\in M_{\ensuremath{\mathrm{ord}}_E(L)}$, and by Lemma \ref{orderDiv} one has that $\frac n{\gcf(n,r)}\big| \ensuremath{\mathrm{ord}}_E(L)\big|\ensuremath{\mathrm{ord}}_{M_j}(L)$.
$\Sigma_j$ is the union of finitely many subschemes $M_k$ that are closed in $M_Y$ by item 4 or because they are empty, hence $\Sigma_j$ is closed in $M_Y$, and thus also in $M_j$.
\item
The number of nonisomorphic sheaves of the form $E\otimes L^{\otimes k}$ is $\ensuremath{\mathrm{ord}}_E(L)$ by Lemma \ref{PreimDiv1}.
As $m\big|\frac n{\gcf(n,r)}$ by assumption and $\frac n{\gcf(n,r)}\big|\ensuremath{\mathrm{ord}}_E(L)$ by Lemma \ref{orderDiv}, one has that $m|\ensuremath{\mathrm{ord}}_E(L)$.
Hence there are $\frac {\ensuremath{\mathrm{ord}}_E(L)}{m}$ nonisomorphic sheaves of the form $E\otimes L^{\otimes km}$, and their classes are the preimages of $\pi$.
If $M_j\neq \Sigma_j$, then $\Sigma_j$ is a proper closed subset. By definition of $\Sigma_j$, for $[E]\in M_j\setminus \Sigma_j$ one has that $\ensuremath{\mathrm{ord}}_E(L)=\ensuremath{\mathrm{ord}}_{M_j}(L)$.
Therefore $M_j\setminus \Sigma_j\to (M_j\setminus \Sigma_j)/\langle \varphi\rangle $ is an unbranched $\frac {\ensuremath{\mathrm{ord}}_{M_j}(L)}{m}:1$ covering, and $\pi$ is an $\frac {\ensuremath{\mathrm{ord}}_{M_j}(L)}{m}:1$ covering branched along $\Sigma_j$.
\qedhere
\end{enumerate}
\end{proof}
\noindent The statement we are most interested in is that if $\Sigma:= \{ [E] \in M_Y^s \;|\; \ensuremath{\mathrm{ord}}_E(L)\neq\ensuremath{\mathrm{ord}}_{M_Y^s}(L)\}$ is a proper subset of $M_Y^s$ then $\pi\colon M_Y^s\to M_Y^s/\langle\varphi\rangle$ is an $\frac {\ensuremath{\mathrm{ord}}_{M_Y^s}(L)}{m}:1$ covering branched along the closed subscheme $\Sigma$.
This follows from the above Proposition \ref{Mdecomp} using $j=\ensuremath{\mathrm{ord}}_{M_Y^s}(L)$.
The case of a prime power covering ensures the assumption if $M_Y^s\neq\emptyset$, as by Lemma \ref{orderDiv} $\ensuremath{\mathrm{ord}}_{M_Y^s}(L)$ is a prime power as well and therefore there is an $[E]\in M_Y^s$ such that $\ensuremath{\mathrm{ord}}_{M_Y^s}(L)=\ensuremath{\mathrm{ord}}_{E}(L)$. Thus we have the
\begin{corollary}\label{ppQuCov}
If $n$ is a prime power and $M_Y^s\neq\emptyset$ then $\pi\colon M_Y^s\to M_Y^s/\langle\varphi\rangle$ is an $\frac {\ensuremath{\mathrm{ord}}_{M_Y^s}(L)}{m}:1$ covering branched along the closed subscheme $\Sigma:= \{ [E] \in M_Y^s \;|\; \ensuremath{\mathrm{ord}}_E(L)\neq\ensuremath{\mathrm{ord}}_{M_Y^s}(L)\} $.
\end{corollary}
\subsection{Conclusions for the pullback morphism}
We now can give a description of the locus of stable sheaves becoming strictly semistable under the pullback morphism $f^*$ between the moduli spaces of sheaves. We still consider the situation of the beginning of Section \ref{arbCC} and have in mind the application to one of the cases in Example \ref{Mexa}.
\begin{theorem}\label{CycStab}
Let $M_Y^s\subseteq M_Y^s(P)$ be a nonempty quasiprojective subscheme containing classes of stable sheaves of rank $r$ such that the morphism $\varphi\colon M_Y^s\to M_Y^s$ induced by $\otimes L^{\otimes m}$ for a suitable $m\big|\frac n{\gcf(n,r)}$ is well-defined,
let $\Sigma:= \{ [E] \in M_Y^s \;|\; \ensuremath{\mathrm{ord}}_E(L)\neq\ensuremath{\mathrm{ord}}_{M_Y^s}(L)\}$, and let $f^*\colon M_Y^s\to M_X(nP)$ be the morphism induced by pullback by $f$.
The following conditions are equivalent:
\begin{enumerate}
\item $\ensuremath{\mathrm{ord}}_{M_Y^s}(L)=\max \{ \ensuremath{\mathrm{ord}}_{E}(L) \;|\; [E]\in M_Y^s\}=n$;
\item there is a stable sheaf of the form $f^*E$ for some $E\in M_Y^s$;
\item $f^*(M_Y^s\setminus\Sigma)=f^*(M_Y^s)\cap M^s_X(nP)\neq \emptyset$;
\item $M_Y^s\setminus\Sigma=(f^*)^{-1}(M^s_X(nP))\neq \emptyset$.
\end{enumerate}
\end{theorem}
\begin{proof}
3 $\Rightarrow$ 2 is trivial.\\
2 $\Rightarrow$ 1: A stable sheaf is simple, hence by Lemma \ref{CycShDec4} $\ensuremath{\mathrm{ord}}_E(L)=n$.
In particular, $\ensuremath{\mathrm{ord}}_{M_Y^s}(L)\ge\max \{ \ensuremath{\mathrm{ord}}_{E}(L) \;|\; [E]\in M_Y^s\}\ge n$.
As $\ensuremath{\mathrm{ord}}_E(L)|\ensuremath{\mathrm{ord}}_{M_Y^s}(L)|n$ by Lemma \ref{orderDiv}, one has that $n=\ensuremath{\mathrm{ord}}_{M_Y^s}(L)$.\\
1 $\Rightarrow$ 4: By Theorem \ref{MHApb} one has $(f^*)^{-1}(M^s_X(nP))\subseteq M_Y^s$. Thus we need to show that $E\not\in\Sigma$ if and only if $f^*E$ is stable for all $E\in M_Y^s$.
Therefore let $E\in M_Y^s$, and assume that $\ensuremath{\mathrm{ord}}_{M_Y^s}(L)=n$ due to item 1.
One has that $E\not\in\Sigma$ if and only if $\ensuremath{\mathrm{ord}}_E(L)=\ensuremath{\mathrm{ord}}_{M_Y^s}(L)=n$, i.e.\ $E\not\cong E\otimes L^{\otimes j}$ for $1\le j<n$.
As $E$ and $E\otimes L$ are both stable and have the same (twisted) Hilbert polynomial, any nontrivial homomorphism is an isomorphism. Thus $E\not\cong E\otimes L^{\otimes j}$ is equivalent to $\ensuremath{\mathrm{Hom}}(E,E\otimes L^{\otimes j})=0$.
In particular, $\ensuremath{\mathrm{ord}}_E(L)=n$ if and only if $\ensuremath{\mathrm{Hom}}(E,E\otimes L^{\otimes j})=0$ for $1\le j<n$.
By Lemma \ref{CycShDec4} below $\ensuremath{\mathrm{Hom}}(E,E\otimes L^{\otimes j})=0$ for $1\le j<n$ if and only if $f^*E$ is simple.
Finally $f^*E$ is simple if and only if it is stable by Proposition \ref{PbStabPStab}.
On the other hand, the assumption $\ensuremath{\mathrm{ord}}_{M_Y^s}(L)=\max \{ \ensuremath{\mathrm{ord}}_{E}(L) \;|\; [E]\in M_Y^s\}$ ensures that $\Sigma\neq M_Y^s$.\\
4 $\Rightarrow$ 3: This follows by taking the image by $f^*$.
\end{proof}
\noindent In the proof we used the following lemma, which will be useful more often.
\begin{lemma}\label{CycShDec4}
Let $E$ be a coherent sheaf on $Y$. Then $f^*E$ is simple if and only if $E$ is simple and $\ensuremath{\mathrm{Hom}}(E,E\otimes L^{\otimes j})=0$ for $1\le j<n$.
In particular, in this case, $E\not\cong E\otimes L^{\otimes j}$ for $1\le j<n$.
\end{lemma}
\begin{proof}
There is a natural isomorphism $\ensuremath{\mathrm{Hom}}(f^*E,f^*E)\cong\ensuremath{\mathrm{Hom}}(E,f_*f^*E)$, and together with Lemma \ref{CycShDec1} one gets $\ensuremath{\mathrm{Hom}}(f^*E,f^*E)\cong\oplus_{j=0}^{n-1}\ensuremath{\mathrm{Hom}}(E,E\otimes L^{\otimes j})$.
The claim now follows because $\ensuremath{\mathrm{Hom}}(E,E)\neq 0$.
\end{proof}
\noindent Coprimeness of rank and $n$ yield the simplest cases for the pullback morphism:
\begin{theorem}\label{gcdnr1}
Let $M_Y^s\subseteq M_Y^s(P)$ be a nonempty quasiprojective subscheme containing classes of stable sheaves of rank $r$ and $m|n$ such that the morphism $\varphi\colon M_Y^s\to M_Y^s$ induced by $\otimes L^{\otimes m}$ is well-defined, and let $f^*\colon M_Y^s\to M_X(nP)$ be the morphism induced by pullback by $f$.
Assume that $\gcf(n,r)=1$. Then $\pi\colon M_Y^s\to M_Y^s/\langle\varphi\rangle$ is an unbranched $\frac {n}{m}:1$ covering and $f^*(M_Y^s)\subseteq M_X^s(nP)$.
If we are in one of the cases of Example \ref{Mexa} then the morphism $f^*\colon M_Y^s\to M_X(nP)$ is $\frac {n}{m}:1$ onto its image.
\end{theorem}
\begin{proof}
By Lemma \ref{PreimDiv4}, one has that $\ensuremath{\mathrm{ord}}_E(L)=n$ for all $[E]\in M_Y^s$ and therefore $\ensuremath{\mathrm{ord}}_{M_Y^s}(L)=n$.
Thus $\Sigma:= \{ [E] \in M_Y^s \;|\; \ensuremath{\mathrm{ord}}_E(L)\neq\ensuremath{\mathrm{ord}}_{M_Y^s}(L)\}=\emptyset$, and by Proposition \ref{Mdecomp} $\pi$ is an unbranched $\frac {n}{m}:1$ covering.
As item 1 of Theorem \ref{CycStab} holds, item 3 of that theorem yields $f^*(M_Y^s)\subseteq M_X^s(nP)$.
If we are in one of the cases of Example \ref{Mexa} then by Proposition \ref{pbFact} the morphism ${M_Y^s/\langle \varphi \rangle}\to M_X(nP)$ induced by $f^*$ is injective. Hence the composition $M_Y^s\to{M_Y^s/\langle \varphi \rangle}\to M_X(nP)$ is $\frac {n}{m}:1$ onto its image.
\end{proof}
\section{Canonical coverings of surfaces}\label{Surfaces}
In this section we apply our results to canonical coverings of surfaces, i.e.\ cyclic coverings given by a torsion canonical bundle.
We start with some definitions in order to fix notations and conventions.
\subsection{Notations and conventions}
Let $Y$ be a nonsingular projective irreducible surface over $\IC$, $H$ an ample divisor on $Y$, and $E$ a coherent sheaf on $Y$.
We associate the element
$$u(E):=(\rk E,\bc_1(E),\chi(E))\in \Hev:=\IN_0\oplus\NS(Y)\oplus\IZ$$
of sheaf invariants to $E$.
We avoid the elegant notion of a Mukai vector in favour of keeping torsion inside $\NS(Y)$.
For an element $u:=(r,c,\chi)\in \Hev$ we define
\begin{eqnarray*}
P(u)&:=&r \frac{H^2}2n^2+\left(c-r\;\frac{K_Y}2\right).Hn+\chi\\
\Delta(u)&:=&c^2 - 2r\chi+2r^2\chi(\cO_Y)-rc.K_Y\\
\chi(u,u)&:=&\chi(\cO_Y) r^2-\Delta(u)\\
f^*u&:=&(r,f^*c,\deg f\;\chi)
\end{eqnarray*}
If $E$ satisfies $u(E)=u$, then, by Riemann-Roch, its Hilbert polynomial is $P(u)$, its discriminant\footnote{Be aware of different conventions of the discriminant's definition.} is $\Delta(u)$, $u(f^*E)=f^*u$ and
$$\chi(E,E):=\sum_{k=0}^2 \ext^k(E,E) = \chi(u,u)\,,$$ where $\ext^k(E,E):=\dim \Ext^k(E,E)$. We will also write $\hom(E,F):=\dim\ensuremath{\mathrm{Hom}}(E,F)$ for two coherent sheaves $E,F$.
\subsection{Symplectic moduli spaces and Lagrangian subvarieties}
We now restrict to canonical coverings, and thus assume that $Y$ has a torsion canonical bundle $K_Y$ of order, say $n$. Let $f\colon X\to Y$ be the covering given by $K_Y$. Then $K_X$ is trivial and either $X$ is a K3 surface, $Y$ is an Enriques surface and $n=2$, or $X$ is an abelian surface, $Y$ is a bielliptic surface and $n=2,3,4$ or $6$, see e.g.\ \cite{BHPV}.
As in Section \ref{arbCC} let $\nu$ be the order of $\bc_1(K_Y)$. If $Y$ is an Enriques surfaces, one has that $\nu=n=2$ due to \cite[Proposition 15.2]{BHPV}.
On the other hand, the situation for an elliptic surface $Y$ is more complicated. By \cite[\S 1]{Ser90} $\bH^2(Y,\IZ)$ may or may not be torsion free depending on the type of $Y$. In particular, $\nu$ may be smaller than $n$, and if $\bH^2(Y,\IZ)$ is torsion free then $\bc_1(K_Y)=0$ and $\nu=1$.
Let $u:=(r,c,\chi)\in \Hev$, and let $H$ and $A$ be two ample divisors on $Y$.
We will consider only Gieseker stability and $(H,A)$-stability in the following, and for the latter we assume $r>0$ for simplicity, as is done in \cite[Section 6]{Zow12}.
Let $M_Y(u)$ be the moduli space of Gieseker $H$- or $(H,A)$-semistable sheaves $E$ with $u(E)=u$. It is projective and a subscheme of $M_Y(P(u))$, hence the results of Section \ref{arbdim} apply. Moreover, by setting $m:=\frac {\nu}{\gcf(\nu,r)}$ we are in the situation of Example \ref{Mexa}.3 with $M_Y=M_Y(u)$.
In case we need to distinguish between Gieseker stability and $(H,A)$-stability, we will use $M_{Y;H}(u)$ and $M_{Y;H,A}(u)$, respectively.
Recall that $M_{Y;H,H}(u)=M_{Y;H}(u)$.
We denote the open subscheme of $M_Y(u)$ of stable sheaves on $Y$ by $M_Y^s(u)$ and the smooth locus of $M_Y^s(u)$ by $M_Y^{sm}(u)$.
$M^s_X(f^*u)$ is nonsingular, each connected component has dimension $2-\chi(f^*u,f^*u)$ and it carries a symplectic form due to Mukai \cite{Muk84}.
The morphism $f^*\colon M_Y(u)\to M_X(\deg f\cdot P)$ induced by the pullback by $f$ which is described in Theorem \ref{MHApb} has image inside $M_X(f^*u)$.
Thus we can replace $M_X(\deg f\cdot P)$ by $M_X(f^*u)$ in the results of Section \ref{arbdim} and consider $f^*$ as a morphism $M_Y(u)\to M_X(f^*u)$.
\begin{proposition}\label{pbsymvan}
The pullback of the symplectic form on $M^s_X(f^*u)$ by the restricted morphism $M_Y^{sm}(u)\cap (f^*)^{-1}(M^s_X(f^*u))\stackrel {f^*}\to M^s_X(f^*u)$ vanishes.
\end{proposition}
\begin{proof}
This is proven in the proof of \cite[item 3 of the main theorem in \S 3]{Kim98a}, for $X$ a K3 surface and $\mu$-stable sheaves. His proof works as well for Gieseker stability and $(H,A)$-stability, and for $X$ an abelian surface.
\end{proof}
\begin{corollary}\label{ressymvan}
The restriction of the symplectic form on $M^s_X(f^*u)$ to $f^*(M_Y^{sm}(u))\cap M^s_X(f^*u)$ vanishes.
\end{corollary}
\noindent This leads to the question whether the subvariety $f^*(M_Y^{sm}(u))\cap M^s_X(f^*u)$ is a Lagrangian subvariety of $M^s_X(f^*u)$. Hence we need to calculate dimensions.
\begin{proposition}\label{CCfEst}
Let $E$ be a coherent sheaf on $Y$ with $u(E)=u$ such that $f^*E$ is stable. Then
$E$ is stable,
$E\not\cong E\otimes K_Y^j$ for $1\le j<n$ and
$\ext^2(E,E)=0$.
$M_Y(u)$ is nonsingular in $[E]$ of expected dimension
\begin{eqnarray*}
\dim_{E} M_Y(u)&=&\ext^1(E,E)\,,\quad\textrm{and}\\
\dim_{f^*E} M_X(f^*u)&=&\ext^1(f^*E,f^*E)=2-n+n \dim_{E} M_Y(u)\,.
\end{eqnarray*}
In particular, one has that
$\ext^1(E,E)=0$ if and only if $\ext^1(f^*E,f^*E)=0$, and that
${\dim_{f^*E} M_X(f^*u)=2\dim_{E} M_Y(u)}$ if and only if $n=2$ or $\ext^1(E,E)=1$.
\end{proposition}
\begin{proof}
$E$ is stable by Proposition \ref{PbStab}. In particular, $E$ is simple and by Lemma \ref{CycShDec4} one has $E\not\cong E\otimes K_Y^j$ for $1\le j<n$.
$f^*E$ is simple as well, thus by the following Lemma \ref{CycCanDim}
\begin{eqnarray}
\ext^1(f^*E,f^*E)=2-n+n\; \ext^1(E,E) \label{ext1fEfE}
\end{eqnarray}
and $\ext^2(E,E)=0$. Therefore $M_Y(u)$ is nonsingular in $[E]$ of expected dimension $\ext^1(E,E)$.
As $X$ is K3 or abelian, $M^s_X(f^*u)$ is nonsingular of expected dimension $2-\chi(f^*u,f^*u)=\ext^1(f^*E,f^*E)$ as stated already above.
Equation (\ref{ext1fEfE}) yields the two equivalences, as $\ext^1$ is always nonnegative and $n\ge 2$.
\end{proof}
\noindent In the proof we used the following
\begin{lemma}\label{CycCanDim}
Let $E$ be a coherent sheaf on $Y$. Then
$$\ext^1(f^*E,f^*E)=2\hom(f^*E,f^*E)-n\left( \hom(E,E)+\hom(E,E\otimes K_Y)\right)+n\; \ext^1(E,E)\,.$$
If $f^*E$ is simple, then
$\ext^1(f^*E,f^*E)=2-n+n\; \ext^1(E,E)\,,$
and
$\ext^2(E,E)=0$.
\end{lemma}
\begin{proof}
We first prove that $\chi(f^*E,f^*F)=\deg f\; \chi(E,F)$ for two coherent sheaves $E$ and $F$, which holds also if $f$ is not cyclic and $Y$ has higher dimension.
If $E$ is locally free, then $f^*E$ is locally free as well, and one has that
\begin{eqnarray*}
\chi(f^*E,f^*F)&=&\chi( \cH om(f^*E,f^*F) )\\
&=&\chi( f^*\cH om(E,F) )\\
&=&\deg f\; \chi( \cH om(E,F) )=\deg f\; \chi(E,F)\,,
\end{eqnarray*}
where we used the canonical isomorphism
$f^*\cH om(E,F) \cong \cH om(f^*E,f^*F)$
\cite{EGA1} (6.7.6) and \cite[\S 12 Theorem 2]{Mum70}.
If $E$ is not locally free, consider a locally free resolution $E_\bullet\twoheadrightarrow E$, which gives a locally free resolution $f^*E_\bullet\twoheadrightarrow f^*E$, and use the additivity of $\chi$.
Thus the above claim holds. As $Y$ is a surface, by Serre duality one has
\begin{eqnarray*}
\ext^1(f^*E,f^*E)&=&\hom(f^*E,f^*E)+\hom(f^*E,f^*E\otimes K_X)\\
&&-n\left( \hom(E,E)+\hom(E,E\otimes K_Y)\right)+n\; \ext^1(E,E)\\
&=&2\hom(f^*E,f^*E)-n\left( \hom(E,E)+\hom(E,E\otimes K_Y)\right)+n\; \ext^1(E,E)\;,
\end{eqnarray*}
whilst the last equation holds due to the fact that the canonical bundle $K_X$ is trivial.\\
If $f^*E$ is simple, then Lemma \ref{CycShDec4} yields that $E$ is simple and $\hom(E,E\otimes K_Y)=0$.
Hence
$$\ext^1(f^*E,f^*E)=2-n+n\; \ext^1(E,E)\,,$$
and by Serre duality one has $\ext^2(E,E)=0$.
\end{proof}
\noindent The two main results of this section now are contained in the following Theorem \ref{CC} and in Theorem \ref{CCsst}.
\begin{theorem}\label{CC}
Let $Y$ be an Enriques or bielliptic surface, $f\colon X\to Y$ the covering given by the torsion canonical bundle $K_Y$ of order $n$ on $Y$,
$\nu$ the order of $\bc_1(K_Y)$,
$u=(r,c,\chi)\in \Hev$, $M_Y(u)$ the moduli space of Gieseker or $(H,A)$-semistable sheaves $E$ with $u(E)=u$ and $M_Y^s(u)$ the open subscheme of $M_Y(u)$ of stable sheaves on $Y$.
Assume that there is a coherent sheaf $E$ on $Y$ with $u(E)=u$ such that $f^*E$ is stable and let
$\Sigma:=\{ [E]\in M^s_Y(u) \;|\; \ensuremath{\mathrm{ord}}_E(K_Y)\neq n\}$\footnote{See Definition \ref{Eorder}.}.
Then
\begin{enumerate}
\item $M^s_Y(u)\setminus\Sigma$ is a nonempty and nonsingular open subset of $M^s_Y(u)$,
\item $f^*(M^s_Y(u)\setminus\Sigma)=f^*(M_Y(u))\cap M^s_X(f^*u)$,
\item $f^*(\Sigma)\cap M^s_X(f^*u)=\emptyset$,
\item $f^*$ induces a $\frac{n\;\gcf(\nu,r)}\nu:1$ covering $M^s_Y(u)\to f^*(M^s_Y(u))$ branched along $\Sigma$, and
\item $\dim M^s_X(f^*u)=2-n+n \dim f^*(M^s_Y(u)\setminus\Sigma)$.
\end{enumerate}
In particular, $f^*(M^s_Y(u)\setminus\Sigma)$ is a Lagrangian subvariety of $M^s_X(f^*u)$ if and only if $n=2$ or $\dim f^*(M^s_Y(u)\setminus\Sigma)=1$.
\end{theorem}
\begin{proof}
A coherent sheaf $E$ on $Y$ with $u(E)=u$ such that $f^*E$ is stable is stable itself by Proposition \ref{PbStab}. In particular, $M^s_Y(u)\neq\emptyset$.
Let $m=\frac \nu{\gcf(\nu,r)}$ as at the beginning of this section, where $\nu$ is still the order of $\bc_1(K_Y)$. Recall that we are in the situation of Example \ref{Mexa}.3 with $M_Y=M_Y(u)$.
Item 2 of Theorem \ref{CycStab} holds by assumption, hence its other items hold as well, where we replace $M_X(nP)$ by $M_X(f^*u)$ as mentioned above:
\begin{itemize}
\item The $M_Y^s$-order of $K_Y$ (Definition \ref{Morder}) is $n$,
\item $f^*(M_Y^s\setminus\Sigma)=f^*(M_Y)\cap M^s_X(f^*u)\neq \emptyset$, and
\item $M_Y^s\setminus\Sigma=(f^*)^{-1}(M^s_X(f^*u))\neq \emptyset$. In particular, $f^*(\Sigma)\cap M^s_X(f^*u)=\emptyset$.
\end{itemize}
Therefore items 1-3 of our claim hold.
As all sheaves with class in $f^*(M^s\setminus\Sigma)$ are stable, Proposition \ref{CCfEst} yields that $M^s\setminus\Sigma$ is nonsingular and $\dim M^s_X(f^*u)=2-n+n \dim (M^s\setminus\Sigma)$.
By Proposition \ref{pbFact} and Proposition \ref{Mdecomp} item 7 with $j=n$ the morphism $f^*$ induces a $\frac{n\;\gcf(\nu,r)}\nu:1$ covering $M^s\to f^*(M^s)$ branched along $\Sigma$.
In particular, $\dim (M^s\setminus\Sigma)=\dim f^*(M^s\setminus\Sigma)$.
By Corollary \ref{ressymvan} the restriction of the symplectic form on $M^s_X(f^*u)$ to $f^*(M^s\setminus\Sigma)$ vanishes, thus
$f^*(M^s\setminus\Sigma)$ is a Lagrangian subvariety if and only if $\dim M^s_X(f^*u)=2\dim f^*(M^s\setminus\Sigma)$.
This is equivalent to $n=2$ or $\dim f^*(M^s_Y(u)\setminus\Sigma)=1$ due to item 5.
\end{proof}
If $f^*(M^s_Y(u))\subseteq M^s_X(f^*u)$ --- e.g.\ if $\gcf(n,r)=1$ (use Theorem \ref{gcdnr1}) or if $f^*u$ is primitive and $f^*H$ (or $f^*A$, respectively) is general (see the appendix) --- and if $M^s_Y(u)$ is nonempty, then the theorem holds with $\Sigma=\emptyset$.
In particular, the covering $M^s_Y(u)\stackrel {f^*} \to f^*(M^s_Y(u))$ is unbranched.
If $f^*(M^s_Y(u))\cap M^s_X(f^*u)=\emptyset$, i.e.\ if any stable sheaf becomes strictly semistable after pullback, then this construction does not seem to yield any Lagrangian subvariety, as $f^*(M^s_Y(u))$ is outside the locus where the symplectic form is defined.
Theorem \ref{CC} is concerned with the case that there is a coherent sheaf $E$ on $Y$ with $u(E)=u$ such that $f^*E$ is stable.
This means that in particular, $E$ is already stable by Proposition \ref{PbStab}.
We now want to consider the opposite case, i.e.\ $E$ is stable but for all such $E$ the pullback $f^*E$ is not stable.
\begin{theorem}\label{CCsst}
Let $Y$ be an Enriques or bielliptic surface, $f\colon X\to Y$ the covering given by the torsion canonical bundle $K_Y$ of order $n$ on $Y$,
$\nu$ the order of $\bc_1(K_Y)$,
$u=(r,c,\chi)\in \Hev$, $M_Y(u)$ the moduli space of Gieseker or $(H,A)$-semistable sheaves $E$ with $u(E)=u$ and $M_Y^s(u)$ the open subscheme of $M_Y(u)$ of stable sheaves on $Y$.
Assume that $M^s_Y(u)\neq\emptyset$ and $f^*(M^s_Y(u))\cap M^s_X(f^*u)=\emptyset$.
Moreover, let $\ell$ be the $M^s_Y(u)$-order of $L$ defined in Definition \ref{Morder}, and $m:=\frac \nu{\gcf(\nu,r)}$ as in Example \ref{Mexa}.3.
Then
\begin{enumerate}
\item $m<n$;
\item for all $E\in M^s_Y(u)$ one has that $E\cong E\otimes K_Y^{\ell}$;
\item if $n$ is a prime power then $\ell\neq n$ and $f^*$ induces an $\frac {\ell}{m}:1$ covering $M^s_Y(u)\to f^*(M^s_Y(u))$ branched along the closed subscheme $\Sigma:= \{ [E] \in M^s_Y(u) \;|\; \ensuremath{\mathrm{ord}}_E(L)\neq\ell\}$.
\end{enumerate}
\end{theorem}
\begin{proof}
\begin{enumerate}
\item By Lemma \ref{PreimDiv4} $m|\ensuremath{\mathrm{ord}}_E(L)|n$ for all $[E]\in M^s_Y(u)$. Assume that $\max \{ \ensuremath{\mathrm{ord}}_{E}(L) \;|\; [E]\in M^s_Y(u)\}=n$. Then $\ell=n$, and item 3 of Theorem \ref{CycStab} contradicts our assumption. Hence $\ensuremath{\mathrm{ord}}_{E}(L)<n$ for all $[E]\in M^s_Y(u)$, and, in particular, $m<n$.
\item This follows from the definition of the $M^s_Y(u)$-order of $L$.
\item Let $n$ be a prime power. By Lemma \ref{orderDiv} $\ell|n$, thus $\ell$ is a prime power as well, and therefore there is an $[E]\in M_Y^s(u)$ such that $\ell=\ensuremath{\mathrm{ord}}_{E}(L)$. Assume that $\ell=n$. Then, as in item 1, item 3 of Theorem \ref{CycStab} contradicts our assumption. Hence $\ell\neq n$. The second statement is Proposition \ref{pbFact} together with Corollary \ref{ppQuCov}.\qedhere
\end{enumerate}
\end{proof}
\noindent If $n$ is not a prime power, which means in our case that $n=6$, it is not ensured that $\Sigma$ is a proper subset.
\subsection{Lagrangian subvarieties in irreducible symplectic manifolds}\label{LIHS}
We keep all assumptions.
For $n=\nu=2$, in particular, for a K3 surface covering an Enriques surface, Theorem \ref{CC} immediately simplifies to
\begin{proposition}\label{CanDouSt}
Let $Y$ be an Enriques or bielliptic surface with torsion canonical bundle $K_Y$ of order $2$ on $Y$, $f\colon X\to Y$ the covering given by $K_Y$,
$u=(r,c,\chi)\in \Hev$, $M_Y(u)$ the moduli space of Gieseker or $(H,A)$-semistable sheaves $E$ with $u(E)=u$ and $M_Y^s(u)$ the open subscheme of $M_Y(u)$ of
stable sheaves on $Y$.
Assume that the order of $\bc_1(K_Y)$ is 2 as well and that there is a coherent sheaf $E$ on $Y$ with $u(E)=u$ such that $f^*E$ is stable.
\begin{enumerate}
\item If the rank $r$ is odd, then one has the following:
\begin{enumerate}
\item $M^s_Y(u)$ is a nonempty and nonsingular,
\item $f^*(M^s_Y(u))=f^*(M_Y(u))\cap M^s_X(f^*u)$,
\item $f^*$ induces an isomorphism $M^s_Y(u)\to f^*(M^s_Y(u))$,
\item $\dim M^s_X(f^*u)=2 \dim f^*(M^s_Y(u))$, and
\item $f^*(M^s_Y(u))$ is a Lagrangian subvariety of $M^s_X(f^*u)$.
\end{enumerate}
\item If the rank $r$ is even and $\Sigma:=\{ E\in M^s_Y(u) \;|\; E\otimes K_Y\cong E \}$, then
\begin{enumerate}
\item $M^s_Y(u)\setminus\Sigma$ is a nonempty and nonsingular open subset of $M^s_Y(u)$,
\item $f^*(M^s_Y(u)\setminus\Sigma)=f^*(M_Y(u))\cap M^s_X(f^*u)$ and $f^*(\Sigma)\cap M^s_X(f^*u)=\emptyset$,
\item $f^*$ induces a $2:1$ covering $M^s_Y(u)\to f^*(M^s_Y(u))$ branched along $\Sigma$,
\item $\dim M^s_X(f^*u)=2 \dim f^*(M^s_Y(u)\setminus\Sigma)$, and
\item $f^*(M^s_Y(u)\setminus\Sigma)$ is a Lagrangian subvariety of $M^s_X(f^*u)$.
\end{enumerate}
\end{enumerate}
\end{proposition}
\noindent If $f^*u$ is primitive and $f^*H$ (or $f^*A$, respectively) is $f^*u$-general, then $M_X(f^*u)=M^s_X(f^*u)$ and $\Sigma=\emptyset$ also for even rank.
Whenever $M_Y(u)$ is nonempty, it yields the Lagrangian subvariety $f^*(M_Y(u))$ in the symplectic manifold $M^s_X(f^*u)$. Note that the latter is an irreducible (holomorphically) symplectic manifold if $X$ is a K3 surface and, when considering $(H,A)$-stability, if a numerical assumption on the invariants holds, see \cite[Corollary 6.7]{Zow12}.
On the other hand, in the case $n=\nu=2$, Theorem \ref{CCsst} simplifies to
\begin{proposition}\label{CanDouSSSt}
Let $Y$ be an Enriques or bielliptic surface with torsion canonical bundle $K_Y$ of order $2$ on $Y$, $f\colon X\to Y$ the covering given by $K_Y$,
$u=(r,c,\chi)\in \Hev$, $M_Y(u)$ the moduli space of Gieseker or $(H,A)$-semistable sheaves $E$ with $u(E)=u$ and $M_Y^s(u)$ the open subscheme of $M_Y(u)$ of
stable sheaves on $Y$.
Assume that the order of $\bc_1(K_Y)$ is 2 as well, that $M^s_Y(u)\neq\emptyset$ and that $f^*(M^s_Y(u))\cap M^s_X(f^*u)=\emptyset$.
Then the rank $r$ is even, for all $E\in M^s_Y(u)$ one has that $E\cong E\otimes K_Y$, and $f^*|_{M^s_Y(u)}$ is injective.
\end{proposition}
Similar results have been proven by Kim in \cite{Kim98a} for $\mu$-stable sheaves of positive rank on Enriques surfaces.\\
As a first question one might ask when the assumption $M^s_Y(u)\neq\emptyset$ is satisfied. Recall that $M^s_Y(u)\neq\emptyset$ is a necessary condition also for Proposition \ref{CanDouSt} due to Proposition \ref{PbStab}.
A general case of odd rank sheaves has been considered in \cite{Yos03}. \cite[Theorem 4.6]{Yos03} implies that
if $Y$ is an unnodal Enriques surface, i.e.\ there is no $-2$-curve, $u$ is primitive with $\chi(u,u)\ge -1$ and $H$ is $u$-general, then $M^s_{Y;H}(u)$ is nonempty and irreducible.
The assumption also implies that $M_{Y;H}(u)=M^s_{Y;H}(u)$, i.e.\ there are no strictly semistable sheaves.\\
If $X$ is a K3 surface, then $\Pic^0(X)=0$, and $M_X(f^*u)$ is an irreducible symplectic manifold if $f^*u$ is primitive and $f^*H$ is $f^*u$-general (or $f^*A$ is $f^*u$-general, respectively, and additionally the above-mentioned numerical condition holds).
If $X$ is an abelian surface, the situation is different: in order to produce higher dimensional irreducible symplectic manifolds one has to get rid of superfluous factors in the Bogomolov decomposition by taking a fibre of the Albanese map. Hence we will now fix the determinant of the considered sheaves and additionally reduce the moduli space to the kernel of a suitable summation map.
Let $Y$ be a bielliptic surface and $f\colon X\to Y$ the canonical covering. In particular, $X$ is an abelian surface and $f$ is cyclic of order $n=2,3,4$ or $6$.
Let still $H$ and $A$ be two ample divisors on $Y$, and we continue considering Gieseker stability and $(H,A)$-stability in the following.
We associate the element $$w(E):=(\rk E,\det(E),\chi(E))\in \Hpev:=\IN_0\oplus\Pic(Y)\oplus\IZ$$ of sheaf invariants to the sheaf $E$.
We fix an element $w:=(r,d,\chi)\in \Hpev$ and define $$P(w):=r \frac{H^2}2n^2+\bc_1(d).Hn+\chi\,.$$
If $E$ satisfies $w(E)=w$, then its Hilbert polynomial is $P(w)$.
Let $M_Y(w)$ be the moduli space of Gieseker or $(H,A)$-semistable sheaves $E$ with $w(E)=w$. It is projective and a subscheme of $M_Y(P(w))$, hence the results of Section \ref{arbdim} apply again. Moreover, by setting $m:=\frac n{\gcf(n,r)}$ we are in the situation of Example \ref{Mexa}.2 with $M_Y=M_Y(w)$.
In case we need to distinguish between Gieseker stability and $(H,A)$-stability, we will use $M_{Y;H}(w)$ and $M_{Y;H,A}(w)$, respectively.
Recall again that $M_{Y;H,H}(w)=M_{Y;H}(w)$.
We denote the open subscheme of $M_Y(w)$ of stable sheaves on $Y$ by $M_Y^s(w)$.
$M^s_X(f^*w)$ is nonsingular, each connected component has dimension $-\chi(f^*w,f^*w)$, where, of course, $\chi((r,d,a),(r,d,a)):=\chi((r,\bc_1(d),a),(r,\bc_1(d),a))$ for $(r,d,a)\in\Hpev$.
The morphism $f^*\colon M_Y(w)\to M_X(\deg f\cdot P(w))$ induced by the pullback by $f$ which is described in Theorem \ref{MHApb} has image inside $M_X(f^*w)$.
Thus we can analogously replace $M_X(\deg f\cdot P(w))$ by $M_X(f^*w)$ in the results of Section \ref{arbdim} and consider $f^*$ as a morphism $M_Y(w)\to M_X(f^*w)$.
\begin{proposition}\label{CCfEst0}
Let $E$ be a coherent sheaf on $Y$ with $w(E)=w$ such that $f^*E$ is stable.
Then $E$ is stable, $E\not\cong E\otimes K_Y^j$ for $1\le j<n$ and $\ext^2(E,E)_0=0$.
$M_Y(w)$ is nonsingular in $[E]$ of expected dimension
\begin{eqnarray*}
\dim_{E} M_Y(w)&=&\ext^1(E,E)_0=\ext^1(E,E)-1\,,\quad\textrm{and}\\
\dim_{f^*E} M_X(f^*w)&=&\ext^1(f^*E,f^*E)_0=\ext^1(f^*E,f^*E)-2=n \dim_{E} M_Y(w)\,.
\end{eqnarray*}
\end{proposition}
\begin{proof}
By Proposition \ref{CCfEst}, $E$ is stable, $E\not\cong E\otimes K_Y^j$ for $1\le j<n$, $\ext^2(E,E)=0$ and
\begin{eqnarray}
\ext^1(f^*E,f^*E)&=&2-n+n \ext^1(E,E)\,. \label{CCfEst0_1}
\end{eqnarray}
Hence, in particular, $\ext^2(E,E)_0=0$.
By \cite[Theorem 4.5.4]{HL10} and its immediate generalisation to $(H,A)$-stability, $M_Y(w)$ is nonsingular in $[E]$ and of expected dimension $\ext^1(E,E)_0$,
and $M_X(f^*w)$ is nonsingular in $[f^*E]$ and of expected dimension $\ext^1(f^*E,f^*E)_0$.
By the formula after \cite[Corollary 4.5.5]{HL10} one has that
\begin{eqnarray*}
\ext^1(E,E)_0&=&\chi(O_Y)-\chi(E,E)=0-1+\ext^1(E,E)-0 \quad\mathrm{and}\quad\\
\ext^1(f^*E,f^*E)_0&=&\chi(O_X)-\chi(f^*E,f^*E)=0-1+\ext^1(f^*E,f^*E)-1\\
&\stackrel{(\ref{CCfEst0_1})}=&n (\ext^1(E,E)-1)= n\ext^1(E,E)_0\,.
\end{eqnarray*}
\[\qedhere\]
\end{proof}
\noindent We can now adjust Theorems \ref{CC} and \ref{CCsst} to the modified situation:
\begin{theorem}\label{CC0}
Assume that there is a coherent sheaf $E$ on $Y$ with $w(E)=w$ such that $f^*E$ is stable and let $\Sigma:=\{ [E]\in M^s_Y(w) \;|\; \ensuremath{\mathrm{ord}}_E(K_Y)\neq n\}$.
Then
\begin{enumerate}
\item $M^s_Y(w)\setminus\Sigma$ is a nonempty and nonsingular open subset of $M^s_Y(w)$,
\item $f^*(M^s_Y(w)\setminus\Sigma)=f^*(M_Y(w))\cap M^s_X(f^*w)$,
\item $f^*(\Sigma)\cap M^s_X(f^*w)=\emptyset$,
\item $f^*$ induces a $\gcf(n,r):1$ covering $M^s_Y(w)\to f^*(M^s_Y(w))$ branched along $\Sigma$, and
\item $\dim M^s_X(f^*w)=n \dim f^*(M^s_Y(u)\setminus\Sigma)$.
\end{enumerate}
\end{theorem}
\begin{proof}
The proof goes analogous to Theorem \ref{CC} using Proposition \ref{CCfEst0} instead of Proposition \ref{CCfEst}.
\end{proof}
\noindent Analogously to Theorem \ref{CCsst} one proves
\begin{theorem}
Assume that $M^s_Y(w)\neq\emptyset$ and $f^*(M^s_Y(w))\cap M^s_X(f^*w)=\emptyset$. Moreover, let $\ell$ be the $M^s_Y(w)$-order of $L$, and $m:=\frac n{\gcf(n,r)}$ as in Example \ref{Mexa}.2. Then
\begin{enumerate}
\item $m<n$ and $\gcf(n,r)\neq 1$,
\item For all $E\in M^s_Y(w)$ one has that $E\cong E\otimes K_Y^{\ell}$, and
\item if $n$ is a prime power then $\ell\neq n$ and $f^*$ induces an $\frac {\ell}{m}:1$ covering $M^s_Y(w)\to f^*(M^s_Y(w))$ branched along the closed subscheme $\Sigma:= \{ [E] \in M^s_Y(w) \;|\; \ensuremath{\mathrm{ord}}_E(L)\neq\ell\}$.
\end{enumerate}
\end{theorem}
\noindent Recall that if $n$ is not a prime power, which means in our case that $n=6$, it is not ensured that $\Sigma$ is a proper subset.
We want to reduce further, which needs some preparation.
A short introduction to bielliptic surfaces can be found e.g.\ in \cite[V.5]{BHPV}. We need the following:
Every bielliptic surface $Y$ admits a finite \'etale covering $B\times C\to Y$ factorising via $X$, where $B$ and $C$ are elliptic curves.
$Y\cong (B\times C)/G$, where $G\subset C$ is a finite subgroup acting on $B$ such that $B/G\cong \IP^1$.
One has that $G\cong \IZ/(\IZ/n)\times \IZ/(\IZ/m)$ with $n$ still the order of $K_Y$ and the possibilities $m=1$ for any $n$, $m=2$ if $n=2$ or $4$, and $m=3$ only for $n=3$.
The generator of the group $\IZ/(\IZ/n)$ acts on $B$ by multiplication with $e^{2\pi i/n}$, and the generator of the group $\IZ/(\IZ/m)$ (in the case $m\neq 1$) by translation by some $a\in B$ with certain properties, see e.g.\ \cite[V.5]{BHPV}.
The covering abelian surface is given by $X\cong (B\times C)\Big/\IZ/(\IZ/m)$, and the group structure on $B\times C$ descends to the group structure of the abelian surface $X$.
This group structure induces a summation map $\sum \colon \CH_0(X)\to X$, where $\CH_0(X)$ is the Chow group of $X$.
The second Chern class associates an element $\bc_2^{CH}(E)\in \CH_0(X)$ to any coherent sheaf $E$ on $X$.
Recall that one considers the kernel $K_X(f^*w)$ of the morphism $$A_{f^*w} \colon M_X(f^*w)\to X, [E]\mapsto \sum \bc_2^{CH}(E)$$ in order to get rid of less interesting factors in the Beauville-Bogomolov decomposition of $M_X(f^*w)$.
In particular, $K_{X;f^*H}(f^*w)$ is an irreducible symplectic manifold if $f^*w$ is primitive, $f^*H$ is $f^*w$-general and $-\chi(f^*w,f^*w)\ge 6$ \cite[Theorem 0.2]{Yos01}.
As $$f^*(M_Y(w))\cap K_X(f^*w) = f^*( (f^*)^{-1}(K_X(f^*w)))\,,$$ we are interested in $K_Y(w):=(f^*)^{-1}(K_X(f^*w))$ and the morphism $K_Y(w)\to K_X(f^*w)$.
For a sheaf class $[E]\in M_Y(w)$ one has that $\bc_2^{CH}(f^*E) = f^*\bc_2^{CH}(E)$.
Let us write $\bc_2^{CH}(E)=\sum_ia_i[b_i,c_i]_Y$ with $a_i\in \IZ$, $b_i\in B$ and $c_i\in C$, where $[\bullet]_Y$ denotes the image of $\bullet\in B\times C$ under the quotient morphism $B\times C\to Y$. This enables us to calculate
\begin{eqnarray*}
\sum \bc_2^{CH}(f^*E) &=& \sum f^*\bc_2^{CH}(E)\\
&=& \sum \sum_ia_i f^*[b_i,c_i]_Y\\
&=&\sum_ia_i \sum_{k=0}^{n-1} [\rho^k b_i,c_i+kg]_X\\
&=&\sum_ia_i [0,nc_i+\frac{n(n-1)}2g]_X\,,
\end{eqnarray*}
where $[\bullet]_X$ denotes the image of $\bullet\in B\times C$ under the quotient morphism $B\times C\to X$.
Therefore the image of $A_{f^*w}\circ f^*$ is at most one-dimensional, and
$$\codim_{M_Y(w)} K_Y(w) = \codim_{f^*(M_Y(w))} f^*(K_Y(w)) \le 1\,.$$
As the pullback of the symplectic structure to the smooth locus of $M_Y(w)$ vanishes by Proposition \ref{pbsymvan}, the corresponding restrictions to $K_Y(w)$ and to $f^*(K_Y(w))$ vanish as well.
We are interested in Lagrangian subvarieties of higher dimensional irreducible symplectic manifolds, so we assume now that $n=2$, $-\chi(f^*w,f^*w)\ge 6$, $f^*w$ is primitive and $f^*H$ is $f^*w$-general.
The codimension of $K_{X;f^*H}(f^*w)$ in $M_{X;f^*H}(f^*w)$ is 2, hence intersecting $f^*(M_{Y;H}(w))$ with $K_{X;f^*H}(f^*w)$ has to reduce the dimension at least by 1, i.e.\
$$\codim_{M_{Y;H}(w)} K_{Y;H}(w) = \codim_{f^*(M_{Y;H}(w))} f^*(K_{Y;H}(w)) \ge 1\,.$$
Thus we have proven the following:
\begin{proposition}\label{CanDouStK}
Let $n=2$, i.e.\ the canonical covering of the bielliptic surface $Y$ by the abelian surface $X$ has degree 2.
If $f^*w$ is primitive, $f^*H$ is $f^*w$-general and $-\chi(f^*w,f^*w)\ge 6$, then the image of the morphism $K_{Y;H}(w)\stackrel{f^*}\to K_{X;f^*H}(f^*w)$ is a Lagrangian subvariety.
\end{proposition}
\noindent This result should generalise to $(H,A)$-stability:
\begin{conjecture}
Let $n=2$. If $f^*w$ is primitive, $f^*A$ is $f^*w$-general and $-\chi(f^*w,f^*w)\ge 6$, then $K_{X;f^*H,f^*A}(f^*w)$ is an irreducible symplectic manifold and the image of the morphism $K_{Y;H,A}(w)\stackrel{f^*}\to K_{X;f^*H,f^*A}(f^*w)$ is a Lagrangian subvariety.
\end{conjecture}
\subsection{Examples}\label{example}
As described so far, canonical double coverings produce Lagrangian subvarieties via pullback.
There are basically two different cases to distinguish: odd and even rank.
We expect that the moduli space of sheaves of odd rank on an Enriques or bielliptic surface $Y$ behave like the rank 1 case.
In particular, Yoshioka proved that $M_H(r,0,\frac{r+1}2)\cong \ensuremath{\mathrm{Hilb}}^{\frac{r+1}2}(Y)$ \cite[Corollary 4.4]{Yos03} for odd rank $r$ and general $H$.
As any moduli space of sheaves of rank 1 and of fixed determinant is isomorphic to a Hilbert scheme of points on the underlying surface, we have a look at these Hilbert schemes.
The easiest case is the morphism $f^*\colon Y\to \ensuremath{\mathrm{Hilb}}^2(X)$ induced by pullback by a canonical double covering $f$, which embeds an Enriques or bielliptic surface into the Hilbert scheme of 2 points of a K3 or abelian surface, respectively. More generally, one has the embedding $f^*\colon \ensuremath{\mathrm{Hilb}}^t(Y)\to \ensuremath{\mathrm{Hilb}}^{2t}(X)$ by Proposition \ref{CanDouSt} item 1 (Enriques case) and Theorem \ref{CC0} (bielliptic case). First one might ask what is the geometry of $\ensuremath{\mathrm{Hilb}}^t(Y)$? Oguiso and Schr\"oer proved in \cite{OS11} the following results:
\begin{theorem}[\cite{OS11} 3.1] Let $Y$ be an Enriques surface and $m\ge 2$. Then $\pi_1(\ensuremath{\mathrm{Hilb}}^m(Y))$ is cyclic of order two, and the universal covering of $\ensuremath{\mathrm{Hilb}}^m(Y)$ is a Calabi-Yau manifold.
\end{theorem}
\begin{theorem}[\cite{OS11} 3.5] Let $Y$ be a bielliptic surface and $m\ge 2$. Then there is an \'etale covering $\cH \to\ensuremath{\mathrm{Hilb}}^m(Y)$ so that $\cH$ is the product of an elliptic curve and a Calabi-Yau manifold of dimension $2m-1$.
\end{theorem}
\noindent Thus the subvariety $f^*(\ensuremath{\mathrm{Hilb}}^t(Y))\subset\ensuremath{\mathrm{Hilb}}^{2t}(X)$, which is isomorphic to $\ensuremath{\mathrm{Hilb}}^t(Y)$, has the corresponding covering from the respective theorem above.
We ask the following
\begin{question}\label{OSgen}
How do these results of Oguiso and Schr\"oer generalise to moduli spaces of sheaves?
\end{question}
On the other hand, we expect that the moduli spaces of sheaves of even rank are less close to the Hilbert scheme case.
Quite recently Hauzer established a connection between certain moduli spaces of sheaves of even rank and certain moduli spaces of sheaves of rank 2 or 4 if the Enriques surface is unnodal \cite[Theorem 2.8]{Hau10}. Moreover, he described particular one-dimensional moduli spaces of rank 2 sheaves on Enriques surfaces:
\begin{theorem}[\cite{Hau10} 0.1]
Let $Y$ be an Enriques surface, and $F_1$ and $F_2$ the two multiple fibres of an elliptic fibration of $Y$.
Then there exists an explicit class of polarisations $H$ such that $M_{Y;H}(2,F_1,1)\cong F_2$.
\end{theorem}
\noindent Be careful that in Hauzer's notation, the last entry of the triple $(2,F_1,1)$ in his article is the second Chern class. However, in this case, Riemann-Roch yields $\chi =1= \bc_2$, hence they look the same in our notation.
This choice of invariants yields the morphism $f^*\colon M_{Y;H}(2,F_1,1)\to M_{X;f^*H}(2,f^*F_1,2)$, which maps $F_2$ 2:1 onto a Lagrangian subvariety $L$ in the surface $M_{X;f^*H}(2,f^*F_1,2)$.
As Hauzer explains in the proof of \cite[Lemma 1.1]{Hau10}, $F_1$ is indivisible in $\bH^2(Y,\IZ)_0$, the torsion free part of $\bH^2(Y,\IZ)$.
The induced map $f^*\colon \bH^2(Y,\IZ)_0\to\bH^2(X,\IZ)$ is injective, hence $f^*F_1$ is primitive in $f^*(\bH^2(Y,\IZ))\subset \bH^2(X,\IZ)$.
By \cite[Proposition 2.3]{Nam85} $f^*(\bH^2(Y,\IZ))$ is the $+1$ eigenspace of the covering involution and therefore a primitive sublattice of $\bH^2(X,\IZ)$.
Hence $f^*F_1$ is primitive in $\bH^2(X,\IZ)$ as well.
If we now choose $H$ such that $f^*H$ is general, then $H$ must already be general by Proposition \ref{fHgenHgen}.
This ensures that we only have stable sheaves, i.e.\ $M_{Y;H}(2,F_1,1)=M^s_{Y;H}(2,F_1,1)$ and $S:=M_{X;f^*H}(2,f^*F_1,2)=M^s_{X;f^*H}(2,f^*F_1,2)$. In particular, $S$ is a projective K3 surface.
Let us consider a general Enriques surface in the sense of \cite[Proposition 5.6]{Nam85}, i.e.\ one has that $f^*(\NS(Y))=\NS(X)$. In particular, hyperplanes in $\NS(X)$ have hyperplanes as preimages in $\NS(Y)$. Going through the proof of \cite[Lemma 1.1]{Hau10} one checks that under this assumption Hauzer's choice of $H$ allows to choose $H$ such that $f^*H$ is general as well.
As $M_{X;f^*H}(2,f^*F_1,2)=M_{X;f^*H}^s(2,f^*F_1,2)$, by Proposition \ref{CanDouSt} item 2 the morphism $f^*$ induces an unramified covering $F_2\to L$ of degree 2 and $M_{Y;H}(2,F_1,1)\cong F_2$ is nonsingular elliptic.
Hence the Lagrangian subvariety $L$ is a nonsingular elliptic curve as well by the Hurwitz formula.
\section{Outlook}\label{Outlook}
Although we have quite general results on the pullback morphism between moduli spaces, the application to particular situations is more interesting if one has relevant results at least on one of these moduli spaces.
One classical example of a cyclic covering is the Godeaux surface covered by the Fermat quintic.
However, not very much is known on the moduli space of semistable sheaves if the underlying surface is of general type. Results of Li \cite{Li94} and, slightly generalised by O'Grady \cite{OGr97}, show that in general the moduli space is of general type as well. General means in particular that the second Chern class of the sheaves is very large.
There are some recent results by Mestrano and Simpson \cite{MS11} on the moduli space $M_{Y;\cO_X(1)}(2,\bc_1(\mathcal O_X(-1)),\chi)$ with arbitrary $\chi$ and $Y\subset\IP^3$ a very general quintic surface. The choice of the first Chern class ensures that the four notions of Gieseker/slope (semi)stability coincide.
By very general the authors mean smooth and at least that $\mathrm{Pic}(X) \cong \mathrm{Pic}(\IP^3) = \IZ$, with further genericity conditions where necessary.
Unfortunately the Fermat quintic is not very general in this sense. The article \cite{Schue11} contains plenty of concrete examples of quintic surfaces in $\IP^3$. The Fermat quintic is contained as Example 3 and is shown to have Picard number 37.
\begin{appendix}
\section{General ample divisors}
In this appendix we recall the notion of general ample divisors and state two results concerning generality and pullback.\\
Let the situation be as in Section \ref{Surfaces}. The ample cone of $Y$ carries a chamber structure for a given triple $u=(r,c,\chi)\in\Hev$ of invariants.
The definition depends on $r$. In the case of $r=1$ we agree that the whole ample cone is the only chamber.
For $r>1$, we follow the definition in \cite[Section 4.C]{HL10}.
Let $\Num(Y):=\Pic(Y)/\equiv$, where $\equiv$ denotes numerical equivalence, and $\Delta:=\Delta(u)>0$.
\begin{definition}
Let
$$W(r,\Delta):=\{ \xi^\perp \cap \ensuremath{\mathrm{Amp}}(Y)_{\IQ} \;|\; \xi\in\Num(Y) \quad\mathrm{with}\quad -\frac {r^2}4 \Delta \le \xi^2 < 0 \}\,,$$
whose elements are called $u$-walls.
The connected components of the complement of the union of all $u$-walls are called $u$-chambers.
An ample divisor is called $u$-general if it is not contained in a $u$-wall.
\end{definition}
\noindent The set $W(r,\Delta)$ is locally finite in $\ensuremath{\mathrm{Amp}}(Y)_{\IQ}$ by \cite[Lemma 4.C.2]{HL10}.
For $r=0$, we follow the definition in \cite[Section 1.4]{Yos01}.
\begin{definition}
Let $c\neq 0$ be effective. For every sheaf $E$ with $u(E)=u$ and every subsheaf $F\subseteq E$ we define $L:=\chi(F)\bc_1(E)-\chi(E)\bc_1(F)$, and for $L\ne 0$ we call
$$W_L:=L^\perp \cap\ensuremath{\mathrm{Amp}}(Y)_\IQ$$ the $u$-wall defined by $L$.
The connected components of the complement of the union of all $u$-walls are called $u$-chambers.
An ample divisor is called $u$-general if it is not contained in a $u$-wall.
\end{definition}
If $r=0=\chi$ then the notion of $H$-(semi)stability for a sheaf $E$ with $u(E)=u$ is independent of the choice of $H$ and one cannot introduce the notion of a $u$-general ample divisor in this particular case.
However, we can move away from this case, as tensoring with the ample line bundle $H$ yields the isomorphism
$M_{Y;H}(0,c,\chi)\cong M_{Y;H}(0,c,\chi+c.H)\,.$
Thus one can assume without loss of generality that $\chi\neq 0$ when investigating the moduli spaces of one-dimensional semistable sheaves on a surface.
\begin{lemma}\label{DeltaPb}
$\Delta(f^*u)=\deg f \Delta(u)$.
\end{lemma}
\begin{proof}\ \\
\begin{eqnarray*}
\Delta(f^*u)
&=& (f^*c)^2 - 2r\deg f\;\chi+2r^2\chi(\cO_X)-rf^*c.K_X\\
&=& \deg f\;c^2 - 2r\deg f\;\chi+2r^2\deg f\;\chi(\cO_Y)-rf^*c.f^*K_Y\\
&=& \deg f(c^2 - 2r\chi+2r^2\chi(\cO_Y)-rc.K_Y)= \deg f\;\Delta(u)
\end{eqnarray*}
\[\qedhere\]
\end{proof}
\begin{proposition}\label{fHgenHgen}
If $f^*H$ is $f^*u$-general, then $H$ is $u$-general.
\end{proposition}
\begin{proof}
According to the definition of a general ample divisor, one has to distinguish by the rank $r$.
\begin{enumerate}
\item \emph{Positive rank $r\ge 2$, i.e.\ twodimensional sheaves:}
Let $\xi\in\Num(Y)$ with $-\frac {r^2}4 \Delta \le \xi^2 < 0$.
Then $f^*\xi\in\Num(X)$, and one has $(f^*\xi)^2=\deg f\;\xi^2$.
Hence $$-\frac {r^2}4 \deg f\;\Delta \le \deg f\; \xi^2=(f^*\xi)^2 < 0\,.$$
By Lemma \ref{DeltaPb} one has $\Delta(f^*u)=\deg f \Delta(u)$, so $f^*\xi$ defines the $f^*u$-wall $(f^*\xi)^\perp \cap \ensuremath{\mathrm{Amp}}(X)_{\IQ}$.
As $f^*H$ is $f^*u$-general, one has $0\neq f^*\xi.f^*H=\deg f\;\xi.H$.
Thus $H$ is $u$-general.
\item \emph{Rank $r=0$ with effective first Chern class $c$, i.e.\ onedimensional sheaves:}
Let $E$ be a coherent sheaf with $u(E)=u$ and $F\subseteq E$ such that $L:=\chi(F)\bc_1(E)-\chi(E)\bc_1(F)\ne 0$.
One has
\begin{eqnarray*}
\tilde L
&:=&\chi(f^*F)\bc_1(f^*E)-\chi(f^*E)\bc_1(f^*F)\\
&=&\deg f\;\chi(F)f^*\bc_1(E)-\deg f\;\chi(E)f^*\bc_1(F)=\deg f\;f^*L\,.
\end{eqnarray*}
As $f^*H$ is $f^*u$-general, one has $0\neq \tilde L.f^*H=\deg f\; f^*L.f^*H=(\deg f)^2\;L.H$.
Thus $H$ is $u$-general.
\qedhere
\end{enumerate}
\end{proof}
\begin{lemma}
If $f^*u$ is primitive, then $u$ is primitive.
\end{lemma}
\begin{proof}
Let $u=mu_0$ with $u_0=(r_0,c_0,\chi_0)\in\Hev$ and $m\in\IN$.
Then $$f^*u=(mr_0,f^*(mc_0),\deg f\;m\chi_0)=mf^*u_0\,.$$
As $f^*u$ is primitive, one has $m=1$, i.e.\ $u$ is primitive.
\end{proof}
\end{appendix}
\end{document} | math |
In Dallas County, less than one in three public school graduates will complete college within six years.
Dallas County Promise is a transformational effort between school districts, colleges, universities, workforce, and communities to increase college completion. At the core, the Promise is a scholarship from the Dallas County Community College District Foundation in partnership with matching university scholarships aligned to high-demand jobs.
Dallas County is committed to developing world-class talent that creates equitable outcomes for students, families, and communities.
To learn more about the College Completion initiative, visit the Dallas County Promise website. | english |
اکہِ دۄہہٕ پرُژھ أمِس زنانہِ امیک وجہ | kashmiri |
Every 17th of March a little bit of Irish comes out in all of us as we celebrate St. Patty’s Day, which has come to be known as an international celebration. Huge parades and parties are held all throughout the world, from New York City to Dublin to Cabo Roig, Spain. What started as an Irish religious holiday of feasting on the anniversary of the Saint’s death has morphed over 1,000 years into the celebration we recognize today.
But let’s take a look at the man behind the day.
St. Patrick himself was born in England during the Roman occupation. Kidnapped, he was brought to Ireland as a slave at the age of 16. He escaped slavery but returned to Ireland in 432. Historically, he is credited with bringing Christianity to Ireland. The legends surrounding the man are much more interesting though, and grew and formed for thousands of years after his death, which is believed to have occurred March 17th of 461.
The most well-known of these legends is regarding the shamrock, which is known throughout the world as a symbol of Irish culture and heritage. Knowing the number three was of importance significant to the pagan Celts of Ireland, St. Patrick used the shamrock’s three cloves to explain the concept of the holy trinity in Christianity, of God as the father, the son, and the holy spirit. In legend, St. Patrick is also credited with the creation of the Celtic cross. In Celtic pagan religion, the circle was symbolic of the sun or moon gods. St. Patrick was preaching near a pagan stone with a circle carved into it. He drew a cross through the circle and blessed the stone, thus creating the now distinct Celtic cross, a cross with a circle within it.
Though he is thought to have died in 461, having a feast on the day of St. Patrick’s death didn’t begin until the 9th or 10th century. Surprisingly, the parade tradition didn’t begin in Ireland, but in the United States in 1762. It was then that Irish soldiers serving in the English army paraded through the streets of New York City to celebrate their country’s patron saint. Since that day, parades and Irish patriotism on March 17th, not only in America but throughout the world, has grown exponentially.
Now people from all different backgrounds and heritage celebrate St. Patrick’s Day, which has become more a celebration of Irish culture than a time to remember St. Patrick and his deeds. Though the celebrations are most popular in North America, everywhere from Japan to Spain to Russia celebrates the holiday.
Surprisingly, Ireland was one of the slowest to take part in such large celebrations. Because it began as a religious holidays, pubs were closed on March 17th in Ireland as recently as the 1970s. However, in the past few decades Ireland has begun to use the interest in St. Patrick’s day to drive tourism and showcase Irish culture and heritage to the world. Today, over a million people partake in St. Patrick’s Day festivities in Dublin, which is now a mutli-day celebration including concerts, parades, fireworks, and more.
In Minnesota we have our own parades and events to celebrate St. Patrick’s Day. The largest events are the St. Patrick’s Day parades, which are held in both Minneapolis and St. Paul on March 17th. Irish bars in the Twin cities take the holiday very seriously, hosting live music, often of Irish origins, tent parties, and sometimes even free beer.
Here is a list of what some local bars are doing to celebrate St. Patrick’s Day. So whether you’re of Irish heritage or not, grab a beer and take part in a worldwide celebration of the history and culture of Ireland this March 17th, I know I will! | english |
१५० वियूज आज बैठूंगा तारों की छाँव में बिछा कर पलकें अपनी इंतज़ार मैं
२९५ वियूज मौन सारे हो गए वाचाल मेरे देश के। बेसुरे से हो गए
२२९ वियूज ये ठंड और धुँध , बारिश की हल्की बूंदाबांदी, गरम गरम चाय
१८१ वियूज सब तरफ मुमकिन नहीं की कुछ अच्छा हो, पर एक सोच मन
१३६ वियूज दे दो, दे दो वोट दो दे, कमल है,पंजा है, साईकिल है,
२५४ वियूज सुना आज एक और बेटी आग के हवाले कर दी गई, बाबा
१६८ वियूज जब तक, दिल में पलती, एक आस है, दूर रहकर भी, तू
२३७ वियूज दोस्ती वही है जिसमें आस हो अपने आप से भी ज्यादा विश्वास
२४० वियूज क्यों सब योद्धा मौन हो गए! अस्त्र- शस्त्र निर्वस्त्र हुए क्यों , | hindi |
/*
md_p.h : physical layout of Linux RAID devices
Copyright (C) 1996-98 Ingo Molnar, Gadi Oxman
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, or (at your option)
any later version.
You should have received a copy of the GNU General Public License
(for example /usr/src/linux/COPYING); if not, write to the Free
Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
*/
#ifndef _MD_P_H
#define _MD_P_H
#include <linux/types.h>
/*
* RAID superblock.
*
* The RAID superblock maintains some statistics on each RAID configuration.
* Each real device in the RAID set contains it near the end of the device.
* Some of the ideas are copied from the ext2fs implementation.
*
* We currently use 4096 bytes as follows:
*
* word offset function
*
* 0 - 31 Constant generic RAID device information.
* 32 - 63 Generic state information.
* 64 - 127 Personality specific information.
* 128 - 511 12 32-words descriptors of the disks in the raid set.
* 512 - 911 Reserved.
* 912 - 1023 Disk specific descriptor.
*/
/*
* If x is the real device size in bytes, we return an apparent size of:
*
* y = (x & ~(MD_RESERVED_BYTES - 1)) - MD_RESERVED_BYTES
*
* and place the 4kB superblock at offset y.
*/
#define MD_RESERVED_BYTES (64 * 1024)
#define MD_RESERVED_SECTORS (MD_RESERVED_BYTES / 512)
#define MD_NEW_SIZE_SECTORS(x) ((x & ~(MD_RESERVED_SECTORS - 1)) - MD_RESERVED_SECTORS)
#define MD_SB_BYTES 4096
#define MD_SB_WORDS (MD_SB_BYTES / 4)
#define MD_SB_SECTORS (MD_SB_BYTES / 512)
/*
* The following are counted in 32-bit words
*/
#define MD_SB_GENERIC_OFFSET 0
#define MD_SB_PERSONALITY_OFFSET 64
#define MD_SB_DISKS_OFFSET 128
#define MD_SB_DESCRIPTOR_OFFSET 992
#define MD_SB_GENERIC_CONSTANT_WORDS 32
#define MD_SB_GENERIC_STATE_WORDS 32
#define MD_SB_GENERIC_WORDS (MD_SB_GENERIC_CONSTANT_WORDS + MD_SB_GENERIC_STATE_WORDS)
#define MD_SB_PERSONALITY_WORDS 64
#define MD_SB_DESCRIPTOR_WORDS 32
#define MD_SB_DISKS 27
#define MD_SB_DISKS_WORDS (MD_SB_DISKS*MD_SB_DESCRIPTOR_WORDS)
#define MD_SB_RESERVED_WORDS (1024 - MD_SB_GENERIC_WORDS - MD_SB_PERSONALITY_WORDS - MD_SB_DISKS_WORDS - MD_SB_DESCRIPTOR_WORDS)
#define MD_SB_EQUAL_WORDS (MD_SB_GENERIC_WORDS + MD_SB_PERSONALITY_WORDS + MD_SB_DISKS_WORDS)
/*
* Device "operational" state bits
*/
#define MD_DISK_FAULTY 0 /* disk is faulty / operational */
#define MD_DISK_ACTIVE 1 /* disk is running or spare disk */
#define MD_DISK_SYNC 2 /* disk is in sync with the raid set */
#define MD_DISK_REMOVED 3 /* disk is in sync with the raid set */
#define MD_DISK_WRITEMOSTLY 9 /* disk is "write-mostly" is RAID1 config.
* read requests will only be sent here in
* dire need
*/
typedef struct mdp_device_descriptor_s {
__u32 number; /* 0 Device number in the entire set */
__u32 major; /* 1 Device major number */
__u32 minor; /* 2 Device minor number */
__u32 raid_disk; /* 3 The role of the device in the raid set */
__u32 state; /* 4 Operational state */
__u32 reserved[MD_SB_DESCRIPTOR_WORDS - 5];
} mdp_disk_t;
#define MD_SB_MAGIC 0xa92b4efc
/*
* Superblock state bits
*/
#define MD_SB_CLEAN 0
#define MD_SB_ERRORS 1
#define MD_SB_BITMAP_PRESENT 8 /* bitmap may be present nearby */
/*
* Notes:
* - if an array is being reshaped (restriped) in order to change the
* the number of active devices in the array, 'raid_disks' will be
* the larger of the old and new numbers. 'delta_disks' will
* be the "new - old". So if +ve, raid_disks is the new value, and
* "raid_disks-delta_disks" is the old. If -ve, raid_disks is the
* old value and "raid_disks+delta_disks" is the new (smaller) value.
*/
typedef struct mdp_superblock_s {
/*
* Constant generic information
*/
__u32 md_magic; /* 0 MD identifier */
__u32 major_version; /* 1 major version to which the set conforms */
__u32 minor_version; /* 2 minor version ... */
__u32 patch_version; /* 3 patchlevel version ... */
__u32 gvalid_words; /* 4 Number of used words in this section */
__u32 set_uuid0; /* 5 Raid set identifier */
__u32 ctime; /* 6 Creation time */
__u32 level; /* 7 Raid personality */
__u32 size; /* 8 Apparent size of each individual disk */
__u32 nr_disks; /* 9 total disks in the raid set */
__u32 raid_disks; /* 10 disks in a fully functional raid set */
__u32 md_minor; /* 11 preferred MD minor device number */
__u32 not_persistent; /* 12 does it have a persistent superblock */
__u32 set_uuid1; /* 13 Raid set identifier #2 */
__u32 set_uuid2; /* 14 Raid set identifier #3 */
__u32 set_uuid3; /* 15 Raid set identifier #4 */
__u32 gstate_creserved[MD_SB_GENERIC_CONSTANT_WORDS - 16];
/*
* Generic state information
*/
__u32 utime; /* 0 Superblock update time */
__u32 state; /* 1 State bits (clean, ...) */
__u32 active_disks; /* 2 Number of currently active disks */
__u32 working_disks; /* 3 Number of working disks */
__u32 failed_disks; /* 4 Number of failed disks */
__u32 spare_disks; /* 5 Number of spare disks */
__u32 sb_csum; /* 6 checksum of the whole superblock */
#ifdef __BIG_ENDIAN
__u32 events_hi; /* 7 high-order of superblock update count */
__u32 events_lo; /* 8 low-order of superblock update count */
__u32 cp_events_hi; /* 9 high-order of checkpoint update count */
__u32 cp_events_lo; /* 10 low-order of checkpoint update count */
#else
__u32 events_lo; /* 7 low-order of superblock update count */
__u32 events_hi; /* 8 high-order of superblock update count */
__u32 cp_events_lo; /* 9 low-order of checkpoint update count */
__u32 cp_events_hi; /* 10 high-order of checkpoint update count */
#endif
__u32 recovery_cp; /* 11 recovery checkpoint sector count */
/* There are only valid for minor_version > 90 */
__u64 reshape_position; /* 12,13 next address in array-space for reshape */
__u32 new_level; /* 14 new level we are reshaping to */
__u32 delta_disks; /* 15 change in number of raid_disks */
__u32 new_layout; /* 16 new layout */
__u32 new_chunk; /* 17 new chunk size (bytes) */
__u32 gstate_sreserved[MD_SB_GENERIC_STATE_WORDS - 18];
/*
* Personality information
*/
__u32 layout; /* 0 the array's physical layout */
__u32 chunk_size; /* 1 chunk size in bytes */
__u32 root_pv; /* 2 LV root PV */
__u32 root_block; /* 3 LV root block */
__u32 pstate_reserved[MD_SB_PERSONALITY_WORDS - 4];
/*
* Disks information
*/
mdp_disk_t disks[MD_SB_DISKS];
/*
* Reserved
*/
__u32 reserved[MD_SB_RESERVED_WORDS];
/*
* Active descriptor
*/
mdp_disk_t this_disk;
} mdp_super_t;
static inline __u64 md_event(mdp_super_t *sb) {
__u64 ev = sb->events_hi;
return (ev<<32)| sb->events_lo;
}
#define MD_SUPERBLOCK_1_TIME_SEC_MASK ((1ULL<<40) - 1)
/*
* The version-1 superblock :
* All numeric fields are little-endian.
*
* total size: 256 bytes plus 2 per device.
* 1K allows 384 devices.
*/
struct mdp_superblock_1 {
/* constant array information - 128 bytes */
__le32 magic; /* MD_SB_MAGIC: 0xa92b4efc - little endian */
__le32 major_version; /* 1 */
__le32 feature_map; /* bit 0 set if 'bitmap_offset' is meaningful */
__le32 pad0; /* always set to 0 when writing */
__u8 set_uuid[16]; /* user-space generated. */
char set_name[32]; /* set and interpreted by user-space */
__le64 ctime; /* lo 40 bits are seconds, top 24 are microseconds or 0*/
__le32 level; /* -4 (multipath), -1 (linear), 0,1,4,5 */
__le32 layout; /* only for raid5 and raid10 currently */
__le64 size; /* used size of component devices, in 512byte sectors */
__le32 chunksize; /* in 512byte sectors */
__le32 raid_disks;
__le32 bitmap_offset; /* sectors after start of superblock that bitmap starts
* NOTE: signed, so bitmap can be before superblock
* only meaningful of feature_map[0] is set.
*/
/* These are only valid with feature bit '4' */
__le32 new_level; /* new level we are reshaping to */
__le64 reshape_position; /* next address in array-space for reshape */
__le32 delta_disks; /* change in number of raid_disks */
__le32 new_layout; /* new layout */
__le32 new_chunk; /* new chunk size (512byte sectors) */
__u8 pad1[128-124]; /* set to 0 when written */
/* constant this-device information - 64 bytes */
__le64 data_offset; /* sector start of data, often 0 */
__le64 data_size; /* sectors in this device that can be used for data */
__le64 super_offset; /* sector start of this superblock */
__le64 recovery_offset;/* sectors before this offset (from data_offset) have been recovered */
__le32 dev_number; /* permanent identifier of this device - not role in raid */
__le32 cnt_corrected_read; /* number of read errors that were corrected by re-writing */
__u8 device_uuid[16]; /* user-space setable, ignored by kernel */
__u8 devflags; /* per-device flags. Only one defined...*/
#define WriteMostly1 1 /* mask for writemostly flag in above */
__u8 pad2[64-57]; /* set to 0 when writing */
/* array state information - 64 bytes */
__le64 utime; /* 40 bits second, 24 btes microseconds */
__le64 events; /* incremented when superblock updated */
__le64 resync_offset; /* data before this offset (from data_offset) known to be in sync */
__le32 sb_csum; /* checksum upto devs[max_dev] */
__le32 max_dev; /* size of devs[] array to consider */
__u8 pad3[64-32]; /* set to 0 when writing */
/* device state information. Indexed by dev_number.
* 2 bytes per device
* Note there are no per-device state flags. State information is rolled
* into the 'roles' value. If a device is spare or faulty, then it doesn't
* have a meaningful role.
*/
__le16 dev_roles[0]; /* role in array, or 0xffff for a spare, or 0xfffe for faulty */
};
/* feature_map bits */
#define MD_FEATURE_BITMAP_OFFSET 1
#define MD_FEATURE_RECOVERY_OFFSET 2 /* recovery_offset is present and
* must be honoured
*/
#define MD_FEATURE_RESHAPE_ACTIVE 4
#define MD_FEATURE_ALL (1|2|4)
#endif
| code |
CHICAGO (WLS) -- A Chicago police officer was hurt after a stolen car crashed near 59th and Wolcott.
Officers attempted to curb a stolen vehicle in the 6000-block of South Winchester around 2:30 p.m. when the suspect rammed a squad car and then struck an officer standing on the street, officials said.
The officer suffered an injury to his left leg and a concussion and was taken to Little Company of Mary with non-life threatening injuries.
One offender is in custody, police are searching for a second person. | english |
Travespera 2016 from La Travespera on Vimeo.
In Asturias there are numerous ride events throughout the year. In Cangas de Onis one of the most outstanding for its originality is La Travespera.
Every year since 2013, La Travespera is a prestigious international event welcoming hundreds of participants on the last Saturday of September. It is a High Mountain Route along 230 km. that motor bikers do on Vespas or Lambrettas, riding along unique spots of the Picos de Europa Natural Park, between Asturias, León and Cantabria.
This circular route leaves Cangas de Onís and passes through Soto de Sajambre, Llánares de la Reina, Potes, Panes, Carreña de Cabrales, Corao to end up in Cangas de Onís, where they are received as a modern bikers heroes.
Asturias, is a Natural Paradise in every sense due to its amazing environment of mountains, valleys, rivers, sea, fauna and flora. The 40% of its territory is designated as natural reserve, the highest percentage of any European region. This makes Asturias ideal for all kind of sport activities including the motorbiking.
Cangas de Onis was the first capital of the Kingdom of Asturias and one of the favourite spot for tourism in the region. It is also the origin of some of the most important symbols of the Principality of Asturias. Cangas de Onís is located at the foot of the Picos de Europa. It is the entrance to the “Royal Site of Covadonga” and the main gateway into this mountainous region.
This place is always opened to the countless motorcycle lovers who meet at the “kilometer zero” to start amazing routes which keep the unevenness between each one of its towns, ideal to enjoy the narrow roads, amazing nature and landscapes… what makes it into a Bikers Paradise.
In Cangas de Onís there is a monument dedicated to motor bikers where many people take photographs, many of them climbing on top of thestatue. | english |
\begin{document}
\title{Tur\'an-type results for intersection graphs of boxes}
\begin{abstract}
In this short note, we prove the following analog of the K\H{o}v\'ari-S\'os-Tur\'an theorem for intersection graphs of boxes. If $G$ is the intersection graph of $n$ axis-parallel boxes in $\mathbb{R}^{d}$ such that $G$ contains no copy of $K_{t,t}$, then $G$ has at most $ctn(\log n)^{2d+3}$ edges, where $c=c(d)>0$ only depends on $d$. Our proof is based on exploring connections between boxicity, separation dimension and poset dimension.
Using this approach, we also show that a construction of Basit et al. of $K_{2,2}$-free incidence graphs of points and rectangles in the plane can be used to disprove a conjecture of Alon et al. We show that there exist graphs of separation dimension 4 having superlinear number of edges.
\end{abstract}
\section{Introduction}
The celebrated K\H{o}v\'ari-S\'os-Tur\'an theorem \cite{KST54} states that if $G$ is a graph on $n$ vertices containing no copy of $K_{t,t}$, then $G$ has at most $O(n^{2-1/t})$ edges. In the past few decades, a great amount of research was dedicated to showing that this bound can be significantly improved in certain restricted families of graphs, many of which are geometric in nature. See e.g. \cite{FPSSZ17} for semi-algebraic graphs, \cite{JP20} for graphs of bounded VC-dimension, and \cite{FP08} for intersection graphs of connected sets in the plane. In particular, Fox and Pach \cite{FP08} proved that if $G$ is the intersection graph of $n$ arcwise connected sets in the plane, and $G$ contains no $K_{t,t}$, then $G$ has at most $cn$ edges, where $c=c(t)>0$ depends only on $t$. In this paper, we are interested in the question that in what meaningful ways can this result be extended in higher dimensions. That is, for which families of geometric objects is it true that if their intersection graph $G$ is $K_{t,t}$-free, then $G$ has at most linear, or almost linear number of edges?
It turns out that already in dimension 3, one must put heavy restrictions on the family for this to hold. As a counterexample to many natural candidates, there exists a family of $n$ lines in $\mathbb{R}^{3}$, whose intersection graph is $K_{2,2}$-free and contains $\Omega(n^{4/3})$ edges. To see this, consider a configuration of $n/2$ points and $n/2$ lines on the plane with $\Omega(n^{4/3})$ incidences, which is the most possible number of incidences by the well known Szemer\'edi-Trotter theorem \cite{SzT83}. To get an intersection graph in $\mathbb{R}^{3}$, replace each point with a line parallel to the $z$-axis containing the point, and replace each line $l$ with a line $l'$ such that the projection of $l'$ to the $xy$-plane is $l$, and the lines $l'$ are pairwise disjoint. This family of $n$ lines in $\mathbb{R}^{3}$ contains no $K_{2,2}$, and has $\Omega(n^{4/3})$ intersections.
One natural family of geometric objects for which the above question becomes interesting is the family of axis-parallel boxes. In this case, we are able to prove an almost linear upper bound on the number of edges.
\begin{theorem}\label{thm:main}
Let $d,t$ be positive integers, then there exists $c=c(d)>0$ such that the following holds. If $G$ is the intersection graph of $n$ $d$-dimensional axis-parallel boxes such that $G$ contains no $K_{t,t}$, then $G$ has at most $ctn(\log n)^{2d+3}$ edges.
\end{theorem}
One might conjecture that the almost linear upper bound in Theorem \ref{thm:main} can be replaced with a linear one. This is true in case $d=2$ by the above mentioned result of Fox an Pach \cite{FP08}. However, much to our surprise, a construction of Basit et al. \cite{BCSTT20} implies that this is not true for $d\geq 3$.
\begin{theorem}[Basit et al. \cite{BCSTT20}]\label{thm:tao}
For every $n$, there exists a $K_{2,2}$-free incidence graph of $n$ points and $n$ rectangles in the plane with $\Omega(n\frac{\log n}{\log\log n})$ edges.
\end{theorem}
One can easily turn this construction into a bipartite intersection graph of boxes in $\mathbb{R}^{3}$. Replace each point $p$ with a box $B\times [0,1]$, where $B$ is a very small square containing $p$. Also, replace each rectangle $R$ with a box $R\times [x,x+\frac{1}{2n}]$, where $0\leq x<1$, and the intervals $[x,x+\frac{1}{2n}]$ are pairwise disjoint. The intersection graph of these $2n$ boxes is the same as the incidence graph of points and rectangles. Therefore, we get the following immediate corollary.
\begin{corollary}\label{cor:tao}
For every $n$, there exists a bipartite $K_{2,2}$-free intersection graph of $n$ boxes in $\mathbb{R}^{3}$ with $\Omega(n\frac{\log n}{\log\log n})$ edges.
\end{corollary}
Our Theorem \ref{thm:main} is almost identical to one of the main results in \cite{BCSTT20}, which was done independently from us. However, the proof we present here is significantly shorter, and is based on exploring connections with other problems in graph theory. Due to our approach, we found out that the construction given by Theorem \ref{thm:tao} can be also used to give a counterexample to a conjecture of Alon et al. \cite{ABCMR18} about the number of edges in a graph of bounded separation dimension.
The \emph{separation dimension} of a graph $G$ is the smallest $d$ for which there exists an embedding $\phi:G\rightarrow \mathbb{R}^{d}$ such that if $\{x,y\}$ and $\{x',y'\}$ are disjoint edges of $G$, then the axis-parallel box spanned by $\phi(x)$ and $\phi(y)$ is disjoint from the axis parallel box spanned by $\phi(x')$ and $\phi(y')$. Alon et al. \cite{ABCMR18} conjectured that for every $d$ there exists a constant $c>0$ such that if $G$ is a graph on $n$ vertices with separation dimension $d$, then $G$ has at most $cn$ edges. They proved this in the case $d=2$. Also, Scott and Wood \cite{SW18} confirmed the conjecture for $d=3$, which also implies the bound $O(n (\log n)^{d-3})$ for $d>3$. However, we show that the conjecture no longer holds for $d\geq 6$.
\begin{theorem}\label{thm:construction}
For every $n$, there exists a graph $G$ on $n$ vertices with $\Omega(n\frac{\log n}{\log \log n})$ edges such that the separation dimension of $G$ is at most 4.
\end{theorem}
Note that dimension 4 cannot be lowered by the aforementioned result of Scott and Wood \cite{SW18}.
The connection between separation dimension and $K_{2, 2}$-free intersection graphs of boxes also implies the following almost matching bound to Theorem \ref{thm:tao}, which improves the corresponding result in \cite{BCSTT20}.
\begin{corollary}\label{cor:K22}
If $G$ is the incidence graph of $n$ points and $n$ rectangles in the plane, and $G$ is $K_{2,2}$-free, then $G$ has at most $O(n\log n)$ edges.
\end{corollary}
\section{Boxicity, poset dimension and separation dimension}
In order to prove Theorem \ref{thm:main} and Theorem \ref{thm:construction}, let us introduce some notation. The \emph{boxicity} of a graph $G$, denoted by $\mbox{box}(G)$ is the smallest $d$ such that $G$ can be realized as the intersection graph of $d$-dimensional boxes.
Given a partially ordered set $P$, the dimension (Duschnik-Miller dimension) of $P$, denoted by $\mbox{dim}(P)$ is the smallest $d$ such that there exists an embedding $\phi:P\rightarrow \mathbb{R}^{d}$ satisfying that $x<_P y$ if and only if $\phi(x)_i<\phi(y)_i$ for $i=1,\dots,d$. (Here, $v_i$ is the $i$-th coordinate of $v$.)
The following connection between boxicity and poset dimension was established by Adiga, Bhowmick and Chandran \cite{ABC11}. Given a graph $G$, define the bipartite poset $(P(G),\prec)$ as follows: let the elements of $P(G)$ be $V(G)\times\{0,1\}$, and let $(u,0)\prec(v,1)$ if $u=v$ or $uv\in E(G)$.
\begin{theorem}\label{dim}\cite{ABC11}
$\frac{1}{2}\mbox{box}(G)\leq\mbox{dim}(P(G))\leq 2\mbox{box}(G)+4.$ Also, if $G$ is bipartite, and $P$ is the underlying partial order, then $\mbox{dim}(P)\leq 2\mbox{box}(G)$.
\end{theorem}
The poset $P(G)$ not only estimates the boxicity of $G$ well, it also (almost) retains the property of being $K_{t,t}$-free (when referring to a poset as a graph, we refer to its comparability graph).
\begin{claim}\label{partite}
If $G$ is $K_{t,t}$-free, then $P(G)$ has a $K_{t,t}$-free induced subgraph with at least $e(G)/2$ edges.
\end{claim}
\begin{proof}
Let $(A,B)$ be a partition of $V(G)$ such that at least half of the edges of $G$ have one endpoint in $A$ and $B$. Then the subgraph of $P(G)$ induced on $\{(a,0):a\in A\}\cup \{(b,1):b\in B\}$ is $K_{t,t}$-free. Indeed, a copy of $K_{t,t}$ in this subgraph would correspond to a copy of $K_{t,t}$ in $G$ in which one of the vertex classes is in $A$, and the other is in $B$.
\end{proof}
Given two points $x$ and $y$ in $\mathbb{R}^{d}$, let $b(x,y)$ denote the box spanned by $x$ and $y$. Let $\prec$ denote the partial ordering on $\mathbb{R}^{d}$ defined as $x\prec y$ if $x_i< y_i$ for $i=1,\dots,d$.
\begin{claim}\label{boxes}
Let $V$ be a set of points in $\mathbb{R}^{d}$ and let $P=(V,\prec)$. If $P$ does not contain $K_{t,t}$, then $P$ contains no matching $\{x^1,y^1\},\dots,\{x^t,y^t\}$ of size $t$ such that $\bigcap_{i=1}^{t} b(x^i,y^i)\neq \emptyset.$
\end{claim}
\begin{proof}
Let us assume that there exists such a matching $\{x^1,y^1\},\dots,\{x^t,y^t\}$, and without loss of generality, assume that $x^i\prec y^i$ for $i=1,\dots,t$. Let $z\in \bigcap_{i=1}^{t} b(x^i,y^i)$, then $x^i\prec z\prec y^i$. But then $x^{i}\prec z\prec y^{j}$ for all $1\leq i,j\leq t$, which means that $x^1,\dots,x^t$ and $y^1,\dots,y^t$ span $K_{t,t}$ in $P$.
\end{proof}
Note that this claim also tells us that if the poset $P$ is $K_{2,2}$-free, then its separation dimension is at most $d$, as $V$ is a suitable embedding of the vertices. Let us use this to prove Theorem \ref{thm:construction}. First, we show a somewhat weaker result.
\begin{theorem}
For every $n$, there exists a graph $G$ on $n$ vertices with $\Omega(n\frac{\log n}{\log \log n})$ edges such that the separation dimension of $G$ is at most 6.
\end{theorem}
\begin{proof}[Proof of Theorem \ref{thm:construction}]
Let $G$ be the bipartite intersection graph of $n$ boxes in $\mathbb{R}^3$ such that $G$ contains no copy of $K_{2,2}$, and $|E(G)|=\Omega(n\frac{\log n}{\log \log n})$. Such a graph exists by Corollary \ref{thm:tao}. But then $\mbox{dim}(P)\leq 2\mbox{box}(G)=6$ by Theorem \ref{dim}, where $P$ is the underlying comparability graph of $G$. We are done as $P$ has separation dimension at most 6 as well.
\end{proof}
In order to improve the dimension from 6 to 4, we just note that if $G$ is the incidence graph given by Theorem \ref{thm:tao} instead of the intersection graph of Corollary \ref{cor:tao}, then $P$ has dimension at most 4. The proof of this follows from a similar argument as the one in \cite{ABC11}, but for the reader's convenience, we present a short proof here as well.
\begin{claim}\label{dim4}
Let $G$ be the incidence graph of points and rectangles in the plane such that no rectangle contains another, and $G$ is $K_{2,2}$-free. Let $P$ be the underlying bipartite poset, then $\mbox{dim}(P)\leq 4$.
\end{claim}
\begin{proof}
We show that if $G$ is $K_{2,2}$-free, then we can assume that no rectangle contains the other. Indeed, suppose that $R\subset Q$ for some rectangles $R$ and $Q$. Then $R$ contains at most one point as $G$ is $K_{2,2}$-free. But then we can replace $R$ with a rectangle $R'$ such that $Q$ is very thin and long, $Q$ contains only the point in $R$, and $Q$ has no containment relation with any other rectangle. But then this configuration has the same incidence graph $G$.
Let $P \subset \mathbb R^2$ be a set of points and $R$ be a set of rectangles in $\mathbb R^2$ such that no rectangle contains the other. Denote by $G$ the corresponding incidence graph. Consider the map $\phi: \mathbb R^2 \rightarrow \mathbb R^4$ defined by $(x, y) \mapsto (x, -x, y, -y)$. Given a rectangle $S = \{ (x, y)~|~a \le x\le b,~ c \le y \le d\}$ on the plane, denote by $\phi(S) \in \mathbb R^4$ the point with coordinates $(a, -b, c, -d)$. Note that a point $p$ is contained in a rectangle $S$ if and only if $\phi(S) \prec \phi(p)$. Clearly, any two points are incomparable, and as no rectangle is contained in another, no two rectangles are comparable.
\end{proof}
This finishes the proof of Theorem \ref{thm:construction}. Let us continue with the proof of Theorem \ref{thm:main}.
\begin{theorem}\label{numedges}
Let $d,t$ be positive integers, then there exists $c=c(d)$ such that the following holds. Let $V$ be a set of $n$ points in $\mathbb{R}^{d}$ and let $G$ be a graph on $V$ such that $G$ contains no matching $\{x^1,y^1\},\dots,\{x^t,y^t\}$ of size $t$ satisfying $\bigcap_{i=1}^t b(x^i,y^i)\neq \emptyset$. Then $e(G)\leq ctn(\log n)^{d-1}$.
\end{theorem}
\begin{proof}
The statement follows from a standard divide and conquer argument.
Let us proceed by induction on $d$. First, consider the base case $d=1$. In this case, $b(x,y)$ is an interval. It is easy to show that if the intersection graph of intervals contains no $K_{2t}$, then it is $(2t-2)$-degenerate. But then $e(G)< 2tn$.
Now suppose that $d\geq 2$. Let $f_d(n)$ denote the minimum $m$ such that any graph $G$ with the desired properties has at most $m$ edges. We show that $f_{d}(n)\leq c_{d}tn(\log n)^{d-1}$, where $c_{d}>0$ depends only on $d$.
Let $G$ be a graph with the desired properties. Without loss of generality, we can assume that no two points in $V$ are on the same axis-parallel hyperplane. Let $H$ be a $(d-1)$-dimensional hyperplane perpendicular to the last coordinate axis such that at most half of the points of $V$ are on each side of $H$. Let $A$ and $B$ be the set of points of $V$ on the two sides of $H$. Let $p(x)$ denote the projection of $x$ into $H$, and let $G'$ be the graph on vertex set $p(V)$ in which $p(x)$ and $p(y)$ are joined by an edge if $xy\in E(G)$ and $x\in A$ and $y\in B$. If $x,x'\in A$ and $y,y'\in B$, then $b(x,y)\cap b(x',y')\neq \emptyset$ if and only if $b(p(x),p(y))\cap b(p(x'),p(y'))\neq \emptyset$. Therefore, $G'$ contains no matching of size $t$ such that the boxes spanned by the edges have a nonempty intersection. Hence, we deduce that
$$e(G)=e(G[A])+e(G[B])+e(G[A,B])\leq 2f_d(n/2)+e(G')\leq 2f_d(n/2)+f_{d-1}(n).$$
From this, we get that $f_d(n)=O(f_{d-1}(n)\log n)=O(tn(\log n)^{d-1})$, where the last equality holds by our induction hypothesis, and the constant hidden in the $O(.)$ notation only depends on $d$.
\end{proof}
In case $t=2$, Theorem \ref{numedges} can be improved. In this case, the graph $G$ has separation dimension at most $d$, which implies that $e(G)\leq cn(\log n)^{d-3}$ by the result of Scott and Wood.
After these preparations, everything is set to prove our main theorem.
\begin{proof}[Proof of Theorem \ref{thm:main}]
Let $G$ be the intersection graph of $n$ boxes in $\mathbb{R}^{d}$, and suppose that $G$ is $K_{t,t}$-free. Then by Theorem \ref{dim} and Claim \ref{partite}, there exists a $K_{t,t}$-free poset $P$ with at least $e(G)/2$ edges, whose dimension is at most $2\mbox{box}(G)+4\leq 2d+4$. But then by Claim \ref{boxes} and Theorem \ref{numedges}, we get $e(P)<ctn(\log n)^{2d+3}$, where $c=c(d)>0$ only depends on $d$. This gives $e(G)<2ctn(\log n)^{2d+3}$.
\end{proof}
Finally, let us prove Corollary \ref{cor:K22}.
\begin{proof}[Proof of Corollary \ref{cor:K22}]
Let $G$ be the incidence graph of $n$ points and $n$ rectangles in the plane, and suppose that $G$ is $K_{2,2}$-free. Then by Claim \ref{dim4}, the underlying poset $P$ of $G$ has dimension at most $4$. But as $P$ is $K_{2,2}$-free, $P$ has separation dimension at most 4, so by the result of Scott and Wood \cite{SW18}, $P$ has at most $O(n\log n)$ edges.
\end{proof}
\end{document} | math |
ژھٹھ ٲس تھچہِ مژٕ پکھہ وایاں | kashmiri |
Bees are responsible for 80% of the pollination of nearly three quarters of the plants that produce 90% of the world’s food, and their numbers are steadily declining. Our cardigan clips make a great conversation starter about the importance of saving the bees!
We are dedicated to making a little bit of a difference in our community, as such $6 from the sale of this product (sold at full price to retail customers) will be donated to a local organisation dedicated to bee conservation. This offer expires at Midnight AEST July 28th, 2018. Please note that products from our 'Busy Bee' range that are purchased with a discount coupon will be excluded from this offer.
Each bee on these cardigan clips measures 3 cms wide x 3 cms high and are made from the eco-friendly wood of the alder tree. We hand paint each individual bee before sealing with a jewellers grade resin to give it a wonderful shine! | english |
\begin{document}
\title{Upper Bounds for Non-Congruent Sphere Packings}
\author{Samuel Reid\thanks{University of Calgary, Centre for Computational and Discrete Geometry (Department of Mathematics \& Statistics), Calgary, AB, Canada. $\mathsf{e-mail: smrei@ucalgary.ca}$}}
\maketitle
\begin{abstract}
We prove upper bounds on the average kissing number $k(\mathcal{P})$ and contact number $C(\mathcal{P})$
of an arbitrary finite non-congruent sphere packing $\mathcal{P}$, and prove an upper bound on the packing density
$\delta(\mathcal{P})$ of an arbitrary infinite non-congruent sphere packing $\mathcal{P}$.
\end{abstract}
\textbf{Keywords:} average kissing number, contact number, packing density, non-congruent sphere packing, linear programming bounds, upper bounds. \\
\text{ \;\; } \textbf{MSC 2010 Subject Classifications:} Primary 52C17, Secondary 51F99.
\section{Lexicon}
Let
$$\mathcal{P} = \bigcup_{i=1}^{k} \bigcup_{j=1}^{n_{i}} \left(x_{ij} + r_{i}\mathbb{S}^{2}\right)$$
be an arbitrary non-congruent sphere packing. Then
$$\|x_{ij} - x_{i'j'}\| \geq r_{i} + r_{i'}, \forall 1 \leq i,i' \leq k, j \neq j'$$
is a necessary condition required for the spheres to be non-overlapping. Hence, the vertex set and edge set of the
sphere packing $\mathcal{P}$ are
\begin{align*}
V(\mathcal{P}) &= \{x_{ij} \in \mathbb{R}^{3} \; \big| \; 1 \leq i \leq k, 1 \leq j \leq n_{i}\} \\
E(\mathcal{P}) &= \{(x_{ij},x_{i',j'}) \; \big| \; \|x_{ij}-x_{i'j'}\| = r_{i} + r_{i'}, x_{ij},x_{i',j'} \in V(\mathcal{P})\}
\end{align*}
The the average kissing number of $\mathcal{P}$ is $k(\mathcal{P}) = 2|E(\mathcal{P})|/n$,
where $n = \displaystyle \sum_{i=1}^{k} n_{i}$, and the contact number of $\mathcal{P}$ is $C(\mathcal{P})=|E(\mathcal{P})|$.
\section{Upper Bounds on Average Kissing Numbers and Contact Numbers of Sphere Packings}
Consider, as in Cohn-Zhao \cite{CohnZhao}, a continuous function $g:[-1,1] \rightarrow \mathbb{R}$ which is positive definite on $\mathbb{S}^2$ with
$g(t) \leq 0, \forall t \in [-1,\cos\theta]$ and
$$\overline{g} = \frac{\int_{-1}^{1} g(t)(1-t^2)dt}{\int_{-1}^{1} (1-t^{2})dt}>0,$$
then $$g \in \mathcal{F}_{\theta}(\mathbb{S}^2).$$
Furthermore, let $A(3,\theta)$ be the maximum size of a spherical $\theta$-code and recall that
$$A^{\text{LP}}(3,\theta) = \inf_{g \in \mathcal{F}_{\theta}(\mathbb{S}^{2})} \frac{g(1)}{\overline{g}}$$
is the best upper bound on $A(3,\theta)$ that could be derived using Theorem 3.1 from Cohn-Zhao \cite{CohnZhao} (which appeals
to the Delsarte-Goethals-Seidel \cite{DGS} and Kabatiansky-Levenshtein \cite{KL} linear programming bounds).
Let $\tau_{r_{j}}(r_{i})$ be the maximum number of radius $r_{i}$ spheres which can touch a radius $r_{j}$ sphere;
$\tau_{r_{i}}(r_{j})$ is defined similarly,
namely the maximum number of radius $r_{j}$ spheres which can touch a radius $r_{i}$ sphere.
\begin{Theorem}\label{contacttheorem}
Let $\mathcal{P}$ be a sphere packing with $n_{i}$ spheres of radius $r_{i}$ for $1 \leq i \leq k$. Then,
$$k(\mathcal{P}) < 12 + \frac{\displaystyle \sum_{i \neq j} \min\left\{n_{i}\min\left\{n_{j},\tau_{r_{i}}(r_{j})\right\},n_{j} \min\left\{n_{i},\tau_{r_{j}}(r_{i})\right\}\right\} - 1.85335 \sum_{i=1}^{k}n_{i}^{2/3}}{\displaystyle \sum_{i=1}^{k}n_{i}}$$
where $\tau_{r_{i}}(r_{j}) \leq A^{\text{LP}}\left(3,\arccos\left(1 - \frac{2r_{j}^{2}}{(r_{i}+r_{j})^{2}}\right)\right)$ and
$\tau_{r_{j}}(r_{i}) \leq A^{\text{LP}}\left(3,\arccos\left(1 - \frac{2r_{i}^{2}}{(r_{i}+r_{j})^{2}}\right)\right)$.
\end{Theorem}
\begin{proof}
Decompose the vertex set and edge set of $\mathcal{P}$ as
\begin{align*}
V(\mathcal{P}) &= \bigcup_{i=1}^{k} V_{i}(\mathcal{P}) := \bigcup_{i=1}^{k} \{ x_{ij} \in V(\mathcal{P}) \; | \; 1 \leq j \leq n_{i} \} \\
|V(\mathcal{P})| &= \sum_{i=1}^{k}|V_{i}(\mathcal{P})| = \sum_{i=1}^{k}n_{i} \\
E(\mathcal{P}) &= \bigcup_{i = 1}^{k} E_{ii}(\mathcal{P}) \cup \bigcup_{i \neq i'} E_{ii'}(\mathcal{P}) :=\bigcup_{i=1}^{k}\{(x_{ij},x_{ij'}) \; | \|x_{ij} - x_{ij'}\| = 2r_{i}, x_{ij},x_{ij'} \in V_{i}(\mathcal{P})\} \\
&\cup \bigcup_{i \neq i'} \{(x_{ij},x_{i'j'}) \; \big| \; \|x_{ij} - x_{i'j'} \| = r_{i} + r_{i'}, x_{ij} \in V_{i}(\mathcal{P}),x_{i'j'} \in V_{i'}(\mathcal{P})\} \\
|E(\mathcal{P})| &= \sum_{i=1}^{k}|E_{ii}(\mathcal{P})| + \frac{1}{2}\sum_{i \neq j}|E_{ij}(\mathcal{P})|
\end{align*}
Apply Theorem 1 (i) of Bezdek and the author \cite{BezdekReid} (Theorem 1.1.6 (i) in \cite{Bezdek}) to bound the cardinality
of each edge set and obtain $|E_{ii}|<6n_{i} - 0.926n_{i}^{2/3}, \forall 1 \leq i \leq k$.
By applying a homothetic transformation with a scaling factor of either $1/r_{i}$ or $1/r_{j}$ to $\mathcal{P}$,
it is clear that $\tau_{r_{j}}(r_{i}) = \tau_{1}(r_{i}/r_{j})$ and $\tau_{r_{i}}(r_{j}) = \tau_{1}(r_{j}/r_{i})$.
Hence, each $|E_{ij}(\mathcal{P})| + |E_{ji}(\mathcal{P})|$
counts the number of edges between spheres of radius $r_{i}$ and $r_{j}$, and thus by the above homothetic transformation
of either type, counts the number of edges between spheres of radius $1$ and radius $r_{j}/r_{i}$ or spheres
of radius $r_{i}/r_{j}$ and radius $1$, respectively.
Therefore, by the law of cosines applied to the geometric embedding of
the contact graph of two spheres of radius $r_{j}/r_{i}$ and a sphere of radius $1$, and the
geometric embedding of the contact graph of two spheres of radius $r_{i}/r_{j}$ and a sphere of radius $1$,
we obtain the $\theta$-code size desired in each case:
\begin{align*}
\theta_{i}^{j} &= \arccos\left(1 - \frac{2r_{j}^{2}}{(r_{i} + r_{j})^{2}}\right) \\
\theta_{j}^{i} &= \arccos\left(1 - \frac{2r_{i}^{2}}{(r_{i} + r_{j})^{2}}\right)
\end{align*}
Hence, $\tau_{1}(r_{i}/r_{j}) = A(3,\theta_{j}^{i}) \leq A^{\text{LP}}(3,\theta_{j}^{i})$ and
$\tau_{1}(r_{j}/r_{i})=A(3,\theta_{i}^{j}) \leq A^{\text{LP}}(3,\theta_{i}^{j})$.
From this we observe, from basic restrictions on the number of spheres of varying radii, that
\begin{align*}
|E_{ij}(\mathcal{P})|=|E_{ji}(\mathcal{P}| &< \min\left\{n_{i}\min\left\{n_{j},\tau_{r_{i}}(r_{j})\right\},n_{j} \min\left\{n_{i},\tau_{r_{j}}(r_{i})\right\}\right\} \\
&=\min\left\{n_{i}\min\left\{n_{j},A^{\text{LP}}(3,\theta_{i}^{j})\right\},n_{j} \min\left\{n_{i},A^{\text{LP}}(3,\theta_{j}^{i})\right\}\right\}
\end{align*}
Cumulatively, these observations combined with the definition of the average kissing number $k(\mathcal{P})$ prove the theorem.
\end{proof}
\pagebreak
In practice, it is difficult to compute $A^{\text{LP}}(3,\theta)$ explicitly, but a weaker bound may be provided by nonnegative
linear combinations of Gegenbauer polynomials $C_{k}^{\frac{n}{2}-1}$ as shown by Schoenbergs characterization
of continuous positive definite functions \cite{Schoenberg}. Gegenbauer polynomials, or ultraspherical polynomials, are
a special case of the Jacobi polynomials, or hypergeometric polynomials, and for algorithmic implementations of the
following theorem we can follow \cite{CohnZhao} and set
$$g(t) = \sum_{k=0}^{\infty} c_{k}C_{k}^{\frac{n}{2}-1}(t), \overline{g}=c_{0}$$
For algorithmic implementation, tighter bounds on $A(n,\theta)$ may be found using de Laat-de Oliveira Filho-Vallentin
semidefinite programming bounds \cite{LOV}, or Cohn-Elkies error-correcting codes bounds \cite{CohnElkies}.
Furthermore, Theorem \ref{contacttheorem} can be considered a packing dependent generalization of the
celebrated Kuperberg-Schramm bound on the supremal average
kissing number $k$ of a sphere packing in $\mathbb{R}^3$ \cite{KuperbergSchramm}, which says that
$12.566 < k := \sup_{\mathcal{P} \hookrightarrow \mathbb{R}^3} k(P) < 8+4\sqrt{3} \approx 14.928$. Future
research goals include the algorithmic implementation of Theorem \ref{contacttheorem} to compare with the Kuperberg-Schramm bound.
We now state Theorem \ref{contacttheorem} in terms of contact numbers which follows directly from the definition
of $k(\mathcal{P})$, thus generalizing Theorem 1 (i) of Bezdek and the author \cite{BezdekReid} (Theorem 1.1.6 (i) in \cite{Bezdek}),
which states that if $\mathcal{P}$ is a packing of $n$ congruent spheres then $C(\mathcal{P}) < 6n - 9.26n^{2/3}$.
\begin{Corollary}
Let $\mathcal{P}$ be a sphere packing with $n_{i}$ spheres of radius $r_{i}$ for $1 \leq i \leq k$. Then,
$$C(\mathcal{P}) < \sum_{i=1}^{k} (6n_{i} - 0.926675n_{i}^{2/3}) + \frac{1}{2}\sum_{i \neq j} \min\left\{n_{i}\min\left\{n_{j},\tau_{r_{i}}(r_{j})\right\},n_{j} \min\left\{n_{i},\tau_{r_{j}}(r_{i})\right\}\right\}$$
where $\tau_{r_{i}}(r_{j}) \leq A^{\text{LP}}\left(3,\arccos\left(1 - \frac{2r_{j}^{2}}{(r_{i}+r_{j})^{2}}\right)\right)$ and
$\tau_{r_{j}}(r_{i}) \leq A^{\text{LP}}\left(3,\arccos\left(1 - \frac{2r_{i}^{2}}{(r_{i}+r_{j})^{2}}\right)\right)$.
\end{Corollary}
\section{Upper Bounds on Infinite Sphere Packings' Densities}
We define the locally maximal tetrahedron $\Delta(r_{i},r_{j},r_{k},r_{l})$ to be the convex hull
of the center points of spheres of radius $r_{i},r_{j},r_{k},r_{l}$, which are maximally contracted;
i.e., there does not exist a non-trivial contractive mapping of the spheres. We can use the geometric
structure of the locally maximal tetrahedron to calculate an upper bound on the density of an infinite sphere
packing of distinct radii $r_{i}, i \in S \subseteq \mathbb{N}$, by defining
$\Delta(r_{i},r_{j},r_{k},r_{l}) = \text{conv}\{\vec{\omega_{i}},\vec{\omega_{j}},\vec{\omega_{k}},\vec{\omega_{l}}\}$
and intersecting spheres of the associated radii at each vertex of the locally maximal tetrahedron.
By connecting spherical geometry and dihedral angles we arrive at the following theorem which holds
for any sphere packing $\mathcal{P}$ in $\mathbb{R}^3$ whether or not it has finitely
many distinct radii, or infinitely many distinct radii, although the theorem does not have a realizable
algorithmic implementation in the case of infinitely many distinct radii.
\begin{Theorem}
Let $\mathcal{P}$ be a sphere packing in $\mathbb{R}^3$ with distinct radii $r_{i}, i \in S \subseteq \in\mathbb{N}$, and let $\delta_{\text{max}}(\mathcal{P}_{\Delta(r_{i},r_{j},r_{k},r_{l})})$
be the maximal packing density of $\Delta(r_{i},r_{j},r_{k},r_{l})$ in $\mathbb{R}^3$. Then,
$$\delta(\mathcal{P}) < 2 \max_{r_{i}\leq r_{j} \leq r_{k} \leq r_{l}} \delta_{\text{max}}(\mathcal{P}_{\Delta(r_{i},r_{j},r_{k},r_{l})}) \left(\left[\sum_{m=i,j,k,l} r_{m}^{3}(A_{m} + B_{m} + C_{m} - \pi)\right] \bigg/ \|\vec{\omega_{2}} \cdot (\vec{\omega_{3}} \times \vec{\omega_{4}})\|\right),$$
where $\Delta(r_{i},r_{j},r_{k},r_{l}) = \text{conv}\{\vec{\omega_{i}},\vec{\omega_{j}},\vec{\omega_{k}},\vec{\omega_{l}}\}$ and
\begin{align*}
U_{ijk} &= \vec{\omega_{j}} \times \vec{\omega_{k}} \\
U_{ikl} &= \vec{\omega_{k}} \times \vec{\omega_{l}} \\
U_{ijl} &= \vec{\omega_{j}} \times \vec{\omega_{l}} \\
U_{jkl} &= (\vec{\omega_{k}} - \vec{\omega_{j}}) \times (\vec{\omega_{l}} - \vec{\omega_{j}})
\end{align*}
\begin{align*}
A_{i}&=A_{l} = \arccos\left(\frac{U_{ijk} \cdot U_{ikl}}{\|U_{ijk}\| \|U_{ikl}\|}\right) \;\;\;\;\;\;\;\;\;\; A_{j}=A_{k} = \arccos\left(\frac{U_{ijk} \cdot U_{jkl}}{\|U_{ijk}\| \|U_{jkl}\|}\right) \\
B_{i}&=B_{j} = \arccos\left(\frac{U_{ijk} \cdot U_{ijl}}{\|U_{ijk}\| \|U_{ijl}\|}\right) \;\;\;\;\;\;\;\;\;\; B_{k}=B_{l} = \arccos\left(\frac{U_{ikl} \cdot U_{jkl}}{\|U_{ikl}\| \|U_{jkl}\|}\right) \\
C_{i}&=C_{k} = \arccos\left(\frac{U_{ikl} \cdot U_{ijl}}{\|U_{ikl}\| \|U_{ijl}\|}\right) \;\;\;\;\;\;\;\;\;\; C_{j}=C_{l} = \arccos\left(\frac{U_{ijl} \cdot U_{jkl}}{\|U_{ijl}\| \|U_{jkl}\|}\right).
\end{align*}
\end{Theorem}
\begin{proof}
Observe that
$$\text{vol}((\vec{\omega_{m}}+r_{m}\mathbb{S}^{2}) \cap \Delta(r_{i},r_{j},r_{k},r_{l})) = \frac{r_{m}}{3} \text{area}(\partial(\vec{\omega_{m}} + r_{m}\mathbb{S}^2) \cap \Delta(r_{i},r_{j},r_{k},r_{l})).$$
is the volume of a spherical wedge intersecting a sphere of radius $r_{m}$ and a locally maximal tetrahedron.
By calculating the volume of each of these spherical wedges in terms of the spherical area of a triangle on $r_{m}\mathbb{S}^2$
and observing that the supremum of $\delta(\mathcal{P})$ is less than the supremal density of a locally maximal tetrahedron $\Delta(r_{i},r_{j},r_{k},r_{l})$, we obtain
$$\delta(\mathcal{P}) < \max_{r_{i} \leq r_{j} \leq r_{k} \leq r_{l}} \frac{\delta_{\text{max}}(\mathcal{P}_{\Delta(r_{i},r_{j},r_{k},r_{l})}) \displaystyle \sum_{m=i,j,k,l} \text{vol}\left((\vec{\omega_{m}} + r_{m} \mathbb{S}^2) \cap \Delta(r_{i},r_{j},r_{k},r_{l})\right)}{\text{vol}(\Delta(r_{i},r_{j},r_{k},r_{l}))}$$
The apparatus for calculating the upper bound is self evident from the definition of dihedral angles,
tetrahedral volumes, and areas of spherical triangles.
\end{proof}
\pagebreak
\end{document} | math |
package config
import (
"github.com/spf13/viper"
)
func genUserManagement() UserManagement {
uMan := UserManagement{}
uMan.Endpoint = viper.GetString("userManagement.endpoint")
switch viper.GetString("userManagement.type") {
case "inmemory":
uMan.Type = ManagementType_inmemory
default:
uMan.Type = ManagementType_inmemory
}
return uMan
}
| code |
#ifndef NETWORK_BASE_CLIENTPOOL_HH_
# define NETWORK_BASE_CLIENTPOOL_HH_
# include <deque>
# include <mutex>
# include <memory>
# include <boost/date_time/posix_time/posix_time.hpp>
# include <types.hh> // libcore
# include "base/Client.hh"
namespace xtd {
namespace network {
namespace base {
template<typename TClient>
class ClientPool
{
private:
class PersistentClient : protected TClient
{
friend class ClientPool<TClient>;
public:
PersistentClient(const string& p_hostname,
const uint32_t p_port);
~PersistentClient(void);
private:
void async_connect(void);
status wait_async_connect(void);
bool isObsolete(uint32_t p_ttlMs) const;
void invalidate(void);
template<typename... Args>
status send(Args&&... p_args);
template<typename... Args>
status receive(Args&&... p_args);
private:
const string& m_hostname;
const uint32_t m_port;
bool m_isConnected;
boost::posix_time::ptime m_lastUsed;
};
public:
typedef sptr<PersistentClient> client_type;
public:
/**
** Constructor
*/
ClientPool(const string& p_hostname,
const uint32_t p_port,
const uint32_t p_ttlMs);
public:
inline const uint32_t& getSendTotal(void) const { return m_sendTotal; }
inline const uint32_t& getSendSuccess(void) const { return m_sendSuccess; }
inline const uint32_t& getSendError(void) const { return m_sendError; }
inline const uint32_t& getSendTimeout(void) const { return m_sendTimeout; }
inline const uint32_t& getRcvTotal(void) const { return m_rcvTotal; }
inline const uint32_t& getRcvSuccess(void) const { return m_rcvSuccess; }
inline const uint32_t& getRcvError(void) const { return m_rcvError; }
inline const uint32_t& getRcvTimeout(void) const { return m_rcvTimeout; }
inline const uint32_t& getCurNbClient(void) const { return m_curNbClient; }
inline const uint32_t& getRecycleHit(void) const { return m_recycleHit; }
inline const uint32_t& getRecycleMiss(void) const { return m_recycleMiss; }
public:
/**
** @brief Réservation d'un client
*/
client_type acquire(void);
/**
** @brief Envoie d'une requete en utilisant l'identifiant de client réservé
*/
template<typename... Args>
status send(client_type& p_client, Args&&... p_args);
/**
** @brief reception d'une reponse en utilisant l'identifiant de client réservé
*/
template<typename... Args>
status receive(client_type& p_client, Args&&... p_args);
/**
** @brief libration d'un client réservé
*/
void release(client_type& p_client);
private:
std::mutex m_mutex;
std::deque<client_type> m_available;
const string m_hostname;
const uint32_t m_port;
const uint32_t m_ttlMs;
private:
uint32_t m_sendTotal;
uint32_t m_sendSuccess;
uint32_t m_sendError;
uint32_t m_sendTimeout;
uint32_t m_rcvTotal;
uint32_t m_rcvSuccess;
uint32_t m_rcvError;
uint32_t m_rcvTimeout;
uint32_t m_curNbClient;
uint32_t m_acquiredClients;
uint32_t m_recycleHit;
uint32_t m_recycleMiss;
};
}}}
#include "base/ClientPool.hxx"
#endif // !NETWORK_BIP_CLIENTPOOL_HH_
| code |
گۄڈٕ گۄڈٕ اوسنہٕ بادشاہ یِمَن کُن وُچھان تہِ | kashmiri |
We are a local plumbing, electrical and remodeling service in Point Clear!
From as early as the 1800s, wealthy families from Mobile, New Orleans and across the United States chose to spend their summers in Point Clear. In the days of yellow fever outbreaks, Pt. Clear residents believed they were escaping to what was deemed as “good air” because of the daily breeze off Mobile Bay. Arrival to the area was traditionally by ferry boat and most people arrived in Pt. Clear at Zundel’s Wharf.
Because of this, the front of the homes (sometimes referred to as cottages) face Mobile Bay and there is a boardwalk between the water and the homes leading to and from Zundel’s Wharf. Remnants of the old pier at the Zundel property were still visible prior to Hurricane Katrina in 2005. | english |
मुंबई गैंगरेप के आरोपी अक्सर शिकार की तलाश में रहते थे
मुंबई(पिट्स प्रतिनिधि) : पिछले दिनों मुंबई के शक्ति मिल्स गैंगरेप मामले में पकड़े गए आरोपियों ने पुलिस के सामने एक बयान दिया है जिससे उनकी विकृत मानसिकता साफ झलकती है. पुलिस को दिए बयान में गिरफ्तार आरोपियों ने बताया कि वे लोग सेक्स करने के
सलमान खान करना चाहते हैं करण जौहर के साथ काम
मुंबई(पिट्स फिल्मप्रतिनिधि) : यूं तो बॉलीवुड के दबंग स्टार सलमान खान के साथ सभी काम करना चाहते हैं. कई अभिनेत्रियां हैं जो सल्लू के साथ फिल्मों में रोमांस करना चाहती हैं. लेकिन कोई है जिसके साथ सलमान काम करने की इच्छा रखते हैं. हालांकि यह
माइक्रोसॉफ्टने खरीदा, नोकिया को ७.1७ अरब डॉलर में
मुंबई(पिट्स प्रतिनिधि) : अमेरिका की दिग्गज साफ्टवेयर कंपनी माइक्रोसाफ्ट ने कहा कि वह स्मार्टफोन बाजार में अपनी स्थिति सुदृढ करने के लिए फिनलैंड की दूरसंचार हार्डवेयर कंपनी नोकिया का मोबाइल हैंडसेट कारोबार ५.४४ अरब यूरो(७.1७ अरबडॉलर) में खरीदेगी. यह सौदा नकद आधार पर निपटाया जाएगा.
नवजोत सिंह सिद्धू कर रहे हैं फिर दमदार राजनीतिक तैयारी
चंडीगढ़ : लगता है नवजोत सिंह सिद्धू को अपनी गलती का एहसास हो गया है इसलिए तो वह पूरी तैयरी के साथ एक फिर राजनीति के मैदान में उतर रहे हैं. अमृतसर से भाजपा सांसद सिद्धू के इस राजनीतिक यू-टर्न को लेकर राजनीतिक गलियारों में
मुगल सम्राट अकबर की छवि खराब करने की कोशिश की जा रही है : नजमा हेपतुल्ला
नई दिल्ली : इन दिनों टी.वी. धारावाहिक भी लोगों की जहन में बना रहता है लेकिन आज कल मुगल सम्राट अकबर पर बनाए गए एक धारावाहिक जोधा अकबर काफी चर्चे में है. गौरतलब है कि इस धारावाहिक का प्रसारण रोकने की मांग करते हुए भाजपा
असमा को भारत में मरवाना चाहते थे पाक सुरक्षा अधिकारी अमेरिकी मीडिया
नई दिल्ली : अमेरिकी मीडिया की रिपोर्ट में दी गई जानकारी के अनुसार पाक के सुरक्षा अधिकारी, मानवाधिकार कार्यकर्ता असमा जहांगीर को भारत में मरवाना चाहते थे लेकिन यह संभव नही हो पाया क्योंकि इसकी भनक अमेरिकी खुफिया एजेंसियोंको लग गई थी. गौरतलब है कि | hindi |
/*
* Copyright (C) 2007-2013 German Aerospace Center (DLR/SC)
*
* Created: 2010-08-13 Markus Litz <Markus.Litz@dlr.de>
* Changed: $Id$
*
* Version: $Revision$
*
* 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.
*/
/**
* @file
* @brief Implementation of CPACS fuselage sections handling routines.
*/
#include "CCPACSFuselageSections.h"
#include "CCPACSFuselageSection.h"
#include "CTiglError.h"
#include <iostream>
#include <sstream>
namespace tigl
{
// Constructor
CCPACSFuselageSections::CCPACSFuselageSections(void)
{
Cleanup();
}
// Destructor
CCPACSFuselageSections::~CCPACSFuselageSections(void)
{
Cleanup();
}
// Cleanup routine
void CCPACSFuselageSections::Cleanup(void)
{
for (CCPACSFuselageSectionContainer::size_type i = 0; i < sections.size(); i++) {
delete sections[i];
}
sections.clear();
}
// Read CPACS fuselage sections element
void CCPACSFuselageSections::ReadCPACS(TixiDocumentHandle tixiHandle, const std::string& fuselageXPath)
{
Cleanup();
ReturnCode tixiRet;
int sectionCount;
std::string tempString;
char* elementPath;
/* Get section element count */
tempString = fuselageXPath + "/sections";
elementPath = const_cast<char*>(tempString.c_str());
tixiRet = tixiGetNamedChildrenCount(tixiHandle, elementPath, "section", §ionCount);
if (tixiRet != SUCCESS) {
throw CTiglError("XML error: tixiGetNamedChildrenCount failed in CCPACSFuselageSections::ReadCPACS", TIGL_XML_ERROR);
}
// Loop over all sections
for (int i = 1; i <= sectionCount; i++) {
CCPACSFuselageSection* section = new CCPACSFuselageSection();
sections.push_back(section);
tempString = fuselageXPath + "/sections/section[";
std::ostringstream xpath;
xpath << tempString << i << "]";
section->ReadCPACS(tixiHandle, xpath.str());
}
}
// Get section count
int CCPACSFuselageSections::GetSectionCount(void) const
{
return static_cast<int>(sections.size());
}
// Returns the section for a given index
CCPACSFuselageSection& CCPACSFuselageSections::GetSection(int index) const
{
index--;
if (index < 0 || index >= GetSectionCount()) {
throw CTiglError("Error: Invalid index in CCPACSFuselageSections::GetSection", TIGL_INDEX_ERROR);
}
return (*sections[index]);
}
} // end namespace tigl
| code |
सरकार की नीतियों को लेकर प्रदर्शन | सरकार की नीतियों को लेकर प्रदर्शन - एलेनाबाद न्यूज,ऐलनाबाद न्यूज़,ऐलनाबाद समाचार
सरकार की नीतियों को लेकर प्रदर्शन
एलेनाबाद न्यूज - ऐलनाबाद | सर्व कर्मचारी संघ हरियाणा से संबंधित नगरपालिका कर्मचारी संघ ऐलनाबाद के कर्मचारियों ने चौ. देवीलाल टाउन...
ऐलनाबाद | सर्व कर्मचारी संघ हरियाणा से संबंधित नगरपालिका कर्मचारी संघ ऐलनाबाद के कर्मचारियों ने चौ. देवीलाल टाउन पार्क में प्रदेश सरकार की कर्मचारी एवं मजदूर विरोधी नीतियों के खिलाफ रोष प्रदर्शन करके प्रदेश की शहरी निकाय मंत्री कविता जैन का पुतला दहन किया गया। जिसकी अध्यक्षता शाखा प्रधान पप्पूराम ने की। सर्व कर्मचारी संघ हरियाणा के ब्लॉक प्रधान लछमन दास चलाना एवं ब्लॉक सचिव राजेश भाकर ने संयुक्त रुप से संबोधित किया। | hindi |
Summer camp can be a rewarding and fun experience for kids of all ages. Whether they attend day camps or overnight camps, experts say fathers should take action to ensure their kids stay safe and healthy while attending.
"Parents should ask camp organizers basic questions about what plans they have in place to keep kids safe, handle medical emergencies, and deal with routine health needs," said Dr. Patricia Hametz, of NewYork-Presbyterian Morgan Stanley Children’s Hospital.
1. Stay hydrated. Remind your child to drink often, even when they don’t feel thirsty while spending time outdoors or in the heat. Pack a refillable water bottle to make staying hydrated easier.
2. Practice sun safety. Kids should be dressed in light-colored, lightweight clothing and should use a sunscreen with an SPF of 30 or more every day.
3. Be safe in and around water. Remind your kids to follow all camp rules in and around pools, lakes and other bodies of water. Children should never be around water without a certified life guard on duty.
4. Protect against bugs. Using an insect repellent with no more than 30 percent DEET and avoiding scented soaps, lotions and hairsprays will help your kids keep the bugs at bay.
Dads should also make sure that the camp has all of their kids’ emergency contact information, including the name and phone number of their pediatrician as well as information on any medical conditions or medications their children have. | english |
A colored music notation system and a method of colorizing music notation using seven colors that are easily distinguishable from one another, representing the seven unique notes in an octave. The color of a note may be chosen from a list of colors in CMYK format, with a tolerance of plus or minus ten points, preferably plus or minus five points in attributes for any of the colors. The colors may be reproduced by any printing method, including using a conventional personal computer. The colors are easily reproducible within acceptable tolerances on a wide variety of media and with a range of printing options, can be read under various lighting conditions, and do not make written music unpleasant to the eye. The colors are also sufficiently different to allow for easy recognition of common chord combinations by the recollection of simple groups of colors. In the preferred embodiment of the invention, notes affected by an accidental are the same color as the natural note to which they are related. The invention also provides a method for applying color to the keys a keyboard instrument.
FIG. 2 is a graphical representation of a colorized musical score according to the method of the invention showing common color combinations.
Symbol No. Corresponding Note Corresponding Note No.
d. Outputting the colorized score to a printer or memory device.
2. The method of claim 1 in which the colors are reproduced by a computer comprising a printer.
3. The method of claim 1 comprising, before step d., the additional step of exporting the image of the colorized score to a graphics program for editing.
4. The method of claim 1 in which a note affected by an accidental symbol is the same color as the corresponding natural note.
5. The method of claim 4 in which the accidental symbol is black.
6. The method of claim 4 in which the accidental symbol of a key signature is colored according to the color of the natural note that is affects.
7. The method of claim 1 in which the keys of a keyboard instrument have colors corresponding to the colors of the notes associated with said keys.
8. The method of claim 7 in which the colors are applied to the keys by adhesive labels.
9. A colorized music score produced according to the method of claim 1.
10. The method of claim 1 in which the tolerance is within plus or minus five points in one or more of the C, M, Y or K attributes.
13. The method of claim 11 comprising, before step d., the additional step of exporting the image of the colorized score to a graphics program for editing.
14. The method of claim 11 in which a note affected by an accidental symbol is the same color as the corresponding natural note.
15. The method of claim 14 in which the accidental symbol is black.
16. The method of claim 14 in which the accidental symbol of a key signature is colored according to the color of the natural note that is affects.
17. The method of claim 11 in which the keys of a keyboard instrument have colors corresponding to the colors of the notes associated with said keys.
18. The method of claim 17 in which the colors are applied to the keys by adhesive labels.
19. A colorized music score produced according to the method of claim 11.
21. The colorized music score of claim 19 in which colors are also assigned to key signature accidental symbols, the colors of the key signature accidental symbols being the same as the colors assigned to the corresponding notes.
22. The method of claim 11 in which the tolerance is within plus or minus five points in one or more of the C, M, Y or K attributes. | english |
अगर बेताबियों के साथ हर पल गुज़रता है, अगर किसी के ना होने का दर्द तुम्हे भी होता है, अगर कुछ कर गुजरने का जज़्बा रखते हो तुम, अगर सच कहने से नहीं...
हम काले क्यों हैं ? बाय श्रुति भार्गव, मैंने देखा हैं हम काले क्यूं है? सात रंगों के इन्द्रधनुष को देख,मुस्कुराहट कान से कान तक बंध जाती है,बारिश और सूरज की किरणों का जादू, सतरंगी एहसास जगाती है।कभी सोचा है,कि...
काश - आ पोम बाय श्रुति काश मैं वो चांद होता जिसे छूने को वो अटल हो सके, काश मैं वो फूल होता जिसकी खुशबू सांसों में बस सके, काश मैं वो भूल होता, जिसे कभी...
हकीकत जैसी भी हो,ख्वाब अद्भुत है। इन ख्वाबों से कहो,मेरी ख्वाइशों से रोज़ मिला करें,जैसे ही पूरी होती है,बदल जाती हैं। बदलती है, अपने रूप,अपने आकार में,अपनी खुशबू में,अपने आकाश में,फिर भी मेरी ही है,बेहद अपनी सी है।बदलती...
नारी हूं,और नर पहले,सफल हूं अपनी परिभाषा में,संतुष्ट भी साकार किए हुए सपनों में।टूटती हूं और चकनाचूर हो जाती हूं,खुद को समेट लेती हूं, अचंभा हो जाती हूं।संक्षेप हूं और विस्तार भी,कश्ती हूं और तूफान भी,बिखरती हूं...
रात को सोते वक़्त, अंधेरे कमरे में,जब आंखे जागती है,ख्याल इतने होते है,अपनी अपनी खुशी तराशती है। किसी को सोते वक़्त अगले दिन का नाश्ता क्या होगा ?ये ख्याल आता है,कोई कल सुबह तक भूक से मर ना...
है मगर, मुझसा दिखता तो है,पर मैं मुझसे कहीं खो गया है।है मगर,देखता सब कुछ है,मेहसूस भी करता है,पर मुझमें से मैं खो गया है।है मगर,सही राह चल रहा है,पर वो खास राह कहीं खो गया है।है मगर,कई...
हैलो की टोन से समझ जाती है, मेरे मन की खलबली को, चुप रहूं, तो भाप लेती है, मेरे दिल में उभर रही तरंगों को,ऊंची आवाज़ में मैं कभी जो बोल देती हूं उससे, समझ जाती...
चलो खामोश रहने का खेल खेलते हैं, हर लव्ज़ को कैद होने देते हैं,एक दूजे से ना कह कर मन की बात,मन को दबोच लेते हैं।खामोश रहने से शायद हमारे बीच का बैर दोस्ती में बदल जाये,खामोश रहने... | hindi |
package com.me.ui.sample.thirdparty.aspect;
import com.me.ui.util.LogUtils;
import org.aspectj.lang.JoinPoint;
import org.aspectj.lang.ProceedingJoinPoint;
import org.aspectj.lang.annotation.After;
import org.aspectj.lang.annotation.Around;
import org.aspectj.lang.annotation.Aspect;
import org.aspectj.lang.annotation.Before;
import org.aspectj.lang.annotation.Pointcut;
/**
* @author kylingo
* @since 2019/03/07 11:01
*/
@Aspect
public class TraceAspect {
public static final String TAG = "TraceAspect";
private static final String POINT_METHOD = "execution(* com.me.ui.sample.thirdparty.aspect.AspectFragment.testAspect(..))";
private static final String POINT_CALLMETHOD = "call(* com.me.ui.sample.thirdparty.aspect.AspectFragment..testAspect(..))";
@Pointcut(POINT_METHOD)
public void methodAnnotated() {
}
@Pointcut(POINT_CALLMETHOD)
public void methodCallAnnotated() {
}
@Around("methodAnnotated()")
public Object aronudWeaverPoint(ProceedingJoinPoint joinPoint) throws Throwable {
Object object = joinPoint.proceed();
String result = "aroundWeaverPoint";
LogUtils.i(TAG, "testAspect aroundWeaverPoint");
// return result;//替换原方法的返回值
return object;
}
@Before("methodCallAnnotated()")
public void beforeCall(JoinPoint joinPoint) {
LogUtils.i(TAG, "testAspect beforeCall");
}
@After("methodCallAnnotated()")
public void afterCall(JoinPoint joinPoint) {
LogUtils.i(TAG, "testAspect afterCall");
}
@Before("call(* com.me.ui.sample.thirdparty.aspect.AspectFragment..testAspect1(..))")
public void executeAspectBefore(JoinPoint joinPoint) {
LogUtils.i(TAG, "testAspect1 beforeCall");
}
@After("call(* com.me.ui.sample.thirdparty.aspect.AspectFragment..testAspect1(..))")
public void executeAspectAfter(JoinPoint joinPoint) {
LogUtils.i(TAG, "testAspect1 afterCall");
}
} | code |
/*
* Copyright (C) 2011 Red Hat, Inc.
*
* This file is released under the GPL.
*/
#include "dm-block-manager.h"
#include "dm-persistent-data-internal.h"
#include "../dm-bufio.h"
#include <linux/crc32c.h>
#include <linux/module.h>
#include <linux/slab.h>
#include <linux/rwsem.h>
#include <linux/device-mapper.h>
#include <linux/stacktrace.h>
#define DM_MSG_PREFIX "block manager"
/*----------------------------------------------------------------*/
#ifdef CONFIG_DM_DEBUG_BLOCK_MANAGER_LOCKING
/*
* This is a read/write semaphore with a couple of differences.
*
* i) There is a restriction on the number of concurrent read locks that
* may be held at once. This is just an implementation detail.
*
* ii) Recursive locking attempts are detected and return EINVAL. A stack
* trace is also emitted for the previous lock acquisition.
*
* iii) Priority is given to write locks.
*/
#define MAX_HOLDERS 4
#define MAX_STACK 10
typedef unsigned long stack_entries[MAX_STACK];
struct block_lock {
spinlock_t lock;
__s32 count;
struct list_head waiters;
struct task_struct *holders[MAX_HOLDERS];
#ifdef CONFIG_DM_DEBUG_BLOCK_STACK_TRACING
struct stack_trace traces[MAX_HOLDERS];
stack_entries entries[MAX_HOLDERS];
#endif
};
struct waiter {
struct list_head list;
struct task_struct *task;
int wants_write;
};
static unsigned __find_holder(struct block_lock *lock,
struct task_struct *task)
{
unsigned i;
for (i = 0; i < MAX_HOLDERS; i++)
if (lock->holders[i] == task)
break;
BUG_ON(i == MAX_HOLDERS);
return i;
}
/* call this *after* you increment lock->count */
static void __add_holder(struct block_lock *lock, struct task_struct *task)
{
unsigned h = __find_holder(lock, NULL);
#ifdef CONFIG_DM_DEBUG_BLOCK_STACK_TRACING
struct stack_trace *t;
#endif
get_task_struct(task);
lock->holders[h] = task;
#ifdef CONFIG_DM_DEBUG_BLOCK_STACK_TRACING
t = lock->traces + h;
t->nr_entries = 0;
t->max_entries = MAX_STACK;
t->entries = lock->entries[h];
t->skip = 2;
save_stack_trace(t);
#endif
}
/* call this *before* you decrement lock->count */
static void __del_holder(struct block_lock *lock, struct task_struct *task)
{
unsigned h = __find_holder(lock, task);
lock->holders[h] = NULL;
put_task_struct(task);
}
static int __check_holder(struct block_lock *lock)
{
unsigned i;
for (i = 0; i < MAX_HOLDERS; i++) {
if (lock->holders[i] == current) {
DMERR("recursive lock detected in metadata");
#ifdef CONFIG_DM_DEBUG_BLOCK_STACK_TRACING
DMERR("previously held here:");
print_stack_trace(lock->traces + i, 4);
DMERR("subsequent acquisition attempted here:");
dump_stack();
#endif
return -EINVAL;
}
}
return 0;
}
static void __wait(struct waiter *w)
{
for (;;) {
set_current_state(TASK_UNINTERRUPTIBLE);
if (!w->task)
break;
schedule();
}
set_current_state(TASK_RUNNING);
}
static void __wake_waiter(struct waiter *w)
{
struct task_struct *task;
list_del(&w->list);
task = w->task;
smp_mb();
w->task = NULL;
wake_up_process(task);
}
/*
* We either wake a few readers or a single writer.
*/
static void __wake_many(struct block_lock *lock)
{
struct waiter *w, *tmp;
BUG_ON(lock->count < 0);
list_for_each_entry_safe(w, tmp, &lock->waiters, list) {
if (lock->count >= MAX_HOLDERS)
return;
if (w->wants_write) {
if (lock->count > 0)
return; /* still read locked */
lock->count = -1;
__add_holder(lock, w->task);
__wake_waiter(w);
return;
}
lock->count++;
__add_holder(lock, w->task);
__wake_waiter(w);
}
}
static void bl_init(struct block_lock *lock)
{
int i;
spin_lock_init(&lock->lock);
lock->count = 0;
INIT_LIST_HEAD(&lock->waiters);
for (i = 0; i < MAX_HOLDERS; i++)
lock->holders[i] = NULL;
}
static int __available_for_read(struct block_lock *lock)
{
return lock->count >= 0 &&
lock->count < MAX_HOLDERS &&
list_empty(&lock->waiters);
}
static int bl_down_read(struct block_lock *lock)
{
int r;
struct waiter w;
spin_lock(&lock->lock);
r = __check_holder(lock);
if (r) {
spin_unlock(&lock->lock);
return r;
}
if (__available_for_read(lock)) {
lock->count++;
__add_holder(lock, current);
spin_unlock(&lock->lock);
return 0;
}
get_task_struct(current);
w.task = current;
w.wants_write = 0;
list_add_tail(&w.list, &lock->waiters);
spin_unlock(&lock->lock);
__wait(&w);
put_task_struct(current);
return 0;
}
static int bl_down_read_nonblock(struct block_lock *lock)
{
int r;
spin_lock(&lock->lock);
r = __check_holder(lock);
if (r)
goto out;
if (__available_for_read(lock)) {
lock->count++;
__add_holder(lock, current);
r = 0;
} else
r = -EWOULDBLOCK;
out:
spin_unlock(&lock->lock);
return r;
}
static void bl_up_read(struct block_lock *lock)
{
spin_lock(&lock->lock);
BUG_ON(lock->count <= 0);
__del_holder(lock, current);
--lock->count;
if (!list_empty(&lock->waiters))
__wake_many(lock);
spin_unlock(&lock->lock);
}
static int bl_down_write(struct block_lock *lock)
{
int r;
struct waiter w;
spin_lock(&lock->lock);
r = __check_holder(lock);
if (r) {
spin_unlock(&lock->lock);
return r;
}
if (lock->count == 0 && list_empty(&lock->waiters)) {
lock->count = -1;
__add_holder(lock, current);
spin_unlock(&lock->lock);
return 0;
}
get_task_struct(current);
w.task = current;
w.wants_write = 1;
/*
* Writers given priority. We know there's only one mutator in the
* system, so ignoring the ordering reversal.
*/
list_add(&w.list, &lock->waiters);
spin_unlock(&lock->lock);
__wait(&w);
put_task_struct(current);
return 0;
}
static void bl_up_write(struct block_lock *lock)
{
spin_lock(&lock->lock);
__del_holder(lock, current);
lock->count = 0;
if (!list_empty(&lock->waiters))
__wake_many(lock);
spin_unlock(&lock->lock);
}
static void report_recursive_bug(dm_block_t b, int r)
{
if (r == -EINVAL)
DMERR("recursive acquisition of block %llu requested.",
(unsigned long long) b);
}
#else /* !CONFIG_DM_DEBUG_BLOCK_MANAGER_LOCKING */
#define bl_init(x) do { } while (0)
#define bl_down_read(x) 0
#define bl_down_read_nonblock(x) 0
#define bl_up_read(x) do { } while (0)
#define bl_down_write(x) 0
#define bl_up_write(x) do { } while (0)
#define report_recursive_bug(x, y) do { } while (0)
#endif /* CONFIG_DM_DEBUG_BLOCK_MANAGER_LOCKING */
/*----------------------------------------------------------------*/
/*
* Block manager is currently implemented using dm-bufio. struct
* dm_block_manager and struct dm_block map directly onto a couple of
* structs in the bufio interface. I want to retain the freedom to move
* away from bufio in the future. So these structs are just cast within
* this .c file, rather than making it through to the public interface.
*/
static struct dm_buffer *to_buffer(struct dm_block *b)
{
return (struct dm_buffer *) b;
}
dm_block_t dm_block_location(struct dm_block *b)
{
return dm_bufio_get_block_number(to_buffer(b));
}
EXPORT_SYMBOL_GPL(dm_block_location);
void *dm_block_data(struct dm_block *b)
{
return dm_bufio_get_block_data(to_buffer(b));
}
EXPORT_SYMBOL_GPL(dm_block_data);
struct buffer_aux {
struct dm_block_validator *validator;
int write_locked;
#ifdef CONFIG_DM_DEBUG_BLOCK_MANAGER_LOCKING
struct block_lock lock;
#endif
};
static void dm_block_manager_alloc_callback(struct dm_buffer *buf)
{
struct buffer_aux *aux = dm_bufio_get_aux_data(buf);
aux->validator = NULL;
bl_init(&aux->lock);
}
static void dm_block_manager_write_callback(struct dm_buffer *buf)
{
struct buffer_aux *aux = dm_bufio_get_aux_data(buf);
if (aux->validator) {
aux->validator->prepare_for_write(aux->validator, (struct dm_block *) buf,
dm_bufio_get_block_size(dm_bufio_get_client(buf)));
}
}
/*----------------------------------------------------------------
* Public interface
*--------------------------------------------------------------*/
struct dm_block_manager {
struct dm_bufio_client *bufio;
bool read_only:1;
};
struct dm_block_manager *dm_block_manager_create(struct block_device *bdev,
unsigned block_size,
unsigned cache_size,
unsigned max_held_per_thread)
{
int r;
struct dm_block_manager *bm;
bm = kmalloc(sizeof(*bm), GFP_KERNEL);
if (!bm) {
r = -ENOMEM;
goto bad;
}
bm->bufio = dm_bufio_client_create(bdev, block_size, max_held_per_thread,
sizeof(struct buffer_aux),
dm_block_manager_alloc_callback,
dm_block_manager_write_callback);
if (IS_ERR(bm->bufio)) {
r = PTR_ERR(bm->bufio);
kfree(bm);
goto bad;
}
bm->read_only = false;
return bm;
bad:
return ERR_PTR(r);
}
EXPORT_SYMBOL_GPL(dm_block_manager_create);
void dm_block_manager_destroy(struct dm_block_manager *bm)
{
dm_bufio_client_destroy(bm->bufio);
kfree(bm);
}
EXPORT_SYMBOL_GPL(dm_block_manager_destroy);
unsigned dm_bm_block_size(struct dm_block_manager *bm)
{
return dm_bufio_get_block_size(bm->bufio);
}
EXPORT_SYMBOL_GPL(dm_bm_block_size);
dm_block_t dm_bm_nr_blocks(struct dm_block_manager *bm)
{
return dm_bufio_get_device_size(bm->bufio);
}
static int dm_bm_validate_buffer(struct dm_block_manager *bm,
struct dm_buffer *buf,
struct buffer_aux *aux,
struct dm_block_validator *v)
{
if (unlikely(!aux->validator)) {
int r;
if (!v)
return 0;
r = v->check(v, (struct dm_block *) buf, dm_bufio_get_block_size(bm->bufio));
if (unlikely(r)) {
DMERR_LIMIT("%s validator check failed for block %llu", v->name,
(unsigned long long) dm_bufio_get_block_number(buf));
return r;
}
aux->validator = v;
} else {
if (unlikely(aux->validator != v)) {
DMERR_LIMIT("validator mismatch (old=%s vs new=%s) for block %llu",
aux->validator->name, v ? v->name : "NULL",
(unsigned long long) dm_bufio_get_block_number(buf));
return -EINVAL;
}
}
return 0;
}
int dm_bm_read_lock(struct dm_block_manager *bm, dm_block_t b,
struct dm_block_validator *v,
struct dm_block **result)
{
struct buffer_aux *aux;
void *p;
int r;
p = dm_bufio_read(bm->bufio, b, (struct dm_buffer **) result);
if (unlikely(IS_ERR(p)))
return PTR_ERR(p);
aux = dm_bufio_get_aux_data(to_buffer(*result));
r = bl_down_read(&aux->lock);
if (unlikely(r)) {
dm_bufio_release(to_buffer(*result));
report_recursive_bug(b, r);
return r;
}
aux->write_locked = 0;
r = dm_bm_validate_buffer(bm, to_buffer(*result), aux, v);
if (unlikely(r)) {
bl_up_read(&aux->lock);
dm_bufio_release(to_buffer(*result));
return r;
}
return 0;
}
EXPORT_SYMBOL_GPL(dm_bm_read_lock);
int dm_bm_write_lock(struct dm_block_manager *bm,
dm_block_t b, struct dm_block_validator *v,
struct dm_block **result)
{
struct buffer_aux *aux;
void *p;
int r;
if (bm->read_only)
return -EPERM;
p = dm_bufio_read(bm->bufio, b, (struct dm_buffer **) result);
if (unlikely(IS_ERR(p)))
return PTR_ERR(p);
aux = dm_bufio_get_aux_data(to_buffer(*result));
r = bl_down_write(&aux->lock);
if (r) {
dm_bufio_release(to_buffer(*result));
report_recursive_bug(b, r);
return r;
}
aux->write_locked = 1;
r = dm_bm_validate_buffer(bm, to_buffer(*result), aux, v);
if (unlikely(r)) {
bl_up_write(&aux->lock);
dm_bufio_release(to_buffer(*result));
return r;
}
return 0;
}
EXPORT_SYMBOL_GPL(dm_bm_write_lock);
int dm_bm_read_try_lock(struct dm_block_manager *bm,
dm_block_t b, struct dm_block_validator *v,
struct dm_block **result)
{
struct buffer_aux *aux;
void *p;
int r;
p = dm_bufio_get(bm->bufio, b, (struct dm_buffer **) result);
if (unlikely(IS_ERR(p)))
return PTR_ERR(p);
if (unlikely(!p))
return -EWOULDBLOCK;
aux = dm_bufio_get_aux_data(to_buffer(*result));
r = bl_down_read_nonblock(&aux->lock);
if (r < 0) {
dm_bufio_release(to_buffer(*result));
report_recursive_bug(b, r);
return r;
}
aux->write_locked = 0;
r = dm_bm_validate_buffer(bm, to_buffer(*result), aux, v);
if (unlikely(r)) {
bl_up_read(&aux->lock);
dm_bufio_release(to_buffer(*result));
return r;
}
return 0;
}
int dm_bm_write_lock_zero(struct dm_block_manager *bm,
dm_block_t b, struct dm_block_validator *v,
struct dm_block **result)
{
int r;
struct buffer_aux *aux;
void *p;
if (bm->read_only)
return -EPERM;
p = dm_bufio_new(bm->bufio, b, (struct dm_buffer **) result);
if (unlikely(IS_ERR(p)))
return PTR_ERR(p);
memset(p, 0, dm_bm_block_size(bm));
aux = dm_bufio_get_aux_data(to_buffer(*result));
r = bl_down_write(&aux->lock);
if (r) {
dm_bufio_release(to_buffer(*result));
return r;
}
aux->write_locked = 1;
aux->validator = v;
return 0;
}
EXPORT_SYMBOL_GPL(dm_bm_write_lock_zero);
void dm_bm_unlock(struct dm_block *b)
{
struct buffer_aux *aux;
aux = dm_bufio_get_aux_data(to_buffer(b));
if (aux->write_locked) {
dm_bufio_mark_buffer_dirty(to_buffer(b));
bl_up_write(&aux->lock);
} else
bl_up_read(&aux->lock);
dm_bufio_release(to_buffer(b));
}
EXPORT_SYMBOL_GPL(dm_bm_unlock);
int dm_bm_flush(struct dm_block_manager *bm)
{
if (bm->read_only)
return -EPERM;
return dm_bufio_write_dirty_buffers(bm->bufio);
}
EXPORT_SYMBOL_GPL(dm_bm_flush);
void dm_bm_prefetch(struct dm_block_manager *bm, dm_block_t b)
{
dm_bufio_prefetch(bm->bufio, b, 1);
}
bool dm_bm_is_read_only(struct dm_block_manager *bm)
{
return bm->read_only;
}
EXPORT_SYMBOL_GPL(dm_bm_is_read_only);
void dm_bm_set_read_only(struct dm_block_manager *bm)
{
bm->read_only = true;
}
EXPORT_SYMBOL_GPL(dm_bm_set_read_only);
void dm_bm_set_read_write(struct dm_block_manager *bm)
{
bm->read_only = false;
}
EXPORT_SYMBOL_GPL(dm_bm_set_read_write);
u32 dm_bm_checksum(const void *data, size_t len, u32 init_xor)
{
return crc32c(~(u32) 0, data, len) ^ init_xor;
}
EXPORT_SYMBOL_GPL(dm_bm_checksum);
/*----------------------------------------------------------------*/
MODULE_LICENSE("GPL");
MODULE_AUTHOR("Joe Thornber <dm-devel@redhat.com>");
MODULE_DESCRIPTION("Immutable metadata library for dm");
/*----------------------------------------------------------------*/
| code |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.