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[6] وٮ۪تھٕ آگر ۶۸۔۷۸۹۱ءمراز ادبی سنگم۔
|
kashmiri
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Part of our ‘Comfort Fit’ collection with soft leather & extra padding.
Available in a 1.5″ wide heel for stability.
Full suede sole & steel shank for added support.
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english
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\begin{document}
\maketitle
\section{Introduction}
Let $L/K$ be an extension of absolutely abelian number fields of equal conductor, $n$.
If $T_{L/K}: L \rightarrow K$ denotes the trace map, then $T_{L/K}(\mathcal{O}_{L})$ is an ideal in $ \mathcal{O}_{K}$. Let $I(L/K)$ denote the norm of $T_{L/K}(\mathcal{O}_{L})$ over $\mathbb{Q}$, i.e.
$[\mathcal{O}_{K} : T_{L/K}(\mathcal{O}_{L})]$. Sharpening the main result of Girstmair in \cite{girstmair}, we determine $I(L/K)$ exactly for any such $L/K$: if $e=v_{2}(n)$ and $m=n/2^{e}$, then
$$
I(L/K) = \left\{
\begin{array}{ll}
2^{[K \cap \mathbb{Q}^{(m)}:\mathbb{Q}]} = 2^{( [K:\mathbb{Q}] / 2^{(e-2)} )} & \textrm{ if $L/K$ is wildly ramified,} \\
1 & \textrm{ otherwise.}
\end{array} \right.
$$
After first determining criteria for wild ramification of $L/K$ (which can only happen at primes above $2$),
the above result
is obtained for $n=2^{e}$ ($e \geq 3$) by computing $T_{L/K}(\mathcal{O}_{L})$ explicitly, and is then extended to the general case. This approach does not rely on Leopoldt's Theorem, in contrast to the techniques used
in \cite{girstmair}. \par
The explicit nature of the calculations used to compute $I(L/K)$ leads to the definition of an ``adjusted trace map'' $\hat{T}_{\mathbb{Q}^{(n)}/K}$ with the property that $\hat{T}_{\mathbb{Q}^{(n)}/K}(\mathcal{O}^{(n)})=\mathcal{O}_{K}$ (here $\mathbb{Q}^{(n)}$ denotes the $n^{\mathrm{th}}$ cyclotomic field and $\mathcal{O}^{(n)}$ its ring of integers). Using this map, we restate Leopoldt's Theorem and show that its proof can be reduced to the (easier) cyclotomic case.
\section{Dirichlet Characters}
We first recall some basic facts about Dirichlet characters. For more details, see Chapter 3 of \cite{wash} and Section 2 of \cite{lettl}.
\begin{definition}
For $n \in \mathbb{N}$, let $\zeta_{n}$ be a primitive $n^{\textrm{th}}$ root of unity and $\mathbb{Q}^{(n)}=\mathbb{Q}(\zeta_{n})$ the $n^{\textrm{th}}$ cyclotomic field. Let $\mathcal{O}^{(n)}=\mathcal{O}_{\mathbb{Q}^{(n)}}=\mathbb{Z}[\zeta_{n}]$ denote the ring of integers of $\mathbb{Q}^{(n)}$, and
$X^{(n)}$ denote the group of Dirichlet characters of conductor dividing $n$. \par
Let $\mathbb{P}$ denote the set of rational primes.
Define $p^*=4$ if $p=2$, and $p^*=p$ if $p \in \mathbb{P}, p \neq 2$.
\end{definition}
\begin{prop}\label{decomp}
Let $p \in \mathbb{P}$ and $e \in \mathbb{N}$, with $e \geq 2$ if $p=2$. Then there exists a natural decomposition
$$(\mathbb{Z} / p^e\mathbb{Z})^{\times} = (\mathbb{Z} / p^{*}\mathbb{Z})^{\times} \times (1+p^{*} \mathbb{Z}) / (1+p^e \mathbb{Z}) $$
where both factors are considered as subgroups of $(\mathbb{Z} / p^e\mathbb{Z})^{\times}$.
Note that we take $(\mathbb{Z} / 4\mathbb{Z})^{\times} = \{ \pm 1 \}$.
\end{prop}
\begin{proof}
Straightforward.
\end{proof}
\begin{definition}
Let $p \in \mathbb{P}$ and $e \in \mathbb{N}$ with $e \geq 2$ if $p=2$. Then dualizing the decomposition of Proposition \ref{decomp} yields the decomposition
$$X^{(p^e)} = \langle \omega_p \rangle \times \langle \psi_{p^e} \rangle$$ with $\langle \omega_p \rangle = X^{(p^*)}$ and $ \langle \psi_{p^e} \rangle$ the group of Dirichlet characters whose conductors divide $p^e$ and which are trivial on the factor $(\mathbb{Z} / p^*\mathbb{Z})^{\times}$.
\end{definition}
\begin{theorem}
Let $n \in \mathbb{N}$. There is an order preserving one-to-one correspondence between subgroups of
$X^{(n)}$ and subfields of $\mathbb{Q}^{(n)}$. Let $X_i$ be the subgroup corresponding to the subfield $K_i$.
Then $|X_i|=[K_i: \mathbb{Q}]$ and the compositum $K_1K_2$ corresponds to $\langle X_1, X_2 \rangle$.
\end{theorem}
\begin{proof}
See Chapter 3 of \cite{wash}.
\end{proof}
\begin{definition}
Let $p \in \mathbb{P}$, $X \subseteq X^{(n)}$ and $e=v_{p}(n)$. Then $X_{p}$ denotes the image of $X$ under the natural projection $\pi_{p}:X^{(n)} \rightarrow X^{(p^{e})}$.
\end{definition}
\begin{theorem}\label{ramindex}
Let $X$ be a group of Dirichlet characters and let $K$ be the associated abelian number field.
Then $p \in \mathbb{P}$ has ramification index $|X_{p}|$ in $K$.
\end{theorem}
\begin{proof}
This is Theorem 3.5 of \cite{wash}.
\end{proof}
\begin{remark}\label{wildtameparts}
When $p$ is odd, $\langle \omega_p \rangle$ and $\langle \psi_{p^e} \rangle$ have orders $p-1$
and $p^{(e-1)}$ respectively. So by considering the decomposition $X^{(p^e)} = \langle \omega_p \rangle \times \langle \psi_{p^e} \rangle$, the field corresponding to $\langle \omega_p \rangle$ can be thought of as the ``tame part'' of $\mathbb{Q}^{(p^{e})}$, and that corresponding to $\langle \psi_{p^e} \rangle$ as the ``wild part''. \par
When $p=2$,
$\langle \omega_{2} \rangle$ and $\langle \psi_{2^{e}} \rangle$ have orders $2$ and $2^{(e-2)}$ respectively, and therefore both correspond to wildly ramified extensions of $\mathbb{Q}$ (namely $\mathbb{Q}(i)$ and the maximal totally real subfield
$\mathbb{Q}(\zeta_{2^{e}}+\zeta_{2^{e}}^{-1})$, respectively). In other words, $\mathbb{Q}^{(2^{e})}$ has no ``tame part''.
\end{remark}
\begin{prop}\label{surjective}
Let $K / \mathbb{Q}$ be an abelian extension of conductor $n=p_1^{e_1} \cdots p_t^{e_t}$ where $p_{1}=2$,
and let $X \subseteq X^{(n)}$ be its associated group of Dirichlet characters.
\begin{enumerate}
\item The natural projection
$ \pi_{\psi}: X \longrightarrow \prod_{i=1}^t \langle \psi_{p_i^{e_i}} \rangle$
is surjective.
\item Let $e=e_{1}=v_{2}(n)$. Then $X_{2}$ is either $X^{(2^{e})}=\langle \omega_{2} \rangle \times \langle \psi_{2^{e}} \rangle$, $\langle \psi_{2^{e}} \rangle$, or $\langle \omega_{2} \psi_{2^{e}} \rangle$. Note that $\psi_{2^{e}}$ is trivial if $e \leq 2$.
\item $\langle X, \prod_{i=2}^{t} \langle \omega_{p_{i}} \rangle \rangle = X_{2} \times
\prod_{i=2}^{t} \langle \omega_{p_i} \rangle \times \prod_{j=2}^{t} \langle \psi_{p_j^{e_j}} \rangle = X_{2} \times
X^{(m)}$ where $m=n/2^{e}$.
\end{enumerate}
\end{prop}
\begin{proof}
Part (a) is essentially part (a) of Lemma 1 in \cite{lettl}. Part (b) follows from the fact that
the natural projection $X \rightarrow \langle \psi_{2^{e}} \rangle$ and thus $X_{2} \rightarrow \langle \psi_{2^{e}} \rangle$ must be surjective.
By part (a), $\langle X, \prod_{i=2}^{t} \langle \omega_{p_{i}} \rangle \rangle$ contains all the Sylow-$p$ subgroups of $X^{(n)}= \prod_{i=1}^t \langle \omega_{p_i} \rangle \times \prod_{j=1}^t \langle \psi_{p_j^{e_j}} \rangle$ for $p$ odd; in particular, it contains
$\prod_{j=2}^t \langle \psi_{p_j^{e_j}} \rangle$. Thus
$\prod_{i=2}^t \langle \omega_{p_i} \rangle \times \prod_{j=2}^t \langle \psi_{p_j^{e_j}} \rangle
\subseteq \langle X , \prod_{i=2}^{t} \langle \omega_{p_{i}} \rangle \rangle$. Part (c) now follows by noting that the image of the natural projection $ \langle X, \prod_{i=2}^{t} \langle \omega_{p_{i}} \rangle \rangle \rightarrow
X^{(2^{e})}$ is $X_{2}$.
\end{proof}
\section{Ramification}
\begin{definition}
Throughout this paper, we take ``tamely ramified'' to mean ``at most tamely ramified'', i.e. ``not wildly ramified''.
\end{definition}
\begin{theorem}\label{wildtrace}
Let $L/K$ be an extension of number fields. Then $T_{L/K}(\mathcal{O}_L)$ is an ideal of $\mathcal{O}_K$. Suppose further that $L/K$ is Galois, and let $\mathfrak{p}$ be a (non-zero) prime of $\mathcal{O}_K$. Then $\mathfrak{p} \mid T_{L/K}(\mathcal{O}_L)$ if and only if $\mathfrak{p}$ is wildly ramified in $L/K$.
\end{theorem}
\begin{proof}
See \cite{mj}. Alternatively, this follows Lemma 2 in section 5 of \cite{lf} and the fact that the extension of residue fields in question is separable.
\end{proof}
\begin{corollary}\label{wildtracecor}
If $L/K$ is a Galois extension of number fields, then $L/K$ is tamely ramified
if and only if $T_{L/K}(\mathcal{O}_L)=\mathcal{O}_K$.
\end{corollary}
\begin{prop}\label{tameaboveodd}
Let $K$ be an abelian number field of conductor $n$. Then $\mathbb{Q}^{(n)}/K$ is tamely ramified at
each prime lying above an odd rational prime.
\end{prop}
\begin{proof}
Let $X$ be the group of Dirichlet characters associated to $K$ and write $n=\prod_{i=1}^t p_i^{e_i}$ where $p_{1}=2$. Let $M$ be the field corresponding to $\prod_{i=2}^{t} \langle \omega_{p_{i}} \rangle$. The extension
$MK/K$ is tamely ramified since the same is true of $M/ \mathbb{Q}$. By parts (b) and (c) of Proposition \ref{surjective} we have $[\mathbb{Q}^{(n)}:MK]=1$ or $2$, and so the result follows.
\end{proof}
\begin{corollary}\label{2subext}
Let $K$ be an abelian number field of conductor $n$. Then wild ramification in $\mathbb{Q}^{(n)}/K$ can only occur in a degree $2$ sub-extension (at primes above $2$).
\end{corollary}
\begin{remark}
The result of Proposition \ref{tameaboveodd} appears to be well-known (it is noted
in \cite{byott_lettl}, for example), but its proof and corollary are not easily found in the literature.
\end{remark}
\begin{prop}\label{no2problems}
Let $K$ be an abelian number field of conductor $n=\prod_{i=1}^t p_i^{e_i}$ with associated
character group $X$. Let $e=e_1=v_2(n)$. Then the following are equivalent:
\begin{enumerate}
\item $X_2 = X^{(2^{e})}$.
\item $X^{(n)} = \langle X, \prod_{i=2}^t \langle \omega_{p_i} \rangle \rangle$.
\item $\mathbb{Q}^{(n)} / K$ is tamely ramified.
\item $T_{\mathbb{Q}^{(n)}/K}(\mathcal{O}^{(n)}) = \mathcal{O}_{K}$, i.e. $I(\mathbb{Q}^{(n)}/K)=1$.
\end{enumerate}
\end{prop}
\begin{proof}
(a) $\Leftrightarrow$ (b) follows from part (c) of Proposition \ref{surjective}. \par
(c) $\Leftrightarrow$ (d) follows from Corollary \ref{wildtracecor}. \par
(a) $\Leftrightarrow$ (c) follows from Proposition \ref{tameaboveodd} and Theorem \ref{ramindex}.
\end{proof}
\begin{remark}
In particular, the equivalent conditions of Proposition \ref{no2problems}
hold when $e \leq 2$. Furthermore, it can be shown that they also hold if there exists $d \in \mathbb{Z}$ with $d \equiv 3 \, (4)$ and $d$ square-free such that $\mathbb{Q}[\sqrt{d}] \subseteq K$.
\end{remark}
\begin{prop}\label{some2problems}
Let $K$ be an abelian number field of conductor $n=\prod_{i=1}^t p_i^{e_i}$ with associated
character group $X$ and let $K_{2}$ be the field corresponding to $X_{2}$. Let $e=e_1=v_2(n)$ and $m=n/2^{e}$. Define $L$ to be the compositum
$K_{2}\mathbb{Q}^{(m)}$, i.e. the field corresponding to $X_{2} \times X^{(m)} \subseteq X^{(n)}$.
When the equivalent conditions of Proposition \ref{no2problems} do \emph{not} hold, the following statements are true:
\begin{enumerate}
\item $X_2$ is either $\langle \psi_{2^{e}} \rangle$ or $\langle \omega_{2} \psi_{2^{e}} \rangle$.
\item $L/K$ is tamely ramified.
\item $\mathbb{Q}^{(n)} = L[i]$, i.e. $\mathbb{Q}^{(n)}$ is the field generated by adjoining a root of $x^{2}+1$ to $L$.
\item $[\mathbb{Q}^{(n)}:L]=[L[i]:L]=2$.
\item $\mathbb{Q}^{(n)}/L$ is wildly ramified at the primes above 2.
\item $T_{L/K}(\mathcal{O}_{L}) = \mathcal{O}_{K}$.
\item $\mathcal{O}_{L}=\mathcal{O}_{K_{2}} \otimes_{\mathbb{Z}} \mathcal{O}^{(m)}$.
\item $I(\mathbb{Q}^{(n)}/L)=2^{r}$ for some $r \geq 1$.
\end{enumerate}
The situation is partially illustrated by the following field diagram.
$$
\xymatrix@1@!0@=48pt {
& & \mathbb{Q}^{(n)} \ar@{-}[d]_{\textrm{wild}}^{2} \ar@{=}[r] & \mathbb{Q}^{(2^{e})} \mathbb{Q}^{(m)} \ar@{=}[r] & L[i] \\
\mathbb{Q}^{(2^{e})} \ar@{-}[urr]^{\phi(m)} \ar@{-}[d]_{\textrm{wild}}^{2} & & L \ar@{-}[d]_{\textrm{tame}}
\ar@{=}[r] & K_{2} \mathbb{Q}^{(m)}\\
K_{2} \ar@{-}[urr]^{\phi(m)} \ar@{-}[d]^{2^{(e-2)}} & & K \\
\mathbb{Q} \ar@{-}[urr]
}
$$
\end{prop}
\begin{proof}
(a) This follows from part (b) of Proposition \ref{surjective} and the hypothesis that part (a) of Proposition \ref{no2problems} does not hold.\par
(b) Since $X_{2} \times X^{(m)} = \langle X, \prod_{i=2}^{t} \langle \omega_{p_{i}} \rangle \rangle$
(Proposition \ref{surjective}, part (c)), the result follows by noting that $L=KM$ in the proof of
Proposition \ref{tameaboveodd}.\par
(c) Since $\langle \omega_{2} \rangle$ corresponds to $\mathbb{Q}[i]$, this follows from part (a).\par
(d) $[\mathbb{Q}^{(n)}:L]=[X^{(n)}:X_{2} \times X^{(m)}] = [X^{(2^{e})}:X_{2}]=2$.\par
(e) This follows from part (b) and the hypothesis that part (c) of Proposition \ref{no2problems} does not hold (i.e. $\mathbb{Q}^{(n)}/K$ is wildly ramified). \par
(f) By Corollary \ref{wildtracecor}, this is equivalent to part (b). \par
(g) Since the discriminants of $\mathcal{O}_{K_{2}}$ and $\mathcal{O}^{(m)}$ are coprime, this follows from III.2.13 in \cite{ft}.\par
(h) This follows from part (e) and Theorem \ref{wildtrace}.
\end{proof}
\begin{prop}\label{ramcriteria}
Let $L/K$ be an extension of absolutely abelian number fields of equal conductor, $n$. Then each prime above an odd rational prime is tamely ramified in $L/K$. Furthermore, $L/K$ is wildly ramified at primes above $2$ if and only if:
\begin{enumerate}
\item the equivalent conditions of Proposition \ref{no2problems} applied to $L$ hold; and
\item the equivalent conditions of Proposition \ref{no2problems} applied to $K$ do \emph{not} hold.
\end{enumerate}
\end{prop}
\begin{proof}
Since $L/K$ is a sub-extension of $\mathbb{Q}^{(n)}/K$, the first statement follows from Proposition \ref{tameaboveodd}.
The second statement holds because wild ramification in $\mathbb{Q}^{(n)}/K$ can only occur in a degree $2$ sub-extension (Corollary \ref{2subext}), so $L/K$ is wildly ramified (at primes above 2)
if and only if $\mathbb{Q}^{(n)}/L$ is tamely ramified and $\mathbb{Q}^{(n)}/K$ is wildly ramified.
\end{proof}
\section{Abelian number fields of conductor $2^{e}$, $e \geq 3$}
In this section, let $e \geq 3$, let $\zeta$ denote a primitive $2^{e}$-th root of unity and let $i = \zeta^{2^{e-2}}$.
\begin{prop}\label{ro2i}
The cyclotomic field $\mathbb{Q}^{(2^{e})}$ has precisely two proper fields of conductor $2^{e}$:
\begin{enumerate}
\item $\mathbb{Q}(\zeta+\zeta^{-1})$, with ring of integers $\mathbb{Z}[\zeta+\zeta^{-1}]$; and
\item $\mathbb{Q}(i(\zeta+\zeta^{-1}))$, with ring of integers $\mathbb{Z}[i(\zeta+\zeta^{-1})]$.
\end{enumerate}
\end{prop}
\begin{proof}
Proposition \ref{surjective} part (b) implies that any proper subfield of $\mathbb{Q}^{(2^{e})}$ of conductor $2^{e}$ has associated character group either $\langle \psi_{2^{e}} \rangle$ or $\langle \omega_{2} \psi_{2^{e}} \rangle$. It is straightforward to check that these correspond to $\mathbb{Q}(\zeta+\zeta^{-1})$ and $\mathbb{Q}(i(\zeta+\zeta^{-1}))$. \par
The ring of integers of $\mathbb{Q}(\zeta+\zeta^{-1})$ is $\mathbb{Z}[\zeta+\zeta^{-1}]$ by Proposition 2.16 of \cite{wash}. A slightly modified version of this argument, keeping track of real and imaginary parts, shows that $\mathbb{Q}(i(\zeta+\zeta^{-1}))$ has ring of integers $\mathbb{Z}[i(\zeta+\zeta^{-1})]$.
\end{proof}
\begin{prop}\label{image2trace}
Let $K_{2}$ be a proper subfield of $\mathbb{Q}^{(2^{e})}$ of conductor $2^{e}$. Let $T=T_{\mathbb{Q}^{(2^{e})}/K_{2}}$.
In the cases of Proposition \ref{ro2i}, we have
\begin{enumerate}
\item $T(\mathbb{Z}[\zeta]) = 2\mathbb{Z} \oplus (\zeta+\zeta^{-1}) \cdot \mathcal{O}_{K_{2}} = 2\mathbb{Z} \oplus
(\zeta+\zeta^{-1}) \cdot \mathbb{Z}[\zeta+\zeta^{-1}] $; and
\item $T(\mathbb{Z}[\zeta]) = 2\mathbb{Z} \oplus i(\zeta+\zeta^{-1}) \cdot \mathcal{O}_{K_{2}} = 2\mathbb{Z} \oplus
i(\zeta+\zeta^{-1}) \cdot \mathbb{Z}[i(\zeta+\zeta^{-1})]$ .
\end{enumerate}
In both cases, $I(\mathbb{Q}^{(2^{e})}/K_{2})=2$.
\end{prop}
\begin{proof}
(a) In this case, $K_{2} = \mathbb{Q}(\zeta+\zeta^{-1})$, $\mathcal{O}_{K_{2}}= \mathbb{Z}[\zeta+\zeta^{-1}]$ and
$\{ 1, \zeta \}$ is a basis for $\mathbb{Q}^{(2^{e})}$ over $K_{2}$. The only non-trivial automorphism of
$\mathbb{Q}^{(2^{e})}$ over $K_{2}$ is induced by complex conjugation, and so for $a,b \in K_{2}$, we have
$$T(a+ \zeta b) = (a+\zeta b) + (a +\zeta^{-1}b) = 2a + (\zeta+\zeta^{-1})b \, .$$
Since $\mathbb{Z} + \zeta \cdot \mathbb{Z}[\zeta+\zeta^{-1}] \subseteq \mathbb{Z}[\zeta]$, we therefore have
$ 2\mathbb{Z} \oplus (\zeta+\zeta^{-1}) \cdot \mathbb{Z}[\zeta+\zeta^{-1}] \subseteq T(\mathbb{Z}[\zeta])$.
However, $\mathbb{Z}[\zeta+\zeta^{-1}] = \mathbb{Z} \oplus (\zeta+\zeta^{-1}) \cdot \mathbb{Z}[\zeta+\zeta^{-1}]$, so
$$\left[ \mathbb{Z}[\zeta+\zeta^{-1}]:2\mathbb{Z} \oplus (\zeta+\zeta^{-1}) \cdot \mathbb{Z}[\zeta+\zeta^{-1}] \right]=2$$
and by part (h) of Proposition \ref{some2problems},
$$\left[\mathbb{Z}[\zeta+\zeta^{-1}]:T(\mathbb{Z}[\zeta])\right]=2^{r}$$ for some $r \geq 1$.
Hence $2\mathbb{Z} \oplus (\zeta+\zeta^{-1}) \cdot \mathbb{Z}[\zeta+\zeta^{-1}] = T(\mathbb{Z}[\zeta])$ (and in fact $r=1$). \par
(b) In this case, $K_{2} = \mathbb{Q}(i(\zeta+\zeta^{-1}))$ and $\mathcal{O}_{K_{2}}= \mathbb{Z}[i(\zeta+\zeta^{-1})]$. The proof is essentially the same as in part (a), noting that $\{ 1 , i\zeta^{-1} = \zeta^{2^{(e-2)}-1}\}$ is a basis for $\mathbb{Q}^{(2^{e})}$ over $K_{2}$ and that the non-trivial Galois conjugate of $i\zeta^{-1}=\zeta^{2^{(e-2)}-1}$
over $K_{2}$ is $i\zeta=\zeta^{2^{(e-2)}+1}$.
\end{proof}
\begin{prop}\label{new2basis}
Consider the cases of Proposition \ref{ro2i}.
\begin{enumerate}
\item Let $A= \{ \zeta+\zeta^{-1}, \zeta^{2}+\zeta^{-2}, \ldots, \zeta^{2^{(e-2)}-1} + \zeta^{-2^{(e-2)}+1}\}$. \\ Then
$T(\mathbb{Z}[\zeta]) = \mathrm{Span}_{\mathbb{Z}}(A \cup \{2\})$, $\mathcal{O}_{K_{2}}=\mathrm{Span}_{\mathbb{Z}}(A \cup \{ 1 \})$
and $\mathrm{Gal}(K_{2}/\mathbb{Q})(A) \subseteq \pm A$.
\item Let $B= \{ i(\zeta+\zeta^{-1}), \zeta^{2}+\zeta^{-2}, i(\zeta^{3}+\zeta^{-3}), \ldots,
i(\zeta^{2^{(e-2)}-1} + \zeta^{-2^{(e-2)}+1})\}$. \\ Then
$T(\mathbb{Z}[\zeta]) = \mathrm{Span}_{\mathbb{Z}}(B \cup \{2\})$, $\mathcal{O}_{K_{2}}=\mathrm{Span}_{\mathbb{Z}}(B \cup \{ 1 \})$
and $\mathrm{Gal}(K_{2}/\mathbb{Q})(B) \subseteq \pm B$.
\end{enumerate}
\end{prop}
\begin{proof}
(a) $T(\mathbb{Z}[\zeta]) = \mathrm{Span}_{\mathbb{Z}}(A \cup \{2\})$ by Proposition \ref{image2trace} and a
straight-forward induction argument; that $\mathcal{O}_{K_{2}}=\mathrm{Span}_{\mathbb{Z}}(A \cup \{ 1 \})$
follows easily. For any $\sigma \in \mathrm{Gal}(K_{2}/\mathbb{Q})$ and any $j \in \{ 1, \ldots, 2^{(e-2)}-1 \} $,
$\sigma(\zeta^{j}+\zeta^{-j}) = \zeta^{jk}+\zeta^{-jk}$
for some $k \in (\mathbb{Z}/2^{e}\mathbb{Z})^{\times}$. However, any such $\zeta^{jk}+\zeta^{-jk}$ can be rewritten as
$\pm (\zeta^{r}+\zeta^{-r})$ for some $r \in \{ 1, \ldots, 2^{(e-2)}-1 \}$ (note $\zeta^{2^{(e-1)}}=-1$).
Part (b) is similar, noting that $\sigma(i) = \pm i$.
\end{proof}
\section{Computing $I(L/K)$}
\begin{prop}\label{ignoretame}
Let $L \subseteq M \subseteq N$ be a tower of Galois number fields such that $N/M$ is tamely ramified. Then $I(N/L) = I(M/L)$.
\end{prop}
\begin{proof}
Since $T_{N/L}(\mathcal{O}_{N}) = T_{M/L}(T_{N/M}(\mathcal{O}_{N}))$ and $T_{N/M}(\mathcal{O}_{N})=\mathcal{O}_{M}$ (by Corollary \ref{wildtracecor}), we have that
$T_{N/L}(\mathcal{O}_{N}) = T_{M/L}(\mathcal{O}_{M})$ and so the result follows from the definition of $I$.
\end{proof}
\begin{corollary}\label{ignoretamecor}
Let $L/K$ be a wildly ramified extension of absolutely abelian number fields of equal conductor, $n$.
Then $I(L/K) = I(\mathbb{Q}^{(n)}/K)$.
\end{corollary}
\begin{proof}
$\mathbb{Q}^{(n)}/L$ is tamely ramified since wild ramification in $\mathbb{Q}^{(n)}/K$ only occurs in a degree $2$ sub-extension (Corollary \ref{2subext}) and $L/K$ is wildly ramified.
\end{proof}
\begin{lemma}\label{tracetensor}
Let $K$ and $M$ be abelian number fields of conductors $k$ and $m$ respectively. Suppose that $k$ and $m$ are relatively prime. Then
$$ T_{\mathbb{Q}^{(k)}M/KM}(\mathcal{O}_{\mathbb{Q}^{(k)}M}) = T_{\mathbb{Q}^{(k)}/K}(\mathcal{O}^{(k)})
\otimes_{\mathbb{Z}} \mathcal{O}_{M} \, .$$
\end{lemma}
\begin{proof}
The proof is straightforward once one observes that by III.2.13 in \cite{ft}, we have
$\mathcal{O}_{KM}=\mathcal{O}_{K} \otimes_{\mathbb{Z}} \mathcal{O}_{M}$ and
$\mathcal{O}_{\mathbb{Q}^{(k)}M}=\mathcal{O}^{(k)} \otimes_{\mathbb{Z}} \mathcal{O}_{M}$.
\end{proof}
\begin{prop}\label{finalcompprop}
Let $K$ be an abelian number field of conductor $n$ such that $\mathbb{Q}^{(n)}/K$ is wildly ramified. Let $m=n/2^{e}$ where $e=v_{2}(n)$ and let $L=K_{2} \otimes_{\mathbb{Q}} \mathbb{Q}^{(m)} = K_{2}\mathbb{Q}^{(m)}$ (as in Proposition \ref{some2problems}). Let $C=A$ or $B$ from Proposition \ref{new2basis}, as appropriate. Define
$$ D = T_{L/K}(\mathcal{O}^{(m)}) \quad \textrm{and} \quad E = T_{L/K}( \mathrm{Span}_{\mathbb{Z}}(C) \otimes_{\mathbb{Z}} \mathcal{O}^{(m)} ) \, . $$
Then
$$\mathcal{O}_{K} = D \oplus E \quad \textrm{and} \quad T_{\mathbb{Q}^{(n)}/K}(\mathcal{O}^{(n)}) = 2D \oplus E \, .$$
\end{prop}
\begin{proof}
Note that $D \subseteq \mathcal{O}^{(m)} = \mathbb{Z} \otimes_{\mathbb{Z}} \mathcal{O}^{(m)}$ and
$E \subseteq \mathrm{Span}_{\mathbb{Z}}(C) \otimes_{\mathbb{Z}} \mathcal{O}^{(m)} $, with the last containment following from Proposition \ref{new2basis} (note $\mathrm{Gal}(L/K)(C) \subseteq \mathrm{Gal}(K_{2}/\mathbb{Q})(C) \subseteq \pm C$). Since $\mathbb{Z} \cap \mathrm{Span}_{\mathbb{Z}}(C) = \{ 0 \}$, we have $D \cap E = \{ 0 \}$, which gives the last equality of
\begin{eqnarray*}
\mathcal{O}_{K} &=& T_{L/K}(\mathcal{O}_{L}) \quad \textrm{(Proposition \ref{some2problems}, part (f))} \\
& = & T_{L/K}(\mathcal{O}_{K_{2}} \otimes_{\mathbb{Z}} \mathcal{O}^{(m)} )
\quad \textrm{(Proposition \ref{some2problems}, part (g))} \\
&=& T_{L/K}((\mathbb{Z} \oplus \mathrm{Span}_{\mathbb{Z}}(C)) \otimes_{\mathbb{Z}} \mathcal{O}^{(m)} )
\quad \textrm{(Proposition \ref{new2basis})} \\
& = & T_{L/K}((\mathbb{Z} \otimes_{\mathbb{Z}} \mathcal{O}^{(m)}) \oplus
(\mathrm{Span}_{\mathbb{Z}}(C) \otimes_{\mathbb{Z}} \mathcal{O}^{(m)})) \\
&=& D+E = D \oplus E \, .
\end{eqnarray*}
Furthermore,
\begin{eqnarray*}
T_{\mathbb{Q}^{(n)}/K}(\mathcal{O}^{(n)}) &=& T_{L/K}(T_{\mathbb{Q}^{(n)}/L}(\mathcal{O}^{(n)}))
= T_{L/K}(T_{\mathbb{Q}^{(n)}/L}(\mathcal{O}^{(2^{e})} \otimes_{\mathbb{Z}} \mathcal{O}^{(m)})) \\
&=& T_{L/K}(T_{\mathbb{Q}^{(2^{e})}/K_{2}}(\mathcal{O}^{(2^{e})}) \otimes_{\mathbb{Z}} \mathcal{O}^{(m)})
\quad \textrm{(Lemma \ref{tracetensor})} \\
&=& T_{L/K}((2\mathbb{Z} \oplus \mathrm{Span}_{\mathbb{Z}}(C)) \otimes_{\mathbb{Z}} \mathcal{O}^{(m)} )
\quad \textrm{(Proposition \ref{new2basis})} \\
&=& 2D \oplus E \quad \textrm{(as above)}.
\end{eqnarray*}
\end{proof}
\begin{remark}
The key point in this proof is the use of Proposition \ref{new2basis} to show that $D \cap E = \{ 0 \}$, and hence that the sums $D+E$ and $2D+E$ are direct.
\end{remark}
\begin{theorem}\label{finalcomp}
Under the hypotheses of Proposition \ref{finalcompprop}, we have
\begin{enumerate}
\item $\mathcal{O}_{K} = \mathcal{O}_{K \cap \mathbb{Q}^{(m)}}
\oplus T_{L/K}( \mathrm{Span}_{\mathbb{Z}}(C) \otimes_{\mathbb{Z}} \mathcal{O}^{(m)} )$;
\item $ T_{\mathbb{Q}^{(n)}/K}(\mathcal{O}^{(n)}) = 2\mathcal{O}_{K \cap \mathbb{Q}^{(m)}}
\oplus T_{L/K}( \mathrm{Span}_{\mathbb{Z}}(C) \otimes_{\mathbb{Z}} \mathcal{O}^{(m)} )$; and
\item $ I(\mathbb{Q}^{(n)}/K) = 2^{[K \cap \mathbb{Q}^{(m)}:\mathbb{Q}]} .$
\end{enumerate}
\end{theorem}
\begin{proof}
Note that $\mathcal{O}_{K \cap \mathbb{Q}^{(m)}} \subseteq \mathcal{O}^{(m)} = \mathbb{Z} \otimes_{\mathbb{Z}} \mathcal{O}^{(m)}$ and, as shown in Proposition \ref{finalcompprop},
$E \subseteq \mathrm{Span}_{\mathbb{Z}}(C) \otimes_{\mathbb{Z}} \mathcal{O}^{(m)} $. Since $\mathbb{Z} \cap \mathrm{Span}_{\mathbb{Z}}(C) = \{ 0 \}$, we have $ \mathcal{O}_{K \cap \mathbb{Q}^{(m)}} \cap E = \{ 0 \}$
(this is essentially the same argument as that used to show $D \cap E = \{ 0 \}$). Furthermore,
$D = T_{L/K}(\mathcal{O}^{(m)}) \subseteq \mathcal{O}_{K \cap \mathbb{Q}^{(m)}}$ and
$\mathcal{O}_{K \cap \mathbb{Q}^{(m)}} \subseteq \mathcal{O}_{K} = D \oplus E$, so $D = \mathcal{O}_{K \cap \mathbb{Q}^{(m)}}$. By Proposition \ref{finalcompprop}, this gives parts (a) and (b).
Now we have
\begin{eqnarray*}
I(\mathbb{Q}^{(n)}/K) &=& [\mathcal{O}_{K} : T_{\mathbb{Q}^{(n)}/K}(\mathcal{O}^{(n)})] =
[ \mathcal{O}_{K \cap \mathbb{Q}^{(m)}}\oplus E: 2\mathcal{O}_{K \cap \mathbb{Q}^{(m)}} \oplus E] \\
&=& [\mathcal{O}_{K \cap \mathbb{Q}^{(m)}}:2\mathcal{O}_{K \cap \mathbb{Q}^{(m)}}] = 2^{\mathrm{rank}_{\mathbb{Z}}(\mathcal{O}_{K \cap \mathbb{Q}^{(m)}})}
=2^{[K \cap \mathbb{Q}^{(m)}:\mathbb{Q}]},
\end{eqnarray*}
giving part (c).
\end{proof}
\begin{theorem}\label{mainresult}
Let $L/K$ be an extension of absolutely abelian number fields of equal conductor, $n$.
Let $e=v_{2}(n)$ and $m=n/2^{e}$. Then
$$
I(L/K) = \left\{
\begin{array}{ll}
2^{[K \cap \mathbb{Q}^{(m)}:\mathbb{Q}]} = 2^{( [K:\mathbb{Q}] / 2^{(e-2)} )} & \textrm{ if $L/K$ is wildly ramified,} \\
1 & \textrm{ otherwise.}
\end{array} \right.
$$
\end{theorem}
\begin{remark}
Recall that criteria for wild ramification of $L/K$ (which can only happen at primes above $2$) are given in Proposition \ref{ramcriteria}.
\end{remark}
\begin{proof}
Suppose $L/K$ is wildly ramified. Then $I(L/K)=I(\mathbb{Q}^{(n)}/K)$ by Corollary \ref{ignoretamecor} and
$I(\mathbb{Q}^{(n)}/K) = 2^{[K \cap \mathbb{Q}^{(m)}:\mathbb{Q}]}$ by Theorem \ref{finalcomp}. Noting that $[K_{2}:\mathbb{Q}]=2^{(e-2)}$
(see Proposition \ref{some2problems}, part (a)) and that $\mathbb{Q}^{(m)}K=\mathbb{Q}^{(m)}K_{2}$, we have
\begin{eqnarray*}
[K \cap \mathbb{Q}^{(m)}:\mathbb{Q}] = \frac{[\mathbb{Q}^{(m)}:\mathbb{Q}]}{[\mathbb{Q}^{(m)}:K \cap \mathbb{Q}^{(m)}]}
= \frac{[\mathbb{Q}^{(m)}:\mathbb{Q}]}{[\mathbb{Q}^{(m)}K:K]}
= \frac{[\mathbb{Q}^{(m)}:\mathbb{Q}]}{[\mathbb{Q}^{(m)}K_{2}:K]} \\
= \frac{[\mathbb{Q}^{(m)}:\mathbb{Q}][K:\mathbb{Q}]}{[\mathbb{Q}^{(m)}K_{2}:\mathbb{Q}]}
= \frac{[\mathbb{Q}^{(m)}:\mathbb{Q}][K:\mathbb{Q}]}{[\mathbb{Q}^{(m)}:\mathbb{Q}][K_{2}:\mathbb{Q}]}
= \frac{[K:\mathbb{Q}]}{[K_{2}:\mathbb{Q}]}
= \frac{[K:\mathbb{Q}]}{2^{(e-2)}} \, .
\end{eqnarray*}
In the case where $L/K$ is tamely ramified, the result follows from Corollary \ref{wildtracecor}.
\end{proof}
\begin{remark}
It is clear that Theorem \ref{mainresult} agrees with the expressions for $I(L/K)$ in \cite{girstmair} (where $K \cap \mathbb{Q}^{(m)}$ is denoted $K_{n/2^{e}}$), and is in fact a sharpening of these results since an exact value for $I(L/K)$ is given for \emph{any} extension of abelian number fields $L/K$ of equal conductor. Furthermore, the above result does not rely on Leopoldt's Theorem.
\end{remark}
\section{The Adjusted Trace Map}
\begin{definition}\label{adjustdef}
Let $K$ be an abelian number field of conductor $n$. We define the ``adjusted trace map'', $\hat{T}_{\mathbb{Q}^{(n)}/K}$.
If $\mathbb{Q}^{(n)}/K$ is tamely ramified, let $\hat{T}_{\mathbb{Q}^{(n)}/K} = T_{\mathbb{Q}^{(n)}/K}$. Otherwise, let $m=n/2^{e}$ where $e=v_{2}(n)$ (recall that $e \geq 3$ in this case). Note that
$\mathcal{O}^{(n)}=\mathcal{O}^{(2^{e})} \otimes_{\mathbb{Z}} \mathcal{O}^{(m)}$ has $\mathbb{Z}$-basis
$\{ \zeta_{2^{e}}^{i} \otimes \zeta_{m}^{j} \, \mid \, 0 \leq i \leq 2^{(e-1)}-1, \, 0 \leq j \leq \phi(m)-1 \}$. Define
$$
\hat{T}_{\mathbb{Q}^{(n)}/K}(\zeta_{2^{e}}^{i} \otimes \zeta_{m}^{j}) = \left\{
\begin{array}{ll}
\frac{1}{2} T_{\mathbb{Q}^{(n)}/K}(\zeta_{m}^{j}) & \textrm{ for } i=0, \\
T_{\mathbb{Q}^{(n)}/K}(\zeta_{2^{e}}^{i} \otimes \zeta_{m}^{j}) & \textrm{ for } 1 \leq i \leq 2^{(e-1)}-1,
\end{array} \right. $$
and extend to a $\mathbb{Q}$-linear map $\mathbb{Q}^{(n)} \rightarrow K$.
\end{definition}
\begin{prop}\label{adjust_surjective}
$\hat{T}_{\mathbb{Q}^{(n)}/K}(\mathcal{O}^{(n)}) = \mathcal{O}_{K}$.
\end{prop}
\begin{proof}
If $\mathbb{Q}^{(n)}/K$ is tamely ramified, this is just Corollary \ref{wildtracecor}. Otherwise,
using the notation of Proposition \ref{finalcompprop}, we see that
$$
\hat{T}_{\mathbb{Q}^{(n)}/K}(\alpha) = \left\{
\begin{array}{ll}
\frac{1}{2} T_{\mathbb{Q}^{(n)}/K}(\alpha) & \textrm{ if } T_{\mathbb{Q}^{(n)}/K}(\alpha) \in D, \\
T_{\mathbb{Q}^{(n)}/K}(\alpha) & \textrm{ if } T_{\mathbb{Q}^{(n)}/K}(\alpha) \in E.
\end{array} \right.
$$
The result now follows immediately from Proposition \ref{finalcompprop}.
\end{proof}
\begin{remark}
It must be noted that the adjusted trace map of Definition \ref{adjustdef} is in fact equivalent to the
map defined in Lemma 3.4 of \cite{lux-pahlings} (page 51), though it is expressed more explicitly here. Furthermore, it is shown to be surjective in \cite{breuer}. However,
the proof given here (Proposition \ref{adjust_surjective}) is very different.
\end{remark}
\begin{lemma}\label{galinvariant}
Let
$\hat{T}_{\mathbb{Q}^{(n)}/K}(\zeta_{n}^{k}) = \varepsilon T_{\mathbb{Q}^{(n)}/K}(\zeta_{n}^{k})$ where $\varepsilon = 1/2$ or $1$. Then
$$\hat{T}_{\mathbb{Q}^{(n)}/K}(\sigma(\zeta_{n}^{k})) = \varepsilon T_{\mathbb{Q}^{(n)}/K}(\sigma(\zeta_{n}^{k})) \quad \forall \sigma \in \mathrm{Gal}(\mathbb{Q}^{(n)}/\mathbb{Q}).$$
\end{lemma}
\begin{proof}
Write $\zeta_{n}^{k} = \zeta_{2^{e}}^{i} \otimes \zeta_{m}^{j}$ and use Definition \ref{adjustdef}.
\end{proof}
\begin{definition}
Let $L /K$ be a finite Galois extension with $G=\mathrm{Gal}(L / K)$.
Then $$\mathcal{A}_{L/K} := \{ \gamma \in K[G] \, | \, \gamma(\mathcal{O}_L) \subseteq \mathcal{O}_L \}$$
is the \emph{associated order} of $L / K$.
\end{definition}
The following is a modified version of Lemma 6 in \cite{byott_lettl}. Note that we use both juxtaposition and the symbol
$\cdot$ to denote the action of a group algebra on a field.
\begin{theorem}
Let $K$ be an abelian number field of conductor $n$, and put $G = \mathrm{Gal}(\mathbb{Q}^{(n)}/\mathbb{Q})$,
$H= \mathrm{Gal}(\mathbb{Q}^{(n)}/K)$. Let $\pi: \mathbb{Q}[G] \rightarrow \mathbb{Q}[G/H]$ denote the $\mathbb{Q}$-linear map induced by the natural projection $G \rightarrow G/H$.
Suppose $\mathcal{O}^{(n)}= \mathcal{A}_{\mathbb{Q}^{(n)}/\mathbb{Q}} \cdot \alpha$ for some $\alpha \in \mathcal{O}^{(n)}$. Then
$\mathcal{A}_{K/\mathbb{Q}}= \pi(\mathcal{A}_{\mathbb{Q}^{(n)}/\mathbb{Q}})$ and $\mathcal{O}_{K}=\mathcal{A}_{K/\mathbb{Q}} \cdot \beta$
where $\beta = \hat{T}_{\mathbb{Q}^{(n)}/K}(\alpha)$.
\end{theorem}
\begin{proof}
Write $G= \{ g_{1}, \ldots, g_{r} \}$ and $H = \{ h_{1}, \ldots, h_{s}\}$.
Let $x \in \mathcal{A}_{\mathbb{Q}^{(n)}/\mathbb{Q}}$ and write
\begin{eqnarray*}
x &=& x_{1}g_{1} + \ldots + x_{r}g_{r} \textrm{ where } x_{i} \in \mathbb{Q} \textrm{ and }g_{i} \in G,\\
\alpha &=& y_{1} + y_{2}\zeta + \ldots + y_{r}\zeta^{r-1} \textrm{ where } y_{i} \in \mathbb{Q} \textrm{ and } \zeta=\zeta_{n}.
\end{eqnarray*}
Then using Lemma \ref{galinvariant}, the $\mathbb{Q}$-linearity of $\hat{T}_{\mathbb{Q}^{(n)}/K}$ and that $G$ is abelian, we have
\begin{eqnarray*}
\hat{T}_{\mathbb{Q}^{(n)}/K}(x \alpha)
&=& \sum_{i=1}^{r} x_{i} \hat{T}_{\mathbb{Q}^{(n)}/K}(g_{i} \alpha )
= \sum_{i=1}^{r} x_{i} \sum_{j=1}^{r} y_{j} \hat{T}_{\mathbb{Q}^{(n)}/K}(g_{i} \zeta^{j-1}) \\
&=& \sum_{i=1}^{r} x_{i} \sum_{j=1}^{r} y_{j} \varepsilon_{j} T_{\mathbb{Q}^{(n)}/K}(g_{i} \zeta^{j-1})
= \sum_{i=1}^{r} x_{i} \sum_{j=1}^{r} y_{j} \varepsilon_{j} \sum_{k=1}^{s} h_{k}g_{i}\zeta^{j-1} \\
&=& \sum_{i=1}^{r} x_{i} g_{i} \sum_{j=1}^{r} y_{j} \varepsilon_{j} \sum_{k=1}^{s} h_{k}\zeta^{j-1}
= \sum_{i=1}^{r} x_{i} g_{i} \sum_{j=1}^{r} y_{j} \varepsilon_{j} T_{\mathbb{Q}^{(n)}/K}(\zeta^{j-1}) \\
&=& \sum_{i=1}^{r} x_{i} g_{i} \sum_{j=1}^{r} y_{j} \hat{T}_{\mathbb{Q}^{(n)}/K}(\zeta^{j-1})
= \sum_{i=1}^{r} x_{i} g_{i} \hat{T}_{\mathbb{Q}^{(n)}/K}(\alpha) \\
&=& x \hat{T}_{\mathbb{Q}^{(n)}/K}(\alpha)
\end{eqnarray*}
where $\varepsilon_{j}=1/2$ or $1$, as appropriate. Thus
\begin{eqnarray*}
\mathcal{O}_{K} &=& \hat{T}_{\mathbb{Q}^{(n)}/K}(\mathcal{O}^{(n)})
\quad \textrm{(Proposition \ref{adjust_surjective})} \\
&=& \hat{T}_{\mathbb{Q}^{(n)}/K}( \mathcal{A}_{\mathbb{Q}^{(n)}/\mathbb{Q}} \cdot \alpha)
= \mathcal{A}_{\mathbb{Q}^{(n)}/\mathbb{Q}} \cdot \hat{T}_{\mathbb{Q}^{(n)}/K}(\alpha) \\
&=& \pi(\mathcal{A}_{\mathbb{Q}^{(n)}/\mathbb{Q}}) \cdot \beta
\quad \textrm{ (since } \beta \in K \textrm{).}
\end{eqnarray*}
\end{proof}
\begin{remark}
Unfortunately, this result cannot be easily extended to the case of relative extensions because
$\hat{T}_{\mathbb{Q}^{(n)}/K}$ is not $K$-linear for $K \neq \mathbb{Q}$.
\end{remark}
\begin{corollary}\label{reducetocyclotomic}
The proof of Leopoldt's Theorem can be reduced to the cyclotomic case.
\end{corollary}
We can now restate Leopoldt's Theorem (see \cite{leopoldt}, \cite{lettl}) with the generator expressed as the image of an element under the adjusted trace map.
\begin{definition}
For $n \in \mathbb{N}$, define the \emph{radical} of $n$ to be $r(n) = \prod_{p|n} p$. \end{definition}
\begin{definition}
For $n \in \mathbb{N}$, define $\mathcal{D}(n) = \{ d \in \mathbb{N} : r(n) | d \textrm{ and } d | n \}$.
\end{definition}
\begin{theorem}[Leopoldt]
Let $K$ be an abelian number field of conductor $n$, let $\zeta_{n}$ be a fixed primitive $n^{\mathrm{th}}$ root of unity, and let
$$\alpha = \hat{T}_{\mathbb{Q}^{(n)}/K} \left( \sum_{d \in \mathcal{D}(n)} \zeta_{n}^{(n/d)} \right) \, .$$
Then we have $\mathcal{O}_K=\mathcal{A}_{K/\mathbb{Q}} \cdot \alpha$, and so $\mathcal{O}_K$
is a free $\mathcal{A}_K$-module of rank $1$.
\end{theorem}
\begin{proof}
By Corollary \ref{reducetocyclotomic}, the proof is reduced to the cyclotomic case, which is relatively straightforward.
\end{proof}
\begin{remark}
In particular, the cyclotomic case follows from the version of Leopoldt's Theorem given in \cite{lettl}.
\end{remark}
\begin{remark}
The definition of $\mathcal{D}(n)$ in \cite{lettl} is different from that given above. However, as noted in
\cite{lettl2}, Leopoldt's Theorem holds in either case. A routine computation shows that when $\mathcal{D}(n)$ is taken to be as in \cite{lettl}, $\alpha$ as defined above is equal to $T$
defined in \cite{lettl}.
\end{remark}
\section{Acknowledgments}
The author is grateful to Steven Chase, Ravi Ramakrishna, and Shankar Sen for useful conversations, and to Kurt Girstmair, Spencer Hamblen, and Jason Martin for looking at an initial draft of this paper. The computational algebra system Magma (\cite{magma}) was used to verify Theorem \ref{mainresult} for abelian number fields of conductor up to $176$. The positive results from this ``experiment'' were psychologically very helpful in proving the theorem.
\end{document}
|
math
|
// -*- C++ -*-
//# VisImagingWeight.cc: Implementation of the VisImagingWeight class
//# Copyright (C) 1997,1998,1999,2000,2001,2002,2003
//# Associated Universities, Inc. Washington DC, USA.
//#
//# This library is free software; you can redistribute it and/or modify it
//# under the terms of the GNU Library General Public License as published by
//# the Free Software Foundation; either version 2 of the License, or (at your
//# option) any later version.
//#
//# This library 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 Library General Public
//# License for more details.
//#
//# You should have received a copy of the GNU Library General Public License
//# along with this library; if not, write to the Free Software Foundation,
//# Inc., 675 Massachusetts Ave, Cambridge, MA 02139, USA.
//#
//# Correspondence concerning AIPS++ should be addressed as follows:
//# Internet email: aips2-request@nrao.edu.
//# Postal address: AIPS++ Project Office
//# National Radio Astronomy Observatory
//# 520 Edgemont Road
//# Charlottesville, VA 22903-2475 USA
//#
//# $Id$
#include <lofar_config.h>
#include <AWImager2/VisibilityIterator.h>
#include <AWImager2/VisImagingWeight.h>
namespace LOFAR {
namespace LofarFT {
VisibilityIterator::VisibilityIterator() : casa::VisibilityIterator() {};
VisibilityIterator::VisibilityIterator(
casa::MeasurementSet & ms,
const casa::Block<casa::Int>& sortColumns,
casa::Double timeInterval)
:
casa::VisibilityIterator(ms, sortColumns, timeInterval),
lofar_imwgt_p(0)
{};
VisibilityIterator &
VisibilityIterator::operator= (const VisibilityIterator & other)
{
// Let the superclass handle its part of the assignment
VisibilityIterator::operator= (other);
lofar_imwgt_p = other.lofar_imwgt_p;
return * this;
}
void
VisibilityIterator::useImagingWeight (casa::CountedPtr<VisImagingWeight> imWgt)
{
lofar_imwgt_p = imWgt;
}
const VisImagingWeight &
VisibilityIterator::getImagingWeightGenerator () const
{
return *lofar_imwgt_p;
}
} // end namespace LofarFT
} // end namespace LOFAR
|
code
|
<?php
require_once('extension/ez_aya_sit/classes/sit_utils.class.php');
$http = eZHTTPTool::instance();
$sitIni = eZINI::instance('ez_aya_sit.ini');
$cheminCacheImages = $sitIni->variable('GlobalSitParameters','ImagesCachePath');
$cheminCacheImages = $_SERVER['DOCUMENT_ROOT']."/".$cheminCacheImages;
if (!file_exists($cheminCacheImages."thumbs/")) {
mkdir($cheminCacheImages."thumbs/", 0777, true);
}
$dureeVieCache = $sitIni->variable('GlobalSitParameters','DureeVieCacheImages');
$fileName = "";
if ($http->hasVariable('img')) {
$fileName = urldecode($http->variable('img'));
}
$width = "";
if ($http->hasVariable('w')) {
$width = urldecode($http->variable('w'));
}
$minWidth = "";
if ($http->hasVariable('mw')) {
$minWidth = urldecode($http->variable('mw'));
}
$fileNameCrypted = sha1($fileName);
$resizedFileNameCrypted = sha1($width."_".$minWidth."_".$fileName);
$cheminFichierCacheImages = $cheminCacheImages.$fileNameCrypted;
$cheminFichierCacheImagesResized = $cheminCacheImages."thumbs/".$resizedFileNameCrypted;
$contenuImagesDistant = false;
$fs = false;
if (file_exists($cheminFichierCacheImagesResized)) {
$fs = stat($cheminFichierCacheImagesResized);
}
$requestHeaders = getallheaders();
$imageType = "";
if (!file_exists($cheminFichierCacheImagesResized) || ($fs && time() > $fs['mtime'] + $dureeVieCache)) {
$contenuImagesDistant = SitUtils::urlGetContentsCurlCustom($fileName, 5, file_exists($cheminFichierCacheImages) ? filemtime($cheminFichierCacheImages) : 60);
if ($contenuImagesDistant) {
file_put_contents($cheminFichierCacheImages, $contenuImagesDistant, LOCK_EX);
}
if (!file_exists($cheminFichierCacheImages)) {
$cheminFichierCacheImages = $_SERVER['DOCUMENT_ROOT']."/extension/ez_aya_sit/design/standard/images/sit/image_fiche_defaut_grande.jpg";
}
//if (($contenuImagesDistant || !array_key_exists('If-Modified-Since', $requestHeaders) || (array_key_exists('Pragma', $requestHeaders) && $requestHeaders['Pragma'] == "no-cache" && array_key_exists('Cache-Control', $requestHeaders) && $requestHeaders['Cache-Control'] == "no-cache"))) {
eZLog::write("Image ".$fileName." appelée", "ez_aya_sit.log");
$imageInfo = getimagesize($cheminFichierCacheImages);
$imageType = $imageInfo[2];
$image = false;
if ( $imageType == IMAGETYPE_JPEG ) {
$image = imagecreatefromjpeg($cheminFichierCacheImages);
} elseif ( $imageType == IMAGETYPE_GIF ) {
$image = imagecreatefromgif($cheminFichierCacheImages);
} elseif ( $imageType == IMAGETYPE_PNG ) {
$image = imagecreatefrompng($cheminFichierCacheImages);
}
if ($image && (!$minWidth || $width < imagesx($image))) {
$ratio = $width / imagesx($image);
$height = imagesy($image) * $ratio;
$newImage = imagecreatetruecolor($width, $height);
imagecopyresampled($newImage, $image, 0, 0, 0, 0, $width, $height, imagesx($image), imagesy($image));
$image = $newImage;
}
if ( $imageType == IMAGETYPE_JPEG ) {
imagejpeg($image, $cheminFichierCacheImagesResized, 90);
} elseif ( $imageType == IMAGETYPE_GIF ) {
imagegif($image, $cheminFichierCacheImagesResized);
} elseif ( $imageType == IMAGETYPE_PNG ) {
imagepng($image, $cheminFichierCacheImagesResized);
}
/*} else {
$cheminFichierCacheImagesResized = false;
}*/
}
/*header("HTTP/1.1 ".($contenuImagesDistant || !array_key_exists('If-Modified-Since', $requestHeaders) || (array_key_exists('Pragma', $requestHeaders) && $requestHeaders['Pragma'] == "no-cache" && array_key_exists('Cache-Control', $requestHeaders) && $requestHeaders['Cache-Control'] == "no-cache") ? "200 OK" : "304 Not Modified"));*/
header("Content-type: image".($imageType == IMAGETYPE_JPEG ? "/jpeg" : ($imageType == IMAGETYPE_GIF ? "/gif" : ($imageType == IMAGETYPE_PNG ? "/png" : ""))));
/*$dateExpirationCacheGMT = filemtime($cheminFichierCacheImages) + $dureeVieCache - ((int)substr(date('O'), 0, 3) * 3600);
header("Expires: ".gmdate("D, d M Y H:i:s", $dateExpirationCacheGMT)." GMT");*/
$dateDerniereModifCacheGMT = filemtime($cheminFichierCacheImages) - ((int)substr(date('O'), 0, 3) * 3600);
header("Last-Modified: ".gmdate("D, d M Y H:i:s", $dateDerniereModifCacheGMT)." GMT");
/*$fs = stat($cheminFichierCacheImages);
header("Etag: ".sprintf('"%x-%x-%s"', $fs['ino'], $fs['size'],base_convert(str_pad($fs['mtime'],16,"0"),10,16)));*/
header_remove("X-Powered-By");
header_remove("Set-Cookie");
header_remove("Served-by");
header_remove("Transfer-Encoding");
header_remove("Content-Language");
header_remove("Pragma");
header_remove("Cache-Control");
if ($cheminFichierCacheImagesResized) {
echo file_get_contents($cheminFichierCacheImagesResized);
}
eZExecution::cleanExit();
?>
|
code
|
मुफ्त डाउनलोड - वॉलपेपर क्रिमिया ३०३५३२
मुफ्त डाउनलोड - वॉलपेपर क्रिमिया, फ़्री-वॉलपेपर.ऑर्ग - सुंदर मुफ्त डेस्कटॉप वॉलपेपर डाउनलोड. ३०३५३२
सुंदर मुफ्त वॉलपेपर डेस्कटॉप - सुंदर मुफ्त वॉलपेपर और पृष्ठभूमि. प्रवीणता का सपना, सुंदर वॉलपेपर डेस्कटॉप. सुंदर मुक्त वॉलपेपर हद वॉलपेपर प्रकृति.
हमारे खूबसूरत ग्रह समझ से बाहर किस्म के द्वारा कल्पना आश्चर्य होता है. छुट्टियों के सभी समावेशी. सस्ते छुट्टी विचारों. रिजर्व - भूमि क्षेत्र और जल क्षेत्र है, जो अपनी प्राकृतिक अवस्था में संरक्षित है, पूरे प्राकृतिक जटिल है, और जहां शिकार और मछली पकड़ने निषिद्ध कर रहे हैं. क्रीमिया प्रायद्वीप खजाना निधि है कि प्रकृति में ही भरता है. मुझे सर्दियां पसंद हैं. प्रकृति का उपहार. वॉलपेपर ३०३५३२: ऐ-पेट्री. सबसे बड़ा यात्रा अनुभव. बर्फ की सुंदरता. परिदृश्य फोटोग्राफी. सस्ती छुट्टियों. इस तरह एक जगह, पूरी दुनिया में नहीं मिल रहा है. याल्टा. क्रीमिया के हर कोने सचमुच साँस लेता इतिहास, किंवदंतियों और परंपराओं. जनवरी. मिस्खोर. सभी समावेशी परिवार सैरगाह. आस दोनों बच्चों और वयस्कों के लिए समान रूप से उपयोगी होते हैं. पर्यटकों के आकर्षण. ताजा ठंडी हवा न केवल उत्थान है, लेकिन यह भी प्रतिरक्षा प्रणाली को मजबूत और भूख बढ़ जाती है और अनिद्रा से राहत मिलती है. लंबी पैदल यात्रा कि छुट्टी. ज्ञान ही शक्ति है. प्रकृति वर्ष के किसी भी समय में आश्चर्यजनक है. लघु सप्ताहांत. प्रकृति वॉलपेपर पृष्ठभूमि. प्रकृति वॉलपेपर पूर्ण हद.
ज्ञान में निवेश का सबसे अच्छा रिटर्न है. जीवन भर के यात्रा के अनुभव. जब बर्फ प्रकृति सुनता गिर जाता है. तौरिका क्रीमिया के शास्त्रीय नाम है. सस्ते छुट्टियों. शिक्षा प्रत्येक व्यक्ति के जीवन में सफलता के लिए महत्वपूर्ण कुंजी है. हर किसी को एक अविस्मरणीय छुट्टी प्यार करता है. स्वास्थ्य सबसे महत्वपूर्ण मानव धन है. एक वर्ष में धूप दिन, समुद्र तटीय समुद्र तटों और पहाड़ के जंगलों, विविध वनस्पति, नदी घाटियों, घाटियों और झरने, सांस्कृतिक स्मारकों की एक बहुत की बहुतायत - यह सब वास्तव में क्रीमिया की एक अमूल्य धन है. भ्रमण की मदद से हम दुनिया के बारे में जानने के लिए. अद्भुत प्रकृति की महिमा, उसके बनाए और असाधारण विचारों अद्भुत और कमाल कर रहे हैं. यात्रा में, जिस तरह से हम सोचते हैं, बैठ वाला, लक्षण, महत्वपूर्ण परिवर्तन से गुजरना कर सकते हैं. जहां एक दिन के लिए जाने के लिए एक जीवन भर के लिए याद करने के लिए? कैसे उपयोगी अपने ख़ाली समय खर्च करने के लिए. अकेले लेने के लिए सर्वश्रेष्ठ छुट्टियां.
एक जीवन भर की यात्राएं जो धरती पर खर्च नहीं करती हैं. झरने, अप्रत्याशित रूपों, चोटियों और रहस्यमय गुफाओं के परिदृश्य: प्रकृति प्राकृतिक मूल के क्रीमिया जगहें संपन्न किया है. डिजिटल फोटोग्राफी पर्यटन - कला फोटोग्राफी दीर्घाओं. डिजिटल ऑनलाइन मीडिया, वेब आर्ट गैलरी - डिजिटल फोटोग्राफी ३०३५३२: यात्रा अनुभव कहानियां. तौरीदा प्रशासनिक रूसी साम्राज्य का एक हिस्सा था. ऐसी प्रकृति आप कहीं और नहीं देखेंगे, यह न केवल आंख, बल्कि आत्मा आकर्षक है.
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\begin{document}
\title{{\Large\bf{A new class of differential quasivariational inequalities with an application to a quasistatic viscoelastic frictional contact problem}}\thanks{This work was supported by the National Natural Science Foundation of China (11671282, 11771067, 12171339, 12171070).}}
\author{Xu Chu$^a$, Tao Chen$^a$, Nan-jing Huang$^a$\footnote{Corresponding author. E-mail address: nanjinghuang@hotmail.com; njhuang@scu.edu.cn}
and Yi-bin Xiao$^b$
\\{\small a. Department of Mathematics, Sichuan University, Chengdu, Sichuan, P.R. China}\\
{ \small b. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, P.R. China}}
\date{ }
\maketitle
\begin{flushleft}
\hrulefill\\
\end{flushleft}
{\bf Abstract}: The overarching goal of this paper is to introduce and investigate a new nonlinear system driven by a nonlinear differential equation, a history-dependent quasivariational inequality, and a parabolic variational inequality in Banach spaces. Such a system can be used to model quasistatic frictional contact problems for viscoelastic materials with long memory, damage and wear. By using the Banach fixed point theorem, we prove an existence and uniqueness theorem of solution for such a system under some mild conditions. As a novel application, we obtain a unique solvability of a quasistatic viscoelastic frictional contact problem with long memory, damage and wear.
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{\bf Keywords}: Differential quasivariational inequality; frictional contact problem; viscoelastic materials; long memory; damage and wear.
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\textbf{2020 AMS Subject Classification:} {49J40; 34G20; 74M10; 74M15}
\section{Introduction}
Assume that $V, X, Y$ and $W$ are separable and reflexive Banach spaces, $V^*$ and $Y^*$ are the dual spaces of $V$ and $Y$, respectively. Let $Y\subset Y_1\subset Y^*$, where $Y_1$ is a separable Hilbert space. Suppose that $K_V$ and $K_Y$ are closed, convex and nonempty subsets of $V$ and $Y$, respectively. Let $I:=[0,T]$, where $T>0$ is a constant. In this paper, we are interested in studying a new class of differential quasivariational inequalities (DQVIs) with the following form: find $u:I\rightarrow K_V$, $\zeta:I\rightarrow K_Y$ and $w:I\rightarrow W$ such that, for all $t\in I$,
\begin{equation}\label{1.1}
\left\{\begin{array}{l}
\dot{w}(t)=F(t,w(t),\dot{u}(t)),\\
\left\langle A(t,u(t)) +\int_0^t B(t-s,u(s),\zeta(s))ds+C(t,\dot{u}(t)),v-\dot{u}(t) \right\rangle_{V^*\times V}\\
\quad \mbox{}+j(w(t) ,\dot{u}(t),v)-j(w(t) ,\dot{u}(t) ,\dot{u}(t))\geq \langle f(t) ,v-\dot{u}(t) \rangle_{V^*\times V},\quad\forall v\in K_V ,\\
\langle \dot{\zeta}(t) ,\eta-\zeta(t) \rangle_{Y_1}+a(\zeta(t) ,\eta-\zeta(t) )\geq\langle\phi(t,u(t) ,\zeta(t) ),\eta-\zeta(t) \rangle_{Y_1},\quad\forall\eta\in K_Y,\\
u(0)=u_0, \;w(0)=w_0, \; \zeta(0)=\zeta_0.\\
\end{array}\right.
\end{equation}
Clearly, if $C,w$ and $\zeta$ are omitted, $K_V=V$, $A(t,u(t))=A(u(t))$ and $j(w(t) ,\dot{u}(t),v)=j({u}(t),v)$, then \eqref{1.1} is reduced to the following problem: find $u:I\rightarrow V$ such that, for all $t\in I$,
\begin{equation*}
\left\{\begin{array}{l}
\left\langle A(u(t)) +\int_0^t B(t-s,u(s))ds,v-\dot{u}(t) \right\rangle_{V^*\times V}\\
\quad+j(u(t),v)-j(u ,\dot{u}(t))\geq \langle f(t) ,v-\dot{u}(t) \rangle_{V^*\times V},\quad\forall v\in V ,\\
u(0)=u_0,\\
\end{array}\right.
\end{equation*}
which is the time-dependent quasivariational inequality with the history-dependent operator considered by Kasri and Touzaline \cite{A2018}.
The research on differential variational inequalities (DVIs) has a long history (see, for example, the excellent survey due to Brogliato and Tanwani \cite{Brog2020} and the references therein). It is well known that Pang and Stewart \cite{Pang2008} are the first systematically to consider DVIs in finite-dimensional Euclidean spaces. Since then, various theoretical results, approximating algorithms and real applications have been investigated extensively for classical DVIs and DQVIs under different conditions in the literature; for instance we refer the reader to \cite{Liu2018,Liu2021,Liu2021+,Weng2021,Weng2021+,Zeng2018,ChenX2014,Gwinner2013,LiXS2010,LiW2015,LiW2017,Wang2017,Zeng2021,Zeng2021+} and the references therein.
It is worth mentioning that contact mechanics has important applications in daily life and industry, such as coupling devices, bearings, ultrasonic welding and many others. After nearly 40 years of research, contact mechanics has formed a relatively complete set of mathematical theory. It is well known that the variational inequalities and quasivariational inequalities as well as hemivariational inequalities are the most important mathematical tools to obtain the existence and uniqueness of the solutions for various problems arising in contact mechanics \cite{Capatina2014,Gasi2015,Han2001,AS2007,MOS2015,Sofonea2012,Kulig2018, T1999,Fr1996,Fr1995}.
In order to describe the long memory property of materials such as rock and rubber, Sofonea and Matei \cite{Sofonea2011} introduced and studied a new class of history-dependent quasivariational inequalities. Moreover, by relaxing the contact condition in \cite{Sofonea2011}, Sofonea and Xiao \cite{Sofonea2016} considered a new class of quasivariational inequalities involving two history-dependent operators. On the other hand, applying Archard's wear laws and Coulomb's friction laws, Andrews et al. \cite{KT1997} obtained the existence and uniqueness of the solution of a dynamic thermoviscoelastic contact problem. Furthermore, Sofonea et al. \cite{SofoneaF2016} proposed a new mathematical model to capture frictional contact problem with wear. Recently, Chen et al. \cite{Chen2020} introduced a hyperbolic quasi-variational inequality to characterize a dynamic viscoelastic contact problem with friction and wear. Very recently, in order to model an elastic frictional contact problem with long memory, damage and wear, Chen et al. \cite{Chen2021} considered a new class of differential nonlinear system driven by a differential equation, a history-dependent hemivariational inequality and a parabolic variational inequality in Banach spaces. Nevertheless, in the study of quasistatic viscoelastic frictional contact problems, it is necessary to consider the properties of viscoelastic materials with long memory, damage and wear (see Section 4 for more details). To the best of authors' knowledge, there are few works considering quasistatic frictional contact problems for viscoelastic materials with long memory, damage and wear. Thus, it would be important and interesting to investigate DQVIs \eqref{1.1} which can be used to model quasistatic frictional contact problems for viscoelastic materials with long memory, damage and wear.
The overarching goal of this paper is to introduce and investigate a new nonlinear system driven by a nonlinear differential equation, a history-dependent quasivariational inequality, and a parabolic variational inequality in Banach spaces, which can be used to describe the quasistatic frictional contact problems for viscoelastic materials with long memory, damage and wear. The main contributions of this paper are twofold. One is to deliver some sufficient conditions for ensuring the existence and uniqueness of solution to DQVIs \eqref{1.1}. The other is to show the unique solvability result for a new quasistatic viscoelastic frictional contact problem with long memory, damage and wear by applying the obtained results for DQVIs \eqref{1.1}.
The rest of the paper is structured as follows. The next section recalls some known definitions and lemmas. After that in Section 3, we show the existence and uniqueness of solution for DQVIs \eqref{1.1} under some mild conditions by employing the Banach fixed point theorem. In Section 4, we provide a novel application of our abstract results to a quasistatic viscoelastic frictional contact problem with long memory, damage and wear.
\section{Preliminaries}
Let $(X,\|\cdot\|_X)$ be a real Banach space with its dual $X^*$ and $\langle\cdot,\cdot\rangle_{X^*\times X}$ denote the duality pairing between $X^*$ and $X$.
In this section, we recall some known definitions and lemmas which will be used to obtain our main results (see \cite{MOS2013,NP1994} for more details).
A functional $j(u):X\rightarrow\mathbb{R}$ is called lower semicontinuous if and only if for any convergence sequence $\{u_n\}_{n=1}^{\infty}\subset X$ satisfying $u_n\rightarrow u\in X$, one has $\lim\inf_{n\rightarrow\infty}{j(u_n)}\geq j(u)$. Let $j: X\rightarrow \mathbb{R}\cup \{+\infty\}$ be a lower semicontinuous convex functional with domain $D(j)=\{u\in X\:|\:j(u)<+\infty\}$ and $u \in D(j)$. Then there exists $u^*\in X^*$ such that $j(v)-j(u)\geq \langle u^*, v-u\rangle_{X^*\times X}$ holds for all $v\in X$.
The set of all such $u^*\in X^*$ is called the convex subdifferential of $j$ at $u$ and we denote it by $\partial j(u)$.
For a set-valued operator $A:X\rightarrow2^{X^*}$, the graph of $A$ is denoted by $G(A)$, i.e.,
$$G(A):=\{(u,u^*)\in X\times X^*\:|\:u^*\in A(u)\}.$$
A set-valued operator $A:X\rightarrow2^{X^*}$ is called monotone if
$$\langle u^*-v^*,u-v\rangle_{X^*\times X}\geq0,\;\forall(u,u^*),\;(v,v^*)\in G(A).$$
Moreover, a monotone operator $A$ is called maximal monotone if for any $(u,u^*) \in X\times X^*$ satisfying
$$\langle u^*-v^*,u-v\rangle_{X^*\times X}\geq0, \quad \forall (v,v^*)\in G(A),$$
one has $(u,u^*)\in G(A)$.
A functional $j: X \rightarrow \mathbb{R} \cup\{\infty\}$ is called proper if $j(v)>-\infty$ for all $v\in X$ and there
exists a point $u\in X$ such that $j(u) < +\infty$. For a proper, convex and lower semicontinuous functional $j: X \rightarrow \mathbb{R} \cup\{\infty\}$, it is well known that $\partial j: X\to 2^{X^*}$ is maximal monotone.
At the end of this section, we recall two known results which will be used to obtain our main results.
\begin{lemma}\label{lemma_hisdep}\cite[Proposition 3.1]{Sofonea2012}
Let $\Lambda: C([0, T] ; X) \rightarrow C([0, T] ; X)$ be an operator satisfying the following property: there exists a constant $h>0$ such that
$$
\begin{array}{l}
\left\|\Lambda u_{1}(t)-\Lambda u_{2}(t)\right\|_{X} \leq
h\int_{0}^{t}
\left\|u_{1}(s)-u_{2}(s)\right\|_X ds, \quad \forall u_{1}, u_{2} \in C([0, T] ; X), \; t \in[0, T].
\end{array}
$$
Then there exists a unique element $u^{*} \in C([0, T] ; X)$ such that $\Lambda u^{*}=u^{*}$.
\end{lemma}
\begin{lemma}\label{l2.2}\cite[Lemma 3.3]{LiuSofonea2019}
Let $V$ be a reflexive Banach space and $K$ be a closed convex nonempty subset of $V$. Suppose that $A: K \to V^{*}$ and $\varphi:K\times K \rightarrow \mathbb{R}$ satisfy the following hypotheses:
\begin{itemize}
\item[H(A):] $A: K \rightarrow V^{*}$ is strongly monotone Lipschitz continuous, i.e.,
\begin{itemize}
\item[(a)] $\langle Au_1-Au_2,u_1-u_2\rangle_{V^*\times V}\ge m\|u_1-u_2\|^2_V$ for all $u_1,u_2 \in K$ with $m>0$;
\item[(b)] $\| Au_1-Au_2\|_{V^*} \le L\|u_1-u_2\|_V$ for all $u_1,u_2 \in K$ with $L>0$.
\end{itemize}
\item[H($\varphi$):] $\varphi: K\times K \rightarrow \mathbb{R} $ is such that
\begin{itemize}
\item[(a)] for any $u \in K, \varphi(u,\cdot)$ is convex and lower semicontinuous on $K$;
\item[(b)] for any $u_1,u_2,v_1,v_2\in K $, there exists $\beta>0$ such that
$$ \varphi(u_1,v_2)-\varphi(u_1,v_1)+\varphi(u_2,v_1)-\varphi(u_2,v_2)\leq\beta\|u_1-u_2\|_V\|v_1-v_2\|_V.$$
\end{itemize}
\end{itemize}
If $m>\beta$, then for each $f\in V^*$, there exists a unique element $u\in K$ such that
$$\langle Au,v-u\rangle_{V^*\times V}+\varphi(u,v)-\varphi(u,u)\geq\langle f,v-u\rangle_{V^*\times V}, \quad \forall v\in K.$$
\end{lemma}
\section{Unique solvability for DQVIs \eqref{1.1}}
\setcounter{equation}{0}
In this section, we provide some sufficient conditions for ensuring the existence and uniqueness of solution for DQVIs \eqref{1.1}. To this end, we consider the Gelfand triplet of Banach spaces $\left(V, H, V^{*}\right)$ which have compact and dense embeddings. We also need the following assumptions.
\begin{itemize}
\item[H(A):]\label{assume:A_2} The operator $A:I\times V\rightarrow V^*$ satisfies
\begin{itemize}
\item[(a)]$A(\cdot, v)$ is continuous on $I$ for any given $v\in V$.
\item[(b)]$A(t,\cdot)$ is Lipschitz continuous with $ L_A>0$ on $V$ for any given $t\in I$, i.e.,
$$\left\|A(t, u_{1})-A(t, u_{2})\right\|_{V^*}\leq L_{A}\left\|u_{1}-u_{2}\right\|_{V}, \quad \forall (t,u_1,u_2)\in I\times V\times V.$$
\end{itemize}
\item[H(B):] The operator $B: I\times V\times Y\rightarrow V^*$ satisfies
\begin{itemize}\label{assume:B_2}
\item[(a)]$B(\cdot, v,\zeta)$ is continuous on $I$ for any given $v \in V$ and $\zeta \in Y$.
\item[(b)]$B(t,\cdot,\cdot)$ is Lipschitz continuous with $L_B>0$ on $V\times Y$ for any given $t\in I$, i.e.,
$$\left\| B(t, u_{1},\zeta_1)-B(t, u_{2},\zeta_2)\right\|_{V^*} \leq L_{B}(\|u_{1}-u_{2}\|_{V}+\|\zeta_1-\zeta_2\|_Y),\quad\forall t\in I,\; \forall u_1,u_2 \in V,\;\forall \zeta_1,\zeta_2\in Y.$$
\item[(c)] There exists $\rho\in L^2(I;\mathbb{R}^+)$ such that
$$\|B(t,u,\zeta)\|_{V^*}\leq \rho(t)(\|\zeta\|_Y+\|u\|_V), \quad \forall (t,u,\zeta)\in I \times V \times Y.$$
\end{itemize}
\item[H(C):]\label{assume:A_2} The operator $C:I\times V\rightarrow V^*$ satisfies
\begin{itemize}
\item[(a)]$C(t,\cdot)$ is Lipschitz continuous with $ L_{C1}>0$ on $V$ for any given $t\in I$, i.e.,
$$\left\|C(t, u_{1})-C(t, u_{2})\right\|_{V^*}\leq L_{C1}\left\|u_{1}-u_{2}\right\|_{V}, \quad \forall (t,u_1,u_2)\in I\times V\times V.$$
\item[(b)]$C(\cdot,u)$ is Lipschitz continuous with $ L_{C2}>0$ on $I$ for any given $u\in V$, i.e.,
$$\left\|C(t_{1}, u)-C(t_{2}, u)\right\|_{V^*}\leq L_{C2}\left\|t_1-t_2\right\|_{V}, \quad \forall (t_1,t_2,u)\in I\times I\times V.$$
\item[(c)]$C(t,\cdot)$ is strong monotone with $ L_C>0$ on $V$ for any given $t\in I$, i.e.,
$$\left\langle C(t, u_{1})-C(t, u_{2}), u_{1}-u_{2}\right\rangle_{V^*\times V} \geq m_{C}\|u_{1}-u_{2}\|_{V}^2,\quad \forall (t,u_1,u_2)\in I\times V\times V.$$
\end{itemize}
\item[H(j):] The functional $j: W\times V\times V\rightarrow \mathbb{R}$ satisfies
\begin{itemize}\label{assume:j_2}
\item[(a)]$j(w, u, \cdot)$ is convex proper and lower semicontinuous on $V$ for any given $(w,u)\in W\times V$.
\item[(b)] There exist $\alpha_0>0$ \ and $\alpha_1>0 $ such that
\begin{align*}
&\; j\left(w_{1}, u_{1}, v_{2}\right)-j\left(w_{1}, u_{1}, v_{1}\right)+j\left(w_{2},u_{2},v_{1}\right)-j\left(w_{2}, u_{2},v_{2}\right)\\
\leq&\; \alpha_0\left\|w_{1}-w_{2}\right\|_{W}\left\|v_{1}-v_{2}\right\|_{V}+\alpha_1\left\|u_{1}-u_{2}\right\|_{V}
\left\|v_{1}-v_{2}\right\|_{V},\quad \forall w_1, w_2\in W,\;\forall u_{1}, u_{2}, v_{1}, v_{2} \in V.
\end{align*}
\end{itemize}
\item[H(F):] The operator $F: I\times W\times V\rightarrow W$ satisfies
\begin{itemize}\label{assume:F_2}
\item[(a)]$F(\cdot, w,v)$ is continuous on $I$ for any given $(w,v)\in W\times V$.
\item[(b)] $F(t,\cdot,\cdot)$ is Lipschitz continuous with $L_F >0$ on $V\times Y$ for any given $t\in I$, i.e.,
$$\left\| F(t,w_1, u_{1})-F(t,w_2, u_{2})\right\|_{W} \leq L_{F}(\|u_{1}-u_{2}\|_{V}+\|w_1-w_2\|_W), \quad \forall t\in I,\; \forall w_1,w_2\in W,\;\forall u_1,u_2 \in V.$$
\end{itemize}
\item[H($\phi$):] The operator $\phi: I\times V\times Y\rightarrow Y_1$ satisfies
\begin{itemize}\label{assume:phi_2}
\item[(a)] $\phi(t,\cdot,\cdot)$ is Lipschitz continuous with $L_\phi>0$ on $V\times Y$ for any given $t\in I$, i.e.,
$$\left\| \phi(t,u, \zeta)-\phi(t,v, \eta)\right\|_{Y_1} \leq L_{\phi}(\|u-v\|_{V}+\|\zeta-\eta\|_{Y_1}), \quad \forall t\in I,\; \forall u,v \in V,\; \forall \zeta,\eta\in Y.$$
\item[(b)]$\phi(\cdot,0_V,0_Y)\in L^2(I; Y_1).$
\end{itemize}
\item[H(a):] The functional $a: Y\times Y\rightarrow \mathbb{R}$ satisfies
\begin{itemize}\label{assume:a_2}
\item[(a)] $a(\cdot,\cdot)$ is a continuous bilinear symmetric coercive functional and there exist $a_1\in\mathbb{R}$ and $a_2>0$ such that
$$ a(\eta, \eta)+a_1\|\eta\|_{Y_1}^2\geq a_2\|\eta\|_Y^2,\quad \forall \eta\in Y.$$
\end{itemize}
\end{itemize}
The main result of this section can be stated as follows.
\begin{theorem}\label{t3.1}
Suppose that assumptions H(A), H(B), H(C), H(j), H(F), H($\phi$), H(a) hold and $ m_C>\alpha_1$. Then DQVIs \eqref{1.1} has a unique solution
$(\zeta,u,w)\in (H^1(I;Y_1)\cap L^2(I;Y))\times C^1(I;K_V)\times C^1(I;W)$.
\end{theorem}
We divide the proof of Theorem \ref{t3.1} into a series of lemmas. To this end, we first consider the following auxiliary problem.
\begin{problem}\label{problem_aux_1}
For any given $w\in C(I;W)$, find $\dot{u}_w:I\rightarrow K_V$ such that, for all $t\in I$ and
\begin{align}\label{eq_pro1}
&\quad\langle C(t,\dot{u}_w(t)),v-\dot{u}_w(t)\rangle_{V^*\times V} +j(w(t),\dot{u}_w(t),v)-j(w(t),\dot{u}_w(t),u\dot{}_w(t))\nonumber\\
&\geq \langle f(t),v-\dot{u}_w(t)\rangle_{V^*\times V}, \quad
\forall v\in K_V.
\end{align}
\end{problem}
\begin{lemma}\label{Lemma_1}
Assume that H(C) and H(j) hold and $ m_C>\alpha_1$. Then for any given $w\in C(I;W)$ and $f\in C(I;V^*)$, Problem \ref{problem_aux_1} has a unique solution $\dot{u}_w\in C(I,K_V)$.
\end{lemma}
\begin{proof}
For any fixed $t$, define an operator $A:V\rightarrow V^*$ and a function $\varphi:V\times V\rightarrow\mathbb{R} $ by setting
$$ Au=C(t,\dot{u}(t)),\quad \varphi(u,v)=j(w(t),\dot{u}(t),v),\quad\forall u,v\in K, t\in I.$$
Clearly, $A$ and $\varphi$ satisfy all the assumptions of Lemma \ref{l2.2}. Thus, it follows from Lemma \ref{l2.2} that Problem \ref{problem_aux_1} has a unique solution $\dot{u}_w(t)\in K_V$. Next we prove that $\dot{u}_w\in C(I,K_V)$. For $t_1,t_2\in I$, denote $\dot{u}_{w}(t_i)=\dot{u}_i,\;f(t_i)=f_i,\;w(t_i)=w_i$ with $i=1,2$. Thus, inequality \eqref{eq_pro1} yields
\begin{eqnarray}\label{test1}
\langle C(t_1,\dot{u}_1),\dot{u}_2-\dot{u}_1\rangle_{V^*\times V} + j(w_1,\dot{u}_1(t),\dot{u}_2)-j(w_1,\dot{u}_1,\dot{u}_1 )\geq\langle f_1,\dot{u}_2-\dot{u}_1\rangle
\end{eqnarray}
and
\begin{eqnarray}\label{test2}
\langle C(t_2,\dot{u}_2),\dot{u}_1-\dot{u}_2\rangle_{V^*\times V} + j(w_2,\dot{u}_2(t),\dot{u}_1)-j(w_2,\dot{u}_2,\dot{u}_2 )\geq\langle f_2,\dot{u}_1-\dot{u}_2\rangle.
\end{eqnarray}
Adding \eqref{test1} to \eqref{test1}, one has
\begin{eqnarray*}
&&\langle C(t_1,\dot{u}_1)-C(t_2,\dot{u}_2),\dot{u}_1-\dot{u}_2\rangle_{V^*\times V} \nonumber\\
&\leq& j(w_1,\dot{u}_1(t),\dot{u}_2)-j(w_1,\dot{u}_1,\dot{u}_1 )+ j(w_2,\dot{u}_2(t),\dot{u}_1)\nonumber\\
&&\mbox{}-j(w_2,\dot{u}_2,\dot{u}_2 )+\langle f_1-f_2,u_1-u_2\rangle
\end{eqnarray*}
and so
\begin{eqnarray}\label{test_sum}
&&\langle C(t_1,\dot{u}_1)-C(t_1,\dot{u}_2),\dot{u}_1-\dot{u}_2\rangle_{V^*\times V} \nonumber\\
&\leq& j(w_1,\dot{u}_1(t),\dot{u}_2)-j(w_1,\dot{u}_1,\dot{u}_1 )+ j(w_2,\dot{u}_2(t),\dot{u}_1)\nonumber\\
&&\mbox{}-j(w_2,\dot{u}_2,\dot{u}_2 )+\langle f_1-f_2,\dot{u}_1-\dot{u}_2\rangle+\langle C(t_2,\dot{u}_2)-C(t_1,\dot{u}_2),\dot{u}_1-\dot{u}_2\rangle_{V^*\times V}.
\end{eqnarray}
By conditions H(j)(b) and H(C)(b)(c), inequality \eqref{test_sum} becomes
\begin{eqnarray}\label{test_sum2}
(m_C-\alpha_1)\|\dot{u}_1-\dot{u}_2\|
\leq \alpha_0\|w_1-w_2\|_W+\|f_1-f_2\|_{V^*} +L_{C2}|t_1-t_2|.
\end{eqnarray}
Since $w\in C(I;W)$ and $f\in C(I;V^*)$, it follows from \eqref{test_sum2} that
$$\lim_{t_1\rightarrow t_2}\|\dot{u}_w(t_1)-\dot{u}_w(t_2)\|_V=0,$$
i.e., $\dot{u}_w\in C(I,K_V)$. This finishes the proof.
\end{proof}
\begin{lemma}\label{Corollary_1}
Suppose that H(A), H(B), H(C), H(j) hold and $ m_C>\alpha_1$. Then for any given $\zeta\in H^1(I;Y_1)\cap L^2(I;Y)$, $w\in C(I;W)$, and $f\in C(I;V^*)$, the following problem
\begin{equation}\label{eq_col}
\left\{\begin{array}{l}
\text { find } u_{w\zeta}:(0, T) \rightarrow K_V \text { such that } \\
\left\langle A(t,u_{w\zeta}(t))+\int_0^tB(t-s,u_{w\zeta}(s),\zeta(s))ds+C(t,\dot{u}_{w\zeta}(t)),v-\dot{u}_{w\zeta}(t)\right\rangle_{V^*\times V}\\ \quad \mbox{}+j(w,\dot{u}_{w\zeta}(t),v)-j(w,\dot{u}_{w\zeta}(t),\dot{u}_{w\zeta}(t))
\geq \langle f(t),v-\dot{u}_{w\zeta}(t)\rangle_{V^*\times V}, \quad \forall t \in I,\; \forall v \in K_V, \\
\text { where } u_{w\zeta}(0)=u_0
\end{array}\right.
\end{equation}
has a unique solution $u_{w\zeta}\in C^1(I;K_V)$.
\end{lemma}
\begin{proof}
Let us fix $\eta \in C(I; K_V )$. We consider the following auxiliary for \eqref{eq_col}.
\begin{equation}\label{eq_col4_1}
\left\{\begin{array}{l}
\text { find } u_{w\zeta\eta}:(0, T) \rightarrow K_V \text { such that } \\
\langle C(t,\dot{u}_{w\zeta\eta}(t)),v-\dot{u}_{w\zeta\eta}(t)\rangle_{V^*\times V} +j(w(t),\dot{u}_{w\zeta\eta}(t),v)-j(w(t),\dot{u}_{w\zeta\eta}(t),\dot{u}_{w\zeta\eta}(t))\\
\quad \mbox{}\geq \langle f_\eta(t),v-\dot{u}_{w\zeta\eta}(t)\rangle_{V^*\times V}, \quad \forall t \in I, \; \forall v \in K_V,\\
\text { where } u_{w\zeta\eta}(0)=u_0,
\end{array}\right.
\end{equation}
where $f_\eta$ is defined by
$$f_\eta(t):=f(t)-\int_0^tB\left(t-s,\int_0^s\eta(s)ds,\zeta(s)\right)-A\left(t,\int_0^t\eta(s)ds\right).$$
We first claim that $f_\eta\in C(I;V^*)$. For any given $ t_1, \;t_2\in I$ with $t_1<t_2$, it follows from the definition of $f_\eta$ that
\begin{eqnarray}
&& \|f_\eta(t_2)-f_\eta(t_1)\|_V^* \nonumber\\
&\leq& \|f(t_1)-f(t_2)\|+ \left\|\int_0^{t_2}B\left(t_2-s,\int_0^s\eta(s)ds,\zeta(s)\right)ds-\int_0^{t_1}B\left(t_1-s,\int_0^s\eta(s)ds,\zeta(s)\right)ds\right\|\nonumber\\
&&\mbox{}+\left\|A\left(t_2,\int_0^{t_2}\eta(s)ds\right)-A\left(t_2,\int_0^{t_1}\eta(s)ds\right)+A\left(t_2,\int_0^{t_1}\eta(s)ds\right)-A\left(t_1,\int_0^{t_1}\eta(s)ds\right)\right\|\nonumber\\
&\leq&\|f(t_1)-f(t_2)\|+\int_{t_1}^{t_2}\left\|B\left(t_2-s,\int_0^s\eta(s)ds,\zeta(s)\right)\right\|ds\nonumber\\
&&\mbox{}+\int_{0}^{t_1}\left\|B\left(t_2-s,\int_0^s\eta(s)ds,\zeta(s)\right)ds-B\left(t_1-s,\int_0^s\eta(s)ds,\zeta(s)\right)\right\|ds\nonumber\\
&&\mbox{}+\left\|A\left(t_2,\int_0^{t_1}\eta(s)ds\right)-A\left(t_1,\int_0^{t_1}\eta(s)ds\right)\right\|+L_A\left\|\int_{t_1}^{t_2}\eta(s)ds\right\|.
\end{eqnarray}
By condition H(B)(c) and H\"{o}lder's inequality, we have
\begin{eqnarray}
&&\int_{t_1}^{t_2}\left\|B\left(t_2-s,\int_0^s\eta(s)ds,\zeta(s)\right)\right\|ds\nonumber\\
&\leq&\int_{t_1}^{t_2}|\rho(t_2-s)|\left\|\int_0^s\eta(s)ds\right\|ds+\int_{t_1}^{t_2}|\rho(t_2-s)|\left\|\zeta(s)\right\|ds\nonumber\\
&\leq&T\|\eta\|_{C^(I;V)}\left(\sqrt{t_2-t_1}\right) \left(\int_{t_1}^{t_2}|\rho(t_2-s)|^2ds\right)^{\frac{1}{2}}+ \left(\int_{t_1}^{t_2}|\rho(t_2-s)|^2ds\right)^{\frac{1}{2}}\left(\int_{t_1}^{t_2}\|\zeta(s)\|^2ds\right)^{\frac{1}{2}}\nonumber\\
&\leq&T^\frac{3}{2}\|\eta\|_{C^(I;V)}\|\rho\|_{L^2(t_1,t_2;\mathbb{R}^+)}+\|\rho\|_{L^2(t_1,t_2;\mathbb{R}^+)}\|\zeta(s)\|_{L^2(t_1,t_2;Y)}\nonumber\\
&&\mbox{}\rightarrow0\;\mbox{as $|t_1-t_2|\rightarrow0$}.
\end{eqnarray}
According to condition H(B)(a), one has
\begin{eqnarray}
\int_{0}^{t_1}\left\|B\left(t_2-s,\int_0^s\eta(s)ds,\zeta(s)\right)ds-B\left(t_1-s,\int_0^s\eta(s)ds,\zeta(s)\right)\right\|ds\rightarrow0\;\mbox{as $|t_1-t_2|\rightarrow0$}.
\end{eqnarray}
By the continuity of $f$ and $A(t,\cdot)$, we know that $\lim_{|t_1-t_2|\rightarrow0}\|f_\eta(t_2)-f_\eta(t_1)\|_V^*=0$ and so $f_\eta\in C(I;V^*)$.
Now it follows from Lemma \ref{Lemma_1} that problem \eqref{eq_col4_1} has a unique solution $\dot{u}_{w\zeta\eta}\in C(I;K_V)$. Thus, we can define an operator $\Lambda:\eta\mapsto \dot{u}_{w\zeta\eta}$, where $\eta\in C(I;V)$ and $\dot{u}_{w\zeta\eta}(t)$ is the unique solution of problem \eqref{eq_col4_1}.
Next we show that $\Lambda$ has a unique fixed point in $C(I;V)$. In fact, let $\dot{u}_0(t)$ and $\dot{u}_1(t)$ be two solutions of \eqref{eq_col4_1}. Then
\begin{align}\label{eq_col4_2}
\langle C(t,\dot{u}_i(t)),v-\dot{u}_i(t)\rangle_{V^*\times V} +j(w,\dot{u}_i(t),v)-j(w,u\dot{}_i(t),\dot{u}_i(t))
\geq \langle f_{\eta_i}(t),v-\dot{u}_i(t)\rangle_{V^*\times V},\; i=0,1
\end{align}
for all $v \in K_V$ and all $t\in I$. Testing \eqref{eq_col4_2} with $\dot{u}_{1-i}(t)$, we obtain
\begin{eqnarray}\label{eq_col4_3}
&&\langle C(t,\dot{u}_0(t))-C(t,\dot{u}_1(t)),\dot{u}_0-\dot{u}_1(t)\rangle_{V^*\times V}\nonumber\\
&\leq& j(w,\dot{u}_0(t),\dot{u}_1(t))+j(w,\dot{u}_1(t),\dot{u}_0(t))-j(w,\dot{u}_1(t),\dot{u}_1(t))\nonumber\\
&&\mbox{}-j(w,\dot{u}_0(t),\dot{u}_0(t))+\langle f_{\eta_1}- f_{\eta_0},\dot{u}_0(t)-\dot{u}_1(t)\rangle_{V^*\times V}
\end{eqnarray}
for all $t \in I$. According to conditions H(A)(b), H(B)(b) and H(j)(b), inequality \eqref{eq_col4_3} yields
\begin{eqnarray}
&& m_C\| \dot{u}_0(t)-\dot{u}_1(t)\|^2_{ V}\nonumber\\
&\leq&\alpha_1\|\dot{u}_0(t)-\dot{u}_1(t)\|^2_V+\|f_{\eta_1}(t)- f_{\eta_0}(t)\|_{V^*}\|\dot{u}_0(t)-\dot{u}_1(t)\|_V\nonumber\\
&\leq&\alpha_1\|\dot{u}_0(t)-\dot{u}_1(t)\|^2_V++\left\|A\left(t,\int_0^t\eta_0(s)ds\right)-A\left(t,\int_0^t\eta_1(s)ds\right)\right\|_V\|\dot{u}_0(t)-\dot{u}_1(t)\|_V\nonumber\\
&&\mbox{}+\left\|\int_0^{t_2}B\left(t-s,\int_0^s\eta_0(s)ds,\zeta(s)\right)ds-\int_0^{t_1}B\left(t-s,\int_0^s\eta_1(s)ds,\zeta(s)\right)ds\right\|_V\|\dot{u}_0(t)-\dot{u}_1(t)\|_V\nonumber\\
&\leq&\alpha_1\|\dot{u}_0(t)-\dot{u}_1(t)\|^2_V+L_A\int_0^t\|\eta_0(s)-\eta_1(s)\|_Vds\|\dot{u}_0(t)-\dot{u}_1(t)\|_V\nonumber\\
&&\mbox{}+L_B\int_0^t\int_0^t\|\eta_0(s)-\eta_1(s)\|_Vdlds\|\dot{u}_0(t)-\dot{u}_1(t)\|_V\nonumber\\
&\leq&\alpha_1\|\dot{u}_0(t)-\dot{u}_1(t)\|^2_V+(L_A+TL_B)\int_0^t\|\eta_0(s)-\eta_1(s)\|_Vds\|\dot{u}_0(t)-\dot{u}_1(t)\|_V
\end{eqnarray}
for all $t \in I$. Hence
\begin{align*}
\| \dot{u}_0(t)-\dot{u}_1(t)\|_{ V}\leq \frac{L_A+TL_B}{ (m_C-\alpha_1)}\int_0^t\|\eta_1(s)-\eta_2(s)\|_{V}ds.
\end{align*}
From Lemma \ref{lemma_hisdep} we can see that $\Lambda$ has a unique fixed point $\dot{u}_{w\zeta}\in C(I;K_V)$. Since $u_{w\zeta}(0)=u_0$, we know that \eqref{eq_col} has a unique solution $u_{w\zeta}\in C^1(I;K_V)$.
This completes the proof.
\end{proof}
\begin{remark}\label{r3.1}
We need some useful inequalities for the solution $u_{w\zeta}$ of problem \eqref{eq_col}. Let $w_1,\;w_2\in C(I;W)$ and $\zeta_1,\;\zeta_2\in H^1(I;Y_1)\cap L^2(I;Y)$. Replacing $w=w_i$ and $\zeta=\zeta_i$ with $i=1,\;2$ in problem \eqref{eq_col} and applying H\"{o}lder's inequality, we have
\begin{eqnarray}
(m_C-\alpha_1)\|\dot{u}_{w_1\zeta_1}(t)-\dot{u}_{w_2\zeta_2}(t)\|
&\leq&
L_B\int_0^t\int_0^t\|\dot{u}_{w_1\zeta_1}(s)-\dot{u}_{w_2\zeta_2}(s)\|ds+L_B\int_0^t\|\zeta_1(s)-\zeta_2(s)\|ds\nonumber\\
&&\mbox{}+\alpha_0\|w_{1}(t)-w_{2}(t)\|+L_A\int_0^t\|\dot{u}_{w_1\zeta_1}(s)-\dot{u}_{w_2\zeta_2}(s)\|ds\nonumber\\
&\leq&
(L_A+L_Bt)\int_0^t\|\dot{u}_{w_1\zeta_1}(s)-\dot{u}_{w_2\zeta_2}(s)\|ds+\mbox{}+\alpha_0\|w_{1}(t)-w_{2}(t)\|\nonumber\\
&&\mbox{}+L_B\sqrt t\left(\int_{0}^{t}\|\zeta_1(s)-\zeta_2(s)\|^2ds\right)^{\frac{1}{2}}ds\
\end{eqnarray}
and so
\begin{eqnarray}
\|\dot{u}_{w_1\zeta_1}(t)-\dot{u}_{w_2\zeta_2}(t)\|&\leq&
\frac{L_A+L_Bt}{m_C-\alpha_1}\int_0^t \|\dot{u}_{w_1\zeta_1}(s)-\dot{u}_{w_2\zeta_2}(s)\|ds+\frac{L_B\sqrt t}{m_C-\alpha_1}\|\zeta_1-\zeta_2\|_{L^2(I;Y_1)}\nonumber\\
&&\mbox{}+\frac{\alpha_0}{m_C-\alpha_1}\|w_{1}(t)-w_{2}(t)\|.
\end{eqnarray}
Applying Gronwall's inequality yields
\begin{eqnarray}
&&\|\dot{u}_{w_1\zeta_1}(t)-\dot{u}_{w_2\zeta_2}(t)\|\nonumber\\
&\leq&\frac{(L_A+L_Bt)\alpha_0}{(m_C-\alpha_1)^2}e^{\frac{2L_At+L_Bt^2}{2(m_C-\alpha_1)}}\int_0^t \|w_{1 }(s)-w_{2 }(s)\|ds+\frac{\alpha_0}{m_C-\alpha_1}\|w_{1}(t)-w_{2}(t)\|\nonumber\\
&&\mbox{}+\left(\frac{L_B \sqrt t}{m_C-\alpha_1}+\frac{2L_B t^\frac{3}{2}(L_A+L_Bt)}{3(m_C-\alpha_1)^2}e^{\frac{2L_At+L_Bt^2}{2(m_C-\alpha_1)}}\right)\|\zeta_1-\zeta_2\|_{L^2(I;Y_1)}.
\end{eqnarray}
\end{remark}
In order to prove Theorem \ref{t3.1}, we also need to introduce the following auxiliary problem.
\begin{problem}\label{problem_aux_2}
Find $u_\zeta:I\rightarrow K_V$ and $w_\zeta:I\rightarrow W$ such that, for all $t\in I$,
\begin{align*}
&\dot{w}_{\zeta}(t)=F(t,w_{\zeta}(t),\dot{u}_{\zeta}(t)),\nonumber\\
&\left\langle A(t,u_{\zeta}(t)) +\int_0^t B(t-s,u_{\zeta}(s)),\zeta(s))ds+C(t,\dot{u}_{\zeta}(t)),v-\dot{u}_{\zeta}(t) \right\rangle_{V^*\times V}\nonumber\\
&\quad \mbox{}+j(w_{\zeta}(t) ,\dot{u}_{\zeta}(t) ,v)-j(w_{\zeta}(t) ,\dot{u}_{\zeta}(t) ,\dot{u}_{\zeta}(t) )\geq \langle f(t) ,v-\dot{u}_{\zeta}(t) \rangle_{V^*\times V},\quad\forall v\in K_V,
\end{align*}
where $u_\zeta(0)=u_0$, $w_\zeta(0)=w_0$ and $\zeta\in C(I;K_Y)$.
\end{problem}
\begin{lemma}\label{lemma_aux_2}
Suppose that H(A), H(B), H(C), H(j) and H(F) hold and $ m_C>\alpha_1$. Then for any given $\zeta\in H^1(I;Y_1)\cap L^2(I;Y)$ and $f\in C(I;V^*)$,
Problem \ref{problem_aux_2} has a unique solution $(u_\zeta,w_\zeta)\in C^1(I;K_V)\times C^1(I;W)$.
\end{lemma}
\begin{proof}
Define an operator $G:C^1(I;W)\rightarrow C(I;K_V)$ by $G(w)(t)=\dot{u}_{w\zeta}(t)$, where $u_{w\zeta}$ is the solution of problem \eqref{eq_col}. By the proof of Lemma \ref{Corollary_1}, we know that $G$ is well-defined.
Then we need to illustrate that there exists a unique $w_\zeta\in C^1(I;W)$ such that
\begin{align}\label{eq_lem4_2}
\dot{w}_{\zeta}(t)=F(t,w_{\zeta}(t),G(w_\zeta)(t)).
\end{align}
Moreover, $(G(w_\zeta),w_\zeta)\in C(I;K_V)\times C^1(I;W)$ is the unique solution of
Problem \ref{problem_aux_2}.
For this purpose, we consider an operator $\Lambda : C(I;W)\rightarrow C^1(I;K_V)$ defined as follows:
$$\Lambda w_{\zeta}(t)=\int_{0}^{t} F(s,w_{\zeta}(s),G(w_\zeta)(s))ds +w_0, \;\forall t \in[0, T].$$
According to the definition of operator $G$ and conditions H(F)(a)(b) that
$\Lambda w_\zeta\in C^1(I;W)$ when $w_\zeta\in C(I;W)$. Because the fixed point of $\Lambda$ is the
solution of \eqref{eq_lem4_2}, we only need to prove that $\Lambda$ has a unique fixed point
in $C(I;W)$.
Let $w_{\zeta1},\;w_{\zeta2}\in C(I;W)$. We can draw a conclusion from $G$, $\Lambda$ and condition H(F)(b) that
\begin{align}\label{eq_lem4_3}
&\quad \| \Lambda w_{\zeta1}(t)- \Lambda w_{\zeta2}(t)\|\nonumber\\
&=\left\|\int_{0}^{t} F(s,w_{\zeta1}(s),G(w_{\zeta1})(s))ds-\int_{0}^{t} F(s,w_{\zeta2}(s),G(w_{\zeta2})(s))ds\right\|\nonumber\\
&\leq\int_{0}^{t}\left\| F(s,w_{\zeta1}(s),G(w_{\zeta1})(s))ds- F(s,w_{\zeta2}(s),G(w_{\zeta2})(s))\right\| ds.\nonumber\\
&\leq\int_0^tL_F\left(\|w_{\zeta1}(s)-w_{\zeta2}(s)\|+\|G(w_{\zeta1})(s)-G(w_{\zeta2})(s)\|\right)ds.
\end{align}
Denote
$$c_q=\frac{(L_A+L_BT)\alpha_0}{(m_C-\alpha_1)^2}e^{\frac{2L_AT+L_BT^2}{2(m_C-\alpha_1)}}, \quad c_p=\frac{\alpha_0}{m_A-\alpha_1}.$$
Then it follows from Remark \ref{r3.1} that
$$\|G(w_{\zeta1})(s)-G(w_{\zeta2})(s)\|
\leq c_q\int_0^s \|w_{\zeta1}(l)-w_{\zeta2}(l)\|dl
+c_p\|w_{\zeta1}(s)-w_{\zeta2}(s)\|.$$
These inequalities give
\begin{align}\label{eq_lem4_4}
&\quad \| \Lambda w_{\zeta1}(t)- \Lambda w_{\zeta2}(t)\|\nonumber\\
&\leq (L_F+L_F c_p)\int_0^t\|w_{\zeta1}(s)-w_{\zeta2}(s)\|ds
+L_Fc_q\int_0^t\int_0^s \|w_{\zeta1}(l)-w_{\zeta2}(l)\|dlds
\end{align}
and so
\begin{align}\label{eq_lem4_5}
\| \Lambda w_{\zeta1}(t)- \Lambda w_{\zeta2}(t)\|_\beta\leq
\left(\frac{L_F+L_F c_p}{\beta}\right)\left\|w_{\zeta1}-w_{\zeta2}\right\|_\beta
+\frac{L_Fc_q}{\beta^2} \left\|w_{\zeta1}-w_{\zeta2}\right\|_\beta
\end{align}
with $\beta>0$ and
$\|w\|_{\beta}=\max _{t \in I} e^{-\beta t}\|w(t)\|_{W}$ for all $w \in C(I ; W).$
Thus, $\|\cdot\|_{\beta}$ is an equivalent norm in $C(I ; W)$.
From inequality \eqref{eq_lem4_5}, we know that $\Lambda$ is a contraction operator on $C(I ; W)$ and $C(I ; W)$
endowed with the norm $\|\cdot\|_{\beta}$. Thus, $\Lambda$ has a unique fixed point $w_\zeta\in C(I;W)$ and so $(w_\zeta,G (w_\zeta))$ solves the Problem \ref{problem_aux_2}, which concludes the proof.
\end{proof}
\begin{remark}\label{r3.2}
We need some useful inequalities for the solution $u_{\zeta}$ of Problem \ref{problem_aux_2}. For any given $\zeta_1,\;\zeta_2\in H^1(I;Y_1)\cap L^2(I;Y)$, let $(\dot{u}_{\zeta_i},w_{\zeta_i})\in C(I;V)\times C^1(I;W)$ be the unique solution of Problem \ref{problem_aux_2} with $\zeta=\zeta_i$ for $i=1,\;2.$
From the proof of Lemma \ref{lemma_aux_2}, we have
\begin{eqnarray}\label{remark_es1}
&&\|w_{\zeta_1}(t)-w_{\zeta_2}(t)\|\nonumber\\
&\leq& (L_F+L_F c_p)\int_0^t \|w_{\zeta_1}(s)-w_{\zeta_2}(s)\|ds
+L_Fc_q\int_0^t\int_0^s \|w_{\zeta_1}(l)-w_{\zeta_2}(l)\|dlds
+L_Fc_r T\|\zeta_1-\zeta_2\|_{L^2(I,Y_1)}\nonumber\\
&\leq& (L_F+L_F c_p+L_Fc_qT)\int_0^t \|w_{\zeta_1}(s)-w_{\zeta_2}(s)\|ds+L_Fc_r T\|\zeta_1-\zeta_2\|_{L^2(I,Y_1)}
\end{eqnarray}
and
\begin{eqnarray}\label{remark_es2}
\|\dot{u}_{\zeta_1}(t)-\dot{u}_{\zeta_2}(t)\|
&\leq& c_q\int_0^t \|w_{\zeta_1}(s)-w_{\zeta_2}(s)\|ds
+c_p\|w_{\zeta_1}(t)-w_{\zeta_2}(t)\|+c_r\|\zeta_1-\zeta_2\|_{L^2(I,Y_1)}.\nonumber\\
&\leq& ( c_qT+c_p) \|w_{\zeta_1}(s)-w_{\zeta_2}(s)\|+c_r\|\zeta_1-\zeta_2\|_{L^2(I,Y_1)},
\end{eqnarray}
where
$$c_r=\frac{L_B \sqrt t}{m_C-\alpha_1}+\frac{2L_B t^\frac{3}{2}(L_A+L_Bt)}{3(m_C-\alpha_1)^2}e^{\frac{2L_At+L_Bt^2}{2(m_C-\alpha_1)}}.$$
By Gronwall's inequality, it follows from \eqref{remark_es1} that
\begin{eqnarray}\label{remark_es3}
\|w_{\zeta_1}(t)-w_{\zeta_2}(t)\|
\leq \left(L_Fc_r T\|\zeta_1-\zeta_2\|_{L^2(I,Y_1)}\right)e^{(L_F+L_F c_p+L_Fc_qT)T}.
\end{eqnarray}
Combining \eqref{remark_es2} and \eqref{remark_es3} yields
\begin{eqnarray*}
\|\dot{u}_{\zeta_1}(t)-\dot{u}_{\zeta_2}(t)\|_V
\leq \left(( L_Fc_qc_rT^2+L_Fc_pc_rT)e^{(L_F+L_F c_p+L_Fc_qT)T}+c_r\right)\|\zeta_1-\zeta_2\|_{L^2(I,Y_1)}.
\end{eqnarray*}
\end{remark}
Moreover, for the unique solution $(u_\zeta,w_\zeta)\in C^1(I;K_V)\times C^1(I;W)$ of Problem \ref{problem_aux_2}, we need to consider the following auxiliary problem.
\begin{problem}\label{problem_last}
Find $\zeta:I\rightarrow K_Y$ such that
\begin{eqnarray}
\langle \dot{\zeta}(t) ,\eta-\zeta(t) \rangle_{Y_1}+a(\zeta(t) ,\eta-\zeta(t) )\geq\langle\phi(t,u_\zeta(t) ,\zeta(t) ),\eta-\zeta(t) \rangle_{Y_1},\quad\forall\eta\in K_Y,
\end{eqnarray}
with $\zeta(0)=\zeta_0\in K_Y.$
\end{problem}
\begin{remark}\label{remark_solu}
We observe that if $\zeta\in H^1(I;Y_1)\cap L^2(I;Y)$ is a unique solution of Problem \ref{problem_last}, then $(\zeta,u_\zeta,w_\zeta)\in H^1(I;Y_1)\cap L^2(I;Y)\times C^1(I;K_Y)\times C^1(I;W)$ is a unique solution of DQVIs \eqref{1.1}.
\end{remark}
In order to prove Lemma \ref{lemma_last}, we need the following Lemma given in \cite{Han2016}.
\begin{lemma} \cite{Han2016} \label{lemma_zeta}
Suppose that condition H(a) holds. Then for any given $\lambda\in L^2(I;Y_1)$, there exists a unique $\zeta\in H^1(I;Y_1)\cap L^2(I;Y)$ such that
\begin{eqnarray}\label{equ_last_theo}
\langle \dot{\zeta}(t),\eta-\zeta(t) \rangle_{Y_1}+a(\zeta(t) ,\eta-\zeta(t) )\geq\langle\lambda(t),\eta-\zeta(t) \rangle_{Y_1},\quad\forall\eta\in K_Y,
\end{eqnarray}
with $\zeta(0)=\zeta_0\in K_Y$. Moreover, if $\zeta_i$ is the unique solution to problem \eqref{equ_last_theo} for $\lambda_i\in L^2(I;Y_1)$ with $i=1,2$, then
\begin{eqnarray}
\left\|\zeta_{1}(t)-\zeta_{2}(t)\right\|_{Y_1}^{2} \leq d_{1} \int_{0}^{t}\left\|\lambda_{1}(s)-\lambda_{2}(s)\right\|_{Y_1}^{2} d s \quad \text { for a.e. } t \in(0, T)
\end{eqnarray}
with $d_1>0.$
\end{lemma}
\begin{lemma}\label{lemma_last}
Suppose that conditions H(A), H(B), H(C), H(j), H(F), H($\phi$), H(a) hold and $ m_C>\alpha_1$. Then Problem \ref{problem_last} has a unique solution $\zeta\in H^1(I;Y_1)\cap L^2(I;Y). $
\end{lemma}
\begin{proof}
For any given $\theta\in L^2(I;Y_1)$, it follows from Lemma \ref{lemma_aux_2} that Problem \ref{problem_aux_2} has a unique solution $(u_\theta,w_\theta)\in C(I;K_V)\times C^1(I;W)$. Let $\phi_{\theta}(t):=\phi(t,u_{\theta}(t),\theta(t))$. We claim that $\phi_{\theta}\in L^2(I;Y_1)$. Indeed, the condition H($\phi$)(a) derives the following inequality
\begin{eqnarray}\label{equ_last1}
\|\phi_\theta(t)\|^2_{Y_1}&\leq&2\|\phi(t,u_\theta(t),\theta(t))-\phi(t,0_V,0_{Y_1})\|^2_{Y_1}
+2\|\phi(t,0_V,0_{Y_1})\|^2_{Y_1}\nonumber\\
&\leq&2(L_{\phi}(\|u_\theta(t)\|_V+\|\theta\|_{Y_1}))^2+2\|\phi(t,0_V,0_{Y_1})\|^2_{Y_1}\nonumber\\
&\leq&4L_{\phi}^2(\|u_\theta(t)\|^2_V+\|\theta\|^2_{Y_1}) +2\|\phi(t,0_V,0_{Y_1})\|^2_{Y_1}.
\end{eqnarray}
Thus inequality \eqref{equ_last1} and condition H($\phi$)(b) show that
\begin{eqnarray}
\|\phi_\theta(t)\|^2_{L^2(I;Y_1)}\leq 4L^2_{\phi}T\|u_\theta(t)\|^2_{C(I:V)}+4L^2_{\phi}\|\theta\|^2_{L^2(I;Y_1)}+2\|\phi(t,0_V,0_{Y_1})\|^2_{L^2(I;Y_1)}
\end{eqnarray}
and so $\phi_{\theta}(t)\in L^2(I;Y_1)$. Taking $\lambda=\phi_\theta$ in Lemma \ref{lemma_zeta}, we know that there exists a unique $\zeta_\theta\in H^1(I;Y_1)\cap L^2(I;Y)$ such that inequality \eqref{equ_last_theo} holds.
Next we define an operator $\Lambda:L^2(I;Y_1)\rightarrow H^1(I;Y_1)\cap L^2(I;Y)$ by $\Lambda(\theta)(t)=\zeta_\theta(t).$ Because $\left(Y, Y_1, Y^{*}\right)$ is a Gelfand triplet with dense and compact embeddings, one has $H^1(I;Y_1)\cap L^2(I;Y)\subset L^2(I;Y_1)$.
The rest is to show that $\Lambda$ has a unique fixed point in $L^2(I;Y_1).$ To this end, let $\theta_i\in L^2(I;Y_1)$ with $i=1,\;2.$ Then Lemma \ref{lemma_zeta} yields
\begin{eqnarray*}
\|\Lambda(\theta_1)(t)-\Lambda(\theta_2)(t)\|^2_{Y_1}&=&\|\zeta_{\theta_1}(t_1)-\zeta_{\theta_2}(t_2)\|^2_{Y_1}\nonumber\\
&\leq&d_{1} \int_{0}^{t}\left\|\phi(s,u_{\theta_1}(s),\theta_1(s))-\phi(s,u_{\theta_2}(s),\theta_2(s))\right\|_{Y_1}^{2} ds.
\end{eqnarray*}
Using condition H($\phi$)(a) to the above inequality obtains
\begin{eqnarray}\label{equ_lem_zeta1}
\|\Lambda(\theta_1)(t)-\Lambda(\theta_2)(t)\|^2_{Y_1}
&\leq&d_{1} \int_{0}^{t}\left\|\phi(s,u_{\theta_1}(s),\theta_1(s))-\phi(s,u_{\theta_2}(s),\theta_2(s))\right\|_{Y_1}^{2} ds\nonumber\\
&\leq&2 d_{1} L_{\phi}\int_{0}^{t}\left(\|u_{\theta_1}(s)-u_{\theta_2}(s)\|_{V}^{2} +\|\theta_1(s)-\theta_2(s)\|_{Y_1}^{2} \right)ds.
\end{eqnarray}
By Remark \ref{r3.2}, we can adapt the inequality \eqref{equ_lem_zeta1} as
\begin{eqnarray}\label{equ_lem_zeta2}
\|\Lambda(\theta_1)(t)-\Lambda(\theta_2)(t)\|^2_{Y_1}
&\leq& 2d_{1} L_{\phi}\int_{0}^{t}\left(\int_{0}^{t}\|\dot{u}_{\theta_1}(s)-\dot{u}_{\theta_2}(s)\|_{V}^{2}ds +\|\theta_1(s)-\theta_2(s)\|_{Y_1}^{2} \right)ds\nonumber\\
&\leq& c\|\theta_1-\theta_2\|^2_{L^2(I;Y_1)},
\end{eqnarray}
where
$$c= 2d_{1} L_{\phi}\left(\left(( L_Fc_qc_rT^2+L_Fc_pc_rT)e^{(L_F+L_F c_p+L_Fc_qT)T}+c_r\right)^2T^2+1\right).$$
Thus inequality \eqref{equ_lem_zeta2} shows that
\begin{eqnarray}\label{equ_lem_zeta3}
\|\Lambda(\theta_1)(t)-\Lambda(\theta_2)(t)\|_{L^(I;Y_1)}\leq \sqrt c\int_{0}^{t}\|\theta_1-\theta_2\|_{L^2(I;Y_1)}.
\end{eqnarray}
Now it follows from Lemma \ref{lemma_hisdep} that there exists a unique element $\theta^*$ in $L^2(I;Y_1)$ such that $\theta^*$ is the unique fixed point of $\Lambda$, i.e., $\theta^*=\Lambda(\theta^*)=\zeta_{\theta^*}\in H^1(I;Y_1)\cap L^2(I;Y)$. Thus, $\theta^*$ is a unique solution of Problem \ref{problem_last}. This completes the proof.
\end{proof}
Finally, we can give the proof of Theorem \ref{t3.1} as follows.
\begin{proof}
By Remark \ref{remark_solu} and Lemma \ref{lemma_last}, it is easy to see that $(\zeta,u_\zeta,w_\zeta)\in (H^1(I;Y_1)\cap L^2(I;Y))\times C^1(I;K_Y)\times C^1(I;W)$ is a unique solution of DQVIs \eqref{1.1}. This completes the proof of Theorem \ref{t3.1}.
\end{proof}
\section{An application to a contact problem for viscoelastic materials with long memory, damage and wear}
\setcounter{equation}{0}
In this section, we will use the results presented in Section 3 to solve a frictional contact problem of viscoelastic materials with long memory, damage and wear. Assume that $\Omega$ is a bounded and open domain in $\mathbb{R}^{d}(d=2,3)$ and its boundary $\Gamma:=\partial \Omega$ is Lipschitz continuous. For any $\bm{x}\in\Omega$, we denote it as $\bm{x}=(x_1,x_2,\cdots,x_d)$. Let $\bm{\nu}$ denote the unit outward normal vector defined $a.e.$ on $\Gamma$. The space of second order symmetric tensors on $\mathbb{R}^d$ is denoted by $\mathbb{S}^d$. Moreover, assume that $\mathbb{R}^{d}$ and $\mathbb{S}^{d}$ are equipped with the following inner products and norms
\begin{align*}
&\bm{u\cdot v}=u_{i}v_{i},\qquad\;\; \|\bm{v}\|_{\mathbb{R}^{d}} =(\bm{v\cdot v})^{\frac{1}{2}}, \qquad\forall \bm{u} ,\bm{v} \in \mathbb{R}^{d}, \\
& \bm{\sigma\cdot \tau}=\sigma_{ij}\tau_{ij},\qquad \|\bm{\tau}\|_{\mathbb{S}^{d}} =(\bm{\tau\cdot \tau})^{\frac{1}{2}}, \qquad\forall \bm{\sigma} ,\bm{\tau} \in \mathbb{S}^{d},
\end{align*}
where repeated indices represent summation convention.
We use notations $\bm{u}=(u_i)$, $\bm{\sigma}=(\sigma_{ij})$ and $\bm{\varepsilon(u)}=(\varepsilon_{ij}(\bm{u}))=(\frac{1}{2}(u_{i,j}+u_{j,i})), i,j=1,2,\cdots,d$ to denote the displacement vector, the stress tensor and the linearized strain tensor, respectively, where $u_{i,j}\equiv \frac{\partial u_i}{\partial x_j}$. Here and below, the spatial derivative is defined in the sense of distribution. The normal and tangential components of stress field $\bm{\sigma}$ on $\Gamma$ are denoted by $\sigma_\nu=(\bm{\sigma\nu})\bm{\cdot}\bm{\nu}$ and $\bm{\sigma_\tau}=\bm{\sigma}\bm{\nu}-\sigma_\nu\bm{\nu}$, respectively. The normal and tangential components of the displacement field on $\Gamma$ are denoted by $u_\nu=\bm{u}\bm{\cdot}\bm{\nu}$ and $\bm{u_\tau}=\bm{u}-u_\nu\bm{\nu}$, respectively.
We are interested in the evolution of the body on the time interval $I:=[0,T]$, where $T>0$ is a constant. The time partial derivative for a function $f(\bm{x},t)$ is denoted by $\dot{f}(\bm{x},t)$. For the sake of simplicity, we usually omit the variable $\bm{x}$ in functions.
Now we are in the position to describe the physical background of viscoelastic materials' quasistatic frictional contact problems with damage, wear and long memory. We consider a viscoelastic body which occupies $\Omega$. The boundary $\partial \Omega$ is composed of three disjoint measurable parts $\Gamma_{1}$, $\Gamma_{2}$ and $\Gamma_{3}$ with $\mbox{meas}(\Gamma_{1})>0$. Volume forces with density $\bm{f}_{0}$ act on $\Omega$ and surface tractions with density $\bm{f}_{2}$ act on $\Gamma_{2}$. On $\Gamma_1$, we suppose that the body is clamped, i.e., the displacement field disappears there. The contact model is established by a normal compliance condition on $\Gamma_3$ with sliding Coulomb's law of dry friction and wear. The foundation is made of a perfectly rigid material. The velocity of the foundation is denoted by $\bm{v}^*(t)\neq\bm{0}$.
Let $\boldsymbol{n}^{*}$ be the direction of the foundation's velocity and $\alpha$ be wear rate which are given respectively by
$$\boldsymbol{n}^{*}(t)=-\boldsymbol{v}^{*}(t) /\left\|\boldsymbol{v}^{*}(t)\right\|, \quad \alpha(t)=k\left\|\boldsymbol{v}^{*}(t)\right\|,$$
where $k$ stands for the wear coefficient. We use the following abbreviations to simplify the notations $\mathcal{Q}=\Omega\times I$, $\Sigma=\Gamma\times I$, $\Sigma_i=\Gamma_i\times I$, $i=1,2,3.$
Thus, we can present the formulation of the contact problem as follows.
\begin{problem}\label{mechanic_problem1}
Find a displacement field $\bm u:\mathcal{Q}\rightarrow\mathbb{R}^d$, a stress field $\bm{\sigma}:\mathcal{Q}\rightarrow\mathbb{S}^d$, a damage field $\zeta:\Sigma_3\rightarrow[0,1]$ and a wear function $w:\Sigma_3\rightarrow\mathbb{R}$ such that
\begin{align}
\bm{\sigma}(t)= \mathcal{A}\left(t, \bm{\varepsilon}(\bm{u}(t))\right)+\int_{0}^{t}\mathcal{B}\left(t-s, \bm{\varepsilon}(\bm{u}(s)), \zeta(s)\right) ds+ \mathcal{C}(t, \bm{\varepsilon}(\dot{\bm{u}}(t)))\quad&\quad \mbox{in}\quad \;\mathcal{Q},\label{eq_mep_1}\\
\dot{\zeta}-\kappa \Delta \zeta+\partial I_{[0,1]}(\zeta) \ni \phi(\bm{\varepsilon}(\bm{u}(t)), \zeta)\quad&\quad \mbox{in}\quad \;\mathcal{Q},\label{eq_dam_1}\\
\frac{\partial \zeta}{\partial \nu}=0\quad&\quad \mbox{on}\quad \;\Sigma,\label{eq_dam_2}\\
-\mbox{Div}\bm{\sigma}(t)=\bm{f}_{0}(t)\quad&\quad \mbox{in}\quad \;\mathcal{Q},\label{eq_mep_2}\\
\bm{u}(t)=\bm{0}\quad&\quad \mbox{on}\quad \Sigma_{1},\label{eq_mep_3}\\
\bm{\sigma}(t)\bm{\nu}=\bm{f}_{2}(t)\quad&\quad \mbox{on}\quad \Sigma_{2},\label{eq_mep_4}\\
\dot{u}_\nu(t)\leq g,\; \sigma_{\nu}(t)+p(w(t),\dot{u}_\nu(t))\leq 0
\quad&\quad\mbox{on}\quad \Sigma_{3},\label{eq_mep_7}\\
(\dot{u}_{\nu}(t)-g)(\sigma_{\nu}(t)+p(w(t),\dot{u}_\nu(t)))= 0 \quad&\quad\mbox{on}\quad \Sigma_{3},\label{eq_mep_8}\\
-\bm{\sigma}_{\tau}(t)= \mu p(w(t),\dot{u}_\nu)\cdot\bm{n^*}(t) \quad&\quad \mbox{on}\quad \;\Sigma_{3},\label{eq_mep_9}\\
\dot{w}(t)=\alpha(t)p(w(t),\dot{u}_\nu(t))\quad&\quad \mbox{on}\quad \;\Sigma_{3},\label{eq_mep_10}\\
\bm{u}(0)=\bm{u}_0,w(0)=0,\zeta(0)=\zeta_0\in(0,1)\quad&\quad \mbox{on}\quad \;\Gamma_{3}\label{eq_mep_12}.
\end{align}
\end{problem}
\begin{remark}\label{rem_subdiff}
The conditions \eqref{eq_mep_7} and \eqref{eq_mep_8} can be rewritten as
$$-\sigma_\nu(t)\in \partial I_K(\dot{u}_\nu)+p(w(t),\dot{u}_\nu),$$
where $K:=\{s\in \mathbb{R}\:|\:s\leq g\}$, $I_K$ is the indication function of $K$ and $\partial I_K$ is the convex subdifferential of $I_K$. From the definition of convex subdifferential, one has
$$(-\sigma_\nu(t)-p(\dot{u}_\nu(t)-w(t)))\cdot(s-\dot{u}_\nu)\leq I_K(s)-I_K(\dot{u}_\nu)=0,\quad \forall s\in K.$$
\end{remark}
Equation \eqref{eq_mep_1} stands for the viscoelastic constitutive law with damage effect and long memory, where $\zeta$ is the damage function, $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$ are elasticity, relaxation and viscosity operators, respectively.
The evolution law of damage function can be modeled from a parabolic differential inclusion \eqref{eq_dam_1}, where $\phi$ is the mechanical source of damage which depends on the damage itself and the strain, $I_{[0,1]}$ is the indication function of the interval $[0,1]$, $\partial I_{[0,1]}$ is the convex subdifferential of $I_{[0,1]}$ and $\kappa >0$ is a constant mircrocrack diffusion coefficient. We select the following damage source function
$$
\phi(\varepsilon(\boldsymbol{u}), \zeta) \equiv \phi_{\mathrm{Fr}}(\|\varepsilon(\boldsymbol{u})\|, \zeta)=\lambda_{D}\left(\frac{1-\zeta}{\zeta}\right)-\frac{1}{2} \lambda_{E}\|\varepsilon(\boldsymbol{u})\|^{2}+\lambda_{w},
$$
where $\lambda_{D}, \lambda_{E},$ and $\lambda_{w}$ are three positive numbers (\cite{Fr1995,Fr1996}).
Equation \eqref{eq_dam_2} represents a homogeneous Neumann condition of $\zeta$ on $\Sigma$ and equation \eqref{eq_mep_2} describes the equilibrium equation. Equations \eqref{eq_mep_3} and \eqref{eq_mep_4} denote the clamped boundary condition and the tractive boundary conditions, respectively.
Equations \eqref{eq_mep_7} and \eqref{eq_mep_8} are called
normal damped condition with unilateral constraint, which have been studied in \cite{Eck2010} without wear. In the previous literature, the normal compliance function $p(\dot{u}_\nu)$ was used to model the
lubricated contact.
In this paper, we assume that the lubrication is incomplete and the normal compliance depends on wear, i.e., $p$ has the form of $p(w,\dot{u}_\nu)$ with $w$ being the wear.
Equation \eqref{eq_mep_9} gives the functional relationship between the wear coefficient $k$ and the friction coefficient which denoted $\mu$. Here we assume that $\bm{v^*}(t)$ is much larger than the tangential body velocity $\bm{\dot{u}_\tau}(t)$. Equation \eqref{eq_mep_10} is related to the development of the wear function which can be obtained from the Archard's law with $\alpha(t)=k\|\bm{v^*}(t)\|$. For more details concerned with equations \eqref{eq_mep_7}-\eqref{eq_mep_10}, we can refer to the literature \cite{SofoneaF2016}.
Finally, equation \eqref{eq_mep_12} gives the initial conditions for the displacement field, the wear function and the damage field.
In order to get the variational formulation of Problem \ref{mechanic_problem1}, we need the following spaces. Let $W^{ k,p}(\Omega;\mathbb{R}^d)$ denote the Sobolev space of all functions. On $\Omega$, the weak derivatives of functions with order less than or equal to $k$ are $p$-integrable. Specially, let $H^1:=W^{ 1,2}(\Omega;\mathbb{R}^d)$ and $H=L^2(\Omega;\mathbb{R}^d).$
Let $V=\{\bm{v}\in H^1|\bm{v}=0\; \mbox{a.e. on} \;\Gamma_3\}$ endowed with the norm $$\|\bm{u}\|_V:=\|\bm{u}\|_{H^1}=\|\bm{u}\|_{L^2(\Omega;\mathbb{R}^d)}+\|\triangledown\bm{u}\|_{L^2(\Omega;\mathbb{R}^{d\times d})},$$
where $\nabla \bm{u}=(\frac{\partial u_{i}}{\partial x_j})$ for $i,j=1,\cdots,d$ with $\bm{u}\in H^1.$
Let $\mbox{Div}\bm{\sigma}=(\sigma_{ij,j})=(\frac{\partial\sigma_{ij}}{\partial x_j})$ with $\bm{\sigma}\in W^{1,2}(\Omega;\mathbb{S}^d)$. Then we have the following Green formula:
$$(Div\bm{\sigma},\bm{v})_H+(\bm{\sigma},\bm{\varepsilon(u)})_{L^2(\Omega;\mathbb{S}^d)}=\int_{\Gamma}\bm{\sigma\nu}\cdot\bm{v} d_{\Gamma},$$
where $(\bm{\sigma},\bm{\varepsilon(u)})_{L^2(\Omega;\mathbb{S}^d)}=\int_{\Omega}\bm{\sigma}\bm{\cdot}\bm{\varepsilon(u)}d\Omega$. From the assumption of $\mbox{meas}(\Gamma_1) > 0$, we know that Korn's inequality holds, i.e.,
$$\|\bm{\varepsilon}(\bm{v})\|_{L^2(\Omega;\mathbb{S}^d)}\geq c\|\bm{v}\|_V,\quad \forall \bm{v}\in V.$$
Here and then, $c$ represents a positive constant which may change from line to line. We can endow with inner product on $ V$ as
$$\langle\bm{u},\bm{v}\rangle_{ V}=\langle\bm{\varepsilon}(\bm{u}),\bm{\varepsilon}(\bm{v})\rangle_{L^2(\Omega;\mathbb{S}^d)}.$$
Let $Y=H^1(\Omega;\mathbb{R})$ and $Y_1=L^2(\Omega;\mathbb{R})$ endowed with the inner products
$$(w,z)_{Y_1}=\int_{\Omega}wzd_{\Omega}, \quad (w,z)_Y=(w,z)_{Y_1}+\int_{\Omega}\nabla w\cdot \nabla zd_{\Omega}.$$
Denote $K_V=\{\bm{v}\in V\:|\:v_\nu\leq g\;\mbox{a.e. on}\;\Gamma_3\}$ and $K_Y=\{u\in Y\:|\:0\leq u\leq 1 \; \mbox{a.e. in}\;\Omega\}$. Then $K_V$ and $K_Y$ are both convex.
Suppose that $\{\bm{u},\bm{\sigma},\zeta,w\}$ are sufficiently smooth functions solving \eqref{eq_mep_1}-\eqref{eq_mep_12} with $t \in I$.
Then, we can employ the following method to derive the variational formulation of Problem \ref{mechanic_problem1}.
Firstly, we use the Green formula and the equation \eqref{eq_mep_1} to obtain
\begin{equation}\label{eq_PV_1}
\langle\boldsymbol{\sigma}(t), \bm{\varepsilon}(\bm{v})-\bm{\varepsilon}(\bm{\bm { \dot{u} }}(t))\rangle_{L^2(\Omega;\mathbb{S}^d)}=\left\langle\bm{f}_{0}(t), \bm{v}-{\bm{\dot{u}}}(t)\right\rangle_{L^2(\Omega;\mathbb{R}^d)}+\int_{\Gamma} \bm{\sigma}(t) \bm{\nu} \cdot(\bm{v}-{\bm{\dot{u}}}(t)) d \Gamma.
\end{equation}
Considering the boundary conditions \eqref{eq_mep_3}, \eqref{eq_mep_4}, \eqref{eq_mep_9} and the formula
$$
\boldsymbol{\sigma}(t) \boldsymbol{\nu} \cdot \boldsymbol{v}=\sigma_{\nu}(t) v_{\nu}+\boldsymbol{\sigma}_{\tau}(t) \cdot \boldsymbol{v}_{\tau},
$$
the equation \eqref{eq_PV_1} can be transformed as
\begin{eqnarray}\label{eq_PV_2}
&&\langle\boldsymbol{\sigma}(t), \bm{\varepsilon}(\bm{v})-\bm{\varepsilon}(\bm{\dot{u}}(t))\rangle_{L^2(\Omega;\mathbb{S}^d)}\nonumber\\
&=&\left\langle\bm{f}_{0}(t), \bm{v}-{\bm{\dot{u}}}(t)\right\rangle_{L^2(\Omega;\mathbb{R}^d)}+\int_{\Gamma_2} \bm{f_2}(t) \cdot(\bm{v}-{\bm{\dot{u}}}(t)) d \Gamma \nonumber-\int_{\Gamma_3}\mu p(w(t),\dot{u}_\nu(t)) \bm{n^*}(t) \cdot(\bm{v}_\tau-{\bm{\dot{u}}_\tau}(t)) d \Gamma\nonumber\\
&&\mbox{}+\int_{\Gamma_3} {\sigma}_\nu(t) \cdot(v_\nu-{\dot{u}_\nu}(t)) d \Gamma.
\end{eqnarray}
From Remark \ref{rem_subdiff} and $\bm{v}\in K_V$, one has
$$\int_{\Gamma_3} {\sigma}_\nu(t) \cdot(v_\nu-{\dot{u}_\nu}(t)) d \Gamma\geq\int_{\Gamma_3}-p(w(t),\dot{u}_\nu(t))(v_\nu-{\dot{u}_\nu}(t)) d \Gamma.$$
Now \eqref{eq_PV_2} becomes
\begin{eqnarray}\label{eq_PV_3}
&&\langle\boldsymbol{\sigma}(t), \bm{\varepsilon}(\bm{v})-\bm{\varepsilon}(\bm{\bm { \dot{u} }}(t))\rangle_{L^2(\Omega;\mathbb{S}^d)}+\int_{\Gamma_3}\mu p(w(t),\dot{u}_\nu(t)) \bm{n^*}(t) \cdot(\bm{v}_\tau-{\bm{\dot{u}}_\tau}(t)) d \Gamma\nonumber \\
&&\mbox{}+\int_{\Gamma_3}p(w(t),\dot{u}_\nu(t))(v_\nu-{\dot{u}_\nu}(t))d \Gamma\nonumber\\
&\geq&\left\langle\bm{f}_{0}(t), \bm{v}-{\bm{\dot{u}}}(t)\right\rangle_{L^2(\Omega;\mathbb{R}^d)}+\int_{\Gamma_2} \bm{f_2}(t) \cdot(\bm{v}-{\bm{\dot{u}}}(t)) d \Gamma.
\end{eqnarray}
Let
$$
j(w,\gamma\bm{\dot{u}},\gamma\bm{v}):={\int_{\Gamma_3}\mu p(w(t),\dot{u}_\nu(t)) \bm{n^*}(t) \cdot\bm{v}_\tau d \Gamma+\int_{\Gamma_3}p(w(t),\dot{u}_\nu(t))v_\nu d \Gamma.}
$$
Then it follows from \eqref{eq_PV_3} that
\begin{eqnarray}\label{eq_PV_4}
&&\langle\boldsymbol{\sigma}(t), \bm{\varepsilon}(\bm{v})-\bm{\varepsilon}(\bm{\bm { \dot{u} }}(t))\rangle_{L^2(\Omega;\mathbb{S}^d)}
+j(w,\gamma\bm{\dot{u}},\gamma\bm v)-j(w,\gamma\bm{\dot{u}},\gamma\bm{\dot{u}})\nonumber\\
&\geq& \left\langle \bm{f_2}(t) ,\gamma\bm{v}-\gamma{\bm{\dot{u}}}(t)\right\rangle_{L^2(\Gamma_3;\mathbb{R}^d)} +\left\langle\bm{f}_{0}(t),\bm{v}-{\bm{\dot{u}}}(t)\right\rangle_{L^2(\Omega;\mathbb{R}^d)},
\end{eqnarray}
where $\gamma :V\rightarrow L^2(\Gamma_3;\mathbb{R}^d)$ is the trace operator. Applying Korn's inequality, the Riesz representation theorem and the trace theorem, we know that there exists $\bm{f}:I\rightarrow V^*$ such that
$$\langle\bm f(t),\bm v-\bm {\dot{u}}\rangle_{V^*\times V}=\left\langle \bm{f_2}(t) ,\gamma\bm{v}-\gamma{\bm{\dot{u}}}(t)\right\rangle_{L^2(\Gamma_3;\mathbb{R}^d)} +\left\langle\bm{f}_{0}(t),\bm{v}-{\bm{\dot{u}}}(t)\right\rangle_{L^2(\Omega;\mathbb{R}^d)}.$$
Similarly, from the integration by parts and the definition of convex subdifferential of $I_{[0,1]}$, it follows that
\begin{eqnarray}\label{eq_PV_5}
\langle\phi(t, \bm{\varepsilon}(\boldsymbol{u}(t)), \zeta(t))-\dot{\zeta}(t), \eta-\zeta(t)\rangle_{L^{2}(\Omega;\mathbb{R})}\leq a(\zeta,\eta-\zeta),
\end{eqnarray}
where $a(\zeta,\eta)=\kappa\int_\Omega\nabla\zeta\cdot\nabla\eta dx$ for all $\zeta,\eta\in Y.$
Now, by integrating these relations and inequalities, the variational formulation of Problem \ref{mechanic_problem1} can be obtained as follows.
\begin{problem}\label{problem_1}
Find $\bm{u}:I\rightarrow K_V$, $\zeta:I\rightarrow K_Y$ and $w:I\rightarrow L^2(\Gamma_3;\mathbb{R})$ such that, for a.e. $t \in I$,
\begin{eqnarray}
\bm{\sigma}(t)= \mathcal{A}(t, \bm{\varepsilon}(\bm{u}(t)))+\int_{0}^{t}\mathcal{B}(t-s, \bm{\varepsilon}(\bm{u}(s)), \zeta(s)) ds+ \mathcal{C}(t, \bm{\varepsilon}(\bm{\dot{u}}(t))) \quad \text { in } \Omega,\quad\\
\langle\boldsymbol{\sigma}(t), \bm{\varepsilon}(\bm{v})-\bm{\varepsilon}(\bm{\bm {\dot{u}(t) }})\rangle_{L^2(\Omega;\mathbb{S}^d)}
+j(w,\gamma\bm{\dot{u}},\gamma\bm v)-j(w,\gamma\bm{\dot{u}},\gamma\bm{\dot{u}})
\geq\left\langle\bm{f}(t), \bm{v}-{\bm{\dot{u}}}(t)\right\rangle_{V^*\times V},\; \forall \bm v\in K_V,\quad\label{eq_u_continuous}\\
\langle(\dot{\zeta}(t), \eta-\zeta(t)\rangle_{Y_1}
+a(\zeta,\eta-\zeta)\geq \langle\phi(t, \bm{\varepsilon}(\boldsymbol{u}(t)),\zeta(t)),\eta-\zeta(t)\rangle_{Y_1}
,\; \forall \eta\in K_Y,\quad\\
\dot{w}(t)=\alpha(t)p(\dot{u}_\nu(t)-w(t))\quad\mbox{ in } \Gamma_3,\quad\label{eq_w_continuous}\\
\bm{u}(0)=\bm{u}_0,w(0)=0,\;\zeta(0)=\zeta_0\in(0,1).\quad
\end{eqnarray}
\end{problem}
To solve Problem \ref{problem_1}, we need the following assumptions.
$H(1):$ The elasticity operator
$\mathcal{A}: \Omega \times I \times\mathbb{S}^{d} \rightarrow \mathbb{S}^{d}$ satisfies
\begin{align}\label{assume:A}
\begin{cases}
(a)\; \mathcal{A}(\cdot, t,\bm{\varepsilon}) \;\mbox{is measurable on}\; \Omega, \mbox{ for all } ( t,\bm{\varepsilon}) \in I\times \mathbb{S}^{d};\\
(b)\; \mathcal{A}(\bm{x},\cdot, \cdot) \;\mbox{is continuous on }I \times\mathbb{S}^{d} \mbox{ for a.e. }\; \bm{x} \in \Omega;\\
(c)\; \mathcal{A}(\bm{x},t, \cdot) \mbox{ is Lipschitz continous with}\; m_{\mathcal{A}}>0\; \mbox{for all }\; t \in I,\; i.e.,\\
\quad\left\|\mathcal{A}\left(\bm{x}, t, \bm{\varepsilon}_{1}\right)-\mathcal{A}\left(\bm{x}, t, \bm{\varepsilon}_{2}\right)\right\|\leq L_{\mathcal{A}}\left\|\bm{\varepsilon}_{1}-\bm{\varepsilon}_{2}\right\|,\;\forall \bm{\varepsilon}_{1}, \bm{\varepsilon}_{2} \in \mathbb{S}^{d}, \mbox{ a.e. } \bm{x} \in \Omega;\\
\end{cases}
\end{align}
$H(2):$ The relaxation operator $\mathcal{B}: \Omega \times I \times \mathbb{S}^{d} \times \mathbb{R} \rightarrow \mathbb{S}^{d}$ satisfies
\begin{align}\label{assume:B}
\begin{cases}
(a) \mathcal{B}(\cdot,t, \bm{\varepsilon},\zeta) \;\mbox{is measurable on}\; \Omega, \mbox{ for all } \bm{\varepsilon} \in \mathbb{S}^{d},\; t\in I \mbox{ and } \zeta\in\mathbb{R};\\
(b) \mathcal{B}(\bm{x}, \cdot,\bm{\varepsilon},\zeta) \mbox{ is continuous on } I\\
\quad\mbox{ for a.e. } \bm{x} \in \Omega \mbox{ and all } (\bm{\varepsilon},\zeta) \in \mathbb{S}^{d}\times \mathbb{R};\\
(c) \mathcal{B}(\bm{x}, t,\cdot,\cdot) \;\mbox{is Lipschitz continuous with}\; L_\mathcal{B}> 0 \mbox{ for all } t\in I \mbox{ and a.e.}\; \bm{x} \in \Omega, i.e.,\\
\quad\|\mathcal{B}(\bm{x},t,\bm{\varepsilon}_{1},\zeta_1)-\mathcal{B}(\bm{x},t,\bm{\varepsilon}_{2},\zeta_2)\| \leq L_{\mathcal{B}}(\|\bm{\varepsilon}_{1}-\bm{\varepsilon}_{2}\|+|\zeta_1-\zeta_2|),\\
\quad\forall \bm{\varepsilon}_{1}, \bm{\varepsilon}_{2} \in \mathbb{S}^{d},\;
\zeta_1,\zeta_2\in\mathbb{R}, \mbox{ a.e. } \bm{x} \in \Omega;\\
(d)\mbox{For all } (t,\bm{\varepsilon},\zeta)\in I\times \mathbb{S}^{d}\times \mathbb{R}\mbox{ and a.e.}\; \bm{x} \in \Omega,\;\mbox{there exixsts a function }\\
\quad\rho_\mathcal{B}\in L^2(I;\mathbb{R}^+) \mbox{ such that } \|\mathcal{B}(\bm{x},t,\bm{\varepsilon},\zeta)\|\leq\rho_\mathcal{B}(t)(|\zeta|+\|\varepsilon\|).
\end{cases}
\end{align}
$H(3):$ The viscosity operator
$\mathcal{C}: \Omega \times I \times\mathbb{S}^{d} \rightarrow \mathbb{S}^{d}$ satisfies
\begin{align}\label{assume:C}
\begin{cases}
(a)\; \mathcal{C}(\cdot, t,\bm{\varepsilon}) \;\mbox{is measurable on}\; \Omega \mbox{ for all } (t,\bm{\varepsilon}) \in I\times \mathbb{S}^{d};\\
(b)\; \mathcal{C}(\bm{x},\cdot, \cdot) \;\mbox{is continuous on }I \times\mathbb{S}^{d} \mbox{ for a.e. }\; \bm{x} \in \Omega;\\
(c)\; \mathcal{C}(\bm{x},t, \cdot) \mbox{ is strongly monotone with}\; m_{\mathcal{C}}>0\; \mbox{for all }\; t \in I,\; i.e.,\\
\quad\left(\mathcal{C}\left(\bm{x},t,\bm{\varepsilon}_{1}\right)-\mathcal{C}\left(\bm{x},t,\bm{\varepsilon}_{2}\right)\right)\cdot\left(\bm{\varepsilon}_{1}-\bm{\varepsilon}_{2}\right) \geq m_{\mathcal{C}}\left\|\bm{\varepsilon}_{1}-\bm{\varepsilon}_{2}\right\|^2,\;\forall \bm{\varepsilon}_{1}, \bm{\varepsilon}_{2} \in \mathbb{S}^{d}, \mbox{ a.e. } \bm{x} \in \Omega;\\
(d)\; \mathcal{C}(\bm{x},t, \cdot) \mbox{ is Lipschitz continous with}\; L_{\mathcal{C}1}>0\; \mbox{for all }\; t \in I,\; i.e.,\\
\quad\left\|\mathcal{C}\left(\bm{x}, t, \bm{\varepsilon}_{1}\right)-\mathcal{C}\left(\bm{x}, t, \bm{\varepsilon}_{2}\right)\right\|\leq L_{\mathcal{C}1}\left\|\bm{\varepsilon}_{1}-\bm{\varepsilon}_{2}\right\|,\;\forall \bm{\varepsilon}_{1}, \bm{\varepsilon}_{2} \in \mathbb{S}^{d}, \mbox{ a.e. } \bm{x} \in \Omega;\\
(e)\; \mathcal{C}(\bm{x},\cdot, \bm{\varepsilon}) \mbox{ is Lipschitz continous with}\; L_{\mathcal{C}2}>0\; \mbox{for all }\; \bm{\varepsilon}\in \mathbb{S}^{d},\; i.e.,\\
\quad\left\|\mathcal{C}\left(\bm{x}, t_1, \bm{\varepsilon}\right)-\mathcal{C}\left(\bm{x}, t_2, \bm{\varepsilon}\right)\right\|\leq L_{\mathcal{C}2}\left\|t_{1}-t_{2}\right\|,\;\forall t_{1}, t_{2} \in I, \mbox{ a.e. } \bm{x} \in \Omega;\\
\end{cases}
\end{align}
$H(4):$ The normal compliance function $p: \Gamma_3 \times\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}^+$ satisfies
\begin{align}\label{assume:p}
\begin{cases}
(a)\; p(\cdot, w,u) \;\mbox{is measurable on}\; \Gamma_3 \mbox{ for all } w,\,u \in \mathbb{R};\\
(b)\; p(\bm{x}, \cdot,\cdot) \;\mbox{is Lipschitz continuous with}\; \tilde{L}_p> 0 \mbox{, a.e.}\; \bm{x} \in \Omega, i.e.,\\
\quad|p(\bm{x},w_1,u_1)-p(\bm{x}, w_2,u_2)| \leq \tilde{L}_p(|w_1-w_2|+|u_1-u_2|),\;\forall w_1,w_2,u_1,u_2 \in \mathbb{R};\\
(c)\; p(\bm{x}, 0,0)=0 \mbox{ for a.e. } \bm{x}\in \Omega \mbox{ and all } r\leq0.
\end{cases}
\end{align}
$H(5):$ The damage source function $\phi: \Omega \times I\times \mathbb{S}^{d} \times \mathbb{R} \rightarrow \mathbb{R}$ satisfies
\begin{align}\label{assume:phi}
\begin{cases}
(a) \phi(\cdot, t,\bm{\varepsilon},\zeta) \;\mbox{is measurable on}\; \Omega,\; \mbox{for all }t\in I,\; \bm{\varepsilon} \in \mathbb{S}^{d}\mbox{ and}\;\zeta\in\mathbb{R};\\
(b) \phi(\bm{x}, t,\cdot,\cdot) \;\mbox{is Lipschitz continuous with}\; L_\phi> 0 \mbox{ for all }t\in I \mbox{ a.e.}\; \bm{x} \in \Omega, i.e.,\\
\quad\|\phi(\bm{x},\boldsymbol{\varepsilon}_{1},\zeta_1)-\phi(\bm{x}, \boldsymbol{\varepsilon}_{2},\zeta_2)\| \leq \tilde{L}_\phi(\|\bm{\varepsilon}_{1}-\boldsymbol{\varepsilon}_{2}\|+|\zeta_1-\zeta_2|)\\
\quad\forall \bm{\varepsilon}_{1}, \bm{\varepsilon}_{2} \in \mathbb{S}^{d},\; \zeta_1,\zeta_2\in\mathbb{R}, \mbox{ a.e. } \bm{x} \in \Omega;\\
(c) \phi(\bm{x},t, \bm{0}_{\mathbb{S}^{d}},0_{\mathbb{R}})\in L^2(I;L^2(\Omega;\mathbb{R})) .
\end{cases}
\end{align}
Moreover, suppose the volume forces and surface tractions satisfy
\begin{equation}\label{assume:f}
\bm{f}_{0} \in C(I;H), \quad \bm{f}_{2} \in C\left(I; L^{2}\left(\Gamma_{2} ; \mathbb{R}^{d}\right)\right).
\end{equation}
To make the sliding condition (\ref{eq_mep_9}) and wear condition (\ref{eq_mep_10}) reasonable, the coefficient of friction, wear coefficient and velocity of the foundation are, respectively, assumed to be
\begin{equation}\label{assume:mu}
\mu\in L^\infty(\Gamma_3;\mathbb{R}),\;\mu(\bm{x})\geq0\;a.e. \;\bm{x}\in \Gamma_3,
\end{equation}
\begin{equation}\label{assume:wear}
k\in L^\infty(\Gamma_3;\mathbb{R}),\;k(\bm{x})\geq0\;a.e. \;\bm{x}\in \Gamma_3,
\end{equation}
and
\begin{equation}\label{assume:v}
\left\{\begin{array}{l}
\boldsymbol{v}^{*} \in C\left(I ; \mathbb{R}^{3}\right) \text { and there exist } v_{1}, v_{2}>0 \text { such that } \\
v_{1} \leq\left\|\boldsymbol{v}^{*}(t)\right\| \leq v_{2}, \;\forall t \in I.
\end{array}\right.
\end{equation}
Clearly, the conditions \eqref{assume:wear} and \eqref{assume:v} suggest that
\begin{equation}\label{assume:n}
\boldsymbol{n}^{*} \in C\left(I ; \mathbb{R}^{3}\right),\;\alpha\in C(I;L^\infty(\Gamma_3;\mathbb{R})).
\end{equation}
Moreover, $g\geq0$ yields that $\bm{0}_V\in K_V$.
For Banach spaces $V=\{\bm{u}\in H^1(\Omega;\mathbb{R}^d)\:|\:\bm{u}=0 \mbox{ on }\Gamma_3\}$ and $Y= H^1(\Omega;\mathbb{R})$, Hilbert spaces $H=L^2(\Omega;\mathbb{R}^d)$ and $Y_1= L^2(\Omega;\mathbb{R})$, by the basic theory of Sobolev spaces, we know that $(V,H,V^*)$ and $(Y,Y_1,Y^*)$ form two Gelfond triplets. Let $X=L^2(\Gamma_3;\mathbb{R}^d) $ and $W=L^2(\Gamma_3;\mathbb{R})$.
Now we give a unique solvability result for Problem \ref{problem_1} as follows.
\begin{theorem}\label{theorem_exist}
Let assumptions \eqref{assume:A}-\eqref{assume:n} hold. If
$$m_{\mathcal{C}}>\|\mu\|_{L^\infty(\Gamma_3;\mathbb{R})}\tilde{L}(p)+\tilde{L}(p),$$
then Problem \ref{problem_1} has a unique solution $(\zeta,\bm{u}_\zeta,w_\zeta)\in (H^1(I;Y_1)\cap L^2(I;Y))\times C(I;K_V)\times C^1(I;W)$.
\end{theorem}
\begin{proof} For any $t\in I$, define operators $A(t,\cdot):V\rightarrow V^*$, $B(t,\cdot,\cdot):V\times Y\rightarrow V^*$, $C(t,\cdot):V\rightarrow V^*$, $F(t,\cdot,\cdot):W\times V\rightarrow W$, $\phi(t,\cdot,\cdot):V\times Y\rightarrow Y^*$, a functional $j(\cdot,\cdot,\cdot):W\times X\times X\rightarrow \mathbb{R}$ and a symmetric bilinear form $a(\cdot,\cdot):Y\times Y\rightarrow \mathbb{R}$ by setting
\begin{align*}
\begin{cases}
\langle A(t,\bm{u}),\bm{v}\rangle_{V^*\times V}=\int_\Omega \mathcal{A}(\bm{x},t,\bm{\varepsilon}(\bm{u}))\bm{\cdot}\bm{\varepsilon}(\bm{v})d\Omega,\\
\langle B(t,\bm{u},\zeta),\bm{v}\rangle_{V^*\times V}=\int_\Omega \mathcal{B}(\bm{x},t,\bm{\varepsilon}(\bm{u}),\zeta)\bm{\cdot}\bm{\varepsilon}(\bm{v})d\Omega,\\
\langle C(t,\bm{u}),\bm{v}\rangle_{V^*\times V}=\int_\Omega \mathcal{C}(\bm{x},t,\bm{\varepsilon}(\bm{u}))\bm{\cdot}\bm{\varepsilon}(\bm{v})d\Omega,\\
F(t,w,\bm{u})=\alpha(t)p(w,u_\nu),\\
\phi(t,\bm{u},\zeta)=\lambda_{D}\left(\frac{1-\zeta}{\zeta}\right)-\frac{1}{2} \lambda_{E}\|\bm{\varepsilon}(\boldsymbol{u})\|^{2}+\lambda_{w},\\
j(w,\bm{r}_1,\bm{r}_2)=\int_{\Gamma_3}\mu p(w,{r_1}_\nu) \bm{n^*}{\bm{r}_2}_{\tau} d\Gamma+\int_{\Gamma_3}p(w,{r_1}_\nu) {r_2}_{\nu} d \Gamma
\end{cases}
\end{align*}
for all $\bm{u},\;\bm{v}\in V$, $\bm{r}_1,\;\bm{r}_2\in X$, $w\in W$ and $\zeta\in Y$. Then Problem \ref{problem_1} can be transformed as follows
\begin{align*}
&\dot{w}(t)=F(t,w(t),\bm{u}(t)),\\
&\left\langle A(t,{\bm{u}}(t)) +\int_0^t B(t-s,{\bm{u}}(s),\zeta(s))ds+C(t,\dot{\bm{u}}(t)),\bm{v}-\dot{\bm{u}}(t) \right\rangle_{V^*\times V}\nonumber\\
& \qquad +j(w(t) ,\dot{\bm{u}},\bm{v})-j(w(t),\dot{\bm{u}}(t) ,\dot{\bm{u}}(t))\geq \langle f(t),\bm{v}-\dot{\bm{u}}(t) \rangle_{V^*\times V},\quad\forall \bm{v}\in K_V ,\\
&\langle \dot{\zeta}(t) ,\eta-\zeta(t) \rangle_{Y_1}+a(\zeta(t) ,\eta-\zeta(t) )\geq\langle\phi(t,u(t) ,\zeta(t) ),\eta-\zeta(t) \rangle_{Y_1},\quad\forall\eta\in K_Y,\\
&\bm{u}(0)=\bm{u}_0,w(0)=0,\zeta(0)=\zeta_0\in(0,1).
\end{align*}
Next we show that assumption conditions H(A)-H(a) are satisfied.
I). It is easy to see that $A(\cdot,\cdot)$ is continuous on $I\times V$ for all $t\in I$. From \eqref{assume:A}(c) and the Holder's inequality, we have
\begin{eqnarray*}
\langle A(t,\bm{u}_1)-A(t,\bm{u}_2),\bm{v}\rangle_{V^*\times V}&\leq&\left(\int_\Omega \|\mathcal{A}(\bm{x},t,\bm{\varepsilon}(\bm{u}_1))
-\mathcal{A}(\bm{x},t,\bm{\varepsilon}(\bm{u}_2))\|^2d\Omega\right)^{\frac{1}{2}}\|\bm{v}\|_V\\
&\leq&L_\mathcal{A}\left(\int_{\Omega}\|\bm{\varepsilon}(\bm{u}_1)-\bm{\varepsilon}(\bm{u}_2)\|^2
d\Omega\right)^{\frac{1}{2}}\|\bm{v}\|_V
\end{eqnarray*}
for all $\bm{u}_1,\;\bm{u}_2,\;\bm{v}\in V$ with $t\in I$. It follows that
$$
\| A(t,\bm{u}_1)-A(t,\bm{u}_2)\|_{V^*}\leq L_\mathcal{A} \|\bm{u}_1-\bm{u}_2\|_V.
$$
This shows that $L_A= L_\mathcal{A} $ in \eqref{assume:A}(c) and so assumption condition H(A) holds.
II). Condition H(B)(a) can be verified from \eqref{assume:B}(b). In addition, by H\"{o}lder's inequality and \eqref{assume:B}(c), one has
\begin{eqnarray*}
\langle B(t,\bm{u}_1,\zeta)-B(t,\bm{u}_2,\eta),\bm{v}\rangle_{V^*\times V}&\leq&\left(\int_\Omega \|\mathcal{B}(\bm{x},t,\bm{\varepsilon}(\bm{u}_1),\zeta)
-\mathcal{B}(\bm{x},t,\bm{\varepsilon}(\bm{u}_2),\eta)\|^2d\Omega\right)^{\frac{1}{2}}\|\bm{v}\|_V\\
&\leq&\sqrt{2}L_\mathcal{B}\left(\int_{\Omega}\|\bm{\varepsilon}(\bm{u}_1)-\bm{\varepsilon}(\bm{u}_2)\|^2
+\|\zeta-\eta\|^2d\Omega\right)^{\frac{1}{2}}\|\bm{v}\|_V
\end{eqnarray*}
and
\begin{eqnarray*}
\langle B(t,\bm{u},\zeta),\bm{v}\rangle_{V^*\times V}&\leq&\left(\int_\Omega\|\mathcal{B}(\bm{x},t,\bm{\varepsilon}(\bm{u}),\zeta)\|^2d\Omega\right)^{\frac{1}{2}}\|\bm{v}\|_V\\
&\leq&\sqrt{2}\rho_{\mathcal{B}}(t)\left(\int_{\Omega}\|\bm{\varepsilon(u)}\|^2+\|\zeta\|^2d\Omega\right)^{\frac{1}{2}}\|\bm{v}\|_V
\end{eqnarray*}
for all $\bm{u}_1,\;\bm{u}_2,\;\bm{u},\;\bm{v}\in V$ and $\zeta,\;\eta\in Y$ with $t\in I$. It follows that
$$
\begin{cases}
\| B(t,\bm{u}_1,\zeta)-B(t,\bm{u}_2,\eta)\|_{V^*}\leq\left(\sqrt{2}L_\mathcal{B}\right) \left(\|\bm{u}_1-\bm{u}_2\|_V+\|\zeta-\eta\|_Y\right),\\
\| B(t,\bm{u},\zeta)\|_{V^*}\leq\left(\sqrt{2}\rho_{\mathcal{B}}(t)\right) \left(\|\bm{u}\|_V+\|\zeta\|_Y\right).
\end{cases}
$$
Thus, $L_B= \sqrt{2}L_\mathcal{B} $ and $\rho(t)= \sqrt{2}\rho_{\mathcal{B}}(t) $ in H(B)(b)(c) and so assumption condition H(B) is true.
III). It is easy to see that $C(\cdot,\cdot)$ is continuous on $I\times V$ for all $t\in I$. From \eqref{assume:C}(c)(d)(e) and H\"{o}lder's inequality, we have
\begin{eqnarray*}
\langle C(t,\bm{u})-C(t,\bm{v}),\bm{u}-\bm{v}\rangle_{V^*\times V}&=&\int_\Omega \left(\mathcal{C}(\bm{x},t,\bm{\varepsilon}(\bm{u}))-\mathcal{C}(\bm{x},t,\bm{\varepsilon}(\bm{v}))\right)
\bm{\cdot}\left(\bm{\varepsilon}(\bm{u})-\bm{\varepsilon}(\bm{v})\right)d\Omega\\
&\geq&m_\mathcal{C}\int_{\Omega}\|\bm{\varepsilon}(\bm{u})-\bm{\varepsilon}(\bm{v})\|^2d\Omega\\
&=&m_\mathcal{C}\|\bm{u}-\bm{v}\|^2_V,\\
\langle C(t,\bm{u}_1)-C(t,\bm{u}_2),\bm{v}\rangle_{V^*\times V}&\leq&\left(\int_\Omega \|\mathcal{C}(\bm{x},t,\bm{\varepsilon}(\bm{u}_1))
-\mathcal{C}(\bm{x},t,\bm{\varepsilon}(\bm{u}_2))\|^2d\Omega\right)^{\frac{1}{2}}\|\bm{v}\|_V\\
&\leq&L_{\mathcal{C}1}\left(\int_{\Omega}\|\bm{\varepsilon}(\bm{u}_1)-\bm{\varepsilon}(\bm{u}_2)\|^2
d\Omega\right)^{\frac{1}{2}}\|\bm{v}\|_V,\\
\langle C(t_1,\bm{u})-C(t_2,\bm{u}),\bm{v}\rangle_{V^*\times V}&\leq&\left(\int_\Omega \|\mathcal{C}(\bm{x},t_1,\bm{\varepsilon}(\bm{u}))
-\mathcal{C}(\bm{x},t_2,\bm{\varepsilon}(\bm{u}))\|^2d\Omega\right)^{\frac{1}{2}}\|\bm{v}\|_V\\
&\leq&L_{\mathcal{C}2}\left(\int_{\Omega}\|t_1-t_2\|^2
d\Omega\right)^{\frac{1}{2}}\|\bm{v}\|_V
\end{eqnarray*}
for all $\bm{u}_1,\;\bm{u}_2,\;\bm{u},\;\bm{v}\in V$ with $t,t_1,t_2\in I$. It follows that
$$
\| C(t,\bm{u}_1)-C(t,\bm{u}_2)\|_{V^*}\leq L_{\mathcal{C}1} \left(\|\bm{u}_1-\bm{u}_2\|_V\right),\quad
\| C(t_1,\bm{u})-C(t_2,\bm{u})\|_{V^*}\leq L_{\mathcal{C}2} \left(\|t_1-t_2\|_V\right).
$$
Thus, $L_{C1}= L_{\mathcal{C}1}$, $L_{C2}= L_{\mathcal{C}2}$ and $m_C=m_{\mathcal{C}}$ in H(C)(a)(b)(c) and so assumption condition H(C) is satisfied.
IV). Since
$$j(w,\bm{r}_1,\bm{r}_2)=\int_{\Gamma_3}\mu p(w, {r_1}_\nu) \bm{n^*}{\bm{r}_2}_\tau + p(w, {r_1}_\nu) {r_2}_\nu d \Gamma,$$
we know that $j(w,\bm{r}_1,\cdot)$ is a convex proper and lower semicontinuous functional with respect to $\bm{r}_2$. Moreover, the convex subdifferential of $j(w,\bm{r}_1,\bm{r}_2)$ with respect to its third variable can be given by
$$\partial j(w,\bm{r}_1,\bm{r}_2)=\int_{\Gamma_3}\mu p(w, {r_1}_\nu) \bm{n^*}+p(w, {r_1}_\nu)\bm{\nu}d\Gamma.$$
This fact combined with \eqref{assume:p}(b)(c) yields that
\begin{eqnarray*}
\|\partial j(w,\bm{r}_1,\bm{r}_2)\|_{X^*}&=&\left(\int_{\Gamma_3}\mu^2 p^2( {w,r_1}_\nu)+ p^2(w, {r_1}_\nu)d\Gamma\right)^\frac{1}{2}\nonumber\\
&\leq&\sqrt{1+\|\mu\|_{L^\infty(\Gamma_3;\mathbb{R})}^2} \left(\int_{\Gamma_3}(p( {w,r_1}_\nu)-p(0,0))^2 d\Gamma\right)^\frac{1}{2}\nonumber\\
&\leq&\sqrt{1+\|\mu\|_{L^\infty(\Gamma_3;\mathbb{R})}^2}\widetilde{L}_p \left(\int_{\Gamma_3}(|w|+|{r_1}_\nu|)^2 d\Gamma\right)^\frac{1}{2}\nonumber\\
&\leq&\sqrt{2+2|\mu\|_{L^\infty(\Gamma_3;\mathbb{R})}^2}\widetilde{L}_p(\|\bm{r}_1\|_X+\|w\|_W).
\end{eqnarray*}
Taking $\sqrt{2+2\|\mu\|_{L^\infty(\Gamma_3;\mathbb{R})}^2}\widetilde{L}_p=c_1$, there exists a constant $c_1>0$ such that
\begin{eqnarray}\label{j}
\|\partial j(w,u,v)\|_X\leq c_1(\|w\|_W+\|u\|_X), \quad \forall (u,v,w)\in X\times X\times W.
\end{eqnarray}
Letting $w_1,\;w_2\in W$ and $\bm{s}_1,\;\bm{s}_2\in X$, one has
\begin{eqnarray}\label{j_ass1}
&&j(w_1,\bm{r}_1,\bm{s}_2)-j(w_1,\bm{r}_1,\bm{s}_1)+j(w_2,\bm{r}_2,\bm{s}_1)-j(w_2,\bm{r}_2,\bm{s}_2)\nonumber\\
&=&\int_{\Gamma_3}\left[\mu p( w_1,{r_1}_\nu) \bm{n^*}({\bm{s}_2}_\tau -{\bm{s}_1}_\tau )+ p(w_1,{r_1}_\nu)( {s_2}_\nu - {s_1}_\nu )+\mu p( w_2,{r_2}_\nu) \bm{n^*}({\bm{s}_1}_\tau -{\bm{s}_2}_\tau )\right.\nonumber\\
&& \mbox{}\left. + p(w_2,{r_2}_\nu)( {s_1}_\nu - {s_2}_\nu )\right]d \Gamma\nonumber\\
&=&\int_{\Gamma_3}\mu \left(p(w_1,{r_1}_\nu)-p(w_2,{r_2}_\nu)\right)\bm{n^*}({\bm{s}_2}_\tau -{\bm{s}_1}_\tau )d\Gamma+\int_{\Gamma_3}( p( w_1,{r_1}_\nu)-p( w_2,{r_2}_\nu))( {s_2}_\nu - {s_1}_\nu )d \Gamma\nonumber\\
&\leq&\int_{\Gamma_3}|\mu| |p(w_1,{r_1}_\nu)-p(w_2,{r_2}_\nu)|\|{\bm{s}_2}_\tau -{\bm{s}_1}_\tau \|d\Gamma+\int_{\Gamma_3} |p( w_1,{r_1}_\nu)-p( w_2,{r_2}_\nu)||{s_2}_\nu -{s_1}_\nu |d\Gamma\nonumber\\
&\leq&\|\mu\|_{L^\infty(\Gamma_3;\mathbb{R})}\int_{\Gamma_3} |p( w_1,{r_1}_\nu)-p( w_2,{r_2}_\nu)|\|{\bm{s}_2}_\tau -{\bm{s}_1}_\tau \|d\Gamma\nonumber\\ &&\mbox{} +\int_{\Gamma_3} |p( w_1,{r_1}_\nu)-p( w_2,{r_2}_\nu)||{s_2}_\nu -{s_1}_\nu |d\Gamma.
\end{eqnarray}
The condition \eqref{assume:p}(b) combined with H\"{o}lder's inequality shows that
\begin{eqnarray*}
&&\int_{\Gamma_3} |p( w_1,{r_1}_\nu)-p( w_2,{r_2}_\nu)|\|{\bm{s}_2}_\tau -{\bm{s}_1}_\tau \|d\Gamma\nonumber\\
&\leq&\tilde{L}_p\int_{\Gamma_3}(|{r_1}_\nu-{r_2}_\nu|+|w_1-w_2|)\|{\bm{s}_2}_\tau -{\bm{s}_1}_\tau \|d\Gamma\nonumber\\
&\leq&\tilde{L}_p(\|{r_1}_\nu-{r_2}_\nu\|_X+|w_1-w_2|_W)\|\bm{s}_2 -\bm{s}_1 \|_X
\end{eqnarray*}
and
\begin{eqnarray}\label{e4.34}
\int_{\Gamma_3} |p( w_1,{r_1}_\nu)-p( w_2,{r_2}_\nu)|\| {s_2}_\nu - {s_1}_\nu \|d\Gamma
\leq\tilde{L}_p(\|{r_1}_\nu-{r_2}_\nu\|_X+|w_1-w_2|_W)\|\bm{s}_2 -\bm{s}_1 \|_X.
\end{eqnarray}
Then inequality \eqref{j_ass1} becomes
\begin{eqnarray}\label{e5.13}
&&j(w_1,\bm{r}_1,\bm{s}_2)-j(w_1,\bm{r}_1,\bm{s}_1)+j(w_2,\bm{r}_2,\bm{s}_1)-j(w_2,\bm{r}_2,\bm{s}_2)\nonumber\\
&\leq&(\|\mu\|_{L^\infty(\Gamma_3;\mathbb{R})}+1)\tilde{L}_p\left(\|w_1-w_2\|_W\|\bm{s}_1-\bm{s}_2\|_X+\|\bm{r}_1-\bm{r}_2\|_X\|\bm{s}_1-\bm{s}_2\|_X\right).
\end{eqnarray}
This implies that H(j)(b) holds with
$$\alpha_0=(\|\mu\|_{L^\infty(\Gamma_3;\mathbb{R})}+1)\tilde{L}_p, \quad \alpha_1=(\|\mu\|_{L^\infty(\Gamma_3;\mathbb{R})}+1)\tilde{L}_p.$$
and so assumption condition H(j) is satisfied.
V). It follows from \eqref{e4.34} that
\begin{eqnarray*}
\|F(t,w_1,\bm{u}_1)-F(t,w_2,\bm{u}_2)\|^2_W&=&\int_{\Gamma_3}\alpha(t)^2(p(w_1,{u_1}_\nu)-p(w_2,{u_2}_\nu))^2d\Gamma\nonumber\\
&\leq& 2 \|\alpha\|_{C(I;L^\infty(\Gamma_3;\mathbb{R}))}^2\tilde{L}_p\left(\|\bm{u}_1-\bm{u}_2\|_X^2+\|w_1-w_2\|_W^2\right)
\end{eqnarray*}
and so
\begin{eqnarray*}
\|F(t,w_1,\bm{u}_1)-F(t,w_2,\bm{u}_2)\|_W\leq\sqrt{2\tilde{L}_p} \|\alpha\|_{C(I;L^\infty(\Gamma_3;\mathbb{R}))} (\|\bm{u}_1-\bm{u}_2\|_V+\|w_1-w_2\|_W).
\end{eqnarray*}
Thus, H(F)(b) holds with
$$L_p=\max\left\{\sqrt{2\tilde{L}_p}\|\alpha\|_{C(I;L^\infty(\Gamma_3;\mathbb{R}))}
,\sqrt{2\tilde{L}_p}\|\alpha\|_{C(I;L^\infty(\Gamma_3;\mathbb{R}))}\right\}.$$
Clearly, H(F)(a) follows from condition \eqref{assume:n} and so assumption condition H(F) is fulfilled.
VI). Based on \eqref{assume:phi}, we can conclude that $H(\phi)$ holds with $L_\phi=\sqrt{2}\tilde{L}_\phi$.
VII). Finally, we can check that H(a) holds. Indeed, from
$$\|\zeta\|_Y^2=\|\zeta\|^2_{Y_1}+\int_\Omega \nabla\zeta\cdot\nabla\zeta d\Omega,$$
it follows that $a(\zeta,\zeta)+\kappa\|\zeta\|^2_{Y_1}=\kappa\|\zeta\|_Y^2$ and so H(a) is satisfied with $a_1=a_2=\kappa.$
Thus, combining I)-VII), we know that Theorem \ref{theorem_exist} is a direct consequence of Theorem \ref{t3.1}.
\end{proof}
\section*{Declarations}
{\bf Conflicts of interest/Competing interests} The authors have no known competing financial interests or personal
relationships that could have appeared to influence the work reported in this manuscript.
\end{document}
|
math
|
An introduction to the Sydney Mining Club members was given in November 2015 to the wavetheflag.org.au website. The aim of the website is to bring all stakeholders into a social media setting to discuss views on the development of Australia’s natural resources.
Additionally the website is to create awareness of the issues that surround the political and governmental processes that help or hinder the development of our resources, and a “voice for reason” exposing issues of “sovereign risk” limiting our economic growth and prosperity.
|
english
|
१०/२३/२०१६ १२:४६:३५ आम
धनतेरस के दिन इस मंत्र का पाठ करने से हर तरह के आर्थिक संकट दूर होते हैं। बाद में इस मंत्र का प्रात:काल प्रतिदिन दीपक जलाकर जाप करने से घोर आर्थिक संकटों से भी राहत मिलती है। मंत्र: ॐ भूरिदा भूरि देहिनो, मा दभ्रं भूर्या भर।<
१०/५/२०१६ ३:५५:0३ प्म
कहतें हैं देवासुर संग्राम के सेनापति भगवान स्कंद की माता होने के कारण मां दुर्गा के पांचवें स्वरूप को स्कंदमाता के नाम से जाना जाता है. आज नवरात्रि का पांचवा दिन है. स्कंदमाता भक्तों को सुख-शांति प्रदान करने वाली देवी है. भगवान कार्तिकेय का एक नाम स्कन्द भी है, जिसके चलते माँ
आजकल समय की कमी के कारण नवरात्रि में कई भक्तगण पूरे विधि-विधान से पूजा नहीं कर सकते. तो ऐसे में आप माँ दुर्गा के नामों का जाप कर सकते हैं. मां दुर्गा के कई रूप हैं. ज्योतिषियों के अनुसार अगर आपको अपनी व्यस्तताओं के चलते मां की आराधना वक्त नहीं मिल पा रहा है तो मां के १०८ नामों
१०/३/२०१६ ११:५१:१७ आम
भगवान ने मानव की रचना करने के बाद उसे धरती पर भेज दिया और अपनेअपने कर्मों के मुताबिक उन्हें परिणाम देने लगे. लेकिन कई बार इन्सान मेहनत करता, बार-बार कोशिश करता है फिर भी सफल नहीं हो पाता. इन नवरात्रि के मौके पर आप अपने जीवन से जुड़ी परेशानियों को दूर भगा सकते हैं. देवी म
या देवी सर्वभूतेषु ब्रह्मचारिणी रूपेण संस्थिता। नमस्तस्यै नमस्तस्यै नमस्तस्यै नमो नम:।। अवतरण वर्णन: पौराणिक मत के अनुरूप नवदुर्गा के दूसरे स्वरूप में मां ब्रह्मचारिणी की पूजा कि जाती है, आदिशक्ति दुर्गा का द्वितीय स्वरूप साधको को अनंत शक्ति देन
नवरात्रि माँ दुर्गा की पूजा शक्ति उपासना का पर्व है. माना जाता है कि नवरात्रि में ब्रह्मांड के सारे ग्रह एकत्रित होकर सक्रिय हो जाते हैं. कई बार इन ग्रहों का दुष्प्रभाव मावन जीवन पर भी पड़ता है. इसी दुष्प्रभाव से बचने के लिए नवरात्रि में माँ दुर्गा की पूजा की जाती है. माँ
भगवान श्री गणेश के इन १० नामों के जाप मात्र से हो जाते हैं सारे दुख दूर
९/२७/२०१६ ५:५7:४६ प्म
भगवान गणेश को देवताओं में सबसे पहले पूजा जाता है यह बात तो हम सभी जानतें हैं। लेकिन आप जानतें हैं कि भगवान गणेश के कुल १०८ नाम हैं। कहतें हैं की अगर आप हर बुधवार सुबह स्नान के बाद गणेश भगवान के १०८ नामों में से सिर्फ १० का भी उच्चारण करतें हैं तो आपके सारे संकट दूर हो जाएंगे।
|
hindi
|
एक सुरक्षित स्थान | हमारी प्रतिदिन की रोटी हिन्दी और डेली ब्रेड
एक सुरक्षित स्थान
पढ़ें: भजन संहिता ४६| एक साल में बाइबिल: यहेजकेल ४२ ; यहेजकेल ४३ ; यहेजकेल ४४ ; १ यूहन्ना १
परमेश्वर हमरा शरणस्थान और बल है, संकट में अति सहज से मिलनेवाला सहायक ल इस कारण हम को कोई भय नहीं ल भजन ४६:१-२
मेरा भाई और मैं पश्चिम वर्जिनिया में एक वृक्षयुक्त पहाड़ी की ढाल पर पले और बड़े हुए जिसने हमारी कल्पनाओं के लिए उपजाऊ परिदृश्य प्रदान किया ल चाहे टार्जन की तरह बेलों से झुलना हो या स्विस परिवार रोबिनसन की तरह ट्री हाउस बनाना हो, हमने उन कहानियों के परिदृश्यों को निभाया जो हमने पढ़ी थीं और जो फिल्मों में देखा था ल हमारे पसंदीदा में से एक था किले बनाकर मान लेना कि हम आक्रमण से सुरक्षित हैं ल वर्षों बाद, मेरे बच्चे काल्पनिक शत्रुओं से बचाव के लिए अपने लिए कम्बलों, चादरों, और तकियों से किले अर्थात् सुरक्षित स्थान बनाए ल एक छिपने का स्थान स्वाभाविक महसूस होता है जहाँ आप सुरक्षित और महफूज़ महसूस कर सकें ल
जब इस्राएल का गायक-कवि राजा दाऊद, एक सुरक्षित स्थान ढूंढता था, वह परमेश्वर के आलावा और कहीं नहीं गया ल भजन ४६:१-२ दावा करते हैं, परमेश्वर हमारा शरणस्थान और बल है, संकट में अति सहज से मिलनेवाला सहायक ल इस कारण हम को कोई भय नहीं ल जब हम दाऊद के जीवन के विषय पुराना नियम के वृतान्त पर विचार करते हैं, और लगभग निरंतर खतरों का सामना करता था, ये शब्द परमेश्वर में भरोसा का एक अद्भुत स्तर प्रगट करते हैं ल उन खतरों के बावजूद, वह निश्चित था उसकी सच्ची सुरक्षा परमेश्वर में ही थी ल
हम भी उस भरोसे को जान सकते हैं ल परमेश्वर जो हमें कभी नहीं छोड़ने या त्यागने का वादा करता है (इब्रानियों १३:५) हम प्रतिदिन अपने जीवन से उसी पर भरोसा रखते हैं ल यद्यपि हम एक खतरनाक संसार में रहते हैं, हमारा परमेश्वर हमें शांति और आश्वासन देता हैं अभी और हमेशा के लिए ल वह हमारा सुरक्षित स्थान है ल
परमेश्वर आज हमारा शरणस्थान है, उसको धन्यवाद दें ल
द्वारा बिल क्राऊडर | दूसरे लेखक को देखें
|
hindi
|
गुंजन मानकतला पति के शो 'भाग' का हिस्सा बनना चाहती है
गुंजन मानकतला पति के शो गुलाम का हिस्सा बनना चाहती है
विक्कास गुलाम में वीर की भूमिका निभा रहे है
गुंजन मानकतला बनना चाहती है गुलाम का हिस्सा
अभिनेत्री गुंजन मानकतला, जिनके पति और अभिनेता विक्कास मानकतला वर्तमान में गुलाम में दिख रहे हैं, को लोकप्रिय टीवी शो का हिस्सा बनने की उम्मीद है। गुंजन का कहना है कि वो गुलाम धारावाहिक का हिस्सा बनना चाहती है| उन्होंने कहा शॉ जिस तरह से उचाईयों को छू रहा है| अब मुझे भी लगता है कि में भी इस शॉ का हिस्सा बनू|
गुंजन ने एक बयान में कहा, मैंने हमेशा विक्कास के काम को पसंद किया है| वह चयनात्मक है और एक परियोजना के लिए हां कहने में कुछ समय लगाते है। उनकी यह आदत मझे काफी अच्छी लगती है| वो जल्दी से किसी भी भूमिका की हाँ नहीं करते| जैसा कि और लोग करते है| विक्कास हर भूमिका के लिए काफी समय लेते है| उसको पूरा रीड करने के बाद ही वो उचित निर्णय लेते है|
जब उन्होंने मुझे गुलाम की कहानी सुनाई तो मैं निश्चित ही प्रभावित हुई थी| लेकिन अब जब मैं इसे गति में देखती हूं| तो मुझे लगता है कि इसके यथार्थवादी अभी तक ताजा सामग्री के साथ है क्योंकि इससे पहले किसी और ने इस तरह की सामग्री को टेलीविजन पर नहीं दिखाया है। वह बोल्ड, साहसी, यथार्थवादी और अभी तक ताजा है| मैं गुलाम जैसे शो का हिस्सा बनना चाहती हूं, उसने कहा। उन्होंने कहा अगर मुझे मौका मिला तो में इस शॉ को जरूर करुँगी|
गुंजन मानकतला पहले से ही सात फेरे-सलोनी का सफर, केसर, घर की लक्ष्मी बेटियां और नागिन जैसी शो में दिखाई दी है। उनकी दमदार एक्टिंग को दर्शको ने काफी पसंद किया है| गुलाम, जिसमें निती टेलर और परम सिंह भी शामिल हैं। शॉ दर्शको को धीरे-धीरे पसंद आ रहा है| उसकी टीआरपी दिन-प्रतिदिन बढ़ रही है|
अम्बेडकर जयंती जुलूस पर सहारनपुर में सांप्रदायिक हिंसा
|
hindi
|
مادئہ تٲریخ بوز جٹھ پٹھ عربی پٲٹھٮ۪ن کٲشُر رَٹھ
|
kashmiri
|
// Copyright (c) Microsoft Corporation. All rights reserved.
// Licensed under the MIT License. See License.txt in the project root for license information.
using System.Reflection;
using System.Resources;
using System.Runtime.InteropServices;
[assembly: AssemblyTitle("Microsoft Azure Management Authorization Library")]
[assembly: AssemblyDescription("Provides Microsoft Management Authorization access.")]
[assembly: AssemblyVersion("2.0.0.0")]
[assembly: AssemblyFileVersion("2.11.0.0")]
[assembly: AssemblyConfiguration("")]
[assembly: AssemblyCompany("Microsoft")]
[assembly: AssemblyProduct("Microsoft Azure .NET SDK")]
[assembly: AssemblyCopyright("Copyright (c) Microsoft Corporation")]
[assembly: AssemblyTrademark("")]
[assembly: AssemblyCulture("")]
[assembly: NeutralResourcesLanguage("en")]
|
code
|
#!/bin/bash
# Copyright 2019 NOKIA
#
# All Rights Reserved
#
# 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.
function configure_nuage_horizon {
local local_settings=$HORIZON_DIR/openstack_dashboard/local/local_settings.py
_horizon_config_set $local_settings "" HORIZON_CONFIG[\'customization_module\'] \"nuage_horizon.customization\"
local horizon_conf=$(apache_site_config_for horizon)
if is_ubuntu || is_fedora || is_suse; then
sudo sh -c "sed -i '/Alias \/dashboard\/static/i \ \ \ \ Alias $HORIZON_APACHE_ROOT/static/nuage /opt/stack/nuage-openstack-horizon/nuage_horizon/static' $horizon_conf"
sudo sh -c "sed -i '/<Directory \/opt\/stack\/horizon/i \ \ \ \ <Directory /opt/stack/nuage-openstack-horizon/>\n Options Indexes FollowSymLinks MultiViews\n AllowOverride None\n Require all granted\n </Directory>' $horizon_conf"
else
exit_distro_not_supported "horizon apache configuration"
fi
}
if [[ "$1" == "stack" ]]; then
if [[ "$2" == "install" ]]; then
echo_summary "Installing Nuage Horizon plugin"
pip_install -r $NUAGE_HORIZON_DIR/requirements.txt -e $NUAGE_HORIZON_DIR -c $REQUIREMENTS_DIR/upper-constraints.txt
elif [[ "$1" == "stack" && "$2" == "post-config" ]]; then
configure_nuage_horizon
fi
elif [[ "$1" == "unstack" ]]; then
# no-op
:
fi
|
code
|
لہذا کوٚرنس زبردست استقبال تہٕ دِیُتنس سؠٹھہٕے زیادٕ عزت
|
kashmiri
|
بُجہِ ووٚنُکھ بۂ کرٕ تہٕ درٛایہِ
|
kashmiri
|
कलेक्टर भाजपा के इशारे पर कर रहीं हैै काम, सुनवाई नहीं कर रहा चुनाव आयोग: पीयूष शर्मा | शिवपुरी न्यूज,
शिवपुरी। मध्यप्रदेश में होने वाले विधानसभा चुनाव को लेकर सभी दल अपने अपने स्तर से तैयारीयां कर रहे है। इसी के चलते आज शिवपुरी विधानसभा से आप पार्टी के प्रत्याशी पीयूष शर्मा ने प्रेसवार्ता कर अपना दर्द मीडिया के सामने रखते हुए निष्पक्ष चुनाव न होने और भाजपा पर चुनाव को प्रभावित कर प्रशासनीय मशीनरी का उपयोग चुनाव में करने का आरोप मढा है।
आप पार्टी के प्रत्याशी ने आरोप लगाया है कि कलेक्टर शिल्पा गुप्ता और महिला एवं बाल विकास विभाग के डीपीओ ओपी पाण्डे शिवपुरी में भाजपा के लिए काम कर रहे है। इस बात की शिकायत वह चुनाव आयोग की ऑनलाईन साईड पर भी कर चुके है। परंतु वहां भी उनकी सुनवाई नही हो रही।
आज प्रेस बार्ता में शिवपुरी से आम आदमी पार्टी के प्रत्याशी पीयूष शर्मा ने आरोप लगाते हए कहा है कि महिला एवं बाल विकास विभाग में डीपीओ के पद पद पदस्थ ओपी पाण्डे पिछले ५ साल से शिवपुरी में पदस्थ है। वह पद का दुरूपयोग करते हुए भाजपा का प्रचार कर रहे है। और सरकारी योजनाओं का प्रचार प्रसार भी कर रहे है।
आरोप लगाते हुए कहा है कि ओपी पाण्डे शिवपुरी में आप पार्टी जिलाध्यक्ष पीयूष शर्मा और प्रदेश संगठन प्रभारी हिमांशु कुलश्रेष्ठ ने एक पत्रकारवार्ता कर आरोप लगाए कि शिवपुरी कलेक्टर शिल्पा गुप्ता और महिला एवं बाल विकास विभाग के डीपीओ ओपी पांडे ने पिछले दिनों खुलेआम भाजपा सरकार की सुकन्या योजना का प्रचार किया। इसकी शिकायत भी की लेकिन अभी तक निर्वाचन आयोग ने कोई कार्रवाई इन दोनों अधिकारियों के खिलाफ नहीं की।
आप पार्टी के जिलाध्यक्ष पीयूष शर्मा ने बताया कि कलेक्टर शिल्पा गुप्ता ने आचार संहिता का उल्लंघन किया है। इसके अलावा महिला बाल विकास विभाग के डीपीओ ओपी पाण्डे जो बीते पांच वर्षों से जिला मुख्यालय पर पदस्थ है और उन्हीं के द्वारा अभी कुछ दिनों पूर्व ही १६ अक्टूबर को आचार संहिता को दरकिनार करते हुए सुकन्या योजना का प्रचार-प्रसार किया गया जो कि भाजपा सरकार की महती योजना है। उन्होंने बताया कि महिला एवं बाल विकास विभाग के डीपीओ ओपी पांडे बीते पांच साल से शिवपुरी में पदस्थ हैं और पूर्व में इन्होंने वर्ष २०१३ का विधानसभा चुनाव शिवपुरी में पदस्थ रहते हुए कराया इसके बाद वर्ष २०१४ का लोकसभा चुनाव कराया और इसके बाद वर्ष २०१८ में जब कोलारस उपचुनाव हुआ तो यह यही पदस्थ रहे।
आप पार्टी का आरोप है कि ओपी पांडे शिवपुरी में भाजपा का प्रचार कर रहे हैं और वर्तमान विधानसभा चुनाव में भी यहां पर इन्हें मुख्य दायित्व व कार्य सौंपा गया है और यह चुनाव को प्रभावित कर रहे हैं। इसलिए इन्हें तत्काल शिवपुरी जिले से मुक्त कर अन्यत्र भेजा जाए और इन पर कार्रवाई हो। आप पार्टी के प्रदेश संगठन प्रभारी हिमांशु कुलश्रेष्ठ ने पूरे प्रदेश में आप पार्टी को प्रचार से संबंधित अनुमतियां देने में आनाकानी की जा रही है। इसके अलावा निर्वाचन से जुड़े अधिकारी भाजपा सरकार के दबाव में काम कर रहे हैं।
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hindi
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/* This is script-generated code. */
/* See fileStageToCache.h for a description of this API call.*/
#include "fileStageToCache.hpp"
int
rcFileStageToCache( rcComm_t *conn, fileStageSyncInp_t *fileStageToCacheInp ) {
int status;
status = procApiRequest( conn, FILE_STAGE_TO_CACHE_AN,
fileStageToCacheInp, NULL, ( void ** ) NULL, NULL );
return ( status );
}
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code
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चीनी मोबाइल कंपनी हुवी ने ६ग इंटरनेट स्पीड सेवा पर काम शुरू किया, ६ग इंटरनेट स्पीड सेवा कब होगी शुरू
चीनी मोबाइल कंपनी हुआवेई ने इंटरनेट स्पीड ६ जी पोस्ट -५ जी पर काम शुरू कर दिया है, चीनी मोबाइल कंपनी हुवी ने ६ग इंटरनेट स्पीड सेवा पर काम शुरू किया
चीनी मोबाइल कंपनी हुवी ने इंटरनेट स्पीड ६ग पोस्ट -५ग पर काम शुरू कर दिया है।
६ग इंटरनेट स्पीड सेवा पर काम शुरू
एक विदेशी समाचार एजेंसी की रिपोर्ट के अनुसार, चीन की मोबाइल कंपनी हुवी ने ओटावा में आर एंड डी सेंटर में टेक्नोलॉजी की जांच शुरू की है।
रिपोर्ट के अनुसार, अमेरिकी उपायों के बाद, ५ग तकनीक हुवी के लिए समस्या खड़ी कर सकती है, यह देखते हुए कि कंपनी ने ६ग सेवा पर काम शुरू किया और इसे जल्द से जल्द पूरा किया जाएगा।
रिपोर्ट के अनुसार, ओटावा में १३ विश्वविद्यालयों और अनुसंधान संस्थानों के साथ हुवी ने अनुसंधान एवं विकास केंद्र में ६ग तकनीक पर शोध शुरू किया। कंपनी के प्रवक्ता के अनुसार, लॉन्च वर्तमान में है क्योंकि ६ग टेक्नोलॉजी २०३० से पहले व्यावसायिक आधार पर उपलब्ध नहीं होगी।
टेक्नोलॉजी -अवलोकन करने वाले विशेषज्ञों का मानना है कि हुवी ६ग नेटवर्क पर अन्य कंपनियों के साथ काम कर रहा है और गूगल के अंड्रॉयड सिस्टम से सबसे अधिक लाभ उठाने के लिए नए एप्लिकेशन।
याद रखें कि उस गवर्नमेंट ने सुरक्षा चिंताओं के कारण हुवी पर प्रतिबंध लगाए थे, जिसके बाद कंपनी ने अंड्रॉयड पर भी भरोसा किया, और हुवी ने अपने ओपरेटिंग सिस्टम को पेश किया।
५ग सेवा कब शुरू की
याद रखें कि अप्रैल में, यूएस और दक्षिण कोरिया में मोबाइल फोन कंपनियों ने दुनिया में पहले ५ग सेवा प्रदाता होने का दावा किया था। वायरलेस टेक्नोलॉजी ५ग की प्रारंभिक रूपरेखा २०१४ में साउथ कोरिया में पेश की गई थी और इसे २०२० तक दुनिया भर में प्रचारित करने की घोषणा की गई थी, हालांकि विशेषज्ञों ने एक साल पहले सफलता हासिल की, जिसके बाद विभिन्न देशों में नियमित रूप से सेवा पेश की गई।
टेक्नोलॉजी विशेषज्ञों के अनुसार, ५ग तकनीक से स्मारफोन पर इंटरनेट की गति ५0 गुना बढ़ जाएगी, जिससे उपयोगकर्ता केवल एक सेकंड में कई मेगाबाइट की फिल्म डाउनलोड कर सकते हैं।
टेक्नोलॉजी मगन: चीनी मोबाइल कंपनी हुवी ने ६ग इंटरनेट स्पीड सेवा पर काम शुरू किया, ६ग इंटरनेट स्पीड सेवा कब होगी शुरू
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hindi
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\begin{document}
\title{Generalized SOR iterative method for a class of complex symmetric
linear system of equations
}
\author{Davod Khojasteh Salkuyeh \and Davod Hezari \and Vahid Edalatpour}
\authorrunning{D. K. Salkuyeh, D. Hezari and V. Edalatpour}
\institute{D. K. Salkuyeh \at
Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran\\
\email{khojasteh@guilan.ac.ir, salkuyeh@gmail.com} \\[2mm]
D. Hezari \at
Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran\\
\email{hezari\_h@yahoo.com}\\[2mm]
V. Edalatpour \at
Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran\\
\email{vedalat.math@gmail.com}
}
\date{Received: date / Accepted: date}
\maketitle
\begin{abstract}
In this paper, to solve a broad class of complex symmetric linear systems, we recast the complex system in a real formulation and apply the generalized successive overrelaxation (GSOR) iterative method to the equivalent real system. We then investigate its convergence properties and determine its optimal iteration parameter as well as its corresponding optimal convergence factor. In addition, the resulting GSOR preconditioner is used to preconditioned Krylov subspace methods such as GMRES for solving the real equivalent formulation of the system. Finally, we give some numerical experiments to validate the theoretical results and compare the performance of the GSOR method with the modified Hermitian and skew-Hermitian splitting (MHSS) iteration.
\keywords{ complex linear systems \and symmetric positive
definite \and SOR \and HSS \and MHSS}
\subclass{65F10 \and 65F15.}
\end{abstract}
\section{Introduction}
\label{SEC1}
Consider the system of linear equations
\begin{equation}\label{1.1}
Au= b, \qquad A\in\mathbb{C}^{n\times n},\quad
u,b\in\mathbb{C}^{n},
\end{equation}
where $A$ is a complex symmetric matrix of the form
\begin{equation}\label{1.2}
A=W+iT, \hspace{.3cm} (i=\sqrt{-1})
\end{equation}
and $W,T\in\mathbb{R}^{n\times n}$ are symmetric matrices with at least one of them being positive
definite. Here, we mention that the assumptions of \cite{Bai-G,MHSS} are stronger than ours, where the author assumed that both of the matrices $W$ and $T$ are symmetric positive semidefinite matrices with at least one of them being positive definite.
Hereafter, without lose of generality, we assume that $W$ is symmetric positive definite. Such systems arise in many problems in scientific
computing and engineering such as FFT-based solution of certain time-dependent PDEs \cite{Bertaccini}, structural dynamics
\cite{Feriani}, diffuse optical tomography \cite{Arridge}, quantum mechanics \cite{Dijk} and molecular scattering
\cite{Poirier}. For more applications of this class of complex symmetric systems, the reader is referred to \cite{Benzi-B} and references
therein.
Bai et al. in \cite{Bai-G} presented the Hermitian/skew-Hermitian splitting (HSS) method to solve non-Hermitian positive definite system of linear equations. After that, this method gains people's attention and proposed different variants of the method. Benzi and Gloub in \cite{Benzi-Golub-SIMAX} and Bai et al. in \cite{Bai-Golub-Pan} have applied the HSS method to solve saddle point problem or as a precondtioner. The normal/skew-Hermitian splitting (NSS) method has been presented by Bai et al. in \cite{Bai-Golub-Ng}. Moreover, Bai et al. \cite{PSS} have presented the positive definite and skew-Hermitian splitting (PSS) method to solve positive definite system of linear equations. Lopsided version of the HSS (LHSS) method has been presented by Li et al. in \cite{LHSS}. More recently Benzi in \cite{GHSS} proposed a generalization of the HSS method to solve positive definite system of linear equation.
We observe that the matrix $A$ naturally possesses a Hermitian/skew-Hermitian (HS) splitting
\begin{equation}\label{1.3}
A=H+S,
\end{equation}
where
\[
H=\frac{1}{2}(A+A^*)=W \quad {\rm and} \quad S=\frac{1}{2}(A-A^*)=iT,
\]
with $A^*$ being the conjugate transpose of $A$. Based on the HS splitting (\ref{1.3}), the HSS method to solve (\ref{1.1}) can be written as:
\noindent{\it \textbf{The HSS iteration method}:} \emph{Given an initial guess} $u^{(0)}
\in \mathbb{C}^{n}$ \emph{and positive constant} $\alpha$, \emph{for} $k=0, 1, 2 \ldots,$ \emph{until} \{$u^{(k)}$\}
\emph{converges, compute}
\begin{equation}\label{1.4}
\left\{
\begin{array}{ll}
(\alpha I+W)u^{(k+\frac{1}{2})}=(\alpha I-iT)u^{(k)}+b, & \\
(\alpha I+iT)u^{(k+1)}=(\alpha I-W)u^{(k+\frac{1}{2})}+b, &
\end{array}
\right.
\end{equation}
\emph{where $I$ is identity matrix.}
Since $W\in \mathbb{R}^{n \times n}$ is symmetric
positive definite, we know from \cite{Bai-G} that if $T$ is positive semidefinite then the HSS
iterative method is convergent for any positive constant $\alpha$.
In each iteration of the HSS method to solve (\ref{1.1}) two
sub-systems should be solved. Since $\alpha I+W$ is symmetric
positive definite, the first sub-system can be solved exactly by
the Cholesky factorization of the coefficient matrix and in the
inexact version, by the conjugate gradient (CG) method. In the
second half-step of an iteration we need to solve a shifted
skew-Hermitian sub-system in each iteration. This sub-system can
be solved by a variant of Bunch-Parlett factorization
\cite{Golub-V} or inexactly by a Krylov subspace iteration scheme
such as the GMRES method \cite{Saad,GMRES} which is the main problem of
HSS method.
Bai et al. \cite{MHSS} recently have presented the following
modified Hermitian and skew-Hermitian splitting (MHSS) method to
iteratively compute a reliable and accurate approximate solution
for the system of linear equations (\ref{1.1}). This method may be run as following:
\noindent{\it \textbf{The MHSS iteration method:}} \emph{Given an initial guess $u^{(0)}
\in \mathbb{C}^{n}$ and positive constant $\alpha$, for $k=0, 1, 2 \ldots,$ until \{$u^{(k)}$\}
converges, compute}
\begin{equation}\label{1.5}
\left \{\begin{array}{ll}
(\alpha I+W)u^{(k+\frac{1}{2})}=(\alpha I-iT)u^{(k)}+b,\\
(\alpha I+T)u^{(k+1)}=(\alpha I+iW)u^{(k+\frac{1}{2})}-ib.
\end{array}\right.
\end{equation}
In \cite{MHSS}, it has been shown that if $T$ is symmetric positive semidefinite then the MHSS iterative method is convergent for any positive constant $\alpha$. Obviously both of the matrices $\alpha I+W$ and $\alpha I+T$ are symmetric positive definite. Therefore, the two sub-systems involved in each step of the MHSS
iteration can be solved effectively by using the Cholesky factorization of the matrices $\alpha I+W$ and $\alpha I+T$. Moreover, to solve both of the sub-systems in the inexact variant of the MHSS method one can use a preconditioned conjugate gradient scheme. It is necessary to mention that the right hand
side of the sub-systems are still complex and the MHSS method involves complex arithmetic. Numerical results presented in \cite{MHSS} show that the MHSS method in general is more effective than the HSS, GMRES, GMRES(10) and GMRES(20) methods in terms of both iteration count and CPU time. More recently Bai et al. in \cite{PMHSS} proposed a preconditioned version of the MHSS method.
The well-known successive overrelaxation (SOR) method is a basic iterative method which is popular in engineering and science applications. For example, it has been used to solve augmented linear systems \cite{SORlike,TSORlike,MSORlike} or as a preconditioner \cite{SORP1,SORP2}. In \cite{SSORlike}, Zheng et al. applied the symmetric SOR-like method to solve saddle point problems. In \cite{GSOR}, Bai et al. have presented a generalization of the SOR method to solve augmented linear systems.
In this paper, by equivalently recasting the complex system of
linear equations (\ref{1.1}) in a 2-by-2 block real linear
system, we define the generalized successive overrelaxation
(GSOR) iterative method to solve the equivalent real system.
Then, for the GSOR method, convergence conditions are derived and
determined its optimal iteration parameter and corresponding
optimal convergence factor. Besides its use as a solver, the GSOR
iteration is also used as a preconditioner to accelerate Krylov
subspace methods such as GMRES. Numerical experiments which use
GSOR as a preconditioner to GMRES, show a well-clustered spectrum
(away from zero) that usually lead to speedy convergence of the
preconditioned iteration. In the new method two sub-systems with
coefficient matrix $W$ should be solved which can be done by the
Cholesky factorization or inexactly by the CG algorithm.
Moreover, the right-hand side of the sub-systems are real.
Therefore, the solution of the system can be obtained by the real
version of the algorithms.
The rest of the paper is organized as follows. In Section 2 we
propose our method and investigate its convergence properties.
Section 3 is devoted to some numerical experiments to show the
effectiveness of the GSOR iteration method as well as the
corresponding GSOR preconditioner. Finally, in Section 4, some
concluding remarks are given .
\section{THE NEW METHOD} \label{SEC3}
Let $u=x+iy$ and $b=p+iq$ where $x,y,p,q\in \mathbb{R}^{n}$. In
this case the complex linear system (\ref{1.1}) can be written as
2-by-2 block real equivalent formulation
\begin{equation}\label{1.6}
\mathcal{A}\pmatrix{x \cr y}=\pmatrix{p \cr q},
\end{equation}
where
\[
\mathcal{A}=\pmatrix{W & -T \cr T & W}.
\]
We split the coefficient matrix of (\ref{1.6}) as
\[
\mathcal{A}=\mathcal{D}-\mathcal{E}-\mathcal{F},
\]
where
\[
\mathcal{D}=\pmatrix{W & 0 \cr 0 & W},\quad \mathcal{E}=\pmatrix{0 & 0 \cr -T & 0} \quad {\rm and} \quad \mathcal{F}=\pmatrix{0 & T \cr 0 &
0}.
\]
In this case the GSOR iterative method to solve (\ref{1.6}) can
be written as
\begin{equation}\label{1.7}
\pmatrix{x^{k+1} \cr y^{k+1}}=\mathcal{G}_{\alpha}\pmatrix{x^{k} \cr
y^{k}}+ \alpha (\mathcal{D}-\alpha \mathcal{E})^{-1}\pmatrix{p \cr q},
\end{equation}
where $0\neq \alpha \in \mathbb{R}$ and
\begin{eqnarray*}
\mathcal{G}_{\alpha}&=&(\mathcal{D}-\alpha \mathcal{E})^{-1}((1-\alpha)\mathcal{D}+\alpha \mathcal{F})\\
&=& \pmatrix{W & 0 \cr \alpha T & W}^{-1}\pmatrix{(1-\alpha)W & \alpha T \cr 0 & (1-\alpha)W}\\
&=& \pmatrix{I & 0 \cr \alpha S & I}^{-1}\pmatrix{(1-\alpha)I & \alpha S \cr 0 & (1-\alpha)I},\\
\end{eqnarray*}
wherein $S=W^{-1}T$. It is easy to see that (\ref{1.7}) is equivalent to
\begin{equation}\label{1.8}
\left \{\begin{array}{ll} Wx^{(k+1)}=(1-\alpha)Wx^{(k)} + \alpha Ty^{(k)}+ \alpha p,\\
Wy^{(k+1)}=-\alpha Tx^{(k+1)} + (1-\alpha)Wy ^{(k)}+ \alpha q,
\end{array}\right.
\end{equation}
where $x^{(0)}$ and $y^{(0)}$ are initial approximations for $x$ and $y$, respectively. As we mentioned, the iterative method (\ref{1.8}) is real valued and the coefficient matrix of both of the sub-systems is $W$.
Since the coefficient matrix ${\cal A}$ is a block two-by-two matrix, then one may imagine that the proposed method is a kind of the block-SOR method (see \cite{HadjiSOR,SongSOR,YoungSOR}). If we introduce
\[
\mathcal{M}_{\alpha}=\frac{1}{\alpha}(\mathcal{D}-\alpha \mathcal{E})\quad {\rm and} \quad \mathcal{N}_{\alpha}=\frac{1}{\alpha}((1-\alpha)\mathcal{D}+\alpha \mathcal{F}),
\]
then it holds that
\[
\mathcal{A}=\mathcal{M}_{\alpha}-\mathcal{N}_{\alpha}\quad {\rm and} \quad \mathcal{G}_\alpha=\mathcal{M}_{\alpha}^{-1} \mathcal{N}_{\alpha}.
\]
Therefore, GSOR is a stationary iterative method obtained by the matrix splitting
$\mathcal{A}=\mathcal{M}_{\alpha}-\mathcal{N}_{\alpha}$. Hence, we deduce that
the matrix $\mathcal{M}_\alpha$ can be used as preconditioner for the system (\ref{1.6}).
Note that, the multiplicative factor $1/\alpha$ can be dropped since it has no effect on
the preconditioned system. Therefore the preconditioned system takes the following form
\begin{equation}\label{1.9}
\mathcal{P}_{\alpha}^{-1}\mathcal{A}\pmatrix{x \cr y}=\mathcal{P}_{\alpha}^{-1}\pmatrix{p \cr q},
\end{equation}
where $\mathcal{P}_\alpha=\mathcal{D}-\alpha \mathcal{E}$. In the
sequel, matrix $\mathcal{P}_\alpha$ will be referred to as the
GSOR preconditioner. Krylov subspace methods such as the GMRES
algorithm to solve (\ref{1.9}) involve only matrix-vector
multiplication of the form
\[
\pmatrix{e \cr f}=\mathcal{P}_{\alpha}^{-1}\mathcal{A}\pmatrix{r \cr s},
\]
which can be done in four steps by the following procedure
\begin{enumerate}
\item $t:=Wr-Ts$;
\item $u:=Tr+Ws$;
\item Solve $We=t$ for $e$;
\item Solve $Wf=u-\alpha Te$ for $f$.
\end{enumerate}
In the steps 3 and 4 of the above procedure one can use the Cholesky factorization of the matrix $W$.
In continuation, we investigate the convergence of the proposed method.
\begin{lemma}\label{l1}
(\cite{Axelson}) Both roots of the real quadratic equation $x^{2}-rx+s=0$ are less than one in modulus if and only if $|s|<1$
and $|r|<1+s$.
\end{lemma}
\begin{lemma}\label{l2}
Let $W,T \in \mathbb{R}^{n \times n} $ be symmetric positive definite and symmetric, respectively. Then, the eigenvalues of
the matrix $S=W^{-1}T$ are all real.
\end{lemma}
\begin{proof}
Since $W$ is a symmetric positive definite matrix, there is a symmetric positive definite matrix $R$ such that
$W=R^{2}$ (see \cite{Golub-V}, page $149$). Therefore, we have
\[
RSR^{-1}=R^{-1}TR^{-1}=R^{-T}TR^{-1}:=Z.
\]
This shows that $S$ is similar to $Z$. On the other hand $Z$ is symmetric and therefore, the eigenvalues of $S$ are real. \qquad $\Box$
\end{proof}
The following theorem presents a necessary and sufficient condition for guaranteeing the convergence of the GSOR method.
\begin{theorem}\label{t1}
Let $W,T \in \mathbb{R}^{n \times n} $ be symmetric positive definite and symmetric, respectively. Then, the GSOR method to solve Eq. (\ref{1.6}) is convergent if and only if
\[
0<\alpha<\frac{2}{1+\rho(S)},
\]
where $S=W^{-1}T$ and $\rho(S)$ is the spectral radius of $S$.
\end{theorem}
\begin{proof}
Let $\lambda \neq 0$ be an eigenvalue of $\mathcal{G}_{\alpha}$ corresponding to the eigenvector $z=(v^{T},w^{T})^T$. Note that for $\lambda = 0$
there is nothing to investigate. Then, we have
\[
\pmatrix{(1-\alpha)I & \alpha S \cr 0 & (1-\alpha)I}\pmatrix{v
\cr w} =\lambda\pmatrix{I & 0 \cr \alpha S & I}\pmatrix{v
\cr w},
\]
or equivalently
\begin{equation}\label{1.10}
\left\{
\begin{array}{ll}
\alpha Sw=(\lambda+\alpha-1)v, \\
-(\lambda+\alpha-1)w=\lambda \alpha S v.
\end{array}
\right.
\end{equation}
If $\lambda=1-\alpha$, for convergence of the GSOR method we must have $|1-\alpha|<1$, or equivalently
\begin{equation}\label{1.11}
0<\alpha<2.
\end{equation}
If $\lambda\neq 1-\alpha$, from (\ref{1.10}), it is easy to verify that
\[
(1-\alpha-\lambda)^{2}w=-\lambda\alpha^{2}S^{2}w.
\]
This shows that for every eigenvalue $\lambda \neq 0$ of $\mathcal{G}_{\alpha}$ there is an eigenvalue $\mu$ of $S$ that satisfies
\begin{equation}\label{1.12}
(1-\alpha-\lambda)^{2}=-\lambda\alpha^{2}\mu^{2}.
\end{equation}
Eq. (\ref{1.12}) is equivalent to
\begin{equation}\label{1.13}
\lambda^{2}+(\alpha^{2}\mu^{2}+2\alpha-2)\lambda+(\alpha-1)^{2}=0.
\end{equation}
According to Lemma \ref{l1}, $|\lambda|<1$ if and only if
\begin{equation}\label{1.14}
\left \{\begin{array}{ll} |\alpha-1|^{2}<1,\\
|\alpha^{2}\mu^{2}+2\alpha-2|<1+(\alpha-1)^{2},
\end{array}\right.
\end{equation}
The first equation in (\ref{1.14}) is equivalent to (\ref{1.11}) and the second equation in (\ref{1.14}) is equivalent to
$(\alpha-2)^{2}>\alpha^{2}\mu^{2}$. Therefore, according to Lemma \ref{l2}, the latter equation holds if and only if
\[
(\alpha-2)^{2}>\alpha^{2}\rho(S)^{2}.
\]
This relation is equivalent to $|\alpha-2|>\alpha\rho(S)$.
From (\ref{1.11}), it is easy to see that the latter inequality
holds if and only if
\[
0<\alpha<\frac{2}{1+\rho(S)},
\]
which completes the proof. \qquad $\Box$
\end{proof}
In the next theorem, we obtain the optimal value of the relaxation parameter $\alpha$ which minimizes the spectral radius
of the iteration matrix of the GSOR method, i.e.,
\[
\rho(\mathcal{G}_{\alpha^{*}})=\min_{0<\alpha<\frac{2}{1+\rho(S)}}\rho(\mathcal{G}_{\alpha}).
\]
\begin{theorem}\label{t2}
Let $W,T \in \mathbb{R}^{n \times n} $ be symmetric positive definite and symmetric, respectively. Then, the optimal value of the relaxation parameter for the GSOR iterative method (\ref{1.8}) is given by
\begin{equation}\label{1.15}
\alpha^*=\frac{2}{1+\sqrt{1+\rho^{2}(S)}},
\end{equation}
and the corresponding optimal convergence factor of the method is given by
\begin{equation}\label{1.16}
\rho(G_{\alpha^*})=1-\alpha^*=1-\frac{2}{1+\sqrt{1+\rho(S)^{2}}},
\end{equation}
where $S=W^{-1}T$ and $\rho(S)$ is the spectral radius of $S$.
\end{theorem}
\begin{proof}
In the proof of Theorem \ref{t1}, we have seen that for every eigenvalue $\lambda \neq 0$ of $\mathcal{G}_{\alpha}$ there is an eigenvalue $\mu$ of $S$ that satisfies (\ref{1.12}). We exploit the roots of this quadratic equation to determine
the optimal parameter $\alpha^*$. The roots of Eq. (\ref{1.12}) are
\[
\lambda_{1,2}(\alpha)=\frac{-(\alpha^{2}\mu^{2}+2\alpha-2)\pm\sqrt{\Delta}}{2},
\]
where
\[
\Delta=\alpha^{2}\mu^{2}(\alpha^{2}\mu^{2}+4\alpha-4).
\]
From (\ref{1.12}), we can write
\begin{equation}\label{1.17}
\frac{\lambda+\alpha-1}{\alpha}=\pm\mu\sqrt{-\lambda}.
\end{equation}
We define the following functions
\[
f_{\alpha}(\lambda)=\frac{\lambda+\alpha-1}{\alpha} \quad {\rm and} \quad g(\lambda)=\pm \mu\sqrt{-\lambda}.
\]
Then $f_{\alpha}(\lambda)$ is a straight line through the point $(1,1)$, whose slope increases monotonically with
decreasing $\alpha$. It is clear that (\ref{1.17}) can be geometrically interpreted as the intersection of the curves
$f_{\alpha}({\lambda})$ and $g({\lambda})$, as illustrated in Fig. 1.
\begin{figure}
\caption{Plot of the curves of $f_{\alpha}
\label{figfg}
\end{figure}
Fig. 1 shows that when $\alpha$ decreases, it is clear that the largest abscissa
of the two points of intersection decreases until $f_{\alpha}({\lambda})$ becomes tangent to
$g({\lambda})$. In this case, we have $\lambda_1=\lambda_2$ and as a result $\Delta=0$, which is equivalent to $\mu=0$ or
$\alpha^{2}\mu^{2}+4\alpha-4=0$. If $\mu\neq 0$, then the nonnegative root of $\alpha^{2}\mu^{2}+4\alpha-4=0$ is equal to
\begin{equation}\label{1.18}
\hat{\alpha}=\frac{2}{1+\sqrt{1+\mu^{2}}},
\end{equation}
and we have $\lambda_{1,2}=1-\hat{\alpha}$. Now, if $\mu=0$, then $|\lambda_{1,2}|=|1-\alpha|$. In this case, $\alpha=1$ is the
best choice, because in this case we have $\lambda_{1,2}=0$. On the other hand, if we set $\mu=0$ in Eq. (\ref{1.18}) we would
have $\hat{\alpha}=1$. For $\alpha<\hat{\alpha}$, the quadratic equation (\ref{1.12}) has two conjugate complex zeroes of modulus
$1-\alpha$, which increases in modulus with decreasing $\alpha$. Thus, for the fixed eigenvalue $\mu$ of $S$, the value $\alpha$,
which minimizes the zero of largest modulus of (\ref{1.12}) is $\hat{\alpha}$. Finally, it is evident that the curve
$g({\lambda})=\pm \sqrt{{-\lambda}}\rho(S)$ is an envelope for all the curves $\pm \sqrt{{-\lambda}}\mu$,
$0\leq \mu\leq \rho(S)$, and we conclude, using the above argument, that
\[
\rho(\mathcal{G}_{\alpha^*})=\min_{0<\alpha<\frac{2}{1+\rho(S)}} \rho(\mathcal{G}_{\alpha})=1-\alpha^*,
\]
where $\alpha^*$ is defined in (\ref{1.15}). \qquad $\Box$
\end{proof}
\begin{corollary}
If $\rho(S)=0$, then according to Eq. (\ref{1.15}) we have $\alpha^*=1$, and therefore by (\ref{1.16}) we deduce $\rho(\mathcal{G}_{\alpha^*})=0$. This means that the method would have the highest speed of convergence. In the simplest case that $T=0$, the method converges in one iteration.
\end{corollary}
\begin{corollary}
Let $W,T \in \Bbb{R}^{n \times n}$ be symmetric positive definite and symmetric positive semidefinite matrices, respectively. Then, from Lemma 2 it can be seen that the eigenvalues of $S$ are all real and nonnegative. Moreover, from Theorem 1 the GSOR method converges if and only if
\[
0<\alpha<\frac{2}{1+\mu_{\max}(S)},
\]
where $\mu_{\max}(S)$ is the largest eigenvalue of $S=W^{-1}T$. Furthermore, by Theorem 2 by replacing $\rho(S)$ by $\mu_{\max}(S)$, the optimal value of the relaxation parameter and the corresponding optimal convergence factor can be computed via (\ref{1.15}) and (\ref{1.16}), respectively.
\end{corollary}
It is noteworthy that, if $W$ and $T$ are symmetric positive semidefinite and symmetric positive definite, respectively, then
the GSOR iteration method can be applied for the equivalent real system that is obtained from $-iAx=-ib$ ($i=\sqrt{-1}$). More
generally, if there exist real numbers $\beta$ and $\delta$ such that both matrices $\widetilde{W}:= \beta W+ \delta T$ and
$\widetilde{T}:= \beta T - \delta W$ are symmetric positive semidefinite with at least one of them positive definite, we can
first multiply both sides of (\ref{1.1}) by the complex number $\beta-i\delta$ to get the equivalent system
\[
(\widetilde{W} + i\widetilde{T})x=\widetilde{b}\quad \rm{with} \quad \widetilde{{\it b}}:=(\beta-i\delta){\it b},
\]
and then employ the GSOR iteration method to the equivalent real
system that is obtained from the above system.
\section{Numerical experiments}\label{SEC4}
In this section, we use three examples of \cite{MHSS} and an example of \cite{LPHSS} to illustrate the feasibility and effectiveness of the GSOR iteration method when it is employed either as a solver or as a preconditioner for GMRES to solve the equivalent real system (\ref{1.6}). In all the examples, $W$ is symmetric positive definite and $T$ is symmetric positive semidefinite. We also compare the performance of the GSOR method with that of the MHSS method, from point of view of both the number of iterations (denoted by IT) and the total computing times (in seconds, denoted by CPU). In each iteration of both the MHSS and GSOR iteration methods, we use the Cholesky factorization of the coefficient matrices to solve the sub-systems. The reported CPU times are the sum of the CPU times for the convergence of the method and the CPU times for computing the Cholesky factorization. It is necessary to mention that to solve symmetric positive definite system of linear equations we have used the sparse Cholesky factorization incorporated with the symmetric approximate minimum degree reordering \cite{Saad}. To do so we have used the \verb"symamd.m" command of M{\small ATLAB} Version 7.
All the numerical experiments were computed in double precision using some M{\small ATLAB} codes on a Pentium 4 Laptop, with a 2.10 GHz CPU
and 1.99GB of RAM. We use a null vector as an initial guess and the stopping criterion
\[
\frac{\|b-Au^{(k)}\|_{2}}{\| b\|_{2}}<10^{-6},
\]
is always used where $u^{(k)}=x^{(k)}+iy^{(k)}.$
\begin{example} (See \cite{MHSS})\label{ex1}
Consider the linear system of equations (\ref{1.1}) as following
\begin{equation}\label{1.19}
\left[\left(K+\frac{3-\sqrt{3}}{\tau}I\right)+i\left(K+\frac{3+\sqrt{3}}{\tau}I\right)\right]x=b,
\end{equation}
where $\tau$ is the time step-size and $K$ is the five-point centered difference matrix approximating the negative Laplacian operator $L=-\Delta$ with homogeneous Dirichlet boundary conditions, on a uniform mesh in the unit square
$[0, 1]\times[0,1]$ with the mesh-size $h=1/(m+1)$. The matrix $K\in\mathbb{R}^{n\times n}$ possesses the tensor-product form
$K=I\otimes V_{m}+V_{m}\otimes I$, with $V_{m}=h^{-2}{\rm tridiag}(-1,2,-1)\in \mathbb{R}^{m\times m}$. Hence, $K$ is an
${n\times n}$ block-tridiagonal matrix, with $n=m^{2}$. We take
\[
W=K+\frac{3-\sqrt{3}}{\tau}I \quad {\rm and} \quad T=K+\frac{3-\sqrt{3}}{\tau}I,
\]
and the right-hand side vector $b$ with its $j$th entry $b_{j}$ being given by
\[
b_{j}=\frac{(1-i)j}{\tau(j+1)^{2}},\quad j=1,2,\ldots,n.
\]
In our tests, we take $\tau=h$. Furthermore, we normalize coefficient matrix and right-hand side by multiplying both by $h^{2}$.
\end{example}
\begin{example}(See \cite{MHSS})\label{ex2}
Consider the linear system of equations (\ref{1.1}) as following
\[
\left[(-\omega^{2}M+K )+i(\omega C_{V}+C_{H}) \right]x=b,
\]
where $M$ and $K$ are the inertia and the stiffness matrices, $C_{V}$ and $C_{H}$ are the viscous and the hysteretic
damping matrices, respectively, and $\omega$ is the driving circular frequency. We take $C_{H}=\mu K$ with $\mu$ a damping coefficient, $M=I$, $C_{V}=10I$, and $K$ the five-point centered difference matrix approximating the negative Laplacian operator with homogeneous Dirichlet boundary conditions, on a uniform mesh in the unit square $[0, 1]\times[0, 1]$ with the mesh-size $h=1/(m+1)$. The matrix $K\in \mathbb{R}^{n\times n}$ possesses the tensor-product form $K=I\otimes V_{m}+V_{m}\otimes I$, with $V_{m}=h^{-2}{\rm tridiag}(-1,2,-1)\in \mathbb{R}^{m\times m}$. Hence, $K$ is an ${n\times n}$ block-tridiagonal matrix, with $n=m^{2}$. In addition, we set $\omega=\pi$, $\mu=0.02$, and the right-hand side vector $b$ to be $b=(1 + i)A{\bf 1}$, with ${\bf 1}$ being the vector of all entries equal to $1$. As before, we normalize the system by multiplying both sides
through by $h^{2}$.
\end{example}
\begin{example}(See \cite{MHSS})\label{ex3}
Consider the linear system of equations $(W+iT)x=b$, with
\[
T=I\otimes V+V\otimes I \quad {\rm and} \quad W=10(I\otimes V_{c}+V_{c}\otimes I)+9(e_{1}e_{m}^{T}+e_{m}e_{1}^{T})\otimes I,
\]
where $V={\rm tridiag}(-1,2,-1)\in \mathbb{R}^{m\times m}$, $V_{c}=V-e_{1}e_{m}^{T}-e_{m}e_{1}^{T}\in \mathbb{R}^{m\times m}$ and $e_{1}$ and $e_{m}$ are the first and last unit vectors in $\mathbb{R}^{m}$, respectively. We take the right-hand side vector $b$ to be $b=(1 + i)A\textbf{1}$, with $\textbf{1}$ being the vector of all entries equal to $1$.
Here $T$ and $W$ correspond to the five-point centered difference matrices approximating the negative Laplacian operator with homogeneous Dirichlet boundary conditions and periodic boundary conditions, respectively, on a uniform mesh in the unit square $[0, 1]\times[0, 1]$ with the mesh-size $h=1/(m+1)$.
\end{example}
\begin{example}\label{ex4} (See \cite{Bertaccini,LPHSS})
We consider the complex Helmholtz equation
\[
-\triangle u+\sigma_1 u + i \sigma_2 u =f,
\]
where $\sigma_1$ and $\sigma_2$ are real coefficient functions, $u$ satisfies Dirichlet boundary conditions
in $D = [0,1] \times [0, 1]$ and $i=\sqrt{-1}$.
We discretize the problem with finite differences on a $m\times m$ grid with mesh size $h = 1/(m + 1)$. This leads to a system of linear equations
\[
\left((K+\sigma_1 I)+i \sigma_2 I\right)x=b,
\]
where $K=I\otimes V_{m}+V_{m}\otimes I$ is the discretization of $-\triangle$ by means of centered differences, wherein $V_{m}=h^{-2}{\rm tridiag}(-1,2,-1)\in \mathbb{R}^{m\times m}$. The right-hand side vector $b$ is taken to
be $b=(1 + i)A\textbf{1}$, with $\textbf{1}$ being the vector of all entries equal to $1$.
Furthermore, before solving the system we normalize the coefficient matrix and the right-hand side vector by multiplying both by $h^{2}$. For the numerical tests we set $\sigma_1=\sigma_2=100$.
\end{example}
In Table \ref{Table1}, we have reported the optimal values of the parameter $\alpha$ (denoted by $\alpha^{*}$) used in both the MHSS and the GSOR iterative methods for different values of $m$ for the four examples. The optimal parameters $\alpha^{*}$ for the MHSS method are those presented in \cite{MHSS} (except for $m=512$). The $\alpha^{*}$ for the GSOR method is obtained from (\ref{1.15}) in which the largest eigenvalue of matrix $S$ $(\mu_{\max}(S))$ has been estimated by a few iterations of the power method.
In Fig. 2 the optimal parameter $\alpha^*$ for the GSOR method versus some values of $m$ has been displayed. From Table \ref{Table1} and Fig. 1 we see that for all the examples $\alpha^*$ decreases with the mesh-size $h$. We also see that for large values of $m$ the value $\alpha^*$ for Examples \ref{ex2} and \ref{ex4} is approximately equal to 0.455 and 0.862, respectively. For Examples \ref{ex1}
we see that $\alpha^*$ is roughly reduced by a factor of 0.96, and for Example \ref{ex3} by a factor of 0.55 as $m$ is doubled.
\begin{figure}
\caption{The optimal parameter $\alpha^*$ for the GSOR method versus some values of $m$; top-left: Example 1, top-right:Example 2, down-left :Example 3, down-right: Example 4.}
\label{Figro}
\end{figure}
\begin{table}\label{Table1}
\caption{ The optimal
parameters $\alpha^{*}$ for MHSS and GSOR iteration
methods.\label{Table1}}
\begin{tabular}{lllllllll}
\\ \hline
\\
Example & Method & Grid\\\cline{3-8}
\\
& & $16\times 16$ & $32\times 32$ & $64\times 64$ & $128\times 128$ & $256\times 256$ &
$512\times 512$
\\ \hline
\\ [0mm]
No. 1 & MHSS & $1.06$ & $0.75$ & $0.54$ & $0.40$ & $0.30$ & $0.21$ \\
& GSOR & $0.550$ & $0.495$ & $0.457$ & $0.432$ & $0.428$ & $0.412$ \\[2mm]
No. 2 & MHSS & $0.21$ & $0.08$ & $0.04$ & $0.02$ & $0.01$ & $0.005$\\
& GSOR & $0.455$ & $0.455$ & $0.455$ & $0.455$ & $0.455$ & $0.457$ \\[2mm]
No. 3 & MHSS & $1.61$ & $1.01$ & $0.53$ & $0.26$ & $0.13$ & $0.07$\\
& GSOR & $0.908$ & $0.776$ & $0.566$ & $0.353$ & $0.199$ & $0.105$\\[2mm]
No. 4 & MHSS & $0.37$ & $0.09$ & $0.021$ & $0.005$ & $0.002$ & $0.0005$\\
& GSOR & $0.862$ & $0.862$ & $0.862$ & $0.862$ & $0.862$ & $0.862$\\ \hline
\end{tabular}
\end{table}
In Figs. 3-6 we depict the eigenvalues distribution of the coefficient matrix $\mathcal{A}$ and the GSOR($\alpha^{*}$)-preconditioned matrix $\mathcal{P}_{\alpha^{*}}^{-1}\mathcal{A}$ with $m=32$, respectively, for Examples 1-4. It is evident that system which is preconditioned by GSOR method is of a well-clustered spectrum around $(1,0)$. These observations imply that when GSOR is applied as a preconditioner for GMRES, the rate of convergence can be improved considerably. This fact is further confirmed by the numerical results presented in Tables 2-5.
\begin{figure}
\caption{Eigenvalues distribution of the original matrix
$\mathcal{A}
\label{Fig1}
\end{figure}
\begin{figure}
\caption{Eigenvalues distribution of the original matrix $\mathcal{A}
\label{Fig2}
\end{figure}
\begin{figure}
\caption{Eigenvalues distribution of the original matrix $\mathcal{A}
\label{Fig3}
\end{figure}
\begin{figure}
\caption{Eigenvalues distribution of the original matrix $\mathcal{A}
\label{Fig4}
\end{figure}
Numerical results for Example \ref{ex1} are listed in Table \ref{Table2}. This table presents IT and CPU times for the MHSS, GSOR, GMRES(10), GSOR-preconditioned GMRES(10) methods. The MHSS iteration is employed to solve the original complex system (\ref{1.1}) and the three other methods
to solve the equivalent real system (\ref{1.6}). As seen, the GSOR method is superior to the MHSS method in terms of both iterations and CPU times. As a preconditioner, we observe that GMRES(10) in conjunction with the GSOR method drastically reduces the number of iterations of the GMRES(10) method. Nevertheless CPU times for the GSOR-preconditioned GMRES(10) is slightly greater than those of the GSOR method.
\begin{table}
\caption{ Numerical results for Example \ref{ex1}. \label{Table2}}
\begin{tabular}{lllllllll}
\\ \hline
\\
Method & $m\times m$ & $16\times 16$ & $32\times
32$&$64\times 64$ & $128\times 128$ & $256\times 256$ & $512\times 512$
\\
\hline
\\
MHSS & IT & $40$ & $54$ &$73$ & $98$ & $133$ & 181 \\
& CPU & $0.11$ & $0.27$ & $1.53$ & $9.66$ & $65.22$ & 546.69 \\
GSOR & IT & $19$ & $22$ &$24$ & $26$ & $27$ & 27 \\
& CPU & $0.05$ & $0.08$ & $0.31$ & $1.47$ & $7.19$ & 44.70 \\
GMRES($10$) & IT & $44$ & $93$ & $163$ & $288$ & $526$ & 974 \\
& CPU & $0.08$ & $0.48$ & $3.27$ & $24.73$ & $222.66$ & 2755.23 \\
GSOR-GMRES($10$) & IT & $3$ & $3$ & $3$ & $4$ & $4$ & 4 \\
& CPU & $0.06$ & $0.13$ & $0.44$ & $2.42$ & $10.97$ & 79.33 \\\hline
\end{tabular}
\end{table}
In Table \ref{Table3}, we show numerical results for Example
\ref{ex2}. In this table, a dagger $(\dag)$ means that the method fails to converge in 2000 iterations. As seen, the GSOR iteration method is more effective than the MHSS iteration method in terms of both iterations and CPU
times. Even with the increase of problem size, we see that the
number of GMRES(10) iterations with the GSOR preconditioner remain
almost constant and are significantly less than those of the GMRES(10) method. Hence, the GSOR
preconditioner can significantly improve the convergence
behavior of GMRES(10).
\begin{table}
\caption{ Numerical results for Example \ref{ex2}. \label{Table3}}
\begin{tabular}{lllllllll}
\\ \hline
\\
Method & $m\times m$ & $16\times 16$ & $32\times 32$ &$64\times 64$ & $128\times 128$ &
$256\times 256$ & $512\times 512$
\\ \hline
\\
MHSS & IT & $34$ & $38$ &$50$ &$81$ & $139$ & 250 \\
& CPU & $0.06$ & $0.20$ & $1.28$ & $8.032$ & $68.25$ & 746.49 \\
GSOR & IT & $26$ & $24$ &$24$ & $23$ & $23$ & 23 \\
& CPU & $0.05$ & $0.08$ & $0.33$ & $1.31$ & $6.23$ & 38.83 \\
GMRES($10$) & IT & $23$ & $117$ & $228$ & $670$ & $\dag$ & $\dag$ \\
& CPU & $0.08$ & $0.64$ & $4.53$ & $58.88$ & $--$ & $--$ \\
GSOR-GMRES($10$) & IT & $2$ & $2$ & $2$ & $2$ & $2$ & 2 \\
& CPU & $0.06$ & $0.11$ & $0.31$ & $1.31$ & $6.25$ & 37.14 \\\hline
\end{tabular}
\end{table}
Numerical results for Example \ref{ex3} are presented in Table \ref{Table4}. In terms of the iteration steps, GSOR-preconditioned GMRES(10) performs much better than GSOR, MHSS and GMRES(10). In terms of computing times, GSOR-preconditioned GMRES(10) costs less CPU than MHSS and GMRES(10) and also less CPU than GSOR expect for $m=32$ and $m=64$. Hence, we find that as a preconditioner for GMRES(10), the GSOR is of high performance, especially when problem size increases.
\begin{table}
\caption{ Numerical results for Example \ref{ex3}. \label{Table4}}
\begin{tabular}{llllllll}
\\ \hline
\\
Method & $m\times m$ & $16\times 16$ & $32\times 32$ &$64\times 64$ & $128\times 128$ &
$256\times 256$ & $512\times 512$
\\ \hline
\\
MHSS & IT & $53$ & $76$ &$130$ &$246$ & $468$ & $869$ \\
& CPU & $0.06$ & $0.39$ & $3.01$ & $26.72$ & $245.26$ & $2792.10$ \\
GSOR & IT & $7$ & $11$ &$20$ & $35$ & $71$ & $131$ \\
& CPU & $0.06$ & $0.07$ & $0.31$ & $2.28$ & $20.39$ & $241.49$ \\
GMRES($10$) & IT & $19$ & $49$ & $91$ & $316$ & $1081$ & $\dag$ \\
& CPU & $0.06$ & $0.25$ & $1.88$ & $27.11$ & $459.95$ & $--$ \\
GSOR-GMRES($10$) & IT & $2$ & $2$ & $2$ & $3$ & $4$ & $8$ \\
& CPU & $0.05$ & $0.09$ & $0.34$ & $2.16$ & $13.73$ & $161.14$ \\\hline
\end{tabular}
\end{table}
\begin{table}
\caption{Numerical results for Example \ref{ex4}. \label{Table5}}
\begin{tabular}{llllllll}
\\ \hline
\\
Method & $m\times m$ & $16\times 16$ & $32\times 32$ &$64\times 64$ & $128\times 128$ &
$256\times 256$ & $512\times 512$
\\ \hline
\\
MHSS & IT & $30$ & $36$ &$39$ &$40$ & $41$ & $41$ \\
& CPU & $0.05$ & $0.14$ & $0.55$ & $2.69$ & $16.5$ & $80.95$ \\
GSOR & IT & $8$ & $8$ &$8$ & $8$ & $7$ & $7$ \\
& CPU & $0.03$ & $0.06$ & $0.11$ & $0.531$ & $2.50$ & $17.63$ \\
GMRES($10$) & IT & $5$ & $12$ & $24$ & $66$ & $219$ & $761$ \\
& CPU & $0.05$ & $0.13$ & $0.36$ & $4.55$ & $99.53$ & $1616.88$ \\
GSOR-GMRES($10$) & IT & $2$ & $2$ & $2$ & $2$ & $2$ & $2$ \\
& CPU & $0.04$ & $0.08$ & $0.31$ & $1.52$ & $8.48$ & $42.67$ \\\hline
\end{tabular}
\end{table}
In Table \ref{Table5}, numerical results of Example \ref{ex4} are presented. All of the comments and observations which we have given for the previous examples can also be posed here. As a preconditioner we see that the GSOR preconditioner drastically reduces the iteration numbers of the GMRES(10) method. For example, the GMRES(10) converges in 761 iterations, while the GMRES(10) in conjunction with the GSOR preconditioner converges only in 2 iterations. In addition, the MHSS method can not compete with the GSOR method in terms of iterations and CPU times.
\section{Conclusion}\label{SEC5}
In this paper we have utilized the generalized successive overrelaxation (GSOR) iterative method to solve the equivalent real formulation of complex linear system (\ref{1.6}), where $W$ is symmetric positive definite and $T$ is symmetric positive semidefinite. Convergence properties of the method have been also investigated. Besides its use as a solver, the GSOR iteration has also been used as a preconditioner to accelerate Krylov subspace methods such as GMRES. Some numerical have been presented to show the effectiveness of the method. Our numerical examples show that our method is quite suitable for such problems. Moreover, the presented numerical experiments show that the GSOR method is superior to MHSS in terms of the iterations and CPU times.
\section*{Acknowledgments}
The authors are grateful to the anonymous referees and the editor of the journal for their valuable comments and suggestions.
\end{document}
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math
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\begin{equation}gin{examp}in{document}
\title{Toeplitz determinants whose elements are the coefficients of univalent functions}
\author{Md Firoz Ali}
\address{Md Firoz Ali,
Department of Mathematics,
Indian Institute of Technology Kharagpur,
Kharagpur-721 302, West Bengal, India.}
\email{ali.firoz89@gmail.com}
\author{D. K. Thomas}
\address{D. K. Thomas,
Department of Mathematics,
Swansea University, Singleton Park,
Swansea, SA2 8PP, United Kingdom.}
\email{d.k.thomas@swansea.ac.uk}
\author{A. Vasudevarao}
\address{A. Vasudevarao,
Department of Mathematics,
Indian Institute of Technology Kharagpur,
Kharagpur-721 302, West Bengal, India.}
\email{alluvasu@maths.iitkgp.ernet.in}
\subjclass[2010]{Primary 30C45, 30C55}
\keywords{univalent functions, starlike functions, convex functions, close-to-convex function, typically real function, Toeplitz determinant.}
\def\@arabic\c@footnote{}
\footnotetext{ {\tiny File:~\jobname.tex,
printed: \number\year-\number\month-\number\day,
\thehours.\ifnum\theminutes<10{0}\fi\theminutes }
} \makeatletter\def\@arabic\c@footnote{\@arabic\c@footnote}\makeatother
\begin{equation}gin{examp}in{abstract}
Let $\mathcal{S}$ denote the class of analytic and univalent functions in $\mathbb{D}:=\{z\in\mathbb{C}:\, |z|<1\}$ of the form $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$. In this paper, we determine sharp estimates for the Toeplitz determinants whose elements are the Taylor coefficients of functions in $\mathcal{S}$ and its certain subclasses. We also discuss similar problems for typically real functions.
\end{abstract}
\thanks{}
\maketitle
\pagestyle{myheadings}
\markboth{Md Firoz Ali, D. K. Thomas and A. Vasudevarao}{Toeplitz determinant}
\section{Introduction and Preliminaries}
Let $\mathcal{H}$ denote the space of analytic functions in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:\, |z|<1\}$ and $\mathcal{A}$ denote the class of functions $f$ in $\mathcal{H}$ with Taylor series
\begin{equation}gin{examp}in{equation}\leftarrowbel{p-001}
f(z)= z+\sum_{n=2}^{\infty}a_n z^n.
\end{equation}
The subclass $\mathcal{S}$ of $\mathcal{A}$, consisting of univalent (i.e., one-to-one) functions has attracted much interest for over a century, and is a central area of research in Complex Analysis. A function $f\in\mathcal{A}$ is called starlike if $f(\mathbb{D})$ is starlike with respect to the origin i.e., $tf(z)\in f(\mathbb{D})$ for every $0\le t\le 1$. Let $\mathcal{S}^*$ denote the class of starlike functions in $\mathcal{S}$. It is well-known that a function $f\in\mathcal{A}$ is starlike if, and only if,
\begin{equation}gin{examp}in{equation*}\leftarrowbel{p-010}
{\rm Re\,}\left(\frac{zf'(z)}{f(z)}\right)>0, \quad z\in\mathbb{D}.
\end{equation*}
An important member of the class $\mathcal{S}^*$ as well as of the class $\mathcal{S}$ is the Koebe function $k$ defined by $k(z)=z/(1-z)^2$. This function plays the role of extremal function in most of the problems for the classes $\mathcal{S}^*$ and $\mathcal{S}$.
A function $f\in\mathcal{A}$ is called convex if $f(\mathbb{D})$ is a convex domain. Let $\mathcal{C}$ denote the class of convex functions in $\mathcal{S}$. It is well-known that a function $f\in\mathcal{A}$ is in $\mathcal{C}$ if, and only if,
\begin{equation}gin{examp}in{equation*}\leftarrowbel{p-015}
{\rm Re\,}\left(1+\frac{zf''(z)}{f'(z)}\right)>0, \quad z\in\mathbb{D}.
\end{equation*}
From the above it is easy to see that $f\in\mathcal{C}$ if, and only if, $zf'\in\mathcal{S}^*$.
A function $f\in\mathcal{A}$ is said to be close-to-convex if there exists a starlike function $g\in\mathcal{S}^*$ and a real number $\alpha\in(-\pi/2,\pi/2)$, such that
\begin{equation}gin{examp}in{equation}\leftarrowbel{p-020}
{\rm Re\,} \left(e^{i\alpha}\frac{zf'(z)}{g(z)}\right)>0, \quad z\in\mathbb{D}.
\end{equation}
Let $\mathcal{K}$ denote the class of all close-to-convex functions. It is well-known that every close-to-convex function is univalent in $\mathbb{D}$ (see \cite{Duren-book}). Geometrically, $f\in\mathcal{K}$ means that the complement of the image-domain $f(\mathbb{D})$ is the union of rays that are disjoint (except that the origin of one ray may lie on another one of the rays).
Let $\mathcal{R}$ denote class of functions $f$ in $\mathcal{A}$ satisfying ${\rm Re\,} f'(z)>0$ in $\mathbb{D}$. It is well-known that functions in $\mathcal{R}$ are close-to-convex, and hence univalent. Functions in $\mathcal{R}$ are sometimes called functions of bounded boundary rotation.
A function $f$ satisfiying the condition $({\rm Im\,}z) ({\rm Im\,}f(z))\ge0$ for $z\in\mathbb{D}$ is called a typically real. Let $\mathcal{T}$ denote the class of all typically real functions. Robertson \cite{Robertson-1935} proved that $f\in\mathcal{T}$ if, and only if, there exists a probability measure $\mu$ on $[-1, 1]$ such that
\begin{equation}gin{examp}in{equation*}\leftarrowbel{p-200}
f(z)=\int_{-1}^{1} k(z,t)\,d\mu(t),
\end{equation*}
where
\begin{equation}gin{examp}in{equation*}\leftarrowbel{p-205}
k(z,t)=\frac{z}{1-2tz+z^2}, \quad z\in\mathbb{D},\quad t\in[-1,1].
\end{equation*}
Hankel matrices and determinants play an important role in several branches of mathematics, and have many applications \cite{Ye-Lim-2016}. The Toeplitz determinants are closely related to Hankel determinants. Hankel matrices have constant entries along the reverse diagonal, whereas Toeplitz matrices have constant entries along the diagonal. For a good summary of the applications of Toeplitz matrices to the wide range of areas of pure and applied mathematics, we refer to \cite{Ye-Lim-2016}. Recently, Thomas and Halim \cite{Thomas-Halim-2017} introduced the concept of the symmetric Toeplitz determinant for analytic functions $f$ of the form (\ref{p-001}), and defined the symmetric Toeplitz determinant $T_q(n)$ as follows
$$
T_q(n):=
\begin{equation}gin{examp}in{vmatrix}
a_n & a_{n+1} & \cdots & a_{n+q-1}\\
a_{n+1} & a_n & \cdots & a_{n+q-2}\\
\vdots & \vdots & \vdots & \vdots &\\
a_{n+q-1} & a_{n+q-2} & \cdots & a_{n}
\end{vmatrix}
$$
where $n,q= 1,2,3\ldots$ with $a_1=1$. In particular,
$$
T_2(2)=
\begin{equation}gin{examp}in{vmatrix}
a_2 & a_3\\
a_3 & a_2
\end{vmatrix},
\quad
T_2(3)=
\begin{equation}gin{examp}in{vmatrix}
a_3 & a_4\\
a_4 & a_3
\end{vmatrix},
\quad
T_3(1)=
\begin{equation}gin{examp}in{vmatrix}
1 & a_2 & a_3\\
a_2 & 1 & a_2\\
a_3 & a_2 & 1
\end{vmatrix}\\,
\quad
T_3(2)=
\begin{equation}gin{examp}in{vmatrix}
a_2 & a_3 & a_4\\
a_3 & a_2 & a_3\\
a_4 & a_3 & a_2
\end{vmatrix}.
$$
For small values of $n$ and $q$, estimates of the Toeplitz determinant $|T_q(n)|$ for functions in $\mathcal{S}^*$ and $\mathcal{K}$ have been studied in \cite{Thomas-Halim-2017}. Similarly, estimates of the Toeplitz determinant $|T_q(n)|$ for functions in $\mathcal{R}$ have been studied in \cite{Radhika-Sivasubramanian-Murugusundaramoorthy-Jahangiri-2016}, when $n$ and $q$ are small.
Apart from \cite{Radhika-Sivasubramanian-Murugusundaramoorthy-Jahangiri-2016} and \cite{Thomas-Halim-2017}, there appears to be little in the literature concerning estimates of Toeplitz determinants. In both
\cite{Radhika-Sivasubramanian-Murugusundaramoorthy-Jahangiri-2016, Thomas-Halim-2017} we observe an invalid assumption in the proofs. It is the purpose of this paper to give estimates for Toeplitz determinants $T_q(n)$ for functions in $\mathcal{S}$, $\mathcal{S}^*$, $\mathcal{C}$, $\mathcal{K}$, $\mathcal{R}$, and $\mathcal{T}$, when $n$ and $q$ are small.
Let $\mathcal{P}$ denote the class of analytic functions $p$ in $\mathbb{D}$ of the form
\begin{equation}gin{examp}in{equation}\leftarrowbel{p-030}
p(z)= 1+\sum_{n=1}^{\infty}c_n z^n
\end{equation}
such that ${\rm\, Re\,} p(z)>0$ in $\mathbb{D}$. Functions in $\mathcal{P}$ are sometimes called Carath\'{e}odory functions. To prove our main results, we need some preliminary results for functions in $\mathcal{P}$.
\begin{equation}gin{examp}in{lem}\cite[p. 41]{Duren-book}\leftarrowbel{p-lemma001}
For a function $p\in\mathcal{P}$ of the form (\ref{p-030}), the sharp inequality $|c_n|\le 2$ holds for each $n\ge 1$. Equality holds for the function $p(z)=(1+z)/(1-z)$.
\end{lem}
\begin{equation}gin{examp}in{lem}\cite[Theorem 1]{Efraimidis-2016}\leftarrowbel{p-lemma010}
Let $p\in\mathcal{P}$ be of the form (\ref{p-030}) and $\mu\in\mathbb{C}$. Then
$$
|c_n-\mu c_kc_{n-k}|\le 2\max\{1,|2\mu-1|\}, \quad 1\le k\le n-1.
$$
If $|2\mu-1|\ge1$ then the inequality is sharp for the function $p(z)=(1+z)/(1-z)$ or its rotations. If $|2\mu-1|<1$ then the inequality is sharp for the function $p(z)=(1+z^n)/(1-z^n)$ or its rotations.
\end{lem}
\section{Main Results}
\begin{equation}gin{examp}in{thm}\leftarrowbel{theorem-001}
Let $f\in\mathcal{S}$ be of the form (\ref{p-001}). Then
\begin{equation}gin{examp}in{enumerate}[(i)]
\item $\displaystyle |T_2(n)|=|a_n^2-a_{n+1}^2|\le 2n^2+2n+1$ for $n\ge2$,\\[-2mm]
\item $\displaystyle |T_3(1)|\le 24$.
\end{enumerate}
Both inequalities are sharp.
\end{thm}
\begin{equation}gin{examp}in{proof}
Let $f\in\mathcal{S}$ be of the form (\ref{p-001}). Then clearly
\begin{equation}gin{examp}in{equation}\leftarrowbel{p-050}
|T_2(n)|=|a_n^2-a_{n+1}^2|\le |a_n^2|+|a_{n+1}^2|\le n^2+(n+1)^2=2n^2+2n+1.
\end{equation}
Equality holds in (\ref{p-050}) for the function $f$ defined by
\begin{equation}gin{examp}in{equation}\leftarrowbel{p-070}
f(z):=\frac{z}{(1-iz)^2}=z+2iz^2-3z^3-4iz^4+5z^5+\cdots.
\end{equation}
Again, if $f\in\mathcal{S}$ is of the form (\ref{p-001}) then by the Fekete-Szeg\"o inequality for functions in $\mathcal{S}$, we have
\begin{equation}gin{examp}in{align}\leftarrowbel{p-080}
|T_3(1)| &=|1-2a_2^2+2a_2^2a_3-a_3^2|\\
&\le 1+2|a_2^2|+|a_3||a_3-2a_2^2|\nonumber\\
&\le 1+8+(3)(5)\nonumber\\
&=24.\nonumber
\end{align}
Equality holds in (\ref{p-080}) for the function $f$ defined by (\ref{p-070}).
\end{proof}
\begin{equation}gin{examp}in{rem}
Since the function $f$ defined by (\ref{p-070}) belongs to $\mathcal{S}^*$, and $\mathcal{S}^*\subset\mathcal{K}\subset\mathcal{S}$, the sharp inequalities in Theorem \ref{theorem-001} also hold for functions in $\mathcal{S}^*$ and $\mathcal{K}$. In particular, the sharp inequalities $|T_2(2)|\le 13$ and $|T_2(3)|\le 25$ hold for functions in $\mathcal{S}^*$, $\mathcal{K}$ and $\mathcal{S}$.
\end{rem}
\begin{equation}gin{examp}in{thm}\leftarrowbel{theorem-005}
Let $f\in\mathcal{S}^*$ be of the form (\ref{p-001}). Then $|T_3(2)|\le 84$.
\noindent The inequality is sharp.
\end{thm}
\begin{equation}gin{examp}in{proof}
Let $f\in\mathcal{S}^*$ be of the form (\ref{p-001}). Then there exists a function $p\in\mathcal{P}$ of the form (\ref{p-030}) such that $zf'(z)=f(z)p(z)$. Equating coefficients, we obtain
\begin{equation}gin{examp}in{equation}\leftarrowbel{p-040}
a_2=c_1,\quad a_3=\frac{1}{2}(c_2+c_1^2)\quad\mbox{ and } \quad a_4=\frac{1}{6}c_1^3+\frac{1}{2}c_1c_2+\frac{1}{3}c_3.
\end{equation}
By a simple computation $T_3(2)$ can be written as $ T_3(2)=(a_2-a_4)(a_2^2-2a_3^2+a_2a_4)$. If $f\in\mathcal{S}^*$ then clearly, $|a_2-a_4|\le |a_2|+|a_4|\le 6$. Thus we need to maximize $|a_2^2-2a_3^2+a_2a_4|$ for functions in $\mathcal{S}^*$, and so writing $a_2, a_3$ and $a_4$ in terms of $c_1, c_2$ and $c_3$ with the help of (\ref{p-040}), we obtain
\begin{equation}gin{examp}in{align*}
|a_2^2-2a_3^2+a_2a_4| &= \left|c_1^2-\frac{1}{3}c_1^4-\frac{1}{2}c_1^2c_2-\frac{1}{2}c_2^2+\frac{1}{3}c_1c_3\right|\\
&\le |c_1|^2+\frac{1}{3}|c_1|^4+\frac{1}{2}|c_2|^2+\frac{1}{3}|c_1|\left|c_3-\frac{3}{2}c_1c_2\right|.
\end{align*}
From Lemma \ref{p-lemma001} and Lemma \ref{p-lemma010}, it easily follows that
\begin{equation}gin{examp}in{equation}
|a_2^2-2a_3^2+a_2a_4| \le 4+\frac{16}{3}+\frac{4}{2}+\frac{2}{3}(4)=14.
\end{equation}
Therefore, $|T_3(2)|\le 84$, and the inequality is sharp for the function $f$ defined by (\ref{p-070}).
\end{proof}
\begin{equation}gin{examp}in{rem}
In \cite{Thomas-Halim-2017}, it was claimed that $|T_2(2)|\le 5$, $|T_2(3)|\le 7$, $|T_3(1)|\le 8$ and $|T_3(2)|\le 12$ hold for functions in $\mathcal{S}^*$, and these estimates are sharp. Similar results were also obtained for certain close-to-convex functions. For the function $f$ defined by (\ref{p-070}), a simple computation gives $|T_2(2)|= 13$ and $|T_2(3)|= 25$, $|T_3(1)|= 24$ and $|T_3(2)|=84$ which shows that these estimates are not correct. In proving these estimates the authors assumed that $c_1>0$ which is not justified, since the functional $|T_q(n)|$ $(n\ge1, q\ge2)$ is not rotationally invariant.
\end{rem}
To prove our next result we need the following results for functions in $\mathcal{S}^*$.
\begin{equation}gin{examp}in{lem}\cite[Theorem 3.1]{Janteng-Halim-Darus-2007}\leftarrowbel{p-lemma020}
Let $g\in\mathcal{S}^*$ and be of the form $g(z)= z+\sum_{n=2}^{\infty}b_n z^n$. Then $|b_2b_4-b_3^2|\le 1$, and the inequality is sharp for the Koebe function $k(z)=z/(1-z)^2$, or its rotations.
\end{lem}
\begin{equation}gin{examp}in{lem}\cite[Lemma 3]{Koepf-1987}\leftarrowbel{p-lemma022}
Let $g\in\mathcal{S}^*$ be of the form $g(z)=z+\sum_{n=2}^{\infty}b_n z^n$. Then for any $\leftarrowmbda\in\mathbb{C}$,
$$
|b_3-\leftarrowmbda b_2^2|\le \max\{1,|3-4\leftarrowmbda|\}.
$$
The inequality is sharp for $k(z)=z/(1-z)^2$, or its rotations if $|3-4\leftarrowmbda|\ge 1$, and for $(k(z^2))^{1/2}$, or its rotations if $|3-4\leftarrowmbda|<1$.
\end{lem}
\begin{equation}gin{examp}in{lem}\cite[Theorem 2.2]{Ma-1999}\leftarrowbel{p-lemma025}
Let $g\in\mathcal{S}^*$ be of the form $g(z)= z+\sum_{n=2}^{\infty}b_n z^n$. Then
$$
|\leftarrowmbda b_nb_m-b_{n+m-1}|\le \leftarrowmbda nm-(n+m-1) \quad\mbox{ for } \leftarrowmbda\ge\frac{2(n+m-1)}{nm},
$$
where $n,m=2,3,\ldots$. The inequality is sharp for the Koebe function $k(z)=z/(1-z)^2$, or its rotations.
\end{lem}
\begin{equation}gin{examp}in{lem}\leftarrowbel{p-lemma030}
Let $f\in\mathcal{K}$ be of the form (\ref{p-001}). Then $|a_2a_4-2a_3^2|\le 21/2$.
\end{lem}
\begin{equation}gin{examp}in{proof}
Let $f\in\mathcal{K}$ be of the form (\ref{p-001}). Then there exists a starlike function $g$ of the form $g(z)= z+\sum_{n=2}^{\infty}b_n z^n$, and a real number $\alpha\in(-\pi/2,\pi/2)$, such that (\ref{p-020}) holds. This implies there exists a Carath\'{e}odory function $p\in\mathcal{P}$ of the form (\ref{p-030}) such that
$$
e^{i\alpha}\frac{zf'(z)}{g'(z)}= p(z){\overline{\operatorname{co}}}s\alpha+i\sin\alpha.
$$
Comparing coefficients we obtain
\begin{equation}gin{examp}in{align*}
2a_2&=b_2+c_1e^{-i\alpha}{\overline{\operatorname{co}}}s\alpha\\
3a_3&=b_3+b_2c_1e^{-i\alpha}{\overline{\operatorname{co}}}s\alpha+c_2e^{-i\alpha}{\overline{\operatorname{co}}}s\alpha\\
4a_4&=b_4+b_3c_1e^{-i\alpha}{\overline{\operatorname{co}}}s\alpha+b_2c_2e^{-i\alpha}{\overline{\operatorname{co}}}s\alpha+c_3e^{-i\alpha}{\overline{\operatorname{co}}}s\alpha,
\end{align*}
and a simple computation gives
\begin{equation}gin{examp}in{align*}
72(a_2a_4-2a_3^2)&= (9b_2b_4-16b_3^2)+(9b_4-23b_2b_3)c_1 e^{-i\alpha}{\overline{\operatorname{co}}}s\alpha\\
&\quad +(9b_3-16b_2^2)c_1^2e^{-2i\alpha}{\overline{\operatorname{co}}}s^2\alpha +(9b_2^2-32b_3)c_2e^{-i\alpha}{\overline{\operatorname{co}}}s\alpha\\
&\quad +(9c_3-23c_1 c_2e^{-i\alpha}{\overline{\operatorname{co}}}s\alpha)b_2e^{-i\alpha}{\overline{\operatorname{co}}}s\alpha +(9c_1c_3-16 c_2^2)e^{-2i\alpha}{\overline{\operatorname{co}}}s^2\alpha.
\end{align*}
Consequently using the triangle inequality, we obtain
\begin{equation}gin{examp}in{align}\leftarrowbel{p-135}
72|a_2a_4-2a_3^2|&\le |9b_2b_4-16b_3^2|+|9b_4-23b_2b_3||c_1| +|9b_3-16b_2^2||c_1^2|\\
&\quad +|9b_2^2-32b_3||c_2| +|9c_3-23c_1 c_2e^{-i\alpha}{\overline{\operatorname{co}}}s\alpha||b_2| +|9c_1c_3-16 c_2^2|.\nonumber
\end{align}
By Lemma \ref{p-lemma020}, Lemma \ref{p-lemma022} and Lemma \ref{p-lemma025}, it easily follows that
\begin{equation}gin{examp}in{align}
|9b_2b_4-16b_3^2| &\le 9|b_2b_4-b_3^2|+7|b_3|^2\le 9+63=72,\leftarrowbel{p-140}\\[2mm]
|9b_4-23b_2b_3| &= 9\left|b_4-\frac{23}{9}b_2b_3\right|\le 9\left(\frac{46}{3}-4\right)=102,\leftarrowbel{p-145}\\
|9b_3-16b_2^2| &= 9\left|b_3-\frac{16}{9}b_2^2\right|\le 9\left(\frac{64}{9}-3\right)=37,\leftarrowbel{p-150}\\
|9b_2^2-32b_3| &= 32\left|b_3-\frac{9}{32}b_2^2\right|\le 32\left(3-\frac{9}{8}\right)=60.\leftarrowbel{p-155}
\end{align}
Again, by Lemma \ref{p-lemma010}, it easily follows that
$$
|9c_3-23c_1 c_2e^{-i\alpha}{\overline{\operatorname{co}}}s\alpha|= 9|c_3-\mu c_1 c_2|\le 18\max\{1,|2\mu-1|\}
$$
where $\mu=\dfrac{23}{9}e^{-i\alpha}{\overline{\operatorname{co}}}s\alpha$. Now note that
\begin{equation}gin{examp}in{align*}
|2\mu-1|^2
&= \left(\frac{23}{9}{\overline{\operatorname{co}}}s2\alpha+\frac{14}{9}\right)^2 + \left(\frac{23}{9}\sin2\alpha\right)^2\\
&= \left(\frac{23}{9}\right)^2+ \left(\frac{14}{9}\right)^2 +2\left(\frac{23}{9}\right)\left(\frac{14}{9}\right) {\overline{\operatorname{co}}}s2\alpha,\\
\end{align*}
and so
$$
1\le |2\mu-1|\le \frac{37}{9}.
$$
Therefore
\begin{equation}gin{examp}in{equation}\leftarrowbel{p-160}
|9c_3-23c_1 c_2e^{-i\alpha}{\overline{\operatorname{co}}}s\alpha| \le 74.
\end{equation}
Again by Lemma \ref{p-lemma010}, it easily follows that
\begin{equation}gin{examp}in{equation}\leftarrowbel{p-165}
|9c_1c_3-16 c_2^2|\le 9|c_1c_3- c_4|+9\left|c_4-\frac{16}{9}c_2^2\right|\le 18+46=64.
\end{equation}
By Lemma \ref{p-lemma001}, and using the inequalities (\ref{p-140}), (\ref{p-145}), (\ref{p-150}), (\ref{p-155}), (\ref{p-160}) and (\ref{p-165}) in (\ref{p-135}), we obtain
$$
|a_2a_4-2a_3^2|\le \frac{1}{72}(72+204 +148 +120 +148 +64)=\frac{21}{2}.
$$
\end{proof}
\begin{equation}gin{examp}in{thm}\leftarrowbel{theorem-020}
Let $f\in\mathcal{K}$ be of the form (\ref{p-001}). Then $|T_3(2)|\le 86$.
\end{thm}
\begin{equation}gin{examp}in{proof}
Let $f\in\mathcal{K}$ be of the form (\ref{p-001}). Then by Lemma \ref{p-lemma030} we have
\begin{equation}gin{examp}in{align*}
|T_3(2)| &=|a_2^3-2a_2a_3^2-a_2a_4^2+2a_3^2a_4|\\
&\le |a_2|^3 + 2|a_2||a_3^2| +|a_4||a_2a_4-2a_3^2|\\
&\le 8+36+42=86.
\end{align*}
\end{proof}
\begin{equation}gin{examp}in{rem}
In Theorem \ref{theorem-005}, we have proved that $|T_3(2)|\le 84$ for functions in $\mathcal{S}^*$, and the inequality is sharp for the function $f$ defined by (\ref{p-070}). Therefore it is natural to conjecture that $|T_3(2)|\le 84$ holds for functions in $\mathcal{K}$ and that equality holds for the function $f$ defined by (\ref{p-070}).
\end{rem}
\begin{equation}gin{examp}in{thm}\leftarrowbel{theorem-015}
Let $f\in\mathcal{C}$ be of the form (\ref{p-001}). Then
\begin{equation}gin{examp}in{enumerate}[(i)]
\item $\displaystyle |T_2(n)|\le 2$ for $n\ge2$.\\[-2mm]
\item $\displaystyle |T_3(1)|\le 4$.\\[-2mm]
\item $\displaystyle |T_3(2)|\le 4$.
\end{enumerate}
All the inequalities are sharp.
\end{thm}
\begin{equation}gin{examp}in{proof}
Let $f\in\mathcal{C}$ be of the form (\ref{p-001}). Then there exists a function $p\in\mathcal{P}$ of the form (\ref{p-030}) such that $f'(z)+zf''(z)=f'(z)p(z)$. Equating coefficients, we obtain
\begin{equation}gin{examp}in{equation}\leftarrowbel{p-115}
2a_2=c_1,\quad 3a_3=\frac{1}{2}(c_2+c_1^2)\quad\mbox{ and }\quad 4a_4=\frac{1}{6}c_1^3+\frac{1}{2}c_1c_2+\frac{1}{3}c_3.
\end{equation}
Clearly
\begin{equation}gin{examp}in{equation}\leftarrowbel{p-120}
|T_2(n)|=|a_n^2-a_{n+1}^2|\le |a_n^2|+|a_{n+1}^2|\le 1+1=2.
\end{equation}
Equality holds in (\ref{p-120}) for the function $f$ defined by
\begin{equation}gin{examp}in{equation}\leftarrowbel{p-125}
f(z):=\frac{z}{1-iz}=z+iz^2-z^3-iz^4+z^5+\cdots.
\end{equation}
Again if $f\in\mathcal{C}$ is of the form (\ref{p-001}) then from Lemma \ref{p-lemma010} and (\ref{p-115}), we obtain
\begin{equation}gin{examp}in{align}\leftarrowbel{p-130}
|T_3(1)| &=|1-2a_2^2+2a_2^2a_3-a_3^2|\\
&\le 1+2|a_2^2|+|a_3||a_3-2a_2^2|\nonumber\\
&\le 1+2+\frac{1}{6}|c_2-2c_1^2|\nonumber\\
&\le 4.\nonumber
\end{align}
It is easy to see that equality holds in (\ref{p-130}) for the function $f$ defined by (\ref{p-125}).
Next note that $ T_3(2)=(a_2-a_4)(a_2^2-2a_3^2+a_2a_4)$. If $f\in\mathcal{C}$ then clearly $|a_2-a_4|\le |a_2|+|a_4|\le 2$. Thus we need to maximize $|a_2^2-2a_3^2+a_2a_4|$ for functions in $\mathcal{C}$.
Writing $a_2, a_3$ and $a_4$ in terms of $c_1, c_2$ and $c_3$ with the help of (\ref{p-115}), we obtain
\begin{equation}gin{examp}in{align*}
|a_2^2-2a_3^2+a_2a_4| &= \frac{1}{144}\left|5c_1^4-36c_1^2+7c_1^2c_2+8c_2^2-6c_1c_3\right|\\
&\le \frac{1}{144}\left(5|c_1|^4+36|c_1|^2+8|c_2|^2+6|c_1||c_3-\frac{7}{6}c_1c_2|\right).
\end{align*}
From Lemma \ref{p-lemma001} and Lemma \ref{p-lemma010}, it easily follows that
\begin{equation}gin{examp}in{equation}
|a_2^2-2a_3^2+a_2a_4| \le \frac{1}{144}\left(80+144+32+32\right)=2.
\end{equation}
Therefore, $|T_3(2)|\le 4$, and the inequality is sharp for the function $f$ defined by (\ref{p-125}).
\end{proof}
\begin{equation}gin{examp}in{thm}\leftarrowbel{theorem-010}
Let $f\in\mathcal{R}$ be of the form (\ref{p-001}). Then
\begin{equation}gin{examp}in{enumerate}[(i)]
\item $\displaystyle |T_2(n)|\le \frac{4}{n^2}+\frac{4}{(n+1)^2}\quad$ for $n\ge2$.\\[1mm]
\item $\displaystyle |T_3(1)|\le \frac{35}{9}$.\\[1mm]
\item $\displaystyle |T_3(2)|\le \frac{7}{3}$.
\end{enumerate}
The inequalities in $(i)$ and $(ii)$ are sharp.
\end{thm}
\begin{equation}gin{examp}in{proof}
Let $f\in\mathcal{R}$ be of the form (\ref{p-001}). Then there exists a function $p\in\mathcal{P}$ of the form (\ref{p-030}) such that $f'(z)=p(z)$. Equating coefficients we obtain $na_n=c_{n-1}$, and so
$$
|a_n|=\frac{1}{n}|c_{n-1}|\le \frac{2}{n}, \quad n\ge 2.
$$
The inequality is sharp for the function $f$ defined by $f'(z)=(1+z)/(1-z)$, or its rotations. Thus
\begin{equation}gin{examp}in{equation}\leftarrowbel{p-090}
|T_2(n)|=|a_n^2-a_{n+1}^2|\le |a_n^2|+|a_{n+1}^2|\le \frac{4}{n^2}+\frac{4}{(n+1)^2}.
\end{equation}
Equality holds in (\ref{p-090}) for the function $f$ defined by
\begin{equation}gin{examp}in{equation}\leftarrowbel{p-100}
f'(z):=\frac{1+iz}{1-iz}.
\end{equation}
Next, if $f\in\mathcal{R}$ is of the form (\ref{p-001}) then
\begin{equation}gin{examp}in{align}\leftarrowbel{p-110}
|T_3(1)| &=|1-2a_2^2+2a_2^2a_3-a_3^2|\\
&\le 1+2|a_2^2|+|a_3||a_3-2a_2^2|\nonumber\\
&\le 1+2+\frac{2}{3}\left|\frac{1}{3}c_2-\frac{1}{2}c_1^2\right|\nonumber\\
&\le 3+\frac{2}{9}\left|c_2-\frac{3}{2}c_1^2\right|\nonumber\\
&\le 3+\frac{8}{9}=\frac{35}{9}.\nonumber
\end{align}
It is easy to see that equality in (\ref{p-110}) holds for the function $f$ defined by (\ref{p-100}).
Again, if $f\in\mathcal{R}$ is of the form (\ref{p-001}) then
\begin{equation}gin{examp}in{align*}
|T_3(2)| &=|a_2^3-2a_2a_3^2-a_2a_4^2+2a_3^2a_4|\\
&\le |a_2|^3 + 2|a_2||a_3^2| +|a_4||a_2a_4-2a_3^2|\\
&\le 1+\frac{8}{9}+\frac{1}{2}|a_2a_4-2a_3^2|\\
&\le \frac{17}{9}+\frac{1}{2}|a_2a_4-2a_3^2|.
\end{align*}
Thus we need to find the maximum value of $|a_2a_4-2a_3^2|$ for functions in $\mathcal{R}$. By (\ref{p-165}), it easily follows that
$$
|a_2a_4-2a_3^2|=\frac{1}{72}|9c_1c_3-16c_2^2|\le \frac{64}{72}=\frac{8}{9}.
$$
Therefore
$$
|T_3(2)|\le \frac{17}{9}+\frac{4}{9}=\frac{7}{3}.
$$
\end{proof}
\begin{equation}gin{examp}in{rem}
The above theorem shows that for $f\in \mathcal{R}$, the sharp inequalities $|T_2(2)|\le 13/9$ and $|T_2(3)|\le 17/36$ hold. In \cite{Radhika-Sivasubramanian-Murugusundaramoorthy-Jahangiri-2016}, it was claimed that $|T_2(2)|\le 5/9$, $|T_2(3)|\le 4/9$, $|T_3(1)|\le 13/9$ and $|T_3(2)|\le 4/9$ hold for functions in $\mathcal{R}$ and these estimates are sharp. For the function $f$ defined by (\ref{p-100}), a simple computation gives $|T_2(2)|= 13/9$, $|T_2(3)|= 17/36$, $|T_3(1)|= 35/9$ and $|T_3(2)|\le 25/12$, showing that the these estimates are not correct. As explained above, the authors assumed that $c_1>0$, which is not justified, since the functional $|T_q(n)|$ $(n\ge1, q\ge2)$ is not rotationally invariant.
\end{rem}
If $f\in\mathcal{T}$ is given by (\ref{p-001}), then the coefficients of $f$ can be expressed by
\begin{equation}gin{examp}in{equation*}\leftarrowbel{p-210}
a_n=\int_{-1}^{1} \frac{\sin(n\arccos t)}{\sin(\arccos t)}\,d\mu(t) =\int_{-1}^{1} U_{n-1}(t)\,d\mu(t), \quad n\ge 1
\end{equation*}
where $U_{n}(t)$ are Chebyshev polynomials of degree $n$ of the second kind.
Let $A_{n,m}$ denote the region of variability of the point $(a_n,a_m)$, where $a_n$ and $a_m$ are coefficients of a given function $f\in\mathcal{T}$ with the series expansion (\ref{p-001}), i.e., $A_{n,m}:=\{(a_n(f),a_m(f)):f\in\mathcal{T}\}$. Therefore, $A_{n,m}$ is the closed convex hull of the curve
$$
\gamma_{n,m}:[-1,1]\ni t\rightarrow (U_{n-1}(t),U_{m-1}(t)).
$$
By the Caratheodory theorem we conclude that it is sufficient to discuss only functions
\begin{equation}gin{examp}in{equation}\leftarrowbel{p-215}
F(z,\alpha,t_1,t_2):=\alpha k(z,t_1)+(1-\alpha)k(z,t_2),
\end{equation}
where $0\le\alpha\le1$ and $-1\le t_1\le t_2\le1$.
Let $X$ be a compact Hausdorff space, and $J_{\mu}=\int_{X} J(t)\,d\mu(t)$. Szapiel \cite{Szapiel-1986} proved the following theorem.
\begin{equation}gin{examp}in{thm}\leftarrowbel{theorem-030}
Let $J:[\alpha,\begin{equation}ta]\rightarrow \mathbb{R}^n$ be continuous. Suppose that there exists a positive integer $k$, such that for each non-zero $\overrightarrow{p}$ in $\mathbb{R}^n$ the number of solutions of any equation $\leftarrowngle \overrightarrow{J(t)},\overrightarrow{p}\rightarrowngle=const$, $\alpha\le t\le \begin{equation}ta$ is not greater than $k$. Then, for every $\mu\in P_{[\alpha,\begin{equation}ta]}$ such that $J_{\mu}$ belongs to the boundary of the convex hull of $J([\alpha,\begin{equation}ta])$, the following statements are true:
\begin{equation}gin{examp}in{enumerate}
\item if $k=2m$, then
\begin{equation}gin{examp}in{enumerate}[(a)]
\item $|supp (\mu)|\le m$, or
\item $|supp (\mu)|=m+1$ and $\{\alpha,\begin{equation}ta\}\subset supp (\mu).$\\
\end{enumerate}
\item if $k=2m+1$, then
\begin{equation}gin{examp}in{enumerate}[(a)]
\item $|supp (\mu)|\le m$, or
\item $|supp (\mu)|=m+1$ and one of the points $\alpha$ and $\begin{equation}ta$ belongs to $supp (\mu).$
\end{enumerate}
\end{enumerate}
\end{thm}
In the above, the symbol $\leftarrowngle \overrightarrow{u},\overrightarrow{v}\rightarrowngle$ means the scalar product of vectors $\overrightarrow{u}$ and $\overrightarrow{v}$, whereas the symbols $P_X$ and $|supp(\mu)|$ describe the set of probability measures on $X$, and the cardinality of the support of $\mu$, respectively.
Putting $J(t)=(U_{1}(t),U_{2}(t))$, $t\in[-1,1]$ and $\overrightarrow{p}=(p_1,p_2)$, we can see that any equation of the form $p_1U_{1}(t)+p_2U_{2}(t)=const$, $t\in[-1,1]$ has at most $2$ solutions. According to Theorem \ref{theorem-030}, the boundary of the convex hull of $J([-1,1])$ is determined by atomic measures $\mu$ for which support consists of at most 2 points. Thus we have the following.
\begin{equation}gin{examp}in{lem}\leftarrowbel{p-lemma055}
The boundary of $A_{2,3}$ consists of points $(a_2,a_3)$ that correspond to the functions $F(z,1,t,0)=k(z,t)$ or $F(z,\alpha,1,-1)$ with $0\le\alpha\le1$ and $-1\le t\le1$ where $F(z,\alpha,t_1,t_2)$ is defined by (\ref{p-215}).
\end{lem}
In a similar way, one can obtain the following:
\begin{equation}gin{examp}in{lem}\leftarrowbel{p-lemma060}
The boundary of $A_{3,4}$ consists of points $(a_3,a_4)$ that correspond to the functions $F(z,\alpha,t,-1)$ or $F(z,\alpha,t,1)$ with $0\le\alpha\le1$ and $-1\le t\le1$ where $F(z,\alpha,t_1,t_2)$ is defined by (\ref{p-215}).
\end{lem}
Before we proceed further, we give some example of typically real functions.
\begin{equation}gin{examp}in{example}\leftarrowbel{example-1}
For each $t\in[-1,1]$, the function $k(z,t)=z/(1-2tz+z^2)$ is a typically real function. For the function $k(z,1)=z/(1-z)^2$, we have $T_2(n)=n^2-(n+1)^2=-(2n+1)$, and $T_3(n)=a_n^3-2 a_{n+1}^2 a_n-a_{n+2}^2 a_n+2 a_{n+1}^2 a_{n+2}=4(n+1)$.
\end{example}
\begin{equation}gin{examp}in{example}\leftarrowbel{example-2}
The function $f(z)=-\log(1-z)=z+\sum_{n=2}^{\infty} (1/n)z^n$ is a typically real function. For this function, we have $T_2(n)=1/n^2-1/(n+1)^2$ and $T_3(n)=4 \left(n^2+3 n+1\right)/(n^3 (n+1)^2 (n+2)^2)$.
\end{example}
\begin{equation}gin{examp}in{lem}\leftarrowbel{p-lemma065}
If $f\in\mathcal{T}$ then $T_2(n)$ attains its extreme values on the boundary of $A_{n,n+1}$.
\end{lem}
\begin{equation}gin{examp}in{proof}
Let $\phi(x,y)=x^2-y^2$, where $x=a_n$ and $y=a_{n+1}$. The only critical point of $\phi$ is $(0,0)$ and $\phi(0,0)=0$. Since $\phi$ may be positive as well as negative for $(x,y)\in A_{n,n+1}$ (see Example \ref{example-1} and Example \ref{example-2}), the extreme values of $\phi$ are attained on the boundary of $A_{n,n+1}$.
\end{proof}
In a similar way, we can prove the following:
\begin{equation}gin{examp}in{lem}\leftarrowbel{p-lemma065}
If $f\in\mathcal{T}$ then $T_3(1)$ attains its extreme values on the boundary of $A_{2,3}$.
\end{lem}
Since all coefficients of $f\in\mathcal{T}$ are real, we look for the lower and the upper bounds of $T_q(n)$ instead of the bound of $|T_q(n)|$. The proof of the following theorem is obvious.
\begin{equation}gin{examp}in{thm}\leftarrowbel{theorem-040}
For every function $f\in\mathcal{T}$ of the form (\ref{p-001}), we have $-(n+1)^2\le T_2(n)\le n^2$.
In particular
\begin{equation}gin{examp}in{enumerate}[(i)]
\item if $n$ is odd then $\max \{T_2(n):f\in\mathcal{T}\}=n^2$ and equality attained for the function $F(z,1/2,1,-1)$.
\item if $n$ is even then $\min \{T_2(n):f\in\mathcal{T}\}=-(n+1)^2$ and equality attained for the function $F(z,1/2,1,-1)$.
\end{enumerate}
\end{thm}
\begin{equation}gin{examp}in{thm}\leftarrowbel{theorem-045}
For $f\in \mathcal{T}$, $\displaystyle \max \{T_2(2):f\in\mathcal{T}\}=\frac{5}{4}$.
\end{thm}
\begin{equation}gin{examp}in{proof}
By Lemma \ref{p-lemma055}, it is enough to consider the functions $F(z,1,t,0)=k(z,t)$ and $F(z,\alpha,1,-1)$ with $0\le\alpha\le1$ and $-1\le t\le1$.
\textbf{Case 1.} For the function $F(z,1,t,0)=k(z,t)=z+2 t z^2+\left(4 t^2-1\right) z^3+\left(8 t^3-4 t\right) z^4+\cdots$, we have
$$
a_2^2-a_3^2=-16 t^4+12 t^2-1\le \frac{5}{4}.
$$
\textbf{Case 2.} For the function $F(z,\alpha,1,-1)=z+ (4 \alpha -2) z^2+3 z^3+(8 \alpha -4) z^4+\cdots$, we have
$$
a_2^2-a_3^2=(2-4 \alpha )^2-9\le -5.
$$
\noindent The conclusion follows from Cases 1 and 2, with the maximum attained for the function $F(z,1,t,0)=k(z,t)$ with $\displaystyle t=\frac{\sqrt3}{2\sqrt2}$.
\end{proof}
\begin{equation}gin{examp}in{cor}
For $f\in \mathcal{T}$, we have the sharp inequality $-9\le T_2(2)\le \frac{5}{4}$.
\end{cor}
\begin{equation}gin{examp}in{thm}\leftarrowbel{theorem-050}
For $f\in \mathcal{T}$, we have $\displaystyle \min \{T_2(3):f\in\mathcal{T}\}=-7$.
\end{thm}
\begin{equation}gin{examp}in{proof}
By Lemma \ref{p-lemma060}, it is enough to consider the functions $F(z,\alpha,t,-1)$ and $F(z,\alpha,t,-1)$ with $0\le\alpha\le1$ and $-1\le t\le1$.
\textbf{Case 1.} For the function $F(z,\alpha,t,-1)=z+2(\alpha +\alpha t-1)z^2+ \left(4\alpha t^2-4\alpha+3\right)z^3+ \left(4\alpha+8\alpha t^3-4 \alpha t-4\right)z^4+\cdots$, we have
$$
T_2(3)=a_3^2-a_4^2=\left(4\alpha t^2-4\alpha+3\right)^2-\left(4\alpha+8\alpha t^3-4 \alpha t-4\right)^2:=\phi(\alpha,t).
$$
By elementary calculus, one can verify that
$$
\min_{0\le\alpha\le1,-1\le t\le1} \phi(\alpha,t)=\phi(0,0)=-7.
$$
\textbf{Case 2.} For the function $F(z,\alpha,t,1)=z+ 2(1-\alpha +\alpha t) z^2+\left(3-4\alpha+4\alpha t^2\right)z^3+\left(4-4\alpha-4 \alpha t+8\alpha t^3\right) z^4+\cdots$, we have $a_2^2-a_3^2=\phi(\alpha,-t)$, and so
$$
\min_{0\le\alpha\le1,-1\le t\le1} \phi(\alpha,-t)=\phi(0,0)=-7.
$$
The conclusion follows from Cases 1 and 2, and the maximum is attained for the function $F(z,0,0,1)$ or $F(z,0,0,-1)$.
\end{proof}
\begin{equation}gin{examp}in{cor}
For $f\in \mathcal{T}$, we have the sharp inequality $-7\le T_2(3)\le 9$.
\end{cor}
\begin{equation}gin{examp}in{thm}\leftarrowbel{theorem-055}
For $f\in \mathcal{T}$, we have $\displaystyle \max \{T_3(1):f\in\mathcal{T}\}=8$, and $\displaystyle \min \{T_3(1):f\in\mathcal{T}\}=-8$.
\end{thm}
\begin{equation}gin{examp}in{proof}
By Lemma \ref{p-lemma055}, it is enough to consider the functions $F(z,1,t,0)=k(z,t)$ and $F(z,\alpha,1,-1)$ with $0\le\alpha\le1$ and $-1\le t\le1$.
\textbf{Case 1.} For the function $F(z,1,t,0)=k(z,t)=z+2 t z^2+\left(4 t^2-1\right) z^3+\left(8 t^3-4 t\right) z^4+\cdots$, we have $T_3(1)=1-2a_2^2+2a_2^2a_3-a_3^2=8 t^2 \left(2 t^2-1\right):=\phi_1(t)$,
and it is easy to verify that
$$
\max_{-1\le t\le1} \phi_1(t)=\phi_1(-1)=8\quad\mbox{and}\quad \min_{-1\le t\le1} \phi_1(t)=\phi_1(-1/2)=-1.
$$
\textbf{Case 2.} For the function $F(z,\alpha,1,-1)=z+ (4 \alpha -2) z^2+3 z^3+(8 \alpha -4) z^4+\cdots$, we have $T_3(1)=8 \left(8 \alpha ^2-8 \alpha +1\right):=\psi_1(\alpha)$,
\noindent and it is again easy to verify that
$$
\max_{0\le\alpha\le1} \psi_1(\alpha)=\psi_1(0)=8\quad\mbox{and}\quad \min_{0\le\alpha\le1} \psi_1(\alpha)=\psi_1(1/2)=-8.
$$
The conclusion follows from Cases 1 and 2, and the maximum is attained for the function $F(z,1,-1,0)=k(z,-1)$, and the minimum is attained for the function $F(z,1/2,1,-1)$.
\end{proof}
\noindent\textbf{Acknowledgement:}
The authors thank Prof. K.-J. Wirths for useful discussion and suggestions.
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{\sc W. Ma}, Generalized Zalcman conjecture for starlike and typically real functions, {\it J. Math. Anal. Appl.} {\bf 234} (1999)(1), 328--339.
\bibitem{Radhika-Sivasubramanian-Murugusundaramoorthy-Jahangiri-2016}
{\sc V. Radhika, S. Sivasubramanian, G. Murugusundaramoorthy} and {\sc J. M. Jahangiri}, Toeplitz matrices whose elements are the coefficients of functions with bounded boundary rotation, {\it J. Complex Anal.}, vol. 2016 (2016), Article ID 4960704, 4 pages.
\bibitem{Robertson-1935}
{\sc M. S. Robertson}, On the coefficients of a typically-real function, {\it Bull. Amer. Math. Soc.}, {\bf 41}, (1935), 565--572.
\bibitem{Szapiel-1986}
{\sc W. Szapiel}, Extremal problems for convex sets. Applications to holomorphic functions, Dissertation Ann. Univ. Mariae Curie-Sk{\l}odowska, Sect. A. {\bf 37} (1986).
\bibitem{Thomas-Halim-2017}
{\sc D. K. Thomas} and {\sc S. A. Halim}, Toeplitz matrices whose elements are the coefficients of starlike and close-to-convex functions,
{\it Bull. Malays. Math. Sci. Soc.}, DOI 10.1007/s40840-016-0385-4, 10 pages.
\bibitem{Ye-Lim-2016}
{\sc K. Ye} and {\sc L.-H. Lim}, Every matrix is a product of Toeplitz matrices, {\it Found. Comput. Math.} {\bf 16} (2016), 577--598.
\end{thebibliography}
\end{document}
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math
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Alison is an experienced practitioner and has been an Occupational Therapist for 20 years. Alison has specialised within the field of Acquired Brain Injury and complex neurological conditions for the past 16 years.
She is a confident proactive therapist with excellent communication skills, who can establish a successful rapport with team members, clients and family alike. Currently Alison works as an independent Occupational Therapist with a variety of professionals to provide a range of high quality Occupational Therapy intervention to clients.
During Alison's career, a broad spectrum of experience has been gained by working with complex neurological conditions, especially brain injury. She has worked within the acute sector of the NHS, Community Rehabilitation, Third Sector provision and Education. This broad level and length of experience means she is very familiar with formulating successful goal oriented treatment programmes, following thorough assessment of physical, cognitive and functional issues.
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english
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Municipalities in the Great Lakes Region are already experiencing the effects of climate change – from flooding, to extreme temperatures, to winter storms, to high winds, Great Lakes cities are at different stages of preparedness for extreme weather associated with climate change.
Through a collaboration with AECOM, the City of Gary and University of Michigan’s Great Lakes Integrated Science and Assessment office (GLISA), the Cities Initiative has developed the Climate Ready Infrastructure and Strategic Sites Protocol (CRISSP), which relies on available data and municipal staff’s own knowledge of their facilities and infrastructure to assess their assets’ vulnerability to extreme weather in a way that is both relatively quick and low-cost.
The CRISSP guides your municipality through a step-by-step process to assemble your CRISSP team, gather relevant information on hazards and climate data, identify municipal infrastructure, facilities and sites located in extreme weather hazard zones, and perform a vulnerability assessment on them. A key aspect of the CRISSP is a helpful, easy to use Risk Matrix tool that takes users through a series of critical questions to assess the vulnerability of municipal facilities, sites or infrastructure.
To access the CRISSP and Risk Matrix, see the links below.
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english
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یہٕ چُھ آمُت ایکِس مشینہِ منز تھاونہٕ یوس ڈسکہِ چُھ ہیمیسیفیریکل شکل منز تراوان۔
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kashmiri
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बैंक होलिडेस २०२०: जनवरी में १० दिन तो साल में इतने दिन बैंक रहेंगे बंद, देखिए पूरी लिस्ट
जनवरी २०२० में कुल १० दिन बंद रहेंगे बैंक।
नई दिल्ली। नए साल २०२० का स्वागत करने के लिए लोगों ने तैयारियां तेज कर दी हैं। इस बार नया साल बुधवार (१ जनवरी २०२०) से शुरू हो रहा है। नए साल पर पहले महीने जनवरी २०२० में बैंक कामकाज को लेकर हम आपको बैंकों की छुट्टी को लेकर पूरी जानकारी दे रहे हैं। नए साल के पहले महीने यानी जनवरी २०२० में कुल मिलाकर १0 दिन बैंक बंद रहेंगे। इन १0 छुट्टियों में दूसरा और चौथा शनिवार सहित अलग-अलग राज्यों में होने वाली छुट्टियां भी शामिल हैं। जानिए जनवरी और पूरे साल में कब-कब बैंक बंद रहेंगे।
भारतीय रिजर्व बैंक ऑफ इंडिया (आरबीआई) ने साल २०२० के लिए बैंक हॉलीडे २०२० कैलेंडर (बैंक होलिडेस २०२० कैलेंडर) जारी कर दिया है। इन दिनों बैंकों का कामकाज जहां बंद रहेगा, इसमें सरकारी और निजी बैंक दोनों की छुट्टियां शामिल हैं।
१ जनवरी २०२०- नए साल २०२० के पहले दिन यानी १ जनवरी २०२० को कुछ राज्यों में बैंक बंद रहेंगे। १ जनवरी २०२० को आइजॉल, चेन्नई, गंगटोक और शिलॉन्ग में बैंकों में अवकाश रहेगा।
२ जनवरी २0२0- जनवरी महीने के दूसरे दिन यानी २ जनवरी २0२0 को आइजॉल और चंडीगढ़ में गुरु गोविंद सिंह जी के जन्मदिन के अवसर पर बैंक बंद रहेंगे।
११ जनवरी २०२०- दूसरा शनिवार होने के कारण बैंक पूरे देश में बंद रहेंगे यानी बैंकों में कामकाज नहीं होगा।
१४ जनवरी २०२०- मकर संक्रांति के अवसर पर १४ जनवरी २०२० को अहमदाबाद क्षेत्र के बैंक बंद रहेंगे।
१५ जनवरी २०२०- इस दिन उतरायण पून्यकाल मकर संक्रांति त्योहार, पोंगल, माघ बिहू और टुसू पूजा की वजह से बेंगलुरू, चेन्नई, गुवाहाटी, हैदराबाद में बैंक बंद रहेंगे।
१६ जनवरी २०२०- १६ जनवरी २०२० को चेन्नई क्षेत्र के बैंक तिरुवल्लूर दिवस के कारण बंद रहेंगे।
१७ जनवरी २०२०- उजहवार थिरुनाल के मौके पर भी चेन्नई में बैंकों में कोई लेन-देन नहीं होगा।
२३ जनवरी २०२०- नेताजी सुभाष चंद्र बोस जयंती के कारण २३ जनवरी २०२० को अगरतला और कोलकाता क्षेत्र के बैंकों में अवकाश रहेगा।
२५ जनवरी २०२०- २५ जनवरी २०२० को महीने का चौथा शनिवार है, जिसके कारण देश भर के बैंक बंद रहेंगे। चौथे शनिवार को देश के सभी सरकारी और प्राइवेट बैंकों में कामकाज नहीं होगा।
३० जनवरी २०२०- वसंत पंचमी/सरस्वती पूजा के कारण ३० जनवरी २०२० को अगरतला, भुवनेश्वर, भोपाल और कोलकाता क्षेत्र के बैंकों में अवकाश रहेगा।
२०२० में राष्ट्रीय बैंक की छुट्टियों की पूरी सूची यहां देखें
१ जनवरी २०२० (बुधवार)- नव वर्ष दिवस
१५ जनवरी २०२० (बुधवार)- पोंगल (दक्षिण राज्यों में)
२६ जनवरी २०२०(रविवार)- गणतंत्र दिवस
३० जनवरी २०२० (गुरुवार)- वसंत पंचमी
२१ फरवरी २०२० (शुक्रवार)- महाशिवरात्रि
१० मार्च २०२० (मंगलवार)- होली
२५ मार्च २०२० (बुधवार)- गुड़ीपड़वा
२ अप्रैल २0२0 (गुरुवार)- राम नवमी
६ अप्रैल २०२० (सोमवार)- महावीर जयंती
१० अप्रैल २०२० (शुक्रवार)- गुड फ्राइडे
१४ अप्रैल २०२० (मंगलवार)- अंबेडकर जयंती
१ मई २०२० (शुक्रवार)- मई दिवस
७ मई २०२० (गुरुवार)- बुद्ध पूर्णिमा
२५ मई २०२० (सोमवार)- ईद अल-फितर
३१ जुलाई (शुक्रवार) या १ अगस्त २०२० (शनिवार)- ईद-उल-अजहा, बकरीद
३ अगस्त २०२० (सोमवार)- रक्षाबंधन
११ अगस्त २०२० (मंगलवार)- कृष्ण जन्माष्टमी
१५ अगस्त २०२० (शनिवार)- स्वतंत्रता दिवस
३० अगस्त २०२० (रविवार)- मुहर्रम
२ अक्टूबर २0२0 (शुक्रवार)- गांधी जयंती
२६ अक्टूबर २०२० (सोमवार)- दशहरा
३० अक्टूबर २०२० (शुक्रवार)- ईद मिलाद-उन-नबी
१४ नवंबर २०२० (शनिवार)- दिवाली
१६ नवंबर २०२० (सोमवार)- भाईदूज
३० नवंबर २०२० (सोमवार)- गुरु नानक जयंती
२५ दिसंबर २०२० (शनिवार)- क्रिसमस डे
नोट: अपने-अपने शहर में बैंक की छुट्टियों की पूरी लिस्ट आप भारतीय रिजर्व बैंक (आरबीआई) की वेबसाइट पर देख सकते हैं। आपको बता दें कि हर राज्य के हिसाब से बैंक में छुट्टियां अलग-अलग होती हैं। यहां क्लिक कर आप छुट्टियों की लिस्ट देख सकते हैं।
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hindi
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یہ چھےٚ سۄ کتاب یۄس مےٚ مسٹر ایوبن دژمٕژ ٲس
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kashmiri
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यम द्वार - एक नए जीवन के आरम्भ का द्वार
यम द्वार, जिसका मतलब है मृत्यु के देव का द्वार, भगवान् शिव के दरबार कैलाश पर्वत की परिक्रमा का आरम्भिक स्थान स्थान है ई तिब्बतन तीथयात्री इसको तरबोचे के नाम से जानते हैं ई शिव के अद्भुत संसार के दर्शनों के लिए तीर्थयात्री इसी द्वार में से निकल कर अपनी कैलाश परिक्रमा का आरम्भ करते हैं ई यम द्वार या तारबोचे, कैलाश पर्वत के सबसे समीप स्थित दारचेन शहर से तकरीबन ३० मिनट की दूरी पर है ई ३० मिनट का दारचेन से यम द्वार या तारबोचे तक का यह सफर गाडी से किया जाता है ई कहते हैं की मृत्यु के देव यमराज के आशीर्वाद से इस द्वार से गुजरने वाली हर जीवात्मा नश्वर संसार के सभी दुखों और जीवन के अन्य सभी पापों से मुक्ति प्राप्त करती है ई
तिब्बत के बौद्ध धर्मावलम्बी कैलाश को अपनी पवित्र देवी 'डेमचोक' का स्थान मानते हैं ई हर वर्ष तिब्बतन तीर्थयात्री तारबोचे पर पुराने प्रार्थना झण्डियों को उतार कर नए प्रार्थना झंडियां लगाते हैं ई यह परिवर्तन की प्रथा को वह एक त्यौहार स्वरुप चौथे लुनार मॉस की पूर्णिमा के दिन मनाते हैं ई इस पवित्र पर्व पर तिब्बत के कोने कोने से यात्री यहां आते हैं और अपने पारम्परिक तरीके से नाच गाना और पूजा करते हैं ई
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hindi
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Quentin Curry (b. 1972 Johnstown, PA) studied at Bard College in New York and The San Francisco Art Institute. In addition to several national group shows, his work has been in solo exhibitions at Stellan Holm Gallery in New York and Kantor Gallery in Los Angeles. Tripoli Gallery first included Curry's work in a group exhibition at Tripoli Gallery East Hampton in 2015 titled "A Walk...”, followed by our 11th Annual Thankgiving Collective in 2015, then Black & White earlier this year, as well as the gallery’s summer group exhibition, Summer Trip. In the past, Curry has used stone dust and oil paint to produce textured, semi-abstract landscapes. His recent works explore a more freehand and primal approach with painting and mark making. While recreating natural forms such as leaves and rocks, Curry’s paintings and sculptures suggest a modern day reinterpretation of ancient pictographs, etched or drawn onto the layered surfaces of paint and material. Curry lives and works in New York City and Sagaponack, NY.
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english
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निक हॉकले को क्रिकेट ऑस्ट्रेलिया का अंतरिम सीईओ नियुक्त किया गया है
केविन रॉबर्ट्स ने मंगलवार को क्रिकेट ऑस्ट्रेलिया (सीए) के मुख्य कार्यकारी पद से इस्तीफा दे दिया था. उनकी जगह टी-२० विश्व कप के मुख्य कार्यकारी निक हॉकले को अंतरिम व्यवस्था के तौर पर सीए का अध्यक्ष पद सौंपा गया है. वह ऐसे समय में यह पद संभाल रहे हैं जब कोविड-१९ महामारी के कारण बोर्ड वित्तीय संकट से जूझ रहा है. अलसो रेड - कोविड-१९ महामारी के बीच भारतीय फुटबॉलर की पत्नी कर रही हैं वो काम जिसे दुनिया कर रही सलाम!
केविन रॉबर्ट्स भरोसा और सम्मान खो चुके थे अलसो रेड - सौरव गांगुली ने टीम इंडिया को जीत की मानसिकता दी, धोनी के बाद विराट ने इसे नई ऊंचाइयों पर पहुंचा
इंटरेनशनल क्रिकेट काउंसिल (आईसीसी) के पूर्व मुख्य कार्यकारी मैलकम स्पीड ने कहा है कि केविन रॉबर्ट्स भरोसा और सम्मान खो चुके थे जिसके कारण उन्हें सीए के शीर्ष पद से हटना पड़ा तथा उनकी जगह कार्यभार संभालने वाले निक हॉकले का काम किसी नए स्पिनर का डेब्यू मैच में ही विराट कोहली का सामना करने जैसा है. अलसो रेड - नासिर हुसैन ने भारतीय टीम के चयन नीति पर उठाए सवाल, बोले-सिर्फ प्लान 'ए' से काम नहीं चलता
स्पीड ने एसईएन रेडियो से कहा, ऐसा लगता है कि वह खिलाड़ियों का भरोसा और सम्मान खो चुके थे. एक बार किसी गुरु ने मुझसे कहा था, सम्मान और भरोसा कौमार्य की तरह है, एक बार खोने पर उन्हें वापस पाना मुश्किल होता है.
उन्होंने कहा, मुझे लगता है कि केविन के साथ भी ऐसा ही हुआ. उन्होंने भरोसा और सम्मान खो दिया. जब उन्होंने पद संभाला था तो उनके पास समय था लेकिन वह ऐसा नहीं कर पाए और वह संदेश को सही तरह से नहीं पहुंचा पाए.
हॉकले का काम आसान नहीं होगा
हॉकले के लिए अब काम आसान नहीं होगा और उन्हें विभिन्न हितधारकों का भरोसा जीतना होगा जिनमें प्रांत, खिलाड़ी और उनके कर्मचारी भी शामिल हैं. इसके अलावा टी-२० विश्व कप को भी लेकर भी अनिश्चितता बनी हुई है जिस पर आईसीसी अगले महीने फैसला कर सकती है.
स्पीड ने कहा, कोई मुश्किल समय नहीं है. यह कुछ हद तक वैसा ही है जैसे किसी नए ऑफ स्पिनर को अपना पहला ओवर विराट कोहली के लिए करने को कहा जाए. उन्होंने कहा, मैं निक हॉकले को नहीं जानता. मुझे लगता है कि वह पिछले कुछ समय से क्रिकेट से जुड़े हैं. उन्हें यहां कई चुनौतियों का सामना करना होगा.
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hindi
|
یہٕ گرٕ یتھ منٛز بہٕ آزٕ ژاوُس گٔژھہِ ہمیشہِ خٲطرٕ خۄشحال آسُن تہٕ زانٛہہ گژھہِ نہٕ کھینٕچ کٔمی گژھنۍ
|
kashmiri
|
package hudson.plugins.warnings;
import java.io.IOException;
import java.util.Collection;
import org.jenkinsci.Symbol;
import org.kohsuke.stapler.Ancestor;
import org.kohsuke.stapler.AncestorInPath;
import org.kohsuke.stapler.QueryParameter;
import org.kohsuke.stapler.StaplerProxy;
import org.kohsuke.stapler.StaplerRequest;
import org.kohsuke.stapler.StaplerResponse;
import jenkins.model.Jenkins;
import net.sf.json.JSONObject;
import hudson.Extension;
import hudson.model.AbstractProject;
import hudson.plugins.analysis.core.NullBuildHistory;
import hudson.plugins.analysis.core.PluginDescriptor;
import hudson.plugins.analysis.graph.DefaultGraphConfigurationView;
import hudson.plugins.analysis.graph.GraphConfiguration;
import hudson.plugins.warnings.parser.ParserRegistry;
import hudson.util.CopyOnWriteList;
import hudson.util.FormValidation;
/**
* Descriptor for the class {@link WarningsPublisher}. Used as a singleton. The
* class is marked as public so that it can be accessed from views.
*
* @author Ullrich Hafner
* @deprecated replaced by classes of io.jenkins.plugins.analysis package
*/
@Deprecated
@Extension(ordinal = 100) @Symbol("warnings")
public final class WarningsDescriptor extends PluginDescriptor implements StaplerProxy {
/** The ID of this plug-in is used as URL. */
static final String PLUGIN_ID = "warnings";
/** The URL of the result action. */
static final String RESULT_URL = PluginDescriptor.createResultUrlName(PLUGIN_ID);
/** Prefix of icons in this plug-in. */
public static final String IMAGE_PREFIX = "/plugin/warnings/icons/";
/** Icon to use for the sidebar links. */
public static final String SMALL_ICON_URL = IMAGE_PREFIX + "warnings-24x24.png";
/** Icon to use for the result summary. */
public static final String LARGE_ICON_URL = IMAGE_PREFIX + "warnings-48x48.png";
private final CopyOnWriteList<GroovyParser> groovyParsers = new CopyOnWriteList<GroovyParser>();
/**
* Returns the URL of the warning results for the specified parser.
*
* @param group
* the parser group
* @return a unique URL
*/
public static String getResultUrl(final String group) {
if (group == null) { // prior 4.0
return RESULT_URL;
}
else {
return PLUGIN_ID + ParserRegistry.getUrl(group) + RESULT_URL_SUFFIX;
}
}
/**
* Returns the URL of the warning project for the specified parser.
*
* @param group
* the parser group
* @return a unique URL
*/
public static String getProjectUrl(final String group) {
if (group == null) { // prior 4.0
return PLUGIN_ID;
}
else {
return PLUGIN_ID + ParserRegistry.getUrl(group);
}
}
/**
* Returns the graph configuration screen.
*
* @param link
* the link to check
* @param request
* stapler request
* @param response
* stapler response
* @return the graph configuration or <code>null</code>
*/
public Object getDynamic(final String link, final StaplerRequest request, final StaplerResponse response) {
if ("configureDefaults".equals(link)) {
Ancestor ancestor = request.findAncestor(AbstractProject.class);
if (ancestor.getObject() instanceof AbstractProject) {
AbstractProject<?, ?> project = (AbstractProject<?, ?>)ancestor.getObject();
return new DefaultGraphConfigurationView(
new GraphConfiguration(WarningsProjectAction.getAllGraphs()), project, "warnings",
new NullBuildHistory(),
project.getAbsoluteUrl() + "/descriptorByName/WarningsPublisher/configureDefaults/");
}
}
return null;
}
/**
* Instantiates a new {@link WarningsDescriptor}.
*/
public WarningsDescriptor() {
this(true);
}
/**
* Instantiates a new {@link WarningsDescriptor}.
*
* @param loadConfiguration
* determines whether the values of this instance should be
* loaded from disk
*/
public WarningsDescriptor(final boolean loadConfiguration) {
super(WarningsPublisher.class);
if (loadConfiguration) {
load();
}
}
@Override
public String getDisplayName() {
return Messages.Warnings_Publisher_Name();
}
@Override
public String getPluginName() {
return PLUGIN_ID;
}
@Override
public String getIconUrl() {
return SMALL_ICON_URL;
}
@SuppressWarnings("rawtypes")
@Override
public boolean isApplicable(final Class<? extends AbstractProject> jobType) {
return true;
}
/**
* Returns the configured Groovy parsers.
*
* @return the Groovy parsers
*/
public GroovyParser[] getParsers() {
return groovyParsers.toArray(new GroovyParser[groovyParsers.size()]);
}
/**
* Adds the given Groovy parser to the configured Groovy parsers.
*
* @param parser the new parser
*/
public void addGroovyParser(final GroovyParser parser) {
groovyParsers.add(parser);
save();
}
/**
* Adds the given collection of Groovy parsers to the configured Groovy parsers.
*
* @param parsers the new parsers
*/
public void addGroovyParsers(final Collection<GroovyParser> parsers) {
groovyParsers.addAll(parsers);
save();
}
/**
* Replaces the configured Groovy parsers with the given collection.
*
* @param parsers the new parsers
*/
public void replaceGroovyParsers(final Collection<GroovyParser> parsers) {
groovyParsers.replaceBy(parsers);
save();
}
@Override
public boolean configure(final StaplerRequest req, final JSONObject formData) {
replaceGroovyParsers(req.bindJSONToList(GroovyParser.class, formData.get("parsers")));
return true;
}
@Override
public FormValidation doCheckPattern(@AncestorInPath final AbstractProject<?, ?> project,
@QueryParameter final String pattern) throws IOException {
FormValidation required = FormValidation.validateRequired(pattern);
if (required.kind == FormValidation.Kind.OK) {
return super.doCheckPattern(project, pattern);
}
else {
return required;
}
}
/**
* Returns whether the current user has the permission to edit the available
* Groovy parsers.
*
* @return {@code true} if the user has the right, {@code false} otherwise
*/
public boolean canEditParsers() {
return Jenkins.getInstance().getACL().hasPermission(Jenkins.RUN_SCRIPTS);
}
@Override
public Object getTarget() {
return this;
}
}
|
code
|
أسۍ چھِ ترجمہ مطالس وُسعت دِنٕکۍ خواہاں "
|
kashmiri
|
using System;
using System.Linq;
using FluentAssertions;
using Paramore.Brighter.Core.Tests.CommandProcessors.TestDoubles;
using Microsoft.Extensions.DependencyInjection;
using Paramore.Brighter.Extensions.DependencyInjection;
using Xunit;
using Paramore.Brighter.Inbox.Handlers;
namespace Paramore.Brighter.Core.Tests.CommandProcessors
{
[Collection("CommandProcessor")]
public class PipelineGlobalInboxContextTests : IDisposable
{
private const string CONTEXT_KEY = "TestHandlerNameOverride";
private readonly PipelineBuilder<MyCommand> _chainBuilder;
private Pipelines<MyCommand> _chainOfResponsibility;
private readonly RequestContext _requestContext;
private readonly InboxConfiguration _inboxConfiguration;
private IAmAnInboxSync _inbox;
public PipelineGlobalInboxContextTests()
{
_inbox = new InMemoryInbox();
var registry = new SubscriberRegistry();
registry.Register<MyCommand, MyGlobalInboxCommandHandler>();
var container = new ServiceCollection();
container.AddTransient<MyGlobalInboxCommandHandler>();
container.AddSingleton<IAmAnInboxSync>(_inbox);
container.AddTransient<UseInboxHandler<MyCommand>>();
container.AddSingleton<IBrighterOptions>(new BrighterOptions() {HandlerLifetime = ServiceLifetime.Transient});
var handlerFactory = new ServiceProviderHandlerFactory(container.BuildServiceProvider());
_requestContext = new RequestContext();
_inboxConfiguration = new InboxConfiguration(
scope: InboxScope.All,
context: (handlerType) => CONTEXT_KEY);
_chainBuilder = new PipelineBuilder<MyCommand>(registry, (IAmAHandlerFactorySync)handlerFactory, _inboxConfiguration);
PipelineBuilder<MyCommand>.ClearPipelineCache();
}
[Fact]
public void When_Building_A_Pipeline_With_Global_inbox()
{
//act
_chainOfResponsibility = _chainBuilder.Build(_requestContext);
var firstHandler = _chainOfResponsibility.First();
var myCommmand = new MyCommand();
firstHandler.Handle(myCommmand);
//assert
var exists = _inbox.Exists<MyCommand>(myCommmand.Id, CONTEXT_KEY, 500);
exists.Should().BeTrue();
}
public void Dispose()
{
CommandProcessor.ClearExtServiceBus();
}
}
}
|
code
|
A passenger gingerly alights from Dallas Car PCC #3332 at Butler Street on the Mattapan High Speed Line.
Photographed by Ron DeFilippo, February, 1969.
Getting off on the wrong side of the train?
|
english
|
ایکسٹرنَل کیروٹِڑ آرٹری چھِ کامَن کیروٹِڑ آرٹری ہٕنٛز اَکھ شاخ یۄس گَردَن نیبرِم طَرفہٕ آسان چھِ۔ اِنسانَن مَنٛز چھِ یِم کامَن کیروٹِڑ آرٹری ہٕنٛز اَکھ شاخ تہٕ بیٛاکھ شاخ چھِ اِنٹرنَل کیروٹِڑ آرٹری. ایکسٹرنَل کیروٹِڑ آرٹری چھِ کَلہٕ کین باقی حِصَن خوٗن واتناوان یِمَن مَنٛز بۄتھ، تال، مینَنجی یِوان چھِ.
ڈانٛچہٕ
نِگار خانہٕ
== حَوالہٕ ==
|
kashmiri
|
The Lancia LC1 has proven to be the car to beat this year, and the Nurburgring 1000km confirmed this trend indeed. However, it was no walk in the park for the Italian team, as the works Porsches issued a serious challenge in this great afternoon of endurance racing.
Coxon snuck past Weber on the start, and things quickly heated up. The pair was to keep a very impressive pace for the first 8 laps, in close company. Quite far behind them, the fight was nothing short of epic. Gaggles of very different Group C cars battling it out on every corner of the green hell is always a sight worth seeing. Kowalski, Tiengou, Martinelli, Whited, Swindells, Vermeersch and Jundt were trading position on every lap, keeping the fans very entertained indeed. Unfortunately, what the Nordschleife gives, the Nordschleife takes back. Soon, as tyres started to wear off, drivers were caught off guard, and we lost several very interesting entries way too early. Tiengou and Kowalski, who had been shining in the early stages of the races, totaled their cars on lap 5 and 6, respectively. Uncharacteristically, Martinelli and his works Porsche, who was going strong in 4th place, suffered an engine failure early in the race. Paulet never got the chance to show his pace with the car.
As Jundt’s Ford was making great progress up the field, eventually securing third place, Rob Swindells and Jason Whited were having impressive runs in their ageing cars, racing very close to each other for ten laps. Unfortunately, on the first driver swap, the Joest Porsche suffered a terrible bug, and retired. The Rondeau driver, after having lost a lap for repairs on lap 10, was to set a formidable come back, lapping very fast and consistently. After having handed over the French car to Bruno Chacon, who had a good stint, this team as well was to suffer one of the dreaded bugs on their last swap. No doubt Swindells and Maycock, already winners in Silverstone this year, and Whited, probably the fastest man on track considering his car, will be forces to be reckoned with in future events.
As Coxon pitted after 8 laps, commentators and fans alike were in shock. Surely the 956 could not compete with the Lancia by stopping five times? As it turned out, when the Lancia stopped for fuel and tyres on lap 12, its stop was 15 seconds longer than the Porsche’s. From then on, the maths were simple: to win, Coxon and Wilks had to beat the Lancia by 1mn15s on track. Unfortunately, as Wilks took over on lap 16 it became clear that he could not keep up with his teammate’s impressive pace. Although he did set great laptimes in the 7mn00 region, Weber meanwhile was still lapping some 15s a lap quicker, and the gap he managed to set on his rivals was to prove decisive. However, there was drama in the Lancia pits, where Jan Titz was nowhere to be found at the start of the race! Eventually, he did show up and took over the LC1 halfway into the event. He managed an excellent double stint with distant pressure from Coxon, who was setting lap records after lap records in an attempt to close the gap. It was not to be though, and when the quick Englishman pitted again with four laps to go, Lancia had the race firmly in their grasp.
In Group 5, Lancia won as well! A commanding performance earned Yves Placais a very fine win. Having been the fastest in the class all year long, it was only fair that he did taste champagne eventually. His main rival Mikula out after a single lap due to technical issues, he found himself in a remote fight with Jukka Maattanen and Jason White, who despite being a tad slower on the track, could make the whole race on three short stops thanks to the M1’s great fuel mileage. However, Yves was not to be denied, and his class drive even allowed him to beat a competitive Ford crew and finish in 4th overall! One of the finest endurance solo drive ever witnessed in the history of the league for sure. Splendid results for Marazzi, Janak and Kalide, who finished in this order, 3rd 4th and 5th in class and 7th, 8th and 9th overall after tough solo drives. Early on in the race, Noack and Canola proved to be serious contenders, but they both crashed out. Hopefully we see them again at the wheel of competitive cars this season. Others interesting performances came from Chapman, Acerlinth, Wattman and Verplanken, but they all had to either retire or make long stops due to several misfortunes. Acerlinth gets the unofficial fair-play award for pushing the Wattman brothers' wounded RX-7 over the line on his 935’s remaining 3 wheels! True Swedish camaraderie for sure.
Zakspeed can boast a solid performance overall, with 3rd and 5th place. Jundt and Jacques showed their pace and reliability, their podium position never being seriously contested after lap 12. Vermeersch, Jereb and Knowles had a slightly more eventful race, finishing without a front bumper, and behind a group 5 car. Tough luck for WM, with a car that’s considerably off the pace, Thiim tried to keep in touch with the rest of the pack but a massive shunt meant he had to spend a long time in the pits. Later on, Dechavanne had to retire the car for the same reason.
Overall the race was a great success, with 23 starting cars and 11 finishers, great sportsmanship, skilled driving all around the track and competitive racing in both classes. Bring on Spa already!
|
english
|
1. Kingtool Aluminum Doors and Windows Machinery Co., Ltd. is a famous automaker in China.
2. Please Contact Us! kingtool aluminium machinery Only Selects High Quality Material To Produce aluminum punching machine, aluminium punching machine, punching machine for aluminium profile To Assure Their Quality. Welcome To Contact Us For Detail Information About It.
3. We present an unmatched range of aluminum hydraulic punching machine, which is precision engineered and valued for their flawless feature and long time superior performance.
|
english
|
بھاگیالکشمی چھِ اَکھ ہِندوستٲنؠ اَداکارہ یۄس فِلمَن مَنٛز چھِ کٲم کَران.
زٲتی زِندگی
فِلمی دور
== حَوالہٕ ==
|
kashmiri
|
<?php
/**
* Copyright 2018 LINE Corporation
*
* LINE Corporation licenses this file to you 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:
*
* https://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.
*/
namespace LINE\LINEBot\MessageBuilder\Flex\ComponentBuilder;
use LINE\LINEBot\Constant\Flex\ComponentAdjustMode;
use LINE\LINEBot\Constant\Flex\ComponentButtonHeight;
use LINE\LINEBot\Constant\Flex\ComponentButtonStyle;
use LINE\LINEBot\Constant\Flex\ComponentGravity;
use LINE\LINEBot\Constant\Flex\ComponentMargin;
use LINE\LINEBot\Constant\Flex\ComponentPosition;
use LINE\LINEBot\Constant\Flex\ComponentSpacing;
use LINE\LINEBot\Constant\Flex\ComponentType;
use LINE\LINEBot\MessageBuilder\Flex\ComponentBuilder;
use LINE\LINEBot\TemplateActionBuilder;
use LINE\LINEBot\Util\BuildUtil;
/**
* A builder class for button component.
*
* @package LINE\LINEBot\MessageBuilder\Flex\ComponentBuilder
*/
class ButtonComponentBuilder implements ComponentBuilder
{
/** @var TemplateActionBuilder */
private $actionBuilder;
/** @var int */
private $flex;
/** @var ComponentMargin */
private $margin;
/** @var ComponentButtonHeight */
private $height;
/** @var ComponentButtonStyle */
private $style;
/** @var string */
private $color;
/** @var ComponentGravity */
private $gravity;
/** @var string */
private $position;
/** @var string */
private $offsetTop;
/** @var string */
private $offsetBottom;
/** @var string */
private $offsetStart;
/** @var string */
private $offsetEnd;
/** @var ComponentAdjustMode */
private $adjustMode;
/** @var array */
private $component;
/**
* ButtonComponentBuilder constructor.
*
* @param TemplateActionBuilder|null $actionBuilder
* @param int|null $flex
* @param ComponentMargin|null $margin
* @param ComponentButtonHeight|null $height
* @param ComponentButtonStyle|null $style
* @param string|null $color
* @param ComponentGravity|null $gravity
*/
public function __construct(
$actionBuilder,
$flex = null,
$margin = null,
$height = null,
$style = null,
$color = null,
$gravity = null
) {
$this->actionBuilder = $actionBuilder;
$this->flex = $flex;
$this->margin = $margin;
$this->height = $height;
$this->style = $style;
$this->color = $color;
$this->gravity = $gravity;
}
/**
* Create empty ButtonComponentBuilder.
*
* @return ButtonComponentBuilder
*/
public static function builder()
{
return new self(null);
}
/**
* Set action.
*
* @param TemplateActionBuilder $actionBuilder
* @return ButtonComponentBuilder
*/
public function setAction($actionBuilder)
{
$this->actionBuilder = $actionBuilder;
return $this;
}
/**
* Set flex.
*
* @param int|null $flex
* @return ButtonComponentBuilder
*/
public function setFlex($flex)
{
$this->flex = $flex;
return $this;
}
/**
* Set margin.
*
* @param ComponentMargin|string|null $margin
* @return ButtonComponentBuilder
*/
public function setMargin($margin)
{
$this->margin = $margin;
return $this;
}
/**
* Set height.
*
* @param ComponentButtonHeight|string|null $height
* @return ButtonComponentBuilder
*/
public function setHeight($height)
{
$this->height = $height;
return $this;
}
/**
* Set style.
*
* @param ComponentButtonStyle|string|null $style
* @return ButtonComponentBuilder
*/
public function setStyle($style)
{
$this->style = $style;
return $this;
}
/**
* Set color.
*
* @param string|null $color
* @return ButtonComponentBuilder
*/
public function setColor($color)
{
$this->color = $color;
return $this;
}
/**
* Set gravity.
*
* @param ComponentGravity|string|null $gravity
* @return ButtonComponentBuilder
*/
public function setGravity($gravity)
{
$this->gravity = $gravity;
return $this;
}
/**
* Set position.
*
* specifiable relative or absolute
*
* @param string|ComponentPosition|null $position
* @return $this
*/
public function setPosition($position)
{
$this->position = $position;
return $this;
}
/**
* Set offsetTop.
*
* specifiable percentage, pixel and keyword.
* (e.g.
* percentage: 5%
* pixel: 5px
* keyword: none (defined in ComponentSpacing)
*
* @param string|ComponentSpacing|null $offsetTop
* @return $this
*/
public function setOffsetTop($offsetTop)
{
$this->offsetTop = $offsetTop;
return $this;
}
/**
* Set offsetBottom.
*
* specifiable percentage, pixel and keyword.
* (e.g.
* percentage: 5%
* pixel: 5px
* keyword: none (defined in ComponentSpacing)
*
* @param string|ComponentSpacing|null $offsetBottom
* @return $this
*/
public function setOffsetBottom($offsetBottom)
{
$this->offsetBottom = $offsetBottom;
return $this;
}
/**
* Set offsetStart.
*
* specifiable percentage, pixel and keyword.
* (e.g.
* percentage: 5%
* pixel: 5px
* keyword: none (defined in ComponentSpacing)
*
* @param string|ComponentSpacing|null $offsetStart
* @return $this
*/
public function setOffsetStart($offsetStart)
{
$this->offsetStart = $offsetStart;
return $this;
}
/**
* Set offsetEnd.
*
* specifiable percentage, pixel and keyword.
* (e.g.
* percentage: 5%
* pixel: 5px
* keyword: none (defined in ComponentSpacing)
*
* @param string|ComponentSpacing|null $offsetEnd
* @return $this
*/
public function setOffsetEnd($offsetEnd)
{
$this->offsetEnd = $offsetEnd;
return $this;
}
/**
* Set adjustMode
*
* @param ComponentAdjustMode|null $adjustMode
* @return $this
*/
public function setAdjustMode($adjustMode)
{
$this->adjustMode = $adjustMode;
return $this;
}
/**
* Builds button component structure.
*
* @return array
*/
public function build()
{
if (isset($this->component)) {
return $this->component;
}
$this->component = BuildUtil::removeNullElements([
'type' => ComponentType::BUTTON,
'action' => $this->actionBuilder->buildTemplateAction(),
'flex' => $this->flex,
'margin' => $this->margin,
'height' => $this->height,
'style' => $this->style,
'color' => $this->color,
'gravity' => $this->gravity,
'position' => $this->position,
'offsetTop' => $this->offsetTop,
'offsetBottom' => $this->offsetBottom,
'offsetStart' => $this->offsetStart,
'offsetEnd' => $this->offsetEnd,
'adjustMode' => $this->adjustMode,
]);
return $this->component;
}
}
|
code
|
کیا تُہۍ ہیٚکوا یِمَن تمام ویکسِنیشن سینٹرن ہُنٛد لسٹ بنٲوِتھ یتؠن کووِشٟلڈ گۄڈنیُک ڈوز آسہِ ۱۸ وُہُر پیٚٹھؠ وٲنٛسہِ خٲطرٕ دٔستِیاب
|
kashmiri
|
/**
* You can nest the object literals within other object literals.
* @type {{bar: number, bas: {bas1: string, bas2: number}}}
*/
var foo = {
bar: 123,
bas: {
bas1: 'some string',
bas2: 345
}
};
console.log(foo);
console.log(foo.bas);
|
code
|
Former princess Ubolratana Rajakanya posted a message on her Instagram after the King prohibited her from standing in the upcoming election as a candidate for prime minister.
BANGKOK - Former princess Ubolratana Rajakanya Sirivadhana Barnavadi posted a message on Saturday morning (Feb 9) to thank her supporters, but did not comment on her candidacy.
The older sister of King Maha Vajiralongkorn posted the message on her Instagram wall hours after the King prohibited her from standing in the upcoming election as a candidate for prime minister of a pro-Thaksin party.
The 67-year-old princess did not directly mention her brother or her political hopes, but thanked supporters for their "love and kindness toward each other over the past day" and expressed gratitude for their support for her.
"I would like to say once again that I want to see Thailand moving forward, being admirable and acceptable by international countries, want to see all Thais have rights, a chance, good living, happiness to all," she said, concluding with"#ILoveYou".
A political bombshell was dropped on Friday morning when the Thai Raksa Chart party - an offshoot of the larger pro-Thaksin party that was ousted from power in the 2014 coup - nominated the 67-year-old as its sole candidate for prime minister in the upcoming elections.
"All royal family members adhere to the same principles... and cannot take any political office, because it contradicts the intention of the Constitution."
The leaders of Thai Raksa Chart have declined to comment on the king's statement. The party cancelled plans to launch election campaigns in Bangkok's China Town on Saturday. It informed reporters without explanation that Thai Raksa Chart leader Preechaphol Pongpanit and party campaign chief Nattawut Saikaur would cancel the visit to Yaowarat.
Topping one of the most dramatic weeks in the nation's political history, Thailand's Election Commission said it will meet on Monday (Feb 11) morning, without specifying the agenda.
All parties contesting the election had to submit their candidate lists to the commission on Friday (Feb 8). It's supposed to check and validate the nominations by Feb 15.
Thailand has been a constitutional monarchy since 1932, but the royal family has wielded great influence. Ubolratana relinquished her royal titles in 1972 when she married an American, a fellow student at the Massachusetts Institute of Technology (MIT), Peter Jensen.
The princess's main opponent in the March general election, if her nomination were to stand, would likely be Prime Minister Prayut Chan-ocha, who was army chief when he led the 2014 coup and now heads the ruling junta, who also announced his candidacy on Friday.
|
english
|
import React from 'react';
import PropTypes from 'prop-types';
import AutoCompleteTextInput from 'common/form/AutoCompleteTextInput';
import { debounce } from 'utils/streamUtils';
import InputTitle from './InputTitle';
import { getCompaniesSearch } from '../../../apis/companySearchApi';
const getItemValue = item => item.label;
const mapToAutocompleteList = l => ({
label: Array.isArray(l.name) ? l.name[0] : l.name,
value: l.id,
});
class CompanyQuery extends React.Component {
constructor(props) {
super(props);
this.handleAutocompleteItems = this.handleAutocompleteItems.bind(this);
this.state = {
autocompleteItems: [],
};
}
search = debounce((e, value) => {
if (value) {
return getCompaniesSearch({ key: value })
.then(r =>
Array.isArray(r)
? this.handleAutocompleteItems(r.map(mapToAutocompleteList))
: this.handleAutocompleteItems([]),
)
.catch(() => this.handleAutocompleteItems([]));
}
return this.handleAutocompleteItems([]);
}, 800);
handleOnChange = (e, value) => {
this.props.onChange(e.target.value);
return this.search(e, value);
};
handleAutocompleteItems(autocompleteItems) {
return this.setState(() => ({
autocompleteItems,
}));
}
render() {
const { autocompleteItems } = this.state;
const {
companyQuery,
onChange,
onCompanyId,
validator,
submitted,
} = this.props;
return (
<div>
<InputTitle text="公司名稱" must />
<AutoCompleteTextInput
placeholder="ＯＯ 股份有限公司"
value={companyQuery}
getItemValue={getItemValue}
items={autocompleteItems}
onChange={this.handleOnChange}
onSelect={(value, item) => {
this.handleAutocompleteItems([]);
onCompanyId(item.value);
return onChange(value);
}}
isWarning={submitted && !validator(companyQuery)}
warningWording="需填寫公司/單位"
/>
</div>
);
}
}
CompanyQuery.propTypes = {
companyQuery: PropTypes.string,
onChange: PropTypes.func,
onCompanyId: PropTypes.func,
validator: PropTypes.func,
submitted: PropTypes.bool,
};
CompanyQuery.defaultProps = {
validator: () => {},
};
export default CompanyQuery;
|
code
|
<div layout="column">
<md-content>
<md-list class="md-default-theme">
<md-list-item class="md-3-line md-with-secondary" ng-click="navigate(org.id)" ng-repeat="org in orgs">
<a href="javascript:void(0)" class="md-list-item-text">
<h3 class="md-primary">{{org.name}}</h3>
<p ng-pluralize count="org.districts.length"
when="{'0': 'No Districts',
'one': '1 District',
'other': '{} Districts'}">
</p>
<p ng-pluralize count="org.households.length"
when="{'0': 'No Households',
'one': '1 Household',
'other': '{} Households'}">
</p>
<div class="md-secondary" layout="row" layout-align="center center">
<md-button ng-click="editOrg($event, $index)" class="md-raised">Edit</md-button>
<md-button ng-click="deleteOrg($event, $index)" class="md-raised md-warn">Delete</md-button>
</div>
<md-divider ng-if="!$last"></md-divider>
</a>
</md-list-item>
</md-list>
</md-content>
</div>
<md-button class="md-accent md-fab md-fab-bottom-right" ng-click="createDialog($event)">+</md-button>
|
code
|
/***************************************************************************
tunerview.cpp - description
-------------------
begin : Mon Jan 10 2005
copyright : (C) 2005 by Philip McLeod
email : pmcleod@cs.otago.ac.nz
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.
Please read LICENSE.txt for details.
***************************************************************************/
#include <qpixmap.h>
#include <qwt_slider.h>
#include <qlayout.h>
#include <qtooltip.h>
//Added by qt3to4:
#include <Q3GridLayout>
#include <QResizeEvent>
#include <QPaintEvent>
#include "tunerview.h"
#include "tunerwidget.h"
#include "ledindicator.h"
#include "useful.h"
#include "gdata.h"
#include "channel.h"
#include "musicnotes.h"
int LEDLetterLookup[12] = { 2, 2, 3, 3, 4, 5, 5, 6, 6, 0, 0, 1 };
TunerView::TunerView( int viewID_, QWidget *parent )
: ViewWidget( viewID_, parent)
{
//setCaption("Chromatic Tuner");
Q3GridLayout *layout = new Q3GridLayout(this, 9, 3, 2);
layout->setResizeMode(QLayout::SetNoConstraint);
// Tuner widget goes from (0, 0) to (0, 8);
//tunerWidget = new TunerWidget(this);
tunerWidget = new VibratoTunerWidget(this);
layout->addMultiCellWidget(tunerWidget, 0, 0, 0, 8);
// Slider goes from (2,0) to (2,9)
//slider = new QwtSlider(this, "slider", Qt::Horizontal, QwtSlider::Bottom, QwtSlider::BgTrough);
#if QWT_VERSION == 0x050000
slider = new QwtSlider(this, Qt::Horizontal, QwtSlider::Bottom, QwtSlider::BgTrough);
#else
slider = new QwtSlider(this, Qt::Horizontal, QwtSlider::BottomScale, QwtSlider::BgTrough);
#endif
slider->setRange(0, 2);
slider->setReadOnly(false);
layout->addMultiCellWidget(slider, 1, 1, 0, 8);
QToolTip::add(slider, tr("Increase slider to smooth the pitch over a longer time period"));
ledBuffer = new QPixmap();
leds.push_back(new LEDIndicator(ledBuffer, this, "A"));
leds.push_back(new LEDIndicator(ledBuffer, this, "B"));
leds.push_back(new LEDIndicator(ledBuffer, this, "C"));
leds.push_back(new LEDIndicator(ledBuffer, this, "D"));
leds.push_back(new LEDIndicator(ledBuffer, this, "E"));
leds.push_back(new LEDIndicator(ledBuffer, this, "F"));
leds.push_back(new LEDIndicator(ledBuffer, this, "G"));
leds.push_back(new LEDIndicator(ledBuffer, this, "#"));
// Add the leds for note names into the positions (1, 0) to (1, 6)
for (int n = 0; n < 7; n++) {
layout->addWidget(leds.at(n), 2, n);
}
// (1, 7) is blank
// Add the flat led
layout->addWidget(leds.at(7), 2, 8);
layout->setRowStretch( 0, 4 );
layout->setRowStretch( 1, 1 );
layout->setRowStretch( 2, 0 );
//connect(gdata->view, SIGNAL(onFastUpdate(double)), this, SLOT(update()));
//connect(gdata, SIGNAL(onChunkUpdate()), tunerWidget, SLOT(doUpdate()));
connect(gdata, SIGNAL(onChunkUpdate()), this, SLOT(doUpdate()));
connect(tunerWidget, SIGNAL(ledSet(int, bool)), this, SLOT(setLed(int, bool)));
}
TunerView::~TunerView()
{
delete slider;
for (uint i = 0; i < leds.size(); i++) {
delete leds[i];
}
delete ledBuffer;
delete tunerWidget;
}
void TunerView::resizeEvent(QResizeEvent *)
{
//tunerWidget->resize(size());
}
void TunerView::resetLeds()
{
for (uint i = 0; i < leds.size(); i++) {
leds[i]->setOn(false);
}
}
void TunerView::slotCurrentTimeChanged(double /*time*/)
{
/*
Channel *active = gdata->getActiveChannel();
if (active == NULL || !active->hasAnalysisData()) {
tunerWidget->setValue(0, 0);
return;
}
ChannelLocker channelLocker(active);
// To work out note:
// * Find the slider's value. This tells us how many seconds to average over.
// * Start time is currentTime() - sliderValue.
// * Finish time is currentTime().
// * Calculate indexes for these times, and use them to call average.
//
double sliderVal = slider->value();
double startTime = time - sliderVal;
double stopTime = time;
//int startChunk = MAX(int(floor(startTime / active->timePerChunk())), 0);
//int stopChunk = MIN(int(floor(stopTime / active->timePerChunk())), active->lookup.size());
int startChunk = active->chunkAtTime(startTime);
int stopChunk = active->chunkAtTime(stopTime)+1;
float pitch;
if (sliderVal == 0) {
pitch = active->dataAtCurrentChunk()->pitch;
} else {
pitch = active->averagePitch(startChunk, stopChunk);
}
float intensity = active->averageMaxCorrelation(startChunk, stopChunk);
if (pitch <= 0) {
tunerWidget->setValue(0, 0);
return;
}
int closePitch = toInt(pitch);
// We can work out how many semitones from A the note is
//int remainder = closeNote % 12;
resetLeds();
leds.at(LEDLetterLookup[noteValue(closePitch)])->setOn(true);
if(isBlackNote(closePitch)) leds.at(leds.size() - 1)->setOn(true);
// Tell the TunerWidget to update itself, given a value in cents
tunerWidget->setValue(100*(pitch - float(closePitch)), intensity);
*/
/* Old code
int toLight = -1;
bool sharp = false;
//A A# B C | C# D D# E | F F# G <-- noteNames
//0 1 2 3 | 4 5 6 7 | 8 9 10 <-- remainder
//------------------------------
//A B C D E F G # <-- leds
//0 1 2 3 4 5 6 7 <-- led index
//
// 3 >= remainder <= 7, the LED is (remainder + 1) / 2, and if even put the sharp on
if (remainder >= 3 && remainder <= 7) {
toLight = (remainder + 1) / 2;
if (remainder % 2 == 0) sharp = true;
} else {
// the LED is remainder / 2 + 1 or just / 2, and if it's odd put the sharp led on.
toLight = (remainder / 2) + 1;
if (remainder < 3) toLight = remainder / 2;
if (remainder % 2 == 1) sharp = true;
}
leds.at(toLight)->setOn(true);
if (sharp) leds.at(leds.size() - 1)->setOn(true);
*/
}
void TunerView::paintEvent( QPaintEvent* )
{
//slotCurrentTimeChanged(gdata->view->currentTime());
}
void TunerView::setLed(int index, bool value)
{
leds[index]->setOn(value);
}
void TunerView::doUpdate()
{
Channel *active = gdata->getActiveChannel();
if (active == NULL || !active->hasAnalysisData()) {
tunerWidget->doUpdate(0.0);
return;
}
ChannelLocker channelLocker(active);
double time = gdata->view->currentTime();
// To work out note:
// * Find the slider's value. This tells us how many seconds to average over.
// * Start time is currentTime() - sliderValue.
// * Finish time is currentTime().
// * Calculate indexes for these times, and use them to call average.
//
double sliderVal = slider->value();
double pitch = 0.0;
if (sliderVal == 0) {
int chunk = active->currentChunk();
if(chunk >= active->totalChunks()) chunk = active->totalChunks()-1;
if(chunk >= 0)
pitch = active->dataAtChunk(chunk)->pitch;
} else {
double startTime = time - sliderVal;
double stopTime = time;
int startChunk = active->chunkAtTime(startTime);
int stopChunk = active->chunkAtTime(stopTime)+1;
pitch = active->averagePitch(startChunk, stopChunk);
}
//float intensity = active->averageMaxCorrelation(startChunk, stopChunk);
tunerWidget->doUpdate(pitch);
}
|
code
|
अंबाला(कमलप्रीत सभ्रवाल): अंबाला कैंट के बीसी बाजार में सुसाइड का एक एेसा मामला सामने आया है, जहां घर पर टीवी देख रही दीपा(३६) ने पंखे से फंदा लगाकर आत्महत्या कर ली। दीपा के परिवार वालों की माने तो उस वक्त टीवी पर फ़िल्म पद्मावती की रिलीज की खबर चल रही थी जिसमें बताया जा रहा था कि एक युवक ने फ़िल्म पद्मावती पर चल रही खबरों से आहत होकर फांसी लगा ली है। इसी दौरान दीपा भी फांसी लगाकर मौत की आगोश में चली गई।
मृतका की छोटी बहन ने कहा कि दीपा कमरे में टीवी देख रही थी और वह बाहर सिलाई मशीन पर कपड़े सिल रही थी। टीवी पर फ़िल्म पद्मावती की रिलीज के विरोध में एक युवक द्वारा आत्महत्या करने की खबर चली तो दीपा ने टीवी की आवाज तेज करके कमरा बंद किया और फांसी के फंदे पर झूल गई। परिवार को समझ नहीं आ रहा कि आखिर एमए तक पढ़ी लिखी और स्कूल में टीचर रह चुकी होनहार बेटी ने अचानक आत्महत्या जैसा जानलेवा कदम क्यों उठा लिया।
घटना के बाद पुलिस मौके पर पहुंची और महिला पुलिस की मदद से शव का पोस्टमार्टम करवा कर परिजनों को सौंप दिया है। मौके से मिले सुसाइड नोट और परिजनों के बयानों के आधार पर कार्रवाई करते हुए जांच जारी है। मौके से मिले दो पन्नों में अलग-अलग तरह की बात लिखी है और दोनों पर ही दीपा के नाम के अंग्रेजी में हस्ताक्षर हैं। इनमें से एक पन्ने पर नीले बॉलपैन से लिखा है कि अपनी शादी तथा घरवालों की परेशानी दूर करने के लिए वह आत्महत्या कर रही है। उसके घर वाले इसके लिए जिम्मेवार नहीं हैं। इसके अलावा एक और पन्ने पर लिखा है कि वह अपनी मर्जी से सुसाइड कर रही है उसके परिवार के लोग इसके जिम्मेवार नहीं हैं।
|
hindi
|
# -*- coding: utf-8 -*-
#
# This file is part of the jabber.at homepage (https://github.com/jabber-at/hp).
#
# This project 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 3 of the
# License, or (at your option) any later version.
#
# This project 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 django-xmpp-account.
# If not, see <http://www.gnu.org/licenses
import logging
import os
import re
import textwrap
from urllib.parse import urljoin
import dns.resolver
import html5lib
from django.conf import settings
from django.core.cache import cache
from django.forms.utils import flatatt
from django.utils.html import format_html
from django.utils.text import normalize_newlines
from django.utils.translation import ugettext as _
from django.utils.translation import ungettext
from .exceptions import TemporaryError
log = logging.getLogger(__name__)
def format_timedelta(delta):
days = delta.days
hours, rem = divmod(delta.seconds, 3600)
minutes, _seconds = divmod(rem, 60)
# Get translated strings for days/hours/minutes
if days:
days = ungettext('one day', '%(count)d days', days) % {'count': days}
if hours:
hours = ungettext('one hour', '%(count)d hours', hours) % {'count': hours}
if minutes: # just minutes
minutes = ungettext('one minute', '%(count)d minutes', minutes) % {'count': minutes}
# Assemble a string based on what we have
if days and hours and minutes:
return _('%(days)s, %(hours)s and %(minutes)s') % {
'days': days, 'hours': hours, 'minutes': minutes, }
elif days and hours:
return _('%(days)s and %(hours)s') % {'days': days, 'hours': hours, }
elif days and minutes:
return _('%(days)s and %(minutes)s') % {'days': days, 'minutes': minutes, }
elif days:
return days
elif hours and minutes:
return _('%(hours)s and %(minutes)s') % {'hours': hours, 'minutes': minutes, }
elif hours:
return hours
elif minutes:
return minutes
else:
return _('Now')
def load_private_key(hostname):
fp = settings.XMPP_HOSTS[hostname].get('GPG_FINGERPRINT')
if fp:
path = os.path.join(settings.GPG_KEYDIR, '%s.key' % fp)
with open(path, 'rb') as stream:
key = stream.read()
path = os.path.join(settings.GPG_KEYDIR, '%s.pub' % fp)
with open(path, 'rb') as stream:
pub = stream.read()
return fp, key, pub
return None, None, None
def load_contact_keys(hostname):
keys = {}
fingerprints = settings.XMPP_HOSTS[hostname].get('CONTACT_GPG_FINGERPRINTS', [])
for fp in fingerprints:
path = os.path.join(settings.GPG_KEYDIR, '%s.pub' % fp)
with open(path, 'rb') as stream:
keys[fp] = stream.read()
return keys
def check_dnsbl(ip):
"""Check the given IP for DNSBL listings.
This method caches results for an hour to improve speed.
"""
cache_key = 'dnsbl_%s' % ip
blocks = cache.get(cache_key)
if blocks is not None:
return blocks
blocks = []
for dnsbl in settings.DNSBL:
reason = None
resolver = dns.resolver.Resolver()
query = '.'.join(reversed(str(ip).split("."))) + "." + dnsbl
try:
resolver.query(query, "A")
except (dns.resolver.NoNameservers, dns.exception.Timeout):
# Nameservers are unreachable
raise TemporaryError(
_("Could not check DNS-based blocklists. Please try again later."))
except dns.resolver.NXDOMAIN: # not blacklisted
continue
try:
reason = resolver.query(query, "TXT")[0].to_text()
except: # reason is optional
pass
blocks.append((dnsbl, reason))
cache.set(cache_key, blocks, 3600) # cache this for an hour
return blocks
def canonical_link(path, host=None):
"""Get the canonical link of a relative URL path.
Uses the ``CANONICAL_BASE_URL`` setting in the default ``XMPP_HOST`` as base URL.
Example::
>>> canonical_link('/foo/bar', {'CANONICAL_BASE_URL': 'https://example.com'})
'https://example.com/foo/bar'
"""
if host is None:
host = settings.XMPP_HOSTS[settings.DEFAULT_XMPP_HOST]
base_url = host['CANONICAL_BASE_URL']
return urljoin(base_url, path)
def absolutify_html(html, base_url):
"""Make relative links in the given html absolute.
This code is copied from `here <http://garethrees.org/2009/10/09/feed/>`_.
Examle::
>>> absolutify_html('<a href="/foobar">test</a>', 'https://example.com')
'<a href="https://example.com/foobar">test</a>'
>>> absolutify_html('<a href="https://example.net/foobar">test</a>', 'https://example.com')
'<a href="https://example.net/foobar">test</a>'
"""
attributes = [
('a', 'href'),
('img', 'src'),
('link', 'href'),
('script', 'src')
]
# Parse SRC as HTML.
tree_builder = html5lib.treebuilders.getTreeBuilder('dom')
parser = html5lib.html5parser.HTMLParser(tree=tree_builder)
dom = parser.parse(html)
# Change all relative URLs to absolute URLs by resolving them relative to
# BASE_URL. Note that we need to do this even for URLs that consist only of
# a fragment identifier, because Google Reader changes href=#foo to
# href=http://site/#foo
for tag, attr in attributes:
for e in dom.getElementsByTagName(tag):
u = e.getAttribute(attr)
if u:
e.setAttribute(attr, urljoin(base_url, u))
# Return the HTML5 serialization of the <BODY> of the result (we don't want
# the <HEAD>: this breaks feed readers).
body = dom.getElementsByTagName('body')[0]
tree_walker = html5lib.treewalkers.getTreeWalker('dom')
html_serializer = html5lib.serializer.HTMLSerializer()
return ''.join(html_serializer.serialize(tree_walker(body)))
def format_link(url, text, **attrs):
"""Format a link. The returned value is marked as safe HTML.
>>> format_link('https://example.com', 'example')
'<a href="https://example.com">example</a>'
>>> format_link('https://example.com', 'example', title='Awesome title')
'<a href="https://example.com" title="Awesome title">example</a>'
"""
attrs = {k: v for k, v in attrs.items() if v is not None}
return format_html('<a href="{}"{}>{}</a>', url, flatatt(attrs), text)
def mailformat(text, width=78):
text = normalize_newlines(text.strip())
text = re.sub('\n\n+', '\n\n', text)
ps = []
for p in text.split('\n\n'):
ps.append(textwrap.fill(p, width=width))
ps.append('')
return '\n'.join(ps).strip()
|
code
|
अपडेट, विकास दुबे के नौकर ने किया खुलासा, थाना पुलिस की लापरवाही से शहीद हुए आठ पुलिस कर्मी
लखनऊ : फर्जी अंक पत्र लगाकर शिक्षक की नौकरी करने वाले चार लोगों पर मुकदमा दर्ज
नेपोटिज्म को लेकर कंगना का तापसी पन्नू पर फूटा गुस्सा, कहा- तुम्हें शर्म आनी चाहिए..
गाजियाबाद: ४ साल का प्यार फिर दोनों ने रचाई शादी, अब ४ दिन में ही दोनों ने दी जान
कानपुर हत्याकांड : थाने से फोन कर कटवाई गई थी गांव की बिजली, एसटीएफ जांच में खुलने लगी है परतें
छोटे पर्दे के मिस्टर बजाज और बॉलीवुड एक्टर करण सिंह ग्रोवर आज अपना ३८वां जन्मदिन सेलिब्रेट कर रहे हैं। इस खास दिन और खास बनाने के लिए करण अपनी पत्नी बिपाशा बसु के साथ मालदीव पहुंचे हुए हैं। बिपाशा ने अपने इंस्टाग्राम पर कुछ फोटोज़ शेयर की हैं जिनमें वो और करण स्वीमिंग पूल में मस्ती करते दिख रहे हैं। किसी फोटो में वो लिप किस कर रहे हैं तो किसी में कुल लुक दे रहे हैं। दोनों की ये फोटोज़ काफी अच्छी हैं।
फोटोज़ शेयर करते हुए बिपाशा ने कैप्शन भी लिखा है जिसमें उन्होंने अपने मंकी लव यानी करण को बर्थडे विशा किया है। एक्ट्रेस ने लिखा, एक और हॉलीड मेरे पार्टनर के साथ मेरे सबकुछ। #मोंकीलव #मालदीव। आपको बता दें कि करण और विपाशा सोशल मीडिया पर काफी एक्टिव हैं और अपनी लविंग फोटोज़ शेयर करते रहते हैं।अगर आप दोनों का इंस्टाग्राम पर देखेंगे तो आपको उनकी कई रोमांटिक फोटोज़ मिल जाएंगी।
दोनों के वर्क फ्रंट की बात करें तो बिपाशा लंबे वक्त से फिल्मों से दूर हैं पर सोशल मीडिया के जरिए वो अपने फैंस से कनैक्ट रहती हैं। एक्ट्रेस साल २०१५ में करण के साथ ही हॉरर फिल्म अलोन में नजर आई थीं। इसके बाद से एक्ट्रेस ने बड़े पर्दे से दूरी बना ली। वहीं करण के हाल ही में एकता कपूर के सीरियल कसौटी जिंदगी के में नजर आए थे। इस सीरियल में उन्होंने मिस्टर बजाज का किरदार निभाया था जो दर्शकों को काफी पसंद आया। इसके अलावा वो हाल ही में बिग बॉस १३ में भी दिखे थे। करण यहां आरती सिंह को सपोर्ट करने आए थे।
अक्षय कुमार ने टीवी क्वीन एकता कपूर के साथ एक नया प्रोजेक्ट किया साइन...
डोनाल्ड ट्रंप अपने पहले भारत दौरे पर पहुंचे अहमदाबाद, अमेरिका ने किया भारत को सलाम
अमिताभ ने जब रेखा को सबके सामने मारा तमाचा, जानिए उस दिन हुआ था क्या ?
इन ४ सितारों की मौत पर किसी को नहीं होता यकीन, एक की तो दौड़ते-दौड़ते निकल गई थी जान!
इस एक्ट्रेस को जब आधी रात मिला मसाज का ऑफर, दिया ऐसा गजब जवाब कि..!
इस एक्ट्रेस की मां पर ऐसे फिदा थे सलमान, बनाना चाहते थे मिसेज खान
इन हिरोइनों ने तो हद ही कर दी पार, सगे बाप-बेटे के साथ किया रोमांस
रामजन्मभूमि पर फिदायीन हमले की बरसी पर रामनगरी की सुरक्षा सख्त
गलवान पर खुलकर भारत के साथ आया जापान, चीन को लेकर कही ऐसी बात
ब्रेकिंग : गलवान नदी में आई तेज़ बाढ़, बह गए चीनी सेना के कैम्प, नदी के तट पर स्थिति हुई खतरनाक
भोपाल में फिर मिले कोरोना के ७४ नये मामले, अब तक १०७ लोगों की मौत
मोती सिंह के बाद योगी सरकार में एक और मंत्री कोरोना +वे, पूरा परिवार हुआ होम क्वारेंटीन
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hindi
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न्यू जीलैंड के पूर्व कप्तान डेनियल विटोरी ने भारत के खिलाफ होने वाले मैच को 'बहुत बड़ा' करार दिया है। साथ ही उन्होंने यह भी कहा कि लगातार तीन जीत के बाद कीवी टीम कुछ स्वच्छंद होकर खेल सकती है। न्यू जीलैंड ने श्री लंका, बांग्लादेश और अफगानिस्तान को हराकर शानदार शुरुआत की लेकिन उसकी असली परीक्षा १३ जून को भारत के खिलाफ होगी, जिसने साउथ अफ्रीका और ऑस्ट्रेलिया जैसी मजबूत टीमों को हराया है।
विटोरी ने आईसीसी के लिए अपने कॉलम में लिखा, 'तीन मैचों में तीन जीत न्यू जीलैंड के लिए मनोबल बढ़ाने वाली हैं लेकिन भारत के खिलाफ अगला मैच बहुत बड़ा होगा।' उन्होंने कहा, 'हम सभी जानते हैं कि भारत के खिलाफ मैच में माहौल पूरी तरह से भिन्न होता है और वह वास्तव में दबाव की स्थिति होती है। भारत संभवत: दुनिया की सर्वश्रेष्ठ टीम है और खचाखच भरे स्टेडियम में मैच खेलना रोमांचक होगा।'
विटोरी ने कहा, 'यह वर्ल्ड कप है और अगर आप खुद को दबाव में रखकर खेलते हो तो स्वयं के लिए मुश्किलें खड़ी करते हो लेकिन अभी तीन जीत के बाद वे कुछ स्वच्छंद होकर खेल सकते हैं।'
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hindi
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झारखंड में मतदाताओं को डराने के मकसद से नक्सलियों ने बम से पुल उड़ाया
रांची। झारखंड में शनिवार को हो रहे विधानसभा चुनाव के पहले चरण के मतदान में मतदाताओं को डराने के उद्देश्य से नक्सलियों ने गुमला में मतदान शुरू होने से ठीक पहले एक पुल उड़ा दिया। पुलिस ने यह जानकारी दी। चुनाव में भाग ना लेने की अपनी धमकी को बेअसर होता देख नक्सलियों ने घाघरा-काठकोथवा राजमार्ग के बीच बना पुल उड़ा दिया। अधिकारियों ने बताया कि धमाके से मतदान पर असर नहीं पड़ा है। झारखंड में सुबह सात बजे से मतदान शुरू हो जाएगा। नक्सलियों ने कुछ दिनों पहले चार पुलिसकर्मियों की हत्या कर दी थी। मतदान अपराह्न तीन बजे तक होगा।
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hindi
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पद्मावती फिल्म पर सुनील शेट्टी का आया विवादित बयान बॉलीवुड में मचा हडकंप देखें विडिओ? | असलीपोस्ट
होम न्यूज पद्मावती फिल्म पर सुनील शेट्टी का आया विवादित बयान बॉलीवुड में मचा...
पद्मावती फिल्म पर सुनील शेट्टी का आया विवादित बयान बॉलीवुड में मचा हडकंप देखें विडिओ?
हम सभी देखते आ रहे है कि जबसे फिल्म पद्मावती कि शूटिंग शुरू हुइ है तभी से यह फिल्म विवादों के घेरे में है और आज भी लोगो का गुस्सा आज कल इस फिल्म को लेकर बहुत ही भरा हुआ है और ऐसे में फिल्म का रिलीज़ होना मुश्किल लग रहा है और वही आपको बता दे कि यह फिल्म १दिसम्बर को रिलीज़ होने वाली थी लेकिन सुप्रीम कोर्ट ने कुछ कारण वश फिल्म को लौटा दिया और जिसके चलते फिल्म कि रिलीज़ डेट टाल दी गयी | अब इसके बाद फिल्म कि रिलीज़ डेट थोडा और लेट कर दी गयी है लेकिन उसमे भी क्या फिल्म रिलीज़ हो पायेगी इसकी भी अभी तक कोई पुष्टि नही हो पायी है |
लेकिन वही आपको बता दे कि अभिनेता सुनील शेट्टी का भी बयान आ चूका है और जो कहा सुनील शेट्टी ने उसका विडिओ आज कल सोशल मिडिया पर जोरो से चल रहा है आपको बता दे कि लोगो का कहना है कि इस फिल्म में इतिहास से छेड़छाड़ कि गयी है जिसके वजह से पूरा राजपूत समाज इसका विरोध कर रहा है और इनके साथ साथ अब पूरा हिन्दू समाज भी एकजुट होते हुए दिख रहा है और लोगो का कहना है कि अगर फिल्म रिलीज़ हुई तो अंजाम अच्छा नही होगा और वही भंसाली को धमकी भी मिल चुकी है |
और वही बात करें अभिनेता सुनील शेट्टी कि तो सुनील शेट्टी ने कहा कि मुझे पता है भंसाली जी कैसी फिल्मे बनाते है और जहाँ तक इस फिल्म का सवाल है उन्होंने ऐसा कुछ नही किया होगा जिससे राजपूत समाज के किसी भी मर्यादा का उल्लंघन हो और इसके साथ साथ यह भी कहा है कि फिल्म को एक बार दिखाई जाये उसके बाद उसे सेंसर बोर्ड के निर्णय पर छोड़ दिया जाय सेंसर बोर्ड तय करेगा कि फिल्म चलनी चाहिए कि नही और पहले एक बार फिल्म को चलाई जाय लोगो को किसी के बह्कावें में नही आना चाहिए पहले एक बार फिल्म को देखे क्या गलत है क्या सही उसके बाद तय करें सेंसर बोर्ड कि फिल्म चलानी चाहिए कि नही ?
प्रेवियस आर्टियलशाहिद कपूर कि पत्नी मीरा ने टैटू के चक्कर में दिखा दिया कुछ ऐसा कि खुद हो गयी शर्म से पानी पानी ?
नेक्स्ट आर्टियलप्रधानमंत्री ने लिया नोटबंदी से भी बड़ा फैसला देश में मचा हड़कंप छोड़ सकते है देश ये दिग्गज नेता ?
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hindi
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स्मिता पाटिल एक ऐसा चेहरा, जिसके सामने आते ही कई किस्से बयां हो जायें. उनका लफ्जों से बयां न कर आंखों से अपनी बात कह जाना वाकई काबिले तारीफ था. ऐसी दमदार अदाकारी कि लोग देखें, तो देखते ही रह जायें. स्मिता पाटिल अपने संवेदनशील किरदारों के लिए खूब चर्चित हुईं. हालांकि, मात्र ३१ साल की उम्र में वे इस दुनियां को अलविदा कह गयीं. स्मिता पाटिल का फिल्मी करियर भले ही १० साल का रहा हो, लेकिन उनकी दमदार अदाकारी आज भी लोगों के जेहन में हैं. मात्र १६ साल की उम्र से ही वे न्यूज रीडर के तौर पर नौकरी करती थीं.
स्मिता पाटिल का जन्म १७ अक्टूबर, १९५५ को हुआ था. १३ दिसंबर, १९८६ को इस दुनिया से चले जाने के बाद उनकी १४ फिल्में रिलीज हुई थी. उनके पिता शिवाजी राय पाटिल महाराष्ट्र सरकार में मंत्री थे, जबकि उनकी मां एक समाज सेविका थी. जानें उनके बारे में ये खास बातें...
जब स्मिता पाटिल न्यूज रीडर के तौर पर काम करती थीं, इसी दौरान उनकी मुलाकात जानेमाने निर्माता निर्देशक श्याम बेनेगल से हुई. उन्होंने स्मिता की प्रतिभा को पहचान कर अपनी फिल्म 'चरण दास चोर' में एक छोटी सी भूमिका निभाने का अवसर दिया. अस्सी के दशक में स्मिता ने व्यावसायिक फिल्मों की ओर रुख किया. कहा जाता है कि जब उन्हें फिल्मों में ब्रेक मिला था, उस समय वे एंकरिंग के साथ-साथ बेहतरीन फोटोग्राफर भी बन चुकी थीं. इस दौरान उन्होंने सुपरस्टार अमिताभ बच्चन के साथ फिल्म 'नमक हलाल' और 'शक्ति' में काम किया. दोनों ही फिल्में कामयाब रहीं. इसके बाद उन्होंने पीछे मुड़कर नहीं देखा.
'कूली' हादसे का अंदेशा था
अमिताभ बच्चन फिल्म कूली की शूटिंग के दौरान गंभीर रूप से घायल हो गये थे, इस बात से सभी वाकिफ हैं, लेकिन इस हादसे को लेकर अमिताभ बच्चन ने एक खुलासा किया था. उन्होंने कहा था कि इस एक्सिडेंट के बारे में स्मिता पाटिल को अंदेशा था. फिल्म की शूटिंग चल रही थी, इस एक्सिडेंट से एक दिन पहले स्मिता पाटिल को अमिताभ बच्चन को लेकर एक सपना आया था. स्मिता पाटिल ने बैंगलुरू में उन्हें आधी रात को फोन करके अपना एक बुरा सपना बताया था, जिसमें अमिताभ घायल हैं. अमिताभ ने हंसकर कहा था, स्मिता जी मैं ठीक हूं, आप परेशान ना हों और सो जाइये.' अगले ही दिन अमिताभ के साथ यह बड़ा हादसा हो गया था.
स्मिता पाटिल की निजी जिंदगी के बारे में चर्चा करें, तो वे राज बब्बर संग अपने अफेयर को लेकर सुर्खियों में रहीं. फिल्म 'आज की आवाज' में राज बब्बर और स्मिता पाटिल ने एकसाथ काम किया था. इसके बाद से ही दोनों के अफेयर की खबरें आने लगी थी, लेकिन स्मिता की मां इस रिश्ते से बिल्कुल खुश नहीं थीं, क्योंकि राज बब्बर की शादी पहले ही नादिरा बब्बर से हुई थी और उनका एक बेटा और बेटी भी थे. राज बब्बर स्मिता के साथ एक्स्ट्रा मैरिटल अफेयर चला रहे थे. मीडिया ने भी उनकी आलोचना करनी शुरू कर दी थी.
कहा जाता है जब स्मिता पाटिल से शादी के बारे में जैसे ही राज बब्बर ने अपने घर में बताया था, उनके माता-पिता ने इस रिश्ते पर ऐतराज जताया था. उन्होंने राज बब्बर को घर और स्मिता पाटिल में से किसी एक को चुनने को कहा था, राज बब्बर ने स्मिता पाटिल को चुनते हुए अपना घर छोड़ दिया था. दोनों के एक बेटा प्रतीक बब्बर हैं.
बेटे के जन्म के दो हफ्ते बाद ही हो गयी थी मौत
घर छोड़ने के वक्त राज बब्बर की शादी फिल्म निर्देशक और थियेटर आर्टिस्ट नादिरा बब्बर से हो चुकी थी और उनके दो बच्चे भी थे. घर छोड़ने के कुछ महीने बाद ही साल १९८६ में राज बब्बर ने स्मिता पाटिल से शादी की, लेकिन बेटे प्रतीक बब्बर को जन्म देने के दो दिन बाद ही स्मिता को वायरल इंफेक्शन हो गया था, जिसकी वजह से उनके ब्रेन में भी इंफेक्शन हो गया था. इंफेक्शन ज्यादा हो जाने की वजह से उन्हें मुंबई के जसलोक हॉस्पिटल में भर्ती कराया गया था. कहा जाता है कि भर्ती कराने के २४ घंटे में ही स्मिता पाटिल के शरीर के कई अंगों ने काम करना बंद कर दिया था. बेटे के जन्म के दो हफ्ते बाद ही १३ दिसंबर, १९८६ को उन्होंने अंतिम सांस ली.
जिंदगी के आखिरी दिनों में स्मिता का राज बब्बर के साथ रिश्ता भी कुछ बहुत सहज नहीं रह गया था. स्मिता पाटिल की एक आखिरी इच्छा थी. उनके मेकअप आर्टिस्ट दीपक सावंत बताते हैं, 'स्मिता कहा करती थीं कि दीपक जब मर जाऊंगी, तो मुझे सुहागन की तरह तैयार करना. निधन के बाद उनकी अंतिम इच्छा के मुताबिक स्मिता के शव को सुहागन की तरह मेकअप किया गया था.
उनकी कुछ चर्चित फिल्मों में 'निशान्त', 'आक्रोश', 'चक्र', 'अर्धसत्य', 'मंथन', 'भूमिका', 'गमन', 'अर्धसत्य', 'अल्बर्ट पिंटो को गुस्सा क्यों आता है', 'अर्थ', 'मिर्च मसाला', 'शक्ति', 'नमक हलाल' और 'अनोखा रिश्ता' शामिल है. फ़िल्म 'भूमिका' और 'चक्र' में दमदार अभिनय के लिए उन्हें दो राष्ट्रीय पुरस्कार के अलावा चार फिल्मफेयर अवॉर्ड भी मिले.
|
hindi
|
A Chinese architectural office has unveiled designs for a tiny shelter that could one day house astronauts on Mars.
The so-called MARS shell has a minimalist structure, which has the shape of a light bulb, with a square base and an inflatable "living bubble".
With purely conceptual renderings, the designs could serve as a helpful basis if humans ever populate the red planet.
A Chinese architectural office has unveiled designs for a tiny shelter that could one day house astronauts on Mars. The MARS case is a minimalist structure with a "living bubble".
WHAT IS THE MARS CASE?
It is a compact, self-sufficient habitat with a length of only 7.8 feet, a width of 7.8 feet and a height of 6.5 feet.
A "living bubble" attached to the base can be inflated, folded and folded in, "like packing and unpacking a suitcase."
Inside, there is a main living area that seems to have a bathroom, a desk, a few chairs and a storage room.
Smartphones, such as those from Xiaomi, can be used to control devices and other features, such as: B. lighting, to be used in the house.
There are even windows on the bubble-like structure that would pop out during inflation.
The MARS case was developed by the Beijing-based design firm Open Architecture in partnership with Chinese technology group Xiaomi.
Open Architecture said it was inspired by Henry David Thoreau's famous novel Walden while designing the structure.
In Walden, Thoreau moves to a cabin in the woods of Massachusetts, where he spends his time without traditional comfort. Instead, he decides to simply live and stay away from society.
"Two hundred years ago, Henry David Thoreau retired from society and moved alone to Walden to reflect on the nature of simple life," said Open Architecture.
"If we live in a world of consumer and environmental crises today and get lost, what are our basic needs?
"MARS Case represents a vision of this ideal home, blending technology, product design and architecture seamlessly," the company added.
The structure is portable and transferable.
Open Architecture wanted to make sure that the structure does not affect the environment, so they are very careful to integrate it into the ecosystem.
"Imagine that humanity has to settle on Mars, our distant lonely planetary neighbor," said Open Architecture.
Mars has become the next big step for humanity's exploration of space.
But before humans come to the red planet, astronauts will take a series of small steps by returning to the moon for a one-year mission.
Details of a mission in lunar orbit were revealed as part of a timeline of events that led to missions to Mars in the 2030s.
In May 2017, Greg Williams, Associate Associate Administrator of Policy and Plans at NASA, outlined the space agency's four-stage plan, which he hopes will one day visit Mars, and the expected timetable.
Phase one and two Several trips are made to the lunar space to facilitate the construction of a habitat that serves as a staging area for the journey.
The last hardware delivered would be the actual Deep Space Transport vehicle, with which later a crew was transported to Mars.
A one-year simulation of life on Mars will be conducted in 2027.
Phases three and four start after 2030 and involve constant occupation of the crew into the Mar system and the Martian surface.
"There we can not rely on natural resources because we have become so used to Earth.
"There is nothing left for us to do but to reduce the excessive consumption of our previous lifestyles and only a minimum of We will survive only through recycling.
"If we let go of the extra, we will think about our lives in a simplified environment," the company added.
According to Open Architecture, the MARS case is able to use and recycle heat, waste gases, condensation and "feed it back into this integrated ecosystem".
"In other words, energy, water and air are fully recycled in the system, minimizing resource consumption," the company said.
They say that it is completely wasteful.
Many scientists and experts have devised their own designs for what human civilization might look like on Mars.
At a recent NASA competition, teams had to submit 3D printed habitats that they believe could be used to colonize the red planet over the next few decades.
|
english
|
न्यूजीलैंड का क्लीन स्वीप, भारत तीसरा मैच भी हारा
माउंट मोंगानुई। लोकेश राहुल (११२) का शानदार शतक भी भारत को तीसरे और आखिरी वनडे में मंगलवार को हार से नहीं बचा सका और मेजबान न्यूजीलैंड ने सलामी बल्लेबाज हेनरी निकोल्स (८०), मार्टिन गुप्तिल (६६) और कॉलिन डी ग्रैंडहोम (नाबाद ५८) के अर्धशतकों की बदौलत यह मुकाबला ५ विकेट से जीतकर ३ मैचों की सीरीज को ३-० से क्लीन स्वीप कर लिया।
न्यूजीलैंड ने इस क्लीन स्वीप से भारत से टी-२० सीरीज में मिली ०-५ की क्लीन स्वीप का बदला चुका दिया और साथ ही भारत के खिलाफ ३ या उससे अधिक मैचों की वनडे सीरीज को पहली बार क्लीन स्वीप कर लिया। भारत ने पिछले न्यूजीलैंड दौरे में वनडे सीरीज ४-१ से जीती थी लेकिन इस बार उसे तीनों ही मैचों में करारी हार का सामना करना पड़ा।
भारत के वनडे इतिहास में यह तीसरा मौका है जब उसे तीन या उससे अधिक मैचों की सीरीज में क्लीन स्वीप का सामना करना पड़ा। वेस्टइंडीज ने १९८३-८४ में भारत को ५-० से और फिर १९८८-८९ में ५-० से हराया था।
भारत को इस तरह ३१ साल बाद क्लीन स्वीप का सामना करना पड़ा। इस दौरान भारत को २००६-०७ में दक्षिण अफ्रीका से पांच मैचों की सीरीज ०-४ से हार का सामना करना पड़ा था हालांकि इस सीरीज का एक मैच रद्द रहा था।
इस सीरीज में भारत की सबसे बड़ी कमजोरी उसकी गेंदबाजी रही और उसके नंबर एक वनडे गेंदबाज जसप्रीत बुमराह को पूरी सीरीज में १ भी विकेट नहीं मिला। बुमराह ने इस मैच में १0 ओवर में ५० रन दिए और खाली हाथ रहे।
तग्गे न्यूजीलैंड का क्लीन स्वीप
महाशिवरात्रि पर बाबा भोलेनाथ की बारात: जाने क्या है इस बार बोल बम सेवा एवं कल्याण समिति का मास्टर प्लान
आखिरकार जीते सिसोदिया, भाजपा प्रत्याशी को कांटे की टक्कर में हराया
और जब खिलाड़ियों के लिए खुद ही ड्रिंक्स लेकर मैदान पहुंचे पीएम
वर्ल्ड कप के बाद एमएस धोनी ले सकते हैं क्रिकेट से संन्यास: आई बड़ी वजह सामने
विश्वकप में आज भिड़ेंगे भारत और बांग्लादेश, जीतने पर सेमीफाइनल में पहुंचेगी टीम इंडिया
|
hindi
|
require 'zabbix_receiver'
require 'serverengine'
require 'slop'
module ZabbixReceiver
class CLI
def self.start(argv)
self.new(argv).start
end
def initialize(argv)
load_options(argv)
end
def start
options = {
worker_type: 'process',
}.merge(@options)
se = ServerEngine.create(Server, Worker, options)
se.run
end
private
def load_options(argv)
output_type = argv.first
output_class = get_output_class(output_type)
if output_class
argv.shift # output type
else
output_type = 'stdout'
output_class = get_output_class(output_type)
end
puts "Using #{output_type} output."
opts = Slop::Options.new
opts.on('--help') { puts opts; exit }
opts.bool '--daemonize', default: false
opts.string '--log'
opts.string '--pid-path'
opts.integer '--workers', default: 1
opts.string '--bind', default: '0.0.0.0'
opts.integer '--port', default: 10051
opts.string '--proxy-to-host'
opts.integer '--proxy-to-port', default: 10051
opts.string '--log-level'
output_class.add_options(opts)
parser = Slop::Parser.new(opts)
@options = Hash[parser.parse(argv).to_hash.map do |k, v|
[k.to_s.tr('-', '_').to_sym, v]
end]
@options[:output_class] = output_class
end
def get_output_class(type)
require "zabbix_receiver/output/#{type}"
class_name = type.split('_').map {|v| v.capitalize }.join
ZabbixReceiver::Output.const_get(class_name)
rescue NameError, LoadError
nil
end
end
end
|
code
|
बॉलीवुड ऐक्टर औरपप्पू पॉलिएस्टर के नाम से मशहूर सैयद बद्र उल हसन का निधन हो गया है। रॉयल फैमिली से रिश्ता रखने वाले सैयद बद्र उल हसन ने यूं तो फिल्म और टीवी में कई रोल किए लेकिन आज भी लोग उन्हें नंदी भगवान के रूप में याद करते हैं। ओम नम: शिवाय सीरियल में नंदी के किरदार को जीवंत बनाने वाले पप्पू पॉलिएस्टर ने मंगलवार की दोपहर इस दुनिया को हमेशा के लिए अलविदा कह दिया।
नवाबी खानदान से था ताल्लुक
इनकी रुह को शान्ती मिले
लखनऊ के गोलागंज और नक्खास की गलियों में पले-बढ़े सैयद बद्र उल हसन खान बहादुर उर्फ पप्पू पॉलिएस्टर का ताल्लुक नवाब अमजद अली शाह की चौथी पीढ़ी से था। यही वजह थी कि जब उन्होंने फिल्मी दुनिया में जाने का फैसला किया तो पूरा खानदान खफा हो गया था। इसके बावजूद पप्पू पॉलिस्टर ने दिल की सुनी और १५० किलो वजन के साथ उन्होंने फिल्मी दुनिया में अपनी पहचान बनाई। टीपू सुल्तान में मैसूर के महाराजा के रूप में उनको बहुत सराहा गया था। 'जोधा-अकबर', 'फिर भी दिल है हिन्दुस्तानी', 'मन', 'तेरे मेरे सपने' जैसी फिल्मों के अलावा कई विज्ञापनों व सीरियल्स में भी काम किया है।
मुंबई में होंगे सुपुर्द-ए-खाक
बेटे के साथ अकेली रह गईं पत्नी तबस्सुम कहती हैं कि पांच दिन से वह अंधेरी के अस्पताल में भर्ती थे। हार्ट अटैक आया और तबीयत बिगड़ती ही गई। सोमवार रात को उन्होंने मुझसे चंद बातें कीं क्योंकि उसके बाद वह वेंटिलेटर पर चले गए थे। वह मुझसे पूछ रहे थे कि यह वेंटिलेटर कब हटेगा, मुझे परेशानी हो रही है। मंगलवार को जैसे ही डॉक्टर ने वेंटिलेटर से उन्हें हटाने की कोशिश की उनकी सांसे रुक गईं। मुंबई में ही उन्हें सुपुर्द-ए-खाक किया जाएगा।
यह इच्छा रह गई अधूरी
सैयद बद्र उल हसन की पत्नी ने बताया कि उनके जाने के साथ लखनऊ में फिल्म बनाने की उनकी ख्वाहिश अधूरी रह गई। जब सैयद लखनऊ से आते थे तो बस फिल्म की ही बात करते थे। वह जब पढ़ाई के लिए लंदन गए थे, तब उन्हें फिल्मों का शौक चढ़ा था। शाही परिवार के थे इसलिए सब फिल्मी दुनिया को बुरा भला कहते थे। सबका यही कहना था कि अगर फिल्मी दुनिया में गए तो हमसे कोई रिश्ता नही रखना लेकिन उन्होंने जिन्दगी अपनी मर्जी से जी। वह बेटे को भी फिल्मी दुनिया में देखना चाहते थे।
|
hindi
|
تاہم ، یہٕ چُھ اکھ عامیت؛ واریہ ریاستن منٛز چٕھ یتھ گام یم زن ریاست کن لۄکٹن شہرن ہنٛدس مقابلس منٛز بھڑس پیمانس پیٹھ چِھ۔
|
kashmiri
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- बंद स्कूलों को शिक्षक अगले माह , गोरखपुर न्यूज इन हिन्दी -अमर उजाला बेहतर अनुभव के लिए अपनी सेटिंग्स में जाकर हाई मोड चुनें।
गोरखपुर। बंद परिषदीय प्राथमिक और पूर्व माध्यमिक विद्यालयों के ताले अक्तूबर में खुल जाएंगे। २९ सितंबर को अंतर जनपदीय स्थानांतरण के बाद ४८५ शिक्षकों की काउंसिलिंग होगी। काउंसिलिंग का प्रारूप को लेकर २७ सितंबर को शासन में बैठक होनी है। लेकिन इस काउंसिलिंग के बाद भी इन विद्यालयों में शिक्षा व्यवस्था को बनाना शिक्षा विभाग के लिए बड़ी चुनौती साबित होगी। जिले में १८८ परिषदीय प्राथमिक विद्यालय और ६९ पूर्व माध्यमिक विद्यालय बंद हैं। इसके अलावा ५२२ प्राथमिक विद्यालय और १६५ पूर्व माध्यमिक विद्यालयों में सिर्फ एक शिक्षक के भरोसे पढ़ाई हो रही है। हालांकि इन विद्यालयों में शिक्षा मित्रों की भी तैनाती है, लेकिन शासन शिक्षामित्रों को शिक्षक नहीं मानता है। ऐसे में जो विद्यालय सिर्फ शिक्षामित्रों के भरोसे हैं, विभाग इन्हें बंद की श्रेणी में रखता है। इस लिहाज से शासन बंद और एकल विद्यालयों में शिक्षकों की तैनाती कर रहा है। अंतर जनपदीय स्थानांतरण के बाद दूसरे जनपदों से ४८५ शिक्षक आए हैं। अभी इन शिक्षकों को कार्यालय में ही ज्वाइन कराया गया है। २७ या २८ सितंबर को अंतर जनपदीय स्थानांतरण से आए शिक्षकों को लेकर शासन में बैठक है। इसमें इनकी पोस्टिंग की रूपरेखा तैयार होगी। इसके बाद २९ सितंबर को बीएसए कार्यालय में इन शिक्षकों की काउंसिलिंग होगी। लेकिन विभाग के सामने एकल या बंद विद्यालयों में पर्याप्त शिक्षकों की तैनाती बड़ी चुनौती है। जिला बेसिक शिक्षा अधिकारी मनिराम सिंह ने कहा कि काउंसिलिंग के बाद सबसे पहले बंद विद्यालयों में शिक्षकों की तैनाती होगी। इसके बाद एकल विद्यालयों में शिक्षक तैनात किए जाएंगे। कैसा लगा
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hindi
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راجستھان ریاستس مَنٛز کر ویکسِنیشن سینٹر ژھانہٕ خٲطرٕ ضلعہٕ چوٗز
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kashmiri
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{\beta}egin{document}
\title[Fiber cones]{
On the Homology and fiber cone of ideals}
{\alpha}uthor{ Clare D'Cruz}
{\alpha}ddress{Chennai Mathematical Institute,
Plot H1, SIPCOT IT Park,
Siruseri,
Paddur PO,
Siruseri, Kelambakkam 603 103,
India}
\email{clare@cmi.ac.in}
{\mathfrak m}ptm
{\mathfrak m}aketitle
{\beta}egin{abstract}
In this paper we give a unified approach for several results concerning the fiber cone. Our novel ideal is to use the complex
${{\mathfrak m}athcal C}1n$. We improve earlier results obtained by several researchers and get some new results. We give a more general definition of ideals of minimal multiplicity and of ideals of almost minimal multiplicity. We also compute the Hilbert series of the fiber cone for these ideals.
\end{abstract}
\section{Introduction}
Throughout this paper we will assume that $(R,{\mathfrak m} )$ is a local ring of positive dimension $d$ and infinite residue field. The associated graded ring $G(I) := {\omega}plus_{n {\gamma}eq 0} I^n/ I^{n+1}$ has been investigated in detail by several researchers. In the last two decades the fiber cone $F(I):= {\omega}plus_{n {\gamma}eq 0} I^n/ {\mathfrak m} I^n$ has been of interest.
Let $I_1$ and $I_2$ be ideals in $(R, {\mathfrak m})$. We call $F_{I_1}(I_2) := {\omega}plus_{n {\gamma}eq 0} I_2^n/ I_1 I_2^n$ the fiber cone of $I_2$ with respect to $I_1$.
Since $G(I)= F_{I}(I)$, it is of interest to know how the properties of these two rings are related. Several recent papers on the fiber cone do imply that it is possible to extend the results known for the associated graded ring to the fiber one.
We begin by recalling a few results on $G({\mathfrak m})$. Let $(R,{\mathfrak m})$ be a Cohen-Macaulay ring. Sally showed that if ${\mathfrak m}$ is an ideal with minimal multiplicity, then $G({\mathfrak m})$ is Cohen-Macaulay and the corresponding Hilbert function $\ell({\mathfrak m}^n / {\mathfrak m}^{n+1})$ can be explicitly described
(\cite{sally1}).
She conjectured that if ${\mathfrak m}$ is an ideal with almost minimal multiplicity, then $G({\mathfrak m})$ has almost maximal depth
(\cite{sally2}). Her conjecture was settled independently by Rossi and Valla in \cite{rossi-valla} and by Wang in \cite{wang}. Sally's work has been generalized in various directions.
Goto gave a more general definition of ideals of minimal multiplicity \cite{goto}.
Inspired by his work, Jayanthan and Verma defined ideals of minimal multiplicity and ideals of almost minimal multiplicity in the case when $I_1$ and $I_2$ are ${\mathfrak m}$-primary ideals satisfying
$I_2 \subseteq I_1$
(\cite{jay-verma}, \cite{jay-verma-almm}). They studied the fiber cone of these ideals in great detail. They generalized Sally's conjecture to ideals of almost minimal multiplicity and showed that if the depth of $G(I_2)$ is at least $d-2$, then the depth of
$F_{I_1}(I_2)$ is atleast $d-1$.
In this paper, we define ideals of minimal multiplicity and ideals of almost minimal multiplicity for any two ${\mathfrak m}$-primary ideals (Definition~\ref{defn-mm-amm}, Lemma~\ref{min-mult-defn}).
We take a new approach in this paper.
Let $1 \leq k \leq d$ and let ${\bf x}_{k}:= x_1, \ldots, x_k$ a system of parameters in $I_2$.
The complex ${{\mathfrak m}athcal C}nn$
has been studied in \cite{anna} and \cite{tom-huc} in connection with the properties of the associated graded ring $G(I)$.
They showed that vanishing of the complex $C({\bf x}_{k}, {{{\mathfrak m}athcal F}i}, (i, n))$ determines
${\mathfrak m}athop{\rm depth} (({\bf x}_k)^{\stackrelr}, G(I))$, where $x_i^{\stackrelr}$ denotes the image of $x_i$ in $I/I^2$.
Associated to the $I_2$-filtration ${{\mathfrak m}athcal F}i = \{I_1I_2^n\}_{n \in {\mathfrak m}athbb Z}$,
we have the
complex ${{\mathfrak m}athcal C}1n$ (defined in Section~\ref{complex}). The corresponding graded ring
$G_{I_1}(I_2)
:= R/I_1 {\omega}plus {\omega}plus_{n {\gamma}eq 1} ({{{\mathfrak m}athcal F}i})_{n-1}/ ({{{\mathfrak m}athcal F}i})_{n}$ has been studied in \cite{rossi-valla-2}.
Note that $G(I) = G_{R}(I)[1]$.
When $({\bf x}_k) \subseteq I_1$, we consider the truncated complex
$D({\bf x}_{k}, {{{\mathfrak m}athcal F}i}, (i, n))$
and one can verify that $D({\bf x}_{k}, {{{\mathfrak m}athcal F}i}, (0, n))=C({\bf x}_{k}, {{{\mathfrak m}athcal F}i}, (0, n))$.
For any element $x \in I_2$, let ${x^{o}}$ denote the image of $x$ in $I_2/ I_1 I_2$.
In this paper, we use the complexes $D({\bf x}_{k}, {{{\mathfrak m}athcal F}i}, (i, n))$ (i=0,1) and the Koszul complex
$K({{\bf x}_{k}}^o, F_{I_1}(I_2))(n)$ to investigate the relation between the properties of the three graded rings
$G(I_2)$, $G_{I_1}(I_2)$ and $F_{I_1}(I_2)$. As a consequence,
we obtain interesting information on the fiber cone.
Huneke's fundamental lemma (\cite[Lemma~2.4]{huneke}) was extended to the filtration ${{\mathfrak m}athcal F}i$ for ${\mathfrak m}$-primary ideals $I_2 \subseteq I_1$ in a two dimensional Cohen-Macaulay local ring
(\cite[Proposition~2.5]{jay-verma-almm}).
In this paper, using the complex
$D({\bf x}_{d}, {{{\mathfrak m}athcal F}i}, (1, n))$ we extend this result to any two ${\mathfrak m}$-primary ideals $I_2 \subseteq I_1$ in a Cohen-Macaulay local ring of dimension $d {\gamma}eq 1$
(Theorem~\ref{thm-fundamental}).
As a consequence we are able to describe the Hilbert coefficients
of the Hilbert polynomial associated to the function $\ell(R/I_{1} I_{2}^{n})$ which we denote by $g_{i, I_1}(I_2)$ ($0 \leq i \leq d$)
(Lemma~\ref{hilb-coef-one}).
A lower bound for $g_{1, I_1}(I_2)$ was given in \cite[Proposition~4.1]{jay-verma} under some assumptions on
$I_2 \subseteq I_1$. In this paper, we improve their bound. We also give an upper bound for $g_{1, I_1}(I_2)$.
As a consequence, we show that when the lower bound is
attained $G_{I_1}(I_2)$ is
Cohen-Macaulay and when the upper bound is attained
${\mathfrak m}athop{\rm depth}~G_{I_1}(I_2) {\gamma}eq d-1$
(Proposition~\ref{prop-al-min-mult-ineq}).
We describe the
Hilbert coefficients of the fiber cone $F_{I_1}(I_2)$,
$f_{i, I_1}(I_2)$, $0 \leq i \leq d-1$
(Lemma~\ref{cor-fiber-coeff}).
The multiplicity of the fiber cone is of interest.
An upper bound for the multiplicity was given in
\cite{cpv} and in \cite{jay-verma} for $I_1={\mathfrak m}$. Using the complex
$D({\bf x}_{d}, {{{\mathfrak m}athcal F}i}, (1, n))$ ($n=1$) obtain an upper bound for the multiplicity in a more general setting (Corollary~\ref{upper-bound-multiplicity}). We also give a lower bound on the multiplicity of the fiber cone. In \cite{cpv} the authors remark that when the upper bound is attained the fiber cone need not be Cohen-Macaulay. We make an interesting observation. We show that when the upper bound is attained then ${\mathfrak m}athop{\rm depth} ( ({\bf x}_d), G_{I_1}(I_2))=d$ and when the lower bound is attained, ${\mathfrak m}athop{\rm depth} (({\bf x}_d), G_{I_1}(I_2)) {\gamma}eq d-1$ (Corollary~\ref{upper-bound-multiplicity}).
One interesting question is: How is the depth of the fiber cone and the associated graded ring related? We give an answer to this question using homological methods.
For ideals of minimal multiplicity and almost minimal multiplicity most of the homologies $H_i(C({\bf x}_{d}{{\mathfrak m}athcal F}, (1, n)))=0$ vanish giving a nice relation between the depth of $G(I_2)$ and $F_{I_1}(I_2)$ (Theorem~\ref{thm-jay-verma-almm}).
We now describe the organization of this paper.
In Section~\ref{complex}, we define the ${{\mathfrak m}athcal C}1n$ and $D({{\beta}f x}_{k}, {\Fi}, (1, n))$. We describe some interesting properties of these complexes. In Section~\ref{def-mm} we define ideals with minimal multiplicity and ideals with almost minimal multiplicity in terms of the homologies of ${{\mathfrak m}athcal C}1n$. In Section~\ref{vanishing-homologies} we compare the depths of the graded rings $F_{I_1}(I_2)$, $G_{I_1}(I_2)$ and $G(I_2)$.
In Section~\ref{hilbert-coefficients} we describe the Hilbert coefficients of $\ell(R/I_1 I_2^n)$. We also state a more general form of the Huneke's Fundamental Lemma. In Section~\ref{hilb-coef-fiber-cone} we describe the Hilbert coefficients of the fiber cone.
In Section~\ref{hilb-fib-cone-mm} we describe Hilbert coefficients of $\ell(R/I_1 I_2^n)$ and of the fiber cone
for ideals
of minimal multiplicity and ideals of almost minimal
multiplicity.
\section{ The complex
${{\mathfrak m}athcal C}1n$, $D({{\beta}f x}_{k}, {\Fi}, (1, n))$ and Hilbert function }
\label{complex}
For any two ideals $I_1$ and $I_2$ of $R$, let
${{\mathfrak m}athcal F}2i$
(resp. ${{{\mathfrak m}athcal F}i}$) be the $I_2$-filtration
$ \{ I_{2}^{n} \}_{n \in {\mathfrak m}athbb Z}$
(resp. $\{I_{1} I_{2}^{n} \}_{n \in {\mathfrak m}athbb Z}$). We use the following convention: For any ideal $I$, $I^n=R$ if $n \leq 0$.
Let $k {\gamma}eq 1$ and
${\bf x}_k:= x_1, \ldots, x_k$ be a sequence of elements in $I_2$.
Using the mapping cone construction, Marley and Huckaba constructed the following complex \cite{tom-huc}:
{\beta}egin{equation}
\label{newdisplay-1}
{{\mathfrak m}athcal C}nn:
0
\rightarrow \frac{R}{I_2^{n-k} }
\rightarrow \cdots
\rightarrow \left( \frac{R}{I_2^{n-i}} \right)^{k \checkmarkoose i}
\rightarrow \cdots
\rightarrow \left( \frac{R}{I_2^{n-1}} \right)^{k \checkmarkoose 1}
\rightarrow \frac{R}{I_2^{n}} \rightarrow 0.
\end{equation}
In a similar way, for the filtration ${{\mathfrak m}athcal F}i$, we get the following complex:
{\beta}egin{equation}
\label{newdisplay-2}
{{\mathfrak m}athcal C}1n:
0
\rightarrow \frac{R}{I_1 I_2^{n-k} }
\rightarrow \cdots
\rightarrow \left( \frac{R}{I_1 I_2^{n-i}} \right)^{k \checkmarkoose i}
\rightarrow \cdots
\rightarrow \left( \frac{R}{I_1 I_2^{n-1}} \right)^{k \checkmarkoose 1}
\rightarrow \frac{R}{I_1 I_2^{n}} \rightarrow 0.
\end{equation}
The maps in (\ref{newdisplay-1}) and (\ref{newdisplay-2}) are induced by the Koszul complex $K.({\bf x}_k;R)$.
Corresponding to the short exact sequence (\ref{newdisplay-2}) we have the short exact sequence of complexes:
{{\mathfrak m}ptmcomx
{\beta}egin{eqnarray*}
0
\rightarrow {{{{\mathfrak m}athcal C}nk1}}
\rightarrow {{{{\mathfrak m}athcal C}1n}}
\rightarrow {{{{\mathfrak m}athcal C}n1k1}}[-1]
\rightarrow 0
\end{eqnarray*}}
and the corresponding long exact sequence of homologies:
{{\mathfrak m}ptmcomx
{\beta}egin{eqnarray}
\label{main-homology} \nonumber
\cdots
\rightarrow & H_i( {{\mathfrak m}athcal C}1n )
\rightarrow & H_{i-1}({{\mathfrak m}athcal C}n1k1)
\\
\rightarrow H_{i-1}({{\mathfrak m}athcal C}nk1)
\rightarrow& H_{i-1}( {{\mathfrak m}athcal C}1n )
\rightarrow& \cdots .
\end{eqnarray}
{\mathfrak m}ptm
{\beta}egin{lemma}
\label{homology-four}
Let $k {\gamma}eq 2$ and $n < k$. If $({\bf x}_k) \subseteq I_1$, then
for all $ i -1 {\gamma}eq n$,
{\beta}egin{eqnarray*}
{ \displaystyle
H_{i}( {{{\mathfrak m}athcal C}1n } )
= \left( \frac{R}{I_1} \right)^{k \checkmarkoose i}
}.
\end{eqnarray*}
\end{lemma}
\proof If $n<k$ and $({\bf x}_k) \subseteq I_1$, then for all $i {\gamma}eq n+1$ all the maps in the complex (\ref{newdisplay-2}) are zero.
\qed
Using Lemma~\ref{homology-four}, when $({\bf x}_k) \subseteq I_1$, we can truncate the complex ${{\mathfrak m}athcal C}1n$ to get the complex
$D({{\beta}f x}_{k}, {\Fi}, (1, n))$ where:
{\beta}egin{equation}
\label{newdisplay-3}
{\beta}egin{array}{rcll}
D({{\beta}f x}_{k}, {\Fi}, (1, n)) :&
0
\rightarrow \left( \frac{R}{I_1 } \right)^{k \checkmarkoose n}
\rightarrow \cdots
\rightarrow \left( \frac{R}{I_1 I_2^{n-i}} \right)^{k \checkmarkoose i}
\rightarrow \cdots
\rightarrow \left( \frac{R}{I_1 I_2^{n-1}} \right)^{k \checkmarkoose 1}
\rightarrow \frac{R}{I_1 I_2^{n}} \rightarrow 0,
& \hspace{.1in} n < k; \\
&
0
\rightarrow \frac{R}{I_1 I_2^{n- k} }
\rightarrow \cdots
\rightarrow \left( \frac{R}{I_1 I_2^{n-i}} \right)^{k \checkmarkoose i}
\rightarrow \cdots
\rightarrow \left( \frac{R}{I_1 I_2^{n-1}} \right)^{k \checkmarkoose 1}
\rightarrow \frac{R}{I_1 I_2^{n}} \rightarrow 0,
& \hspace{.1in} n {\gamma}eq k;
\end{array}
\end{equation}
By our construction,
$H_i(D({{\beta}f x}_{k}, {\Fi}, (1, n)))= 0$ for $i >n$. In this section, we will show that the complex (\ref{newdisplay-3}) and the complex
(\ref{newdisplay-1}) do share some similar properties.
We recall some basic facts and results.
We say that an element
$x \in I_2$ is superficial for
$I_2$ and
$I_1$ if there exists a positive integer
$r_0$ such that for all
$r {\gamma}eq r_0$ and all $ s {\gamma}eq 0$,
$xR \cap I_1^s I_2^r =x I_1^sI_2^{r-1}$.
Let $k {\gamma}eq 2$. We say that
${\bf x}_k$
is a superficial sequence if for each $i=1, \ldots,k$,
${\omega}verline{x_i} \in {\omega}verline{I_2}$ is superficial for ${\omega}verline{I_2}$ and ${\omega}verline{I_1}$ where
${\beta}ar{\hphantom{sp}}$ denotes the image in $R/(x_1, \ldots, x_{i-1})$.
Rees also showed that we can choose a minimal reduction
$({\bf x}_{d}) \in I_2$ which is superficial sequence for $I_2$ and $I_1$ \cite{rees}. Hence, we can assume that $({\bf x}_k)$ is a minimal reduction of $I_2$ and is generated by a superficial sequence for $I_2$ and $I_1$.
As the homologies of ${{\mathfrak m}athcal C}1n$ and $D({{\beta}f x}_{k}, {\Fi}, (1, n))$ can be computed using techniques similar to those in \cite{anna}, with slight modification, we state the theorem without proof.
{\beta}egin{thm}
\label{homology}
Let $ k {\gamma}eq 1$ and let ${\bf x}_{k} \in I_2$ be a superficial sequence for $I_2$ and $I_1$. Then
{\beta}egin{enumerate}
\item
\label{homology-one}
For all $n \in {\mathfrak m}athbb Z$,
${\displaystyle
H_0( {{\mathfrak m}athcal C}1n )
\cong \frac{R}
{I_1I_2^{n} + ({\bf x}_{k})}.
}$\\
Let $({\bf x}_d) \subseteq I_1$. Then ${\displaystyle
H_0(D({{\beta}f x}_{k}, {\Fi}, (1, n))) =
\left\{
{\beta}egin{array}{ll}
H_0({{\mathfrak m}athcal C}1n) & \hspace{.1in} {\mathfrak m}box{ for } n {\gamma}eq 0,\\
0 & \hspace{.1in} {\mathfrak m}box{ for } n < 0.\\
\end{array} \right.
}$.
\item
\label{homology-two}
Let ${\bf x}_k$ be a regular sequence. Then for all $n {\gamma}eq 2$,
${\displaystyle
H_1( {{{{\mathfrak m}athcal C}1n }} )
\cong \frac{({\bf x}_{k}) \cap I_1 I_2^{n}}
{ ({\bf x}_{k}) I_1 I_2^{n-1}}}.
$\\
Let $({\bf x}_d) \subseteq I_1$. Then ${\displaystyle
H_1(D({{\beta}f x}_{k}, {\Fi}, (1, n))) =
\left\{
{\beta}egin{array}{ll}
\frac{({\bf x}_{k}) \cap I_1 I_2^{n}}
{ {\bf x}_{k} I_1 I_2^{n-1}} & \hspace{.1in} {\mathfrak m}box{ for } n {\gamma}eq 1,\\
0 & \hspace{.1in} {\mathfrak m}box{ for } n < 1.\\
\end{array} \right.
}.$
\item
\label{homology-three}
For all $n \in {\mathfrak m}athbb Z$,
${\displaystyle
H_{k}( {{{{\mathfrak m}athcal C}1n }} )
\cong \frac{ I_1 I_2^{n-k+1} : ( {\bf x}_{k}) }
{I_1 I_2^{n-k}}}.$\\
Let $({\bf x}_k) \subseteq I_1$. Then ${\displaystyle
H_k(D({{\beta}f x}_{k}, {\Fi}, (1, n))) =
\left\{
{\beta}egin{array}{ll}
H_k({{\mathfrak m}athcal C}1n) & \hspace{.1in} {\mathfrak m}box{ for } n {\gamma}eq k,\\
0 & \hspace{.1in} {\mathfrak m}box{ for } n < k\\
\end{array} \right.
}$.
\end{enumerate}
\end{thm}
The depth of the graded ring $G_{I_1}(I_2)$ and the vanishing of the homologies of the complex $D({{\beta}f x}_{k}, {\Fi}, (1, n))$ are related as follows:
{\beta}egin{pro}
\label{grade}
Let $I_2 \subseteq I_1$ be ${\mathfrak m}$-primary ideals in $(R, {\mathfrak m})$. Assume that $({\bf x}_k)$ is a superficial sequence for $I_2$ and $I_1$ and that $({\bf x}_k) \subseteq I_1$.
For the filtration ${{\mathfrak m}athcal F}i$,
{\beta}egin{eqnarray*}
{\mathfrak m}athop{\rm depth}(( {\bf x}{_k})^{\stackrelr}, G_{I_1}(I_2)) = {\mathfrak m}in \{ H_{j-k}(D({{\beta}f x}_{k}, {\Fi}, (1, n))) \not = 0
{\mathfrak m}box { for some } n\}.
\end{eqnarray*}
\end{pro}
\proof The proof is similar to the proof of Proposition~3.3
of \cite{tom-huc}. \qed
In the next lemma we show that the complex
$D({{\beta}f x}_{k}, {\Fi}, (1, n))$ satisfies a certain rigidity similar to that of the complex ${{\mathfrak m}athcal C}nn$.
{\beta}egin{lemma}
\label{vanishing-rigidity}
Let $I_1, I_2$ and ${\bf x}_k$ be as in Proposition~\ref{grade}.
If $H_j(D({{\beta}f x}_{k}, {\Fi}, (1, n))) = 0$ for some $j {\gamma}eq 1$ and for all $n$, then
$H_i(D({{\beta}f x}_{k}, {\Fi}, (1, n))) = 0
$ for all $i {\gamma}eq j$ and for all $n$.
\end{lemma}
\proof The proof follows by induction on $k$. \qed
We state a crucial property satisfied by ${{\mathfrak m}athcal C}1n$ and $D({{\beta}f x}_{k}, {\Fi}, (1, n))$.
{\beta}egin{lemma}
\label{vanishing}
Let $1 \leq i \leq k$. Let ${\bf x}_k \in I_2$ be a regular sequence in $R$ which is superficial for $I_2$ and $I_1$. Then for all $n {\gamma}g 0$ we have
$H_i({{\mathfrak m}athcal C}1n)=H_i(D({{\beta}f x}_{k}, {\Fi}, (1, n))) = 0$.
\end{lemma}
\proof Note that for $n {\gamma}g 0$, $H_i({{\mathfrak m}athcal C}1n) = H_i(D({{\beta}f x}_{k}, {\Fi}, (1, n)))$.
The proof follows by applying induction on $k$ for the complex $D({{\beta}f x}_{k}, {\Fi}, (1, n))$. \qed
{\beta}egin{notn}
{\beta}egin{eqnarray*}
{\beta}egin{array}{lllllll}
h_i ( C_{{\beta}ullet},{\bf x}_{k})(1,n)
&:=& \ell
\left( H_i ({{\mathfrak m}athcal C}1n)
\right).
&&
h_i (C_{{\beta}ullet}, {\bf x}_{k})(0,n)
&:=& \ell \left( H_i ({{\mathfrak m}athcal C}nn) \right).\\
h_i ( D_{{\beta}ullet}, {\bf x}_{k})(1,n)
&:=& \ell
\left( H_i (D({{\beta}f x}_{k}, {\Fi}, (1, n)))
\right).
&&
h_i (D_{{\beta}ullet},{\bf x}_{k}) (1, *)
&:=& \sum_{n {\gamma}eq 0} h_i (D_{{\beta}ullet},{\bf x}_{k}) (1, n).\\
h_i (C_{{\beta}ullet},{\bf x}_{k}) (1, *)
&:=& \sum_{n \in {\mathfrak m}athbb Z} h_i (C_{{\beta}ullet},{\bf x}_{k}) (1, n).
&& h_i (D_{{\beta}ullet},{\bf x}_{k}) (1, *)
&:=& \sum_{n {\gamma}eq 0} h_i (D_{{\beta}ullet},{\bf x}_{k}) (1, n).
\end{array}
\end{eqnarray*}
\end{notn}
{\beta}egin{thm}
\label{rigidity}
Let $(R, {\mathfrak m})$ be a local ring. Let $I_1$ and $I_2$ be ${\mathfrak m}$-primary ideals of $R$. Let $k {\gamma}eq 1$ and
${\bf x}_{k}$ be a regular sequence which is superficial for $I_2$ and $I_1$.
Let $i {\gamma}eq 1$. If $({\bf x}_k) \subseteq I_1$, then
{\beta}egin{eqnarray}
\label{rigidity-two}
\sum_{j {\gamma}eq i} (-1)^{j-i} h_j(D_{{\beta}ullet},{\bf x}_{k})(1, *) {\gamma}eq 0.
\end{eqnarray}
Equality holds if and only if
{\beta}egin{eqnarray*}
{\mathfrak m}athop{\rm depth}(({\bf x}_k)^{\stackrelr}, G_{I_1}(I_2)) {\gamma}eq k-i+1.
\end{eqnarray*}
\end{thm}
\proof The proof is similar to the proof of Theorem~3.7
of \cite{tom-huc}. \qed
{\beta}egin{notn}
{\beta}egin{eqnarray}
{\beta}egin{array}{lllllll}
\label{notation-hilbert}
H_{{{\mathfrak m}athcal F}i}(1,n)
&:=& \ell \left( \frac{R}{I_1 I_2^n} \right)
&\hphantom{sp}&
H_{{{\mathfrak m}athcal F}i}^{\prime}(1,n)&:=&
\left\{
{\beta}egin{array}{ll}
\ell
\left( \frac{R}{I_1 I_2^n}
\right)
& {\mathfrak m}box{ if } n {\gamma}eq 0\\
0
& {\mathfrak m}box{ if } n <0.
\end{array}
\right.
\end{array}
\end{eqnarray}
\end{notn}
We now give a relation between the homology modules and the Hilbert function $H_{{{\mathfrak m}athcal F}i}(1,n)$.
{\beta}egin{lemma}
\label{lemma-extension-fiber}
Let $(R, {\mathfrak m})$ be a Cohen-Macaulay ring of dimension $d {\gamma}eq 2$.
Let $I_1$ and $I_2$ be an ${\mathfrak m}$-primary ideals of $R$ and
$({\bf x}_{d}) \in I_2$ a minimal reduction which is a superficial sequence for $I_2$ and $I_1$.
{\beta}egin{enumerate}
\item
For all $n {\gamma}eq 2$,
{\beta}egin{eqnarray}
\label{alt-sum-new-1}
\sum_{i=0}^{d} (-1)^i{d\checkmarkoose i}
H_{{{\mathfrak m}athcal F}i}(1,n-i) = e_0( I_2)
- \ell
\left(
\frac{I_1 I_2^{n}}
{({\bf x}_{d}) I_1 I_2^{n-1}}
\right)
+\sum_{i=2}^{d} (-1)^i h_i(C_{{\beta}ullet},{\bf x}_{d})(1, n).
\end{eqnarray}
\item
Let $({\bf x}_d) \subseteq I_1$. The for all $n \in {\mathfrak m}athbb Z$,
{\beta}egin{eqnarray}
\label{alt-sum-new-d}
\sum_{i=0}^{d} (-1)^i{d\checkmarkoose i}
H_{{{\mathfrak m}athcal F}i}^{\prime}(1,n-i) = e_0( I_2)
- \ell
\left(
\frac{I_1 I_2^{n}}
{({\bf x}_{d}) I_1 I_2^{n-1}}
\right)
+\sum_{i=2}^{d} (-1)^i h_i(D_{{\beta}ullet},{\bf x}_{d})(1, n).
\end{eqnarray}
\end{enumerate}
\end{lemma}
\proof
From Theorem~\ref{homology}, for all $n {\gamma}eq 2$,
{\beta}egin{eqnarray}
\label{alt-sum-1} \nonumber
h_0(C_{{\beta}ullet}, {\bf x}_{d})(1, n)
- h_1(C_{{\beta}ullet}, {\bf x}_{d})(1, n)
&=& \ell \left(
\frac{R}{I_1 I_2^{n} + ({\bf x}_{d})}
\right)
- \ell \left(
\frac{( {\bf x}_{d}) \cap I_1 I_2^{n}}
{({\bf x}_{d}) I_1 I_2^{n-1}}
\right)\\ \nonumber
&=& \ell \left(
\frac{R}{ ( {\bf x}_{d})}
\right)
- \ell \left( \frac{I_1 I_2^{n} + ({\bf x}_{d})}
{( {\bf x}_{d})}
\right)
- \ell \left(
\frac{( {\bf x}_{d}) \cap I_2^{n}}
{({\bf x}_{d}) I_1 I_2^{n-1}}
\right)\\
&=& e_0(I_{2})
- \ell
\left(
\frac{ I_1 I_2^{n}}
{ ({\bf x}_{d}) I_1 I_2^{n-1}}
\right).
\end{eqnarray}
For the complex $D({{\beta}f x}_{k}, {\Fi}, (1, n))$, (\ref{alt-sum-1}) holds true for all $n \in {\mathfrak m}athbb Z$.
\qed
Lemma~\ref{homology-negative} gives us insight for the behaviour of homology modules for the complex ${{\mathfrak m}athcal C}1n$ for $n \leq 0$.
{\beta}egin{lemma}
\label{homology-negative}
Let $I_1$ and $I_2$ be ${\mathfrak m}$-primary ideals in a Cohen-Macaulay local ring $R$. Let ${\bf x}_k$ be a superficial sequence in $R$.
{\beta}egin{enumerate}
\item
\label{homology-negative-one}
For $n \leq 0$,
$
{\displaystyle
\sum_{i=0}^{k} (-1)^i {k \checkmarkoose i}
h_i (C_{{\beta}ullet},{\bf x}_k)(1, n-i)=0.
}$
\item
\label{homology-negative-two}
We have a surjective map:
{\beta}egin{eqnarray}
\label{homology-negative-two-eqn}
\phi: \left( \frac{R}{I_1} \right)^k
\longrightarrow \frac{({\bf x}_k)}{I_1 ({\bf x}_k)}.
\end{eqnarray}
\item
\label{homology-negative-three}
Let $2 {\gamma}eq i {\gamma}eq k$. Then
{\beta}egin{eqnarray*}
\sum_{i=2}^k
(-1)^i
\left[ h_i (C_{{\beta}ullet},{\bf x}_k)(1, 1)
- {k \checkmarkoose i} \ell \left( \frac{R}{I_1} \right)
\right] \leq 0
\end{eqnarray*}
Let $k=d$, then equality holds if and only if $({\bf x}_d) \subseteq I_1 $.
Moreover if equality holds, then the map $\phi$ in
(\ref{homology-negative-two})
is an isomorphism.
\end{enumerate}
\end{lemma}
\proof For
$n \leq 0$, the complex (\ref{newdisplay-2})
is of the form
{\beta}egin{eqnarray*}
0
\rightarrow \frac{R}{I_1 }
\rightarrow \cdots
\rightarrow \left( \frac{R}{I_1 } \right)^{k \checkmarkoose i}
\rightarrow \cdots
\rightarrow \left( \frac{R}{I_1 } \right)^{k \checkmarkoose 1}
\rightarrow \frac{R}{I_1 } \rightarrow 0.
\end{eqnarray*}
Hence
{\beta}egin{eqnarray*}
\sum_{i=0}^{k} (-1)^i {k \checkmarkoose i}
h_i(C_{{\beta}ullet}, {\bf x}_k) (1, n-i)
= \sum_{i=0}^{k} (-1)^i {k \checkmarkoose i}
\ell \left( \frac{R}{I_1} \right)=0.
\end{eqnarray*}
This proves(\ref{homology-negative-one}).
(\ref{homology-negative-two}) was proved in \cite[Lemma~3.4]{jay-verma-almm}.
We prove (\ref{homology-negative-three}).
The first part of (\ref{homology-negative-three}) follows from the fact that
$
{\displaystyle
h_i (C_{{\beta}ullet},{\bf x}_k)(1, 1)
\leq {k \checkmarkoose i} \ell \left( \frac{R}{I_1} \right)
}.
$
We have
{\beta}egin{eqnarray}
\label{homology-negative-four}
&&
\ell \left( \frac{R} {I_1 I_2} \right)
- \ell \left( \frac{{\bf x}_d} {I_1 ({\bf x}_d)} \right)
+ \sum_{i=2}^{d} (-1)^i {d \checkmarkoose i}
\left( \frac{R} {I_1} \right)\\ \nonumber
&{\gamma}eq& \ell \left( \frac{R} {I_1I_2} \right)
- d~ \left( \frac{R} {I_1 } \right)
+ \sum_{i=2}^d (-1)^i {d \checkmarkoose i}
\left( \frac{R} {I_1} \right)
\hspace{.5in} {\mathfrak m}box{by (\ref{homology-negative-two})}\\ \nonumber
&=& e({\bf x}_d)
- \ell \left( \frac{I_1 I_2}{I_1 ({\bf x}_d)} \right)
+ \sum_{i=2}^d (-1)^i
h_i(C_{{\beta}ullet}, {\bf x}_d) (1, 1)
\hspace{.5in} {\mathfrak m}box{[putting $n=1$ in
Theorem~\ref{lemma-extension-fiber}(1)].}
\end{eqnarray}
If $({\bf x}_d)\subseteq I_1$,
then equality holds by Lemma~\ref{homology-four}.
Conversely, suppose equality holds in
(\ref{homology-negative-four}), then
{\beta}egin{eqnarray*}
h_i(C_{{\beta}ullet}, {\bf x}_d)(1,1) = {d \checkmarkoose i} \ell \left( \frac{R}{I_1} \right)
\hspace{.2in}
{\mathfrak m}box{for all }i=2, \ldots ,d.
\end{eqnarray*}
In particular,
{\beta}egin{eqnarray*}
\frac{I_1 : ({\bf x}_d)}{I_1} =
h_d(1,1) = \ell \left( \frac{R}{I_1} \right)
\end{eqnarray*}
This implies that $I_1 : ({\bf x}_d) = R$.
Finally, if equality holds, in
(\ref{homology-negative-four}), then
$d~\ell \left( \frac{R}{I_1} \right)
= \ell \left( \frac{({\bf x}_d)}{I_1 ({\bf x}_d)} \right)$
and hence the map in (\ref{homology-negative-two}) is an isomorphism.
\qed
{\beta}egin{remark}
If $({\bf x}_d) \not \subseteq I_1$, then
$
\sum_{i=2}^d
(-1)^i
\left[ h_i (C_{{\beta}ullet}, {\bf x}_d)(1, 1) - \ell (R/I_1) \right]
$ may be non-zero (Example~\ref{countex}).
\end{remark}
{\beta}egin{example}
\label{countex}
Let $R= k[x,y,z]_{{\mathfrak m}}$, where ${\mathfrak m} = (x,y,z)$. Let
$I_2 = (x^3, y^3,z^3, xy, xz, yz)$
and $({\bf x}_3)= (x^3 + yz, y^3+ z^3+ xz, xz+ xy)$.
Let $n {\gamma}eq 4$ and $I_1 = (x^n, y^3,z^3, xy, xz, yz)$.
Here $({\bf x}_3)$ is generated by a superficial sequence and $({\bf x}_3)\not \subseteq I_1$.
The map in (\ref{homology-negative-two-eqn}) is not an isomorphism.
\qed
\end{example}
\section{Minimal and almost minimal multiplicity and vanishing results}
\label{def-mm}
Goto \cite{goto} defined ideals of minimal multiplicity. Following Goto, Jayanthan and Verma defined ideals of minimal (resp. almost minimal) multiplicity in \cite{jay-verma} (resp. \cite{jay-verma-almm})
when $I_2 \subseteq I_1$.
In this paper, we generalize these definitions for any
two ${\mathfrak m}$-primary ideals $I_2$ and $I_1$.
We also show that, under some mild assumptions, the homologies of the complex ${{\mathfrak m}athcal C}1n$ for these ideals have nice vanishing properties.
{\beta}egin{defn}
\label{defn-mm-amm}
Let $(R, {\mathfrak m})$ be a Cohen-Macaulay local ring and let $I_1$ and $I_2$
be ${\mathfrak m}$-primary ideals in $R$. $I_2$ has minimal multiplicity (resp. almost minimal multiplicity) with respect to $I_1$ if for some minimal reduction $({\bf x}_d)$ of
$I_2$
{\beta}egin{eqnarray*}
\ell \left( \frac{I_1 I_2}{I_1 ({\bf x}_d)} \right)
= 0
\hphantom{sp}\left( resp. \hphantom{sp} \ell \left( \frac{I_1 I_2}{I_1 ({\bf x}_d)} \right)
= 1 \right).
\end{eqnarray*}
\end{defn}
{\beta}egin{lemma}
\label{min-mult-defn}
Let $(R, {\mathfrak m})$ be a Cohen-Macaulay local ring of positive dimension. Then $I_2$ has minimal multiplicity (resp. almost minimal multiplicity) with respect to $I_1$ if and only if for some minimal reduction $({\bf x}_d)$ of $I_2$,
{\beta}egin{eqnarray*}
e_0(I_2)
- \ell \left( \frac{R}{I_1I_2} \right)
+ d \cdot \ell \left( \frac{R}{I_1} \right)
&=& - \sum_{i=2}^d
(-1)^i
\left[ h_i(C_{{\beta}ullet}, {\bf x}_d) (1, 1)
- {d \checkmarkoose i} \ell \left( \frac{R}{I_1} \right)
\right]\\
\left(
{\mathfrak m}box{resp. }e_0(I_2)
- \ell \left( \frac{R}{I_1I_2} \right)
+ d \cdot \ell \left( \frac{R}{I_1} \right)
\right.
&=&
\left.
1 - \sum_{i=2}^d
(-1)^i
\left[ h_i(C_{{\beta}ullet}, {\bf x}_d)(1, 1)
- {d \checkmarkoose i} \ell \left( \frac{R}{I_1} \right)
\right]\right).
\end{eqnarray*}
\end{lemma}
\proof We have:
{\beta}egin{equation}
\label{newdisplay-4}
{{\mathfrak m}athcal C}d11:
0
\rightarrow \left( \frac{R}{I_1} \right)^{d \checkmarkoose d}
\rightarrow \left( \frac{R}{I_1} \right)^{d \checkmarkoose d-1}
\rightarrow \cdots
\rightarrow \left( \frac{R}{I_1} \right)^{d}
\rightarrow \frac{R}{I_1 I_2}
\rightarrow 0
\end{equation}
By Theorem~\ref{homology} we have
we have
{\beta}egin{eqnarray*}
\ell \left( \frac{R}{I_1 I_2} \right)
- d~\ell \left( \frac{R}{I_1 } \right)
+ \sum_{i=2}^d
(-1)^i {d \checkmarkoose i} \ell \left( \frac{R}{I_1} \right)
&=& e_0(I_2)
- \ell \left(
\frac{ I_1 I_2}
{({\bf x}_{d}) I_1}\right)
+ \sum_{i=2}^d(-1)^i h_i (C_{{\beta}ullet}, {\bf x}_d)(1, 1).
\end{eqnarray*}
Hence
{\beta}egin{eqnarray*}
e_0(I_2)
- \ell \left( \frac{R}{I_1 I_2} \right)
+ d~\ell \left( \frac{R}{I_1 } \right)
= - \sum_{i=2}^d(-1)^i \left[
h_i (C_{{\beta}ullet}, {\bf x}_d)(1, 1)
- \ell \left( \frac{R}{I_1} \right)
\right]
+ \ell \left(
\frac{ I_1 I_2}
{({\bf x}_{d}) I_1}\right).
\end{eqnarray*}
\qed
Example~\ref{countex-1} reveals that
$I_1 I_2= I_1 ({\bf x}_d)$ while
$e_0(I_2) - \ell (I_1I_2)+ d~\ell (R/I_1) >1$.
Therefore, the results in \cite{jay-verma} and
\cite{jay-verma-almm} can be extended to ideals
$I_2 \not \subseteq I_1$ if one takes into account the homologies
$H_i(C_{{\beta}ullet}, {\bf x}_d) (1, 1)$, $i=2, \ldots, d$.
{\beta}egin{example}
\label{countex-1}
Let $R= k[x,y,z]_{{\mathfrak m}}$, where ${\mathfrak m} = (x,y,z)$. Let
$I_2 = (x^2, y^2,z^2, xy, xz, yz)$
and $({\bf x}_3)= (x^2 + yz, y^2+ z^2+ xz, xz+ xy)$.
Let $n {\gamma}eq 4$ and $I_1 = (x^n, y^2,z^2, xy, xz, yz)$.
Here $({\bf x}_3)$ is a minimal reduction of $I_2$, is generated by a superficial sequence and $({\bf x}_3)\not \subseteq I_1$.
\qed
\end{example}
Using Lemma~\ref{homology-negative}
and Lemma~\ref{min-mult-defn} one can verify that if $({\bf x}_d) \subseteq I_1$ for some minimal reduction of $I_2$,
then $I_2$ has minimal (resp. almost minimal) multiplicity with respect to $I_1$ if and only if
{\beta}egin{eqnarray*}
e_0(I_2)
- \ell \left( \frac{R}{I_1I_2} \right)
+ d \cdot \ell \left( \frac{R}{I_1} \right)
= 0\\
\left(resp. \hphantom{sp} e_0(I_2)
- \ell \left( \frac{R}{I_1I_2} \right)
+ d \cdot \ell \left( \frac{R}{I_1} \right)
=1\right).
\end{eqnarray*}
The following lemma is useful as it describes the vanishing of certain homology modules as well as describe the Hilbert coefficients of the fiber cone.
{\beta}egin{lemma}
\label{mm-amm}
Let $(R,{\mathfrak m})$ be a Cohen-Macaulay local ring of dimension $d$. Let $I_1$ and $I_2$ be ${\mathfrak m}$-primary ideals of $R$. Let $({\bf x}_{d})$ be a minimal reduction of $I_{2}$.
{\beta}egin{enumerate}
\item
\label{lemma-mm-van}
Suppose $I_1I_2 = I_1 ({\bf x}_d)$. Then for all $n {\gamma}eq 1$
{\beta}egin{enumerate}
\item
\label{mm-1}
$
{\displaystyle
\ell \left( \frac{I_1 I_2^n}{({\bf x}_{d})I_1 I_2^{n-1}} \right)=0
}
$.
\item
\label{mm-3}
$
({\bf x}_{d}) \cap I_1 I_2^n
= ({\bf x}_{d}) I_1 I_2^{n-1}.
$
\end{enumerate}
\item
\label{lemma-amm-van}
Suppose
$
{\displaystyle
\ell \left( \frac{I_1 I_2}{({\bf x}_{d})I_1 } \right)
= 1
}
$
{\beta}egin{enumerate}
\item
\label{amm-1}
$
{\displaystyle
\ell
\left(
\frac{I_1 I_2^n}{({\bf x}_{d})I_1 I_2^{n-1}}
\right)
\leq 1
}
$
for all $n {\gamma}eq 1$.
\item
\label{amm-2}
For all $n {\gamma}eq 1$,
{\beta}egin{eqnarray*}
({\bf x}_{d}) \cap I_1 I_2^n
&=& \left\{
{\beta}egin{array}{ll}
({\bf x}_{d}) I_1 I_2^{n-1}
+ (y_1 y_2^n)
& {\mathfrak m}box{ if } ({\bf x}_{d}): (y_1 y_2^n) = (1)\\
({\bf x}_{d}) I_1 I_2^{n-1}
& {\mathfrak m}box{ if } ( {\bf x}_{d}): (y_1 y_2^n) \subseteq {\mathfrak m}
\end{array} \right.
\end{eqnarray*}
for some $y_{1}\in I_{1}$ and $y_{2} \in I_{2}$.
\end{enumerate}
\end{enumerate}
\end{lemma}
\proof (\ref{mm-1}) follows from the definition. Using
(\ref{mm-1}) we get
$
\label{al-mm-2}
({\bf x}_{d}) \cap I_1 I_2^n
= ({\bf x}_{d}) \cap({\bf x}_{d}) I_1 I_2^{n-1}
= ({\bf x}_{d}) I_1 I_2^{n-1}.
$
for all $n {\gamma}eq 1$.
\noindent
To prove (\ref{amm-1}) it is enough to show that for all $n{\gamma}eq 1$
{\beta}egin{eqnarray}
\label{star-two}
I_2I_2^n
= ({\bf x}_{d}) I_1 I_2^{n-1}
+ (y_1y_2^n),
\hspace{.2in}
{\mathfrak m}box{and }
\hspace{.2in}
{\mathfrak m} y_1 y_2
\subseteq ({\bf x}_{d}) I_1 I_2^{n-1}.
\end{eqnarray}
Since
$
\ell (I_1I_2/ ({\bf x}_{d})I_1)
= 1$,
there exists $y_1 \in I_1$ and $y_2 \in I_2$ such
that
{\beta}egin{eqnarray}
\label{star-one}
I_1 I_2
&=& ( {\bf x}_{d})I_1 + (y_1 y_2)\hspace{.2in} {\mathfrak m}box{ and } \hspace{.2in}
{\mathfrak m} y_1 y_2
\subseteq ({\bf x}_{d}) I_1.
\end{eqnarray}
If $n = 1$, then we are done by our assumption.
Let $n >1$. Clearly
$({\bf x}_{d}) I_1 I_2^{n-1} + (y_1 y_2^{n}) \subseteq I_1 I_2^n$
then by induction hypothesis
{\beta}egin{eqnarray} \nonumber
I_1 I_2^n
&=& (I_1 I_2^{n-1}) I_2
= ({\bf x}_{d}) I_1 I_2^{n-2} + (y_1 y_2^{n-1}) I_2
= ({\bf x}_{d}) I_1 I_2^{n-1} + (y_1 y_2^{n-1})I_2\\
\nonumber
(y_1 y_2^{n-1}) I_2
&\subseteq& (y_2) I_1 I_2^{n-1}
= (y_2) (({\bf x}_{d}) I_1I_2^{n-2}
+ (y_1 y_2^{n-1}))
\subseteq ({\bf x}_{d}) I_1 I_2^{n-1} + (y_1 y_2^{n})\\
\label{anni}
{\mathfrak m} (y_1 y_2^n)
&= & {\mathfrak m} (y_1 y_2) (y_2^{n-1})
\subseteq ( ({\bf x}_{d}) I_1 )(y_2^{n-1})
\subseteq ( {\bf x}_{d}) I_1 I_2^{n-1}.
\end{eqnarray}
This proves (\ref{amm-1}).
We now prove (\ref{amm-2}).
From (\ref{amm-1}) we have
{\beta}egin{eqnarray}
\label{proof-amm-2}
({\bf x}_{d}) \cap I_1 I_2^n
&=& ({\bf x}_{d})
\cap (({\bf x}_{d}) I_1 I_2^{n-1}
+ (y_1 y_2^n))\\ \nonumber
&=& ({\bf x}_{d}) I_1 I_2^{n-1}
+ ({\bf x}_{d}) \cap (y_1 y_2^n)\\ \nonumber
&=& ({\bf x}_{d} )I_1 I_2^{n-1}
+ (y_1 y_2^n)
(({\bf x}_{d}): (y_1 y_2^n))
\end{eqnarray}
Now two cases arise. \\
{\ncent Case(i):}$ ({\bf x}_{d}): (y_1 y_2^n)
= (1)$. In this case
$({\bf x}_{d}) \cap I_1 I_2^n
= ({\bf x}_{d}) I_1 I_2^{n-1}
+ (y_1 y_2^n)
$.\\
{\ncent Case(ii):}
$ ({\bf x}_{d}): (y_1 y_2^n) \subseteq {\mathfrak m}$. Then by
(\ref{amm-1}),
$({\bf x}_{d}) \cap I_1 I_2^n
\subseteq ({\bf x}_{d} )I_1 I_2^{n-1}
+ {\mathfrak m} (y_1 y_2^n)
\subseteq ({\bf x}_{d}) I_1 I_2^{n-1}
$ by (\ref{anni}).
This proves (\ref{amm-2}).
\qed
The homologies of the complex ${{\mathfrak m}athcal C}dnn$ and $ D({{\beta}f x}_{d}, {\Fi}, (1, n))$ for ideals of minimal multiplicity and almost minimal multiplicity do satisfy some nice interesting vanishing properties. We list them in this section.
{\beta}egin{pro}
\label{prop-van-almm}
Let $(R,{\mathfrak m})$ be a Cohen-Macaulay local ring, $I_1$ and $I_2$-primary ideals of $R$. Let ${\bf x}_{d} \in I_2$ be a superficial sequence for $I_2$ and $I_1$. Suppose $({\bf x}_d) \subseteq I_1$.
{\beta}egin{enumerate}
\item
\label{prop-van-almm-mm}
Let $I_2$ be an ideal of minimal multiplicity with respect to $I_1$.
Then
{\beta}egin{eqnarray*}
h_i(D_{{\beta}ullet}, {\bf x}_{d})(1, *) = 0,
\hspace{.5in} {\mathfrak m}box{ for all }i {\gamma}eq 1.
\end{eqnarray*}
\item
\label{lemma-vanishing-ammm-2}
Let $I_2$ be an ideal of almost minimal multiplicity with respect to $I_1$.
{\beta}egin{enumerate}
\item
For all $n {\gamma}eq 1$,
$
{\displaystyle
h_1(D_{{\beta}ullet},{\bf x}_{d})(1, n) \leq 1. }
$
\item
\label{lemma-vanishing-ammm-b}
If
$({\bf x}_{d}) \cap I_1 I_2= ({\bf x}_d)I_1$,
then
$h_i(D_{{\beta}ullet}, {\bf x}_{d})(1, *)=0$ for all
$i {\gamma}eq 2$.
\end{enumerate} \end{enumerate}
\end{pro}
\proof
It is enough to show that $h_1(D_{{\beta}ullet}, {\bf x}_{d})(1, *) = 0$
[Theorem~\ref{vanishing-rigidity}].
By Theorem~\ref{homology}(\ref{homology-two}) and Lemma~\ref{mm-amm},
for all $n {\gamma}eq 1$,
{\beta}egin{eqnarray*}
h_1( D_{{\beta}ullet},{\bf x}_{d})(1, n)
\cong \frac{({\bf x}_{d}) \cap I_1 I_2^n}
{ ({\bf x}_{d}) I_1 I_2^{n-1}}
= \frac{({\bf x}_{d}) \cap I_1 I_2^n}
{ I_1 ({\bf x}_d)^{n}}
= 0,
\end{eqnarray*}
if $I_2$ is an ideal of minimal multiplicity with respect to $I_1$.
If $I_2$ is an ideal of minimal multiplicity with respect to $I_1$, then by Theorem~\ref{homology}(\ref{homology-two}) and Lemma~\ref{mm-amm}, for all $n {\gamma}eq 1$,
{\beta}egin{eqnarray*}
h_1(D_{{\beta}ullet},{\bf x}_{d})(1, n)
\cong \frac{({\bf x}_{d}) \cap I_1 I_2^n}
{({\bf x}_{d}) I_1 I_2^{n-1}}
\cong \frac{( {\bf x}_{d}) I_1 I_2^{n-1} + (({\bf x}_{d}) \cap (y_1 y_2^n))}
{{\bf x}_{d} I_1 I_2^{n-1}}
\end{eqnarray*}
for some $y_1\in I_1$ and $y_2 \in I_2$.
By (\ref{anni}),
$
{\mathfrak m} (y_1 y_2^n \cap {\bf x}_d)
\subseteq {\mathfrak m} y_1 y_2^n
\in ({\bf x}_d) I_1 I_2^{n-1}
$.
(\ref{lemma-vanishing-ammm-b}) follows from
Theorem~\ref{homology}(\ref{homology-two}) and Lemma~\ref{mm-amm}(\ref{lemma-amm-van}), Proposition~\ref{grade} and \cite[Theorem~4.2.1]{rossi-valla-2}.
\qed
\section{ Comparing depth of $F_{I_1}(I_2)$, $G_{I_1}(I_2)$ and $G(I_2)$}
\label{vanishing-homologies}
Throughout this section we will assume that
$({\bf x}_d)$ is a minimal reduction of $I_2$ and $({\bf x}_d)\subseteq I_1$.
For all $n {\gamma}eq 0$ we have the
exact sequence of complexes:
{\beta}egin{eqnarray}
\label{koszul-fiber}
0 &
\rightarrow & K({{\bf x}_{d}^o}, {{\mathfrak m}athcal F}i)(n)
\rightarrow D({{\beta}f x}_{k}, {\Fi}, (1, n))
\rightarrow {{\mathfrak m}athcal C}nn
\rightarrow 0
\end{eqnarray}
where $K({{\bf x}_{d}}^o, {{\mathfrak m}athcal F}i)$ is the Koszul complex of the fiber cone
$F_{I_1}(I_2)$ with respect to the sequence ${\bf x}_{d}^o = x_{1}^o, \ldots, x_{d}^o$. We have the corresponding long exact sequence of complexes:
{\mathfrak m}ptmcomx
{\beta}egin{eqnarray}
\label{exact-depth-2}
{\beta}egin{array}{lrrl}
& \cdots
\rightarrow & H_{i+1}(D( {\bf x}_{d}, {{\mathfrak m}athcal F}, (1, n)))
\rightarrow & H_{i+1}(C( {\bf x}_{d}, {{\mathfrak m}athcal F}, (0, n)))\\
\rightarrow & H_{i} (K( {\bf x}_{d}^o, F_{I_1}(I_2)))(n)
\rightarrow & H_{i} (D({{\beta}f x}_{k}, {\Fi}, (1, n)))
\rightarrow & H_{i} ({{\mathfrak m}athcal C}nn)
\rightarrow \cdots
\end{array}.
\end{eqnarray}
{\mathfrak m}ptm
{\beta}egin{thm}
[Depth Lemma]
\label{depth-lemma}
Let $I_1$ and $I_2$ be ideals in a local ring $(R, {\mathfrak m})$.
If ${\mathfrak m}athop{\rm depth}~G(I_2) < {\mathfrak m}athop{\rm depth}~G_{I_1}(I_2)$, then
${\mathfrak m}athop{\rm depth}~F_{I_1}(I_2) = {\mathfrak m}athop{\rm depth}~G(I_2)+1$.
\end{thm}
\proof
Use (\ref{koszul-fiber}), Proposition~\ref{grade} and Proposition~3.1 of \cite{tom-huc}.
\qed
Using the results in this paper we generalize Theorem~4.4 of \cite{jay-verma-almm} and give a simple proof of Proposition ~5.4 of \cite{jay-verma-almm}.
{\beta}egin{notn}
Let $s({\bf x}_d):= {\mathfrak m}in\{ n | ({\bf x}_d)I_1 I_2^n = I_1 I_2^{n+1} \}$.
\end{notn}
{\beta}egin{thm}
\label{thm-jay-verma-almm}
Let $(R,{\mathfrak m})$ be a Cohen-Macaulay local ring, $I_1$ an $I_2$-primary ideals of $R$.
Let $({\bf x}_{d})$ be a minimal reduction of $I_2$ which is generated by a superficial sequence.
{\beta}egin{enumerate}
\item
\label{thm-jay-verma-almm-a}
Let $I_{2}$ be an ideal of minimal multiplicity with respect to $I_{1}$.
{\beta}egin{enumerate}
\item
\label{thm-jay-verma-almm-a-i}
$G_{I_1}(I_2)$ is Cohen-Macaulay;
\item
\label{thm-jay-verma-almm-a-ii}
For all $i=0, \ldots, d-1$,
${\mathfrak m}athop{\rm depth}( F_{I_1}(I_2))= i+1$ if and only if
${\mathfrak m}athop{\rm depth}( G(I_2)) =i$.
\end{enumerate}
\item
\label{thm-jay-verma-almm-b}
Let $I_{2}$ be an ideal of almost minimal multiplicity with respect to $I_{1}$.
Suppose $({\bf x}_d) \cap I_1 I_2 = I_1 ({\bf x}_d)$.
{\beta}egin{enumerate}
\item
\label{thm-jay-verma-almm-b-i}
For all $i=1, \ldots, d-2$, ${\mathfrak m}athop{\rm depth}(({\bf x}_{d}), F_{I_1}(I_2))= i+1$ if and only if
${\mathfrak m}athop{\rm depth}(({\bf x}_{d}), G(I_2)) =i$.
\item
\label{thm-jay-verma-almm-c}
$F_{I_{1}}(I_{2})$ is Cohen-Macaulay if and only if
${\mathfrak m}athop{\rm depth}(({\bf x}_{d}), G(I_2)) {\gamma}eq d-1$
and
$\ell(I_{1}I_{2}^{n} + ({\bf x}_d) I_{2}^{n-1}/
({\bf x}_d)I_{2}^{n-1})=1$
for all
$n = 1,\ldots, s({\bf x}_d)$.
\end{enumerate}
\end{enumerate}
\end{thm}
\proof
(\ref{thm-jay-verma-almm-a-i} )
follows from
Proposition~\ref{prop-van-almm}.
(\ref{thm-jay-verma-almm-a-ii}) follows form
(\ref{thm-jay-verma-almm-a-i}) and Theorem~\ref{depth-lemma}(1).
(\ref{thm-jay-verma-almm-b-i})
follows form \cite[4.2.1]{rossi-valla-2},
Proposition~\ref{prop-van-almm} and Lemma~\ref{depth-lemma}(1) .
We now prove (\ref{thm-jay-verma-almm-c}).
Applying Theorem~\ref{homology} and Lemma~3.2 in \cite{tom-huc} to the above sequence we get
{\beta}egin{eqnarray*}
\cdots
\rightarrow H_2(C( {\bf x}_{d}, G(I_2)))(n)
\rightarrow H_{1} (K( {\bf x}_{d}^o, F_{I_1}(I_2)))(n)
\rightarrow \frac{({\bf x}_d) \cap I_{1} I_{2}^{n}}
{({\bf x}_d)I_{1}I_{2}^{n-1}}
\rightarrow \frac{({\bf x}_d) \cap I_{2}^{n}}
{({\bf x}_d)I_{2}^{n-1}}
\rightarrow \cdots.
\end{eqnarray*}
Thus we have the exact sequence
{\beta}egin{eqnarray*}
\cdots
\rightarrow H_2(C( {\bf x}_{d}, G(I_2)))(n)
\rightarrow H_{1} (K( {\bf x}_{d}^o, F_{I_1}(I_2)))(n)
\rightarrow \frac{({\bf x}_d) I_2^{n-1 }\cap I_{1} I_{2}^{n}}
{({\bf x}_d)I_{1}I_{2}^{n-1}}
\rightarrow 0
\end{eqnarray*}
Hence $H_{1} (K( {\bf x}_{d}^o, F_{I_1}(I_2)))(n)=0$ if and only if $H_2(C( {\bf x}_{d}, G(I_2)))(n)$
and
${\displaystyle
\ell \left(
\frac{({\bf x}_d)I_{2}^{n-1} \cap I_{1} I_{2}^{n}}
{({\bf x}_d)I_{1}I_{2}^{n-1}}
\right)=0.}$
Moreover for $n=1, \ldots, s({\bf x}_d)$,
{\beta}egin{eqnarray*}
1
&=& \ell \left(
\frac{I_{1} I_{2}^{n}}
{({\bf x}_d)I_{1}I_{2}^{n-1}}
\right)\\
&=& \ell
\left(
\frac{I_{1} I_{2}^{n}}
{({\bf x}_d)I_{2}^{n-1} \cap I_{1} I_{2}^{n}}
\right)
+ \ell \left(
\frac{({\bf x}_d)I_{2}^{n-1} \cap I_{1} I_{2}^{n}}
{({\bf x}_d) I_{2}^{n-1}}
\right)\\
&=& \ell
\left(
\frac{I_{1} I_{2}^{n} + ({\bf x}_d)I_{2}^{n-1} }
{(x_d) I_{2}^{n-1}}\right)
+ \ell \left(
\frac{({\bf x}_d)I_{2}^{n-1} \cap I_{1} I_{2}^{n}}
{({\bf x}_d)I_{1}I_{2}^{n-1}}
\right)\\
&=& \ell
\left(
\frac{I_{1} I_{2}^{n} + ({\bf x}_d)I_{2}^{n-1} }
{({\bf x}_d) I_{2}^{n-1}}\right)
+ \ell (H_{1} (K( {\bf x}_{d}^o, F_{I_1}(I_2)))(n) ) .
\end{eqnarray*}
This gives
$ \ell (H_{1} (K( {\bf x}_{d}^o, F_{I_1}(I_2)))(n) ) =0$
if and only if
${\displaystyle
\ell
\left(
\frac{I_{1} I_{2}^{n} + ({\bf x}_d) I_{2}^{n-1} }
{({\bf x}_d) I_{2}^{n-1}}\right)=1}$.
\qed
\section{ Hilbert coefficients}
\label{hilbert-coefficients}
Throughout this section we will assume that $(R, {\mathfrak m})$ is a
Cohen-Macaulay local ring.
Let $I$ be an ${\mathfrak m}$-primary ideal of $R$. It is well known that for $n {\gamma}g 0$, the function $H(n):= \ell(R/I^n)$ is a polynomial in $n$ and we will denote by $P(n)$. For a two dimensional Cohen-Macaulay local ring and an ${\mathfrak m}$-primary ideal $I$, Huneke gave a relation between $\Delta^2[P(n) - H(n)]$ and the multiplicity of the ideal $I$ which is known as
Huneke's fundamental (\cite[Lemma~2.4]{huneke}).
This was generalized for an ${\mathfrak m}$-primary ideal in a d-dimensional Cohen-Macaulay local ring in \cite{huc} and for an Hilbert filtration in \cite{tom-huc}.
In \cite[Proposition~2.5]{jay-verma-almm}, Huneke's fundamental lemma was generalized for the filtration ${{\mathfrak m}athcal F} = \{I_1I_2^n\}_{n {\gamma}eq 0}$ for a two dimensional Cohen-Macaulay ring. We generalize Huneke's fundamental lemma for the filtration
${{\mathfrak m}athcal F}i$ and for any dimension $d {\gamma}eq 1$ (Theorem~\ref{thm-fundamental}). If we put $I = I_1 = I_2$ then we can recover Huneke's result as well as the result of Huckaba and Marley.
For all $n \in {\mathfrak m}athbb Z$ let
$H_{{{\mathfrak m}athcal F}i}(1,n):= \ell (R/ I_1 I_2^n)$ be the Hilbert function of ${{\mathfrak m}athcal F}2i$ and let $P_{{{\mathfrak m}athcal F}i}(1,n)$ denote the corresponding
Hilbert polynomial. This polynomial can be written in the form
{\beta}egin{eqnarray*}
\label{g-i-2}
P_{{{\mathfrak m}athcal F}i}(1,n)
= \sum_{j=0}^{d} (-1)^j
g_{j, I_1}(I_2)
{n + d-j -1 \checkmarkoose d-j}.
\end{eqnarray*}
We now describe the coefficients $g_{i, I_1}(I_2)$, $0 \leq i \leq d$. Our result is analogous to \cite[Lemma~2.8, Proposition~2.9]{huc}.
{\beta}egin{pro}
\label{proposition-alt-sum-coeff}
Let $I_1$ and $I_2$ be ${\mathfrak m}$-primary ideals in a local ring $(R, {\mathfrak m})$ of dimension $d {\gamma}eq 1$. Let $({\bf x}_{d})$ be a minimal reduction which is generated by a superficial sequence for $I_2$ and $I_1$. Let $1 \leq i \leq d$.
{\beta}egin{enumerate}
\item
\label{proposition-alt-sum-coeff-1}
$
{\displaystyle
\Delta^{d-i}
\left[ P_{{{\mathfrak m}athcal F}i}(1,0)
\right]
= (-1)^i g_{i, I_1}(I_2).
}$
\item
\label{proposition-alt-sum-coeff-2}
${\displaystyle
g_{i, I_1}(I_2)
= \sum_{n {\gamma}eq i-1} {n \checkmarkoose i-1}
\Delta^{d}
\left[ P_{{{\mathfrak m}athcal F}i}(1,n+1)
- H_{{{\mathfrak m}athcal F}i}(1,n+1) \right]
+ (-1)^d \ell \left( \frac{R}{K_i} \right), }$
where $K_i= R$ if $i< d$ and $K_d = I_1$.
\end{enumerate}
\end{pro}
\proof
Let $d {\gamma}eq 1$. If $i=d$, then we have
$ \Delta^{d-i} P_{{{\mathfrak m}athcal F}i}(1,0)
= \Delta^0 P_{{{\mathfrak m}athcal F}i}(1,0)
= P_{{{\mathfrak m}athcal F}i}(1,0)
= (-1)^dg_{d,{I_1}}(I_2)$.
Now let $i<d$ and let ${\bf x} \in I_2$ be superficial for $I_1$ and $I_2$.
If ${\omega}verline{\hphantom{xx}}$ denotes the image in $R/x$, then
{\beta}egin{eqnarray*}
\Delta^{d-i}
P_{{{\mathfrak m}athcal F}i}(1,0)
= \Delta^{d-i-1}
[P_{{\mathcal F}_{{\omega}verline{I_1};{\omega}verline{I_2}}}(1,0)]
= (-1)^i g_{i, {\omega}verline{I_1}}({\omega}verline{I_2})
= (-1)^i g_{i,{I_1}}({I_2}) .
\end{eqnarray*}
This proves (\ref{proposition-alt-sum-coeff-1})
We now prove (\ref{proposition-alt-sum-coeff-2}).
{\beta}egin{eqnarray*}
&& \sum_{n {\gamma}eq i-1} {n \checkmarkoose i-1}
\Delta^{d}
\left[ P_{{{\mathfrak m}athcal F}i}(1,n+1)
- H_{{{\mathfrak m}athcal F}i}(1,n+1) \right] \\
&=& (-1)^{i-1}
\sum_{n {\gamma}eq 0}
\Delta^{d-i+1}
\left[ P_{{{\mathfrak m}athcal F}i}(1,n+1)
- H_{{{\mathfrak m}athcal F}i}(1,n+1)
\right]
{\mathfrak m}box{ \cite[Lemma~2.7]{tom-huc} } \\
&=& (-1)^{i}
\Delta^{d-i}
\left[ P_{{{\mathfrak m}athcal F}i}(1,0)
- H_{{{\mathfrak m}athcal F}i}(1,0)
\right] \\
&=& g_{i, I_1}(I_2)
+ (-1)^{i+1}
\Delta^{d-i} H_{{{\mathfrak m}athcal F}i}(1,0)
\hspace{3.1in}
[{\mathfrak m}box{ by } (\ref{proposition-alt-sum-coeff-1})].
\end{eqnarray*}
One can verify that
{\beta}egin{eqnarray*}
\Delta^{d-i} H_{{{\mathfrak m}athcal F}i}(1,0)
&=& \left\{
{\beta}egin{array}{ll}
\ell \left( \frac{R}{I_1} \right) & {\mathfrak m}box { if } i=d\\
0
& {\mathfrak m}box { if } i< d\\
\end{array}
\right.
\end{eqnarray*}
\qed
{\beta}egin{lemma}
\label{hilb-coef}
Let $(R, {\mathfrak m})$ be a Cohen-Macaulay
ring of dimension $d {\gamma}eq 1$.
Let $I_1$ and $I_2$ be an ${\mathfrak m}$-primary ideals of $R$ and
$({\bf x}_{d})$ be a minimal reduction of $I_2$ which is generated by a superficial sequence for $I_2$ and $I_1$. Then
$
\displaystyle
{g_{0, I_1}(I_2)
= e ( I_2).}$
\end{lemma}
\proof From Lemma~\ref{vanishing} we get
$h_i( {\bf x}_{d})(1, n) = 0$ for all $i {\gamma}eq 1$ and for all $n {\gamma}g 0$. Since $H_{{\mathfrak m}athcal F}(1,n)$ is a polynomial $P_{{\mathfrak m}athcal F}(1,n)$ for all $n {\gamma}g 0$,
from Lemma~\ref{lemma-extension-fiber}
we have
{\beta}egin{eqnarray}
\label{coefficient-fiber-g1}
&& g_{0, I_1}(I_2) \\ \nonumber
&=& \limn
\sum_{i=0}^{d-1} (-1)^i{d-1 \checkmarkoose i}
P_{{{\mathfrak m}athcal F}i}(1,n-i)\\ \nonumber
&=& \limn
\sum_{i=0}^{d-1} (-1)^i{d-1 \checkmarkoose i}
H_{{{\mathfrak m}athcal F}i}(1,n-i)
= e_0(I_2)
- \limn~\ell
\left(
\frac{I_1 I_2^{n}}
{({\bf x}_{d}) I_1 I_2^{n-1}}
\right)
= e_0( I_2).
\end{eqnarray}
The last equality follows from the fact that $({\bf x}_{d})$ is a minimal reduction of $I_2$ and hence $I_1 I_2^{n}
=({\bf x}_{d}) I_1 I_2^{n-1}$ for all $n {\gamma}g 0$.
This proves the lemma.
\qed
{\beta}egin{lemma}
[Huneke's fundamental lemma for ${{{\mathfrak m}athcal F}i}$]
\label{thm-fundamental}
Let $(R, {\mathfrak m})$ be a Cohen-Macaulay ring of dimension $d {\gamma}eq 2$.
Let $I_1$ and $I_2$ be an ${\mathfrak m}$-primary ideals of $R$ and
$({\bf x}_{d})$ be a minimal reduction of $I_2$ which is generated by a superficial sequence for $I_2$ and $I_1$. For all
$n {\gamma}eq 1$,
{\beta}egin{eqnarray*}
\Delta^d \left[ P_{{{\mathfrak m}athcal F}i}(1,n) - H_{{{\mathfrak m}athcal F}i}(1,n) \right]
&=& \ell
\left(
\frac{ I_1 I_2^{n} + ({\bf x}_d)}
{ ({\bf x}_{d})}
\right)
- \sum_{i = 1}^d (-1)^i
h_i(C_{{\beta}ullet}, {\bf x}_{d})(1, n) \\
&=& \ell
\left(
\frac{ I_1 I_2^{n}}
{ ({\bf x}_{d}) I_1 I_2^{n-1}}
\right)
- \sum_{i = 2}^d (-1)^i
h_i(C_{{\beta}ullet}, {\bf x}_{d})(1, n) .
\end{eqnarray*}
\end{lemma}
\proof For all $n {\gamma}eq 0$,
$ \Delta^d \left[P_{{\mathfrak m}athcal F}(1,n) \right]
= g_{0, I_1}(I_2)$.
Now use Lemma~\ref{hilb-coef}
and Lemma~\ref{lemma-extension-fiber}.
\qed
We are ready to describe the coefficients
$g_{i, I_1}(I_2)$ explicitly, in terms of the homology modules of the complex ${{\mathfrak m}athcal C}1n$.
{\beta}egin{lemma}
\label{hilb-coef-one}
Let $(R, {\mathfrak m})$ be a Cohen-Macaulay ring of dimension $d {\gamma}eq 1$. Let $I_1$ and $I_2$ be ${\mathfrak m}$-primary ideals
and
$({\bf x}_{d}) $ a minimal reduction of $I_2$ which is generated by a superficial sequence for $I_2$ and $I_1$. Let $1 \leq i \leq d$. We can write:
{\mathfrak m}ptmcomx
{\beta}egin{eqnarray}
\label{hilb-coef-one-eq}
g_{i, I_1}(I_2)
&=&
\sum_{n {\gamma}eq i-1} {n \checkmarkoose i-1}
\left[
\ell
\left(
\frac{I_1 I_2^{n+1} +({\bf x}_{d}) }
{({\bf x}_{d}) }
\right)
- \sum_{j {\gamma}eq 1} (-1)^j
h_j( C_{{\beta}ullet}, {\bf x}_{d})(1, n+1)\right]
+ (-1)^d\ell \left( \frac{R}{K_i} \right)
\\
\label{hilb-coef-one-eq-two}
&=&
\sum_{n {\gamma}eq i-1} {n \checkmarkoose i-1}
\left[
\ell
\left(
\frac{I_1 I_2^{n+1}}
{({\bf x}_{d}) I_1 I_2^{n}}
\right)
- \sum_{j {\gamma}eq 2} (-1)^i
h_j( C_{{\beta}ullet}, {\bf x}_{d})(1, n+1)\right]
+ (-1)^d\ell \left( \frac{R}{K_i} \right)
\end{eqnarray}
where $K_i = R$ for $1 \leq i \leq d-1$ and $K_d= I_1$.
\end{lemma}
{\mathfrak m}ptm
\proof
The proof follows from Proposition~\ref{proposition-alt-sum-coeff}(\ref{proposition-alt-sum-coeff-2}) and Lemma~\ref{thm-fundamental}.
\qed
A formula for $g_{1, I_1}(I_2)$ plays a very important role in analyzing the depth of $G_{I_1}(I_2)$. When $({\bf x}_d) \subseteq I_1$, we have nice formulas in terms of the homology modules.
{\beta}egin{lemma}
\label{formula-g1}
Let $({\bf x}_d)\subseteq I_1$. With the assumptions as in
Lemma~\ref{hilb-coef-one}
{\beta}egin{eqnarray}
\label{formula-g1-one}
g_{1, I_1}(I_2)
&=& \sum_{n {\gamma}eq 1}
\ell
\left( \frac{I_1 I_2^{n} + ({\bf x}_d)}
{({\bf x}_{d}) }
\right)
- \sum_{j {\gamma}eq 1} (-1)^{j} h_j(D_{{\beta}ullet},{\bf x}_{d})(1, *)
- \left( \frac{R}{I_1} \right)\\
\label{formula-g1-two}
&=& \sum_{n {\gamma}eq 1}
\ell
\left( \frac{I_1 I_2^{n}}
{({\bf x}_{d}) I_1 I_2^{n-1}}
\right)
- \sum_{j {\gamma}eq 2} (-1)^{j} h_j(D_{{\beta}ullet},{\bf x}_{d})(1, *)
- \left( \frac{R}{I_1}
\right).
\end{eqnarray}
\end{lemma}
\proof Put $i=1$ in equation (\ref{hilb-coef-one-eq}) of Lemma~\ref{hilb-coef-one}. We get
{\beta}egin{eqnarray*}
g_{1, I_1}(I_2)
&=& \sum_{n {\gamma}eq 1}
\left[ \ell
\left(
\frac{I_1 I_2^{n} +({\bf x}_{d}) }
{({\bf x}_{d}) } \right)
- \sum_{j {\gamma}eq 1} (-1)^j
h_j( C_{{\beta}ullet}, {\bf x}_{d})(1, n)\right]\\
&=& \sum_{n {\gamma}eq 1}
\left[ \ell
\left( \frac{I_1 I_2^{n} +({\bf x}_{d})}{({\bf x}_{d})}\right)
- \sum_{j {\gamma}eq 1} (-1)^j
\left[ \sum_{n {\gamma}eq j}
h_j( C_{{\beta}ullet}, {\bf x}_{d})(1, n)
+ \sum_{n=1}^{j-1} h_j( C_{{\beta}ullet}, {\bf x}_{d})(1, n)
\right] \right]
\end{eqnarray*}
For all $i {\gamma}eq 2$,
{\beta}egin{eqnarray*}
h_j(C_{{\beta}ullet},{\bf x}_{d})(1, n)
= \left\{
{\beta}egin{array}{ll}
h_j(D_{{\beta}ullet},{\bf x}_{d})(1, n) & {\mathfrak m}box{ if } n {\gamma}eq j\\
{d \checkmarkoose j} \ell \left( \frac{R}{I_1} \right) & {\mathfrak m}box{ if } 1 \leq n \leq j-1.
\end{array}
\right.
\end{eqnarray*}
Therefore,
{\beta}egin{eqnarray*}
g_{1, I_1}(I_2)
&=& \sum_{n {\gamma}eq 1}
\ell
\left( \frac{I_1 I_2^{n} + ({\bf x}_d)}
{({\bf x}_{d})}
\right)
- \sum_{j {\gamma}eq 1} (-1)^{j}
h_j(D_{{\beta}ullet},{\bf x}_{d})(1, *)
- \left[
\sum_{j=2}^d (-1)^j (j-1) {d \checkmarkoose j}
\left( \frac{R}{I_1}
\right)
\right]
\\
&=& \sum_{n {\gamma}eq 1}\ell
\left( \frac{I_1 I_2^{n} + ({\bf x}_d)}
{({\bf x}_{d})}
\right)
- \sum_{j {\gamma}eq 1} (-1)^{j} h_j(D_{{\beta}ullet},{\bf x}_{d})(1, *)
- \left( \frac{R}{I_1}
\right).
\end{eqnarray*}
In a similar way, we can prove (\ref{formula-g1-two}).
\qed
If $({\bf x}_d)$ is a minimal reduction of $I_2$ and
$({\bf x}_d) \subseteq I_1$, then we can give bounds on
$g_{1, I_1}(I_2)$. In \cite[Proposition~4.1]{jay-verma} only a lower bound was given. We also give an upper bound.
{\beta}egin{pro}
\label{prop-al-min-mult-ineq}
Let $({\bf x}_d)\subseteq I_1$. With the assumptions as in
Lemma~\ref{hilb-coef-one}:
{\beta}egin{enumerate}
\item
${\displaystyle
\label{al-min-mult-eq-one}
g_{1, I_1}(I_2)
{\gamma}eq
\sum_{n {\gamma}eq 1}
\left( \frac{I_1 I_2^n+ ({\bf x}_{d})}
{({\bf x}_{d}) }
\right)
- \ell \left( \frac{R}{I_1} \right) }
$
and equality holds if and only if $G_{I_1}(I_2)$ is Cohen-Macaulay.
\item
\label{al-min-mult-eq-two}
${\displaystyle
g_{1, I_1}(I_2)
\leq \sum_{n {\gamma}eq 1}
\left(
\frac{I_1 I_2^n}
{({\bf x}_{d}) I_1 I_2^{n-1}} \right)
- \ell \left( \frac{R}{I_1} \right) }
$
and equality holds if and only if ${\mathfrak m}athop{\rm depth}~G_{I_1}(I_2) {\gamma}eq d-1$.
\end{enumerate}
\end{pro}
\proof The proof follows from Theorem~\ref{rigidity},
Proposition~\ref{grade} and
Proposition~\ref{prop-al-min-mult-ineq}
\qed
{\beta}egin{cor}
Let $({\bf x}_d)\subseteq I_1$. With the assumptions as in
Lemma~\ref{hilb-coef-one}:
{\beta}egin{enumerate}
\item
$g_{1, I_1}(I_2) {\gamma}eq - \ell ( R/I_1)$ and equality holds if and only if $G_{I_1}(I_2)$ is Cohen-Macaulay and $I_1 I_2^n + ({\bf x}_d)= ({\bf x}_d)$ for all $ {\gamma}eq 1$.
\item
${\displaystyle
\ell
\left(
\frac{R}{I_1I_2} \right)
{\gamma}eq e_0(I_2) - g_{1, I_1}(I_2)
+ \ell
\left(
\frac{I_1I_2+ ({\bf x}_d)}{I_1 I_2}
\right)}$
and equality holds if and only if $G_{I_1}(I_2)$ and
$I_1 I_2^n+({\bf x}_d) = ({\bf x}_d)$ for all $n {\gamma}eq 2$.
\end{enumerate}
\end{cor}
\proof Both (1) and (2) follows from
Proposition~\ref{prop-al-min-mult-ineq}(1).
\section{ Hilbert coefficients of the fiber cone}
\label{hilb-coef-fiber-cone}
Throughout this section we will assume $I_1$ and $I_2$ are ${\mathfrak m}$-primary ideals in a Cohen-Macaulay local ring $(R,{\mathfrak m})$In this section we describe the Hilbert coefficients of the fiber cone in terms of the length of the homologies of the complex ${{\mathfrak m}athcal C}1n$ and ${{\mathfrak m}athcal C}nn$.
{\beta}egin{notn}
We denote the Hilbert function (resp. polynomial) of the fiber cone by
{\beta}egin{eqnarray*}
H ( F_{I_{1}}(I_{2}), n)
= \ell \left( \frac{I_{2}^{n}}{I_{1} I_{2}^{n}}\right)
\hphantom{space}
\left( {\mathfrak m}box{resp. }
P ( F_{I_{1}}(I_{2}), n)
= \sum_{i=0}^{d-1} (-1)^i
f_{i, I_1}(I_2)
{n + d-1-i \checkmarkoose d-1-i} \right).
\end{eqnarray*}
\end{notn}
{\beta}egin{remark}
\label{hilbert-fiber cone}
Since
{\beta}egin{eqnarray*}
\ell \left( \frac{I_2^n}{I_1 I_2^n}\right)
&=& \ell \left( \frac{R}{I_1 I_2^n}\right)
- \ell \left( \frac{R}{I_2^n}\right)\\
&=& \sum_{i=0}^{d-1} (-1)^{i}
\left[e_{i+1}(I_{2}) - g_{i+1, I_1}{(I_2)}
+ e_{i}(I_{2}) - g_{i, I_1}(I_2) \right]
{n + d-1 -i\checkmarkoose d-1-i},
\end{eqnarray*}
we have
{\beta}egin{eqnarray*}
f_{i, I_1}(I_2)
= e_{i+1}(I_{2}) - g_{i+1, I_1}(I_2)
+ e_{i}(I_{2}) - g_{i, I_1}(I_2) \hspace{.2in}
{\mathfrak m}box{ for all }
\hphantom{space} i=0, \ldots, d-1.
\end{eqnarray*}
\end{remark}
{\beta}egin{lemma}
\label{cor-fiber-coeff}
Let $d {\gamma}eq 1$
and
$({\bf x}_{d})$ a minimal reduction of $I_2$ which is generated by a superficial sequence for $I_2$ and $I_1$. Let $1 \leq i \leq d$. We can write:
{\mathfrak m}ptmcomx
{\beta}egin{eqnarray*}
f_{i, I_1}(I_2)
= \sum_{n {\gamma}eq i} {n \checkmarkoose i}
\left[
\ell
\left(
\frac{I_2^{n}}
{I_1 I_2^{n} + ({\bf x}_d) \cap I_2^n }
\right)
- \sum_{j {\gamma}eq 1} (-1)^j \left[
h_j( C_{{\beta}ullet}, {\bf x}_{d})(0, n)
- h_j( C_{{\beta}ullet}, {\bf x}_{d})(1, n)
\right]
\right]
- (-1)^d \ell \left( \frac{R}{K_{i}} \right)
\end{eqnarray*}
{\beta}egin{eqnarray*}
f_{i, I_1}(I_2)
= \sum_{n {\gamma}eq i} {n \checkmarkoose i}
\left[ \left[
\ell
\left(
\frac{I_2^{n}}
{I_1 I_2^{n}}
\right)
-\ell
\left(
\frac{({\bf x}_d)I_2^{n-1}}
{({\bf x}_d)I_1 I_2^{n-1}}
\right)
\right]
- \sum_{j {\gamma}eq 2} (-1)^j \left[
h_j( C_{{\beta}ullet}, {\bf x}_{d})(0, n)
- h_j( C_{{\beta}ullet}, {\bf x}_{d})(1, n)
\right]
\right]
- (-1)^d \ell \left( \frac{R}{K_{i}} \right)
\end{eqnarray*}
where $K_i = R$ for $i=0, \ldots, d-2$ and $K_{d-1} = I_1$
\end{lemma}
\proof The proof follows from (4.5) of \cite{tom-huc} and
Lemma~\ref{hilb-coef-one}.
{\mathfrak m}ptm
We give bounds for the multiplicity of the fiber cone. This is an improvement of \cite{cpv} and \cite[Corollary~4.2]{jay-verma}.
{\beta}egin{cor}
\label{upper-bound-multiplicity}
Let $I_1$ and $I_2$ be ${\mathfrak m}$-primary ideals in a local ring $(R, {\mathfrak m})$ of dimension at least two. Let $({\bf x}_{d})$ be a minimal reduction which is a superficial sequence for $I_1$ and $I_2$. Suppose
$({\bf x}_d) \subseteq I_1$ and $({\bf x}_d) \cap I_1 I_2 = ({\bf x}_d) I_1$.
{\beta}egin{enumerate}
\item
\label{upper-bound-0}
${\displaystyle
f_{0, I_1}(I_2)
\leq e_{1}(I_{2}) - e_{0}(I_{2})
+ \ell \left( \frac{I_2}{I_1 I_2} \right)
+ \ell \left( \frac{R}{I_2} \right)
- (d-1)~\left( \frac{R}{I_1} \right)} $
and equality holds if and only if
$G_{I_1}(I_2)$ is Cohen-Macaulay and for all $n {\gamma}eq 2$,
$I_1 I_2^n + ({\bf x}_d) = ({\bf x}_d)$.
\item
\label{upper-bound-1}
${\displaystyle
f_{0, I_1}(I_2)
{\gamma}eq e_{1}(I_{2})
- \left(
\frac{I_1 I_2}
{({\bf x}_{d}) I_1 }
\right)
- \sum_{n {\gamma}eq 2}
\left(
\frac{I_1 I_2^n}
{({\bf x}_{d}) I_1 I_2^{n-1} }
\right)
+ \ell \left(
\frac{R}{I_1} \right)} $
and equality holds if and only if
${\mathfrak m}athop{\rm depth}~G_{I_1}(I_2) {\gamma}eq d-1$.
\end{enumerate}
\end{cor}
\proof By Remark~\ref{hilbert-fiber cone} and Proposition~\ref{prop-al-min-mult-ineq} we get
{\beta}egin{eqnarray*}
f_{0, I_1}(I_2)
&=& e_{1}(I_{2}) - g_{1, I_1}(I_2)\\
&\leq& e_{1}(I_{2})
- \sum_{n {\gamma}eq 1}
\left( \frac{I_1 I_2^n + ({\bf x}_d)}
{({\bf x}_{d}) }
\right)
+ \ell \left( \frac{R}{I_1} \right)\\
&\leq& e_{1}(I_{2})
- \left(
\frac{I_1 I_2 +({\bf x}_d)}
{({\bf x}_{d}) }
\right)
+ \ell \left( \frac{R}{I_1} \right)\\
&\leq& e_{1}(I_{2})
- \left(
\frac{I_1 I_2 }
{({\bf x}_{d} )I_1 }
\right)
+ \ell \left( \frac{R}{I_1} \right).
\end{eqnarray*}
Using Lemma~\ref{homology-negative} we get:
{\beta}egin{eqnarray*}
\left( \frac{I_1 I_2}{({\bf x}_{d}) I_1 } \right)
= e({\bf x}_d)
- \left( \frac{R}{I_1 I_2} \right)
+ d~\left( \frac{R}{ I_1} \right)
= e({\bf x}_d)
- \left( \frac{I_2}{I_1 I_2} \right)
- \left( \frac{R}{I_2} \right)
+ d~\left( \frac{R}{ I_1} \right).
\end{eqnarray*}
If equality holds if and only if
${\displaystyle
g_{1, I_1}(I_2)
= \left(
\frac{I_1 I_2}
{({\bf x}_{d}) I_1}
\right)
- \ell
\left(
\frac{R}{I_1}
\right)
}$ and
$I_1 I_2^n + ({\bf x}_d) = ({\bf x}_d)$ for all $n {\gamma}eq 2$.
Now apply Proposition~\ref{prop-al-min-mult-ineq}(1).
This proves (\ref{upper-bound-0}).
Once again by Remark~\ref{hilbert-fiber cone} and Proposition~\ref{prop-al-min-mult-ineq} we get
{\beta}egin{eqnarray*}
f_{0, I_1}(I_2)
&=& e_{1}(I_{2}) - g_{1, I_1}(I_2)\\
&{\gamma}eq& e_{1}(I_{2})
- \sum_{n {\gamma}eq 1}
\left( \frac{I_1 I_2^n}
{({\bf x}_{d})I_1 I_2^{n-1} }
\right)
+ \ell \left( \frac{R}{I_1} \right)\\
&=& e_{1}(I_{2})
- \left(
\frac{I_1 I_2}
{({\bf x}_{d}) I_1 }
\right)
- \sum_{n {\gamma}eq 2}
\left(
\frac{I_1 I_2^n}
{({\bf x}_{d}) I_1 I_2^{n-1} }
\right)
+ \ell \left(
\frac{R}{I_1} \right).
\end{eqnarray*}
Applying
Proposition~\ref{prop-al-min-mult-ineq}(2) we conclude that
equality holds if and only if ${\mathfrak m}athop{\rm depth}(({\bf x}_d), G_{I_1}(I_1)) {\gamma}eq d-1$.
\qed
{\beta}egin{cor}
With the assumptions as in
Corollary~\ref{upper-bound-multiplicity} we have:
{\beta}egin{enumerate}
\item
If ${\displaystyle
f_{0, I_1}(I_2)
= e_{1}(I_{2}) - e_{0}(I_{2})
- \ell \left( \frac{I_2}{I_1 I_2} \right)
- \ell \left( \frac{R}{I_2} \right)
- (d-1)~\left( \frac{R}{I_1} \right)}$, then for $0 \leq depth~G_{I_1}(I_2)$\\$ \leq d-1$,
$
{\mathfrak m}athop{\rm depth}~F_{I_1}(I_2)= {\mathfrak m}athop{\rm depth}~G_{I_1}(I_2) + 1.
$
\item
If ${\displaystyle
f_{0, I_1}(I_2)
= e_{1}(I_{2})
- \left(
\frac{I_1 I_2}
{({\bf x}_{d}) I_1 }
\right)
- \sum_{n {\gamma}eq 2}
\left(
\frac{I_1 I_2^n}
{({\bf x}_{d}) I_1 I_2^{n-1} }
\right)
+ \ell \left(
\frac{R}{I_1} \right)}$, then for $0 \leq depth~G_{I_1}(I_2) \leq d-2$,
$
{\mathfrak m}athop{\rm depth}~F_{I_1}(I_2)= {\mathfrak m}athop{\rm depth}~G_{I_1}(I_2) + 1
$.
\end{enumerate}
\end{cor}
\proof The proof follows from Corollary~\ref{upper-bound-multiplicity} and by Theorem~\ref{grade} and Theorem~\ref{depth-lemma}.
\qed
\section{ Hilbert series for ideals of minimal and almost minimal multiplicity}
\label{hilb-fib-cone-mm}
In this section we describe the Hilbert series of the fiber cone for ideals of minimal multiplicity and ideals of almost minimal multiplicity.
{\beta}egin{notn}
{\beta}egin{eqnarray*}
H_{{{\mathfrak m}athcal F}i}(i, t)
&=& \sum_{n {\gamma}eq 0}H_{{{\mathfrak m}athcal F}i}(i, n)t^{n}
= \sum_{n {\gamma}eq 0}\ell \left(
\frac{R}{I_1^i I_2^n} \right) t^n, \hspace{.5in} i=0,1.\\
H(F_{I_{1}}(I_{2}), t)
&=& \sum_{n {\gamma}eq 0}H ( F_{I_{1}}(I_{2}), n)t^{n}.
\end{eqnarray*}
\end{notn}
Recall that $s({\bf x}_d):= {\mathfrak m}in\{ n | ({\bf x}_d)I_1 I_2^n = I_1 I_2^{n+1} \}$.
{\beta}egin{thm}
\label{hilb-series-one}
Let $(R, {\mathfrak m})$ be a Cohen-Macaulay local ring of dimension
$d {\gamma}eq 1$.
Let $I_2$ be an ideal of minimal multiplicity with respect to $I_1$. Let $({\bf x}_d)$ be a minimal reduction of $I_1$ and assume that $({\bf x}_d)\subseteq I_1$.
{\beta}egin{enumerate}
\item
${\displaystyle
H_{{{\mathfrak m}athcal F}i}(1,t)
= \frac{ \ell (R/I_1)
- t \left[ \ell (R/I_1) -e_0(I_2) \right] }
{(1-t)^{d}}.}$
\item
$
g_{1, I_1}(I_2) = e_0(I_2)
- \ell \left( \frac{R}{I_1} \right).
$
For $2 \leq i \leq d$,
$g_{i, I_1}(I_2)=0.$
\item
$
{\displaystyle
H( F_{I_1}(I_2),t)
= \frac{ \ell (R/I_1) - t \left[ \ell (R/I_1) -e_0(I_2) \right] }
{(1-t)^{d+1}}
- H_{{{\mathfrak m}athcal F}i}(0, t).
}$
\item
$f_{0, I_1}(I_2)= e_{1}(I_{2})
-e_{0}(I_2)
+ \ell \left( \frac{R}{I_1} \right).$
For $1 \leq i \leq d-1$,
$f_{i, I_1}(I_2)=e_{i+1}(I_{2}).$
\end{enumerate}
\end{thm}
\proof
From Theorem~\ref{homology},
Lemma~\ref{mm-amm} and
and Theorem~\ref{thm-fundamental},
by induction on $n$ we get
{\beta}egin{eqnarray}
\label{alt-sum-new-g}
H_{{{\mathfrak m}athcal F}i}(1,n)
= e_0(I_2) {n+ d \checkmarkoose d}
- \left[ e_0(I_2)
- \ell \left( \frac{R}{I_1} \right) \right]
{n + d-1 \checkmarkoose d-1}.
\end{eqnarray}
Summing over all $n {\gamma}eq 0$ we get (1) and (2).
(3) and (4) are an immediate consequence of (1), (2) and Remark~\ref{hilbert-fiber cone}. \qed
To prove Theorem~\ref{final-hilb-ser} we need the following combinatorial lemma:
{\beta}egin{lemma}
\label{combin}
For all $n {\gamma}eq s$ and for all $d {\gamma}eq 1$ we have:
{\beta}egin{eqnarray*}
{n-s + d-1 \checkmarkoose d}
= \sum_{i=0}^{d} (-1)^i{s + 1 \checkmarkoose i} {n+ d-i \checkmarkoose d-i}.
\end{eqnarray*}
\end{lemma}
\proof The proof follows by induction on $d$. The case $d=1$
can be verified easily. If $d>1$, then
{\beta}egin{eqnarray*}
{n-s + d-1 \checkmarkoose d}
&=& {n+1 -s + d-1 \checkmarkoose d} - {n+1 -s + d-2 \checkmarkoose d-1}\\
&=& \sum_{i=0}^{d} (-1)^i{s + 1 \checkmarkoose i} {n+1+ d-i \checkmarkoose d-i}
- \sum_{i=0}^{d-1}(-1)^i{s + 1 \checkmarkoose i} {n+1+ d-1-i \checkmarkoose d-1-i}\\
&=& \sum_{i=0}^{d} (-1)^i {s + 1 \checkmarkoose i} {n+ d-i \checkmarkoose d-i}.
\end{eqnarray*}
{\beta}egin{thm}
\label{final-hilb-ser}
Let $(R, {\mathfrak m})$ be a Cohen-Macaulay ring of dimension $d {\gamma}eq 2$.
Let $I_1$ and $I_2$ be an ${\mathfrak m}$-primary ideals of $R$ and
$({\bf x}_{d}) \in I_2$ be a minimal reduction which is a superficial sequence for $I_1$ and $I_2$. Assume that
$(x_d) \subseteq I_1$ and $({\bf x}_d) \cap I_1 I_2 = I_1({\bf x}_d)$.
{\beta}egin{enumerate}
\item
${\displaystyle
H_{{{\mathfrak m}athcal F}i}(1,t)
= \frac{ \ell (R/I_1)
- t \left[ \ell (R/I_1) -e_0(I_2) \right]
}
{(1-t)^{d+1}}
+ \frac{t^{s+1}}{(1-t)^{d+1}}
}$
\item
$\displaystyle
{
g_{i, I_1}(I_2)=
\left\{ {\beta}egin{array}{ll}
e_0(I_2)
- \ell \left( \frac{R}{I_1} \right)
+ s & i=1\\
{s + 1 \checkmarkoose i} & i =2, \ldots, d\\
\end{array}
\right.}.
$
\item
${\displaystyle
{\beta}egin{array}{rl}
H(F_{I_{1}}(I_{2}), t)
=& \frac{ \ell (R/I_1)
- t \left[ \ell (R/I_1) -e_0(I_2) \right] }
{(1-t)^{d+1}}
+ \frac{t^{s+1}}{(1-t)^{d+1}}
- H_{{{\mathfrak m}athcal F}i}(0, t).
\end{array}
}$
\item
$\displaystyle{
f_{i, I_1}(I_2)
= \left\{
{\beta}egin{array}{ll}
e_{1}(I_{2}) -e_0(I_2)
+ \ell \left( \frac{R}{I_1} \right)
- s
& i=0\\
e_{i+1}(I_{2}) - {s+1 \checkmarkoose i}
& i=1, \ldots,d-1
\end{array}
\right.
}
$.
\end{enumerate}
\end{thm}
\proof Put $s = s({\bf x}_d)$. Then
from Theorem~\ref{homology},
Lemma~\ref{mm-amm} and Theorem~\ref{thm-fundamental},
{\beta}egin{eqnarray*}
H_{{{\mathfrak m}athcal F}i}(1,n)
&=& \left\{
{\beta}egin{array}{ll}
\left[ e_0( I_2)-1 \right] {n+ d -1\checkmarkoose d}
+ \ell \left( \frac{R}{I_1} \right){n + d-1 \checkmarkoose d-1}
& 0 \leq n \leq s
\\
\hphantom{s}
[e_0( I_2)-1] {n+ d-1\checkmarkoose d}
+ \ell \left( \frac{R}{I_1} \right)
{n + d-1 \checkmarkoose d-1}
+ {d + n - (s+1) \checkmarkoose d}
& n {\gamma}eq s+1\\
\end{array}
\right.
\end{eqnarray*}
For all $n {\gamma}eq s+1$ we have
{\beta}egin{eqnarray*}
H_{{{\mathfrak m}athcal F}i}(1,n)
&=& e_0( I_2) {n+ d \checkmarkoose d}
- \left[ e_0( I_2)-1-\ell \left( \frac{R}{I_1} \right) + (s+1)\right]
+ \sum_{i {\gamma}eq 2} (-1)^i {s+1 \checkmarkoose i} {n + d-i \checkmarkoose d-i}.
\end{eqnarray*}
This proves (2).
(3) and (4) are an immediate consequence of (1), (2), Remark~\ref{hilbert-fiber cone} and Lemma~\ref{combin}. \qed
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\end{thebibliography}
\end{document}
\end{document}
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math
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विवादों के निस्तारण के लिए विवाद निवारण समिति का हुआ गठन
रूद्रपुर, खबर संसार। मा० मुख्यमंत्री हेल्पालाईन में, जनसुनवाई दिवसों में, तहसील दिवसों मे विभिन्न विवादों के निस्तारण हेतु प्रत्येक परगने मे परगनाधिकारी की अध्यक्षता मे विवाद निवारण समिति का गठन किया गया है। समिति मे सम्बन्धित पुलिस क्षेत्राधिकारी सदस्य सचिव तथा सम्बन्धित चकबन्दी अधिकारी सदस्य नामित है।
उक्त जानकारी देते हुए जिलाधिकारी डा० नीरज खैरवाल ने बताया मा० मुख्यमंत्री हेल्पालाईन में, जनसुनवाई दिवसों में, तहसील दिवसों मे, जनता दरबार, बहुउद्देशीय शिविरों मे प्राप्त होने वाली ऐसी शिकायते जिनमे दो पक्षो के बीच विभिन्न विवादों का होना पाया जाता है। ऐसे शिकायतकर्ताओ को तहसील या थाने के बार-बार चक्कर काटने पडते है।
उन्होने बताया इन शिकायतो मे अधिकांश शिकायते इस प्रकार पाई जाती है जिनमे उभय पक्षो को विधि व अभिलेखो की वास्तविक स्थिति की जानकारी उपलब्ध हो जाये तो शिकायतकर्ता शान्तिपूर्ण, सौहार्दपूर्ण व त्वरित न्यायोचित समाधान के लिए तैयार रहता है।
जिलाधिकारी ने बताया ऐसी शिकायतों को संयुक्त रूप से पुलिस/राजस्व विभाग के अधिकारियों द्वारा सुने जाने व समाधान का प्रयास किये जाने पर बहुत से ऐसे विवादो का समाधान किया जा सकता है, जो लम्बे समय से न्यायालयो मे विचाराधीन है। उन्होने बताया गठित समिति जिलाधिकारी कार्यालय, वरिष्ठ पुलिस अधीक्षक कार्यालय, तहसील, परगना, पुलिस क्षेत्राधिकारी कार्यालयों मे व मा० मुख्यमंत्री हेल्पलाईन से प्राप्त होने वाले विभिन्न विवाद सम्बन्धित शिकायतो का प्रत्येक मंगलवार को आयोजित होने वाले तहसील दिवस के दिन तहसील कार्यालय मे सम्बन्धित पक्षो को बुलाकर सुनवाई कर विवाद का निस्तारण शान्तिपूर्ण ढंग से कराने का प्रयास करेंगे।
जिलाधिकारी ने बताया प्राप्त शिकायतो का अनिवार्य रूप से एक माह के अन्दर सुनवाई कर निस्तारण किया जाना आवश्यक होगा। उन्होने बताया विशेष परिस्थितियो में १५ दिन का समय वरिष्ठ पुलिस अधीक्षक के अनुमोदन से बढाया जा सकेगा। परगना स्तरीय समितियों के कार्यवाहियों के पर्यवेक्षण हेतु जनपद स्तर पर दो समितियो का गठन किया गया है
परगना क्षेत्र रूद्रपुर, किच्छा, सितारगंज व खटीमा हेतु अपर जिलाधिकारी (प्रशासन) व अपर पुलिस अधीक्षक (शहर) तथा परगना क्षेत्र बाजपुर, काशीपुर व जसपुर हेतु अपर जिलाधिकारी (वि०/रा०) व अपर पुलिस अधीक्षक (काशीपुर) का गठन किया गया है। ऐसे गम्भीर विवाद जिनका परगना स्तरीय समिति द्वारा समाधान सम्भव न हो, जनपद स्तर पर जिलाधिकारी व एसएसपी की संयुक्त समिति द्वारा पक्षकारो को सुने जाने एवं विवाद का समाधान किये जाने का प्रयास किया जायेगा।
विधायक गणेश जोशी ने ३० मीटर स्पान पुल का शिलान्यास किया
घटना की निष्पक्ष जांच के लिए मुख्यमंत्री ने डीएम को दिया निर्देश
|
hindi
|
\begin{document}
\begin{frontmatter}
\title{Pathwise optimal transport bounds between a~one-dimensional
diffusion and its Euler scheme}
\runtitle{Pathwise optimal transport bounds}
\begin{aug}
\author[A]{\fnms{A.} \snm{Alfonsi}\corref{}\ead[label=e1]{alfonsi@cermics.enpc.fr}\thanksref{t1}},
\author[A]{\fnms{B.} \snm{Jourdain}\ead[label=e2]{jourdain@cermics.enpc.fr}\thanksref{t1}}
\and
\author[B]{\fnms{A.} \snm{Kohatsu-Higa}\ead[label=e3]{arturokohatsu@gmail.com}\thanksref{t2}}
\runauthor{A. Alfonsi, B. Jourdain and A. Kohatsu-Higa}
\affiliation{Universit\'e Paris-Est, Universit\'e Paris-Est, and
Ritsumeikan University and~Japan Science and Technology Agency}
\address[A]{A. Alfonsi\\
B. Jourdain\\
CERMICS\\
Projet MathFi ENPC-INRIA-UMLV\\
Universit\'e Paris-Est\\
6-8 Avenue Blaise Pascal\\
77455 Marne La Vall\'ee, Cedex 2\\
France\\
\printead{e1}\\
\phantom{E-mail: }\printead*{e2}}
\address[B]{A. Kohatsu-Higa\\
Department of Mathematical Sciences\\
Ritsumeikan University\\
1-1-1 Nojihigashi\\
Kusatsu, Shiga 525-8577\\
Japan\\
and\\
Japan Science and Technology Agency\\
\printead{e3}}
\thankstext{t1}{Benefited from the support of the ``Chaire Risques
Financiers,'' Fondation du
Risque and of the French National Research Agency (ANR) under the program
ANR-08-BLAN-0218 BigMC and the Labex Bezout.}
\thankstext{t2}{Supported by grants of the Japanese goverment.}
\end{aug}
\received{\smonth{9} \syear{2012}}
\revised{\smonth{5} \syear{2013}}
\begin{abstract}
In the present paper, we prove that the Wasserstein distance on the
space of continuous sample-paths equipped with the supremum norm
between the laws of a uniformly elliptic one-dimensional diffusion
process and its Euler discretization with $N$ steps is smaller than
$O(N^{-2/3+\varepsilon})$ where $\varepsilon$ is an arbitrary positive
constant. This rate is intermediate between the strong error estimation
in $O(N^{-1/2})$ obtained when coupling the stochastic differential
equation and the Euler scheme with the same Brownian motion and the
weak error estimation $O(N^{-1})$ obtained when comparing the
expectations of the same function of the diffusion and of the Euler
scheme at the terminal time $T$. We also check that the supremum over
$t\in[0,T]$ of the Wasserstein distance on the space of probability
measures on the real line between the laws of the diffusion at time $t$
and the Euler scheme at time $t$ behaves like
$O(\sqrt{\log(N)}N^{-1})$.
\end{abstract}
\begin{keyword}[class=AMS]
\kwd{65C30}
\kwd{60H35}
\end{keyword}
\begin{keyword}
\kwd{Euler scheme}
\kwd{Wasserstein distance}
\kwd{weak trajectorial error}
\kwd{diffusion bridges}
\end{keyword}
\end{frontmatter}
\setcounter{section}{-1}
\section{Introduction}
For $\sigma\dvtx \mathbb{R}\to\mathbb{R}$ and $b\dvtx
\mathbb{R}\to\mathbb {R}$, we are interested in the simulation of the
stochastic differential equation
\begin{equation}
dX_t=\sigma(X_t)\,dW_t+b(X_t)\,dt,
\label{sde}
\end{equation}
where $X_0=x_0\in\mathbb{R}$ and $W=(W_t)_{t\geq0}$ is a standard Brownian
motion. We make the standard Lipschitz assumptions on the coefficients,
\[
\exists K\in(0,+\infty), \forall x,y\in\mathbb{R}\qquad
\bigl|\sigma(x)-\sigma(y)\bigr|+\bigl|b(x)-b(y)\bigr|\leq K|x-y|.
\]
For $T>0$, we are interested in the approximation of
$X=(X_t)_{t\in[0,T]}$ by its Euler scheme
$\bar{X}=(\bar{X}_t)_{t\in[0,T]}$ with $N\geq1$ time-steps. We consider
the regular grid $\{0=t_0<t_1<t_2<\cdots<t_N=T\}$ of the interval
$[0,T]$ with $t_k=\frac{kT}{N}$ and define inductively $\bar{X}_0=x_0$
and
\begin{equation}
\bar{X}_t=\bar{X}_{t_k}+\sigma(\bar{X}_{t_k})
(W_t-W_{t_k})+b(\bar{X}_{t_k})
(t-t_k)\qquad\mbox{for }t\in[t_k,t_{k+1}].
\label{eul}
\end{equation}
It is well known that the order of convergence of the strong error of
discretization is $N^{-1/2}$. Indeed, we have (see~\cite{Ka})
\begin{equation}
\forall p\geq1, \exists C<+\infty, \forall N\geq1\qquad
\mathbb{E}^{1/p}
\Bigl[\sup_{t\leq T}|X_t-\bar{X}_t|^p
\Bigr]\leq\frac{C}{\sqrt{N}}.
\end{equation}
See Section~\ref{Sec_res_std} for a more precise statement. This
upper-bound gives the correct order of convergence since according to
Remark 3.6 \cite{kurtz5}, when $\sigma$ and $b$ are continuously
differentiable, $(\sqrt{N}(X_t-\bar{X}_t))_{t\leq T}$ converges in law
as $N$ goes to $\infty$ to some diffusion limit which is nonzero as
soon as $\sigma$ is positive and nonconstant (see also \cite
{kurtz6} and \cite{jac1} where stable convergence is also proved). When
$\sigma$ is constant, then the Euler scheme coincides with the Milstein
scheme, and the strong order of convergence is $N^{-1}$.
On the other hand, the order of convergence of the weak error of
discretization is always $N^{-1}$. For example, according to \cite{tt},
when $\sigma$ and $b$ are $C^\infty$ with bounded derivatives of all
orders and $f\dvtx \mathbb{R}\to\mathbb{R}$ is $C^\infty$ with
polynomial growth together with its derivatives, then for each integer
$L\geq1$, the expansion
\begin{equation}
\mathbb{E} \bigl[f(X_T) \bigr]-\mathbb{E} \bigl[f(
\bar{X}_T) \bigr]=\sum_{l=1}^L
\frac
{a_l}{N^l}+{\mathcal O} \bigl(N^{-(L+1)} \bigr)\label{deverrfaib}
\end{equation}
in powers of $N^{-1}$ holds for the weak error. The bound
$|\mathbb{E}[f(\bar{X}_T)]-\mathbb{E}[f(X_T)]|\leq\frac{C}{N}$ holds
when $\sigma,b$ and $f$ are $C^4$ with the same growth assumptions.
When $f$ is only assumed to be measurable and bounded, it is proved in
\cite{bt1,bt2} that the expansion (\ref{deverrfaib}) still holds for
$L=1$ if $b$ and $\sigma$ are smooth functions satisfying an
hypoellipticity condition. Under uniform ellipticity, Guyon
\cite{gu} even extends this expansion by only assuming that $f$ is a
tempered distribution acting on the densities of both $X_T$ and
$\bar{X}_T$.
In view of financial applications, the weak error analysis gives the
convergence rate to $0$ of the discretization bias introduced when
replacing $X$ by its Euler scheme $\bar{X}$ for the computation of the
price $\mathbb{E}[f(X_T)]$ of a vanilla European option with payoff $f$
and maturity $T$ written on $X$. Let $\mathcal{C}$ denote the space
$C([0,T],\mathbb{R})$ of continuous paths endowed with the sup norm.
When dealing with exotic options with payoff $F\dvtx
\mathcal{C}\to\mathbb{R}$ Lipschitz continuous,
\[
\bigl\vert\mathbb{E} \bigl[F(X) \bigr]-\mathbb{E} \bigl[F(\bar{X})
\bigr]
\bigr\vert\leq\mathbb{E}\bigl|F(X)-F(\bar{X})\bigr|\leq\frac{C}{\sqrt{N}},
\]
where the second inequality follows from the strong error estimate. But
the first inequality is very rough and prevents us from taking
advantage of the cancellations in the mean which occur and permit us to
obtain the upper-bound $\frac{C}{N}$ for vanilla options. The weak
error analysis has been performed for specific path-dependent payoffs,
typically when $F(X)=f(X_T,Y_T)$ with $Y_t$ a function of $(X_s)_{0\leq
s\leq t}$ such that $((X_t,Y_t))_{0\leq t\leq T}$ is a Markov process.
The cases $Y_t=\int_0^tX_s\,ds$ and $Y_t=\max_{0\leq s\leq t}X_s$,
respectively, correspond to Asian \cite{thesestemam} and barrier \cite
{gob1,gob2,gobmen} or lookback options \cite{thess}. But no general
theory has been developped so far to analyze the weak trajectorial
error. The Wasserstein distance between the laws $\mathcal{L}(X)$ and
$\mathcal{L}(\bar{X})$ of $X$ and $\bar{X}$ defined by
\[
\mathcal{W}_1 \bigl(\mathcal{L}(X),\mathcal{L}(\bar{X}) \bigr)=\sup
_{F\dvtx \mathcal{C}\to\mathbb
{R}\dvtx \mathrm{Lip}(F)\leq1}\bigl|\mathbb{E} \bigl[F(\bar{X}) \bigr]-\mathbb{E} \bigl[F(X)
\bigr]\bigr|,
\]
where $\mathrm{Lip}(F)$ denotes the Lipschitz constant of $F$ is the
appropriate measure to deal with the whole class of exotic Lipschitz
payoffs. Notice that this distance has already been used in the context
of discretization schemes for SDEs: in the multidimensional setting, by
a clever rotation of the driving Brownian motion, Cruzeiro, Malliavin
and Thalmaier~\cite{cruz} construct a modified Milstein scheme which
does not involve the simulation of iterated Brownian integrals and with
order of convergence $N^{-1}$ for the Wasserstein distance. A simpler
scheme with the same convergence properties is exhibited in
\cite{js} for usual stochastic volatility models.
The weak and strong error estimations recalled above imply that
\begin{equation}
\exists c, C<+\infty, \forall N\geq1\qquad \frac{c}{N}\leq\mathcal{W}_1
\bigl(\mathcal{L}(X),\mathcal{L}(\bar{X}) \bigr)\leq\frac{C}{\sqrt{N}}.
\label{minmajw1}
\end{equation}
A very nice feature of the Wasserstein distance is its primal
representation in the Kantorovitch duality theory. This representation
is obtained by choosing $p=1$, $E=\mathcal{C}$ and
$(\mu,\nu)=(\mathcal{L}(X),\mathcal{L}(\bar{X}))$ in the general
definition
\begin{equation}
\mathcal{W}_p(\mu,\nu)= \biggl(\inf_{\pi\in\Pi(\mu,\nu)}\int
_{E\times E}|x-y|^p\pi(dx,dy) \biggr)^{1/p},
\label{defwas}
\end{equation}
where $p\in[1,+\infty)$, $(E,|~|)$ is a normed vector space, $\mu$ and
$\nu$ are two probability measures on $E$ endowed with its Borel
sigma-field and the infimum is computed on the set $\Pi(\mu,\nu)$ of
probability measures on $E\times E$ with respective marginals $\mu$ and
$\nu$; see, for instance, Remark 6.5 page 95 \cite{villani}. When one
is able to exhibit some coupling $(Y,\bar{Y})$ with
$Y\stackrel{\mathcal{L}}{=}X$ and
$\bar{Y}\stackrel{\mathcal{L}}{=}\bar{X}$, then the law of
$(Y,\bar{Y})$ belongs\vspace*{1pt} to
$\Pi(\mathcal{L}(X),\mathcal{L}(\bar{X}))$ and necessarily
$\mathcal{W}_p(\mathcal{L}(X),\mathcal{L}(\bar{X}))\leq
\mathbb{E}^{1/p} [\sup_{t\in[0,T]}|Y_t-\bar{Y}_t|^p ]$. For the obvious
coupling $(Y,\bar{Y})=(X,\bar{X})$ obtained by choosing the same
driving Brownian motion for the diffusion and its Euler scheme, one
recovers the upper bound in (\ref{minmajw1}) from the strong error
analysis. The main result of the present paper is the construction of a
better coupling which leads to the upper bound
\[
\forall p\geq1, \forall\varepsilon>0, \exists C<+\infty, \forall N\geq1
\qquad
\mathcal{W}_p \bigl(\mathcal{L}(X),\mathcal{L}(\bar{X}) \bigr)\leq
\frac{C}{N^{2/3-\varepsilon}}
\]
proved in Section~\ref{sec_pathwise} under additional regularity
assumptions on the coefficients and uniform ellipticity. To construct
this coupling, we first obtain in Section~\ref{sec_marginal} a
time-uniform estimation of the Wasserstein distance between the
respective laws $\mathcal{L}(X_t)$ and $\mathcal{L}(\bar{X}_t)$ of
$X_t$ and $\bar{X}_t$,
\[
\forall p\geq1, \exists C<+\infty, \forall N\geq1\qquad
\sup_{t\in[0,T]} \mathcal{W}_p \bigl(\mathcal{L}(X_t),\mathcal{L}(
\bar{X}_t) \bigr)\leq\frac{C\sqrt{\log(N)}}{N}.
\]
Previously, in Section~\ref{Sec_res_std}, we recalled well-known
results concerning the moments and the dependence on the initial
condition of the solution to the SDE~(\ref{sde}) and its Euler scheme.
Also, we make explicit the dependence of the strong error estimations
$\mathbb{E} [\sup_{s\le t}|\bar{X}_s-X_s|^p]$ with respect to
$t\in[0,T]$, which will play a key role in our analysis.
\section{Basic estimates on the SDE and its Euler scheme}\label{Sec_res_std}
We recall some well-known results concerning the flow defined
by~(\ref{sde}) (see, e.g., Karatzas and Shreve~\cite{KS}, page 306) and
its Euler approximation.
\begin{aprop}
Let us denote by $(X^{x}_t)_{t\in[0,T]}$ the solution of (\ref{sde}),
starting from $x\in\mathbb{R}$. One has that for any $p\geq1$, the existence
of a positive constant $C\equiv C(p,T)$ such that
\begin{eqnarray}
\forall x \in\mathbb{R}\qquad
\mathbb{E} \Bigl[\sup_{t\in[0,T]}\bigl|X^{x}_t\bigr|^p\Bigr]&\leq& C\bigl(1+|x|\bigr)^p, \label{momenteds}
\\
\qquad\quad \forall x\in\mathbb{R}, \forall s \leq t \leq T
\qquad \mathbb{E} \Bigl[\sup _{u\in[s,t]}\bigl|X^{x}_{u}-X^{x}_{s}\bigr|^p
\Bigr] &\leq& C\bigl(1+|x|\bigr)^p(t-s)^{p/2},\label{accroisseds}
\\
\forall x, y\in\mathbb{R}\qquad \mathbb{E} \Bigl[\sup_{t\in
[0,T]}\bigl|X^{x}_t-X^{y}_t\bigr|^p
\Bigr] &\leq& C|y-x|^p.\label{cieds}
\end{eqnarray}
\end{aprop}
\begin{aprop}\label{vitfort_prop}
Let $(\bar{X}^{x}_t)_{t\in[0,T]}$
denote the
Euler scheme~(\ref{eul}) starting from~$x$.
For any $p\in[1,\infty)$, there exists a positive constant $C\equiv
C(p,T)$ such that
\begin{eqnarray}
\forall N\geq1, \forall x\in\mathbb{R}\qquad \mathbb{E} \Bigl[\sup
_{t\in[0,T]}\bigl|\bar{X}^{x}_t\bigr|^p
\Bigr]&\leq& C\bigl(1+|x|\bigr)^p,\label{momenteul}
\\
\hspace*{30pt}\forall N\geq1, \forall x\in\mathbb{R}, \forall t\in[0,T]\qquad \mathbb{E} \Bigl[\sup_{r\in[0,t]}\bigl|\bar{X}^{x}_r-X^{x}_r\bigr|^p
\Bigr]&\leq&\frac{C t^{p/2}(1+|x|)^p}{N^{p/2}}.\label{vitfort}
\end{eqnarray}
\end{aprop}
The moment bound~(\ref{momenteul}) for the Euler scheme holds in fact
as soon as the drift and the diffusion coefficients have a sublinear
growth. The strong convergence order is established in
Kanagawa~\cite{Ka} for Lipschitz and bounded coefficients. In fact, it
is straightforward to extend Kanagawa's proof to merely Lipschitz
coefficients by using the estimates~(\ref{momenteds})
and~(\ref{momenteul}) and obtain
\begin{equation}
\label{vitfort2}
\qquad\forall N\geq1, \forall x\in\mathbb{R}, \forall t\in[0,T]\qquad \mathbb{E} \Bigl[\sup_{r\in[0,t]}\bigl|\bar{X}^{x}_r-X^{x}_r\bigr|^p
\Bigr]\leq\frac{C (1+|x|)^p}{N^{p/2}}.
\end{equation}
The estimate~(\ref{vitfort}) precises the dependence on $t$. This
slight improvement will in fact play a crucial role in constructing the
coupling between the diffusion and the Euler scheme. We prove it for
the sake of completeness, even though the arguments are standard.
\begin{pf*}{Proof of~(\ref{vitfort})}
Let $\tau_s=\sup\{t_i, t_i\le s\}$ denote the last discretization time
before~$s$. We have $\bar{X}^{x}_t-X^x_t=\int_0^t
b(\bar{X}^{x}_{\tau_s})-b(X^x_s) \,ds + \int_0^t
\sigma(\bar{X}^{x}_{\tau_s})-\sigma(X^x_s) \,dW_s$. By the Jensen and
Burkholder--Davis--Gundy inequalities,
\begin{eqnarray*}
&& \mathbb{E} \Bigl[ \sup_{r \in[0,t]} \bigl|\bar{X}^{x}_r-X^x_r\bigr|^p
\Bigr]
\\
&&\qquad \le 2^p \biggl( \mathbb{E} \biggl[ \biggl(\int
_0^t \bigl|b \bigl(\bar{X}^{x}_{\tau_s}
\bigr)-b \bigl(X^x_s \bigr)\bigr| \,ds \biggr)^p
\biggr]
\\
&&\hspace*{48pt}{} + C_p\mathbb{E} \biggl[ \biggl(\int_0^t
\bigl(\sigma\bigl(\bar{X}^{x}_{\tau_s} \bigr)-\sigma
\bigl(X^x_s \bigr) \bigr)^2 \,ds
\biggr)^{p/2} \biggr] \biggr)
\\
&&\qquad \le 2^p \biggl( t^{p-1}\int_0^t
\mathbb{E} \bigl[ \bigl|b \bigl(\bar{X}^{x}_{\tau
_s} \bigr)-b
\bigl(X^x_s \bigr)\bigr|^p \bigr] \,ds
\\
&&\hspace*{48pt}{}+ C_pt^{p/2-1}\int_0^t \mathbb{E} \bigl[
\bigl|\sigma\bigl(\bar{X}^{x}_{\tau_s} \bigr)- \sigma\bigl(X^x_s
\bigr)\bigr|^p \bigr]\,ds \biggr).
\end{eqnarray*}
Denoting by $\mathrm{Lip}(\sigma)$ the finite Lipschitz constant of
$\sigma$, we have $|\sigma(\bar{X}^{x}_{\tau_s})-\sigma(X^x_s)|\le
\mathrm{Lip}(\sigma)(|\bar{X}^{x}_{\tau_s}-X^x_{\tau_s}|+|X^x_{\tau_s}-X^x_s|)$.
Thus, (\ref{accroisseds}) and~(\ref{vitfort2}) yield\break
$\mathbb{E}[|\sigma(\bar{X}^{x}_{\tau_s})-\sigma(X^x_s)|^p]\le \frac{C
(1+|x|)^p}{N^{p/2}}, $ and the same bound holds for $b$
replacing~$\sigma$. Since $t^p\leq T^{p/2}t^{p/2}$, we easily conclude.
\end{pf*}
\section{The Wasserstein distance between the marginal laws}\label{sec_marginal}
In this section, we are interested in finding an upper bound for the
Wasserstein distance between the marginal laws of the SDE~(\ref{sde})
and its Euler scheme. It is well known that the optimal coupling
between two one-dimensional random variables is obtained by the inverse
transform sampling. Thus, let $F_t$ and $\bar{F}_t$ denote the
respective cumulative distribution functions of $X_t$ and $\bar{X}_t$.
The $p$-Wasserstein distance between the time-marginals of the
solution\vadjust{\goodbreak}
to the SDE and its Euler scheme is given by (see Theorem~3.1.2
in~\cite{raru})
\begin{equation}
\mathcal{W}_p \bigl(\mathcal{L}(X_t),\mathcal{L}(
\bar{X}_t) \bigr)= \biggl(\int_0^1\bigl|F_t^{-1}(u)-
\bar{F}_t^{-1}(u)\bigr|^p\,du \biggr)^{1/p}.
\label{wpinv}
\end{equation}
Let us state now the main result of this section. We set
\begin{eqnarray*}
C^k_b&=& \bigl\{f\dvtx \mathbb{R} \rightarrow\mathbb{R}\ k \mbox{
times continuously differentiable s.t. }
\\
&&\hspace*{115pt}\bigl\|f^{(i)} \bigr\|_\infty<
\infty, 0\le i\le k \bigr\}.
\end{eqnarray*}
\begin{ahyp}\label{hyp_wass_marginal}
Let $a=\sigma^2$. We assume that $a, b \in C^2_b$, $a''$
is globally \mbox{$\gamma$-}H\"older continuous with $\gamma>0$ and
\[
\exists\underline{a}>0, \forall x\in\mathbb{R}, a(x)\geq\underline
{a} \mbox{ (uniform ellipticity)}.
\]
\end{ahyp}
Since $\sigma$ is Lipschitz continuous, under
Hypothesis~\ref{hyp_wass_marginal}, we have either
$\sigma\equiv\sqrt{a}$ or $\sigma\equiv-\sqrt{a}$. From now on, we
assume without loss of generality that $\sigma\equiv\sqrt{a}$ which is
a $C^2_b$ function bounded from below by the positive constant
$\underline{\sigma}=\sqrt{\underline{a}}$.
\begin{theorem}\label{wasun}
Under Hypothesis~\ref{hyp_wass_marginal}, we have for
any $p\ge1$,
\[
\forall N\geq1\qquad \sup_{t\in[0,T]}\mathcal{W}_p \bigl(
\mathcal{L}(X_t),\mathcal{L}(\bar{X}_t) \bigr)\leq
\frac{C\sqrt{\log(N)}}{N},
\]
where $C$ is a positive constant that only depends on $p$, $T$,
$\underline{a}$ and ($\|a^{(i)}\|_\infty$, $\|b^{(i)}\|_\infty$, $0\le
i\le2$) and does not depend on the initial condition~$x\in\mathbb{R}$.
\end{theorem}
\begin{arem}\label{w1unif}
When $p=1$, the slightly better bound
$\sup_{t\in[0,T]}\mathcal{W}_1(\mathcal{L}(X_t),\allowbreak \mathcal{L}(\bar
{X}_t))\leq\frac{C}{N}$ holds if $\sigma$ is uniformly elliptic,
according to~\cite{thesesbai}, Chapter~3. This is proved in a
multidimensional setting for $C^\infty$ coefficients $\sigma$ and $b$
with bounded derivatives by extending the results of \cite{gu} but can
also be derived from a result of Gobet and Labart~\cite{goblab} only
supposing that $b,\sigma\in C^{3}_b$. Let $p_t(x,y)$ and
$\bar{p}_t(x,y)$ denote, respectively, the density of~$X^{0,x}_t$ and
$\bar{X}^{0,x}_t$. Then Theorem~2.3 in~\cite{goblab} gives the
existence of a constant $c>0$ and a finite nondecreasing function $K$
(depending on the upper bounds of $\sigma$ and $b$ and their
derivatives) such that
\[
\forall(t,x,y) \in(0,T]\times\mathbb{R}^2\qquad
\bigl|p_t(x,y)-\bar{p}_t(x,y)\bigr|\leq\frac{TK(T)}{N t}\exp\biggl(-\frac
{c|x-y|^2}{t}\biggr).
\]
As remarked in~\cite{thesesbai}, Chapter~3, for $f\dvtx \mathbb{R}\to
\mathbb{R}$ a Lipschitz continuous function with Lipschitz constant not
greater than one, one deduces that
\begin{eqnarray*}
\bigl|\mathbb{E} \bigl[f(X_t) \bigr]-\mathbb{E} \bigl[f(
\bar{X}_t) \bigr]\bigr|&=& \biggl\vert\int_{\mathbb
{R}}
\bigl(f(y)-f(x) \bigr) \bigl(p_t(x,y)-\bar{p}_t(x,y)
\bigr)\,dy \biggr\vert
\\
&\leq&\frac
{K(T)T}{N t}\int_{\mathbb{R}}|y-x|\exp\biggl(-
\frac{c|x-y|^2}{t} \biggr)\,dy
\\
&=&\frac{K(T)T}{cN},
\end{eqnarray*}
which gives $\sup_{t\leq
T}\mathcal{W}_1(\mathcal{L}(X_t),\mathcal{L}(\bar{X}_t))\leq
\frac{CK(T)T}{N}$ by the dual formulation of the $1$-Wasserstein
distance.
\end{arem}
Our approach consists of controlling the time evolution of the
Wasserstein distance. To do so, we need to compute the evolution of
both $F_t^{-1}(u)$ and $\bar{F}_t^{-1}(u)$. In the two next
propositions, we derive partial differential equations satisfied by
these functions by integrating in space the Fokker--Planck equations
and then applying the implicit function theorem.
\begin{aprop}\label{propevolftm1}
Assume that
Hypothesis~\ref{hyp_wass_marginal} holds.
Then for any $t\in(0,T]$, the cumulative distribution function
$x\mapsto F_t(x)$ is invertible with inverse denoted by $F_t^{-1}(u)$.
Moreover, the function $(t,u)\mapsto F_t^{-1}(u)$ is $C^{1,2}$ on
$(0,T]\times(0,1)$ and satisfies
\begin{equation}
\partial_t F_t^{-1}(u)=-\frac{1}{2}
\partial_u \biggl(\frac
{a(F_t^{-1}(u))}{\partial_u F_t^{-1}(u)} \biggr)+b \bigl(F_t^{-1}(u)
\bigr).\label{fpinvfr}
\end{equation}
\end{aprop}
\begin{aprop}\label{propevolbarftm1}
Assume that $\sigma$ and $b$ have linear growth $\exists C>0$, $\forall
x\in\mathbb{R}$, $|\sigma(x)|+|b(x)|\leq C(1+|x|)$ and that uniform
ellipticity holds, $\exists\underline{a}>0$, $\forall x\in\mathbb{R}$,
$a(x)\geq\underline{a}$. Then for any $t\in(0,T]$, $\bar{X}_t$ admits a
density $\bar{p}_t(x)$ with respect to the Lebesgue measure and its
cumulative distribution function $x\mapsto\bar{F}_t(x)$ is invertible
with inverse denoted by $\bar{F}_t^{-1}(u)$. Moreover, for each
$k\in\{0,\ldots,N-1\}$, the function $(t,u)\mapsto\bar{F}^{-1}_t(u)$ is
$C^{1,2}$ on $(t_k,t_{k+1}]\times(0,1)$ and, on this set, it is a
classical solution of
\begin{eqnarray}
\label{eqevolbarftm1}
\partial_t \bar{F}_t^{-1}(u)&=&-
\frac{1}{2}\partial_u \biggl(\frac
{\alpha_t(u)}{\partial_u \bar{F}_t^{-1}(u)} \biggr)+
\beta_t(u),
\end{eqnarray}
where $\alpha_t(u)=\mathbb{E}[a(\bar{X}_{t_k})|\bar{X}_t=\bar
{F}_t^{-1}(u)]$ and
$\beta_t(u)=\mathbb{E}[b(\bar{X}_{t_k})|\bar{X}_t=\bar{F}_t^{-1}(u)]$.
\end{aprop}
The proofs of these two propositions are postponed to
Appendix~\ref{App_Sec1}. Let us mention here that
Proposition~\ref{propevolftm1} also holds when $b'$ is only H\"older
continuous: the Lipschitz assumption on~$b'$ is needed later to prove
Theorem~\ref{wasun}. The PDEs (\ref{fpinvfr})~and~(\ref{eqevolbarftm1})
enable us to compute the time derivative of the $p$th power of the
Wasserstein distance (\ref{wpinv}) and prove, again in
Appendix~\ref{App_Sec1} the following key lemma.
\begin{alem}\label{lemmajoderwp}
Under Hypothesis~\ref{hyp_wass_marginal}, for $p\geq2$, the function
$t\mapsto\mathcal{W}_p^p(\mathcal{L}(X_t),\allowbreak
\mathcal{L}(\bar{X}_t))$ is continuous on $[0,T]$, and its first order
distribution\vadjust{\goodbreak} derivative\break
$\partial_t\mathcal{W}_p^p(\mathcal{L}(X_t),\mathcal{L}(\bar{X}_t))$ is
an integrable function on $[0,T]$. Moreover, $dt$ a.e.,
\begin{eqnarray}\label{majoderwp}
&& \partial_t\mathcal{W}_p^p \bigl(\mathcal{L}(X_t),\mathcal{L}(\bar{X}_t) \bigr)\nonumber
\\[-1pt]
&&\qquad \leq C \biggl(\mathcal{W}_p^p \bigl(\mathcal{L}(X_t),
\mathcal{L}(\bar{X}_t) \bigr)
\nonumber\\[-9pt]\\[-9pt]
&&\hspace*{45pt}{} +\int_0^1\bigl|F_t^{-1}(u)-
\bar{F}_t^{-1}(u)\bigr|^{p-1}\bigl|b \bigl(
\bar{F}_t^{-1}(u) \bigr)-\beta_t(u)\bigr|\,du\nonumber
\\[-1pt]
&&\hspace*{45pt}{}+\int_0^1\bigl|F_t^{-1}(u)-
\bar{F}_t^{-1}(u)\bigr|^{p-2} \bigl(a \bigl(
\bar{F}_t^{-1}(u) \bigr)-\alpha_t(u)
\bigr)^2\,du \biggr),\nonumber
\end{eqnarray}
where $C$ is a positive constant that only depends on $p$,
$\underline{a}$, $\|a'\|_\infty$ and $\|b'\|_\infty$.
\end{alem}
The last ingredient of the proof of Theorem~\ref{wasun} is the next
lemma, the proof of which is also postponed in Appendix~\ref{App_Sec1}.
\begin{alem}\label{malcal} Let $\tau_t=\sup\{t_i, t_i\le t\}$ denote
the last discretization time before~$t$. Under
Hypothesis~\ref{hyp_wass_marginal}, we have for all $p\geq1$,
\[
\exists C<+\infty, \forall N\geq1, \forall t\in[0,T]\qquad \mathbb{E} \bigl[
\bigl\vert\mathbb{E} [ W_{t}-W_{\tau_{t}
}|\bar{X}_{t} ]
\bigr\vert^{p} \bigr] \leq C \biggl(\frac{1}{N\vee(N^2t)}
\biggr)^{p/2}.
\]
\end{alem}
\begin{pf*}{Proof of Theorem~\ref{wasun}}
Since
$\mathcal{W}_p(\mathcal{L}(X_t),\mathcal{L}(\bar{X}_t))\le\mathcal
{W}_{p'}(\mathcal{L}(X_t),\mathcal{L}(\bar{X}_t)) $ for $p\le p'$, it
is enough to prove the estimation for $p\geq2$. Therefore we suppose
without loss of generality that $p\ge2$. Let
$\psi_p(t)=\mathcal{W}^2_p(\mathcal{L}(X_t),\mathcal{L}(\bar{X}_t))$
and\looseness=-1
\begin{eqnarray}
\mbox{for any integer } k\geq1\qquad h_k(x)=k^{-2/p}h(kx)\nonumber
\\
\eqntext{\mbox{where }h(x)= \cases{x^{2/p}, &\quad if $x\geq1$,
\vspace*{2pt}\cr
1+\dfrac{2}{p}(x-1), &\quad otherwise.}}
\end{eqnarray}\looseness=0
Since $h_k$ is $C^1$ and nondecreasing, Lemma~\ref{lemmajoderwp} and
H\"older's inequality imply that
\begin{eqnarray*}
&& h_k \bigl(\psi^{p/2}_p(t) \bigr)
\\[-2pt]
&&\qquad = h_k
\bigl(\mathcal{W}_p^p \bigl(\mathcal{L}(X_0),
\mathcal{L}(\bar{X}_0) \bigr) \bigr)+\int_0^th_k'
\bigl(\psi^{p/2}_p(s) \bigr)\partial_s
\mathcal{W}_p^p \bigl(\mathcal{L}(X_s),
\mathcal{L}(\bar{X}_s) \bigr)\,ds
\\[-2pt]
&&\qquad \leq h_k(0)
+C\int_0^th_k'
\bigl(\psi^{p/2}_p(s) \bigr)
\\[-2pt]
&&\hspace*{91pt} {}\times\biggl[\psi ^{p/2}_p(s)
\\[-2pt]
&&\hspace*{107pt}{}
+\psi^{(p-1)/2}_p(s)
\biggl(\int _0^1\bigl|b \bigl(\bar{F}_s^{-1}(u)
\bigr)-\beta_s(u)\bigr|^p\,du \biggr)^{1/p} \nonumber
\\
&&\hspace*{107pt}
{}+\psi^{(p-2)/2}_p(s) \biggl(\int_0^1\bigl|a
\bigl(\bar{F}_s^{-1}(u) \bigr)-\alpha
_s(u)\bigr|^p\,du \biggr)^{2/p} \biggr]\,ds.
\end{eqnarray*}
Since for fixed $x\geq0$, the sequence $(h'_k(x))_k$ is nondecreasing
and converges to $\frac{2}{p}x^{(2/p)-1}$ as $k\to\infty$, one may take
the limit in this inequality thanks to the monotone convergence theorem
and remark that the image of the Lebesgue measure on $[0,1]$ by
$\bar{F}_s^{-1}$ is the distribution of $\bar{X}_s$ to deduce
\begin{eqnarray}\label{pregron}
\psi_p(t)&\leq&\frac{2C}{p}\int_0^t
\psi_p(s)+\psi^{1/2}_p(s)\mathbb{E}
^{1/p} \bigl(\bigl|b(\bar{X}_s)-\mathbb{E} \bigl(b(
\bar{X}_{\tau_s})|\bar{X}_s \bigr)\bigr|^p \bigr)
\nonumber\\[-8pt]\\[-8pt]
&&{} + \mathbb{E}^{2/p} \bigl(\bigl|a(\bar{X}_s)-\mathbb{E} \bigl(a(
\bar{X}_{\tau _s})|\bar{X}_s \bigr)\bigr|^p \bigr) \,ds.\nonumber
\end{eqnarray}
One has
\begin{eqnarray*}
a(\bar{X}_{\tau_s})-a(\bar{X}_s)&=&a'(
\bar{X}_s)\sigma(\bar{X}_{s}) (W_{\tau_s}-W_s)
\\[-1pt]
&&{} -a'(\bar{X}_s) \bigl[ \bigl(\sigma(\bar{X}_{\tau
_s})-\sigma(
\bar{X}_{s}) \bigr) (W_s-W_{\tau_s})+b(
\bar{X}_{\tau
_s}) (s-\tau_s) \bigr]
\\[-2pt]
&&{}+(\bar{X}_{\tau_s}-\bar{X}_s)\int_0^1a'
\bigl(v \bar{X}_{\tau_s}+(1-v)\bar{X}_s
\bigr)-a'( \bar{X}_s)\,dv.
\end{eqnarray*}
Using Jensen's inequality, the boundedness assumptions on $a,b$ and
their derivatives and Lemma \ref{malcal}, one gets
\begin{eqnarray*}
&& \mathbb{E} \bigl(\bigl|a(\bar{X}_s)-
\mathbb{E} \bigl(a( \bar{X}_{\tau_s})|\bar{X}_s
\bigr)\bigr|^p \bigr)
\\
&&\qquad \leq C\mathbb{E} \bigl(\bigl|\sigma a'(
\bar{X}_s)\bigr|^p\bigl| \mathbb{E}\bigl((W_s-W_{\tau_s})|
\bar{X}_s \bigr)\bigr|^p \bigr)
\\
&&\quad\qquad{}+C\mathbb{E} \bigl((s-\tau_s)^p+\bigl| \bigl(\sigma(\bar
{X}_{\tau
_s})-\sigma(\bar{X}_{s}) \bigr) (W_s-W_{\tau_s})\bigr|^p+|
\bar{X}_{\tau
_s}-\bar{X}_s|^{2p} \bigr)
\\
&&\qquad \leq\frac{C}{N^{p/2}\vee(N^ps^{p/2})}.
\end{eqnarray*}
The same bound holds with $a$ replaced by $b$. With (\ref{pregron}) and
Young's inequality, one deduces
\begin{eqnarray*}
\psi_p(t)&\leq& C\int_0^t
\psi_p(s)+\frac{\psi^{1/2}_p(s)}{\sqrt{N}\vee(N\sqrt{s})}+\frac
{1}{N\vee(N^2s)}\,ds
\\
&\leq& C\int _0^t\psi_p(s)+
\frac{1}{N\vee(N^2s)}\,ds.
\end{eqnarray*}
One concludes by Gronwall's lemma.
\end{pf*}
\begin{arem} When $a(x)\equiv a$ is constant, the term $\mathbb
{E}^{2/p}
(|a(\bar{X}_s)-\break\mathbb{E}(a(\bar{X}_{\tau_s})|\bar{X}_s)|^p )$ in
(\ref{pregron}) vanishes and the above reasoning ensures that $\bar
{\psi}_p(t)$ defined as $\sup_{s\in[0,T]}\psi_p(s)$ satisfies
\begin{eqnarray*}
\bar{\psi}_p(t)&\leq& C\int_0^t \bar{\psi}_p(s) \,ds
+C\bar{\psi}^{1/2}_p(t) \int
_0^t\frac{1}{\sqrt{N}\vee(N\sqrt{s})}\,ds
\\
&\le& C\int _0^t\bar{\psi}_p(s)
\,ds + \frac{1}{2}\bar{\psi}_p(t) +\frac{C^2 (T+1)^2}{2N}.
\end{eqnarray*}
By Gronwall's lemma, we recover the estimation
$\sup_{t\in[0,T]}\mathcal{W}_p(\mathcal{L}(X_t),\break \mathcal{L}(\bar
{X}_t))\leq\frac{C}{N}$
which is also a consequence of the strong order of convergence of the
Euler scheme when the diffusion coefficient is constant.
\end{arem}
\section{The Wasserstein distance between the pathwise laws}\label{sec_pathwise}
We now state the main result of the paper.
\begin{ahyp}\label{hyp_wass_pathwise}
We assume that $a \in C^4_b, b \in C^3_b$ and
\[
\exists\underline{a}>0, \forall x\in\mathbb{R}\qquad a(x)\geq\underline{a}
\mbox{ (uniform ellipticity)}.
\]
\end{ahyp}
Clearly, Hypothesis~\ref{hyp_wass_pathwise} implies
Hypothesis~\ref{hyp_wass_marginal}.
\begin{theorem}\label{main_thm}
Under Hypothesis~\ref{hyp_wass_pathwise}, we have
\[
\forall p\geq1, \forall\varepsilon>0, \exists C<+\infty, \forall N\geq1
\qquad \mathcal{W}_p \bigl(\mathcal{L}(X),\mathcal{L}(\bar{X}) \bigr)\leq
\frac{C}{N^{2/3-\varepsilon}}.
\]
\end{theorem}
Before proving the theorem, let us state some of its consequences for
the pricing of lookback options. It is well known (see, e.g.,
\cite{glas} page 367) that if $(U_k)_{0\leq k\leq N-1}$ are independent
random variables uniformly distributed on $[0,1]$ and independent from
the Brownian increments $(W_{t_{k+1}}-W_{t_{k}})_{0\leq k\leq N-1}$
then
$\bar{\hspace*{-1.2pt}\bar{X}}\stackrel{\mathrm{def}}{=}\frac{1}{2}\max_{0\leq
k\leq N-1} (\bar{X}_{t_k}+\bar{X}_{t_{k+1}}+\sqrt{(\bar
{X}_{t_{k+1}}\,{-}\,\bar{X}_{t_k})^2-2\sigma^2(\bar{X}_{t_k})t_1\ln(U_k)} ) $
is such\vspace*{-2pt} that $
(\bar{X}_0,\bar{X}_{t_1},\ldots,\bar{X}_{T},\bar{\hspace*{-1.2pt}\bar{X}}
)\stackrel{\mathcal{L}}{=}(\bar{X}_0,\bar{X}_{t_1},\ldots,\bar
{X}_{T},\max_{t\in [0,T]}\bar{X}_t)$.
\begin{acor}
If $f\dvtx \mathbb{R}^2\to\mathbb{R}$ is Lipschitz continuous, then,
under Hypothesis~\ref{hyp_wass_pathwise},
\begin{eqnarray}\label{vitfaiblook}
&& \forall\varepsilon>0, \exists C<+\infty, \forall N\geq1
\nonumber\\[-6pt]\\[-12pt]
&&\qquad \Bigl\vert \mathbb{E} \Bigl[f \Bigl(X_T,\max_{t\in[0,T]}X_t
\Bigr) \Bigr]-\mathbb{E}
\bigl[f(\bar{X}_T,\bar{\hspace*{-1.2pt}\bar{X}}) \bigr]
\Bigr\vert\leq\frac{C}{N^{2/3-\varepsilon}}.\nonumber
\end{eqnarray}
\end{acor}
To our knowledge, this result appears to be new. Of course, when $f$ is
also differentiable with respect to its second variable, one has
\begin{eqnarray*}
&& \mathbb{E} \Bigl[f \Bigl(X_T,\max_{t\in[0,T]}X_t \Bigr)
\Bigr]
\\
&&\qquad =\mathbb{E} \bigl[f (X_T,x_0 ) \bigr]+
\int_{x_0}^{+\infty}\mathbb{E} \bigl[
\partial_2f(X_T,x)1_{\{\max_{t\in[0,T]}X_t\geq x\}} \bigr]\,dx.
\end{eqnarray*}
One could try to combine the weak error analysis for the first term on
the right-hand side with Theorem 2.3 \cite{gob2} devoted to barrier
options to obtain the order $N^{-1}$ instead on $N^{-2/3+\varepsilon}$
in (\ref{vitfaiblook}). Unfortunately, one cannot succeed for two main
reasons. First, it is not clear whether the estimation in Theorem 2.3
\cite{gob2} is preserved by integration over $[x_0,+\infty)$. More
importantly, for this estimation to hold, a structure condition on the
payoff function implying that $\partial_2f(x,x)=0$ for all $x\geq x_0$
is needed.
\begin{pf*}{Proof of Theorem \ref{main_thm}}
We first deduce from Theorem \ref{wasun} some bound on the Wasserstein
distance between the finite-dimensional marginals of the diffusion $X$
and its Euler scheme $\bar{X}$ on a coarse time-grid. For
$m\in\{1,\ldots,N-1\}$, we set $n=\lfloor N/m\rfloor$ and define
\[
s_l=\frac{lmT}{N}\qquad\mbox{for } l\in\{0,\ldots,n-1\}\mbox{ and } s_n=T.
\]
We will use this coarse time-grid $(s_l)_{1\leq l\leq n}$ to
approximate the supremum norm on $\mathcal{C}$ and therefore we endow
consistently $\mathbb{R}^n$ with the norm $|(x_1,\ldots,\break x_n)|=\max
_{1\leq l\leq n}|x_l|$. Combining the next proposition, the proof of
which is postponed in Appendix~\ref{App_Sec2} with Theorem \ref{wasun},
one obtains that
\begin{equation}
\mathcal{W}_p \bigl(\mathcal{L}(X_{s_1},\ldots,X_{s_n}),\mathcal{L}(\bar{X}_{s_1},\ldots,
\bar{X}_{s_n}) \bigr)\leq\frac{C\sqrt{\log
N}}{m},\label{wasfindim}
\end{equation}
where the constant $C$ does not depend on $(m,N)$.
\begin{aprop}\label{prop_wass_multi}
Let $\mathbb{R}^n$ be endowed
with the norm $|(x_1,\ldots,x_n)|=\max_{1\leq l\leq n}|x_l|$. For any
$p\geq1$, there is a constant $C$ not depending on $n$ such that
\[
\mathcal{W}_p \bigl(\mathcal{L}(X_{s_1},\ldots,X_{s_n}),\mathcal{L}(\bar{X}_{s_1},\ldots,
\bar{X}_{s_n}) \bigr)\leq Cn\sup_{0\leq t\leq T,x\in
\mathbb{R}}
\mathcal{W}_p \bigl(\mathcal{L} \bigl(\bar{X}^{x}_t
\bigr),\mathcal{L} \bigl(X^{x}_t \bigr) \bigr).
\]
\end{aprop}
There is a probability measure
$\pi(dx_1,\ldots,dx_n,d\bar{x}_1,\ldots,d\bar{x}_n)$ in $\Pi
(\mathcal{L}(X_{s_1},\allowbreak\ldots,
X_{s_n}),\mathcal{L}(\bar{X}_{s_1},\ldots,\bar{X}_{s_n}))$ which
attains the Wasserstein distance in the left-hand side of
(\ref{wasfindim}); see, for instance, Theorem 3.3.11 \cite{raru},
according to which $\pi$ is the law of
$(X_{s_1},\ldots,X_{s_n},\xi_{s_1},\ldots,\xi_{s_n})$ with
$(\xi_{s_1},\ldots,\xi_{s_n})\in\partial_{|~|}\varphi
(X_{s_1},\ldots,\break X_{s_n})$ where $\partial_{|~|}\varphi$ is the
subdifferential, for the above defined norm $|~|$ on $\mathbb{R}^n$, of
some \mbox{$|~|$-}convex function $\varphi$. Let
$\tilde{\pi}(x_1,\ldots,x_n,d\bar{x}_1,\ldots,d\bar{x}_n)$ denote a
regular conditional probability of $(\bar{x}_1,\ldots,\bar{x}_n)$ given
$(x_1,\ldots,x_n)$ when $\mathbb{R}^{2n}$ is endowed with $\pi$ and
$(\bar{Y}_{s_1},\ldots,\bar{Y}_{s_n})$ be distributed according to
$\tilde{\pi}(X_{s_1},\ldots,X_{s_n},\break d\bar{x}_1,\ldots,d\bar{x}_n)$. The
vector $(X_{s_1},\ldots,X_{s_n},\bar{Y}_{s_1},\ldots,\bar {Y}_{s_n})$
is distributed according to $\pi$ so that
\begin{eqnarray}\label{xybar}
(\bar{Y}_{s_1},\ldots,\bar{Y}_{s_n})&\stackrel{
\mathcal{L}} {=}&(\bar{X}_{s_1},\ldots,\bar{X}_{s_n})\quad\mbox{and}
\nonumber\\[-8pt]\\[-8pt]
\mathbb{E}^{1/p} \Bigl[\max_{1\leq l\leq n}|X_{s_l}-
\bar{Y}_{s_l}|^p \Bigr]&\leq&\frac{C\sqrt{\log N}}{m}.\nonumber
\end{eqnarray}
Let $p_t(x,y)$ denote the transition density of the SDE~(\ref{sde}) and
$\ell_t(x,y)=\log(p_t(x,y))$. According to Appendix \ref{diff_bridge}
devoted to diffusion bridges, the processes
\[
\biggl(W^l_t=\int
_{s_l}^t \bigl(dW_s-
\sigma(X_s)\partial_x\ell_{s_{l+1}-s}(X_s,X_{s_{l+1}})
\,ds \bigr),t \in[s_l,s_{l+1}) \biggr) _{0\leq l\leq n-1}
\]
are independent Brownian motions independent from
$(X_{s_1},\ldots,X_{s_n})$. We suppose from now on that the vector
$(\bar{Y}_{s_1},\ldots,\bar{Y}_{s_n})$ has been generated independently
from these processes and so will be all the random variables and
processes needed in the remaining of the proof (see in particular the
construction of $\beta$ below). Moreover, again by Appendix
\ref{diff_bridge}, the solution of
\begin{equation}\label{defz}
\cases{
\displaystyle Z^{x,y}_t=x+
\int_{s_l}^t\sigma\bigl(Z^{x,y}_s
\bigr)\,dW^l_s
\vspace*{2pt}\cr
\displaystyle\hspace*{31pt}{} +\int_{s_l}^t \bigl[b \bigl(Z^{x,y}_s \bigr)+\sigma^2 \bigl(Z^{x,y}_s
\bigr)\partial_x\ell _{s_{l+1}-s} \bigl(Z^{x,y}_s,y \bigr) \bigr]\,ds,
\vspace*{2pt}\cr
\hspace*{200pt} t\in[s_l,s_{l+1}),
\vspace*{4pt}\cr Z^{x,y}_{s_{l+1}}=y}
\end{equation}
is distributed according to the conditional law of
$(X_t)_{t\in[s_l,s_{l+1}]}$ given $(X_{s_l},\break X_{s_{l+1}})=(x,y)$ and for
each $l\in\{0,\ldots,n-1\}$, one has $(Z^{X_{s_l},X_{s_{l+1}}}_t)_{t\in
[s_l,s_{l+1}]}=(X_t)_{t\in[s_l,s_{l+1}]}$.
In order to construct a good coupling between $\mathcal{L}(X)$ and
$\mathcal{L}(\bar{X})$, a natural idea would be to extend
$(\bar{Y}_{s_1},\ldots,\bar{Y}_{s_n})$ to a process
$(\bar{Y}_t)_{t\in[0,T]}$ with law $\mathcal{L}(\bar{X})$ by defining
for each $l\in\{0,\ldots,n-1\}$, $(\bar{Y}_t)_{t\in[s_l,s_{l+1}]}$ as
the process obtained by inserting the Brownian motion $W^l$, the
starting point $\bar{Y}_{s_l}$ and the ending point $\bar{Y}_{s_{l+1}}$
in the It\^{o}'s decomposition of the conditional dynamics of
$(\bar{X}_t)_{t\in[s_l,s_{l+1}]}$ given $\bar{X}_{s_l}=x$ and
$\bar{X}_{s_{l+1}}=y$. Unfortunately, even if this Euler scheme bridge
is deduced by a simple transformation of the Brownian bridge on a
single time-step, it becomes a complicated process\vspace*{1pt} when
the difference between the starting and ending times is larger than
$\frac{T}{N}$ because of the lack of Markov property. At the end of the
proof, we will choose the difference $s_{l+1}-s_l$ of order
$\frac{T}{N^{1/3}}$ and, therefore,\vspace*{-2pt} much larger than the
time-step $\frac{T}{N}$. In addition, it is not clear how to compare
the paths of the diffusion bridge and the Euler scheme bridge driven by
the same Brownian motion. That is, why we are going to introduce some
new process $(\tilde{\chi}_t)_{t\in[0,T]}$ such that the comparison
will be performed at the diffusion bridge level, which is not so
easy~yet.
To construct $\tilde{\chi}$, we are going to exhibit a Brownian motion
$(\beta_t)_{t\in[0,T]}$ such that $\bar{Y}_{s_1},\ldots,\bar{Y}_{s_n}$
are the values on the coarse time-grid $(s_l)_{1\leq l\leq n}$ of the
Euler scheme~(\ref{eul}) driven by $\beta$ instead of $W$. The
extension $(\bar{Y}_t)_{t\in[0,T]}$ with law $\mathcal{L}(\bar{X})$ is
then simply defined as the whole Euler scheme driven by $\beta$:
\begin{eqnarray}
\bar{Y}_t=\bar{Y}_{t_k}+\sigma(\bar{Y}_{t_k}) (
\beta_t-\beta_{t_k})+b(\bar{Y}_{t_k})
(t-t_k),\nonumber
\\
\eqntext{ t\in[t_{k},t_{k+1}], 0\le k\le N-1.}
\end{eqnarray}
The construction of $\beta$ is postponed at the end of the
present proof. One then defines
\[
\chi_t=\bar{Y}_{s_l}+\int_{s_{l}}^t
\sigma(\chi_{s})\,d\beta_s+\int_{s_{l}}^tb(
\chi_s)\,ds,\qquad t\in[s_{l},s_{l+1}), 0\le l\le
n-1.
\]
Notice that the process $\chi=(\chi_t)_{t\in[0,T]}$ which evolves
according to the SDE~(\ref{sde}) with $\beta$ replacing $W$ on each
time-interval $[s_l,s_{l+1})$ is c\`adl\`ag: discontinuities may arise
at the points $\{s_{l+1}, 0\leq l\leq n-1\}$. We denote by
$\chi_{s_{l+1}-}$ its left-hand limit at time $s_{l+1}$ and set
$\chi_{T}=\chi_{s_n-}$. The strong error estimation (\ref{vitfort})
will permit us to estimate the difference between the processes
$\bar{Y}$ and $\chi$. For the subsequent choice of $\beta$, we do not
expect the processes $\chi$ and $X$ to be close. Nevertheless, the
process $\tilde{\chi}$ obtained by setting
\[
\forall l\in\{0,\ldots,n-1\}, \forall t\in[s_l,s_{l+1})
\qquad
\tilde{\chi}_t=Z^{\chi_{s_l},\chi
_{s_{l+1}-}}_t\quad\mbox{and}\quad\tilde{\chi}_T=\chi_T,
\]
where $Z^{x,y}$ is defined in (\ref{defz}) is such that
$\mathcal{L}(\tilde{\chi})=\mathcal{L}(\chi)$ by
Propositions~\ref{prop_bridge} and~\ref{prop_bridge2}. On each coarse
time-interval $[s_l,s_{l+1})$ the diffusion bridges associated with $X$
and $\tilde{\chi}$ are driven by the same Brownian motion $W^l$.
Moreover the differences $|X_{s_l}-Y_{s_l}|$ between the starting
points and $|X_{s_{l+1}}-\chi_{s_{l+1}-}|\leq
|X_{s_{l+1}}-Y_{s_{l+1}}|+| Y_{s_{l+1}}-\chi_{s_{l+1}-}|$ between the
ending points is controlled by (\ref{xybar}) and the above mentionned
strong error estimation. That is, why one may expect to obtain a good
estimation of the difference between the processes $X$ and
$\tilde{\chi}$. By the triangle inequality and since
$\mathcal{L}(\bar{X})=\mathcal{L}(\bar{Y})$ and
$\mathcal{L}(\tilde{\chi})=\mathcal{L}(\chi)$,
\begin{eqnarray}\label{triangle}
\qquad\mathcal{W}_p \bigl(\mathcal{L}(\bar{X}),\mathcal{L}(X) \bigr)&\leq&
\mathcal{W}_p \bigl(\mathcal{L}(\bar{X}),\mathcal{L}(\chi) \bigr)+
\mathcal{W}_p \bigl(\mathcal{L}(\chi),\mathcal{L}(X) \bigr)
\nonumber
\\
&\leq&\mathbb{E}^{1/p} \Bigl[\sup_{t\in[0,T]}|
\bar{Y}_t-\chi_t|^p \Bigr]+\mathbb{E}
^{1/p} \Bigl[\sup_{t\in[0,T]}|X_t-\tilde{
\chi}_t|^p \Bigr],
\end{eqnarray}
where, for the definition of
$\mathcal{W}_p(\mathcal{L}(\bar{X}),\mathcal{L}(\chi))$ and
$\mathcal{W}_p(\mathcal{L}(\chi),\mathcal{L}(X))$, the space of
c\`adl\`ag sample-paths from $[0,T]$ to $\mathbb{R}$ is endowed with
the supremum norm. Let us first estimate the first term on the
right-hand side. From~(\ref{vitfort}), we get
\[
\mathbb{E} \Bigl[\sup_{t\in[s_l,s_{l+1})}|\bar{Y}_t-\chi
_t|^{p} \big|\bar{Y}_{s_l} \Bigr]\leq C
\frac{m^{p/2}(1+|\bar{Y}_{s_l}|)^{p}}{N^{p}},
\]
where the constant $C$ does not depend on $(N,m)$. We deduce that
\begin{eqnarray*}
\mathbb{E} \Bigl[\sup_{t\in[0,T]}|\bar{Y}_t-
\chi_t|^{p} \Bigr]
&=&\mathbb{E} \Bigl[\max _{0\leq l\leq n-1}\sup_{t\in[s_l,s_{l+1})}|\bar{Y}_t-\chi
_t|^{p} \Bigr]
\\
&\leq& \sum_{l=0}^{n-1}
\mathbb{E} \Bigl[\mathbb{E} \Bigl[\sup_{t\in
[s_l,s_{l+1})}|
\bar{Y}_t-\chi_t|^{p} \big|\bar{Y}_{s_l}
\Bigr] \Bigr]
\\
&\leq& C\frac{m^{p/2}}{N^{p}}\sum_{l=0}^{n-1}
\mathbb{E} \bigl[\bigl(1+|\bar{Y}_{s_l}|\bigr)^{p} \bigr]
\\
&\leq& C
\frac{m^{p/2-1}}{N^{p-1}},
\end{eqnarray*}
where we used (\ref{momenteul}) for the last inequality. As a
consequence,
\begin{equation}
\label{Wass_chi_se} \mathbb{E}^{1/p} \Bigl[
\sup_{t\in
[0,T]}|\bar{Y}_t-\chi_t|^p
\Bigr] \le C\frac{m^{1/2-1/p}}{N^{1-1/p}}.
\end{equation}
Let us now estimate the second term on the right-hand side of
(\ref{triangle}). By Proposition~\ref{prop_bridge2} and since for
$l\in\{0,\ldots,n-1\}$, $\chi_{s_l}=\bar{Y}_{s_l}$,
\begin{eqnarray*}
\sup_{t\leq T}|X_t-\tilde{\chi}_t|&=&\max _{0\leq l\leq
n-1}\sup_{t\in
[s_l,s_{l+1})}\bigl|Z^{X_{s_l},X_{s_{l+1}}}_t-Z^{\chi_{s_l},\chi
_{s_{l+1}-}}_t\bigr|
\\
&\leq& C\max_{0\leq l\leq n-1}|X_{s_l}-\bar{Y}_{s_l}| \vee|X_{s_{l+1}}-\chi_{s_{l+1}-}|.
\end{eqnarray*}
Since, by the triangle inequality and the continuity of $\bar{Y}$,
\begin{eqnarray*}
|X_{s_{l+1}}-\chi_{s_{l+1}-}|&\leq&|X_{s_{l+1}}-\bar
{Y}_{s_{l+1}}|+|\bar{Y}_{s_{l+1}}-\chi_{s_{l+1}-}|
\\
&\leq&
|X_{s_{l+1}}-\bar{Y}_{s_{l+1}}|+\sup_{t\in[0,T]}|
\bar{Y}_{t}-\chi_{t}|,
\end{eqnarray*}
one deduces that
\[
\sup_{t\leq T}|X_t-\tilde{\chi}_t|\leq C
\Bigl(\max_{1\leq l\leq
n}|X_{s_l}-\bar{Y}_{s_l}|+
\sup_{t\in[0,T]}|\bar{Y}_{t}-\chi_{t}| \Bigr).
\]
Combined with (\ref{xybar}) and (\ref{Wass_chi_se}), this implies
\begin{eqnarray*}
\mathbb{E}^{1/p} \Bigl[\sup_{t\leq T}|X_t- \tilde{\chi}_t|^p
\Bigr]&\leq& C\mathbb{E} ^{1/p} \Bigl[\max_{1\leq l\leq
n}|X_{s_l}-\bar{Y}_{s_l}|^p \Bigr]+C\mathbb{E}^{1/p}
\Bigl[\sup_{t\in[0,T]}| \bar{Y}_{t}-\chi_{t}|^p \Bigr]
\\
&\leq& C \biggl(
\frac{\sqrt{\log N}}{m}+\frac{m^{1/2-1/p}}{N^{1-1/p}} \biggr).
\end{eqnarray*}
Plugging this inequality together with (\ref{Wass_chi_se}) in
(\ref{triangle}), we deduce that
\[
\mathcal{W}_p \bigl(\mathcal{L}(X),\mathcal{L}(\bar{X}) \bigr) \leq
C \biggl(\frac{\sqrt{\log N}}{m}+\frac{m^{1/2-1/p}}{N^{1-1/p}} \biggr)
\]
and conclude by choosing $m=\lfloor N^{2/3} \rfloor$ that for
$p\geq\frac{1}{3\varepsilon}$,
$\mathcal{W}_p(\mathcal{L}(X),\mathcal{L}(\bar{X}))\leq\frac{C}{N^{2/3-\varepsilon}}$.
When $\frac{1}{3\varepsilon}>1$, the conclusion follows for
$p\in[1,\frac{1}{3\varepsilon})$ since
$\mathcal{W}_p(\mathcal{L}(X),\allowbreak\mathcal{L}(\bar{X}))\leq\mathcal
{W}_{1/3\varepsilon}(\mathcal{L}(X),\mathcal{L}(\bar{X}))$.
To complete the proof, we still have to construct the Brownian motion
$\beta$. We first reconstruct on the fine time grid $(t_k)_{1\leq k\leq
N}$ an Euler scheme $(\bar{Y}_{t_k},0\le k\le N)$ interpolating the
values on the coarse grid $(s_l)_{1\leq l\leq n}$. Let us denote by
$\bar{p}(x,y)$ the density of the law
$\mathcal{N}(x+b(x)T/N,\sigma(x)^2T/N)$ of the Euler scheme starting
from~$x$ after one time step~$T/N$. Thanks to the ellipticity
assumption, we have $\bar{p}(x,y)>0$ for any $x,y\in\mathbb{R}$.
Conditionally on $(\bar{Y}_{s_1},\ldots,\bar{Y}_{s_n})$, we generate
independent random vectors
\[
(\bar{Y}_{s_{l-1}+t_1},\ldots,\bar{Y}_{s_{l-1}+t_{m-1}})_{1\leq
l\leq n-1}\quad\mbox{and}\quad (\bar{Y}_{s_{n-1}+t_1},\ldots,\bar{Y}_{t_{N-1}})
\]
with respective densities
\[
\frac{\bar{p}(\bar{Y}_{s_{l-1}},x_1)\bar{p}(x_1,x_2)\cdots\bar
{p}(x_{n-1},\bar{Y}_{s_{l}})
}{\int_{\mathbb{R}^{n-1}}\bar{p}(\bar{Y}_{s_{l-1}},y_1)\bar
{p}(y_1,y_2)\cdots\bar{p}(y_{n-1},\bar{Y}_{s_{l}})\,dy_1\cdots
dy_{n-1}}
\]
and
\[
\frac{\bar{p}(\bar{Y}_{s_{n-1}},x_1)\bar{p}(x_1,x_2)\cdots \bar
{p}(x_{N-1-m(n-1)},\bar{Y}_{s_{n}}) }{\int_{\mathbb
{R}^{N-1-m(n-1)}}\bar{p}(\bar{Y}_{s_{n-1}},y_1)\bar{p}(y_1,y_2)\cdots
\bar{p}(y_{N-1-m(n-1)},\bar{Y}_{s_{n}})\,dy_1\cdots dy_{N-1-m(n-1)}}.
\]
This ensures that $(\bar{Y}_{t_k})_{0\leq k\leq
n}\stackrel{\mathcal{L}}{=}(\bar{X}_{t_k})_{0\leq k\leq n}$. Then we
get, thanks to the ellipticity condition, that
$ (\frac{1}{\sigma(\bar{Y}_{t_{k-1}})}(\bar{Y}_{t_k}-\bar
{Y}_{t_{k-1}}-b(\bar{Y}_{t_{k-1}})) )_{1\le k\le N}$~are
independent centered Gaussian variables with variance~$T/N$. By using
independent Brownian bridges, we can then construct a Brownian motion
$(\beta_t)_{t\in[0,T]}$ such that
\[
\beta_{t_k}-\beta_{t_{k-1}}= \frac{1}{\sigma(\bar{Y}_{t_{k-1}})} \bigl(
\bar{Y}_{t_k}-\bar{Y}_{t_{k-1}}-b(\bar{Y}_{t_{k-1}})
\bigr),
\]
which completes the construction.
\end{pf*}
\section*{Conclusion}
In this paper, we prove that the order of convergence of the
Wasserstein distance $\mathcal{W}_p$ on the space of continuous paths
between the laws of a uniformly elliptic one-dimensional diffusion and
its Euler scheme with $N$-steps is not worse that
$N^{-2/3+\varepsilon}$. In view of a possible extension to
multidimensional settings, two main difficulties have to be resolved.
First, we take advantage of the optimality of the inverse transform
coupling in dimension one to obtain a uniform bound on the Wasserstein
distance between the marginal laws with optimal rate $N^{-1}$ up to a
logarithmic factor. In dimension $d>1$, the optimal coupling between
two probability measures on $\mathbb{R}^d$ is not available, which
makes the estimation of the Wasserstein distance between the marginal
laws much more complicated even if, for $\mathcal{W}_1$, the order
$N^{-1}$ may be deduced from the results of \cite{goblab}; see Remark
\ref{w1unif}. Next, one has to generalize the estimation on diffusion
bridges given by Proposition~\ref{prop_bridge2} which we deduce from
the Lamperti transform in dimension $d=1$.
In the perspective of the multi-level Monte Carlo method introduced by
Giles~\cite{giles}, coupling with order of convergence
$N^{-2/3+\varepsilon}$ the Euler schemes with $N$ and $2N$ steps would
also be of great interest for variance reduction, especially in
multidimensional situations where the Milstein scheme is not feasible;
see \cite{js} for the implementation of this idea in the example of a
discretization scheme devoted to usual stochastic volatility models.
But this does not seem obvious from our nonconstructive coupling
between the Euler scheme and its diffusion limit. For both the
derivation of the order of convergence of the Wasserstein distance on
the path space and the explicitation of the coupling, the limiting step
in our approach is Proposition \ref{prop_wass_multi}. In this
proposition, we bound the dual formulation of the Wasserstein distance
between $n$-dimensional marginals by the Wasserstein distance between
one-dimensional marginals multiplied by $n$.
Even if the order of convergence of the Wasserstein distance on the
path space obtained in the present paper may not be optimal, it
provides the first significant step from the order $N^{1/2}$ obtained
with the trivial coupling where the diffusion and the Euler scheme are
driven by the same Brownian motion.
\begin{appendix}
\section{Proofs of Section~\lowercase{\protect\texorpdfstring{\ref{sec_marginal}}{2}}}\label{App_Sec1}
\begin{pf*}{Proof of Proposition \ref{propevolftm1}}
According to \cite{friedman}, Theorems 5.4 and 4.7, for any
$t\in(0,T]$, the solution $X_t$ of (\ref{sde}) starting from $X_0=x_0$
admits a density $p_t(x)$ w.r.t. the Lebesgue measure on the real line,
the function $(t,x)\mapsto p_t(x)$ is $C^{1,2}$ on
$(0,T]\times\mathbb{R}$, and on this set,\vspace*{-1pt} it is a
classical solution of the Fokker--Planck equation
\begin{equation}
\partial_t p_t(x)=\tfrac{1}{2}\partial
_{xx} \bigl(a(x)p_t(x) \bigr)-\partial_x
\bigl(b(x)p_t(x) \bigr). \label{fp}
\end{equation}
Moreover, the following Gaussian bounds hold:
\begin{eqnarray}\label{gb}
\bigl|p_t(x)\bigr|+\sqrt{t}\bigl|\partial_x p_t(x)\bigr|\leq\frac{C}{\sqrt{t}}e^{-(x-x_0)^2/Ct}
\nonumber\\[-10pt]\\[-10pt]
\eqntext{\exists C>0, \forall t\in(0,T], \forall x\in\mathbb{R}.}
\end{eqnarray}
The partial derivatives $\partial_x F_t(x)=p_t(x)$ and
$\partial_{xx}F_t(x)=\partial_xp_t(x)$ exist and are continuous on
$(0,T]\times\mathbb{R}$. For $0<s<t\leq T$ and $y\leq x$, integrating
(\ref{fp}) over $[s,t]\times[y,x]$, then letting $y\to-\infty$ thanks
to (\ref{gb}), one obtains
$F_t(x)-F_s(x)=\int_{s}^t\frac{1}{2}\partial_{x}(a(x)p_r(x))-b(x)p_r(x)\,dr$.
By continuity of the integrand w.r.t. $(r,x)$ one deduces that the
partial derivative $\partial_tF_t(x)$ exists and is continuous on
$(0,T]\times\mathbb{R}$. So, $(t,x)\mapsto F_t(x)$ is $C^{1,2}$ on
$(0,T]\times\mathbb{R}$ and solves
\begin{equation}
\label{fpfr}\partial_t F_t(x)=\tfrac{1}{2}
\partial_{x} \bigl(a(x)\partial_x F_t(x)
\bigr)-b(x)\partial_x F_t(x).
\end{equation}
According to Aronson \cite{aron}, the density is also bounded from
below by some Gaussian kernel $\exists c>0, \forall
(t,x)\in(0,T]\times\mathbb{R}, |p_t(x)|\geq
\frac{c}{\sqrt{t}}e^{-(x-x_0)^2/ct}$. This enables us to apply the
implicit function theorem to $(t,x,u)\mapsto F_t(x)-u$ to deduce that
the inverse $u\mapsto F_t^{-1}(u)$ of $x\mapsto F_t(x)$ is $C^{1,2}$ in
the variables $(t,u)\in(0,T]\times(0,1)$ and solves
\begin{eqnarray*}
\partial_t F_t^{-1}(u)&=&-\frac{\partial_t F_t}{\partial_x
F_t}
\bigl(F_t^{-1}(u) \bigr)
\\
&=&-\frac{1}{2}\partial_{x} \bigl(a(x)\partial_x
F_t(x) \bigr)\big|_{x=F_t^{-1}(u)}\partial_u
F_t^{-1}(u)+b \bigl(F_t^{-1}(u)
\bigr)
\\
&=&-\frac{1}{2}\partial_{u} \biggl(\frac{a(F_t^{-1}(u))}{\partial_u
F_t^{-1}(u)}
\biggr)+b \bigl(F_t^{-1}(u) \bigr),
\end{eqnarray*}
where we used\vspace*{-1pt} (\ref{fpfr}) for the second equality and
$\partial_u F_t^{-1}(u)=\frac{1}{\partial_xF_t( F_t^{-1}(u))}$ for both
the second and the third equalities.
\end{pf*}
\begin{pf*}{Proof of Proposition \ref{propevolbarftm1}}
For $t\in(0,t_1]$,
$\bar{X_t}$ admits the Gaussian density with mean $x_0+b(x_0)t$ and
variance $a(x_0)t$. By induction on $k$ and independence of
$W_t-W_{t_k}$ and $\bar{X}_{t_k}$ in (\ref{eul}), one checks that for
$k\in\{1,\ldots,\allowbreak n-1\}$, $\bar{X}_{t_k}$ admits a density
$\bar{p}_{t_k}(x)$ and that for $t\in(t_{k},t_{k+1}]$, $(\bar
{X}_{t_k},\bar{X_t})$ admits the density
\[
\rho(t_k,t,y,x)=\bar{p}_{t_k}(y)\frac{\exp({-
{(x-y-b(y)(t-t_k))^2}/{2a(y)(t-t_k)}})}{\sqrt{2\pi a(y)(t-t_k)}}.
\]
The marginal density $\bar{p}_t(x)=\int_\mathbb{R}\bar
{p}_{t_k}(y)\frac{\exp({-({x-y-b(y)(t-t_k)^2})/{2a(y)(t-t_k)}})}{\sqrt{2\pi
a(y)(t-t_k)}}\,dy$ of $\bar{X}_t$ is continuous on
$(t_k,t_{k+1}]\times\mathbb{R}$ by Lebesgue's theorem and positive.
Let $N(x)=\int_{-\infty}^xe^{-y^2/2}\frac{dy}{\sqrt{2\pi}}$ denote the
cumulative distribution function of the standard Gaussian law and
$k\in\{0,\ldots,N-1\}$. Again by the independence structure in
(\ref{eul}), for $(t,x)\in(t_k,t_{k+1}]\times\mathbb{R}$,
$\bar{F}_t(x)=\break \mathbb{E} (N (\frac{x-\bar{X}_{t_k}-b(\bar
{X}_{t_k})(t-t_k)}{\sqrt{a(\bar{X}_{t_k})(t-t_k)}} ) )$. One~has
\begin{eqnarray*}
\partial_t N \biggl(\frac{x-y-b(y)(t-t_k)}{\sqrt{a(y)(t-t_k)}} \biggr
)&=& - \biggl(\frac{x-y-b(y)(t-t_k)}{2\sqrt{2\pi a(y)(t-t_k)^3}}+\frac
{b(y)}{\sqrt{2\pi a(y)(t-t_k)}} \biggr)
\\
&&{}\times \exp\biggl(-\frac
{(x-y-b(y)(t-t_k))^2}{2a(y)(t-t_k)}\biggr).
\end{eqnarray*}
By the growth assumption on $\sigma$ and $b$, one easily checks that
$\forall k\in\{0,\ldots,N\}$, $\mathbb{E}(\bar{X}^2_{t_k})<+\infty$.
With the uniform ellipticity assumption, one deduces by a standard
uniform integrability argument that $\bar{F}_t(x)$ is differentiable
w.r.t. $t$ with partial\vadjust{\goodbreak} derivative
\begin{eqnarray}\label{evolfbart}
\qquad \partial_t\bar{F}_t(x)&=&-\mathbb{E} \biggl[ \biggl(
\frac{x-\bar{X}_{t_k}-b(\bar{X}_{t_k})(t-t_k)}{2\sqrt{2\pi a(\bar
{X}_{t_k})(t-t_k)^3}}+\frac{b(\bar{X}_{t_k})}{\sqrt{2\pi a(\bar
{X}_{t_k})(t-t_k)}} \biggr)
\nonumber\\[-8pt]\\[-8pt]
&&\hspace*{70pt}
{}\times
\exp \biggl(-\frac{(x-\bar{X}_{t_k}-b(\bar
{X}_{t_k})(t-t_k))^2}{2a(\bar{X}_{t_k})(t-t_k)}\biggr) \biggr]\nonumber
\end{eqnarray}
continuous in $(t,x)\in(t_k,t_{k+1}]\times\mathbb{R}$. In the same way,
one checks smoothness of $\bar{F}_t(x)$ in the spatial variable $x$ and
obtains that this function is $C^{1,2}$ on \mbox{$(t_k,t_{k+1}]\times
\mathbb{R}$.}
When $k\geq1$,
\begin{eqnarray*}
&& \mathbb{E} \biggl[b(\bar{X}_{t_k})\frac{\exp (-{(x-\bar
{X}_{t_k}-b(\bar{X}_{t_k})(t-t_k))^2}/{(2a(\bar{X}_{t_k})(t-t_k)}))}{\sqrt{2\pi a(\bar
{X}_{t_k})(t-t_k)}} \biggr]
\\
&&\qquad =\int_{\mathbb{R}}b(y)\rho(t_k,t,y,x)\,dy
\\
&&\qquad =\mathbb{E} \bigl[b(
\bar{X}_{t_k})|\bar{X}_t=x \bigr]\bar{p}_t(x).
\end{eqnarray*}
For $k=0$, even if $(\bar{X}_0,\bar{X}_t)$ has no density, the
equality between the opposite sides of this equation remains true.
Combining Lebesgue's theorem and a similar reasoning, one checks that
\begin{eqnarray*}
&& -\mathbb{E} \biggl[\frac{x-\bar{X}_{t_k}-b(\bar
{X}_{t_k})(t-t_k)}{\sqrt{2\pi a(\bar{X}_{t_k})(t-t_k)^3}}\exp\biggl({-\frac
{(x-\bar{X}_{t_k}-b(\bar
{X}_{t_k})(t-t_k))^2}{2a(\bar{X}_{t_k})(t-t_k)}}\biggr) \biggr]
\\
&&\qquad =\partial
_x\mathbb{E} \biggl[a(\bar{X}_{t_k})\frac{\exp({-{(x-\bar
{X}_{t_k}-b(\bar
{X}_{t_k})(t-t_k))^2}/({2a(\bar{X}_{t_k})(t-t_k)})})}{\sqrt{2\pi a(\bar
{X}_{t_k})(t-t_k)}}
\biggr]
\\
&&\qquad =\partial_x \bigl[\mathbb{E} \bigl(a(\bar{X}_{t_k})|
\bar{X}_t=x \bigr)\bar{p}_t(x) \bigr].
\end{eqnarray*}
With (\ref{evolfbart}), one deduces that
\begin{equation}
\label{gyongyfbart}
\qquad\partial_t\bar{F}_t(x)=
\tfrac{1}{2}\partial_x \bigl(\mathbb{E} \bigl[a(\bar
{X}_{t_k})|\bar{X}_t=x \bigr]\partial_x
\bar{F}_t(x) \bigr)-\mathbb{E} \bigl[b(\bar{X}_{t_k})|
\bar{X}_t=x \bigr]\partial_x\bar{F}_t(x).
\end{equation}
One checks that the function $(t,u)\mapsto\bar{F}_t^{-1}(u)$ is smooth
and satisfies the partial differential equation (\ref{eqevolbarftm1})
by arguments similar to the ones given at the end of the proof of
Proposition \ref{propevolftm1}.
\end{pf*}
\begin{arem}
In the same way, for $k\in\{0,\ldots,N-1\}$, one could prove that on
$(t_k,t_{k+1}]\times\mathbb{R}$, $(t,x)\mapsto\bar{p}_t(x)$ is
$C^{1,2}$ and satisfies the partial differential
\[
\partial_t\bar{p}_t(x)=\tfrac{1}{2}
\partial_{xx} \bigl(\mathbb{E} \bigl[a(\bar{X}_{t_k})|
\bar{X}_t=x \bigr]\bar{p}_t(x) \bigr)-
\partial_x \bigl(\mathbb{E} \bigl[b(\bar{X}_{t_k})|
\bar{X}_t=x \bigr] \bar{p}_t(x) \bigr)\vadjust{\goodbreak}
\]
obtained by spatial derivation of (\ref{gyongyfbart}). This shows that
$(\bar{X}_t)_{t\in[0,T]}$ has the same marginal distributions as the
diffusion process with coefficients given by the above conditional
expectations, which is also a consequence of \cite{gyongy}.
\end{arem}
\begin{pf*}{Proof of Lemma \ref{lemmajoderwp}}
By the continuity of
the paths of $X$ and $\bar{X}$ and the finiteness of
$\mathbb{E} [\sup_{t\leq T}(|X_t|^{p+1}+|\bar
{X}_t|^{p+1}) ]$, one easily checks that
$t\mapsto\mathcal{W}_p^p(\mathcal{L}(X_t),\mathcal{L}(\bar{X}_t))$ is
continuous.
Let $k\in\{0,\ldots,N-1\}$ and $s,t\in(t_k,t_{k+1}]$ with $s\leq t$.
Combining Propositions \ref{propevolftm1}~and~\ref{propevolbarftm1}
with a spatial integration by parts, one obtains for $\varepsilon\in
(0,1/2)$,
\begin{eqnarray}\label{preipp}
\hspace*{12pt}&& \int_\varepsilon^{1-\varepsilon}\bigl|F_t^{-1}(u)-
\bar{F}_t^{-1}(u)\bigr|^p\,du\nonumber
\\[-2pt]
&&\qquad =\int_\varepsilon^{1-\varepsilon}\bigl|F_s^{-1}(u)-\bar
{F}_s^{-1}(u)\bigr|^p\,du\nonumber
\\[-2pt]
&&\quad\qquad{} +p\int_s^t\int_\varepsilon
^{1-\varepsilon}\bigl|F_r^{-1}(u)-\bar{F}_r^{-1}(u)\bigr|^{p-2}
\bigl(F_r^{-1}(u)-\bar{F}_r^{-1}(u)\bigr)\nonumber
\\[-2pt]
&&\hspace*{92pt}
{}\times\bigl(b \bigl(F_r^{-1}(u) \bigr)-
\beta_r(u) \bigr)\,du\,dr\nonumber
\\[-2pt]
&&\quad\qquad{} +\frac{p(p-1)}{2}\int_s^t\int
_\varepsilon^{1-\varepsilon
}\bigl|F_r^{-1}(u)-
\bar{F}_r^{-1}(u)\bigr|^{p-2} \bigl(
\partial_u F_r^{-1}(u)-\partial_u
\bar{F}_r^{-1}(u) \bigr)
\nonumber\\[-9pt]\\[-9pt]
&&\hspace*{128pt}
{}\times \biggl(\frac
{a(F_r^{-1}(u))}{\partial_u F_r^{-1}(u)}-
\frac{\alpha_r(u)}{\partial
_u \bar{F}_r^{-1}(u)} \biggr)\,du\,dr\nonumber
\\[-2pt]
&&\quad\qquad{}+\frac{p}{2}\int_s^t\bigl|F_r^{-1}(1-
\varepsilon)-\bar{F}_r^{-1}(1-\varepsilon)\bigr|^{p-2}
\bigl(F_r^{-1}(1-\varepsilon)-\bar{F}_r^{-1}(1-
\varepsilon) \bigr)\nonumber
\\[-2pt]
&&\hspace*{68pt}
{}\times\biggl(\frac{\alpha_r(1-\varepsilon
)}{\partial_u \bar{F}_r^{-1}(1-\varepsilon)}-\frac
{a(F_r^{-1}(1-\varepsilon))}{\partial_u F_r^{-1}(1-\varepsilon
)} \biggr)\,dr\nonumber
\\[-2pt]
&&\quad\qquad{} -\frac{p}{2}\int_s^t\bigl|F_r^{-1}(
\varepsilon)-\bar{F}_r^{-1}(\varepsilon)\bigr|^{p-2}
\bigl(F_r^{-1}(\varepsilon)-\bar{F}_r^{-1}(
\varepsilon) \bigr)\nonumber
\\[-2pt]
&&\hspace*{68pt}
{}\times \biggl(\frac{\alpha_r(\varepsilon
)}{\partial_u \bar{F}_r^{-1}(\varepsilon)}-\frac
{a(F_r^{-1}(\varepsilon))}{\partial_u F_r^{-1}(\varepsilon)} \biggr)\,dr.\nonumber
\end{eqnarray}
We are now going to take the limit as $\varepsilon\to0$. We will check
at the end of the proof that
\begin{eqnarray}\label{IPP_terms}
&& \lim_{u\rightarrow0^+\ \mathrm{or}\ 1^-}\ \sup
_{r\in[s,t]}\frac{a(F_t^{-1}(u))}{\partial_u
F_t^{-1}(u) } \bigl|F_t^{-1}(u)-
\bar{F}_t^{-1}(u) \bigr|^{p-1}
\nonumber\\[-8pt]\\[-8pt]
&&\quad{} + \sup _{r\in [s,t]}\frac{\alpha_t(u) }{\partial_u \bar{F}_t^{-1}(u) }
\bigl|F_t^{-1}(u)- \bar{F}_t^{-1}(u) \bigr|^{p-1}=0,\nonumber
\end{eqnarray}
which enables us to get rid of the two last boundary terms.\vadjust{\goodbreak}
Combining Young's inequality with the uniform ellipticity assumption
and the positivity of $\partial_uF_t^{-1}(u)$ and $\partial_u
\bar{F}_t^{-1}(u)$, one obtains
\begin{eqnarray*}
&& \bigl(\partial_u F_r^{-1}(u)-
\partial_u \bar{F}_r^{-1}(u) \bigr) \biggl(
\frac{a(F_r^{-1}(u))}{\partial_u F_r^{-1}(u)}-\frac{\alpha_r(u)}{\partial
_u \bar{F}_r^{-1}(u)} \biggr)
\\
&&\qquad = \bigl(a \bigl(F_r^{-1}(u) \bigr)-
\alpha_r(u) \bigr)\frac{\partial_u
F_r^{-1}(u)-\partial_u \bar{F}_r^{-1}(u)}{\partial_u F_r^{-1}(u)\vee
\partial_u \bar{F}_r^{-1}(u)}
\\
&&\quad\qquad{} -a \bigl(F_r^{-1}(u) \bigr)\frac{((\partial_u
\bar{F}_r^{-1}(u)-\partial_u F_r^{-1}(u))^+)^2}{\partial_u
F_r^{-1}(u)\partial_u \bar{F}_r^{-1}(u)}
\\
&&\quad\qquad{}-\alpha_r(u)\frac
{((\partial_u F_r^{-1}(u)-\partial_u \bar
{F}_r^{-1}(u))^+)^2}{\partial_u F_r^{-1}(u)\partial_u \bar
{F}_r^{-1}(u)}
\\
&&\qquad\leq\frac{1}{4\underline{a}} \bigl(a \bigl(F_r^{-1}(u)
\bigr)- \alpha_r(u) \bigr)^2+\underline{a}
\frac{(\partial_u F_r^{-1}(u)-\partial
_u \bar{F}_r^{-1}(u))^2}{(\partial_u F_r^{-1}(u)\vee\partial_u \bar
{F}_r^{-1}(u))^2}
\\
&&\quad\qquad{} - \bigl(a \bigl(F_r^{-1}(u) \bigr)\wedge
\alpha_r(u) \bigr)\frac{(\partial
_u \bar{F}_r^{-1}(u)-\partial_u F_r^{-1}(u))^2}{\partial_u
F_r^{-1}(u)\partial_u \bar{F}_r^{-1}(u)}
\\
&&\qquad\leq\frac{1}{4\underline{a}} \bigl(a \bigl(F_r^{-1}(u)
\bigr)- \alpha_r(u) \bigr)^2.
\end{eqnarray*}
Hence, up to the factor $\frac{p(p-1)}{2}$, the third term on the
right-hand side of (\ref{preipp}) is equal to
\begin{eqnarray*}
&&\int_s^t\int_\varepsilon^{1-\varepsilon}\bigl|F_r^{-1}(u)-
\bar{F}_r^{-1}(u)\bigr|^{p-2}
\\[-2pt]
&&\hspace*{38pt}{}\times \biggl[ \bigl(
\partial_u F_r^{-1}(u)-\partial_u
\bar{F}_r^{-1}(u) \bigr) \biggl(\frac{a(F_r^{-1}(u))}{\partial_u
F_r^{-1}(u)}-
\frac{\alpha_r(u)}{\partial_u \bar{F}_r^{-1}(u)} \biggr)
\\
&&\hspace*{174pt}{} -\frac{ (a(F_r^{-1}(u))-\alpha_r(u) )^2}{4\underline
{a}} \biggr]\,du\,dr
\\[-2pt]
&&\quad{} +\frac{1}{4\underline{a}}\int_s^t
\int_\varepsilon^{1-\varepsilon}\bigl|F_r^{-1}(u)-
\bar{F}_r^{-1}(u)\bigr|^{p-2} \bigl(a
\bigl(F_r^{-1}(u) \bigr)-\alpha_r(u)
\bigr)^2\,du\,dr,
\end{eqnarray*}
where the integrand in the first integral is nonpositive. Since
\begin{eqnarray*}
\hspace*{-5pt}&&\int_s^t\hspace*{-1pt}\int_0^1\bigl|F_r^{-1}(u)-
\bar{F}_r^{-1}(u)\bigr|^{p-2}
\\[-1pt]
\hspace*{-5pt}&&\hspace*{25pt}{} \times\hspace*{-0.3pt} \bigl(\bigl|F_r^{-1\hspace*{-0.3pt}}(u)-\hspace*{-0.2pt}
\bar{F}_r^{-1\hspace*{-0.3pt}}(u)\bigr|\bigl|b \bigl(F_r^{-1\hspace*{-0.3pt}}(u)
\bigr)-\beta_r(u)\bigr|
+ \bigl(a \bigl(F_r^{-1\hspace*{-0.3pt}}(u)
\bigr)-\alpha_r(u) \bigr)^2 \bigr)\,du\,dr
\\[-1pt]
\hspace*{-5pt}&&\qquad \leq2\|b\|_\infty\int_s^t
\mathcal{W}_{p}^{p-1} \bigl(\mathcal{L}(X_r),
\mathcal{L}(\bar{X}_r) \bigr)\,dr
\\[-1pt]
\hspace*{-5pt}&&\quad\qquad{}+4\|a\|^2_\infty
\int_s^t\mathcal{W}_{p}^{p-2}
\bigl(\mathcal{L}(X_r),\mathcal{L}(\bar{X}_r)
\bigr)\,dr<+\infty,
\end{eqnarray*}
one can take the
limit $\varepsilon\to0$ in (\ref{preipp}) using Lebesgue's theorem for
the second term on the right-hand side and combining Lebesgue's theorem
with monotone convergence for the third term to obtain
\begin{eqnarray}\label{vipp}
&& \mathcal{W}_{p}^{p} \bigl(\mathcal{L}(X_t),
\mathcal{L}(\bar{X}_t) \bigr)\nonumber
\\
&&\qquad =\mathcal{W}_{p}^{p}
\bigl(\mathcal{L}(X_s),\mathcal{L}(\bar{X}_s) \bigr)
\nonumber
\\
&&\quad\qquad{}+p\int_s^t\int_0^{1}\bigl|F_r^{-1}(u)-
\bar{F}_r^{-1}(u)\bigr|^{p-2} \bigl(F_r^{-1}(u)-
\bar{F}_r^{-1}(u) \bigr)
\nonumber\\[-8pt]\\[-8pt]
&&\hspace*{82pt}{}\times \bigl(b \bigl(F_r^{-1}(u)
\bigr)-\beta_r(u) \bigr)\,du\,dr
\nonumber
\\
&&\quad\qquad{}+\frac{p(p-1)}{2}\int_s^t\int
_0^{1}\bigl|F_r^{-1}(u)-\bar
{F}_r^{-1}(u)\bigr|^{p-2} \bigl(\partial_u
F_r^{-1}(u)-\partial_u \bar
{F}_r^{-1}(u) \bigr)\nonumber
\\
&&\hspace*{118pt}
{}\times \biggl(\frac{a(F_r^{-1}(u))}{\partial_u
F_r^{-1}(u)}-
\frac{\alpha_r(u)}{\partial_u \bar{F}_r^{-1}(u)} \biggr)\,du\,dr.\nonumber
\end{eqnarray}
The last term which belongs to $[-\infty,+\infty)$ is finite since so
are all the other terms. We deduce the integrability of
\begin{eqnarray*}
(r,u)&\mapsto&\bigl|F_r^{-1}(u)-\bar{F}_r^{-1}(u)\bigr|^{p-2}
\bigl(\partial_u F_r^{-1}(u)-
\partial_u \bar{F}_r^{-1}(u) \bigr)
\\
&&\times{} \biggl(\frac{a(F_r^{-1}(u))}{\partial_u
F_r^{-1}(u)}-\frac{\alpha_r(u)}{\partial_u \bar{F}_r^{-1}(u)} \biggr)
\end{eqnarray*}
on $[s,t]\times(0,1)$. Similar arguments show that the integrability
property and (\ref{vipp}) remain true for $s=t_k$. By summation, they
remain true for $0\leq s\leq t\leq T$. So the integrability holds on
$[0,T]$ for the derivative in the distributional sense
\begin{eqnarray*}
&& \partial_t\mathcal{W}_{p}^{p} \bigl(
\mathcal{L}(X_t),\mathcal{L}(\bar{X}_t) \bigr)
\\
&&\qquad =p\int_0^{1}\bigl|F_t^{-1}(u)-
\bar{F}_t^{-1}(u)\bigr|^{p-2} \bigl(F_t^{-1}(u)-
\bar{F}_t^{-1}(u) \bigr) \bigl(b \bigl(F_t^{-1}(u)
\bigr)-\beta_t(u) \bigr)\,du
\\
&&\quad\qquad{}+\frac{p(p-1)}{2}\int_0^{1}\bigl|F_t^{-1}(u)-
\bar{F}_t^{-1}(u)\bigr|^{p-2} \bigl(
\partial_u F_t^{-1}(u)-\partial_u
\bar{F}_t^{-1}(u) \bigr)
\\
&&\hspace*{103pt}
{}\times \biggl(\frac
{a(F_t^{-1}(u))}{\partial_u F_t^{-1}(u)}-
\frac{\alpha_t(u)}{\partial
_u \bar{F}_t^{-1}(u)} \biggr)\,du
\\
&&\qquad \leq p\int_0^{1}\bigl|F_t^{-1}(u)-
\bar{F}_t^{-1}(u)\bigr|^{p-2}
\\
&&\hspace*{53pt}
{}\times \biggl[
\bigl(F_t^{-1}(u)-\bar{F}_t^{-1}(u)
\bigr) \bigl(b \bigl(F_t^{-1}(u) \bigr)-
\beta_t(u) \bigr)
\\
&&\hspace*{106pt}{} +\frac{(p-1) (a(F_t^{-1}(u))-\alpha_t(u) )^2}{8\underline {a}} \biggr]\,du.
\end{eqnarray*}
Equation (\ref{majoderwp}) follows by remarking that
\begin{eqnarray*}
&& \bigl(a \bigl(F_t^{-1}(u) \bigr)-\alpha_t(u)
\bigr)^2
\\
&&\qquad \leq2 \bigl(\bigl\|a'\bigr\|_\infty
^2\bigl|F_t^{-1}(u)-\bar{F}_t^{-1}(u)\bigr|^2+
\bigl(a \bigl(\bar{F}_t^{-1}(u) \bigr)-
\alpha_t(u) \bigr)^2 \bigr)
\end{eqnarray*}
and using a similar idea for $|b(F_t^{-1}(u))-\beta_t(u)|$.
To prove (\ref{IPP_terms}) for $0<s\leq t\leq T$, we use the Aronson
estimates recalled in the proof of Proposition \ref{propevolftm1} for
$X_t$ and deduced from Theorem 2.1 \cite{lemairemenozzi}, for the Euler
scheme
\begin{eqnarray}\label{aronson}
&& \frac{c}{\sqrt{r}}\exp\biggl(-\frac{(x-x_0)^2}{cr} \biggr)
\nonumber\\[-8pt]\\[-8pt]
&&\qquad \le
p_r(x)\wedge\bar{p}_r(x)\le p_r(x)\vee
\bar{p}_r(x)
\le \frac{C}{\sqrt{r}}\exp\biggl(-\frac{(x-x_0)^2}{C r}\biggr).\nonumber
\end{eqnarray}
Setting $K_1=\frac{c}{\sqrt{t}}$, $c_1=cs/2$, $K_2=\frac{C}{\sqrt{s}}$
and $c_2=Ct/2$, one has
\begin{eqnarray}\label{aronson2}
K_1 \exp\biggl(-\frac{(x-x_0)^2}{2c_1 } \biggr) \le\rho_r(x) \le
K_2 \exp\biggl(-\frac{(x-x_0)^2}{2c_2 } \biggr)
\nonumber\\[-8pt]\\[-8pt]
\eqntext{\forall r\in[s,t], \forall x \in\mathbb{R},}
\end{eqnarray}
where $\rho_r$ denotes either $p_r$ or $\bar{p}_r$. The four limits in
(\ref{IPP_terms}) can be obtained similarly, and we focus on the one of
$\sup_{r\in[s,t]}\frac{a(F_r^{-1}(u))}{\partial_u
F_r^{-1}(u) } |F_r^{-1}(u)-\bar{F}_r^{-1}(u) |^{p-1}$.
Up to modifying $K_1>0$ and decreasing $c_1>0$, we get
from~(\ref{aronson2}) that
\begin{eqnarray}
K_1(x_0-x)\exp\biggl(-\frac{(x-x_0)^2}{2c_1 } \biggr) \le\rho_r(x) \le
K_2 (x_0-x) \exp\biggl(-\frac
{(x-x_0)^2}{2c_2 } \biggr)\nonumber
\\
\eqntext{\forall r\in[s,t], \forall x \le x_0-1,}
\end{eqnarray}
which leads to
\[
\forall x \le x_0-1\qquad K_1c_1\exp\biggl(-
\frac{(x-x_0)^2}{2c_1 } \biggr) \le G_r(x) \le K_2c_2
\exp\biggl(-\frac{(x-x_0)^2}{2c_2 } \biggr),
\]
where $G_r$ denotes either $F_r$ or $\bar{F}_r$. Thus, the inverse
function satisfies
\begin{equation}
x_0- \sqrt{-2c_2 \log\biggl(\frac{u}{K_2c_2}
\biggr)} \le\bar{F}_r^{-1}(u)\le x_0-
\sqrt{-2c_1 \log\biggl(\frac{u}{K_1c_1} \biggr)}\label{continvrep}
\end{equation}
for $u$ small
enough. The two last inequalities imply that when $x\rightarrow-
\infty$,
\[
\forall r\in[s,t]\qquad \bar{F}_r^{-1} \bigl(F_r(x)
\bigr) \ge x_0 - \sqrt{-2 c_2 \biggl[ \log\biggl(
\frac{K_1c_1}{K_2c_2} \biggr) -\frac{(x-x_0)^2}{2c_1} \biggr]}
\]
and $\sup_{r\in[s,t]}|x-\bar{F}_r^{-1}(F_r(x))
|\mathop{=}\limits_{x\rightarrow- \infty}O(x)$. With the boundedness
of $a$\vspace*{-1pt} and (\ref{aronson2}), we easily deduce that
\[
\lim_{x\rightarrow- \infty} \sup_{r\in[s,t]}a(x)
p_r(x) \bigl|x-\bar{F}_r^{-1}
\bigl(F_r(x) \bigr) \bigr|^{p-1}=0.
\]
Since, by (\ref{continvrep}), $\bar{F}_r^{-1}(u)$
converges to $-\infty$ uniformly in $r\in[s,t]$ as $u$ tends to $0$,
we conclude that
\[
\lim_{u\rightarrow0^+}\sup_{r\in[s,t]}\frac
{a(F_r^{-1}(u))}{\partial_u
F_r^{-1}(u) }
\bigl|F_r^{-1}(u)-\bar{F}_r^{-1}(u)
\bigr|^{p-1}=0.
\]\upqed
\end{pf*}
\begin{pf*}{Proof of Lemma \ref{malcal}}
By Jensen's inequality,
\begin{eqnarray*}
\mathbb{E} \bigl[\bigl|\mathbb{E}(W_t-W_{\tau_t}|
\bar{X}_t)\bigr|^p \bigr]&\leq&\mathbb{E} \bigl[|W_t-W_{\tau_t}|^{p}
\bigr]\leq\frac{C}{N^{p/2}}.
\end{eqnarray*}
Let us now check that the left-hand side is also smaller than
$\frac{C}{t^{p/2}N^{p}}$. To do this, we will study
\[
{\mathbb{E}} \bigl[ (W_{t}-W_{\tau_{t}})g(\bar{X}_{t})
\bigr],
\]
where $g$ is any smooth real valued function.
In order to continue, we need to do various estimations on the Euler
scheme and its Malliavin derivative, which we denote by $D_u\bar{X}_t$.
Let $\eta_{t}=\min\{t_{i};t\leq t_{i}\}$ denote the discretization
time just after $t$. We have $D_u\bar{X}_t=0$ for $u>t$, and
\begin{eqnarray}
D_{u}\bar{X}_{t}&=&1_{\{t\leq
\eta_u\}}\sigma(\bar{X}_{\tau_{t}})\nonumber
\\
&&{} +1_{\{t>\eta_u\}} \bigl(1+\sigma'(
\bar{X}_{\tau_{t}}) (W_t-W_{\tau_t})+b'(\bar{X}_{\tau_{t}}) (t-\tau_t) \bigr)D_u
\bar{X}_{\tau_{t}}\nonumber
\\
\eqntext{\mbox{for }u\leq t.}
\end{eqnarray}
Then by induction, one clearly obtains that for $u\le t$,
\begin{eqnarray*}
D_{u}\bar{X}_{t} & =&\sigma(\bar{X}_{\tau_{u}})
\bar{\mathcal{E}}_{u,t},
\\
\bar{\mathcal{E}}_{u,t} & =& \cases{ 1, &\quad if $\tau_{t}
\leq\eta_{u}$,
\vspace*{3pt}
\cr
\bigl( 1+b^{\prime}(
\bar{X}_{\tau_{t}}) (t-\tau_{t})+\sigma^{\prime}(
\bar{X}_{\tau_{t}}) (W_{t}-W_{\tau_{t}}) \bigr), &\quad if $
\eta_{u}=\tau_{t}$,
\vspace*{4pt}
\cr
\displaystyle\prod
_{i=N\eta_{u}/T}^{N\tau_{t}/T-1} \bigl( 1+b^{\prime}(
\bar{X}_{t_{i}}) (t_{i+1}-t_{i})+\sigma^{\prime}(\bar{X}_{t_{i}}) (W_{t_{i+1}}-W_{t_{i}})
\bigr)
\cr
\hspace*{33pt}{}\times\bigl( 1+b^{\prime}(\bar{X}_{\tau_{t}}) (t-\tau_{t})+
\sigma^{\prime}(\bar{X}_{\tau_{t}}) (W_{t}-W_{\tau_{t}})
\bigr),&\quad if $\eta_{u}<\tau_{t}$.}
\end{eqnarray*}
Note that $\bar{\mathcal{E}}$ satisfies the following properties: (1)
$\bar{\mathcal{E}}_{u,t}=\bar{\mathcal{E}}_{\eta(u),t}$ and\break (2)~$\bar{\mathcal{E}}_{t_i,t_j}\bar{\mathcal{E}}_{t_j,t}=\bar{\mathcal
{E}}_{t_i,t}$ for $t_i\le t_j\le t$. We also introduce the process
$\mathcal{E}$ defined by
\[
\mathcal{E}_{u,t}=\exp\biggl( \int_{u}^{t}b^{\prime}(X_{s})-
\frac
{1}{2}
\sigma^{\prime}(X_{s})^{2}\,ds+
\int_{u}^{t}\sigma^{\prime
}(X_{s})\,dW_{s}
\biggr).
\]
The next lemma, the proof of which is postponed at the end of the
present proof states some useful properties of the processes
$\mathcal{E}$ and $\bar{\mathcal{E}}$.
\begin{alem}\label{lemme_majorations} Let us assume that $b,\sigma\in
C^2_b$. Then we have
\begin{eqnarray}
\sup_{0\leq s\leq t \le T}{\mathbb{E}} \bigl[
\mathcal{E}_{s,t}^{-p} \bigr]+{\mathbb{E}} \bigl[
\mathcal{E}_{s,t}^{p} \bigr] &\leq& C, \label{eq:propE}
\\
\sup_{0\leq s\leq t \le T}{\mathbb{E}} \bigl[ \bar{\mathcal{E}}_{s,t}^{p}
\bigr] &\leq& C,\label{eqA.13}
\\
\sup_{0\leq s,u\leq t \le T}{\mathbb{E}} \bigl[ |
D_u\bar{\mathcal{E}}_{s,t}|^p+|
D_u \mathcal{E}_{s,t}|^p \bigr] &\leq& C,
\label{eq:A13}
\\
\sup_{0\leq t \le T}{\mathbb{E}} \bigl[ \vert\mathcal
{E}_{0,t}-\bar{\mathcal{E}}_{0,t}\vert
^{p} \bigr] &\leq&\frac
{C}{N^{p/2}}, \label{vitesse_forte}
\end{eqnarray}
where $C$ is a positive constant depending only on $p$ and $T$.
\end{alem}
We next define the localization given by
\[
\psi=\varphi\bigl( \mathcal{E}_{0,t}^{-1} (
\mathcal{E}_{0,t}
-\bar{\mathcal{E}}_{0,t} ) \bigr).
\]
Here $\varphi\dvtx \mathbb{R\rightarrow}[0,1]$ is a~$C^\infty$
symmetric function so that
\[
\varphi(x)=\cases{ 0, &\quad if $|x|>\frac{1}{2}$, \vspace*{2pt}
\cr
1, &
\quad if $|x|<\frac{1}{4}$.}
\]
One has
\begin{eqnarray*}
\mathbb{E} \bigl[ (W_{t}-W_{\tau_{t}})g(\bar{X}_{t})
\bigr] &=&\mathbb{E} \bigl[ (W_{t}-W_{\tau_{t}})g(
\bar{X}_{t})\psi\bigr]+\mathbb{E} \bigl[ (W_{t}-W_{\tau_{t}})g(
\bar{X}_{t}) (1-\psi) \bigr]
\\
&=&\int_{\tau_t}^t\mathbb{E} \bigl[\psi
g'(\bar{X}_{t})D_u\bar{X}_{t}
\bigr] \,du+\mathbb{E} \biggl[g(\bar{X}_{t})\int_{\tau_t}^tD_u
\psi \,du \biggr]
\\
&&{}+\mathbb{E} \bigl[ (W_{t}-W_{\tau_{t}})g(
\bar{X}_{t}) (1-\psi) \bigr],
\end{eqnarray*}
where the second equality follows from the duality formula; see, for
example, Definition 1.3.1 in \cite{N}. Since for $\tau_{t}\leq u\leq t$
\begin{eqnarray*}
{\mathbb{E}} \bigl[ \psi g^{\prime}(\bar{X}_{t})D_{u}
\bar{X}_{t} \bigr] & =& {\mathbb{E}} \bigl[ \psi g^{\prime}(
\bar{X}_{t})\sigma(\bar{X}_{\tau_{t}
}) \bigr]
\\
&=&t^{-1}
\mathbb{E} \biggl[\int_0^t D_sg(
\bar{X}_{t})\frac
{\psi
\sigma(\bar{X}_{\tau_{t}
})}{D_s\bar{X}_t}\,ds \biggr]
\\
& =&t^{-1}{\mathbb{E}} \biggl[ g(\bar{X}_{t})\int
_{0}^{t}\psi\sigma(\bar{X}_{\tau_{t}})
\sigma^{-1} ( \bar{X}_{\tau_{s}} ) \bar{\mathcal{E}
}_{s,t}^{-1}\delta W_{s} \biggr],
\end{eqnarray*}
one deduces
\begin{eqnarray}
\qquad\qquad\mathbb{E} [ W_{t}-W_{\tau_{t}}\vert\bar{X}_{t} ]
&=&t^{-1}\int_{\tau_{t}}^{t} \mathbb{E}
\biggl[ \int_{0}^{t}\psi\sigma(
\bar{X}_{\tau_{t}})\sigma^{-1} ( \bar{X}_{\tau
_{s}} )\bar{\mathcal{E}}_{s,t}^{-1}\delta W_{s}\big|
\bar{X}_{t} \biggr] \,du
\nonumber\\[-8pt]\label{espcond}\\[-8pt]
&&{}+ \mathbb{E} \biggl[\int_{\tau_t}^t
D_u\psi \,du \big| \bar{X}_{t} \biggr]+\mathbb{E} \bigl[ (
W_{t}-W_{\tau_{t}} ) (1-\psi)\vert\bar{X}_{t}
\bigr].\nonumber
\end{eqnarray}
Here $\delta W$ denotes the Skorohod integral. In order to obtain the
conclusion of the lemma, we need to bound the $L^p$-norm of each term
on the right-hand side of~(\ref{espcond}). In particular, we will use
the following estimate (which also proves the existence of the Skorohod
integral on the left-hand side below) which can be found in Proposition
1.5.4 in \cite{N}:
\begin{equation}
\label{controle_normep} \biggl\Vert\int_{0}^{t}
\psi\sigma(\bar{X}_{\tau_{t}})\sigma^{-1} ( \bar{X}_{\tau_{s}}
) \bar{\mathcal{E}}_{s,t}^{-1}\delta W_{s} \biggr
\Vert_{p}\leq C(p) \bigl\Vert\psi\sigma(\bar{X}_{\tau_{t}})
\sigma^{-1} (\bar{X}_{\tau_{\cdot}} )\bar{\mathcal{E}}_{\cdot,t}^{-1}
\bigr\Vert_{1,p},\hspace*{-35pt}
\end{equation}
where $\|F_\cdot\|_{1,p}^p =\mathbb{E} [ (\int_0^t F_s^2 \,ds )^{p/2}+
(\int_0^t \int_0^t (D_uF_s)^2 \,ds\,du )^{p/2} ]$. By Jensen's
inequality for $p\ge2$, we have
\begin{equation}
\label{upper_bound_1p} \qquad\|F_\cdot
\|_{1,p}^p \le t^{p/2-1} \int_0^t
\mathbb{E} \bigl[|F_s|^p \bigr] \,ds + t^{p-2}
\int_0^t\int_0^t
\mathbb{E} \bigl[|D_uF_s|^p \bigr]\,ds\,du
\end{equation}
and we will use this inequality to upper bound~(\ref{controle_normep}).
When $1\le p\le2$, we will use alternatively the following upper bound
$\|F_\cdot\|_{1,p}^p \le( \int_0^t \mathbb{E}[F_s^2] \,ds )^{p/2}+
(\int_0^t \int_0^t \mathbb{E}[(D_uF_s)^2] \,ds\,du )^{p/2}$ that comes
from H\"older's inequality.
For $\psi>0$, we have $\mathcal{E}_{0,t}^{-1} ( \mathcal
{E}_{0,t}
-\bar{\mathcal{E}}_{0,t} )\leq\frac{1}{2}$ so that
$\bar{\mathcal{E}}_{0,t}\geq\frac{1}{2}\mathcal{{E}}_{0,t}>0$. From
Hypothesis~\ref{hyp_wass_pathwise}, there are constants
$0<\underline{\sigma}\le\bar{\sigma}<\infty$ such that
$0<\underline{\sigma}\leq\sigma\leq\bar{\sigma}$, and one has
\begin{eqnarray*}
&& \int_{0}^{t}{\mathbb{E}} \bigl[ \bigl( \psi
\sigma(\bar{X}_{\tau_{t}} )\sigma^{-1} ( \bar{X}_{\tau_{s}} )
\bar{\mathcal{E}}_{s,t} ^{-1} \bigr) ^{p} \bigr]\,ds
\\
&&\qquad \leq \biggl(\frac{\bar{\sigma
}}{\underline{\sigma}} \biggr)^p\int_{0}^{t}{
\mathbb{E}} \bigl[ \psi^{p}\bar{\mathcal{E}}_{0,t}^{-p}
\bar{\mathcal{E}}_{0,\eta
(s)}^{p} \bigr]\,ds
\\
&&\qquad \leq \biggl(\frac{2\bar{\sigma}}{\underline{\sigma}} \biggr)^p\sqrt
{\mathbb{E}
\bigl[\mathcal{{E}}_{0,t}^{-2p} \bigr]} \int
_0^t \sqrt{\mathbb{E} \bigl[|\bar{\mathcal{
E}}_{0,\eta
(s)}|^{2p} \bigr]}\,ds \leq C t,
\end{eqnarray*}
by using the estimates~(\ref{eq:propE}) and (\ref{eqA.13}).
Next, we focus on getting an upper bound for
\begin{equation}
\int_0^t \int_{0}^{t}{
\mathbb{E}} \bigl[ \bigl\vert D_{u} \bigl( \psi\sigma(
\bar{X}
_{\tau_{t}})\sigma^{-1} ( \bar{X}_{\tau_{s}}
) \bar{\mathcal{E}
}_{s,t}^{-1} \bigr) \bigr
\vert^{p} \bigr] \,ds \,du.\label{eq:Dloc}
\end{equation}
To do so, we compute the derivative using basic derivation rules, which
gives
\begin{eqnarray}\label{eq:ft}
&& D_{u} \bigl( \psi\sigma(\bar{X}_{\tau_{t}})
\sigma^{-1} ( \bar{X}
_{\tau_{s}} ) \bar{\mathcal{
E}}_{s,t}^{-1} \bigr)\nonumber
\\
&&\qquad =D_u\psi\sigma(
\bar{X}_{\tau_{t}})\sigma^{-1} ( \bar{X}_{\tau
_{s}} )\bar{\mathcal{E}}_{s,t}^{-1}+\psi\sigma^{\prime}(
\bar{X}_{\tau_{t}})D_{u}\bar{X}_{\tau_{t}}
\sigma^{-1} ( \bar{X}_{\tau_{s}} ) \bar
{\mathcal{E}}_{s,t}^{-1}
\nonumber\\[-8pt]\\[-8pt]
&&\quad\qquad{}-\psi\sigma(\bar{X}_{\tau_{t}}) \sigma^{-2}\sigma'
( \bar{X}_{\tau_{s}} )\sigma(\bar{X}_{\tau_u}){\mathcal{
\bar E}} _{u,\tau_s}\bar{\mathcal{E}}_{s,t}^{-1}
\mathbf{1}_{u\le\tau_s}\nonumber
\\
&&\quad\qquad{}-\psi\sigma(\bar{X}_{\tau_{u}})\sigma^{-1} (
\bar{X}_{\tau_{s}} ) \bar{\mathcal{E}}_{s,t}^{-2}D_{u}
\bar{\mathcal{E}}_{s,t}.\nonumber
\end{eqnarray}
One has then to get an upper bound for the $L^p$-norm of each term. As
many of the arguments are repetitive, we show the reader only some of
the arguments that are involved. Let us start with the first term. We
have
\begin{eqnarray*}
D_u\psi&=&\varphi^{\prime} \bigl( \mathcal{E}_{0,t}^{-1}
( \mathcal{E}
_{0,t}-\bar{\mathcal{E}}_{0,t} )
\bigr) D_{u} \bigl[ \mathcal{E}
_{0,t}^{-1}
( \mathcal{E}_{0,t}-\bar{\mathcal{E}}_{0,t} ) \bigr]
\end{eqnarray*}
and $D_{u} [ \mathcal{E}
_{0,t}^{-1} ( \mathcal{E}_{0,t}-\bar{\mathcal{E}}_{0,t} ) ]=
\mathcal{E}
_{0,t}^{-2}D_{u}\mathcal{E}
_{0,t}\bar{\mathcal{E}}_{0,t}
-\mathcal{E}
_{0,t}^{-1}D_u\bar{\mathcal{E}}_{0,t} $. From the estimates
in~(\ref{eq:propE}), (\ref{eqA.13}) and~(\ref{eq:A13}), we obtain
\begin{equation}
\label{eq:Dest} \sup_{u\in[0,t]}\Vert D_u\psi
\Vert_{p}\leq\bigl\|\varphi^{\prime}\bigr\|_\infty C(p).
\end{equation}
Since $\bar{\mathcal{E}}_{s,t}^{-1}=\bar{\mathcal{E}}_{0,\eta(s)}
\bar{\mathcal{E}}_{0,t}^{-1}$ and $\bar{\mathcal{E}}_{0,t}\geq
\frac{1}{2}\mathcal{{E}}_{0,t}>0$ if $\varphi^{\prime} (
\mathcal{E}_{0,t}^{-1} ( \mathcal{E}
_{0,t}-\bar{\mathcal{E}}_{0,t} ) ) \neq0$, we have
\[
\mathbb{E} \bigl[ \bigl\vert D_u\psi\sigma(
\bar{X}_{\tau_{t}})\sigma^{-1} ( \bar{X}_{\tau
_{s}} )
\bar{\mathcal{E}}_{s,t}^{-1} \bigr\vert^p
\bigr] \le\biggl( \frac{2\bar{\sigma}}{\underline{\sigma}} \biggr)^p
\Vert
D_u\psi\Vert_{2p}^p \mathbb{E} \bigl[
\bigl\vert\mathcal{{E}}_{0,t}^{-1}\bar{\mathcal{
E}}_{0,\eta(s)} \bigr\vert^{2p} \bigr]^{1/2}.
\]
Similar bounds hold for the three other terms. Note that the highest
re\-quirements on the derivatives of~$b$ and~$\sigma$ will come from the
terms involving $D_u\bar{\mathcal{E}}$ in~(\ref{eq:ft}). Gathering all
the upper bounds,\vspace*{-2pt} we get that\break $\Vert
\psi\sigma(\bar{X}_{\tau_{t}})\sigma^{-1} (\bar {X}_{\tau_{\cdot}}
)\bar{\mathcal{E}}_{\cdot,t}^{-1}\Vert _{1,p}^p \le C(t^{p/2}+t^p)
\le Ct^{p/2}$ since $0\le t\le T$. From~(\ref{controle_normep}), we
finally obtain
\[
\biggl\Vert\int_{0}^{t}\psi\sigma(
\bar{X}_{\tau_{u}})\sigma^{-1} ( \bar{X}_{\tau_{s}} )
\bar{\mathcal{E}}_{s,t}^{-1}\delta W_{s} \biggr
\Vert_{p}\leq C(p)t^{1/2}.
\]
We are now in position to conclude. Using Jensen's inequality, the
results (\ref{eq:propE}), (\ref{vitesse_forte}), (\ref{espcond}),
(\ref{eq:Dest}) and the definition of $\varphi$ together with
Chebyshev's inequality, we have for any $k>0$ that
\begin{eqnarray*}
&& \mathbb{E} \bigl[ \bigl\vert\mathbb{E} \bigl[ W_{t}-W_{\tau_{t}}
|\bar{X}_{t} \bigr] \bigr|^{p} \bigr]
\\[-2pt]
&&\qquad \leq C \biggl( t^{-p}(t-\tau_{t})^{p}
\biggl\Vert\int_{0}^{t}\psi\sigma(
\bar{X}_{\tau_{t}}
)\sigma^{-1} ( \bar{X}_{\tau_{s}}
) \bar{\mathcal{E}}_{s,t}^{-1}\delta
W_{s} \biggr\Vert_{p}^p
\\[-2pt]
&&\hspace*{45pt}{} +(t-\tau_{t})^{p-1}\int
_{\tau_{t}}^{t}\Vert D_u\psi\Vert
_{p}^p\,du
\\[-2pt]
&&\hspace*{45pt}{}+\sqrt{\mathbb{E} \bigl(|W_t-W_{\tau_t}|^{2p}
\bigr)} 4^{k/2} \bigl(\mathbb{E} \bigl(|\mathcal{{E}}_{0,t}-
\bar{\mathcal{E}}_{0,t}|^{2k} \bigr)\mathbb{E} \bigl(\mathcal
{{E}}_{0,t}^{-2k} \bigr) \bigr)^{1/4} \biggr)
\\[-2pt]
&&\qquad \leq C \biggl( t^{-p/2}(t-\tau_{t})^{p}+(t-\tau
_{t})^{p}+ \biggl(\frac{1}{N} \biggr)^{ ( 2p+k ) /4}
\biggr)
\\[-2pt]
&&\qquad \leq C \biggl( \frac{1}{t^{p/2}N^{p}}+\frac{1}{N^{p/2+k/4}} \biggr).
\end{eqnarray*}\upqed
\end{pf*}\eject
\begin{pf*}{Proof of Lemma~\ref{lemme_majorations}}
The upper bounds~(\ref{eq:propE}) and (\ref{eqA.13}) on $\mathcal{{E}}$ and $\bar{\mathcal
{E}}$ are obvious since $b'$ and $\sigma'$ are bounded. Now, let us
remark that $\bar{\mathcal{E}}$ and $\mathcal{{E}}$ satisfy
\begin{eqnarray*}
\mathcal{{E}}_{u,t}&=&1+\int_u^t
\sigma' ({X}_{s})\mathcal{{E}}_{u,s}
\,dW_s+\int_u^tb'
({X}_{s})\mathcal{{E}}_{u,s} \,ds,
\\
\bar{\mathcal{E}}_{\eta_u,t}&=&1+\int_{\eta_u}^t
\sigma' (\bar{X}_{\tau_s})\bar{\mathcal{E}}_{\eta_u,\tau_s}
\,dW_s+\int_{\eta_u}^tb' (
\bar{X}_{\tau_s})\bar{\mathcal{E}}_{\eta
_u,\tau_s} \,ds.
\end{eqnarray*}
Thus, (\ref{vitesse_forte}) can be easily obtained by noticing that
$(\bar{X}_t,\bar{\mathcal{E}}_{0,t})$ is the Euler scheme for the SDE
$(X_t,\mathcal{E}_{0,t})$ which has Lipschitz coefficients, and by
using the strong convergence order of $1/2$; see, for
example,~\cite{Ka}.\vadjust{\goodbreak}
The estimate (\ref{eq:A13}) on $D_u{\mathcal{E}}$ is given, for
example, by Theorem 2.2.1 in \cite{N}. On the other hand, we have for
$\eta(s)\le u\le t$
\begin{eqnarray*}
D_u\bar{\mathcal{E}}_{\eta_s,t}&=&\sigma'(
\bar{X}_{\tau_u}) \bar{\mathcal{E}}_{\eta_s,\tau_u}
\\
&&{} +\int _{\eta_u}^t \bigl[ \sigma''(
\bar{X}_{\tau_r}) \sigma(\bar{X}_{\tau_u}) \bar{
\mathcal{E}}_{\eta_u,\tau_r} \bar{\mathcal{E}}_{\eta_s,\tau_r} +
\sigma'(\bar{X}_{\tau_r}) D_u\bar{
\mathcal{E}}_{\eta_s,\tau_r} \bigr] \,dW_r
\\
&&{}+\int_{\eta_u}^t \bigl[ b''(
\bar{X}_{\tau_r}) \sigma(\bar{X}_{\tau_u}) \bar{
\mathcal{E}}_{\eta_u,\tau_r} \bar{\mathcal{E}}_{\eta_s,\tau_r} +b'(
\bar{X}_{\tau_r}) D_u\bar{\mathcal{E}}_{\eta_s,\tau_r}
\bigr]\,dr.
\end{eqnarray*}
In order to obtain a $L^p(\Omega)$ estimate, we then
use~(\ref{eqA.13}), $b,\sigma\in C^2_b$ and Gronwall's lemma.
\end{pf*}\vspace*{-15pt}
\section{Proofs of Section~\lowercase{\protect\texorpdfstring{\ref{sec_pathwise}}{3}}}\vspace*{-5pt}\label{App_Sec2}
\begin{pf*}{Proof of Proposition \ref{prop_wass_multi}}
We use the dual representation of the Wasserstein distance
(\ref{defwas}) deduced from Kantorovitch duality theorem (see, e.g.,
Theorem 5.10, page 58 \cite{villani}),
\[
\mathcal{W}^p_p(\mu,\nu)=\sup_{\phi\in L^1(\nu)}
\biggl(\int_E\tilde{\phi}(x)\mu(dx)-\int
_E\phi(x)\nu(dx) \biggr),
\]
where $\tilde{\phi}(x)=\inf_{y\in E} (\phi(y)+|y-x|^p )$.
We also denote by $(X^{s,x}_t)_{t\in[s,T]}$ the solution to (\ref{sde})
starting from $x\in\mathbb{R}$ at time $s\in[0,T]$ and by
$(\bar{X}^{t_j,x}_t)_{t\in[t_j,T]}$ the Euler scheme starting from $x$
at time $t_j$ with $j\in\{0,\ldots,N\}$. It is enough to check that
\begin{eqnarray*}
w_k&\stackrel{\mathrm{def}}{=}&\mathcal{W}_p \bigl(\mathcal{L}
\bigl(\bar{X}_{s_1},\ldots,\bar{X}_{s_k},X^{s_k,\bar
{X}_{s_k}}_{s_{k+1}},\ldots,X^{s_k,\bar{X}_{s_k}}_{s_{n}} \bigr),
\\
&&\hspace*{21pt}
\mathcal{L} \bigl(
\bar{X}_{s_1},\ldots,\bar{X}_{s_{k-1}},X^{s_{k-1},\bar
{X}_{s_{k-1}}}_{s_{k}},\ldots,X^{s_{k-1},\bar{X}_{s_{k-1}}}_{s_{n}} \bigr) \bigr)
\end{eqnarray*}
is smaller\vspace*{1pt} than $C\sup_{0\leq t\leq
T,x\in\mathbb{R}}\mathcal{W}_p(\mathcal{L}(\bar{X}^{x}_t),\mathcal{L}(X^{x}_t))$
since $\mathcal{W}_p(\mathcal{L}(\bar{X}_{s_1},\ldots,\bar{X}_{s_n}),\allowbreak \mathcal
{L}(X_{s_1},\ldots,X_{s_n}))\leq\sum_{k=1}^nw_k$. For $f\dvtx
\mathbb{R}^n\rightarrow\mathbb{R}$ a bounded measurable function and
\[
\tilde{f}(x_1,\ldots,x_n)=\inf_{(y_1,\ldots,y_n)\in\mathbb{R}^n}
\Bigl\{ f(y_1,\ldots,y_n)+\max_{1\leq
j\leq n}|y_j-x_j|^p
\Bigr\},
\]
we set
$f_k(x_1,\ldots,x_k)=\mathbb{E}(f(x_1,\ldots,x_k,X^{s_k,x_k}_{s_{k+1}},\ldots,X^{s_k,x_k}_{s_n}))$.
First choosing
\[
(y_1,\ldots,y_{k-1},y_{k+1},\ldots,y_n)= \bigl(\bar{X}_{s_1},\ldots,\bar
{X}_{s_{k-1}},X^{s_k,y_k}_{s_{k+1}},\ldots,X^{s_k,y_k}_{s_{n}}
\bigr),
\]
then conditioning to $\sigma(W_s,s\leq s_{k})$ and using (\ref{cieds}),
next conditioning to $\sigma(W_s,s\leq s_{k-1})$ and using the dual
formulation of the Wasserstein distance, one gets
\begin{eqnarray*}
&&\mathbb{E} \bigl(\tilde{f} \bigl(\bar{X}_{s_1},\ldots,\bar
{X}_{s_k},X^{s_k,\bar
{X}_{s_k}}_{s_{k+1}},\ldots,X^{s_k,\bar{X}_{s_k}}_{s_{n}}
\bigr)
\\[-2pt]
&&\hspace*{9pt}
{}-f \bigl(\bar{X}_{s_1},\ldots,\bar{X}_{s_{k-1}},X^{s_{k-1},\bar
{X}_{s_{k-1}}}_{s_{k}},\ldots,X^{s_{k-1},\bar
{X}_{s_{k-1}}}_{s_{n}} \bigr) \bigr)
\\[-2pt]
&&\qquad \leq\mathbb{E} \Bigl(\inf_{y_k\in\mathbb{R}} \Bigl\{f \bigl(\bar
{X}_{s_1},\ldots,\bar{X}_{s_{k-1}},y_k,X^{s_k,y_k}_{s_{k+1}},\ldots,X^{s_k,y_k}_{s_{n}} \bigr)
\\[-2pt]
&&\hspace*{125pt}{}+\max_{k\leq j\leq
n}\bigl|X^{s_k,y_k}_{s_{j}}-X^{s_k,\bar{X}_{s_k}}_{s_{j}}\bigr|^p
\Bigr\}
\\[-2pt]
&&\hspace*{10pt}\quad\qquad{}-f \bigl(\bar{X}_{s_1},\ldots,\bar{X}_{s_{k-1}},X^{s_{k-1},\bar
{X}_{s_{k-1}}}_{s_{k}},\ldots,X^{s_{k-1},\bar{X}_{s_{k-1}}}_{s_{n}} \bigr) \Bigr)
\\[-2pt]
&&\qquad\leq\mathbb{E} \Bigl(\inf_{y_k\in\mathbb{R}} \bigl\{f_k(\bar
{X}_{s_1},\ldots,\bar{X}_{s_{k-1}},y_k)+C|y_k-
\bar{X}_{s_k}|^p \bigr\}
\\[-2pt]
&&\hspace*{95pt}{}
-f_k \bigl(\bar
{X}_{s_1},\ldots,\bar{X}_{s_{k-1}},X^{s_{k-1},\bar
{X}_{s_{k-1}}}_{s_{k}}
\bigr) \Bigr)
\\[-2pt]
&&\qquad\leq C\mathbb{E} \bigl(\mathcal{W}_p^p \bigl(\mathcal{L}
\bigl(X^{s_{k-1},x}_{s_k} \bigr),\mathcal{L} \bigl(\bar
{X}^{s_{k-1},x}_{s_k} \bigr) \bigr)\big|_{x=\bar{X}_{s_{k-1}}} \bigr)
\\[-2pt]
&&\qquad \leq C\sup_{x\in\mathbb{R}}\mathcal{W}^p_p \bigl(
\mathcal{L} \bigl(\bar{X}^{x}_{s_k-s_{k-1}} \bigr),\mathcal{L}
\bigl(X^{x}_{s_k-s_{k-1}} \bigr) \bigr)
\\[-2pt]
&&\qquad\leq C\sup_{0\leq t\leq T,x\in\mathbb{R}}\mathcal{W}^p_p
\bigl( \mathcal{L} \bigl(\bar{X}^{x}_t \bigr),\mathcal{L}
\bigl(X^{x}_t \bigr) \bigr).
\end{eqnarray*}\upqed
\end{pf*}
\section{Some properties of diffusion bridges}\label{diff_bridge}
Let us suppose that the SDE $dX_t=b(X_t)\,dt+\sigma(X_t)\,dW_t$,
$X_0=x$ has a~transition density $p_t(x,y)$ which is positive and of
class $\mathcal{C}^{1,2}$ with respect to $(t,x)\in\mathbb{R}_+^*
\times \mathbb{R}$. We check later in this section that this holds
under Hypothesis~\ref{hyp_wass_pathwise}. Then, the law of the
diffusion bridge with deterministic time horizon~$\mathcal{T}$ is given
by (see, e.g., Fitzsimmons, Pitman and Yor~\cite{FPY})
\begin{eqnarray}
\mathbb{E} \bigl[F(X_u,0\le u\le t)|X_\mathcal{T}=y \bigr]=
\mathbb{E} \biggl[F(X_u,0\le u\le t)\frac{
p_{\mathcal{T}-t}(X_t,y)}{p_\mathcal{T}(x,y)} \biggr],\nonumber
\\
\eqntext{0\le t<\mathcal{T},}
\end{eqnarray}
where $F\dvtx C([0,t],\mathbb{R})\rightarrow\mathbb{R}$ is a bounded
measurable function. Indeed for\allowbreak $g\dvtx \mathbb{R}\to\mathbb{R}$
measurable and bounded, using that $X_\mathcal{T}$ has the density
$p_\mathcal{T}(x,y)$, then the Markov property at time $t$, one checks
that
\begin{eqnarray*}
&& \mathbb{E} \biggl[\mathbb{E} \biggl[F(X_u,0\le u\le t)
\frac{
p_{\mathcal{T}-t}(X_t,y)}{p_\mathcal{T}(x,y)} \biggr] \bigg|_{y=X_\mathcal{T}
}g(X_\mathcal{T}) \biggr]
\\
&&\qquad =
\mathbb{E} \biggl[F(X_u,0\le u\le t)\int_\mathbb
{R}g(y)p_{\mathcal{T}-t}(X_t,y)\,dy \biggr]
\\
&&\qquad =\mathbb{E} \bigl[F(X_u,0\le u\le t)\mathbb{E}
\bigl[g(X_\mathcal{T})|X_t \bigr] \bigr]
\\
&&\qquad =\mathbb{E}
\bigl[F(X_u,0\le u\le t)g(X_\mathcal{T}) \bigr].
\end{eqnarray*}
We thus focus on the change of probability measure
\[
\frac{d\mathbb{P}^y}{d\mathbb{P}} \bigg|_{\mathcal{F}_t}=\frac
{p_{\mathcal{T}-t}(X_t,y)}{p_\mathcal{T}(x,y)}=:M_t,
\]
so that $\mathbb{E}[F(X_u,0\le u\le t)|X_\mathcal{T}=y]=\mathbb
{E}^y[F(X_u,0\le u\le t)]$ where $\mathbb{E}^y$ denotes the expectation
with respect to $\mathbb{P}^y$. We define $\ell_t(x,y)=\log p_t(x,y)$.
The process $(M_t)_{t\in[0,\mathcal {T})}$ is a martingale, and by
It\^o's formula, we get $dM_t=M_t \partial_x
\ell_{\mathcal{T}-t}(X_t,y) \sigma(X_t)\,dW_t$, which gives
\[
M_t=\exp\biggl( \int_0^t
\partial_x\ell_{\mathcal{T}-s}(X_s,y) \sigma
(X_s)\,dW_s -\frac{1}{2} \int
_0^t \partial_x
\ell_{\mathcal{T}
-s}(X_s,y)^2 \sigma(X_s)^2\,ds
\biggr).
\]
Girsanov's theorem then gives that for all $y\in\mathbb{R}$,
$(W^y_t=W_t-\int_0^t \partial_x\ell_{\mathcal{T}-s}(X_s,\break y)
\*\sigma(X_s)\,ds)_{t\in [0, \mathcal{T})}$ is a Brownian motion
under~$\mathbb{P}^y$, so that $(W^{X_\mathcal{T} }_t)_{t\in [0,
\mathcal{T})}$ is a Brownian motion independent of~$X_\mathcal{T}$.
Moreover, we have
\begin{equation}
\label{bridge_dyn} dX_t= \bigl[b(X_t)+
\partial_x\ell_{\mathcal{T}-t}(X_t,y)
\sigma(X_t)^2 \bigr]\,dt +\sigma(X_t)\,dW_t^y,
\end{equation}
which gives precisely the diffusion bridge dynamics.
Conversely, we would like now to reconstruct the diffusion from the
initial and the final value by using diffusion bridges. The following
result, stated in dimension one, may be generalized to higher
dimensions.
\begin{aprop}\label{prop_bridge} We consider an SDE
$dX_t=b(X_t)\,dt+\sigma(X_t)\,dW_t$, $X_0=x$ with a transition density
$p_t(x,y)$ positive and of class $\mathcal{C}^{1,2}$
with respect to $(t,x)\in\mathbb{R}_+^* \times\mathbb{R}$. Let $(B_t,t\ge0)$ be a
standard Brownian motion and $Z_\mathcal{T}$ be a~random variable with
density $p_\mathcal{T}(x,y)$ drawn independently from~$B$. We assume
that pathwise uniqueness holds for the SDE
\begin{eqnarray}\label{eds_pont}
dZ^{x,y}_t &=& \bigl[b
\bigl(Z^{x,y}_t \bigr)+\partial_x
\ell_{\mathcal{T}-t} \bigl(Z^{x,y}_t,y \bigr) \sigma
\bigl(Z^{x,y}_t \bigr)^2 \bigr]\,dt +\sigma
\bigl(Z^{x,y}_t \bigr)\,dB_t,\quad t\in[0,\mathcal{T}),\hspace*{-25pt}
\nonumber\\[-4pt]\\[-4pt]
Z^{x,y}_0 &=& x,\nonumber\hspace*{-25pt}
\end{eqnarray}
for any $x,y\in\mathbb{R}$, and set $Z_t= Z^{x,Z_\mathcal{T}}_t$ for $t
\in
[0,\mathcal{T})$. Then, $(Z_t)_{t \in[0,\mathcal{T}]}$ and
$(X_t)_{t\in[0,\mathcal{T}]}$ have the same law.
\end{aprop}
A consequence of this result is that $(Z_t,t \in[0,\mathcal{T}])$ has
continuous paths, which gives that $\lim_{t\rightarrow
\mathcal{T}-}Z^{x,y}_t = y$ a.s., $dy$-a.e.
\begin{pf}
Let $t\in[0,\mathcal{T})$ and $F\dvtx C([0,t],\mathbb
{R})\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be bounded and
measurable functions. Since pathwise uniqueness for the
SDE~(\ref{eds_pont}) implies weak uniqueness, we get
\begin{eqnarray*}
\mathbb{E} \bigl[F \bigl(Z_{u}^{x,y},0\leq u\leq t \bigr)
\bigr]&=&\mathbb{E}^y \bigl[F(X_u,0\leq u\leq t) \bigr]
\\
&=& \mathbb{E} \biggl[F(X_u,0\leq u\leq t)\frac
{p_{\mathcal{T}
-t}(X_t,y)}{p_\mathcal{T}(x,y)} \biggr].
\end{eqnarray*}
Thus we have
\begin{eqnarray*}
\mathbb{E} \bigl[F(Z_{u},0\leq u\leq t)g(Z_\mathcal{T})
\bigr]&=&\mathbb{E} \biggl[F(X_u,0\leq u\leq t)\int
_\mathbb{R}p_{\mathcal{T}
-t}(X_t,y)g(y)\,dy \biggr]
\\
&=&
\mathbb{E} \bigl[F(X_u,0\leq u\leq t)g(X_\mathcal{T} )
\bigr].
\end{eqnarray*}
Hence the finite-dimensional marginals of the two processes are equal.
Since $(X_t)_{t\in[0,\mathcal{T}]}$ has continuous paths and
$(Z_t)_{t\in[0,\mathcal{T}]}$ has c\`adl\`ag paths (continuous on
$[0,\mathcal{T})$ with a possible jump at $\mathcal{T}$), this
completes the proof.
\end{pf}
From now on, we assume that Hypothesis~\ref{hyp_wass_pathwise} holds.
We introduce the Lamperti transformation of the stochastic process
$(X_t,t\ge0)$. We\vspace*{-4pt} define $\varphi(x)=\int_0^x\frac{dy}{\sigma(y)}$ and
$\alpha(y)= (\frac{b}{\sigma}-\frac{\sigma'}{2} )\circ
\varphi^{-1}(y)$, $\hat{X}_t\stackrel{\mathrm{def}}{=}\varphi(X_t)$ so
that we have
\begin{equation}
\label{sde_lamperti} d\hat{X}_t= \alpha(
\hat{X}_t)\,dt + dW_t,\qquad t\in[0,T].
\end{equation}
By\vspace*{-2pt} Hypothesis~\ref{hyp_wass_pathwise}, $\varphi$ is a $C^5$ bijection,
$\alpha\in C^3_b$ and both $\varphi$ and $\varphi^{-1}$ are Lipschitz
continuous. We denote by $\hat{p}_t(\hat{x},\hat{y})$ the transition density
of~$\hat{X}$ and $\hat{\ell}_t(\hat{x},\hat{y})=\log(\hat
{p}_t(\hat{x},\hat{y}))$.
\begin{alem}\label{lem_densite} The density $\hat{p}_t(\hat{x},\hat
{y})$ is
$C^{1,2}$ with respect to $(t,\hat{x})\in\mathbb{R}_+^*\times
\mathbb{R}$. Besides, we have
\[
\partial_{\hat{x}} \hat{\ell}_t(\hat{x},\hat{y})=
\frac{\hat
{y}-\hat{x}}{t}-\alpha(\hat{x})+ g_t(\hat{x},\hat{y}),
\]
where $g_t(\hat{x},\hat{y})$ is a continuous function on $\mathbb
{R}_+\times
\mathbb{R}^2$ such that
$\partial_{\hat{x}} g_t(\hat{x},\hat{y})$ and $\partial_{\hat{y}}
g_t(\hat{x},\hat{y})$
exist and
\[
\forall T>0\qquad
\sup_{t\in[0,T], \hat{x},\hat{y}\in\mathbb
{R}}\bigl|\partial_{\hat{x}}
g_t(\hat{x},\hat{y})\bigr|+\bigl|\partial_{\hat{y}} g_t(\hat
{x},\hat{y})\bigr|<\infty.
\]
\end{alem}
\begin{pf}
It is well known that we can express the transition
density~$\hat{p}_t(\hat{x},\hat{y})$ by using Girsanov's theorem as an
expectation on a Brownian bridge between $\hat{x}$~and~$\hat{y}$.
Namely, since $\alpha$ and its derivatives are bounded, we can apply a
result stated in Gihman and Skorohod~\cite{gs} (Theorem~1, Chapter~3,
Section~13) or in Rogers \cite{rog} to get that $\hat{p}_t(\hat{x},\hat{y})$ is positive and
\begin{eqnarray*}
\hat{\ell}_t(\hat{x},\hat{y})&=&-\frac{(\hat{x}-\hat
{y})^2}{2t}+\int
_{\hat{x}}^{\hat{y}
}\alpha(z)\,dz
\\
&&{} +\log\mathbb{E}
\biggl(\exp\biggl({-\frac{1}{2}\int_0^t\bigl(\alpha
'+\alpha
^2\bigr)\biggl(\hat{x}+W_s+\frac{s}{t}(\hat{y}-\hat{x}-W_t)\biggr)\,ds} \biggr)\biggr)
\\
&&{} -\frac{1}{2}\log(2\pi t).
\end{eqnarray*}
Clearly, $\hat{\ell}_t(\hat{x},\hat{y})$ is $C^{1,2}$ in
$(t,\hat{x})\in\mathbb{R}_+^*\times
\mathbb{R}$ (we can use carefree the dominated convergence theorem for
the third
term since $\alpha\in C^3_b$), and we have
\begin{eqnarray*}
g_t(\hat{x},\hat{y})
&=& -\frac{1}{2}
\biggl(\mathbb{E} \biggl[\exp\biggl({-\frac
{1}{2}\int_0^t\bigl(\alpha'+\alpha^2\bigr)\biggl(\hat{x}+W_s+\frac{s}{t}(\hat{y}-\hat
{x}-W_t)\biggr)\,ds}\biggr)
\\
&&\hspace*{33pt}{}\times \int_0^t\frac{t-s}{t}\bigl(\alpha''+2\alpha\alpha'\bigr)\biggl(\hat
{x}+W_s+\frac
{s}{t}(\hat{y}-\hat{x}-W_t)\biggr)\,ds \biggr]\biggr)
\\
&&\hspace*{9pt}{}\bigg/\biggl({\mathbb{E} \biggl[\exp\biggl({-\frac
{1}{2}\int
_0^t\bigl(\alpha'+\alpha^2\bigr)\biggl(\hat{x}+W_s+\frac{s}{t}(\hat{y}-\hat
{x}-W_t)\biggr)\,ds}\biggr) \biggr]}\biggr).
\end{eqnarray*}
This is a continuous function on~$\mathbb{R}_+\times\mathbb{R}^2$,
and we easily
conclude by using the dominated convergence theorem and~$\alpha\in
C^3_b$.
\end{pf}
By straightforward calculations, we have
\[
p_t(x,y)=\frac{1}{\sigma(y)}\hat{p}_t \bigl(\varphi(x),
\varphi(y) \bigr)
\]
and $p_t(x,y)$ is thus positive and $C^{1,2}$ with respect to $(t,x)$.
The diffusion bridge~(\ref{bridge_dyn}) is thus well defined. Since
$\partial_x \ell_t(x,y) =\frac{1}{\sigma(x)}\partial_{\hat x}
\hat{\ell}_t(\varphi(x),\varphi(y))$, we get by It\^{o} formula
from~(\ref{bridge_dyn})
\begin{eqnarray*}
d \hat{X}_t&=& \bigl[\alpha(\hat{X}_t)+
\partial_{\hat{x}} \hat{\ell}_{\mathcal{T}-t} \bigl(\hat{X}_t,
\varphi(y) \bigr) \bigr]\,dt+dW^y_t,
\\
dW^y_t &=&dW_t-
\partial_{\hat{x}} \hat{\ell}_{\mathcal{T}-t} \bigl(\hat{X}_t,
\varphi(y) \bigr)\,dt.
\end{eqnarray*}
Therefore, as one could expect, the Lamperti transform on the diffusion
bridge coincides with the diffusion bridge on the Lamperti transform.
\begin{aprop}\label{prop_bridge2}
Let Hypothesis~\ref{hyp_wass_pathwise} hold. There exists a
deterministic constant~$C$ such that
\[
\forall\mathcal{T}\in(0,T], x,x',y,y'\in\mathbb{R}\qquad
\sup_{t\in[0,\mathcal{T})} \bigl|Z^{x,y}_t-Z^{x',y'}_t\bigr|
\le C \bigl(\bigl|x-x'\bigr|\vee\bigl|y-y'\bigr| \bigr)
\]
and in particular, pathwise uniqueness holds for~(\ref{eds_pont}).
\end{aprop}
\begin{pf}
For $\hat{x},\hat{y}\in\mathbb{R}$, we consider the following SDE:
\begin{eqnarray}\label{pont_Z2}
d \hat{Z}^{\hat{x},\hat
{y}}_t&=&dB_t+
\biggl[\frac{\hat{y}-\hat{Z}^{\hat{x},\hat{y}}_t}{\mathcal
{T}-t}+g_{\mathcal{T}-t} \bigl(\hat{Z}^{\hat{x},\hat{y}}_t,
\hat{y} \bigr) \biggr]\,dt,\qquad t\in[0, \mathcal{T}),
\nonumber\\[-20pt]\\
\hat{Z}^{\hat{x},\hat{y}}_{0}&=&\hat{x},\nonumber
\end{eqnarray}
which corresponds to the diffusion bridge on the Lamperti
transform~$\hat{X}$. We set
$\Delta_t=\hat{Z}^{\hat{x},\hat{y}}_t-\hat{Z}^{\hat{x}',\hat
{y}'}_t$ for
$t\in[0,\mathcal{T})$ and $\hat{x}',\hat{y}' \in\mathbb{R}$. We have
\[
d \Delta_t = \biggl[\frac{\hat{y}-\hat{y}'-\Delta_t}{\mathcal{T}-t}
+g_{\mathcal{T}-t} \bigl(
\hat{Z}^{\hat{x},\hat{y}}_t,\hat{y} \bigr)-g_{\mathcal{T}-t} \bigl(\hat
{Z}^{\hat{x}',\hat{y}'}_t,\hat{y}' \bigr) \biggr]\,dt
\]
and thus $d(|\Delta_t| \vee|\hat{y}-\hat{y}'|)=
\operatorname{sign}(\Delta_t)\mathbf{1}_{|\Delta_t|\ge|\hat{y}-\hat{y}'|}\,d\Delta_t$.
On the one hand, we observe that
$\mathbf{1}_{|\Delta_t|\ge|\hat{y}-\hat{y}'|}[\operatorname{sign}(\Delta_t)
(\hat{y}-\hat{y}')-|\Delta_t|]\le0$. On the other hand, $g_t$ is
uniformly Lipschitz w.r.t. $(\hat{x},\hat{y})$ on $t\in[0,T]$ by
Lemma~\ref{lem_densite}, which leads to
\[
d \bigl(|\Delta_t| \vee\bigl|\hat{y}-\hat{y}'\bigr| \bigr)\le C
\bigl(|\Delta_t| \vee\bigl|\hat{y}-\hat{y}'\bigr| \bigr)
\]
for some positive constant~$C$. Gronwall's lemma gives then
$|\Delta_t|\le e^{CT}(|\hat{x}-\hat{x}'|\vee|\hat{y}-\hat{y}'|)$.
This gives in
particular pathwise uniqueness for~(\ref{pont_Z2}).
Now, let us\vspace*{-1pt} assume that $(Z^{x,y}_t)_{t\in[0,\mathcal{T})}$
solves~(\ref{eds_pont}). Then
$\varphi(Z^{x,y}_t)$ solves~(\ref{pont_Z2}) with $\hat{x}=\varphi
(x)$ and
$\hat{y}=\varphi(y)$, and we necessarily have
$Z^{x,y}_t=\varphi^{-1}(\hat{Z}_t^{\varphi(x),\varphi(y)})$ by pathwise
uniqueness. Both $\varphi$ and $\varphi^{-1}$ are Lipschitz, and we
denote by $K$ a~common Lipschitz constant. Then we get
\begin{eqnarray*}
\bigl|Z^{x,y}_t-Z^{x',y'}_t\bigr| &=&\bigl|
\varphi^{-1} \bigl(\hat{Z}_t^{\varphi(x),\varphi(y)} \bigr) -
\varphi^{-1} \bigl(\hat{Z}_t^{\varphi(x'),\varphi(y')} \bigr) \bigr|
\\
&\le&
K^2 e^{CT} \bigl(\bigl|x-x'\bigr|\vee\bigl|y-y'\bigr|
\bigr),
\end{eqnarray*}
which gives the desired result.
\end{pf}
\end{appendix}
\printaddresses
\end{document}
|
math
|
मुंबई। बॉलीवुड अभिनेता इरफान खान के प्रवक्ता ने कहा है कि अदाकार की सेहत के बारे में सोशल मीडिया पर आई खबरें पूरी तरह से झूठी हैं। इस तरह की खबरें घूम रही हैं कि ५१ वर्षीय अभिनेता की सेहत लगातार बिगड़ रही है। यह खबर तब फैलनी शुरू हुई जब एक पत्रकार ने इरफान के स्वास्थ्य के बारे में ट्वीट किया। हालांकि उन्होंने जल्द ही अपना पोस्ट डिलीट कर दिया।
प्रवक्ता ने एक बयान में कहा कि पिछले कुछ दिनों से इरफान की सेहत को लेकर सोशल मीडिया पर खबरें फैल रही हैं जो पूरी तरह से झूठी हैं और उनमें जरा भी सच्चाई नहीं है। हम उनके परिवार और दोस्त के तौर पर एक बार फिर से मीडिया के सदस्यों से समर्थन और प्रार्थनाओं की गुजारिश करते हैं।
पिछले महीने, इरफान ने बताया था कि वह न्यूरोएंडोक्राइन ट्यूमर से ग्रस्त हैं। यह एक दुर्लभ प्रकार का कैंसर है जो शरीर के विभिन्न हिस्सों को निशाना बना सकता है। वह इलाज के लिए देश से बाहर हैं।
मशहूर 'स्वान लेक' बैले दिल्ली में देगा परफॉर्मेंस
कॉमेडियन सिद्धार्थ सागर का चौंकाने वाला खुलासा, खाने में ड्रग्स मिलाकर देत...
१२-०४-२०१८ १०:३३:०४ आम
|
hindi
|
This entry was posted on Monday, March 23rd, 2009 at 17:51 and is filed under Путешествия. 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
|
सीबीएसई की अध्यक्ष अनिता करवाल ने बोर्ड परीक्षाओं से पहले छात्रों को अनूठे अंदाज में संदेश दिया है। उन्होंने छात्रों से कहा कि पढ़ाई पर ध्यान...
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परीक्षा में जातिसूचक सवाल पूछने पर मामला दर्ज, बोर्ड चेयरमैन पर कार्रवाई की मांग
दिल्ली अधीनस्थ सेवा चयन बोर्ड (डीएसएसएसबी) की शनिवार १३ अक्तूबर को शिक्षक पात्रता परीक्षा आयोजित की गई थी। इस परीक्षा में हिंदी भाषा और बोध अनुभाग के...
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एक आइडिए ने बदल दी इस शख्स की जिंदगी, बना करोड़पति!
सभापति शुक्ला ने अपने हुनर और जज्बे के बल पर अपने ही गांव में एक इंडस्ट्री खोल दी और साथ ही पूरे गांव को ...
|
hindi
|
At Elevate Audiology, we believe in patient-centered pricing. There is no need to sacrifice quality care for low prices. For that reason, we will bill insurance for diagnostic services as a courtesy and offer affordable private pay options including a $50 diagnostic hearing evaluation and consultation and $195 for initial tinnitus assessment. In addition, we offer itemized pricing including bundled and unbundled treatment plans.
Most offices still operate in a “bundled” only structure, whereas your total investment includes your hearing aids, professional services, batteries, warranties, and supplies for a defined period of time. This is a good option for patients that do not want to pay individually for each appointment or supply item, need more than two or three appointments a year, need regular earwax removal, etc.
If this is the better option for you, you will still see an itemized list of your costs so you know exactly what you are paying for and what is included in your treatment plan.
Our service plans are $435 per year with bundled options including pre-determined 2 and 3 years service plans.
We are proud to offer unbundled pricing, as well. The initial financial investment is lower and you pay for your services as you need them. This is a great option for individuals that have worn hearing aids with fairly stable hearing loss and are confident in their abilities to take care of them on a regular basis between maintenance appointments.
In this model, the total cost includes the device(s) and the first month of evidence-based professional services (including Real-Ear Measure) and initial supplies. After your adjustment period, you pay for your services and supplies as you go.
It is still recommended we see you twice a year for a deep cleaning of your devices, visual inspection of your ear canals, follow-up with your treatment plan, and any programming adjustments, as needed.
If we determine that it is in your best interest to have a service plan as that is going to save you money, you are able to purchase this at any time.
Our batteries come in packs of 8. We offer size 10, 312, 13, and 675.
Battery packs are $6 each.
Join our free battery club and receive 1 free pack for every 4 packs purchased.
If you are motivated to improve your quality of life, we are motivated to serve you. Elevate Audiology is committed to helping everyone that needs our services regardless of their financial means.
Please do not let fear of cost stop you from making an appointment with us.
Financing options are available for those that qualify.
In addition, we are proud HearNow and Vocational Rehabilitation providers, as well. These options are only available to those that qualify.
We are committed to our community and believe EVERYONE deserves to hear. Please contact us for more information.
|
english
|
पीएससी की तैयारी में जुटी स्वाती ने किया कॉलेज में टॉप
कॉलेज की छात्रा स्वाती शुक्ला ने बीएससी अंतिम वर्ष की परीक्षा ८५.५ फीसदी
अंक लेकर कॉलेज में टॉप किया है। जूनियर छात्राओं को नि:शुल्क कोाचिंग
सेवा देकर स्वाति ने यह मुकाम हांसिल किया है। साधारण लिपिक परिवार की बेटी
नपा जनता वाचनालय में पीएससी की तैयारी कर रही है। पिपरियाशासकीय पीजी कॉलेज बीएससी अंतिम वर्ष छटवें सेमेस्टर परीक्षा में छात्रा स्वाती शुक्ला ने कॉलेज में अव्वल स्थान प्राप्त किया। स्वाती को ८५.५ फीसदी अंक हांसिल हुए जो कॉलेज में सर्वाधिक है। नपा जनता वाचनालय की नियमित पाठक रही स्वाती ने पढ़ाई के साथ नि:शुल्क कोचिंग में सहपाठी छात्राओं को शिक्षण भी दिया। प्रारंभ से ही स्वाती पढ़ाई में अव्वल रही बोर्ड परीक्षाएं भी उत्कृष्ट अंको से उत्तीर्ण की है। स्वाती अपनी सफलता का श्रेय माता पिता तीरथ शुक्ला एवं अर्चना शुक्ला के साथ गुरुजनों को देती है। साधारण लिपिक परिवार की बेटी स्वाति का लक्ष्य उच्च प्रशासनिक अधिकारी बन कर देश सेवा करने का है। नपा का जनता वाचनालय मेंवह नियमित प्रतियोगी परीक्षा की तैयारियों में अपना समय देना नही भू्रलती।वर्तमान में स्वाती वाचनालय के माध्यम से वह पीएससी प्रतियोगी परीक्षा की तैयारी कर रही है। प्राचार्य राजीव माहेश्वरी,सीएमओ कमलेश पाटीदार, नपाध्यक्ष राजीव जायसवाल, उपाध्यक्ष राजेन्द्र उपाध्याय,महिला सशक्तीकरण अधिकारी अनामिका तिवारी, वाचनालय के रुपेश मौर्य, लाइब्रेरियन गजेन्द्र बेमन ने छात्रा को उत्कृष्ट सफलता पर शुभकामनाए दी है। अपने विवाह के सपने को भारत मैट्रीमोनी पर साकार करे।- निःशुल्क रजिस्ट्रेशन करे! टैग्स: होशंगाबाद
|
hindi
|
\begin{document}
\begin{abstract}
Using twisted nearby cycles, we define a new notion of slopes for complex holonomic $\mathcal{D}$-modules. We prove a boundedness result for these slopes, study their functoriality and use them to characterize regularity. For a family of (possibly irregular) algebraic connections $\mathcal{E}_t$ parametrized by a smooth curve, we deduce under natural conditions an explicit bound for the usual slopes of the differential equation satisfied by the family of irregular periods of the $\mathcal{E}_t$. This generalizes the regularity of the Gauss-Manin connection proved by Katz and Deligne. Finally, we address some questions about analogues of the above results for wild ramification in the arithmetic context.
\end{abstract}
\title{A Boundedness theorem for nearby slopes of holonomic $\mathcal{D}
Let $V$ be a smooth algebraic variety over a finite field of characteristic $p>0$, and let $U$ be an open subset in $V$ such that $D:=V\setminus U$ is a normal crossing divisor. Let $\ell$ be a prime number different from $p$. Using restriction to curves, Deligne defined \cite{Delignebornee} a notion of $\ell$-adic local system on $U$ with bounded ramification along $D$. Such a definition is problematic to treat functoriality questions: the direct image of a local system is not a local system any more, duality does not commute with restriction in general. In this paper, we investigate the characteristic 0 aspect of this problem, that is the
\begin{question}
Let $X$ be a complex manifold. Can one define a notion of holonomic $\mathcal{D}_X$-module with bounded irregularity which has good functoriality properties?
\end{question}
In dimension 1, to bound the irregularity number of a $\mathcal{D}$-module with given generic rank amounts to bound its slopes. Let $\mathcal{M}$ be a holonomic $\mathcal{D}_X$-module and let $Z$ be a hypersurface of $X$. Mebkhout \cite{Mehbgro} showed that the \textit{irregularity complex} $\Irr_{Z}(\mathcal{M})$ of $\mathcal{M}$ along $Z$ is a perverse sheaf endowed with a $\mathds{R}_{>1}$ increasing locally finite filtration by sub-perverse sheaves $\Irr_{Z}(\mathcal{M})(r)$. In dimension one, this construction gives back the usual notion of slope modulo the change of variable $r\longrightarrow 1/(r-1)$. The \textit{analytic slopes of $\mathcal{M}$ along $Z$} are the $r>1$ for which the supports of the graded pieces of $(\Irr_{Z}(\mathcal{M})(r))_{r>1}$ are non empty. \\ \indent
The existence of a uniform bound in $Z$ is not clear a priori. We thus formulate the following
\begin{conjecture}\label{conj0}
Locally on $X$, the set of analytic slopes of a holonomic
$\mathcal{D}_X$-module is bounded.
\end{conjecture}
This statement means that for a holonomic $\mathcal{D}_X$-module $\mathcal{M}$, one can find for every point in $X$ a neighbourhood
$U$ and a constant $C>0$ such that the analytic slopes of $\mathcal{M}$ along any germ of hypersurface in $U$ are $\leq C$.\\ \indent
On the other hand, Laurent defined \textit{algebraic slopes} using his theory of micro-characteristic varieties \cite{ LaurentPolygone}. From Laurent and Mebkhout work \cite{LM}, we know that the set of analytic slopes of a holonomic $\mathcal{D}$-module $\mathcal{M}$ along $Z $ is equal to the set of algebraic slopes of $\mathcal{M}$ along $Z$. Since micro-characteristic varieties are invariant by duality, we deduce that analytic slopes are invariant by duality.\\ \indent
The aim of this paper is to define a third notion of slopes and to investigate some of its properties. The main idea lies in the observation that for a germ $\mathcal{M}$ of $\mathcal{D}_{\mathds{C}}$-module at $0\in \mathds{C}$, \textit{the slopes of $\mathcal{M}$ at 0 are encoded in the vanishing of certain nearby cycles}. We show in \ref{propdim1} that $r\in \mathds{Q}_{\geq 0}$ is a slope for $\mathcal{M}$ at 0 if and only if one can find a germ $N$ of meromorphic connection at 0 with slope $r$ such that $\psi_{0}(\mathcal{M}\otimes N)\neq 0$.\\ \indent
We thus introduce the following definition. Let $X$ be a complex manifold and let $\mathcal{M}$ be an object of the derived category $\mathcal{D}_{\hol}^b(X)$ of complexes of $\mathcal{D}_X$-modules with bounded and holonomic cohomology. Let $f\in \mathcal{O}_X$. We denote by $\psi_f$ the nearby cycle functor\footnote{For general references on the nearby cycle functor, let us mention \cite{Kashpsi},\cite{Malpsi},\cite{MS} and \cite{MM}.} associated to $f$. We define the \textit{nearby slopes of $\mathcal{M}$ associated to $f$} to be the set $\Sl_{f}^{\nb}(\mathcal{M})$ complement in $\mathds{Q}_{\geq 0}$ of the set of rationals $r\geq 0$ such that for every germ $N$ of meromorphic connection at 0 with slope $r$, we have
\begin{equation}\label{annulation0}
\psi_{f}(\mathcal{M}\otimes f^{+}N)\simeq 0
\end{equation}
Let us observe that the left-hand side of \eqref{annulation0} depends on $N$ only via $\hat{\mathcal{O}}_{\mathds{C},0}\otimes_{\mathcal{O}_{\mathds{C},0}} N$, and that nearby slopes are sensitive to the reduced structure of $\divi f$, whereas the analytic and algebraic slopes only see the support of $\divi f$. \\ \indent
Twisted nearby cycles appear for the first time in the algebraic context in \cite{DeligneLettreMalgrange}. Deligne proves in \textit{loc. it.} that for a given function $f$, the set of $\hat{\mathcal{O}}_{\mathds{C},0}$-differential modules $N$ such that
$\psi_{f}(\mathcal{M}\otimes f^{+}N)\neq 0$ is finite.\\ \indent
The main result of this paper is an affirmative answer to conjecture \ref{conj0} for nearby slopes, that is the
\begin{theorem}\label{theoremprincipal}
Locally on $X$, the set of nearby slopes of a holonomic
$\mathcal{D}$-module is bounded.
\end{theorem}
This statement means that for a holonomic $\mathcal{D}_X$-module $\mathcal{M}$, one can find for every point in $X$ a neighbourhood
$U$ and a constant $C>0$ such that the nearby slopes of $\mathcal{M}$ associated to any $f\in \mathcal{O}_U$ are $\leq C$.\\ \indent
For meromorphic connections with good formal structure, we show the following refinement
\begin{theorem}\label{theoremprincipalraffiné}
Let $\mathcal{M}$ be a meromorphic connection with good formal structure. Let $D$ be the pole locus of $\mathcal{M}$ and let $D_1, \dots, D_n$ be the irreducible components of $D$. We denote by $r_{i}(\mathcal{M})\in \mathds{Q}_{\geq 0}$
the highest generic slope of $\mathcal{M}$ along $D_i$. Then, the nearby slopes of $\mathcal{M}$ are $\leq r_{1}(\mathcal{M})+\cdots +r_{n}(\mathcal{M})$.
\end{theorem}
The main tool used in the proof of theorem \ref{theoremprincipal} is a structure theorem for formal meromorphic connections first conjectured in \cite{CastroSabbah}, studied by Sabbah \cite{Sabbahdim} and proved by Kedlaya \cite{Kedlaya1}\cite{Kedlaya2} in the context of excellent schemes and analytic spaces, and independently by Mochizuki \cite{Mochizuki2}\cite{Mochizuki1} in the algebraic context.\\ \indent
Let us give some details on the strategy of the proof of theorem \ref{theoremprincipal}. A dévissage carried out in \ref{reduction} allows one to suppose that $\mathcal{M}$ is a meromorphic connection. Using Kedlaya-Mochizuki theorem, one reduces further the proof to the case where
$\mathcal{M}$ has good formal structure. We are thus left to prove theorem \ref{theoremprincipalraffiné}. We resolve the singularities of $Z$. The problem that occurs at this step is that a randomly chosen embedded resolution $p:\tilde{X}\longrightarrow X$ will increase the generic slopes of $\mathcal{M}$ in a way that cannot be controlled. We show in \ref{propositionprincipale} that a fine version of embedded resolution \cite{BMUniformization} allows to control the generic slopes of $p^{+}\mathcal{M}$ in terms of the sum $ r_{1}(\mathcal{M})+\cdots +r_{n}(\mathcal{M})$ and the multiplicities of $p^{\ast}Z$. A crucial tool for this is a theorem
\cite[I 2.4.3]{Sabbahdim} proved by Sabbah in dimension 2 and by Mochizuki \cite[2.19]{MochStokes} in any dimension relating the good formal models appearing at a given point with the generic models on the divisor locus. Using a vanishing criterion \ref{lemmeannulation}, one finally proves \eqref{annulation0} for $r> r_{1}(\mathcal{M})+\cdots +r_{n}(\mathcal{M})$.\\ \indent
Let $\mathcal{M}\in \mathcal{D}_{\hol}^b(X)$ and let us denote by $\mathds{D}\mathcal{M}$ the dual complex of $\mathcal{M}$. The nearby slopes satisfy the following functorialities
\begin{theorem}\label{theoremprincipal-1}
$(i)$ For every $f\in \mathcal{O}_X$, we have $$\Sl_{f}^{\nb}(\mathds{D}\mathcal{M})=\Sl_{f}^{\nb}(\mathcal{M})$$
$(ii)$ Let $p:X\longrightarrow Y$ be a proper morphism and let $f\in \mathcal{O}_Y$ such that $p(X) $ is not contained in $f^{-1}(0)$. Then
$$\Sl_{f}^{\nb}(p_{+}\mathcal{M})\subset \Sl_{fp}^{\nb}(\mathcal{M})$$
\end{theorem}
Let us observe that $(ii)$ is a direct application of the compatibility of nearby cycles with proper direct image \cite{MS}.\\ \indent
It is an interesting problem to try to compare nearby slopes and analytic slopes. This question won't be discussed in this paper, but we characterize regular holonomic $\mathcal{D}$-modules using nearby slopes.
\begin{theorem}\label{comparaisonreg}
A complex $\mathcal{M}\in \mathcal{D}_{\hol}^b(X)$ is regular if and only if for every quasi-finite morphism $\rho: Y\longrightarrow X$ with $Y$ a complex manifold, the set of nearby slopes of $\rho^{+}\mathcal{M}$ is contained in $\{ 0\}$.
\end{theorem}
For an other characterization of regularity (harder to deal with in practice) using $R\mathcal{H}om$ and the solution functor, we refer to \cite{carmodreg}.\\ \indent
Let us give an application of the preceding results. Let $U$ be a smooth complex algebraic variety and let $\mathcal{E}$ be an algebraic connection on $U$. We denote by $H^{k}_{\dR}(U,\mathcal{E})$ the $k^{th}$ de Rham cohomology group of $\mathcal{E}$, and by $\mathcal{V}$ the local system of horizontal sections of $\mathcal{E}^{\an}$ on $U^{\an}$. If $\mathcal{E}$ is regular, Deligne proved \cite{Del} that the canonical comparison morphism
\begin{equation}\label{accperiode}
H^{k}_{\dR}(U,\mathcal{E} )\longrightarrow H^{k}(U^{\an},\mathcal{V} )
\end{equation}
is an isomorphism. If $\mathcal{E}$ is the trivial connection, this is due to Grothendieck \cite{GroDR}. In the irregular case, \eqref{accperiode} is no longer an isomorphism. It can happen that $H^{k}_{\dR}(U,\mathcal{E} )$ is non zero and $H^{k}(U^{\an},\mathcal{V})$ is zero, which means that there are not enough topological cycles in $U^{\an}$. The \textit{rapid decay homology} $H_{k}^{\rd}(U,\mathcal{E}^{\ast} )$ needed to remedy this problem appears in dimension one in \cite{HB} and in higher dimension in \cite{Hiendimdeux}\cite{HienInv}. It includes cycles drawn on a compactification of $U^{\an}$ taking into account the asymptotic at infinity of the solutions of the dual connection $\mathcal{E}^{\ast}$. By Hien duality theorem, we have a perfect pairing
\begin{equation}\label{accperiodeirr}
\int : H^{k}_{\dR}(U,\mathcal{E} )\times H_{k}^{\rd}(U,\mathcal{E}^{\ast} )\longrightarrow \mathds{C}
\end{equation}
For $\omega\in H^{k}_{\dR}(U,\mathcal{E})$ and $\gamma\in H_{k}^{\rd}(U,\mathcal{E}^{\ast})$, the complex number $\int_\gamma \omega$ is a \textit{period for $\mathcal{E}$}.\\ \indent
Let $f:X\longrightarrow S$ be a proper and generically smooth morphism, where $X$ denotes an algebraic variety and $S$ denotes a neighbourhood of 0 in $\mathds{A}^{1}_{\mathds{C}}$. Let $U$ be the complement of a normal crossing divisor $D$ of $X$ such that for every $t\neq 0$ close enough to 0, $D_t$ is a normal crossing divisor of $X_t$. Let $\mathcal{E}$ be an algebraic connection on $U$. Let us denote by $D_1, \dots, D_n$ the irreducible components of $D$ meeting $f^{-1}(0)$ and let $r_i(\mathcal{E})$ be the highest generic slope of $\mathcal{E}$ along $D_i$.\\ \indent
As an application of theorem \ref{theoremprincipalraffiné},
we prove the following
\begin{theorem}\label{GMirr}
If $\mathcal{E}$ has good formal structure along $D$ and if the fibers $X_t$, $t\neq 0$ of $f$ are non characteristic at infinity\footnote{this is for example the case if $D$ is smooth and if the fibers of $f$ are transverse to $D$.} for $\mathcal{E}$, then the periods of the family $(\mathcal{E}_t)_{t\neq 0}$ are solutions of a system of linear polynomial differential equations whose slopes at $0$ are
$\leq r_1(\mathcal{E})+\cdots +r_n(\mathcal{E})$.
\end{theorem}
In the case where $\mathcal{E}$ is the trivial connection, we recover that the periods of a proper generically smooth family of algebraic varieties are solutions of a regular singular differential equation with polynomial coefficients \cite{KatzIHES}\cite{Del}. \\ \indent
The role played in this paper by nearby cycles has Verdier specialization \cite{SpeVerdier} and moderate nearby cycles \cite[XIII]{SGA7-2} as $\ell$-adic counterparts. Let $V$ be an algebraic scheme over a perfect field $k$ of characteristic $p>0$, and let $\overline{V}$ be a compactification of $V$. Let $j: V\longrightarrow \overline{V}$ be the canonical inclusion, and $f\in \mathcal{O}_{\overline{V}}$. Let $S$ be the strict
henselianization of
$\mathcal{O}_{\mathds{A}^{1}_{k},0}$, denote by $\eta$ the generic point of $S$ and let us choose a geometric point $\overline{\eta}$ over $\eta$. Let $P$ be the wild ramification group of $\pi_1(\eta, \overline{\eta})$. Define $\overline{V}_S:=\overline{V}\times_{\mathds{A}^{1}_{k}} S$, $f_S:\overline{V}_S\longrightarrow S$ the base change of $f$ to $S$, $f_\eta: \overline{V}_\eta\longrightarrow \eta$ the restriction of $f_S$ over $\eta$ and $\iota: \overline{V}_S\longrightarrow \overline{V}$ the canonical morphism. Let $\mathcal{F}$ be a $\ell$-adic complex on $V$ with bounded constructible cohomology.
We say that $r\in \mathds{Q}_{\geq 0}$ is a nearby slope\footnote{Or a Verdier slope if Verdier specialization is used instead of moderate nearby cycles.} for $\mathcal{F}$ associated to $(\overline{V},f)$ if one can find a constructible $\ell$-adic sheaf $N$ on $\eta_{\et}$ with slope $r$ such that
$$
(\psi_f(\iota^{\ast}j_{!}\mathcal{F}\otimes f_\eta^{\ast}N))^{P}\neq 0
$$
Hence, conjecture \ref{conj0} has a $\ell$-adic analogue that may be worth investigating. This is the following
\begin{question}
Is it true that the set of nearby slopes of $\mathcal{F}$ is bounded?
\end{question}
That nearby slopes do not depend on the choice of a compactification is not clear to the author. Nor that a smoothness assumption on $V$ is needed. In any case, this leads to a notion of $\ell$-adic tame complex in the sense of Verdier specialization or moderate nearby cycles. As an analogue of the regularity of $\mathcal{O}_X$ in the theory of $\mathcal{D}$-modules, we raise the following
\begin{question}
Is it true that the constant sheaf $\overline{\mathds{Q}}_{\ell}$ on $V$ is tame in the sense of moderate nearby cycles? That is, that for every couple $(\overline{V},f)$ as above and for every constructible $\ell$-adic sheaf $N$ on $\eta_{\et}$ with slope $>0$, we have
$$
(\psi_f(\iota^{\ast}j_{!}\overline{\mathds{Q}}_{\ell}\otimes f_\eta^{\ast}N))^{P}\simeq 0 \text{ ?}
$$
\end{question}
Conjecture \ref{conj0} first appears in \cite{ProgCNRS}. This paper grew out an attempt to prove it. I thank Pierre Deligne, Zoghman Mebkhout and Claude Sabbah for valuable comments on this manuscript and Marco Hien for mentioning \cite{HR}, which inspired me a statement in the spirit of theorem \ref{GMirr} and reignited my interest for a proof of theorem \ref{theoremprincipal}.
This work has been achieved with the support of Freie Universität/Hebrew University of Jerusalem joint post-doctoral program and ERC 226257 program. I thank Hélène Esnault and Yakov Varshavsky for their support.
\section{Notations}
We collect here a few definitions used all along this paper. The letter $X$ will denote a complex manifold.
\subsection{}
For a morphism $f:Y\longrightarrow X$ with $Y$ a complex manifold, we denote by $f^{+}:D^{b}_{\hol}(\mathcal{D}_X)\longrightarrow D^{b}_{\hol}(\mathcal{D}_Y)$ and $f_{+}:D^{b}_{\hol}(\mathcal{D}_Y)\longrightarrow D^{b}_{\hol}(\mathcal{D}_X)$ the inverse image and direct image functors for $\mathcal{D}$-modules. We note $f^{\dag}$ for $f^{+}[\dim Y-\dim X]$.
\subsection{}
Let $\mathcal{M}\in \mathcal{D}_{\hol}^b(X)$ and $f\in \mathcal{O}_X$.
From
$
\mathcal{H}^{k}\psi_{f}(\mathcal{M}\otimes f^{+}N)\simeq
\psi_{f}(\mathcal{H}^{k}\mathcal{M}\otimes f^{+}N)
$
for every $k$, we deduce
\begin{equation}\label{slopecomplex}
\Sl_{f}^{\nb}(\mathcal{M})=\bigcup_{k}\Sl_{f}^{\nb}(\mathcal{H}^{k}\mathcal{M})
\end{equation}
Let us define
$
\Sl^{\nb}(\mathcal{M}):=\bigcup_{f\in \mathcal{O}_X} \Sl_{f}^{\nb}(\mathcal{M})
$. The elements of $\Sl^{\nb}(\mathcal{M})$ are the \textit{nearby slopes} of $\mathcal{M}$.
For $S\subset \mathds{Q}_{\geq 0}$, we denote by $ \mathcal{D}_{\hol}^b(X)_{S}$ the full subcategory of $\mathcal{D}_{\hol}^b(X)$ of complexes whose nearby slopes are in $S$.
\subsection{}\label{DRetSol}
Let us denote by $\DR:D^{b}_{\hol}(\mathcal{D}_X) \longrightarrow D^{b}_{c}(X,\mathds{C})$ the \textit{de Rham functor}\footnote{In this paper, we follow Hien's convention \cite{HienInv} according to which for a holonomic module $\mathcal{M}$, the complex $\DR \mathcal{M}$ is concentrated in degrees $0,\dots, \dim X$.} and by $\Sol:D^{b}_{\hol}(\mathcal{D}_X) \longrightarrow D^{b}_{c}(X,\mathds{C})$ the \textit{solution functor} for holonomic $\mathcal{D}_X$-modules.
\subsection{}\label{localisation}
For every analytic subspace $Z$ in $X$, we denote by $i_Z:Z\hookrightarrow X$ the canonical inclusion. The \textit{local cohomology triangle} for $Z$ and $\mathcal{M}\in \mathcal{D}_{\hol}^b(X)$ reads
\begin{equation}\label{ocholocal}
\xymatrix{
R\Gamma_{[Z]}\mathcal{M}\ar[r]& \mathcal{M} \ar[r]& R\mathcal{M}(\ast Z)\ar[r]^-{+1}&
}
\end{equation}
It is a distinguished triangle in $D^{b}_{\hol}(\mathcal{D}_X)$.
The complex $R\Gamma_{[Z]}\mathcal{M}$ is \textit{the local algebraic cohomology} of $\mathcal{M}$ along $Z$ and $R\mathcal{M}(\ast Z)$ is the \textit{localization} of $\mathcal{M}$ along $Z$.
\subsection{}\label{bonnedecompositionformelle}
Let $\mathcal{M}$ be a germ of meromorphic connection at the origin of $\mathds{C}^{n}$. Let $D$ be the pole locus of $\mathcal{M}$. For $x\in D$, we define $\hat{\mathcal{M}}_x:=\hat{\mathcal{O}}_{\mathds{C}^{n},x}\otimes_{\mathcal{O}_{\mathds{C}^{n},x}}\mathcal{M}$. We say that $\mathcal{M}$ has \textit{good formal structure} if
\begin{enumerate}
\item $D$ is a normal crossing divisor.
\item For every $x\in D$, one can find coordinates $(x_1,\dots, x_n)$ centred at $x$ with $D$ defined by $x_1\cdots x_i=0$, and an integer $p\geq 1$ such that if $\rho$ is the morphism $(x_1,\dots, x_n)\longrightarrow (x_1^p,\dots, x_i^p,x_{i+1}, \dots, x_n)$, we have a decomposition
\begin{equation}\label{decomposition}
\rho^{+}\hat{\mathcal{M}}_x\simeq \displaystyle{\bigoplus_{\varphi \in \mathcal{O}_{\mathds{C}^{n}}(\ast D)/\mathcal{O}_{\mathds{C}^{n}}}}
\mathcal{E}^{\varphi}\otimes \mathcal{R}_{\varphi}
\end{equation}
where $\mathcal{E}^{\varphi}=(\hat{\mathcal{O}}_{\mathds{C}^{n},x}(\ast D),d+d\varphi)$ and $\mathcal{R}_{\varphi}$ is a meromorphic connection with regular singularity along $D$.
\item\label{conditionphi} For all $\varphi\in \mathcal{O}_{\mathds{C}^{n}}(\ast D)/\mathcal{O}_{\mathds{C}^{n}}$ contributing to \eqref{decomposition}, we have $\divi \varphi \leq 0$.
\end{enumerate}
Let us remark that classically, one asks for condition \eqref{conditionphi} to be also true for the differences of two $\varphi$ intervening in \eqref{decomposition}. We won't impose this extra condition in this paper.
\subsection{} \label{divplusgrandepente}
Let $\mathcal{M}$ be a meromorphic connection on $X$ such that the pole locus $D$ of $\mathcal{M}$ has only a finite number of irreducible components $D_1,\dots, D_n$. For every $i=1,\dots, n$, we denote by $r_{D_i}(\mathcal{M})$ the highest generic slope of $\mathcal{M}$ along $D_i$. We define the \textit{divisor of highest generic slopes of $\mathcal{M}$} by
$$
r_{D_1}(\mathcal{M})D_1+\dots +r_{D_n}(\mathcal{M})D_n\in Z(X)_{\mathds{Q}}
$$
\section{Preliminaries on nearby cycles in the case of good formal structure}
\subsection{}
Let $n$ be an integer and take $i\in \mathds{N}^{\llbracket 1,n\rrbracket}$. The \textit{support of $i$} is the set of $k\in \llbracket 1,n\rrbracket$ such that $i_k\neq 0$. If $E\subset \llbracket 1,n\rrbracket$, we define $i_E$ by $i_{Ek}=i_k$ for $k\in E$ and $i_{Ek}=0$ if $k\notin E$.
\subsection{}\label{moduleelementaire}
Let $R$ be a regular $\mathds{C}((t))$-differential module, and take $\varphi\in \mathds{C}[t^{-1}]$. For every $n\geq 1$, we define $\rho: t\longrightarrow t^{p}=x$ and
$$
\El(\rho, \varphi, R):=\rho_{+}(\mathcal{E}^{\varphi}\otimes R)
$$
If $R$ is the trivial rank 1 module, we will use the notation $\El(\rho, \varphi)$.
In general, $\El(\rho, \varphi, R)$ has slope $\ord \varphi/p$. The $\mathds{C}((x))$-modules of type $\El(\rho, \varphi, R)$ for variable $(\rho, \varphi, R)$ are called \textit{elementary modules}. From \cite[3.3]{SabbahpourLE}, we know that every $\mathds{C}((x))$-differential module can be written as a direct sum of elementary modules.
\subsection{Dimension 1}
In this paragraph, we work in a neighbourhood of the origin $0\in \mathds{C}$. Let $x$ be a coordinate on $\mathds{C}$. Take $p\geq 1$ and define $\rho: x\longrightarrow t=x^{p}$.
\begin{proposition}\label{propdim1}
Let $\mathcal{M}$ be a germ of holonomic $\mathcal{D}$-module at the origin. Let $r>0$ be a rational number. The following conditions are equivalent
\begin{enumerate}
\item The rational $r$ is not a slope for $\mathcal{M}$ at $0$.
\item For every germ $N$ of meromorphic connection of slope $r/p$, we have
$$
\psi_{\rho}(\mathcal{M}\otimes \rho^{+}N)\simeq 0
$$
\end{enumerate}
\end{proposition}
\begin{proof}
Since $\psi$ is not sensitive to localization and formalization, one can work formally at 0 and suppose that $\mathcal{M}$ and $N$ are differential $\mathds{C}((x))$-modules. \\ \indent
Let us prove $(2)\Longrightarrow (1)$ by contraposition. Define $\rho^{\prime}: u\longrightarrow u^{p^{\prime}}=x$, $\varphi(u)\in \mathds{C}[u^{-1}]$ with $q=\ord\varphi(u)$ and $R$ a $\mathds{C}((u))$-regular module such that $\El(\rho^{\prime}, \varphi(u), R)$ is a non zero elementary factor \ref{moduleelementaire} of $\mathcal{M}$ with slope $r=q/p$. Define
$$
N:=\rho_+\El(\rho^{\prime}, -\varphi(u))=\El(\rho\rho^{\prime}, -\varphi(u))
$$
The module $N$ has slope $q/pp^{\prime}=r/p$. A direct factor of $\psi_{\rho}(\mathcal{M}\otimes \rho^{+}N)$ is
\begin{align*}
\psi_{\rho}(\rho_{+}^{\prime}(\mathcal{E}^{\varphi}\otimes R)\otimes \rho^{+} N))&\simeq
\psi_{\rho}(\rho_{+}^{\prime}(\mathcal{E}^{\varphi}\otimes R)\otimes \rho^{+}\El(\rho\rho^{\prime}, -\varphi(u)))
\\
& \simeq \psi_{\rho}(\rho_{+}^{\prime}(\mathcal{E}^{\varphi}\otimes R\otimes (\rho\rho^{\prime})^{+}\El(\rho\rho^{\prime}, -\varphi(u)))\\
&\simeq
\psi_{\rho\rho^{\prime}}(\mathcal{E}^{\varphi}\otimes R\otimes (\rho\rho^{\prime})^{+}\El(\rho\rho^{\prime}, -\varphi(u)))
\end{align*}
where the last identification comes from the compatibility of $\psi$ with proper direct image. By \cite[2.4]{SabbahpourLE}, we have
$$
(\rho\rho^{\prime})^{+}\El(\rho\rho^{\prime}, -\varphi(u))\simeq \displaystyle{\bigoplus_{\zeta^{pp^{\prime}}=1}}\mathcal{E}^{-\varphi(\zeta u)}
$$
So $\psi_{\rho\rho^{\prime}} R$ is a direct factor of $\psi_{\rho}(\mathcal{M}\otimes \rho^{+}N)$ of rank $np(\rg R)>0$, and $(2)\Longrightarrow (1)$ is proved. \\ \indent
Let us prove $(1)\Longrightarrow (2)$. Let $N$ be a $\mathds{C}((t))$-differential module of slope $r/p$. Then $\rho^{+}N$ has slope $r$. Thus, the slopes of $\mathcal{M}\otimes \rho^{+}N$ are $> 0$. Hence, it is enough to show the following
\begin{lemme}
Let $M$ be a $\mathds{C}((x))$-differential module whose slopes are $>0$. Then, we have $\psi_{\rho}M\simeq 0$.
\end{lemme}
By Levelt-Turrittin decomposition, we are left to study the case where $M$ is a direct sum of modules of type $\mathcal{E}^{\varphi}\otimes R$, where $\varphi\in \mathds{C}[x^{-1}]$ and where $R$ is a regular $\mathds{C}((x))$-module. The hypothesis on the slopes of $M$ implies $\varphi\neq 0$, and the expected vanishing is standard.
\end{proof}
\subsection{A vanishing criterion}\label{cycleprochbonne}
Let $\mathcal{M}$ be a germ of meromorphic connection at the origin $0\in \mathds{C}^{n}$. We suppose that $\mathcal{M}$ has good formal structure at $0$. Let $D$ be the pole locus of $\mathcal{M}$. Let $\rho_p$ be a ramification of degree $p$ along the components of $D$ as in \eqref{decomposition}.
\begin{proposition}\label{lemmeannulation}
Let $f\in \mathcal{O}_{\mathds{C}^{n},0}$. Let us define $Z:=\divi f$ and suppose that $|Z|\subset D$. We suppose that for every irreducible component $E$ of $|Z|$, we have $$r_E(\mathcal{M})\leq r v_E(f)$$
Then for every germ $N$ of meromorphic connection at 0 with slopes $>r$, we have
\begin{equation}\label{psinulle}
\psi_{f}(\mathcal{M}\otimes f^{+}N)\simeq 0
\end{equation}
in a neighbourhood of $0$.
\end{proposition}
\begin{proof}
Let us choose local coordinates $(x_1,\dots, x_n)$ and $a \in \mathds{N}^{n}$ such that $f$ is the function $x\longrightarrow x^a$.
Take $N$ with slopes $> r$. One can always suppose that $N$ is a $\mathds{C}((t))$-differential module and $p=qk$ where $\rho^{\prime}:t\longrightarrow t^{k}$ decomposes $N$.\\ \indent
The morphism $\rho_p$ is a finite cover away from $D$, so the canonical adjunction morphism
\begin{equation}\label{morphsurjectif}
\xymatrix{
\rho_{p+}\rho^{+}_p \mathcal{M} \ar[r]& \mathcal{M}
}
\end{equation}
is surjective away from $D$. So the cokernel of \eqref{morphsurjectif} has support in $D$. From \cite[3.6-4]{Mehbsmf}, we know that both sides of \eqref{morphsurjectif} are localized along $D$. So \eqref{morphsurjectif} is surjective. We thus have to prove
\begin{equation}\label{annulationvoulue}
\psi_{f\rho_p}(\rho^{ +}_p\mathcal{M}\otimes (f\rho_p)^{+}N)\simeq 0
\end{equation}
Since $|Z|\subset D$, we have $f\rho_p=\rho^{\prime}f\rho_q$. So the left hand side of \eqref{annulationvoulue} is a direct sum of $k$ copies of
\begin{equation}\label{kcopies}
\psi_{f\rho_q}(\rho^{ +}_p\mathcal{M}\otimes (f\rho_p)^{+}N)
\end{equation}
We thus have to prove that \eqref{kcopies} is 0 in a neighbourhood of 0. We have
$$
(f\rho_p)^{+}N\simeq (f\rho_q )^{+}\rho^{\prime +}N
$$
with $\rho^{\prime +}N$ decomposed with slopes $>rk$. The zero locus of $f\rho_q$ is $|Z|$, and if $E$ is an irreducible component of $|Z|$, the highest generic slope of $\rho^{+}_p\mathcal{M}$ along $E$ is $$r_{E}(\rho^{+}_p\mathcal{M})=p \cdot r_{E}(\mathcal{M})\leq r k \cdot q \cdot v_{E} (f)=rk \cdot v_{E} (f\rho_q)$$
Hence we can suppose that $\rho_p=\id$ and that $N$ is decomposed. \\ \indent
Take
$$
N=\mathcal{E}^{P(t)/t^{m}}\otimes R
$$
with $P(t)\in \mathds{C}[t]$ satisfying $P(0)\neq 0$, with $m>r$ and with $R$ regular. Since $\psi$ is insensitive to formalization, one can suppose
$$
\mathcal{M}=\mathcal{E}^{\varphi(x)}\otimes \mathcal{R}
$$
with $\varphi(x)$ as in \ref{bonnedecompositionformelle} $(3)$ and $\mathcal{R}$ regular. By Sabbah-Mochizuki theorem, the multiplicity of $-\divi\varphi(x)$ along a component $D^{\prime}$ of $D$ is a generic slope of $\mathcal{M}$ along $D^{\prime}$. Thus, one can write $\varphi(x)=g(x)/x^b$ where $g(0)\neq 0$ and where the $b_i$ are such that if $i\in \Supp a$, we have $b_i\leq r a_i<ma_i$.
We thus have to prove the
\begin{lemme}\label{lemmeannulation2}
Take $g, h\in \mathcal{O}_{\mathds{C}^{n},0}$ such that $g(0)\neq 0$ and $h(0)\neq 0$. Let $\mathcal{R}$ be a regular meromorphic connection with poles contained in $x_1\cdots x_n=0$. Take $a,b\in \mathds{N}^{\llbracket 1,n\rrbracket}$ such that $A:=\Supp a$ is non empty and $b_i< a_i$ for every $i\in A$. Then
$$
\psi_{x^a}(\mathcal{E}^{g(x)/x^b+h(x) /x^{a}}\otimes \mathcal{R})\simeq 0
$$
in a neighbourhood of 0.
\end{lemme}
\end{proof}
\subsection{Proof of \ref{lemmeannulation2}}
We define $\mathcal{M} :=\mathcal{E}^{g(x)/x^b+h(x) /x^{a}}\otimes \mathcal{R}$. Since $A$ is not empty, a change of variable allows one to suppose $h=1$. If $\Supp b\subset A$, a change of variable shows that \ref{lemmeannulation2} is a consequence of \ref{dernierlemmeannulation}. Let $i\in \Supp b$ be an integer such that $i\notin A$. Using $x_i$, a change of variable allows one to suppose $g=1$.
Let $p_1,\dots, p_n\in \mathds{N}^{\ast}$ such that $a_j p_j$ is independent from $j$ for every $j\in A$ and $p_j=1$ if $j\notin A$. Let $\rho_p$ be the morphism $x\longrightarrow x^p$. Like in \eqref{morphsurjectif}, we see that
$$
\xymatrix{
\rho_{p+}\rho^{+}_p \mathcal{M} \ar[r]& \mathcal{M}
}
$$
is surjective. We are thus left to prove that \ref{lemmeannulation2} holds for multi-indices $a$ such that $a_j$ does not depend on $j$ for every $j\in A$. Let us denote by $\mathds{1}_A$ the characteristic function of $A$. From \cite[3.3.13]{PTM}, it is enough to prove
$$
\psi_{x^{\mathds{1}_A}}(\mathcal{E}^{1/x^b+1 /x^{a}}\otimes \mathcal{R})\simeq 0
$$
Using the fact that $\mathcal{R}$ is a successive extension of regular modules of rank 1, one can suppose that $\mathcal{R}=x^{c}$, where $c\in \mathds{C}^{\llbracket 1,n\rrbracket}$. Let
\[
\xymatrix{
\mathds{C}^{n} \ar@{^{(}->}[r]^-{i} \ar[rd]_-{x^{\mathds{1}_A}}& \mathds{C}^{n} \times \mathds{C}\ar[d] \\
& \mathds{C}
}
\]
be the inclusion given by the graph of $x\longrightarrow x^{\mathds{1}_A}$. Let $t$ be a coordinate on the second factor of $\mathds{C}^{n} \times \mathds{C}$. We have to prove
$$
\psi_{t}(i_+(x^{c}\mathcal{E}^{1/x^b+1 /x^{a}}))\simeq 0
$$
Define $\delta:=\delta(t-x^{\mathds{1}_A})\in i_+(x^{c}\mathcal{E}^{1/x^b+1 /x^{a}})$ and let $(V_k)_{k\in \mathds{Z}}$ be the Kashiwara-Malgrange filtration on $\mathcal{D}_{\mathds{C}^{n} \times \mathds{C}}$ relative to $t$.
For $d\in \mathds{N}^{\llbracket 1,n\rrbracket}$ such that $x^d=0$ is the pole locus of $x^{c}\mathcal{E}^{1/x^b+1 /x^{a}}$, the family of sections $x^d$ generates $x^{c}\mathcal{E}^{1/x^b+1 /x^{a}}$. For such $d$, the family $s:=x^d\delta$ generates $i_+(x^{c}\mathcal{E}^{1/x^b+1 /x^{a}})$. We are left to prove $s\in V_{-1}s$. One can always suppose that $1\in A$.
$$
x_1\partial_1 s=(d_1+c_1)s-\frac{b_1}{x^{b}}s-\frac{a_1}{x^{a}}s- x^{\mathds{1}_A}\partial_t s
$$
We define $M\in \mathds{N}^{\llbracket 1,n\rrbracket}$ by $M_k=\max(a_k, b_k)$ for every $k\in \llbracket 1,n\rrbracket$. We thus have
\begin{equation}\label{eq0}
x^{M}x_1\partial_1 s=(d_1+c_1)x^{M}s-b_1x^{M-b}s-a_1x^{M-a}s- x^{M} x^{\mathds{1}_A}\partial_t s
\end{equation}
We have $M=a+b_{A^{c}}=\mathds{1}_A+ (a-\mathds{1}_A)+b_{A^{c}}=\mathds{1}_A+b+m$ with $m\in \mathds{N}^{\llbracket 1,n\rrbracket}$. So $$x^{M-b}s=x^m ts\in V_{-1}s$$
Moreover, we have
$$
x^{M}x_1\partial_1 s=x_1\partial_1 x^M s-M_1 x^M s=x_1\partial_1 x^{m+b} ts-M_1 x^{m+b} ts\in V_{-1}s
$$
and
$$
x^{M} x^{\mathds{1}_A}\partial_t s=x^{m+b}\partial_t x^{2\times\mathds{1}_A}s=x^{m+b}\partial_t t^{2}s =2x^{m+b} ts+x^{m+b}t(t\partial_t) s \in V_{-1}s
$$
So \eqref{eq0} gives
\begin{equation}\label{eq1}
x^{M-a}s \in V_{-1}s
\end{equation}
Let us recall that $i$ is such that $i\notin A$ and $i\in \Supp b$. In particular $(M-a)_i=b_i\neq 0$ and $\partial_i \delta=0$. Applying $x_i\partial_i$ to \eqref{eq1}, we obtain
$$
(d_i+c_i+b_i) x^{M-a}s -b_i\frac{ x^{M-a}}{x^b}s\in V_{-1}s
$$
so from \eqref{eq1}, we deduce $x^{M-a-b}s\in V_{-1}s$ . We have $M-a-b=-b_A$, so by multiplying $x^{M-a-b}s$ by $x^{b_A}$, we get $s\in V_{-1}s$.
\subsection{}
The aim of this paragraph is to prove the following
\begin{lemme}\label{dernierlemmeannulation}
Let $\alpha,a\in \mathds{N}^{\llbracket 1,n\rrbracket}$ such that $\Supp \alpha$ is not empty and $\Supp \alpha\subset \Supp a$. Let $\mathcal{R}$ be a regular meromorphic connection with poles contained in $x_1\cdots x_n=0$. We have
$$
\psi_{x^\alpha}(\mathcal{E}^{1/x^{a}}\otimes \mathcal{R})\simeq 0
$$
\end{lemme}
\begin{proof}
Let $p_1,\dots, p_n$ be integers such that $\alpha_i p_i$ does not depend of $i$ for every $i\in \Supp \alpha$ (we denote by $m$ this integer) and $p_i=1$ if $i\neq \Supp\alpha$. Let $\rho_p$ be the morphism $x\longrightarrow x^p$. Like in \eqref{morphsurjectif}, the morphism
$
\rho_{p+}\rho^{+}_p \mathcal{M} \longrightarrow \mathcal{M}
$
is surjective. We are left to prove \ref{dernierlemmeannulation} for $\alpha$ such that $\alpha_i$ does not depend of $i$ for every $i\in \Supp \alpha$. From \cite[3.3.13]{PTM}, one can suppose $\alpha_i=1$ for every $i\in \Supp \alpha$. So $\alpha\leq a$.\\ \indent
One can suppose $\mathcal{R}=x^{b}$ where $b\in \mathds{N}^{\llbracket 1,n\rrbracket}$.
Let
\[
\xymatrix{
\mathds{C}^{n} \ar@{^{(}->}[r]^-{i} \ar[rd]_-{x^{\alpha}}& \mathds{C}^{n} \times \mathds{C}\ar[d] \\
& \mathds{C}
}
\]
be the inclusion given by the graph of $x\longrightarrow x^{\alpha}$. Let $t$ be a coordinate on the second factor of $\mathds{C}^{n} \times \mathds{C}$. We have to show
$$
\psi_{t}(i_+(x^{b}\mathcal{E}^{1/x^{a}}))\simeq 0
$$
Define $\delta:=\delta(t-x^{\alpha})\in i_+(x^{b}\mathcal{E}^{1/x^{a}})$.
For $c\in \mathds{N}^{\llbracket 1,n\rrbracket}$ such that $\Supp c\subset \Supp a\cup \Supp b$, the family of sections $x^c$ generates $x^{b}\mathcal{E}^{1/x^{a}}$. For such $c$, the family $s:=x^c\delta$ generates $i_+(x^{b}\mathcal{E}^{1/x^{a}})$. It is thus enough to show $s\in V_{-1}s$. Let us choose $i\in \Supp \alpha$. We have
$$
x_i\partial_i s=(c_i+b_i)s-\frac{a_i}{x^{a}}s- x^{\alpha}\partial_t s
$$
We have $\alpha\leq a$. Define $a=\alpha+a^{\prime}$. From
$$
x^{\alpha}x_i\partial_i s =x_i\partial_i x^{\alpha} s-x^{\alpha} s=x_i\partial_i t s-ts\in V_{-1}s
$$
we deduce that $a_i s+x^{a^{\prime}}x^{2\alpha}\partial_t s \in V_{-1}s$.
We also have $x^{2\alpha}\partial_t s=\partial_t x^{2\alpha} s=\partial_t t^{2} s=2ts+t (t\partial_t)s\in V_{-1}s$. Since $a_i\neq 0$, we deduce $s\in V_{-1}s$ and \ref{dernierlemmeannulation} is proved.
\end{proof}
\section{Proof of theorem \ref{theoremprincipal} }
\subsection{Dévissage to the case of meromorphic connections }\label{reduction}
Suppose that theorem \ref{theoremprincipal} is true for meromorphic connections for every choice of ambient manifold. Let us show that theorem \ref{theoremprincipal} is true for $\mathcal{M}\in \mathcal{D}_{\hol}^{b}(X)$. We argue by induction on $\dim X$. The case where $X$ is a point is trivial. Let us suppose that $\dim X>0$. We define $Y:=\Supp \mathcal{M}$ and we argue by induction on $\dim Y$.
\\ \indent
Let us suppose that $Y$ is a strict closed subset of $X$. We denote by $i:Y\longrightarrow X$ the canonical inclusion. Let $\pi:\tilde{Y}\longrightarrow Y$ be a resolution of the singularities of $Y$ \cite{AHV} and $p:=i\pi$. The regular locus $\Reg Y$ of $Y$ is a dense open subset in $Y$ and $\pi$ is an isomorphism above $\Reg Y$. By Kashiwara theorem, we deduce that the cone $\mathcal{C}$ of the adjunction morphism
$$
\xymatrix{
p_{+}p^{\dag}\mathcal{M} \ar[r]& \mathcal{M}
}
$$
has support in $\Sing Y$, with $\Sing Y$ a strict closed subset in $Y$. Let $x\in X$ and let $B$ be a neighbourhood of $x$ with compact closure $\overline{B}$. Then, $p^{-1}(\overline{B})$ is compact. Since $\dim \tilde{Y}<\dim X$, theorem \ref{theoremprincipal} is true for $p^{\dag}\mathcal{M}\in \mathcal{D}_{\hol}^{b}(\tilde{Y})$. Let $(U_i)$ be a finite family of open sets in $\tilde{Y}$ covering $p^{-1}(\overline{B})$ and such that for every $i$, the set $\Sl^{\nb}((p^{\dag}\mathcal{M})_{|U_i})$ is bounded by a rationnal $r_i$. Define $R=\max_i r_i$. \\ \indent
By induction hypothesis applied to $\mathcal{C}$, one can suppose at the cost of taking a smaller $B$ containing $x$
that the set $\Sl^{\nb}(\mathcal{C}_{|B})$ is bounded by a rational $R^{\prime}$. Take $f\in \mathcal{O}_B$. We have a distinguished triangle
\begin{equation}\label{trianglede psi}
\xymatrix{
\psi_{f}(p_{+}p^{\dag}\mathcal{M} \otimes f^{+}N) \ar[r]& \psi_{f}( \mathcal{M}\otimes f^{+}N)\ar[r]& \psi_{f}(\mathcal{C}\otimes f^{+}N) \ar[r]^-{+1}&
}
\end{equation}
By projection formula and compatibility of $\psi$ with proper direct image, \eqref{trianglede psi} is isomorphic to
$$
\xymatrix{
p_{+}\psi_{fp}(p^{\dag}\mathcal{M} \otimes (pf)^{+}N) \ar[r]& \psi_{f}( \mathcal{M}\otimes f^{+}N)\ar[r]& \psi_{f}(\mathcal{C}\otimes f^{+}N) \ar[r]^-{+1}&
}
$$
So we have the desired vanishing on $B$ for $r>\max(R,R^{\prime})$.
\\ \indent
We are left with the case where $\dim \Supp \mathcal{M}=\dim X$. Let $Z$ be a hypersurface containing $\Sing \mathcal{M}$. We have a triangle
$$
\xymatrix{
R\Gamma_{[Z]}\mathcal{M} \ar[r]& \mathcal{M} \ar[r]& \mathcal{M}(\ast Z) \ar[r]^-{+1}&
}
$$
By applying the induction hypothesis to $R\Gamma_{[Z]}\mathcal{M}$, we are left to prove theorem \ref{theoremprincipal} for $ \mathcal{M}(\ast Z) $. The module $ \mathcal{M}(\ast Z) $ is a meromorphic connection, which concludes the reduction step.
\subsection{The case of meromorphic connections}
At the cost of taking an open cover of $X$, let us take a resolution of turning points $p:\tilde{X}\longrightarrow X$ for $\mathcal{M}$ as given by Kedlaya-Mochizuki theorem. Let $D$ be the pole locus of $\mathcal{M}$. Since $p$ is an isomorphism above $X\setminus D$, the cone of
\begin{equation}\label{conemorphs0}
\xymatrix{
p_+ p^{+ }\mathcal{M} \ar[r]& \mathcal{M}
}
\end{equation}
has support in the pole locus $D$ of $\mathcal{M}$.
From \cite[3.6-4]{Mehbsmf}, the left hand side of \eqref{conemorphs0} is localized along $D$. So \eqref{conemorphs0} is an isomorphism. We thus have a canonical isomorphism
$$
p_{+ }\psi_{fp}(p^{+ }\mathcal{M}\otimes (fp)^{+}N)\simeq
\psi_{f}(\mathcal{M}\otimes f^{+}N)
$$
Since $p$ is proper, we see as in \ref{reduction} that we are left to prove theorem \ref{theoremprincipal} for $p^{+ }\mathcal{M}$. We thus suppose that $\mathcal{M}$ has a good formal structure. At the cost of taking an open cover, we can suppose that $D$ has only a finite number of irreducible components. Let $S$ be the divisor of highest generic slopes \ref{divplusgrandepente} of $\mathcal{M}$. Let $S_1, \dots, S_m$ be the irreducible components of $|S|$. Let us prove that $\Sl^{\nb}(\mathcal{M})$ is bounded by $\deg S$. It is is a local statement. Let $f\in \mathcal{O}_X$ and define $Z:=\divi f$. Let us denote by $|Z|$ (resp. $|S|$) the support of $Z$ (resp. $S$) and let us admit for a moment the validity of the following
\begin{proposition}\label{propositionprincipale}
Locally on $X$, one can find a proper birationnal morphism $\pi :\tilde{X}\longrightarrow X$ such that
\begin{enumerate}
\item\label{cond1prop} $\pi$ is an isomorphism above $X\setminus |Z|$.
\item\label{cond2prop} $\pi^{-1}(|Z|)\cup \pi^{-1}(|S|)$ is a normal crossing divisor.
\item\label{cond3prop} for every valuation $v_E$ measuring the vanishing order along an irreducible component $E$ of $\pi^{-1}(|Z|)$, we have
$$
v_E(S)\leq (\deg S) v_E(f)
$$
\end{enumerate}
\end{proposition}
Let us suppose that \ref{propositionprincipale} is true. At the cost of taking an open cover, let us take a morphism $\pi:\tilde{X}\longrightarrow X$ as in \ref{propositionprincipale}. Since condition \eqref{cond1prop} is true, the cone of the canonical comparison morphism
\begin{equation}\label{conemorphs}
\xymatrix{
\pi_+ \pi^{+ }\mathcal{M} \ar[r]& \mathcal{M}
}
\end{equation}
has support in $|Z|$. Since $f^{+}N$ is localized along $|Z|$, we deduce that \eqref{conemorphs} induces an isomorphism
$$
\xymatrix{
(\pi_+ \pi^{+ }\mathcal{M}) \otimes f^{+}N \ar[r]^-{\sim}& \mathcal{M}\otimes f^{+}N
}
$$
Applying $\psi_f$ and using the fact that $\pi$ is proper, we see that it is enough to prove
\begin{equation}\label{ptitequation}
\psi_{f\pi}(\pi^{+}\mathcal{M}\otimes (f\pi)^{+}N)\simeq 0
\end{equation}
for every germ $N$ of meromorphic connection at the origin with slope $r >\deg S$. Since $(f\pi)^{+}N$ is localized along $\pi^{-1}(|Z|)$, the left-hand side of \eqref{ptitequation} is
\begin{equation}\label{psiapresresolution}
\psi_{f\pi}((\pi^{+}\mathcal{M})(\ast \pi^{-1}(|Z|))\otimes (f\pi)^{+}N)
\end{equation}
The vanishing of \eqref{psiapresresolution} is a local statement on $\tilde{X}$. Since \eqref{cond2prop} and \eqref{cond3prop} are true, \ref{lemmeannulation} asserts that it is enough to show that for every irreducible component $E$ of $\pi^{-1}(|Z|)$, we have
$$r_E((\pi^{+}\mathcal{M})(\ast \pi^{-1}(|Z|)))\leq (\deg S) v_E(f\pi)$$
Let us notice that $v_E(f\pi)=v_E(f)$. Let $P$ be a point in the smooth locus of $E$. Let
$\varphi$ as in \eqref{decomposition} for $\mathcal{M}$ at the point $Q:=\pi(P)$. For $i=1,\dots, n$, let $t_i=0$ be an equation of $S_i$ in a neighbourhood of $Q$. Modulo a unit in $\mathcal{O}_{X,Q}$, we have $\varphi=1/t_1^{r_1}\cdots t_n^{r_n}$ where $r_i\in \mathds{Q}_{\geq 0}$. If $u=0$ is a local equation for $E$ in a neighbourhood of $P$, we have modulo a unit in $\mathcal{O}_{\tilde{X},P}$
$$\varphi \pi=\frac{1}{u^{r_1v_E(t_1)}\cdots u^{r_n v_E(t_n)}}$$
So the slope of $\mathcal{E}^{\varphi \pi}(\ast \pi^{-1}(|Z|))$ along $E$ is $r_1v_E(t_1)+\cdots +r_n v_E(t_n)$. By Sabbah-Mochizuki theorem, $r_i$ is a slope of $\mathcal{M}$ generically along $S_i$, so $r_i\leq r_{S_i}( \mathcal{M})$. We deduce that
$$
r_E(\pi^{+}\mathcal{M}(\ast \pi^{-1}(|Z|)))\leq
\sum_i r_{S_i}( \mathcal{M}) v_E(t_i)=v_E(S)\leq (\deg S) v_E(f)
$$
This concludes the proof of theorem \ref{theoremprincipal} and theorem \ref{theoremprincipalraffiné}.
\subsection{Proof of \ref{propositionprincipale}}
At the cost of taking an open cover of $X$, let us take a finite sequence of blow-up
\begin{equation}\label{resolution}
\xymatrix{
\pi_n:X_n\ar[r]^-{p_{n-1}} & X_{n-1}\ar[r]^-{p_{n-2}} & \cdots \ar[r]& X_1\ar[r]^-{p_0} & X_0=X
}
\end{equation}
given by 3.15 and 3.17 of \cite{BMUniformization} for $Z$ relatively to the normal crossing divisor $|S|$. Let $|Z|_i$ be the strict transform of $|Z|$ in $X_i$ and let $C_i$ be the center of $p_i$. We define inductively $H_0=H$ and $H_{i+1}=p_{i}^{-1}(H_{i})\cup p^{-1}_i(C_i)$ for $i=1,\dots, n$, where $p_{i}^{-1}$ denotes the set theoretic inverse image. In particular $H_{i+1}$ is a closed subset of $X_{i+1}$. We will endow it with its canonical reduced structure. Then, \eqref{resolution} satisfies \\ \indent
$(i)$ $C_i$ is a smooth closed subset of $|Z|_i$.\\ \indent
$(ii)$ $C_i$ is nowhere dense in $|Z|_i$.\\ \indent
$(iii)$ $C_i$ and $H_i$ have normal crossing for every $i$.\\ \indent
$(iv)$ $|Z|_n\cup H_n$ is a normal crossing divisor.\\ \noindent
Since $C_i$ and the components of $H_i$ are reduced and smooth, condition $(iii)$ means that locally on $X_i$, one can find coordinates $(x_1,\dots, x_k)$ such that $H_i$ is given by the equation $x_1\cdots x_l=0$ and the ideal of $C_i$ is generated by some $x_j$ for $j=1,\dots, k$.
Using condition $(i)$, we see by induction that $\pi^{-1}_n(|Z|)\cup \pi^{-1}_n(|S|)=|Z|_n\cup H_n$. Proposition \ref{propositionprincipale} is thus a consequence of
\begin{proposition}\label{casparticulier}
Let
$$
\xymatrix{
\pi_n:X_n\ar[r]^-{p_{n-1}} & X_{n-1}\ar[r]^-{p_{n-2}} & \cdots \ar[r]& X_1\ar[r]^-{p_0} & X_0=X
}
$$
be a sequence of blow-up satisfying $(i)$,$(ii)$ and $(iii)$. For every irreducible component $E$ of $\pi^{-1}_n(|Z|)$, we have
\begin{equation}\label{inegalitevoulue}
v_E(S)\leq (\deg S) v_E(f)
\end{equation}
\end{proposition}
\begin{proof}
Let $S_1, \dots, S_m$ be the irreducible components of $|S|$ and let $Z_1, \dots, Z_{m^{\prime}}$ be the irreducible components of $Z$. Note that some $Z_i$ can be in $|S|$. We define $a_i=v_{Z_i}(f)>0$ and let $Z_{ji}$ (resp. $S_{ji}$) be the strict transform of $Z_j$ (resp. $S_j$) in $X_i$.\\ \indent
We argue by induction on $n$. If $n=0$, $E$ is one of the $Z_i$ and then \eqref{inegalitevoulue} is obvious. We suppose that \eqref{inegalitevoulue} is true for a composite of $n$ blow-up and we prove that \eqref{inegalitevoulue} is true for a composite of $n+1$ blow-up. \\ \indent
Let $\mathcal{C}_n$ be the set of irreducible components of $$\displaystyle{\bigcup_{i=0}^{n-1} (p_{n-1}\cdots p_i)^{-1}(C_i)}$$
Each element $E\in \mathcal{C}_n$ will be endowed with its reduced structure. Condition $(i)$ implies that the irreducible components of $\pi_n^{\ast}Z$ are the $Z_{in}$ and the elements of $\mathcal{C}_n$. Condition $(ii)$ implies that none of the $Z_{in}$ belongs to $\mathcal{C}_n$. Thus, we have
$$
\pi_n^{\ast}Z=\divi f\pi_n=a_1 Z_{1n}+\cdots +a_{m^{\prime}}Z_{m^{\prime}n}+\displaystyle{\sum_{E\in \mathcal{C}_n}} v_E(f) E
$$
On the other hand, we have
$$
\pi_n^{\ast}S=r_{S_1}( \mathcal{M}) S_{1n}+\cdots+ r_{S_m}( \mathcal{M}) S_{mn}+\displaystyle{\sum_{E\in \mathcal{C}_n}} v_{E}(S) E
$$
Let us consider the last blow-up
$p_{n}: X_{n+1}\longrightarrow X_{n}$. Let us denote by $P$ the exceptionnal divisor of $p_{n}$ and let $E_{n+1}$ be the strict transform of $E\in \mathcal{C}_n$ in $X_{n+1}$.
We have
$$
p_{n}^{\ast} Z_{in}=Z_{in+1}+\alpha_i P \text{\quad \quad with $\alpha_i\in \mathds{N}$}
$$
Since
$$
H_n=\displaystyle{\bigcup_{j=0}^{m}} S_{jn} \cup \displaystyle{\bigcup_{E\in \mathcal{C}_n}} E
$$
we deduce from condition $(iii)$ and smoothness of $C_n$ that
$$
p_{n}^{\ast} E=E_{n+1}+ \epsilon_E P
\text{\quad \quad with $\epsilon_E\in \{0,1\}$}
$$
and
$$
p_{n}^{\ast} S_{in}=S_{in+1}+ \epsilon_i P \text{\quad \quad with $\epsilon_i\in \{0,1\}$}
$$
Hence, we have
$$
\pi_{n}^{\ast}Z=\sum a_iZ_{in+1}+\displaystyle{\sum_{E\in \mathcal{C}_n}} v_E(f) E_{n+1}+(\sum a_i\alpha_i+\sum_{E\in \mathcal{C}_n} \epsilon_E v_E(f))P
$$
and
$$
\pi_{n}^{\ast}S=\sum r_{S_i}( \mathcal{M}) S_{in+1}+\displaystyle{\sum_{E\in \mathcal{C}_n}} v_E(S) E_{n+1}+ (\sum r_{S_i}( \mathcal{M}) \epsilon_i+\sum_{E\in \mathcal{C}_n} \epsilon_E v_{E}(S))P
$$
Formula \eqref{inegalitevoulue} is true for the $Z_{in+1}$. By induction hypothesis, formula \eqref{inegalitevoulue} is true for $E_{n+1}$, where $E\in \mathcal{C}_n$. We are left to prove that \eqref{inegalitevoulue} is true for $P$. Conditions $(i)$ and $(ii)$ imply that one of the $\alpha_i$ is non zero, so
\begin{align*}
(\deg S)\left(\sum a_i\alpha_i+\sum \epsilon_E v_E(f)\right) &\geq (\deg S)+(\deg S)\sum \epsilon_E v_E(f) \\
&\geq \sum r_{S_i}( \mathcal{M}) \epsilon_i+\sum \epsilon_E (\deg S)v_E(f) \\
&\geq \sum r_{S_i}( \mathcal{M}) \epsilon_i+\sum \epsilon_E v_E(S)
\end{align*}
\end{proof}
\section{Duality}
We prove theorem \ref{theoremprincipal-1} $(i)$. Let us denote by $\mathds{D}$ the duality functor for $\mathcal{D}$-modules. There is a canonical comparison morphism
\begin{equation}\label{comparaison}
\xymatrix{
\mathds{D}(\mathcal{M}\otimes f^{+}N)\ar[r]& \mathds{D}\mathcal{M} \otimes f^{+}\mathds{D} N
}
\end{equation}
On a punctured neighbourhood of $0\in \mathds{C}$, the module $N$ is isomorphic to a finite sum of copies of the trivial connection. Thus, there is a neighbourhood $U$ of $Z$ such that the restriction of \eqref{comparaison} to $U\setminus Z$ is an isomorphism. Hence, the cone of \eqref{comparaison} has support in $Z$. We deduce that
$$
\xymatrix{
(\mathds{D}(\mathcal{M}\otimes f^{+}N))(\ast Z)\ar[r]^-{\sim}&\mathds{D} \mathcal{M} \otimes f^{+}((\mathds{D}N)(\ast 0))
}
$$
We have $(\mathds{D}N)(\ast 0)\simeq N^{\ast}$, where $\ast$ is the duality functor for meromorphic connection. Note that $\ast$ is a slope preserving involution. Since nearby cycles are insensitive to localization and commute with duality for $\mathcal{D}$-modules, we have
$$
\psi_f(\mathds{D}\mathcal{M} \otimes f^{+}N^{\ast})\simeq \mathds{D}(\psi_f(\mathcal{M}\otimes f^{+}N))
$$
and theorem \ref{theoremprincipal-1} $(i)$ is proved.
\section{Regularity and nearby cycles}
The aim of this section is to prove theorem \ref{comparaisonreg}.
\subsection{} We will use the following
\begin{lemme}\label{lemmeevitement}
Let $F$ be a germ of closed analytic subspace at the origin $0\in \mathds{C}^{n}$. Let $Y_1, \dots, Y_k$ be irreducible closed analytic subspaces of $\mathds{C}^{n}$ containing 0 and such that $F\cap Y_i$ is a strict closed subset of $Y_i$ for every $i$. Then, there exists a germ of hypersurface $Z$ at the origin containing $F$ and such that $Z\cap Y_i$ has codimension 1 in $Y_i$ for every $i$.
\end{lemme}
\begin{proof}
Denote by $\mathcal{I}_{F}$ (resp. $\mathcal{I}_{Y_i}$) the ideal sheaf of $F$ (resp. $Y_i$). By irreducibility, $\mathcal{I}_{Y_i, 0}$ is a prime ideal in $\mathcal{O}_{\mathds{C}^{n},0}$. The hypothesis say $\mathcal{I}_{F}\nsubseteq \mathcal{I}_{Y_i}$ for every $i$. From \cite[1.B]{Mat2}, we deduce
$$\mathcal{I}_{F}\nsubseteq \bigcup_i\mathcal{I}_{Y_i}$$
Any function $f\in \mathcal{I}_{F}$ not in $\bigcup_i\mathcal{I}_{Y_i}$ defines a hypersurface as wanted.
\end{proof}
\subsection{}\label{lemmemeb}
We say that a holonomic module $\mathcal{M}$ is \textit{smooth} if the support $\Supp\mathcal{M}$ of $\mathcal{M}$ is smooth equidimensional and if the characteristic variety of $\mathcal{M}$ is equal to the conormal of $\Supp \mathcal{M}$ in $X$. We denote by $\Sing \mathcal{M}$ the complement of the smooth locus of $\mathcal{M}$. It is a strict closed subset of $\Supp \mathcal{M}$.\\ \indent
Let $x\in X$ and let us define $F$ as the union of $\Sing \mathcal{M}$ with the irreducible components of $\Supp \mathcal{M}$ passing through $x$ which are not of maximal dimension. Define $Y_1, \dots, Y_k$ to be the irreducible components of $\Supp \mathcal{M}$ of maximal dimension passing through $x$. From \ref{lemmeevitement}, one can find a hypersurface $Z$ passing through $x$ such that
\begin{enumerate}
\item $Z\cap \Supp \mathcal{M}$ has codimension $1$ in $\Supp \mathcal{M}$.
\item The cohomology modules of $\mathcal{H}^{k}\mathcal{M}$ are smooth away from $Z$.
\item $\dim \Supp R\Gamma_{[Z]}\mathcal{M}<\dim \Supp \mathcal{M}$.
\end{enumerate}
\subsection{}\label{implicationdirecte} The direct implication of theorem \ref{comparaisonreg} is a consequence of the preservation of regularity by inverse image and the following
\begin{proposition}\label{premierinclusion}
We have $\mathcal{D}_{\hol}^b(X)_{\reg}\subset \mathcal{D}_{\hol}^b(X)_{\{0\}}$.
\end{proposition}
\begin{proof}
Take $\mathcal{M}\in\mathcal{D}_{\hol}^b(X)_{\reg}$. We argue by induction on $\dim X$. The case where $X$ is a point is trivial. By arguing on $\dim \Supp \mathcal{M}$ as in \ref{reduction}, we are left to prove \ref{premierinclusion} in the case where $\mathcal{M}$ is a regular meromorphic connection. Let $D$ be the pole locus of $\mathcal{M}$. Take $f\in \mathcal{O}_X$ and let $N$ with slope $>0$. To prove
$$
\psi_{f}(\mathcal{M}\otimes f^{+}N)\simeq 0
$$
one can suppose using embedded desingularization that $D+\divi f$ is a normal crossing divisor. We then conclude with \ref{lemmeannulation}.
\end{proof}
\subsection{}
To prove the reverse implication of theorem \ref{comparaisonreg}, we argue by induction on $\dim X\geq 1$. The case of curves follows from
\ref{propdim1}. We suppose that $\dim X\geq 2$ and we take
$\mathcal{M}\in \mathcal{D}_{\hol}^b(X)_{\{0\}}$. We argue by induction on $\dim \Supp \mathcal{M}$. The case where $\Supp \mathcal{M}$ is punctual is trivial.\\ \indent
Suppose that $0<\dim\Supp \mathcal{M}<\dim X$. Since $\Supp \mathcal{M}$ is a strict closed subset of $X$, one can always locally write $X=X^{\prime}\times D$ where $D$is the unit disc of $\mathds{C}$ and where the projection $X^{\prime}\times D\longrightarrow X^{\prime}$ is finite on $\Supp \mathcal{M}$. Let $i:X^{\prime}\times D\longrightarrow X^{\prime}\times \mathds{P}^1$ be the canonical immersion. There is a commutative diagram
\begin{equation}\label{diagcom}
\xymatrix{
\Supp \mathcal{M} \ar[r] \ar[rd] &X^{\prime}\times \mathds{P}^1\ar[d]^-{p}\\
& X^{\prime}
}
\end{equation}
The oblique arrow of \eqref{diagcom} is finite, and $p$ is proper. So the horizontal arrow is proper. Thus, $\Supp \mathcal{M}$ is a closed subset in $X^{\prime}\times \mathds{P}^1$. Hence, $\mathcal{M}$ can be extended by $0$ to $X^{\prime}\times \mathds{P}^1$. We still denote by $\mathcal{M}$ this extension. It is an object of $\mathcal{D}_{\hol}^b(X^{\prime}\times \mathds{P}^1)_{\{0\}}$ and we have to show that it is regular.\\ \indent
Let $Z$ be a divisor in $X^{\prime}$ given by the equation $f=0$ and let $\rho: Y\longrightarrow X^{\prime}$ be a finite morphism. Since $p$ is smooth, the analytic space $Y^{\prime}$ making the following diagram
$$
\xymatrix{
Y^{\prime} \ar[r]^-{\rho^{\prime}} \ar[d]_-{p^{\prime}} &X^{\prime}\times \mathds{P}^1\ar[d]^-{p}\\
Y \ar[r]_-{\rho} & X^{\prime}
}
$$
cartesian is smooth. Moreover $\rho^{\prime}$ is finite. By base change \cite[1.7.3]{HTT}, projection formula and compatibility of $\psi$ with proper direct image, we have for every germ $N$ of meromorphic connection with slope $>0$
\begin{align*}
\psi_{f}(\rho^{+}p_+\mathcal{M}\otimes f^{+}N)&\simeq \psi_{f}(p_+^{\prime}\rho^{\prime +}\mathcal{M}\otimes f^{+}N)\\
&\simeq
\psi_{f}(p_+^{\prime}(\rho^{\prime +}\mathcal{M}\otimes (fp^{\prime})^{+}N))\\
&\simeq
p_+^{\prime}\psi_{fp^{\prime}}(\rho^{\prime +}\mathcal{M}\otimes (fp^{\prime})^{+}N)\\
&\simeq 0
\end{align*}
By induction hypothesis $p_+\mathcal{M}$ is regular. Let $Y_1, \dots, Y_n$ be the irreducible components of $\Supp \mathcal{M}$ with maximal dimension. Since $\Sing \mathcal{M}\cap Y_i$ is a strict closed subset of $Y_i$ and since a finite morphism preserves dimension, $p(\Sing \mathcal{M})\cap p(Y_i)$ is a strict closed subset of the irreducible closed set $p(Y_i)$. In a neighbourhood of a given point of $p(\Sing \mathcal{M})$, one can find from \ref{lemmemeb} a hypersurface $Z$ containing $p(\Sing \mathcal{M})$ such that $Z\cap p(Y_i)$ has codimension 1 in $p(Y_i)$ for every $i$. So $p^{-1}(Z)$ contains $\Sing \mathcal{M}$ and
$$\dim p^{-1}(Z)\cap Y_i=\dim Z \cap p(Y_i)=\dim p(Y_i)-1=\dim Y_i-1$$
Since $\Irr^{\ast}_{Z}$ is compatible with proper direct image \cite[3.6-6]{Mehbsmf}, we have
$$
\Irr^{\ast}_{Z}p_+\mathcal{M}\simeq Rp_{\ast}\Irr^{\ast}_{p^{-1}(Z)}\mathcal{M}\simeq 0
$$
Since $p$ is finite over $\Supp \mathcal{M}$, we have
$$
Rp_{\ast}\Irr^{\ast}_{p^{-1}(Z)}\mathcal{M} \simeq p_{\ast}\Irr^{\ast}_{p^{-1}(Z)}\mathcal{M}
$$
So for every $x\in p^{-1}(Z)$, the germ of $\Irr^{\ast}_{p^{-1}(Z)}\mathcal{M}$ at $x$ is a direct factor of the complex $(p_{\ast}\Irr^{\ast}_{Z}p_+\mathcal{M})_{p(x)}\simeq 0$. Thus $\Irr^{\ast}_{p^{-1}(Z)}\mathcal{M}\simeq 0$. From \cite[4.3-17]{Mehbsmf}, We deduce that $\mathcal{M}(\ast p^{-1}(Z))$ is regular. \\ \indent
To show that $\mathcal{M}$ is regular, we are left to prove that $R\Gamma_{[p^{-1}(Z)]}\mathcal{M}$ is regular. From \ref{implicationdirecte}, the nearby slopes of all quasi-finite inverse images of $\mathcal{M}(\ast p^{-1}(Z))$ are contained in $\{0\}$. Thus, this is also the case for $R\Gamma_{[p^{-1}(Z)]}\mathcal{M}$. By construction of $Z$, $$\dim \Supp R\Gamma_{[p^{-1}(Z)]}\mathcal{M} < \dim \Supp \mathcal{M}$$
We conclude by applying the induction hypothesis to $R\Gamma_{[p^{-1}(Z)]}\mathcal{M}$. \\ \indent
Let us suppose that $\Supp \mathcal{M}$ has dimension $\dim X$, and let $Z$ be a hypersurface as in \ref{lemmemeb}. Then $\mathcal{M}(\ast Z)$ is a meromorphic connection with poles along $Z$. Let us show that $\mathcal{M}(\ast Z)$ is regular. By \cite[4.3-17]{Mehbsmf}, it is enough to prove regularity generically along $Z$. Hence, one can suppose that $Z$ is smooth. By Malgrange theorem \cite{Reseaucan}, one can suppose that $Z$ is smooth and that $\mathcal{M}(\ast Z)$ has good formal structure along $Z$. Let $(x_1,\dots, x_n, t)$ be coordinates centred at $0\in Z$ such that $Z$ is given by $t=0$ and let
$\rho: (x,u)\longrightarrow (x,u^{p})$ be as in \ref{bonnedecompositionformelle} for $\mathcal{M}(\ast Z)$. Let
$ \mathcal{E}^{g(x,u)/u^{k}}\otimes \mathcal{R}$ be a factor of $\rho^{+}(\hat{\mathcal{M}}_0(\ast Z))$
where $g(0,0)\neq 0$ and where $\mathcal{R}$ is a regular meromorphic connection with poles along $Z$. For a choice of $k$-th root in a neighbourhood of $g(0,0)$, we have
$$
\psi_{u/\sqrt[k]{g}}(\rho^{+}\mathcal{M}\otimes (u/\sqrt[k]{g})^{+}\mathcal{E}^{-1/u^{k}})\simeq 0
$$
Since nearby cycles commute with formalization, we deduce
$$
\psi_{u}(\rho^{+}(\hat{\mathcal{M}}_0(\ast Z))\otimes \mathcal{E}^{-g/u^{k}})\simeq \psi_{u}(\rho^{+}\hat{\mathcal{M}}_0\otimes \mathcal{E}^{-g/u^{k}})\simeq 0
$$
Thus $\psi_u \mathcal{R}\simeq 0$, so $\mathcal{R}\simeq 0$. Hence, the only possibly non zero factor of $\rho^{+}(\hat{\mathcal{M}}_0(\ast Z))$ is the regular factor. So $\mathcal{M}(\ast Z)$ is regular. We obtain that $\mathcal{M}$ is regular by applying the induction hypothesis to $R\Gamma_{[Z]}\mathcal{M}$.
\section{Slopes and irregular periods}
\subsection{}\label{DRrapide}
Let $X$ be a smooth complex manifold and let $D$ be a normal crossing divisor in $X$. Define $U:= X\setminus D$ and let $j:U \longrightarrow X$ be the canonical inclusion. Let $\mathcal{M}$ be a meromorphic connexoin on $X$ with poles along $D$.\\ \indent
We denote by $p:\tilde{X}\longrightarrow X$ the real blow-up of $X$ along $D$. Let $\mathcal{A}_{\tilde{X}}^{<D}$ be the sheaf \cite[II]{Sabbahdim} of differentiable functions on $\tilde{X}$ whose restriction to $X$ are holomorphic and whose asymptotic development along $p^{-1}(D)$ are zero. We define the \textit{de Rham complex with rapid decay} by
$$
\DR^{<D}_{\tilde{X}}\mathcal{M}:=\mathcal{A}_{\tilde{X}}^{<D}\otimes_{p^{-1}\mathcal{O}_{X}}p^{-1}\DR_{X}\mathcal{M}
$$
\subsection{}
With the notations in \ref{DRrapide}, if $\mathcal{M}$ has good formal structure along $D$, we define \cite[Prop 2]{HienInv}
$$
H^{\rd}_k(X,\mathcal{M}):=H^{2d-k}(\tilde{X},\DR^{<D}_{\tilde{X}}\mathcal{M})
$$
The left-hand side is the space of \textit{cycles with rapid decay for $\mathcal{M}$}. For a topological description justifying the terminology, we refer to \cite[5.1]{HienInv}.
\subsection{Proof of theorem \ref{GMirr}}
We denote by $j:U\longrightarrow X$ the canonical immersion, $d:=\dim X$ and $\Sl_0(\mathcal{H}^{k}f_+\mathcal{E})$ the slopes of $\mathcal{H}^{k}f_+\mathcal{E}$ at 0. We will also use the letter $f$ for the restriction of $f$ to $U$. From \cite[4.7.2]{HTT}, we have a canonical identification
\begin{equation}\label{isoan}
\xymatrix{
(f_+\mathcal{E})^{\an}\simeq (f_{+}(j_+\mathcal{E}))^{\an}\ar[r]^-{\sim} & f_{+}^{\an}(j_+\mathcal{E})^{\an}
}
\end{equation}
We deduce
$$
\Sl_0(\mathcal{H}^{k}f_+\mathcal{E})=\Sl_0(\mathcal{H}^{k}f_{+}^{\an}(j_+\mathcal{E})^{\an})
$$
Let $x$ be a local coordinate on $S$ centred at the origin. From \ref{propdim1}, we have
$$
\Sl_0(\mathcal{H}^{k}f_{+}^{\an}(j_+\mathcal{E})^{\an})=\Sl_x^{\nb}(\mathcal{H}^{k}f_{+}^{\an}(j_+\mathcal{E})^{\an})
$$
Since $\Sl_x^{\nb}(\mathcal{H}^{k}f_{+}^{\an}(j_+\mathcal{E})^{\an})\subset \Sl_x^{\nb}(f_{+}^{\an}(j_+\mathcal{E})^{\an})$, we deduce from theorem \ref{theoremprincipalraffiné} and theorem \ref{theoremprincipal-1}
$$
\Sl_0(\mathcal{H}^{k}f_+\mathcal{E})\subset \Sl_{f(x)}^{\nb}((j_+\mathcal{E})^{\an})\subset [0,r_1+\cdots +r_n]
$$
We are thus left to relate $\Sol(\mathcal{H}^{k}f_{+}^{\an}(j_+\mathcal{E})^{\an})$ to the periods of $\mathcal{E}_{t}$, for $t\neq 0$ close enough to $0$. Such a relation appears for a special type of rank 1 connections in \cite{HR}. We prove more generally the following
\begin{proposition}\label{derniereprop}
For every $k$, we have a canonical isomorphism
\begin{equation}\label{isocannonique}
\xymatrix{
R^{k}f_{\ast}^{\an}\Sol (j_+\mathcal{E})^{\an}\ar[r]^-{\sim}& R^{k}(f^{\an}p)_{\ast}\DR^{<D}_{\tilde{X}}(j_+\mathcal{E}^{\ast})^{\an}
}
\end{equation}
For $t\neq 0$ close enough to 0, the fiber of the right-hand side of \eqref{isocannonique} at $t$ is canonically isomorphic to $H_{2d-2-k}^{\rd}(U_t, \mathcal{E}^{\ast}_t)$.
\end{proposition}
\begin{proof}
Set
$
\mathcal{M}:=(j_+\mathcal{E})^{\an}
$. Since $\mathcal{M}$ has good formal structure, we know from
\cite[5.3.1]{MemoireMochizuki} that
$$
Rp_{\ast}\DR^{<D}_{\tilde{X}}\mathcal{M}\simeq \DR (\mathcal{M}(!D))
$$
where $\mathcal{M}(!D):=\mathds{D}((\mathds{D}\mathcal{M})(\ast D))$. We have $(\mathds{D}\mathcal{M})(\ast D)=\mathcal{M}^{\ast}$ where
$\mathcal{M}^{\ast}$ is the dual connection of $\mathcal{M}$. Thus we have
$$
\Sol\mathcal{M}\simeq \DR\mathds{D}\mathcal{M}\simeq Rp_{\ast}\DR^{<D}_{\tilde{X}}\mathcal{M}^{\ast}
$$
By applying $Rf_{\ast}^{\an}$, we obtain for every $k$ and every $t\neq 0$ close enough to 0
$$
\xymatrix{
(R^{k}f_{\ast}^{\an}\Sol \mathcal{M})_{t}\ar[d]_-{(1)} \ar[r]^-{\sim}& (R^{k}(f^{\an}p)_{\ast}\DR^{<D}_{\tilde{X}}\mathcal{M}^{\ast})_t \ar[d]^{(6)}
\\
H^{k}(X_t^{\an},(\Sol\mathcal{M})_t) \ar[d]_-{(2)}&
H^{k}(X_t^{\an}, (\DR^{<D}_{\tilde{X}}\mathcal{M}^{\ast})_t) \ar[dd]^-{(7)}
\\
H^{k}(X_t^{\an},\Sol\mathcal{M}_t) \ar[d]_-{(3)} &
\\
H^{-k}(X_t^{\an}, \mathds{D}\Sol\mathcal{M}_t)^{\ast} \ar[d]_-{(4)} & H^{k}(X_t^{\an}, \DR^{<D_t}_{\tilde{X_t}}\mathcal{M}^{\ast}_t) \ar[d]^-{\wr} \\
H^{2d-2-k}(X_t^{\an}, \DR\mathcal{M}_t)^{\ast} \ar[d]_-{(5)} &
H_{2d-2-k}^{\rd}(X_t^{\an}, \mathcal{M}_t^{\ast})\ar[d]^-{|}\\
H^{2d-2-k}(U_t, \DR\mathcal{E}_t)^{\ast} \ar[r]_-{(8)}&H_{2d-2-k}^{\rd}(U_t, \mathcal{E}^{\ast}_t)
}
$$
By proper base change theorem, the morphisms $(1)$ and $(6)$ are isomorphisms. The morphism $(2)$ is an isomorphism by non charactericity hypothesis. The morphism $(3)$ is an isomorphism by Poincaré-Verdier duality. The morphism $(4)$ is an isomorphism by duality theorem for $\mathcal{D}$-modules \cite{TheseMeb}\cite{KashKawai}. The morphism $(5)$ is an isomorphism by GAGA and exactness of $j_{t\ast}$ where $j_t:U_t\longrightarrow X_t$ is the inclusion morphism. The morphism $(8)$ is an isomorphism by Hien duality theorem. We deduce that $(7)$ is an isomorphism.
\end{proof}
Let $\mathbf{e}:=(e_1, \dots, e_n)$ be a local trivialization of $\mathcal{H}^{k}(f_{+}\mathcal{E})(\ast 0)$ in a neighbourhood of 0. One can suppose that $f$ is smooth above $S^{\ast}:=S\setminus \{ 0\}$. Set $U^{\ast}:=U\setminus \{f^{-1}(0)\}$. From \cite[1.4]{DiMaSaSai}, we have an isomorphism of left $\mathcal{D}_S$-modules
$$
\mathcal{H}^{k}(f_{+}\mathcal{E})_{|S^{\ast }}\simeq
R^{k+d-1}f_{\ast}\DR_{U^{\ast}/S^{\ast}}\mathcal{E}
$$
where the right hand side is endowed with the Gauss-Manin connection as defined in \cite{KatzOda}. We deduce that $(\mathbf{e}_t)_{t\in S^{\ast}}$ is an algebraic family of bases for the $H^{k+d-1}_{\dR}(X_t, \mathcal{E}_t)$. \\ \indent
At the cost of shrinking $S$, Kashiwara perversity theorem \cite{Ka2} shows that the only possibly non zero terms of the hypercohomology spectral sequence
$$
E_{2}^{pq}=\mathcal{H}^{p}\Sol \mathcal{H}^{-q}(f_+\mathcal{E})^{\an}_{|S^{\ast }} \Longrightarrow \mathcal{H}^{p+q}\Sol (f_+\mathcal{E})^{\an}_{|S^{\ast }}
$$
sit on the line $p=0$. Hence, at the cost of shrinking $S$ again, we have
\begin{equation}\label{derniereequation}
\Sol \mathcal{H}^{k}(f_+\mathcal{E})^{\an}_{|S^{\ast }}\simeq \mathcal{H}^{0}\Sol \mathcal{H}^{k}(f_+\mathcal{E})^{\an}_{|S^{\ast }}\simeq \mathcal{H}^{-k}\Sol (f_+\mathcal{E})^{\an}_{|S^{\ast }}
\end{equation}
Since $\Sol$ is compatible with proper direct image, we deduce from \eqref{isoan} and \eqref{derniereequation}
\begin{equation}\label{derniereequationbis}
\Sol \mathcal{H}^{k}(f_+\mathcal{E})^{\an}_{|S^{\ast }}\simeq R^{-k+d-1}f_\ast\Sol (j_{+}\mathcal{E})^{\an}
\end{equation}
Let $\mathbf{s}:=(s_1, \dots, s_n)$ be a local trivialization of $\Sol \mathcal{H}^{k}(f_{+}\mathcal{E})^{\an}$ over an open subset in $S^{\ast \an}$. From \eqref{derniereequationbis} and \ref{derniereprop}, we deduce that $(\mathbf{s}_t)_{t\in S^{\ast}}$ is a continuous family of basis for the spaces $H_{k+d-1}^{\rd}(U_t, \mathcal{E}^{\ast}_t)$. The periods of $\mathcal{E}$ are by definition the coefficients of $\mathbf{s}$ in $\mathbf{e}$, and theorem \ref{GMirr} is proved.
\end{document}
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math
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मुंबई। रविवार शाम को मुंबई में विद्याविहार और कुर्ला स्टेशन के बीच लोकल ट्रेन पटरी से उतर गई। जानकारी के अनुसार कल्याण और सीएसएमटी के बीच चलने वाली लोकल ट्रेन पटरी से उतर गई थी, जिसकी वजह से काफी मध्य रेलवे की उपनगरी ट्रेनों का संचालन बाधित हुआ था। रेलवे के एक अधिकारी ने बताया कि ट्रेन का डिब्बा जब पटरी से उतरा उससे पहले शाम को करीब सात बजे डिब्बे में शॉर्ट सर्किट देखा गया था। यह उस वक्त हुआ जब ट्रेन विद्याविहार पहुंचने वाली थी। ट्रेन में सवार कुछ यात्रियों ने इस बारे में स्टेशन अधिकारियों को जानकारी दी थी, जिसके बाद ट्रेन को रोक दिया गया था। जिसके बाद एक बार फिर से ट्रेन के भीतर हादसा सामने आया जब रात तकरीबन ८.५५ पर ट्रेन का डिब्बा पटरी से उतर गया। इस हादसे में किसी के घायल होने की खबर नहीं है।
|
hindi
|
अभिनेता आमिर खान के पानी फाउंडेशन की टीम पर आदिवासियों ने बोला हमला, ६ लोग जख्मी नेटवर्क महानगर
नासिक ब्रेकिंग न्यूज़ महाराष्ट्र मुंबई शहर शहर और राज्य सामाजिक खबरें
अभिनेता आमिर खान के पानी फाउंडेशन की टीम पर आदिवासियों ने बोला हमला, ६ लोग जख्मी
नासिक, मशहूर अभिनेता आमिर खान और उनकी पत्नी किरण राव द्वारा संचालित पानी फाउंडेशन की टीम पर आदिवासियों ने अचानक हमला कर दिया। इसमें टीम के ६ लोग जख्मी हुए हैं। घटना शुक्रवार को नासिक के पास चांदवड तालुका में मतेवाडी में हुई है।
पानी फाउंडेशन की टीम और स्थानीय लोग वॉटर कप स्पर्धा में जलयुक्त योजना के तहत बारिश का पानी जमा करने के लिए खुदाई करने गए थे। जिस जमीन पर काम चल रहा था, वह वन विभाग की सीमा में है। इस जमीन पर वहां के आदिवासी खेती करते हैं। हालांकि, प्रशासन ने दावा किया है कि आदिवासियों ने इस भूमि पर अतिक्रमण किया है।
जैसे ही टीम ने वहां खुदाई शुरू की, आदिवासियों ने लाठी-डंडों और पत्थरों से हमला कर दिया। हमले के कारण लोगों में भगदड़ मच गई,जिसमें छह लोग जख्मी हो गए। सभी को चांदवड के उप जिला अस्पताल में भर्ती कराया गया है। घटना के बाद पुलिस मौके पर पहुँच कर मामले की तहकीकात कर रही है।
हमले में जख्मी लोगों का हॉस्पिटल में इलाज किया जा रहा है
बिल्ली के बच्चे को केरोसीन डालकर जलाया,गिरफ्तार
पैसे का लालच देकर करता था नाबालिग बच्चों का यौन शोषण,रंगेहाथों पकड़ा गया
उद्धव ठाकरे के जन्मदिन पर देर रात तक लगा रहा, बधाई देने वालों का तांता
हैदराबाद गैंगरेप केस के चारों आरोपियों का एनकाउंटर, पिता बोले- अब मेरी बच्ची की आत्मा को शांति मिलेगी
लॉकडाउन: एयर इंडिया ने शुरू की टिकट बुकिंग, १ जून से शुरू होगी इंटरनैशनल फ्लाइट की सेवा
|
hindi
|
Last week TEDMED 2013 came to an official close. At HIBC.tv we were among a deep and wide network of affiliates who were empowered to stream the event live for the benefit of the global community at large, as well as the local, regional, or national constituents we serve.
This has been another amazing TEDMED organized, hosted and curated event that leveraged the power of ideas, media and technology to ‘carry the message’ to as wide an audience as possible while subtracting any barriers to this message whether physical, financial or psychological. In our view, they succeeded and in a BIG way.
This is SESSION 5: What Happens When We Mix Up the Models?. We will post each session and attempt to add some of the key ideas that resonated with the audience as reflected on twitter.
The complete social media ‘digital footprint’ can be accessed here. You can pull an transcript for the entire session, or set the parameters for the day, time or session you have an interest in.
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english
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<div class="wrap" id="coupon_data">
<div id="add_coupon_box">
<h2><?php _e( 'Add Coupon', 'wpsc' ); ?></h2>
<form name='add_coupon' method="post" action="<?php echo admin_url( 'edit.php?post_type=wpsc-product&page=wpsc-edit-coupons' ); ?>">
<table class="form-table">
<tbody>
<?php do_action( 'wpsc_coupon_add_top' ); ?>
<tr class="form-field">
<th scope="row" valign="top">
<label for="add_coupon_code"><?php _e( 'Coupon Code', 'wpsc' ); ?></label>
</th>
<td>
<input name="add_coupon_code" id="add_coupon_code" type="text" style="width: 300px;"/>
<p class="description"><?php _e( 'The code entered to receive the discount', 'wpsc' ); ?></p>
</td>
</tr>
<tr class="form-field" id="discount_amount">
<th scope="row" valign="top">
<label for="add-coupon-code"><?php _e( 'Discount', 'wpsc' ); ?></label>
</th>
<td>
<input name="add_discount" id="add-coupon-code" type="number" class="small-text" min="0" />
<span class="description"><?php _e( 'The discount amount', 'wpsc' ); ?></span>
</td>
</tr>
<tr class="form-field">
<th scope="row" valign="top">
<label for="add_discount_type"><?php _e( 'Discount Type', 'wpsc' ); ?></label>
</th>
<td>
<select name='add_discount_type' id='add_discount_type'>
<option value='0'><?php _e( 'Fixed Amount', 'wpsc' ); ?></option>
<option value='1'><?php _e( 'Percentage', 'wpsc' ); ?></option>
<option value='2'><?php _e( 'Free shipping', 'wpsc' ); ?></option>
</select>
<p class="description"><?php _e( 'The discount type', 'wpsc' ); ?></p>
</td>
</tr>
<tr class="form-field">
<th scope="row" valign="top">
<label for="add_start"><?php _e( 'Start and End', 'wpsc' ); ?></label>
</th>
<td>
<span class="description"><?php _e( 'Start: ', 'wpsc' ); ?></span>
<input name="add_start" id="add_start" type="text" class="regular-text pickdate" style="width: 100px"/>
<span class="description"><?php _e( 'End: ', 'wpsc' ); ?></span>
<input name="add_end" id="add_end" type="text" class="regular-text pickdate" style="width: 100px"/>
</td>
</tr>
<tr>
<th scope="row" valign="top">
<?php _e( 'Active', 'wpsc' ); ?>
</th>
<td>
<input type='hidden' value='0' name='add_active' />
<input type="checkbox" value='1' checked='checked' name='add_active' id="add_active" />
<label for="add_active"><?php _e( 'Activate coupon on creation.', 'wpsc' ) ?></label>
</td>
</tr>
<tr>
<th scope="row" valign="top">
<?php _e( 'Use Once', 'wpsc' ); ?>
</th>
<td>
<input type='hidden' value='0' name='add_use-once' />
<input type='checkbox' value='1' name='add_use-once' id="add_use-once" />
<label for="add_use-once"><?php _e( 'Deactivate coupon after it has been used.', 'wpsc' ) ?></label>
</td>
</tr>
<tr>
<th scope="row" valign="top">
<?php _e( 'Apply On All Products', 'wpsc' ); ?>
</th>
<td>
</span><input type='hidden' value='0' name='add_every_product' />
<input type="checkbox" value="1" name='add_every_product' id="add_every-product"/>
<label for="add_every-product"><?php _e( 'This coupon affects each product at checkout.', 'wpsc' ) ?></label>
</td>
</tr>
<tr class="form-field coupon-conditions">
<th scope="row" valign="top">
<label><strong><?php _e( 'Conditions', 'wpsc' ); ?></strong></label>
</th>
<td>
<input type="hidden" name="rules[operator][]" value="" />
<div class='coupon-condition'>
<select class="ruleprops" name="rules[property][]">
<option value="item_name" rel="order"><?php _e( 'Item name', 'wpsc' ); ?></option>
<option value="item_quantity" rel="order"><?php _e( 'Item quantity', 'wpsc' ); ?></option>
<option value="total_quantity" rel="order"><?php _e( 'Total quantity', 'wpsc' ); ?></option>
<option value="subtotal_amount" rel="order"><?php _e( 'Subtotal amount', 'wpsc' ); ?></option>
<?php echo apply_filters( 'wpsc_coupon_rule_property_options', '' ); ?>
</select>
<select name="rules[logic][]">
<option value="equal"><?php _e( 'Is equal to', 'wpsc' ); ?></option>
<option value="greater"><?php _e( 'Is greater than', 'wpsc' ); ?></option>
<option value="less"><?php _e( 'Is less than', 'wpsc' ); ?></option>
<option value="contains"><?php _e( 'Contains', 'wpsc' ); ?></option>
<option value="not_contain"><?php _e( 'Does not contain', 'wpsc' ); ?></option>
<option value="begins"><?php _e( 'Begins with', 'wpsc' ); ?></option>
<option value="ends"><?php _e( 'Ends with', 'wpsc' ); ?></option>
<option value="category"><?php _e( 'In Category', 'wpsc' ); ?></option>
<?php echo apply_filters( 'wpsc_coupon_rule_logic_options', '' ); ?>
</select>
<input type="text" name="rules[value][]" style="width: 150px;"/>
<a title="<?php esc_attr_e( 'Delete condition', 'wpsc' ); ?>" class="button-secondary wpsc-button-round wpsc-button-minus" href="#"><?php echo _x( '–', 'delete item', 'wpsc' ); ?></a>
<a title="<?php esc_attr_e( 'Add condition', 'wpsc' ); ?>" class="button-secondary wpsc-button-round wpsc-button-plus" href="#"><?php echo _x( '+', 'add item', 'wpsc' ); ?></a>
</div>
</td>
</tr>
<?php do_action( 'wpsc_coupon_add_bottom' ); ?>
</tbody>
</table>
<?php submit_button( __( 'Add Coupon', 'wpsc' ), 'primary', 'add_coupon' ); ?>
</form>
</div>
</div><!--end .wrap-->
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code
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میانہِ ایٚم سٕوایپ ویلٹ پؠٹھٕ کَر اَبھِشِک ٹرانسفر ۹۹۳ پۄنٛسہٕ
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kashmiri
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मार्च १९, 20१९ वेटलॉस?
मोटापा कम करना आज लोगों के लिए बड़ी चुनौती बना हुआ है क्योंकि इसे कंट्रोल में करना काफी मुश्किल काम है लेकिन हैल्दी लाइफस्टाइल के लिए वजन कम करना बहुत जरूरी है क्योंकि बढ़ा वजन सिर्फ फिगर ही खराब नहीं करता बल्कि बीमारी को खतरा भी बढ़ाता है इसलिए तो लोग आजकल वेट कम करने के लिए तरह-तरह के नुस्खे अपना रहे हैं। लोग डाइट, वर्कआउट व सैर पर ज्यादा फोक्स करते हैं। सेलेब्स की तरह लोग कीटो, वेगन और ना जाने किन-किन डाइट का सहारा ले रहे हैं लेकिन इन सब चीजों के लिए आपके पास पर्याप्त समय और पैसा भी होना चाहिए। इसी चक्कर में कई लोग वेट कम नहीं कर पाते लेकिन आज हम आपको सेलिब्रेटीज न्यूट्रिशनिस्ट रिजुता दिवेकर के वेट लूज टिप्स के बारे में आपको बताते हैं जिसकी मदद से आप एक आइडल मील प्लान बनाकर वजन कम कर सकते हैं।
रिजुता दिवेकर का कहना है कि वजन कम करने वाले सबसे पहले तो वजन जांचने वाली मशीन पर खड़े होने से बचें। यह चीजें आपको तनाव में ले आती हैं।
चलिए आपको बताते हैं ब्रेकफास्ट से लेकर डिनर तक का डाइट प्लान
सुबह का खाना खाना ना भूलें।
नाश्ता घर का बना ही खाएं
समय पर करना भी जरूरी देरी से किया नाश्ता बीमारियों की वजह बनता है।
लंच कैसा होना चाहिए और कब खाएं
एक निर्धारित समय पर ही करें लंच
रोटी-सब्जी और दाल लंच के लिए बेस्ट
पोषक तत्वों से भरपूर हो डाइट
आप फ्रैश जूस (बिना चीनी) भी ले सकते हैं।
डिनर हो हल्का-फुल्का
डिनर का समय जल्दी का रखें।
सोने से ठीक पहले खाना ना खाएं।
हल्का फुलका खाना खाएं। दाल को जरूर शामिल रखें
डिनर में घी जरूर शामिल करें।
डाइट में हो खास बातः
खाना घर का देसी होगा तो सबसे बढ़िया है। पैकेज्ड फूड में वो बात नहीं होती जो घर के खाने में होती है
फ्रूट्स हमारे बेस्ट फ्रेंड्स होते हैं। फ्रूट चांट किसे पसंद नहीं। हेल्थी भी है और डाइजेस्ट करने में आसान भी। लेकिन इसके पूरे फायदे लेने के लिए जरूरी है कि हम विपरीत दो विपरीत प्रकृति के फलों को एक-साथ न खाएं। जानिए, कौन-से फ्रूट्स साथ नहीं खाने चाहिए...
सुबह-सुबह हर चीज की जल्दी रहती है। स्कूल, कॉलेज, ट्यूशन और ऑफिस। इन सबकी भागदौड़ के बीच जब हम नाश्ते के लिए वक्त नहीं निकाल पाते हैं तो फ्रूट्स हमारे बेस्ट फ्रेंड्स होते हैं।
फ्रूट चांट किसे पसंद नहीं। हेल्थी भी है और डाइजेस्ट करने में आसान भी। लेकिन इसके पूरे फायदे लेने के लिए जरूरी है कि हम विपरीत दो विपरीत प्रकृति के फलों को एक-साथ न खाएं। जानिए, कौन-से फ्रूट्स साथ नहीं खाने चाहिए...
गर्मियों का पसंदीदा फल है यह। इसमें पानी की मात्रा सबसे अधिक होती है। बाकी जितने भी फ्रूट्स हम इस सीजन में यह उनकी तुलना में सबसे जल्दी डाइजेस्ट होता है। अगर इसके साथ किसी और फल को मिक्स करके खाया जाए तो डाइजेशन में दिक्कत हो सकती है। तो अगली बार तरबूज और खरबूजे की हर वरायटी को किसी और फ्रूट के साथ मिक्स करके नहीं खाएं।
ऐसिडिक और सब ऐसिडिक फ्रूट्स विद स्वीट फ्रूट्स
अगर आप चाहते हैं कि आपका पाचन ठीक से चलता रहे तो ऐसिडिक और सब ऐसिडिक फलों को एक-साथ न खाएं। ऐसिडिक फ्रूट्स जैसे अंगूर और स्ट्रॉबेरी और सब ऐसिडिक फ्रूट्स जैसे सेब, अनार, अडू को मीठे फल केला और किशमिश या मुनक्का के साथ मिक्स करके नहीं खाना चाहिए।
ऐसे ही अगर आप जीमिचलाना, ऐसिड बढ़ना और सिरदर्द जैसी दिक्कतों से दूर रहना चाहते हैं तो अमरूद और केले को मिक्स करके न खाएं।
फल और सब्जियों के पचने का वक्त अलग होता है। कुछ न्यूट्रिशियन्स का कहना है कि फल खाने के दौरान जब वह पेट में पहुंचते हैं, तब तक आंशिक रूप से पच चुके होते हैं। साथ ही फलों में सब्जियों की तुलना में अधिक शुगर कंटेंट ज्यादा होता है, जिसके चलते सब्जियों के पाचन में दिक्कत हो सकती है।
अगर आप सीने में चुभन और पित्त बढ़ने की समस्या से बचना चाहते हैं तो गाजर और संतरे को कभी एक-साथ न खाएं।
स्टार्च फ्रूट्स और हाई प्रोटीन फ्रूट्स
नैचुरली बहुत ही कम फूर्टस में स्टार्च होता है। इनमें ग्रीन बनाना भी शामिल है। लेकिन बहुत सारी सब्जियों में स्टार्च की भरपूर मात्रा होती है। जैसे, मक्का, आलू, चावल, राजमा, लोबिया और सिंघाड़ा। इन सब्जियों को कभी भी हाई प्रोटीन फलों जैसे, अमरूद, किशमिश, मुनक्का, पालक और ब्राकली के साथ नहीं खाना चाहिए। ऐसा इसलिए क्योंकि बॉडी को प्रोटीन डाइजेस्ट करने के लिए ऐसिडिक बेस चाहिए होता है और स्टार्च को डाइजेस्ट करने के लिए ऐल्कलाइन बेस चाहिए होता है।
फल खाने से जुड़ी जरूरी बातें...
- एक बार में ४ या ६ फल से अधिक न खाएं।
- अगर आपने प्रोटीन अधिक कंज्यूम कर लिया है तो अगली सुबह पपीता खाएं। यह आपके पेट को सही रहने में मददगार होगा।
-अगर आपने नमक का अधिक सेवन कर लिया है तो वॉटर बेस्ड फल जैसे तरबूज खाएं। इससे अगले दिन फ्लश ईजी होगा।
- अगर आपने कार्बोहाइड्रेट आ अधिक सेवन कर लिया है या पास्ता अधिक खा लिया है, तो अगली सुबह ऐपल खाएं।
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hindi
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مینٛگروو کلہِ چِھ نوُن ژالن وٲلہِ پیچیدٕ تہٕ زبردست ماحولِیٲتی نِظام یِم گرم تہٕ نیم گرم بین مدو جزر خطن منٛز چھِ
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kashmiri
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इडुक्की: महिला आईएएस अधिकारी पर सार्वजनिक तौर पर टिप्पणी को लेकर माकपा विधायक विवादों में घिर गए हैं. विधायक ने मन्नार में एक व्यवसायिक कॉम्प्लेक्स में "अनधिकृत" निर्माण रोकने का प्रयास कर रही युवा महिला आईएएस अफसर के लिए कहा कि "उनके पास दिमाग नहीं" हैं. यह घटना शुक्रवार को प्रकाश में आई. कुछ टेलीविजन चैनलों ने देवीकुलम विधायक एस राजेंद्रन के एक वीडियो फुटेज को प्रसारित किया. इसमें सब कलेक्टर डॉक्टर रेणु राज के लिए विधायक दिमाग न होने होने वाली टिप्पणी कर रहे हैं.
राजेंद्रन ने कहा, ''उनके पास (सब कलेक्टर) दिमाग नहीं है. उन्होंने सिर्फ कलेक्टर बनने के लिए पढ़ाई की है. इन लोगों के पास काफी कम दिमाग होता है. क्या उन्हें स्केच और प्लान की जानकारी नहीं हासिल करनी चाहिए थी.'' विधायक ने वहां मौजूद लोगों से कहा, ''एक कलेक्टर पंचायत के निर्माण में दखल नहीं दे सकती है. यह एक लोकतांत्रिक देश है.''
वहीं, सब कलेक्टर ने बताया कि कि अवैध निर्माण को लेकर पंचायत को छह फरवरी को एक मेमो (ज्ञापन) जारी किया गया था. लेकिन यहां काम अब भी चल रहा है. इसी बीच इडुक्की जिले की माकपा इकाई ने कहा है कि वह विधायक से इस संबंध में स्पष्टीकरण मांगेगी.
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hindi
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یِتھہٕ کٕنۍ بنیٛاو دوٚب جلدٕے سؠٹھا امیر
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kashmiri
|
अगस्ता दोषियों की खैर नहीं- पर्रिकर
शारेंनई दिल्ली | समाचार डेस्क: रक्षा मंत्री पर्रिकर ने ने संसद में कहा अगस्ता के दोषियों की खैर नहीं. उन्होंने कहा अगस्ता के दोषियों को कटघरे में खड़ा किया जायेगा. सदन में जोर देकर रक्षा मंत्री ने कहा अगस्ता का हाल बोफोर्स जैसा नहीं होने देंगे. अगस्तावेस्टलैंड सौदे को लेकर पूर्ववर्ती संप्रग सरकार पर तीखा प्रहार करते हुए रक्षा मंत्री मनोहर पर्रिकर ने कहा कि तत्कालीन कांग्रेस नीत सरकार ने अगस्तावेस्टलैंड को हेलीकाप्टर का ठेका देने के लिए हर तरह की रियायत दी और पूर्व वायु सेना प्रमुख एसपी त्यागी, गौतम खेतान तो बहती गंगा में हाथ धोने वाले छोटे नाम है, हम बड़े नामों का पता लगा रहे हैं, जिन्होंने रिश्वत ली.
पर्रिकर ने कहा, मैं आप सबको आश्वस्त कर सकता हूं कि हम नाकाम नहीं होंगे. हम जो बोफोर्स कांड में नहीं कर सके, वह अगस्ता वेस्टलैंड मामले में करेंगे.
नियम १९७ के तहत ध्यानाकर्षण प्रस्ताव पर भारतीय जनता पार्टी के सांसदों अनुराग ठाकुर व निशिकांत दूबे, सौगत रॉय (तृणमूल कांग्रेस) व ज्योतिरादित्य सिंधिया (कांग्रेस) द्वारा पूछे गए सवालों के जवाब में पर्रिकर की यह प्रतिक्रिया सामने आई.
स्वीडन की बोफोर्स एबी कंपनी पर १५५ मिलीमीटर की होवित्जर तोपों का ठेका पाने के लिए रिश्वत देने का आरोप लगा. इस विवाद के कारण सन् १९८९ में राजीव गांधी के नेतृत्व वाली सरकार को लोकसभा चुनाव में हार का मुंह देखना पड़ा था. कांग्रेस नेता सोनिया गांधी, पूर्व प्रधानमंत्री मनमोहन सिंह तथा राहुल गांधी संसद के बाहर सड़क पर उतरे और नरेंद्र मोदी सरकार को चेताया कि कांग्रेस कोई कमजोर पार्टी नहीं है.
लोकसभा में कांग्रेस नेता मल्लिकार्जुन खड़गे ने मामले की सर्वोच्च न्यायालय की निगरानी में जांच कराने की मांग की. मांग नहीं माने जाने पर कांग्रेस अध्यक्ष सोनिया गांधी, उपाध्यक्ष राहुल गांधी, खड़गे व अन्य कांग्रेस सदस्य सदन से बहिर्गमन कर गए.
जंतर-मंतर पर हजारों कार्यकर्ताओं को संबोधित करते हुए सोनिया गांधी ने सरकार पर राष्ट्रीय स्वयंसेवक संघ के निर्देश पर चलने का आरोप लगाया.
समर्थकों द्वारा सोनिया गांधी जिंदाबाद के नारों के बीच उन्होंने कहा, मैं सरकार को चेताना चाहती हूं कि वह कांग्रेस को एक कमजोर पार्टी समझने की भूल न करे.
उन्होंने अपने संक्षिप्त भाषण में कहा, यहां से एक कड़ा संदेश जाना चाहिए और इसे केवल रायसीना हिल्स ही नहीं, बल्कि नागपुर में उन लोगों तक पहुंचना चाहिए, जिनके निर्देश पर मोदी सरकार काम करती है.
उधर, सदन में कांग्रेस सरकार पर हमला करते हुए पर्रिकर ने एंटनी के प्रति सद्भाव जताते हुए उन्हें बेचारा करार दिया.
पर्रिकर ने कहा, बेचारे एंटनी साहब के हाथ बंधे थे. उन्होंने कहा कि इस मामले में इटली में साल २०१२ में एक व्यक्ति की गिरफ्तारी भी हुई थी.
रक्षा मंत्री ने कहा, ..क्योंकि एंटनी अपनी छवि बचाए रखना चाहते थे. इसलिए दो से तीन घंटे के भीतर उन्होंने फाइल को आगे बढ़वाया, दस्तावेजों को मंजूरी दी और सीबीआई जांच का आदेश दे दिया.
रक्षा मंत्री ने कहा, हेलीकॉप्टर के फील्ड ट्रायल का एंटनी ने विरोध किया था, लेकिन बाद में उन्हें अपना रुख बदलने के लिए समझा लिया गया.
रक्षा मंत्री ने हालांकि कहा कि सीबीआई ने जनवरी २०१४ तक मामले में कुछ नहीं किया.
वहीं, कांग्रेस पार्टी के नेता ज्योतिरादित्य सिंधिया ने पार्टी अध्यक्ष सोनिया गांधी को सिंहनी करार दिया और कहा कि अगस्ता मामले में उनका नाम किसी भी प्रामाणिक दस्तावेज में नहीं आया है.
लोकसभा में सिंधिया ने भाजपा नेताओं द्वारा अपने भाषणों में इशारों में सोनिया गांधी पर निशाना साधने के लिए भाजपा पर हमला किया. सिंधिया ने सत्ता पक्ष की ओर इशारा करते हुए कहा, सोनिया गांधी एक सिंहनी हैं, जिनसे उन्हें डर लगता है.
अगस्ता वेस्टलैंड के भारत स्थित कार्यालय में इटली के एक अधिकारी पीटर हुलेट के एक पत्र का संदर्भ देते हुए सिंधिया ने कहा, हुलेट ने लिखा है कि सोनिया गांधी तथा उनके सलाहकार ऐसे लोग हैं, जिनका उच्चायुक्त को सम्मान करना चाहिए.
उन्होंने कहा, बिना हस्ताक्षर वाले अप्रमाणिक कागज के टुकड़ों के अलावा, गांधी का नाम किसी भी प्रामाणिक दस्तावेज में कहीं नहीं है.
लोकसभा में शुक्रवार का दिन बेहद हंगामेदार रहा, जहां अगस्ता वेस्टलैंड मामले को लेकर सत्ता तथा विपक्ष के बीच आरोप-प्रत्यारोप का दौर चलता रहा.
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hindi
|
package main
import (
"bytes"
"io"
"net/http"
"os"
"strings"
)
var (
// 文件根目录
RootDir = "/data/pet/"
// 用于文件上传认证
Token = "123"
// SecretKey string = os.Getenv(FILE_SERVER_SECRET_KEY)
)
type File struct {
Name string
Path string
Force bool
Rename bool
}
// 请求处理
func Index(w http.ResponseWriter, req *http.Request) {
defer func() {
req.Body.Close()
}()
req.Header.Add("Access-Control-Allow-Origin", "*")
err := req.ParseForm()
if err != nil {
http.Error(w, `{"error":"ParseFrom Failed"}`, http.StatusBadRequest)
log.Error(err)
return
}
log.Info("func Get:", req.Method)
w.Write([]byte(`{"code":0}`))
// switch strings.ToUpper(req.Method) {
// case "POST":
// Push(w, req)
// return
// case "GET":
// default:
// http.Error(w, `{"error":"Method Error"}`, http.StatusMethodNotAllowed)
// return
// }
// path := RootDir + req.URL.Path
// if !filter(path) {
// http.Error(w, `{"error":"Not Exists"}`, http.StatusNotFound)
// return
// }
// fif, err := os.Stat(path)
// if err != nil || fif.IsDir() {
// http.Error(w, `{"error":"Not Exists"}`, http.StatusNotFound)
// return
// }
// f, err := os.OpenFile(path, os.O_RDONLY, 0444)
// if err != nil {
// http.Error(w, `{"error":"Not Exists"}`, http.StatusNotFound)
// return
// }
// io.Copy(w, f)
}
// 上传文件
func Receive(w http.ResponseWriter, req *http.Request) {
defer req.Body.Close()
err := req.ParseForm()
if err != nil {
http.Error(w, `{"code":-1,"error":"ParseForm Failed"}`, http.StatusBadRequest)
return
}
if strings.ToUpper(req.Method) != "POST" {
http.Error(w, `{"code":-1,"error":"Method Error"}`, http.StatusMethodNotAllowed)
return
}
buf := new(bytes.Buffer)
dir := RootDir
// 接收文件
for k, v := range req.Form {
log.Debug(k, v)
}
f, fh, err := req.FormFile("file")
if err != nil {
http.Error(w, `{"error":"ParseFileForm Failed"}`, http.StatusBadRequest)
log.Error(err.Error())
return
}
//
size, err := io.Copy(buf, f)
if err != nil {
log.Error(err.Error())
return
}
hsha1 := req.Header.Get("X-Ckeyer-Sha1")
hsha2 := HmacSha1(buf.Bytes(), Token)
if hsha2 != hsha1 {
http.Error(w, `{"error":"Auth failed"}`, http.StatusNotAcceptable)
return
}
path := dir + fh.Filename
if !filter(path) {
http.Error(w, `{"error":"Not Exists"}`, http.StatusNotFound)
return
}
log.Notice(fh.Filename, ", ", FmtSize(size))
// finfo, err := os.Stat(dir)
// if err != nil {
// os.MkdirAll(dir, 0644)
// } else if !finfo.IsDir() {
// if force {
// os.Remove(dir)
// os.MkdirAll(dir, 0644)
// } else {
// http.Error(w, `{"error":"Dir is a exists File"}`, http.StatusBadRequest)
// return
// }
// }
// if _, err = os.Stat(path); err == nil {
// if force {
// os.Remove(path)
// } else {
// http.Error(w, `{"error":"File Exists"}`, http.StatusBadRequest)
// return
// }
// }
newf, err := os.Create(path)
if err != nil {
http.Error(w, `{"error":"Create File Failed"}`, http.StatusBadRequest)
return
}
defer newf.Close()
_, err = io.Copy(newf, buf)
if err != nil {
http.Error(w, `{"error":"Upload File Failed"}`, http.StatusBadRequest)
return
}
w.Write([]byte(`{"success":"ok"}`))
}
func filter(url string) bool {
err_seps := []string{"..", "~", ".go", "--"}
for _, sep := range err_seps {
if strings.Count(url, sep) > 0 {
return false
}
}
return true
}
|
code
|
یِتھہٕ کٕنۍ زن گۄڈَے وَننہٕ آو زِ طوطس اوس تہِ سٲلِم پتاہ آسان یہِ دُنیاہس منٛز سپدان اوس
|
kashmiri
|
Discover phentermine 7 Phentermine the safest, most effective weight loss drug on the market. Learn where to buy phentermine and Adipex online. Read about side effects of ... Phentermine 37.5 is available as capsules, tablets and orally disintegrating tablets. Where to buy Phentermine 37.5mg online? Phentermine 37.5. The drug Phentermine 37.5 mg is the most popular dosage because it is extremely potent and prescribed by many doctors. The drug is used to treat ... Phentermine is an appetite suppressant used together with diet and exercise to treat obesity. Learn about side effects, interactions and indications. Phentermine (contracted from phenyl-tertiary-butyl amine), also known as α,α-dimethylphenethylamine, is a psychostimulant drug of the substituted amphetamine ... Weight Loss With Phentermine 37 5 Mg - Juice Recipes That Detox Pancreas Weight Loss With Phentermine 37 5 Mg Best Drug Detox Cleanse To Pass Urine Test Detox Drinks ... Send text and picture messages for free The messages that you send and receive using our website are free.
|
english
|
ACCT - Occupational Safety majors will fulfill ACCT with OSH 349 or 495.
OSH 110, 200, 225, 261, 262, 305, 349, 366, 367, 370, 379, 390, 410W, 412, 420, 450, and 495.
PHY 101 (Element 4), TRS 225, and 395.
ᴳ = Course also satisfies a General Education element.Some supporting course hours are therefore included within the 36 hr. General Education requirement above.
|
english
|
/**
* Copyright 2013 Facebook, Inc.
*
* 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.
*/
/*jslint node: true*/
/**
* Desugars ES6 Object Literal short notations into ES3 full notation.
*
* // Easier return values.
* function foo(x, y) {
* return {x, y}; // {x: x, y: y}
* };
*
* // Destrucruting.
* function init({port, ip, coords: {x, y}}) { ... }
*
*/
var Syntax = require('esprima-fb').Syntax;
var utils = require('../src/utils');
/**
* @public
*/
function visitObjectLiteralShortNotation(traverse, node, path, state) {
utils.catchup(node.key.range[1], state);
utils.append(':' + node.key.name, state);
return false;
}
visitObjectLiteralShortNotation.test = function(node, path, state) {
return node.type === Syntax.Property &&
node.kind === 'init' &&
node.shorthand === true;
};
exports.visitorList = [
visitObjectLiteralShortNotation
];
|
code
|
Periodic Table
==============
On a flat periodic table, the brilliant and interactive periodic table is augmented which can be arranged in table, sphere, helix, and grid modes. It is based on a CSS example of Three.js.
|
code
|
var objectMerge = require('object-merge');
module.exports.parse = function(data) {
'use strict';
var keyExpressionStatements = [];
var keyConditionExpression;
var expressionAttributeNames = {};
var expressionAttributeValues = {};
var raw = {};
if(!Array.isArray(data)) {
Object.keys(data).forEach(function (k) {
if(k.substring(0,4) == 'raw.') {
raw[k.substring(4,k.length)] = data[k]
} else {
var expressionNameKey = '#' + k;
var expressionValueKey = ':' + k;
expressionAttributeNames[expressionNameKey] = k;
expressionAttributeValues[expressionValueKey] = data[k];
keyExpressionStatements.push(expressionNameKey + ' = ' + expressionValueKey);
}
});
} else if (Array.isArray(data)) {
data.forEach(function (k) {
var expressionNameKey = '#' + k.attribute;
var expressionValueKey = ':' + k.attribute;
expressionAttributeNames[expressionNameKey] = k.attribute;
expressionAttributeValues[expressionValueKey] = k.value;
if(k.operator == 'begins_with') {
keyExpressionStatements.push('begins_with(' + expressionNameKey + ',' + expressionValueKey + ")");
} else {
keyExpressionStatements.push(expressionNameKey + ' ' + k.operator + ' ' + expressionValueKey);
}
});
}
keyConditionExpression = keyExpressionStatements.join(' and ');
var expression = {
KeyConditionExpression: keyConditionExpression,
ExpressionAttributeNames: expressionAttributeNames,
ExpressionAttributeValues: expressionAttributeValues,
};
return objectMerge(expression, raw);
};
|
code
|
The basic functionality is found in the COM object hnetcfg.dll. So the first thing to-do is to get a firewall manager.
Getting the current profile The profile is used for all firewall rules interactions.
Is firewall on or of?
I´m about to try your code. I need to test an open port on the windows firewall.
This is the same for every windows version ?
And thank you ! Helped a lot !
|
english
|
The administration of Mayor Greg Fischer has already spent over $121,000 on tickets, lodging and transportation for prospective business guests who will come to Louisville for the Kentucky Derby this May, according to records obtained by Insider Louisville through an open records request.
The continuation of annual spending exceeding $100,000 on Derby guests — the identities of which are likely to remain secret — is occurring despite the looming budget crunch of Metro Government, which faces a $35 million shortfall in the next fiscal year beginning July 1.
In recent years, Fischer and Mary Ellen Wiederwohl, the chief of the administration’s economic development team at Louisville Forward, have defended such Derby spending as a vital component of their strategy to lure prospective businesses to invest or expand in Louisville, which ends up paying for itself many times over in new jobs and tax revenue.
Despite refusing to reveal the identity of prospective business guests — as well as specifics on subsequent investments made in Louisville by the businesses of such guests — Wiederwohl said last year that their 2017 Derby guests had already located over a dozen expansion or attraction projects with $911 million in new investment and 125 new jobs in Louisville, with 11 actively pursuing expansion projects.
Wiederwohl added that the money for these Derby expenses was allocated for this use and approved by Metro Council back in June 2018 and planning for Derby events began in the fall, with save the dates sent out, tickets purchased and contracts for hotel rooms signed before the end of last year. Like last year, the Derby Eve party at Metro Hall will be privately funded through corporate sponsors.
A proposed ordinance that would have forced the administration to reveal the names of its Derby guests within five years was rejected by a 12-12 vote of Metro Council in December, which Wiederwohl said at the time would have a “chilling effect” on the administration’s ability to attract such prospective business guests.
Last year, the administration spent nearly $111,000 on Derby-related expenses for guests, including $72,441 on Churchill Downs tickets and nearly $30,000 on Omni Hotel rooms. The total bill for entertaining guests came to $108,618, as the city took in $20,000 of sponsorships for the mayor’s Derby Eve party at Metro Hall.
Wiederwohl signed an agreement to reserve the hotel rooms for this year’s Derby in December, while the city’s invoice for the Derby tickets indicates a payment was due by Jan. 18. The invoice for parking indicates that a payment was due last Friday, while the invoice to reserve transportation for guests is dated Feb. 11.
On Feb. 7, Fischer first warned of “devastating” cuts to city services that would be needed if new tax revenue was not created, citing a $35 million budget shortfall in the next fiscal year and $65 million shortfall over the next four years, mostly due to rapidly increasing pension payments mandated by the board of the Kentucky Retirement System.
While Fischer supported the eventual tripling of the tax rate on most types of insurance premiums to cover the shortfall without any cuts or layoffs, a compromise ordinance that would have nearly doubled those tax rates next year — but still requiring $15 million in cuts — was rejected by an 11-15 vote of Metro Council last month.
The mayor is scheduled to present his proposed budget for the next fiscal year to Metro Council at its meeting on April 25, and the council must pass a final budget on June 25.
When announcing the shortfall in February, Fischer released a spreadsheet listing the potential budget cuts he may propose if Metro Council did not raise new tax revenue, which included 317 layoffs of city workers and 246 positions lost through attrition across many departments in the next fiscal year alone. It also included the possible closure of libraries, community centers, swimming pools and golf courses, in addition to privatizing the Louisville Zoo and ceasing the operation of the Belle of Louisville.
While Louisville Tourism announced last week that it is kicking in $500,000 to keep the Belle of Louisville open, Fischer is already taking cost-cutting measures to prepare for the budget shortfall, including canceling the new recruiting class of the Louisville Metro Police Department, increasing the health insurance premiums for city workers by 3% and closing the city’s four outdoor pools this summer.
Fischer is also considering ending the city’s contract with ShotSpotter — a new technology that LMPD has used in recent years to detect and quickly respond to possible gunshots — and asked the unions representing city workers to agree to a pay freeze next year to avoid layoffs. Leaders of those local unions have already rejected that request.
The mayor’s original spreadsheet of potential cuts for Louisville Forward listed $146,300 in savings created by one position lost through attrition, in addition to $600,000 in reduced funding for SummerWorks and external agency grants for the arts, which are both distributed through that economic development department.
While the Fischer administration has spent just shy of $400,000 on prospective business guests at the Derby over the last four years, state records show that the administration of Gov. Matt Bevin has been an even more prodigious spender, totaling $529,528 in the last three years alone. The governor’s administration also withholds the names of such guests, citing an exemption in the Kentucky Open Records Act regarding prospective business.
In December, the Fischer administration released the names of 14 Derby guests over the past two years who were called “cheerleaders” for the city, as they were local business leaders who spoke to prospective business guests about the positives of operating in Louisville.
Wiederwohl also told Insider on Wednesday that while the administration “will release the names of industry hosts and economic development staff after Derby, as we did last year,” they “will not be releasing the names of our business guests.” She added that Derby guests “have an expectation of privacy when attending,” with some requiring a non-disclosure agreement.
This story has been updated with comments from Wiederwohl.
|
english
|
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english
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प्रधानमंत्री नरेंद्र मोदी और फ्रांस के राष्ट्रपति इमैनुएल मैक्रों को संयुक्त रूप से संयुक्त राष्ट्र के उत्कृष्ट पर्यावरण पुरस्कार यूनाइटेड नेशन्स चैम्पियंस ऑफ द अर्थ से सम्मानित किया गया है. संयुक्त राष्ट्र महासचिव एंतोनिया गुतेरेज ०३ अक्टूबर २०१८ में नई दिल्ली में पीएम मोदी को यह पुरस्कार प्रदान किया. संयुक्त राष्ट्र पर्यावरण संरक्षण की दिशा में बेहतरीन काम करने वाली हस्तियों को इस प्रतिष्ठित पुरस्कार से सम्मानित करता है.
मोदी - मैक्रों को चैंपियंस ऑफ़ अर्थ अवार्ड
प्राप्त जानकारी के अनुसार अंतर्राष्ट्रीय सौर गठबंधन को लेकर अग्रणी एवं उत्साही कार्यों के लिए और पर्यावरणीय कार्यों में सहयोग के नये क्षेत्रों को बढ़ावा देने के लिए नरेंद्र मोदी और इमैनुएल मैक्रों को 'नीति नेतृत्व' श्रेणी के तहत यह पुरस्कार प्रदान किया गया.
प्रधानमंत्री को २०२२ तक भारत में प्लास्टिक के उपयोग को खत्म करने का संकल्प लेने और मैक्रों को पर्यावरण के लिए वैश्विक समझौते को लेकर सराहना की गयी.
दोनों देशों के राष्ट्र प्रमुखों ने पूरे विश्व में इस बारे में जाग्रति फैलाई तथा अन्य देशों को भी सौर गठबंधन में शामिल करने हेतु सफलतापूर्वक प्रोत्साहित किया.
भारत के लिए दूसरी सम्मान की बात है कि केरल के कोच्चि इंटरनैशनल एयरपोर्ट को भी नवीकरणीय ऊर्जा की दिशा में आगे बढ़ते हुए उद्यमी दृष्टि दिखाने के लिए अवॉर्ड दिया गया है. कोच्चि इंटरनैशनल एयरपोर्ट विश्व को यह सन्देश देता है कि पर्यावरण को नुकसान पहुंचाए बिना भी वैश्विक विकास में अग्रणी भूमिका निभाई जा सकती है.
इस दौरान कहा गया कि समाज की गति में वृद्धि जारी है, ऐसे में दुनिया का पहला पूर्ण सौर संचालित हवाई अड्डा इस बात का प्रमाण है कि ग्रीन बिजनस ही अच्छा बिजनस है. कोच्चि अंतरराष्ट्रीय हवाई अड्डे को 'नवीकरणीय ऊर्जा के उपयोग में नेतृत्व' के लिए पुरस्कार प्रदान किया गया.
चैंपियंस ऑफ़ अर्थ अवार्ड
संयुक्त राष्ट्र पर्यावरण कार्यक्रम ने वर्ष २००५ में सार्वजनिक और निजी क्षेत्रों से उत्कृष्ट पर्यावरण हितैषी कार्य करने वाले व्यक्तियों को पहचानने के लिए वार्षिक पुरस्कार कार्यक्रम के रूप में चैंपियंस ऑफ़ अर्थ पुरस्कार की स्थापना की.
आम तौर पर प्रतिवर्ष पांच से सात विजेताओं का चयन किया जाता है. प्रत्येक पुरस्कार विजेता को एक ट्रॉफी प्राप्त करने के लिए एक पुरस्कार समारोह में आमंत्रित किया जाता है. विजेता पुरस्कार प्राप्ति हेतु भाषण देते हैं और एक प्रेस कॉन्फ्रेंस में भाग लेते हैं.
वर्ष २०१७ में यंग चैंपियंस ऑफ़ अर्थ श्रेणी की भी शुरुआत की गई. इसके अंतर्गत १८ से ३० वर्ष के उन व्यक्तियों को पुरस्कृत किया जाता है जिन्होंने पर्यावरण संरक्षण की दिशा में सकारात्मक एवं उत्कृष्ट पहल की हो.
|
hindi
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#include <asm/octeon/octeon.h>
#include "cvmx-config.h"
#include "cvmx-helper.h"
#include "cvmx-pko-defs.h"
#include "cvmx-gmxx-defs.h"
#include "cvmx-pcsxx-defs.h"
void __cvmx_interrupt_gmxx_enable(int interface);
void __cvmx_interrupt_pcsx_intx_en_reg_enable(int index, int block);
void __cvmx_interrupt_pcsxx_int_en_reg_enable(int index);
int __cvmx_helper_xaui_probe(int interface)
{
int i;
union cvmx_gmxx_hg2_control gmx_hg2_control;
union cvmx_gmxx_inf_mode mode;
/*
* Due to errata GMX-700 on CN56XXp1.x and CN52XXp1.x, the
* interface needs to be enabled before IPD otherwise per port
* backpressure may not work properly.
*/
mode.u64 = cvmx_read_csr(CVMX_GMXX_INF_MODE(interface));
mode.s.en = 1;
cvmx_write_csr(CVMX_GMXX_INF_MODE(interface), mode.u64);
__cvmx_helper_setup_gmx(interface, 1);
/*
* Setup PKO to support 16 ports for HiGig2 virtual
* ports. We're pointing all of the PKO packet ports for this
* interface to the XAUI. This allows us to use HiGig2
* backpressure per port.
*/
for (i = 0; i < 16; i++) {
union cvmx_pko_mem_port_ptrs pko_mem_port_ptrs;
pko_mem_port_ptrs.u64 = 0;
/*
* We set each PKO port to have equal priority in a
* round robin fashion.
*/
pko_mem_port_ptrs.s.static_p = 0;
pko_mem_port_ptrs.s.qos_mask = 0xff;
/* All PKO ports map to the same XAUI hardware port */
pko_mem_port_ptrs.s.eid = interface * 4;
pko_mem_port_ptrs.s.pid = interface * 16 + i;
cvmx_write_csr(CVMX_PKO_MEM_PORT_PTRS, pko_mem_port_ptrs.u64);
}
/* If HiGig2 is enabled return 16 ports, otherwise return 1 port */
gmx_hg2_control.u64 = cvmx_read_csr(CVMX_GMXX_HG2_CONTROL(interface));
if (gmx_hg2_control.s.hg2tx_en)
return 16;
else
return 1;
}
int __cvmx_helper_xaui_enable(int interface)
{
union cvmx_gmxx_prtx_cfg gmx_cfg;
union cvmx_pcsxx_control1_reg xauiCtl;
union cvmx_pcsxx_misc_ctl_reg xauiMiscCtl;
union cvmx_gmxx_tx_xaui_ctl gmxXauiTxCtl;
union cvmx_gmxx_rxx_int_en gmx_rx_int_en;
union cvmx_gmxx_tx_int_en gmx_tx_int_en;
union cvmx_pcsxx_int_en_reg pcsx_int_en_reg;
/* (1) Interface has already been enabled. */
/* (2) Disable GMX. */
xauiMiscCtl.u64 = cvmx_read_csr(CVMX_PCSXX_MISC_CTL_REG(interface));
xauiMiscCtl.s.gmxeno = 1;
cvmx_write_csr(CVMX_PCSXX_MISC_CTL_REG(interface), xauiMiscCtl.u64);
/* (3) Disable GMX and PCSX interrupts. */
gmx_rx_int_en.u64 = cvmx_read_csr(CVMX_GMXX_RXX_INT_EN(0, interface));
cvmx_write_csr(CVMX_GMXX_RXX_INT_EN(0, interface), 0x0);
gmx_tx_int_en.u64 = cvmx_read_csr(CVMX_GMXX_TX_INT_EN(interface));
cvmx_write_csr(CVMX_GMXX_TX_INT_EN(interface), 0x0);
pcsx_int_en_reg.u64 = cvmx_read_csr(CVMX_PCSXX_INT_EN_REG(interface));
cvmx_write_csr(CVMX_PCSXX_INT_EN_REG(interface), 0x0);
/* (4) Bring up the PCSX and GMX reconciliation layer. */
/* (4)a Set polarity and lane swapping. */
/* (4)b */
gmxXauiTxCtl.u64 = cvmx_read_csr(CVMX_GMXX_TX_XAUI_CTL(interface));
/* Enable better IFG packing and improves performance */
gmxXauiTxCtl.s.dic_en = 1;
gmxXauiTxCtl.s.uni_en = 0;
cvmx_write_csr(CVMX_GMXX_TX_XAUI_CTL(interface), gmxXauiTxCtl.u64);
/* (4)c Aply reset sequence */
xauiCtl.u64 = cvmx_read_csr(CVMX_PCSXX_CONTROL1_REG(interface));
xauiCtl.s.lo_pwr = 0;
xauiCtl.s.reset = 1;
cvmx_write_csr(CVMX_PCSXX_CONTROL1_REG(interface), xauiCtl.u64);
/* Wait for PCS to come out of reset */
if (CVMX_WAIT_FOR_FIELD64
(CVMX_PCSXX_CONTROL1_REG(interface), union cvmx_pcsxx_control1_reg,
reset, ==, 0, 10000))
return -1;
/* Wait for PCS to be aligned */
if (CVMX_WAIT_FOR_FIELD64
(CVMX_PCSXX_10GBX_STATUS_REG(interface),
union cvmx_pcsxx_10gbx_status_reg, alignd, ==, 1, 10000))
return -1;
/* Wait for RX to be ready */
if (CVMX_WAIT_FOR_FIELD64
(CVMX_GMXX_RX_XAUI_CTL(interface), union cvmx_gmxx_rx_xaui_ctl,
status, ==, 0, 10000))
return -1;
/* (6) Configure GMX */
gmx_cfg.u64 = cvmx_read_csr(CVMX_GMXX_PRTX_CFG(0, interface));
gmx_cfg.s.en = 0;
cvmx_write_csr(CVMX_GMXX_PRTX_CFG(0, interface), gmx_cfg.u64);
/* Wait for GMX RX to be idle */
if (CVMX_WAIT_FOR_FIELD64
(CVMX_GMXX_PRTX_CFG(0, interface), union cvmx_gmxx_prtx_cfg,
rx_idle, ==, 1, 10000))
return -1;
/* Wait for GMX TX to be idle */
if (CVMX_WAIT_FOR_FIELD64
(CVMX_GMXX_PRTX_CFG(0, interface), union cvmx_gmxx_prtx_cfg,
tx_idle, ==, 1, 10000))
return -1;
/* GMX configure */
gmx_cfg.u64 = cvmx_read_csr(CVMX_GMXX_PRTX_CFG(0, interface));
gmx_cfg.s.speed = 1;
gmx_cfg.s.speed_msb = 0;
gmx_cfg.s.slottime = 1;
cvmx_write_csr(CVMX_GMXX_TX_PRTS(interface), 1);
cvmx_write_csr(CVMX_GMXX_TXX_SLOT(0, interface), 512);
cvmx_write_csr(CVMX_GMXX_TXX_BURST(0, interface), 8192);
cvmx_write_csr(CVMX_GMXX_PRTX_CFG(0, interface), gmx_cfg.u64);
/* (7) Clear out any error state */
cvmx_write_csr(CVMX_GMXX_RXX_INT_REG(0, interface),
cvmx_read_csr(CVMX_GMXX_RXX_INT_REG(0, interface)));
cvmx_write_csr(CVMX_GMXX_TX_INT_REG(interface),
cvmx_read_csr(CVMX_GMXX_TX_INT_REG(interface)));
cvmx_write_csr(CVMX_PCSXX_INT_REG(interface),
cvmx_read_csr(CVMX_PCSXX_INT_REG(interface)));
/* Wait for receive link */
if (CVMX_WAIT_FOR_FIELD64
(CVMX_PCSXX_STATUS1_REG(interface), union cvmx_pcsxx_status1_reg,
rcv_lnk, ==, 1, 10000))
return -1;
if (CVMX_WAIT_FOR_FIELD64
(CVMX_PCSXX_STATUS2_REG(interface), union cvmx_pcsxx_status2_reg,
xmtflt, ==, 0, 10000))
return -1;
if (CVMX_WAIT_FOR_FIELD64
(CVMX_PCSXX_STATUS2_REG(interface), union cvmx_pcsxx_status2_reg,
rcvflt, ==, 0, 10000))
return -1;
cvmx_write_csr(CVMX_GMXX_RXX_INT_EN(0, interface), gmx_rx_int_en.u64);
cvmx_write_csr(CVMX_GMXX_TX_INT_EN(interface), gmx_tx_int_en.u64);
cvmx_write_csr(CVMX_PCSXX_INT_EN_REG(interface), pcsx_int_en_reg.u64);
cvmx_helper_link_autoconf(cvmx_helper_get_ipd_port(interface, 0));
/* (8) Enable packet reception */
xauiMiscCtl.s.gmxeno = 0;
cvmx_write_csr(CVMX_PCSXX_MISC_CTL_REG(interface), xauiMiscCtl.u64);
gmx_cfg.u64 = cvmx_read_csr(CVMX_GMXX_PRTX_CFG(0, interface));
gmx_cfg.s.en = 1;
cvmx_write_csr(CVMX_GMXX_PRTX_CFG(0, interface), gmx_cfg.u64);
__cvmx_interrupt_pcsx_intx_en_reg_enable(0, interface);
__cvmx_interrupt_pcsx_intx_en_reg_enable(1, interface);
__cvmx_interrupt_pcsx_intx_en_reg_enable(2, interface);
__cvmx_interrupt_pcsx_intx_en_reg_enable(3, interface);
__cvmx_interrupt_pcsxx_int_en_reg_enable(interface);
__cvmx_interrupt_gmxx_enable(interface);
return 0;
}
cvmx_helper_link_info_t __cvmx_helper_xaui_link_get(int ipd_port)
{
int interface = cvmx_helper_get_interface_num(ipd_port);
union cvmx_gmxx_tx_xaui_ctl gmxx_tx_xaui_ctl;
union cvmx_gmxx_rx_xaui_ctl gmxx_rx_xaui_ctl;
union cvmx_pcsxx_status1_reg pcsxx_status1_reg;
cvmx_helper_link_info_t result;
gmxx_tx_xaui_ctl.u64 = cvmx_read_csr(CVMX_GMXX_TX_XAUI_CTL(interface));
gmxx_rx_xaui_ctl.u64 = cvmx_read_csr(CVMX_GMXX_RX_XAUI_CTL(interface));
pcsxx_status1_reg.u64 =
cvmx_read_csr(CVMX_PCSXX_STATUS1_REG(interface));
result.u64 = 0;
/* Only return a link if both RX and TX are happy */
if ((gmxx_tx_xaui_ctl.s.ls == 0) && (gmxx_rx_xaui_ctl.s.status == 0) &&
(pcsxx_status1_reg.s.rcv_lnk == 1)) {
result.s.link_up = 1;
result.s.full_duplex = 1;
result.s.speed = 10000;
} else {
/* Disable GMX and PCSX interrupts. */
cvmx_write_csr(CVMX_GMXX_RXX_INT_EN(0, interface), 0x0);
cvmx_write_csr(CVMX_GMXX_TX_INT_EN(interface), 0x0);
cvmx_write_csr(CVMX_PCSXX_INT_EN_REG(interface), 0x0);
}
return result;
}
int __cvmx_helper_xaui_link_set(int ipd_port, cvmx_helper_link_info_t link_info)
{
int interface = cvmx_helper_get_interface_num(ipd_port);
union cvmx_gmxx_tx_xaui_ctl gmxx_tx_xaui_ctl;
union cvmx_gmxx_rx_xaui_ctl gmxx_rx_xaui_ctl;
gmxx_tx_xaui_ctl.u64 = cvmx_read_csr(CVMX_GMXX_TX_XAUI_CTL(interface));
gmxx_rx_xaui_ctl.u64 = cvmx_read_csr(CVMX_GMXX_RX_XAUI_CTL(interface));
/* If the link shouldn't be up, then just return */
if (!link_info.s.link_up)
return 0;
/* Do nothing if both RX and TX are happy */
if ((gmxx_tx_xaui_ctl.s.ls == 0) && (gmxx_rx_xaui_ctl.s.status == 0))
return 0;
/* Bring the link up */
return __cvmx_helper_xaui_enable(interface);
}
extern int __cvmx_helper_xaui_configure_loopback(int ipd_port,
int enable_internal,
int enable_external)
{
int interface = cvmx_helper_get_interface_num(ipd_port);
union cvmx_pcsxx_control1_reg pcsxx_control1_reg;
union cvmx_gmxx_xaui_ext_loopback gmxx_xaui_ext_loopback;
/* Set the internal loop */
pcsxx_control1_reg.u64 =
cvmx_read_csr(CVMX_PCSXX_CONTROL1_REG(interface));
pcsxx_control1_reg.s.loopbck1 = enable_internal;
cvmx_write_csr(CVMX_PCSXX_CONTROL1_REG(interface),
pcsxx_control1_reg.u64);
/* Set the external loop */
gmxx_xaui_ext_loopback.u64 =
cvmx_read_csr(CVMX_GMXX_XAUI_EXT_LOOPBACK(interface));
gmxx_xaui_ext_loopback.s.en = enable_external;
cvmx_write_csr(CVMX_GMXX_XAUI_EXT_LOOPBACK(interface),
gmxx_xaui_ext_loopback.u64);
/* Take the link through a reset */
return __cvmx_helper_xaui_enable(interface);
}
|
code
|
\begin{document}
\title{Quantum tomographic cryptography with Bell diagonal states:
non-equivalence of classical and quantum distillation protocols}
\author{Dagomir Kaszlikowski}
\affiliation{Department of Physics, National University of Singapore,
Singapore 117\,542, Singapore}
\author{Jenn Yang, Lim}
\affiliation{Department of Physics, National University of Singapore,
Singapore 117\,542, Singapore}
\author{D. K. L. Oi}
\affiliation{Centre for Quantum Computation, Department of Applied Mathematics
and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, U.K.}
\author{Frederick H.\ Willeboordse}
\affiliation{Department of Physics, National University of Singapore,
Singapore 117\,542, Singapore}
\author{Ajay Gopinathan}
\affiliation{National Institute of Education,
Nanyang Technological University, Singapore 639\,798, Singapore}
\affiliation{Department of Physics, National University of Singapore,
Singapore 117\,542, Singapore}
\author{L. C. Kwek}
\affiliation{National Institute of Education,
Nanyang Technological University, Singapore 639\,798, Singapore}
\affiliation{Department of Physics, National University of Singapore,
Singapore 117\,542, Singapore}
\begin{abstract}
We present a generalized tomographic quantum key distribution protocol in
which the two parties share a Bell diagonal mixed state of two qubits. We
show that if an eavesdropper performs a coherent measurement on many quantum
ancilla states simultaneously, classical methods of secure key distillation
are less effective than quantum entanglement distillation protocols. We also
show that certain Bell diagonal states are resistant to any attempt of
incoherent eavesdropping.
\end{abstract}
\pacs{03.67. -a,89.70. +c}
\maketitle
\section{Introduction}
The security of quantum key distribution (QKD)~\cite{BB84, Ekert91} is an
important consequence of the application of the laws of physics to information
and communication theory. A one-time pad provides perfect cryptographic
security for sending messages between two parties but relies on being able to
distribute a shared secret key~\cite{vernam,walsh}. Classically, it is
impossible to amplify a set of shared randomness, but quantum mechanics allows
this to be done by the transmission of quantum states~\cite{bennett92}. The
full power of quantum cryptography rests on the ability to place upper bounds
on the knowledge of a potential eavesdropper (Eve) about the distributed key
shared by the legitimate parties (Alice and Bob). In this paper we present a
generalization of the so-called tomographic quantum key distribution
protocol~\cite{Cherng}. We consider the situation where Alice and Bob use qubits in a maximally entangled state distributed by a central source. The qubits undergo a quantum channel that converts the state to a Bell diagonal mixed state.
We analyze the security of this protocol under two broad scenarios. In the first
scenario, Alice and Bob agree on a cryptographic key if the correlations
between their measurement results are stronger than any possible correlations
between one of them and Eve, under the assumption
that Eve has full control over the source of entangled qubits but she can only
perform incoherent measurements. The tomographic element of the protocol
allows Alice and Bob to compute the maximal strength of correlations between
Eve and any one of them. The Csisz{\'a}r-K{\"o}rner~\cite{CK} theorem then
guarantees that if the correlations between Alice and Bob are stronger than
those between Eve and either of them, a secure key can be established through
\emph{one-way} error correcting codes.
In the second scenario, we examine the situation when Eve's
correlations are initially stronger than Alice and Bob's. It was
shown that in some cases it is still possible to obtain a secure
key \cite{Maurer}. The idea is that by means of two-way
communication Alice and Bob can strengthen their correlations with
respect to Eve's so that the CK theorem can be applied again. This
procedure is called \emph{advantage distillation} (AD).
There are two possible strategies for Eve within the second scenario:
incoherent and coherent measurements. The first case was examined
in~\cite{ACIN} where it was shown that advantage distillation is possible as
long as the two-qubit state shared by Alice and Bob is entangled. We re-derive
this result using different reasoning than the one presented in~\cite{ACIN}.
In the second case, we show that the above result no longer holds in the case
of coherent measurements by Eve. Indeed, if the qubits are affected by too many
errors (caused by Eve's actions), advantage distillation fails despite Alice and
Bob still sharing an entangled state. In such cases the only way for Alice and
Bob to obtain a secure key is to revert to quantum entanglement distillation.
\section{Tomographic QKD}
In a tomographic QKD scheme, a central source distributes
entangled qubits to Alice and Bob. They independently and randomly
choose to measure three tomographically complete observables
$\sigma_x$, $\sigma_y$ and $\sigma_z$ (Pauli matrices) on each
qubit. At the end of the transmission, they publicly announce
their choice of observables for each qubit pair. They then proceed
to divide their measurement results according to those for which
their measurement bases match, and those for which their
measurement bases do not match. Exchanging a subset of their
measurements allows Alice and Bob to tomographically reconstruct
the density operator of the two-qubit state they share.
Ideally, in the absence of noise in the source or channels, they
expect to receive the maximally entangled state
\begin{equation}\label{ideal}
\ket{\psi_{ideal}}=\frac{1}{\sqrt{2}}\left(\ket{z_0,z_0}+\ket{z_1,z_1}\right),
\end{equation}
where $\ket{z_{k}}$ is the eigenstate of $\sigma_z$ with the
eigenvalue $(-1)^{k}$, and Alice (Bob) possesses the left (right)
qubit.
The results for matching bases can then be used to generate a
cryptographic key as they are either perfectly correlated ( for
$x$ and $z$ bases) or anti-correlated (for $y$ basis).
However, Alice and Bob cannot realistically expect to obtain the maximally
entangled state Eq.~(\ref{ideal}) because either the source is not ideal, the
channel conveying the qubits is noisy, or there is an eavesdropper tampering
with the source. For security analysis, we assume that Eve has total control
over the source and that all the errors are caused by her when she tries to
extract information about the key.
To constrain Eve's information, Alice and Bob use part of their measurements to
perform full tomography on the state distributed by the source. The protocol
we consider here is such that Alice and Bob agree to communicate if and only
if they see the Bell diagonal state
\begin{eqnarray}\label{theform}
\varrho_{AB} &=& \Sum{a,b=0}{1}p_{ab}\ket{z_{ab}}\bra{z_{ab}},
\end{eqnarray}
where
\begin{eqnarray}
\ket{z_{ab}} &=& \frac{1}{\sqrt{2}}\Sum{k=0}{1}\om{kb} \ket{z_k,
z_{k+a}}
\end{eqnarray}
and $\Sum{a,b=0}{1}p_{ab}=1, \om{}=-1$. Following the nomenclature
of~\cite{Hashing} we call $a$ the amplitude bit and $b$ the phase
bit. Here, we assume that $p_{00} > \frac{1}{2}$ \cite{p00}.
The above state can be obtained from the maximally entangled state
Eq.~(\ref{ideal}) assuming that the travelling qubits undergo bit
and phase flips. The so-called Werner state, i.e., the maximally
entangled state with white noise, is a special case where
$p_{01}=p_{10}=p_{11}$. Therefore, the protocol presented here is
more general than the one studied in~\cite{cadIN, Cherng} where
only Werner states were considered.
As Alice and Bob perform their measurements in the three bases
$x,y$ and $z$, it is convenient to express the state
$\varrho_{AB}$ in the $x$ and $y$ bases. This can easily be done
using the transformation rules on the Bell states,
\begin{equation}
\ket{z_{ab}}=\om{ab}\ket{x_{ba}}=(-\ensuremath{\mathrm{i}})^a\om{ab}\ket{y_{a+b+1\;
a}}.
\end{equation}
Writing out (\ref{theform}) in the other bases, we have
\begin{equation}
\varrho_{AB}=\Sum{a,b=0}{1}p_{ba}\ket{x_{ab}}\bra{x_{ab}}=
\Sum{a,b=0}{1}p_{b\;a+b+1}\ket{y_{ab}}\bra{y_{ab}}.
\end{equation}
We can then compute the probability of Alice and Bob obtaining
correlated results conditional on a particular choice of basis:
\begin{eqnarray}
\pr{correlation$|$$x$ basis} &=& p_{00}+p_{11} \nonumber\\
\pr{correlation$|$$y$ basis} &=& p_{01}+p_{10} \nonumber\\
\pr{correlation$|$$z$ basis} &=& p_{00}+p_{11},
\end{eqnarray}
and also the probability of getting anti-correlated results:
\begin{eqnarray}
\pr{anti-correlation$|$$x$ basis} &=& p_{01}+p_{10} \nonumber\\
\pr{anti-correlation$|$$y$ basis} &=& p_{00}+p_{11} \nonumber\\
\pr{anti-correlation$|$$z$ basis} &=& p_{01}+p_{10}.
\end{eqnarray}
Since $p_{00}>\frac{1}{2}$, Alice and Bob are more likely to
obtain correlated results when they measure in the $x$ and $z$
bases, and anti-correlated results in the $y$ basis; Alice and Bob
will thus make use of correlation to generate their key when they
measure in the $x$ and $z$ bases, and anti-correlation to generate
their key when in the $y$ basis.
\section{Eavesdropping}
In order to obtain as much information as possible about the key generated by
Alice and Bob, Eve entangles their qubits with ancilla states $\ket{e_{ab}}$
in her possession. The best she can do is to prepare the following tripartite
pure state
\begin{equation}
\ket{\psi_{ABE}} =\sum_{a,b=0}^{1} \sqrt{p_{ab}} |z_{ab}\rangle
|e_{ab}\rangle, \label{eq:eavestate}
\end{equation}
where $\langle e_{ab}|e_{cd}\rangle = \delta_{a,c}\delta_{b,d}$.
Tracing out Eve gives the mixed state Eq.~(\ref{theform}) that
Alice and Bob measures, and this purification is the most general
one as far as incoherent attacks are concerned.
Eve's purifications, when expressed in different bases, read
\begin{eqnarray}\label{evezstate1}
\ket{\psi_{ABE}}&=& \frac{1}{\sqrt{2}}
\Sum{k,a=0}{1}\ket{z_k,z_{k+a}}\left(\Sum{b=0}{1}\sqrt{p_{ab}}\om{kb}\ket{e_{ab}}\right)\nonumber\\
&=&
\frac{1}{\sqrt{2}}\Sum{k,a=0}{1}\ket{x_k,x_{k+a}}\left(\Sum{b=0}{1}\sqrt{p_{ba}}\om{kb}\om{ab}\ket{e_{ba}}\right)
\nonumber\\
&=& \frac{1}{\sqrt{2}}\Sum{k,a=0}{1}\ket{y_k,y_{k+a}} \nonumber\\
&& \;\; \left(\Sum{b=0}{1}\sqrt{p_{b\;a+b+1}}(-\ensuremath{\mathrm{i}})^b\om{kb}\om{b(a+b+1)}\ket{e_{b\; a+b+1}}\right), \nonumber\\
\end{eqnarray}
We can express Eq.~(\ref{evezstate1}) more conveniently as
\begin{eqnarray}
|\psi_{ABE}\rangle&=&\Sum{k,a=0}{1}\sqrt{\frac{p_a}{2}}\ket{z_k,z_{k+a}}\ket{f_{ka}^{z}}
\nonumber\\
&=&
\Sum{k,a=0}{1}\sqrt{\frac{q_a}{2}}\ket{x_k,x_{k+a}}\ket{f_{ka}^{x}}
\nonumber\\
&=&
\Sum{k,a=0}{1}\sqrt{\frac{r_a}{2}}\ket{y_k,y_{k+a}}\ket{f_{ka}^{y}},
\end{eqnarray}
where
\begin{eqnarray}
p_a &=& \Sum{b=0}{1}p_{ab} \nonumber\\
q_a &=& \Sum{b=0}{1}p_{ba} \nonumber\\
r_a &=& \Sum{b=0}{1} p_{b \; a+b+1}
\end{eqnarray}
and the normalized kets
\begin{eqnarray}
\ket{f_{ka}^{z}} &=&
\frac{1}{\sqrt{p_a}}\Sum{b=0}{1}\sqrt{p_{ab}}\om{kb}\ket{e_{ab}}
\nonumber\\
\ket{f_{ka}^{x}} &=&
\frac{1}{\sqrt{q_a}}\Sum{b=0}{1}\sqrt{p_{ba}}\om{kb}\om{ab}\ket{e_{ba}}
\nonumber\\
\ket{f_{ka}^{y}} &=& \frac{1}{\sqrt{r_a}}\Sum{b=0}{1}\sqrt{p_{b\;
a+b+1}}(-\ensuremath{\mathrm{i}})^b\om{kb}\om{b(a+b+1)}\ket{e_{b\; a+b+1}}
\nonumber\\
\end{eqnarray}
are such that their inner products are given by
\begin{eqnarray}
\langle f_{0a}^{z}|f_{1a}^{z} \rangle &=&
\frac{p_{a0}-p_{a1}}{p_{a0}+p_{a1}}\equiv \lambda^{z}_a \nonumber\\
\langle f_{0a}^{x}|f_{1a}^{x} \rangle &=&
\frac{p_{0a}-p_{1a}}{p_{0a}+p_{1a}} \equiv\lambda^{x}_a \nonumber\\
\langle f_{0a}^{y}|f_{1a}^{y} \rangle &=&
\frac{p_{0\;a+1}-p_{1a}}{p_{0\;a+1}+p_{1a}} \equiv\lambda^{y}_a.
\end{eqnarray}
The ancillas with different $a$'s are orthogonal.
Eve's eavesdropping strategy proceeds as follows. After Alice and
Bob announce their measurement bases, Eve knows on which pairs of
qubits they measured the same observables and that her ancilla is
a mixture of four possible states. Formally this can be viewed as
a transmission of information from Alice and Bob to Eve encoded in
the quantum state of Eve's ancilla. To find the optimal
eavesdropping strategy, she has to maximize this information
transfer by a choice of a suitable generalized measurement known
as a Positive Operator Value Measure (POVM). For example, if
Alice and Bob measured in the $x$ basis, Eve will obtain the
following mixed state of her ancilla,
\begin{equation}
\varrho_E^x=\Sum{k,a=0}{1} \frac{q_a}{2} |f^{x}_{ka}\rangle\langle
f^{x}_{ka}|.
\end{equation}
This is equivalent to Alice and Bob ``communicating'' to Eve that
they measured $\{00,01,10,11\}$ by sending her the quantum states
$\{\ket{f^{x}_{00}},\ket{f^{x}_{01}},\ket{f^{x}_{11}},
\ket{f^{x}_{10}}\}$ with prior probabilities
$\{\frac{q_0}{2},\frac{q_1}{2},\frac{q_1}{2},\frac{q_0}{2}\}$
respectively. Eve has to find the optimal measurement that will
extract from the transmission as much information as possible,
called the \emph{accessible information}. Note that this is not
equivalent to finding a measurement that minimizes the error of
distinguishing between these states~\cite{SHOR}.
\section{Incoherent Attack}
We first assume that Eve carries out an \emph{incoherent} attack
in which she performs measurements on her ancillas one at a time.
In contrast, in a coherent attack, she would measure joint
observables of more than one ancilla, or construct her initial
state Eq.~(\ref{eq:eavestate}) so that more than one pair of
qubits were entangled with each ancilla.
The ancilla states for each basis can be divided into two groups. The first
group corresponds to $a=0$ and refers to the case when Alice and Bob obtain
correlated results. The second group corresponds to the case $a=1$ and refers
to the case when Alice and Bob obtain anti-correlated results.
For example, if Alice and Bob both measure in the $y$ basis, Eve
will have the state
\begin{equation}
\varrho_E^y=\Sum{k,a=0}{1} \frac{r_a}{2} |f^{y}_{ka}\rangle\langle
f^{y}_{ka}|.
\end{equation}
The first group $a=0$ occurs with probability $r_0$ and the second
group $a=1$ occurs with probability $r_1$. Similarly, if they
measure in the $x$ ($z$) basis, the first group occurs with
probability $q_0$ ($p_0$) while the second group occurs with
probability $q_1$ ($p_1$). The ancillas in the first group
$\ket{f_{k0}^{m}}$ ($m=x,y,z$) are orthogonal to those in the
second group $\ket{f_{k1}^{m}}$.
For the purpose of applying the Csisz{\'a}r-K{\"o}rner theorem, we
need only to compute the mutual information between Eve and Bob
and compare this with the mutual information between Alice and
Bob; Eve would have to optimize her measurements on her ancilla so
that it maximizes the information she gains about Bob's
measurement results.
Let us now present the POVM measurement that maximizes the
information transferred by Bob to Eve.
In the first step, Eve sorts the mixture of the ancillas into two
sub-ensembles according to the index $a$. This can easily be done
using a projective measurement. This sorting is an auxiliary step
as, at this stage, she does not gain any more information about
the result of Bob's measurement. After that, depending on the
outcome of the projection ($a=0$ or $a=1$), Eve has an
equiprobable mixture of two non-orthogonal ancilla states each
corresponding to Alice and Bob's result. For example, if the
chosen measurement basis was the $z$ basis, Eve will receive the
mixed state
\begin{eqnarray}
\varrho^z_E &=&
\Sum{k,a=0}{1}\frac{p_a}{2}\ket{f^z_{ka}}\bra{f^z_{ka}}.
\end{eqnarray}
Projecting into either the $a=0$ or $a=1$ orthogonal subspaces
(depending on Alice and Bob's measurement outcomes), she will
obtain one of the two equiprobable mixtures of non-orthogonal
ancilla states
\begin{eqnarray}
\varrho_0^z = \frac{1}{2} \ket{f^z_{00}}\bra{f^z_{00}}+\frac{1}{2}
\ket{f^z_{10}}\bra{f^z_{10}} &\mbox{if Alice and Bob obtained}&
\nonumber\\
&\mbox{correlated results, $a=0$;}& \nonumber\\
\varrho_1^z = \frac{1}{2} \ket{f^z_{01}}\bra{f^z_{01}}+\frac{1}{2}
\ket{f^z_{11}}\bra{f^z_{11}} &\mbox{if Alice and Bob obtained}&
\nonumber\\
&\mbox{anti-correlated results, $a=1$.}& \nonumber\\
\end{eqnarray}
Next, she applies the measurement that maximizes the accessible
information encoded in the mixture of the two ancilla states given
by the outcome of her projective measurement. In the case of two
equally likely states, this optimum measurement is given by the
so-called {\it square-root measurement}~\cite{sqm1, sqm2}. The
outcome probabilities of the square-root measurement are
\begin{eqnarray}
\eta_a^{x} &=& \frac{1}{2}\left(1+\sqrt{1-(\lambda_a^{x})^2}\right)\nonumber\\
\eta_a^{y} &=& \frac{1}{2}\left(1+\sqrt{1-(\lambda_a^{y})^2}\right)\nonumber\\
\eta_a^{z} &=&
\frac{1}{2}\left(1+\sqrt{1-(\lambda_a^{z})^2}\right),
\end{eqnarray}
where $\eta_a^{m}$ is the probability of correctly inferring a
given ancilla state in the $m$ basis ($m=x,y,z$). The index $a$
refers to the correlation/anti-correlation subspace in which the
ancilla lies.
It is straightforward to compute the mutual information between
Bob and Eve:
\begin{eqnarray}
I_{BE} &=&
\frac{1}{3}I_{BE}^{x}+\frac{1}{3}I_{BE}^{y}+\frac{1}{3}I_{BE}^{z},
\end{eqnarray}
where $I_{BE}^{m}$ is the mutual information when Alice and Bob
measure in the same basis $m$. We have
\begin{eqnarray}
I_{BE}^{x} &=& q_0 \left(1-H(\eta_0^{x})\right)+q_1
\left(1-H(\eta_1^{x})\right) \nonumber\\
I_{BE}^{y} &=& r_0 \left(1-H(\eta_0^{y})\right)+r_1
\left(1-H(\eta_1^{y})\right) \nonumber\\
I_{BE}^{z} &=& p_0 \left(1-H(\eta_0^{z})\right)+p_1
\left(1-H(\eta_1^{z})\right).
\end{eqnarray}
Here, $H(\eta_a^{m}) =
-\eta_a^{m}\log_2{\eta_a^{m}}-(1-\eta_a^{m})\log_2{(1-\eta_a^{m})}$
is the \emph{binary entropy} of the respective probability
distributions. Also, the mutual information between Alice and Bob
is given by
\begin{eqnarray}
I_{AB} &=& 1-\frac{1}{3}\left(H(p_0)+H(q_0)+H(r_0)\right).
\end{eqnarray}
We are interested in the conditions for which our protocol is secure against Eve's incoherent eavesdropping attack.
Now, even if Eve obtains some information about the transmitted key through her incoherent measurement,
Alice and Bob can still obtain a secure key with a few additional
steps. According to the Csisz{\'a}r-K{\"o}rner (CK) theorem, a
secure key can be generated from a raw key sequence by means of a
suitably chosen error-correcting code and classical one-way
communication between Alice and Bob if the mutual information
between Alice and Bob exceeds that between Eve and either one of
them (the CK regime). For the protocol considered, the mutual
information between Alice and Eve, and Bob and Eve, are the same
so that security is assured as long as
\begin{eqnarray}\label{MI}
I_{AB} &>& I_{BE}.
\end{eqnarray}
\section{Quantum Entanglement Distillation}
If there is too much noise in the two-qubit state, the CK theorem is
not immediately applicable. Instead, Alice and Bob need to either select a
subsequence of their bit values in a systematic way or pre-process their
two-qubit state before measuring, so that the CK theorem is applicable once
more. One method of doing this is \emph{quantum entanglement distillation} (QED), a quantum
procedure by which many weakly entangled qubit pairs are distilled into a
smaller number of more strongly entangled qubit pairs by means of local
operations and classical communication.
Alice and Bob's two-qubit state Eq.~(\ref{theform}) can be
distilled successfully using local operations and classical
communication as long as they satisfy the Peres--Horodecki partial
transposition criterion \cite{HOR-PER}: A two-qubit state
$\varrho$ is quantum distillable if and only if it is a
\emph{non-positive partial transposed} (NPPT) state. A state
$\varrho$ is NPPT if $\varrho^{T_B}\not\ge 0$ so that it has at
least one negative eigenvalue. Here, $\varrho^{T_B}$ denotes the
transposition with respect to Bob's basis only. The partial
transpose of each of our Bell states gives,
\begin{eqnarray}
\ket{z_{kl}}\bra{z_{kl}} &\longrightarrow&
\frac{1}{2}-\ket{z_{k+1\; l+1}}\bra{z_{k+1\; l+1}}.
\end{eqnarray}
Applying the Peres--Horodecki criterion to our Bell diagonal
mixture, we find that the state Eq.~(\ref{theform}) is quantum
distillable provided that
\begin{eqnarray}\label{qd}
\max_{ab} p_{ab} &>& \frac{1}{2}.
\end{eqnarray}
\section{Advantage Distillation}
Instead of manipulating their qubits in QED, Alice and Bob can
process the raw key sequence they have established in the protocol
in order to obtain a more secure key sequence. One such procedure
is known as \emph{advantage distillation} (AD).
In the AD protocol, Alice and Bob divide their raw key sequence
into blocks of length $L$. For each block, Alice generates a
random bit and adds this, modulo $2$, to each bit of the block.
She then sends this processed block to Bob via a public channel.
After receiving the block, Bob subtracts his corresponding block
from it (modulo $2$). If all the bit values are the same, it is a
deemed a good block. Otherwise it is a bad block. Bob then informs
Alice whether the block he received was good or bad. If it is a
good block, Alice will record the random bit she initially
generated into her distilled bit sequence while Bob enters into
his distilled sequence the common bit value he found after
subtraction. If it is a bad block, they will both reject the bits
and it plays no further part in the distillation procedure.
Now for a good block, two cases can occur:
\begin{description}
\item[(I)] Alice and Bob's distilled bits are the same;
\item[(II)] Alice and Bob's distilled bits are different.
\end{description}
Case (I) occurs when Alice and Bob started out with an identical
raw block (i.e.\ their length $L$ blocks are perfectly
correlated). On the other hand, Case (II) occurs when Alice and
Bob start out with raw $L$-blocks that are anti-correlated with
each other.
Now, for large $L$, there will be approximately $\frac{L}{3}$ bits
in the good block that result from Alice and Bob's $z$ basis
measurement. For these, $p_0$ is the probability that Alice and
Bob obtain correlated results while $p_1$ is the probability that
they obtain anti-correlated results. Similarly, $\frac{L}{3}$
bits will result from $x$ ($y$) basis measurement
--- $q_0$ ($r_0$) is the probability that Alice and Bob obtain
correlated results while $q_1$ ($r_1$) is the probability that
they obtain anti-correlated results. Thus for a good block, Case
(I) occurs with probability $\frac{p_0^{L/3} q_0^{L/3}
r_1^{L/3}}{p_0^{L/3} q_0^{L/3}
r_1^{L/3}+p_1^{L/3}q_1^{L/3}r_0^{L/3}}$ while Case (II) occurs
with probability $\frac{p_1^{L/3} q_1^{L/3} r_0^{L/3}}{p_0^{L/3}
q_0^{L/3} r_1^{L/3}+p_1^{L/3}q_1^{L/3}r_0^{L/3}}$ (remember that
for the $y$ basis, Alice and Bob generate their raw key from
anti-correlation, which corresponds to probability $r_1$). The
error rate for Alice and Bob (the proportion of Case (II) blocks)
is given by
\begin{eqnarray}
E_{AB} &=& \frac{p_1^{L/3} q_1^{L/3} r_0^{L/3}}{p_0^{L/3}
q_0^{L/3} r_1^{L/3}+p_1^{L/3}q_1^{L/3}r_0^{L/3}},
\end{eqnarray}
which for $L\gg 1$, $p_1<p_0$ and $q_1<q_0$ (since $p_{00} > \frac{1}{2}$) is approximately
\begin{eqnarray}
E_{AB} &\approx& \left(\frac{p_1 q_1 r_0}{p_0 q_0
r_1}\right)^{L/3}.
\end{eqnarray}
Eve is able to intercept the processed blocks that Alice sends to
Bob via the classical channel. From their public communication, she
will also be able to know which of the blocks are accepted or rejected. For
the good blocks, she has to deduce the distilled bit for each block.
To do this, she can either resort to incoherent or coherent measurements on
her ancillas.
\subsection{Incoherent Attack on Advantage Distillation}
In the incoherent attack, Eve performs a square root measurement to distinguish
her ancillas one by one and, from her results, tries to deduce what Alice and Bob measured for each entry in an $L$-block. She then subtracts Alice's transmitted
block from her own corresponding block, as Bob does. Typically, Eve's block
will be inhomogeneous after subtraction so she decides by majority voting which
bit value to assign to a particular block -- she bets on the value which
occurs most frequently in her block, and if there are the same number of $0$s
as $1$s, she picks one of them at random.
Consider Case (I) blocks, i.e. Alice and Bob start out with
correlated raw blocks. From her square-root measurement, Eve
guesses each entry in the block correctly with the following
probabilities:
\begin{itemize}
\item $\eta_0^{x}$ if Alice and Bob measured in the $x$ basis;
\item $\eta_1^{y}$ if they measured in the $y$ basis;
\item $\eta_0^{z}$ if they measured in the $z$-basis.
\end{itemize}
She guesses an entry incorrectly with probabilities
\begin{itemize}
\item $1-\eta_0^{x}$ if Alice and Bob measured in the $x$ basis;
\item $1-\eta_1^{y}$ if they measured in the $y$ basis;
\item $1-\eta_0^{z}$ if they measured in the $z$-basis.
\end{itemize}
Because Eve applies majority voting, she makes errors whenever she
guesses more than half of the entries in a block wrongly. If the
same number of $0$s and $1$s appear in her guesses, she picks one
of them at random and makes errors half of the time. We can thus
compute Eve's error rate:
\begin{eqnarray}\label{whaoC}
E^{(I)}_{BE} &=& \Sum{\sum_i
e_i>\frac{L}{2}}{}\ch{\frac{L}{3}}{e_x} (1-\eta^{x}_0)^{e_x}
(\eta^{x}_0)^{\frac{L}{3}-e_x} \nonumber\\
&&\times \ch{\frac{L}{3}}{e_y} (1-\eta^{y}_1)^{e_y}
(\eta^{y}_1)^{\frac{L}{3}-e_y}\nonumber\\
&&\times\ch{\frac{L}{3}}{e_z} (1-\eta^{z}_0)^{e_z}
(\eta^{z}_0)^{\frac{L}{3}-e_z}
\nonumber\\
&&+\frac{1}{2}\Sum{\sum_i e_i=\frac{L}{2}}{}\ch{\frac{L}{3}}{e_x}
(1-\eta^{x}_0)^{e_x} (\eta^{x}_0)^{\frac{L}{3}-e_x} \nonumber\\
&&\times \ch{\frac{L}{3}}{e_y}
(1-\eta^{y}_1)^{e_y} (\eta^{y}_1)^{\frac{L}{3}-e_y}\nonumber\\
&&\times \ch{\frac{L}{3}}{e_z}
(1-\eta^{z}_0)^{e_z} (\eta^{z}_0)^{\frac{L}{3}-e_z},
\end{eqnarray}
where $e_i$ is the number of errors made in the $i^{th}$ basis.
The second summation arises from the situation when Eve has to assign $0$
or $1$ at random to the block because the number of $0$s and $1$s in the block
are equal.
For $L\gg 1$, we can lower bound the summations in Eq.~(\ref{whaoC}) by
approximating them with the main contributing terms, i.e., terms for which the
binomial factor $\ch{\frac{L}{3}}{e_m}, (m=x,y,z)$ has its peak:
\begin{eqnarray}
E^{(I)}_{BE} &\sim& \ch{\frac{L}{3}}{\frac{L}{6}}
(1-\eta^{x}_0)^{\frac{L}{6}} (\eta_0^{x})^{\frac{L}{6}}
\nonumber\\
&&\times\ch{\frac{L}{3}}{\frac{L}{6}}
(1-\eta^{y}_1)^{\frac{L}{6}} (\eta_1^{y})^{\frac{L}{6}}\nonumber\\
&&\times\ch{\frac{L}{3}}{\frac{L}{6}}
(1-\eta^{z}_0)^{\frac{L}{6}} (\eta_0^{z})^{\frac{L}{6}}.
\end{eqnarray}
By applying Stirling's approximation we have
\begin{equation}
E^{(I)}_{BE} \sim 2^L \left(\eta_0^{x}\eta_1^{y}\eta_0^{z}
(1-\eta_0^{x})(1-\eta_1^{y})(1-\eta_0^{z})\right)^{\frac{L}{6}}.
\end{equation}
Similarly for Case (II) blocks in which Alice and Bob start out with
anti-correlated raw blocks, we can obtain the error rate for Eve:
\begin{equation}
E^{(II)}_{BE} \sim 2^L \left(\eta_1^{x}\eta_0^{y}\eta_1^{z}
(1-\eta_1^{x})(1-\eta_0^{y})(1-\eta_1^{z})\right)^{\frac{L}{6}}.
\end{equation}
Finally, the total error rate for Eve is given by
\begin{eqnarray}
E_{BE} &\sim&
\frac{p_0^{\frac{L}{3}}q_0^{\frac{L}{3}}r_1^{\frac{L}{3}}}{p_0^{\frac{L}{3}}
q_0^{\frac{L}{3}}r_1^{\frac{L}{3}}+p_1^{\frac{L}{3}}q_1^{\frac{L}{3}}r_0^{\frac{L}{3}}}
E^{(I)}_{BE} \nonumber\\
&& +\frac{p_1^{\frac{L}{3}}
q_1^{\frac{L}{3}}r_0^{\frac{L}{3}}}{p_0^{\frac{L}{3}}
q_0^{\frac{L}{3}}r_1^{\frac{L}{3}}+p_1^{\frac{L}{3}}q_1^{\frac{L}{3}}r_0^{\frac{L}{3}}}
E^{(II)}_{BE}
\end{eqnarray}
Since the coefficient in front of $E^{(I)}_{BE}$ goes to 1 while the coefficient in
front of $E^{(II)}_{BE}$ goes to 0, we are left with
\begin{equation}
E_{BE} \approx 2^L \left(\eta_0^{x}\eta_1^{y}\eta_0^{z}
(1-\eta_0^{x})(1-\eta_1^{y})(1-\eta_0^{z})\right)^{\frac{L}{6}}.
\end{equation}
By comparing the error rates \cite{Maurer}, we can obtain the condition for AD
to be successful under an incoherent attack:
\begin{equation}
\lim_{L\rightarrow\infty}\frac{E_{AB}}{E_{BE}} < 1
\label{eq:CKcond}
\end{equation}
which reduces to
\begin{equation}\label{cadIncoh}
\frac{p_1}{p_0}\frac{q_1}{q_0}\frac{r_0}{r_1} <
8\sqrt{\eta_0^{x}\eta_1^{y}\eta_0^{z}
(1-\eta_0^{x})(1-\eta_1^{y})(1-\eta_0^{z})}.
\end{equation}
For the special case of Werner states
($p_{01}=p_{10}=p_{11}=\frac{1-p_{00}}{3}$, so that $p_0=q_0=r_1$,
$p_1=q_1=r_0$ and $\eta_0^x=\eta_1^y=\eta_0^z$), we find that
Eq.~(\ref{cadIncoh}) reduces to
\begin{eqnarray}
\frac{p_1}{p_0} &<& 2\sqrt{\eta_0^z (1-\eta_0^z)}.
\end{eqnarray}
A similar result was obtained by Bru$\ss$ et al.\ \cite{cadIN}.
\subsection{Coherent Attack on Advantage Distillation}
We consider a particularly simple scheme of coherent attack that
is similar to that presented in \cite{K}. Eve's strategy is as
follows.
For each good block, Eve has a corresponding set of ancilla states and rather than measuring her ancillas one-by-one (an incoherent attack), she performs a joint
measurement on \emph{all} $L$ of them to acquire knowledge about the value that
Alice assigned to the block. By also making use of the classical information
that is exchanged between Alice and Bob during the distillation process, Eve
can learn a lot more than if she were to measure her ancillas one by one.
Consider first a Case (I) block.
As an example, suppose that Alice and Bob start out with the same
block $01001$ for $L=5$, and Alice's random bit is $1$. After
addition (modulo $2$), she sends the processed block $10110$ to
Bob via the public channel which Eve is able to intercept. Eve can
also project her block of ancilla states into the orthogonal
subspace corresponding to Alice and Bob having a correlated or
anti-correlated block. Doing this, she can know that Alice and Bob
started out with the same raw blocks (i.e.\ Case (I) blocks). From
this, Eve can then deduce the following possibilities:
\begin{enumerate}
\item If Alice's random bit is `$0$', Alice and Bob must have started out with
raw blocks $10110$. If Alice and Bob had measured in the bases $x,y,y,z,x$ for the respective entries in the block, the \
ancilla state that she holds will be
$\ket{f^{x}_{11}}\ket{f^{y}_{01}}\ket{f^{y}_{10}}
\ket{f^{z}_{11}}\ket{f^{x}_{00}}$.
\item If Alice's random bit is `$1$', Alice and Bob must have started out with
raw blocks $01001$. If $x,y,y,z,x$ is the order of basis measurements for the entries,
the ancilla state that she holds will then be
$\ket{f^{(x)}_{00}}\ket{f^{y}_{10}}\ket{f^{y}_{01}}
\ket{f^{z}_{00}}\ket{f^{x}_{11}}$.
\end{enumerate}
The mutual inner product between the two ancilla states is
$(\lambda^{x}_0)^{n_x}
(\lambda^{y}_1)^{n_y}(\lambda^{z}_0)^{n_z}$, where $n_a$ is the
number of times the basis $a$ was measured. The optimal
measurement to distinguish these two states is again the square
root measurement. In general, for each Case (I) block of length
$L$, Eve needs to distinguish just $2$ possible $L$-ancilla states
with mutual inner product $(\lambda^{x}_0)^{n_x}
(\lambda^{y}_1)^{n_y} (\lambda^{z}_0)^{n_z}$.
Now, for large $L$, we have $n_x, n_y, n_z \approx \frac{L}{3}$. Eve's
probability of correctly inferring a particular $L$-ancilla state is
given by
\begin{equation}
\frac{1}{2}\left(
1+\sqrt{1-(\lambda^{x}_0\lambda^{y}_1\lambda^{z}_0)^{\frac{2L}{3}}}\right)
\approx 1-\frac{1}{4}
(\lambda^{x}_0\lambda^{y}_1\lambda^{z}_0)^{\frac{2L}{3}}.
\end{equation}
Her error rate for Case (I) blocks is thus
\begin{equation}
E_{BE}^{(I)}\approx
\frac{1}{4}(\lambda^{x}_0\lambda^{y}_1\lambda^{z}_0)^{\frac{2L}{3}}.
\end{equation}
Similarly when we consider Case (II) blocks,
Eve's error rate is
\begin{equation}
E_{BE}^{(II)}\approx \frac{1}{4}
(\lambda^{x}_1\lambda^{y}_0\lambda^{z}_1)^{\frac{2L}{3}}.
\end{equation}
Eve's \emph{total} error rate is thus
\begin{eqnarray}
E_{BE} &=&
\frac{p_0^{\frac{L}{3}}q_0^{\frac{L}{3}}r_1^{\frac{L}{3}}}{p_0^{\frac{L}{3}}
q_0^{\frac{L}{3}}r_1^{\frac{L}{3}}+p_1^{\frac{L}{3}}q_1^{\frac{L}{3}}r_0^{\frac{L}{3}}}
E_{BE}^{(I)} \nonumber\\
&&
+\frac{p_1^{\frac{L}{3}}q_1^{\frac{L}{3}}r_0^{\frac{L}{3}}}{p_0^{\frac{L}{3}}
q_0^{\frac{L}{3}}r_1^{\frac{L}{3}}+p_1^{\frac{L}{3}}q_1^{\frac{L}{3}}r_0^{\frac{L}{3}}}E_{BE}^{(II)} \nonumber\\
&\approx& \frac{1}{4}
(\lambda^{x}_0\lambda^{y}_1\lambda^{z}_0)^{\frac{2L}{3}}
\end{eqnarray}
since once again, the coefficient in front of $E^{(I)}_{BE}$ goes to 1 while the coefficient in front of $E^{(II)}_{BE}$ goes to 0.
Finally by comparing error rates (Eq.~(\ref{eq:CKcond})), we obtain the
condition for AD to be possible under a coherent attack by Eve:
\begin{equation}\label{cadCo}
\frac{p_1}{p_0}\frac{q_1}{q_0}\frac{r_0}{r_1} <
\left(\lambda^{x}_0\lambda^{y}_1\lambda^{z}_0\right)^2.
\end{equation}
\section{Discussion}
We now analyze the above results. A Bell diagonal density matrix
is characterized by four real parameters and a normalization
condition so we can parameterize such a state by the probability
$p_{00}$ (the amount of the state $|z_{00}\rangle$ in the Bell
mixture) and two angles $\theta,\phi$ characterizing the remaining
three probabilities $p_{01},p_{10},p_{11}$:
\begin{eqnarray}
p_{01} &=& (1-p_{00})\cos^2\theta\cos^2\phi\nonumber\\
p_{10} &=& (1-p_{00})\sin^2\theta\cos^2\phi\nonumber\\
p_{11} &=& (1-p_{00})\sin^2\phi.
\end{eqnarray}
This means that for a fixed $p_{00}$, all the quantities such as
$I_{AB},I_{BE},E_{AB},E_{BE}$ for incoherent and coherent attacks
are two-argument functions.
First, for each $p_{00}$ we can plot a region characterizing all
the Bell diagonal states which lead to secure \emph{raw} keys. As
long as $p_{00}$ is greater than around $0.765$ all corresponding
states are secure. Below this, fewer and fewer states are secure
(white regions in Fig.~\ref{fig:fig1}) until, for
$p_{00}=\frac{1}{2}$, the Bell diagonal mixture becomes separable
and no secret bits can be obtained. Even then we can still
identify certain states that are resistant against incoherent
eavesdropping as long as $p_{00}$ is greater than half. These are
states of the form
$p_{00}\ket{z_{00}}\bra{z_{00}}+p_{01}\ket{z_{01}}\bra{z_{01}}$,
$p_{00}\ket{z_{00}}\bra{z_{00}}+p_{10}\ket{z_{10}}\bra{z_{10}}$
and
$p_{00}\ket{z_{00}}\bra{z_{00}}+p_{11}\ket{z_{11}}\bra{z_{11}}$.
It is interesting to note that this threshold of $0.765$, below
which it is no longer possible to generate secure keys for every
state, is the same threshold as that for the Werner state ---
this means that the Werner state will be the first state to become
insecure as the $p_{00}$ threshold is exceeded.
Second, using Eq.~(\ref{cadIncoh}) we verified the results
presented in~\cite{ACIN}, namely that QED is equivalent to AD if
Eve can only perform incoherent attacks. In other words, as long
as $p_{00}$ is greater than $\frac{1}{2}$, Alice and Bob do not
need QED because AD works equally well and does not require
collective operations on qubits, which are difficult to realize
experimentally.
However, if Eve is capable of carrying out a coherent attack, QED
is much more powerful than AD (Fig.~\ref{fig:fig2}). We see that
as $p_{00}\rightarrow\frac{1}{2}$, more states fall into the black
regions where AD fails and only QED is possible. As before, the
same states that are resistant to incoherent attack in the CK
regime are resistant to the above coherent attack on AD.
\begin{figure*}
\caption{Comparison of secure regions for the protocol for
different values of $p_{00}
\label{fig:fig1}
\end{figure*}
\begin{figure*}
\caption{Comparison of secure regions in advantage distillation
for different values of $p_{00}
\label{fig:fig2}
\end{figure*}
\section{Conclusion}
We have generalized the tomographic QKD scheme to Bell diagonal states and
analyzed its resistance to various eavesdropping attacks, both in the CK
regime and when Alice and Bob perform advantage distillation. We have shown the
inequivalence of advantage distillation and entanglement distillation in the
presence of coherent measurement by a potential eavesdropper. It still remains
to be seen whether Eve can further increase her information gain by entangling
more than one pair of Alice and Bob's qubits with her ancilla.
DKLO is supported by the Cambridge-MIT Institute project on
quantum information and Sidney Sussex College Cambridge, and
acknowledges EU grants RESQ (IST-2001-37559) and TOPQIP
(IST-2001-39215). DK, LCK and AG wish to acknowledge support from
A*STAR Grant R-144-000-071-305. DK wishes to acknowledge NUS Grant
R-144-000-089-112. DK and AG also wish to thank Antonio Ac\'{\i}n
for valuable discussions.
\end{document}
|
math
|
In the past it was acceptable for marketers to be either creative or analytical. But today’s dynamic competitive marketplace requires marketers that can bridge the divide and leverage both creativity and analytics to deliver sustainable growth.
Unfortunately it is rare for a human being to possess an acute aptitude for analytics and a high capacity for creativity. It’s simply not the way most of us are wired.
In general, the left and right hemispheres of our brain process information in different ways. And while the two sides do in fact work together, most of us have an inherent tendency to favor one or the other.
But that doesn’t change the marketing ecosystem that we live in.
Today’s top marketers are recognizing that while creativity is important, creativity used in conjunction with analytics will measurably heighten marketing success.
It’s time to stop thinking like just a Chief Marketing Officer, and time to start contributing like a Chief Profit Officer.
To make the jump from a Chief Marketing Officer to a “Chief Profit Officer” takes both a strong creative approach and a savvy analytical base.
You’ve got to be able to see around corners and know when and where your competition is connecting with your customers, so that you can intercept them in real time.
Just because a customer formerly found your company and may have purchased from you previously doesn’t necessarily mean they will follow the same route back to your business when they are ready to buy again.
With the constant proliferation of media, marketing technologies and competitors, the likelihood that a customer will be hijacked somewhere along the path to purchase is all too likely.
Understanding your analytics and data necessitates a different way of thinking. Much of creativity comes from the inside out. It’s strategic but it also requires you to create something out of nothing.
Data-driven marketers take a different approach. They accumulate information from all around them and try to make sense of it.
There is much more to it than sifting through reports and analyzing what has happened in the past. The real value comes from the ability to identify key insights and map out a course of action that puts you in front of customers when and where they are ready to buy next.
Most platforms and algorithms are built on math principles that are decades old.
The "Holy Grail" for today's most astute marketers is the ability to ensure the most efficient marketing mix, optimize marketing spend, and deliver the most profitable, converting customers. All of which is essential to creating long-term shareholder value. It’s important to recognize that marketing dollars & budgets can impact enterprise EBIT by as much as 20%.
This requires current, insightful data, which allows you to not only adjust campaigns and spend, but also to recognize and capture opportunities directly as they arise.
In the past this has been a challenge for marketers.
Forget analyzing data from a “rear view” perspective. Smart marketers will soon be looking to tools like DemandJump that provide insight towards what businesses should do next.
At a fundamental level, DemandJump uses predictive analytics to synthesize data in a way that provides marketers with specific action steps. These steps can then be executed on to capture opportunities and seize share of voice from competitors.
Just think of it as your own personal data scientist.
Through complex algorithms, graph theory, machine learning, and more, DemandJump helps show marketers the greatest opportunities for growth.
Modern marketing is a complex science. The exponential scale and complexity of data has created needs that many may have never thought possible.
The good news is that with every challenge comes opportunity. Utilizing an effective, data-driven approach to help guide talented creative teams in the right direction provides yet another chance to differentiate your brand from the competition.
The future of marketing isn’t just about thinking like a Chief Marketing Officer, it’s about thinking like a Chief Profit Officer.
|
english
|
#!/bin/bash
set -e
branch=$1
version=$2
echo "Building salmon [branch = ${branch}]. Tagging version as ${version}"
# Activate Holy Build Box environment.
source /hbb_exe/activate
set -x
# Install things we need
yum install -y --quiet wget
wget http://download.fedoraproject.org/pub/epel/5/x86_64/epel-release-5-4.noarch.rpm
rpm -i --quiet epel-release-5-4.noarch.rpm
#yum install -y --quiet git
yum install -y --quiet unzip
yum install -y --quiet bzip2-devel.x86_64
yum install -y --quiet xz-devel.x86_64
curl -k -L https://github.com/COMBINE-lab/salmon/archive/${branch}.zip -o ${branch}.zip
unzip ${branch}.zip
mv salmon-${branch} salmon
cd salmon
mkdir build
cd build
cmake -DFETCH_BOOST=TRUE ..
make
make install
make test
cd ../scripts
bash make-release.sh -v ${version} -n linux_x86_64
cd ../RELEASES
cp *.tar.gz /io/
|
code
|
समाचार अब तक: महासती विहुला पूजा शुरू, मेला का उद्घाटन
महासती विहुला पूजा शुरू, मेला का उद्घाटन
भागलपुर के सांसद शहनवाज हुसैन ने सोमवार को महासती विहुला मेला का उद्घाटन करते हुए कहा कि इस क्षेत्र में बनने वाले बिजय घाट के पुल का नाम महा सती विहुला के नाम से कराने की कोशिश करूँगा ।इस सम्बन्ध में मुख्य मंत्री से बात करूँगा। अब बकवास नहीं विकास का ज़माना है। नवगछिया को जितना मशहूर होना चाहिए उतना अभी तक नहीं हुआ है। मैं इसे देश स्तर पर मशहूर करने कार्य कर रहा हूँ। भागलपुर कि तरह ही नवगछिया भी चमकेगा और इसका नाम भी भारत के लोग जानेंगे।
इस समारोह को जदयू नवगछिया जिला अध्यक्ष वीरेंद्र सिंह, भाजपा तकनीक मंच के प्रदेश अध्यक्ष शैलेन्द्र कुमार, सांसद प्रवक्ता मृणाल शेखर, सांसद प्रतिनिधि प्रवीन भगत सहित कई क्षेत्रीय नेता एवं कार्यकर्त्ता उपस्थित थे।
सांसद ने नाबार्ड एवं दिशा ग्रामीण मंच के सौजन्य से लगाई गई मञ्जूषा चित्र कला प्रदर्शनी का भी अवलोकन किया ।
महासती विहुला की पूजा नवगछिया शहर , भागलपुर, नाथनगर, चंपा नगर सहित पुरे बिहार में कई जगह शुरू हो गई है।
|
hindi
|
Is your favourite floor rug looking a little under the weather? Maybe its time to arrange rug steam cleaning in Windale. The team at Carpet Cleaning Direct can have your favourite floor rug looking its best in no time at all. Our expert team who perform all rug steam cleaning in Windale and surroundings areas are highly trained in oil, soil and stain removal on a wide range of fabric types.
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|
english
|
Elevation exists "so that people far from God will be filled with life in Christ." That's the vision that was planted in the heart of Pastor Steven Furtick at the age of 16. In 2005, God assembled a group of families around Pastor Furtick to begin seeing this vision come to life. In February of 2006, Elevation Church launched its first worship experience. Two and a half short years later Elevation Church is meeting in multiple locations with a weekly attendance that has topped 4,000.
In February of 2006, Elevation Church launched its first worship experience. Two and a half short years later Elevation Church is meeting in multiple locations with a weekly attendance that has topped 4,000. Through God's favor and a relentless pursuit of His vision we have watched over 2,000 people profess faith in Jesus Christ during our second year of ministry.
Commitment to the vision of life transformation is what drives every corner of this ministry. These songs were birthed from the heart of this movement. Our hope is that through these songs, you'll experience a glimpse of this revolutionary move of God.
|
english
|
In the title compound, (C7H11N2)[Er(H2O)8]Cl4·H2O, the asymmetric unit consists of one 4-(dimethylamino)pyridinium and one octaaquaerbium cation balanced by four Cl− anions, and one water molecule. The 4-(dimethylamino)pyridinium cation is protonated at the pyridine N atom. The dimethylamino group (C/N/C) lies close to the plane of the pyridinium ring, making a dihedral angle of 4.5 (3)°. In the crystal, the [Er(H2O)8]3+ cations are linked via O—H⋯O and O—H⋯Cl hydrogen bonds, forming two-dimensional networks propagating in the ab plane. These networks are linked via O—H⋯O and O—H⋯Cl hydrogen bonds, forming a three-dimensional network. The 4-(dimethylamino)pyridinium cations are located in the cavities and are linked to the framework via N—H⋯Cl, C—H⋯O and C—H⋯Cl hydrogen bonds.
For similar structures in this series involving 4-(dimethylamino)pyridinium, see: Benslimane et al. (2012a ,b ). For details of the Cambridge Structural Database, see: Allen (2002 ). For hydrogen-bond motifs see: Bernstein et al. (1995 ).
Symmetry codes: (i) x-1, y, z; (ii) -x+2, -y+1, -z+1; (iii) x, y-1, z; (iv) x+1, y, z; (v) -x+2, -y+1, -z; (vi) -x+1, -y, -z; (vii) -x+1, -y+1, -z; (viii) -x+2, -y, -z+1.
The title compound is part of a series of lanthanide complexes with the organic cation 4-(dimethylamino)pyridinium, for example: (C7H10N2)2.LaCl(H2O)8.Cl4.3H2O (I) (Benslimane et al., 2012a) and (C7H10N2)3.[Nd2Cl4(H2O)10].Cl5.2H2O (II) (Benslimane et al., 2012b).
The title compound (III) contains an inorganic [Er(H2O)8]3+ and an organic (C7H10N2)+ cation equilibrated by four Cl anions, and one lattice water molecule (Fig. 1). Atom Er1 is coordinated by eight water molecules with Er-O bond distances ranging from 2.2989 (15) to 2.3807 (15) Å. The [Er(H2O)8]3+ cations are linked to the organic cations via Cl- anions through intermolecular O-H···Cl and N-H···Cl hydrogen bonds. Each Cl- anion acts as an acceptor of hydrogen bonds from the pyridinium groups and the water molecules. The water molecules, which act as bridging units between the cations, form cooperative infinite chains parallel to the (100) plane through O-H···Cl hydrogen bonds generating centrosymmetric R24(8) ring motives (Bernstein et al., 1995), as shown in Fig. 2 and Table 1.
In the three compounds, (I) - (III), there is a decrease in the bond lengths of the metal-O(water) bonds, from 2.5101 (15) - 2.5632 (15) Å in (I), 2.404 (3) - 2.479 (4) Å in (II) and 2.2989 (15) - 2.3807 (15) Å in (III). This trend corresponds to the decreasing metallic radius of the lanthanide ion involved; La3+, Nd3+ and Er3+, respectively. In addition, the 4-(dimethylamino)pyridinium cation in the three compounds is protonated at the pyridine N atom. The N-C bond linking the dimethylamino substituent to the pyridinium ring is short, 1.321 (3), 1.324 (3)Å for (I), 1.330 (5), 1.2855 (2) Å for (II) and 1.331 (3) Å for (III), suggesting some delocalization in the cation. A search of the Cambridge Structural Database (CSD, V5.33, Update 4, August 2012; Allen, 2002) reveals similar structures incorporating the 4-(dimethylamino)pyridinium cation for which the corresponding mean N-C distance is 1.34 (1) Å. The dimethylamino group lies close to the plane of the pyridinium ring, with dihedral angles of 3.5 (3) and 2.0 (3)° for (I), 1.6 (6)° and 6.5 (3)° for (II) and 4.5 (3)° for (III).
In conclusion, on the structural level the atomic arrangement in all three compounds, (I) - (III), consists of networks of alternating organic–inorganic layers. The chloride anions are located between these entities forming hydrogen bonds with the NH atoms of the 4-(dimethylamino)pyridinium cations and the water molecules. There are also C—H···Cl interactions present involving one of the 4-(dimethylamino)pyridinium cations. The result is the formation of three-dimensional supramolecular architectures.
For similar structures in this series involving 4-(dimethylamino)pyridinium, see: Benslimane et al. (2012a,b). For details of the Cambridge Structural Database, see: Allen (2002). For hydrogen-bond motifs see: Bernstein et al. (1995).
4-(Dimethylamino)pyridine (1 mmol, 0.051g) and hydrochloric acid (1M) was added slowly to a solution of ErCl3.6H2O (1mmol, 0.08g). The mixture was refluxed at 353 K for about 1 h and then cooled to room temperature. Slow evaporation of the solvent at room temperature lead to the formation of pink plate-like crystals of the title compound.
The H-atoms of the coordinated water molecules were located in difference Fourier syntheses and were initially refined using distance restraints: O-H = 0.85 (2) Å, and H···H= 1.40 (2) Å, with Uiso(H) = 1.5Ueq(O). In the last cycles of refinement they were constrained to ride on their parent O atoms. The N-bound H atom was located in a difference Fourier map but like the C-bound H atoms it was included in calculated positions and treated as riding: N-H=0.86 Å, C-H = 0.93 (aromatic), 0.96 (methyl), with Uiso(H) = 1.5Ueq(C) for the methyl groups and 1.2Ueq(N,C) for the other H atoms.
Fig. 1. The molecular structure of the title compound, showing the atom-numbering. Displacement ellipsoids are drawn at the 50% probability level. The O-H···Cl and N-H···Cl hydrogen bonds are shown as double dashed lines.
Fig. 2. A view of part of the crystal structure of the title compound lying parallel to (100), showing the formation of rings via O-H···Cl and N-H···Cl hydrogen-bonds. Hydrogen bonds are drawn as dashed lines [symmetry codes: (i) x-1, y, z; (ii) -x+2, -y+1, -z+1; (iii) x+1, y, z].
Symmetry codes: (i) x−1, y, z; (ii) −x+2, −y+1, −z+1; (iii) x, y−1, z; (iv) x+1, y, z; (v) −x+2, −y+1, −z; (vi) −x+1, −y, −z; (vii) −x+1, −y+1, −z; (viii) −x+2, −y, −z+1.
Technical support (X-ray measurements) from Laboratory of Coordination Chemistry, UPR-CNRS 8241, Toulouse, are acknowledged.
|
english
|
سہ اوس یژھان کشمیر اکھ یژھ جاے یتیٚتھ لوکھ بیٚیہ ہیٚچھتھ
|
kashmiri
|
#!/usr/bin/env python
#
# Copyright 2013 The Chromium Authors. All rights reserved.
# Use of this source code is governed by a BSD-style license that can be
# found in the LICENSE file.
"""Runs all types of tests from one unified interface."""
import argparse
import collections
import contextlib
import itertools
import logging
import os
import shutil
import signal
import sys
import threading
import traceback
import unittest
# Import _strptime before threaded code. datetime.datetime.strptime is
# threadsafe except for the initial import of the _strptime module.
# See http://crbug.com/724524 and https://bugs.python.org/issue7980.
import _strptime # pylint: disable=unused-import
from pylib.constants import host_paths
if host_paths.DEVIL_PATH not in sys.path:
sys.path.append(host_paths.DEVIL_PATH)
from devil import base_error
from devil.utils import reraiser_thread
from devil.utils import run_tests_helper
from pylib import constants
from pylib.base import base_test_result
from pylib.base import environment_factory
from pylib.base import test_instance_factory
from pylib.base import test_run_factory
from pylib.results import json_results
from pylib.results import report_results
from pylib.utils import logdog_helper
from pylib.utils import logging_utils
from py_utils import contextlib_ext
_DEVIL_STATIC_CONFIG_FILE = os.path.abspath(os.path.join(
host_paths.DIR_SOURCE_ROOT, 'build', 'android', 'devil_config.json'))
def _RealPath(arg):
if arg.startswith('//'):
arg = os.path.abspath(os.path.join(host_paths.DIR_SOURCE_ROOT,
arg[2:].replace('/', os.sep)))
return os.path.realpath(arg)
def AddTestLauncherOptions(parser):
"""Adds arguments mirroring //base/test/launcher.
Args:
parser: The parser to which arguments should be added.
Returns:
The given parser.
"""
parser.add_argument(
'--test-launcher-retry-limit',
'--test_launcher_retry_limit',
'--num_retries', '--num-retries',
dest='num_retries', type=int, default=2,
help='Number of retries for a test before '
'giving up (default: %(default)s).')
parser.add_argument(
'--test-launcher-summary-output',
'--json-results-file',
dest='json_results_file', type=os.path.realpath,
help='If set, will dump results in JSON form '
'to specified file.')
parser.add_argument(
'--test-launcher-shard-index',
type=int, default=os.environ.get('GTEST_SHARD_INDEX', 0),
help='Index of the external shard to run.')
parser.add_argument(
'--test-launcher-total-shards',
type=int, default=os.environ.get('GTEST_TOTAL_SHARDS', 1),
help='Total number of external shards.')
return parser
def AddCommandLineOptions(parser):
"""Adds arguments to support passing command-line flags to the device."""
parser.add_argument(
'--device-flags-file',
type=os.path.realpath,
help='The relative filepath to a file containing '
'command-line flags to set on the device')
parser.set_defaults(allow_unknown=True)
parser.set_defaults(command_line_flags=None)
def AddTracingOptions(parser):
# TODO(shenghuazhang): Move this into AddCommonOptions once it's supported
# for all test types.
parser.add_argument(
'--trace-output',
metavar='FILENAME', type=os.path.realpath,
help='Path to save test_runner trace json output to.')
parser.add_argument(
'--trace-all',
action='store_true',
help='Whether to trace all function calls.')
def AddCommonOptions(parser):
"""Adds all common options to |parser|."""
default_build_type = os.environ.get('BUILDTYPE', 'Debug')
debug_or_release_group = parser.add_mutually_exclusive_group()
debug_or_release_group.add_argument(
'--debug',
action='store_const', const='Debug', dest='build_type',
default=default_build_type,
help='If set, run test suites under out/Debug. '
'Default is env var BUILDTYPE or Debug.')
debug_or_release_group.add_argument(
'--release',
action='store_const', const='Release', dest='build_type',
help='If set, run test suites under out/Release. '
'Default is env var BUILDTYPE or Debug.')
parser.add_argument(
'--break-on-failure', '--break_on_failure',
dest='break_on_failure', action='store_true',
help='Whether to break on failure.')
# TODO(jbudorick): Remove this once everything has switched to platform
# mode.
parser.add_argument(
'--enable-platform-mode',
action='store_true',
help='Run the test scripts in platform mode, which '
'conceptually separates the test runner from the '
'"device" (local or remote, real or emulated) on '
'which the tests are running. [experimental]')
parser.add_argument(
'-e', '--environment',
default='local', choices=constants.VALID_ENVIRONMENTS,
help='Test environment to run in (default: %(default)s).')
class FastLocalDevAction(argparse.Action):
def __call__(self, parser, namespace, values, option_string=None):
namespace.verbose_count = max(namespace.verbose_count, 1)
namespace.num_retries = 0
namespace.enable_device_cache = True
namespace.enable_concurrent_adb = True
namespace.skip_clear_data = True
namespace.extract_test_list_from_filter = True
parser.add_argument(
'--fast-local-dev',
type=bool, nargs=0, action=FastLocalDevAction,
help='Alias for: --verbose --num-retries=0 '
'--enable-device-cache --enable-concurrent-adb '
'--skip-clear-data --extract-test-list-from-filter')
# TODO(jbudorick): Remove this once downstream bots have switched to
# api.test_results.
parser.add_argument(
'--flakiness-dashboard-server',
dest='flakiness_dashboard_server',
help=argparse.SUPPRESS)
parser.add_argument(
'--gs-results-bucket',
help='Google Storage bucket to upload results to.')
parser.add_argument(
'--output-directory',
dest='output_directory', type=os.path.realpath,
help='Path to the directory in which build files are'
' located (must include build type). This will take'
' precedence over --debug and --release')
parser.add_argument(
'--repeat', '--gtest_repeat', '--gtest-repeat',
dest='repeat', type=int, default=0,
help='Number of times to repeat the specified set of tests.')
parser.add_argument(
'-v', '--verbose',
dest='verbose_count', default=0, action='count',
help='Verbose level (multiple times for more)')
AddTestLauncherOptions(parser)
def ProcessCommonOptions(args):
"""Processes and handles all common options."""
run_tests_helper.SetLogLevel(args.verbose_count, add_handler=False)
if args.verbose_count > 0:
handler = logging_utils.ColorStreamHandler()
else:
handler = logging.StreamHandler(sys.stdout)
handler.setFormatter(run_tests_helper.CustomFormatter())
logging.getLogger().addHandler(handler)
constants.SetBuildType(args.build_type)
if args.output_directory:
constants.SetOutputDirectory(args.output_directory)
def AddDeviceOptions(parser):
"""Adds device options to |parser|."""
parser = parser.add_argument_group('device arguments')
parser.add_argument(
'--adb-path',
type=os.path.realpath,
help='Specify the absolute path of the adb binary that '
'should be used.')
parser.add_argument(
'--blacklist-file',
type=os.path.realpath,
help='Device blacklist file.')
parser.add_argument(
'-d', '--device', nargs='+',
dest='test_devices',
help='Target device(s) for the test suite to run on.')
parser.add_argument(
'--enable-concurrent-adb',
action='store_true',
help='Run multiple adb commands at the same time, even '
'for the same device.')
parser.add_argument(
'--enable-device-cache',
action='store_true',
help='Cache device state to disk between runs')
parser.add_argument(
'--skip-clear-data',
action='store_true',
help='Do not wipe app data between tests. Use this to '
'speed up local development and never on bots '
'(increases flakiness)')
parser.add_argument(
'--target-devices-file',
type=os.path.realpath,
help='Path to file with json list of device serials to '
'run tests on. When not specified, all available '
'devices are used.')
parser.add_argument(
'--tool',
dest='tool',
help='Run the test under a tool '
'(use --tool help to list them)')
parser.add_argument(
'--upload-logcats-file',
action='store_true',
dest='upload_logcats_file',
help='Whether to upload logcat file to logdog.')
logcat_output_group = parser.add_mutually_exclusive_group()
logcat_output_group.add_argument(
'--logcat-output-dir', type=os.path.realpath,
help='If set, will dump logcats recorded during test run to directory. '
'File names will be the device ids with timestamps.')
logcat_output_group.add_argument(
'--logcat-output-file', type=os.path.realpath,
help='If set, will merge logcats recorded during test run and dump them '
'to the specified file.')
def AddGTestOptions(parser):
"""Adds gtest options to |parser|."""
parser = parser.add_argument_group('gtest arguments')
parser.add_argument(
'--app-data-file',
action='append', dest='app_data_files',
help='A file path relative to the app data directory '
'that should be saved to the host.')
parser.add_argument(
'--app-data-file-dir',
help='Host directory to which app data files will be'
' saved. Used with --app-data-file.')
parser.add_argument(
'--delete-stale-data',
dest='delete_stale_data', action='store_true',
help='Delete stale test data on the device.')
parser.add_argument(
'--enable-xml-result-parsing',
action='store_true', help=argparse.SUPPRESS)
parser.add_argument(
'--executable-dist-dir',
type=os.path.realpath,
help="Path to executable's dist directory for native"
" (non-apk) tests.")
parser.add_argument(
'--extract-test-list-from-filter',
action='store_true',
help='When a test filter is specified, and the list of '
'tests can be determined from it, skip querying the '
'device for the list of all tests. Speeds up local '
'development, but is not safe to use on bots ('
'http://crbug.com/549214')
parser.add_argument(
'--gtest_also_run_disabled_tests', '--gtest-also-run-disabled-tests',
dest='run_disabled', action='store_true',
help='Also run disabled tests if applicable.')
parser.add_argument(
'--runtime-deps-path',
dest='runtime_deps_path', type=os.path.realpath,
help='Runtime data dependency file from GN.')
parser.add_argument(
'-t', '--shard-timeout',
dest='shard_timeout', type=int, default=120,
help='Timeout to wait for each test (default: %(default)s).')
parser.add_argument(
'--store-tombstones',
dest='store_tombstones', action='store_true',
help='Add tombstones in results if crash.')
parser.add_argument(
'-s', '--suite',
dest='suite_name', nargs='+', metavar='SUITE_NAME', required=True,
help='Executable name of the test suite to run.')
parser.add_argument(
'--test-apk-incremental-install-json',
type=os.path.realpath,
help='Path to install json for the test apk.')
filter_group = parser.add_mutually_exclusive_group()
filter_group.add_argument(
'-f', '--gtest_filter', '--gtest-filter',
dest='test_filter',
help='googletest-style filter string.')
filter_group.add_argument(
'--gtest-filter-file',
dest='test_filter_file', type=os.path.realpath,
help='Path to file that contains googletest-style filter strings. '
'See also //testing/buildbot/filters/README.md.')
def AddInstrumentationTestOptions(parser):
"""Adds Instrumentation test options to |parser|."""
parser.add_argument_group('instrumentation arguments')
parser.add_argument(
'--additional-apk',
action='append', dest='additional_apks', default=[],
type=_RealPath,
help='Additional apk that must be installed on '
'the device when the tests are run')
parser.add_argument(
'-A', '--annotation',
dest='annotation_str',
help='Comma-separated list of annotations. Run only tests with any of '
'the given annotations. An annotation can be either a key or a '
'key-values pair. A test that has no annotation is considered '
'"SmallTest".')
# TODO(jbudorick): Remove support for name-style APK specification once
# bots are no longer doing it.
parser.add_argument(
'--apk-under-test',
help='Path or name of the apk under test.')
parser.add_argument(
'--coverage-dir',
type=os.path.realpath,
help='Directory in which to place all generated '
'EMMA coverage files.')
parser.add_argument(
'--delete-stale-data',
action='store_true', dest='delete_stale_data',
help='Delete stale test data on the device.')
parser.add_argument(
'--disable-dalvik-asserts',
dest='set_asserts', action='store_false', default=True,
help='Removes the dalvik.vm.enableassertions property')
parser.add_argument(
'--enable-java-deobfuscation',
action='store_true',
help='Deobfuscate java stack traces in test output and logcat.')
parser.add_argument(
'-E', '--exclude-annotation',
dest='exclude_annotation_str',
help='Comma-separated list of annotations. Exclude tests with these '
'annotations.')
parser.add_argument(
'-f', '--test-filter', '--gtest_filter', '--gtest-filter',
dest='test_filter',
help='Test filter (if not fully qualified, will run all matches).')
parser.add_argument(
'--gtest_also_run_disabled_tests', '--gtest-also-run-disabled-tests',
dest='run_disabled', action='store_true',
help='Also run disabled tests if applicable.')
parser.add_argument(
'--render-results-directory',
dest='render_results_dir',
help='Directory to pull render test result images off of the device to.')
parser.add_argument(
'--enable-relocation-packing',
dest='enable_relocation_packing',
action='store_true',
help='Whether relocation packing is enabled.')
def package_replacement(arg):
split_arg = arg.split(',')
if len(split_arg) != 2:
raise argparse.ArgumentError(
'Expected two comma-separated strings for --replace-system-package, '
'received %d' % len(split_arg))
PackageReplacement = collections.namedtuple('PackageReplacement',
['package', 'replacement_apk'])
return PackageReplacement(package=split_arg[0],
replacement_apk=_RealPath(split_arg[1]))
parser.add_argument(
'--replace-system-package',
type=package_replacement, default=None,
help='Specifies a system package to replace with a given APK for the '
'duration of the test. Given as a comma-separated pair of strings, '
'the first element being the package and the second the path to the '
'replacement APK. Only supports replacing one package. Example: '
'--replace-system-package com.example.app,path/to/some.apk')
parser.add_argument(
'--runtime-deps-path',
dest='runtime_deps_path', type=os.path.realpath,
help='Runtime data dependency file from GN.')
parser.add_argument(
'--screenshot-directory',
dest='screenshot_dir', type=os.path.realpath,
help='Capture screenshots of test failures')
parser.add_argument(
'--shared-prefs-file',
dest='shared_prefs_file', type=_RealPath,
help='The relative path to a file containing JSON list of shared '
'preference files to edit and how to do so. Example list: '
'[{'
' "package": "com.package.example",'
' "filename": "ExampleSettings.xml",'
' "set": {'
' "boolean_key_in_xml": true,'
' "string_key_in_xml": "string_value"'
' },'
' "remove": ['
' "key_in_xml_to_remove"'
' ]'
'}]')
parser.add_argument(
'--store-tombstones',
action='store_true', dest='store_tombstones',
help='Add tombstones in results if crash.')
parser.add_argument(
'--strict-mode',
dest='strict_mode', default='testing',
help='StrictMode command-line flag set on the device, '
'death/testing to kill the process, off to stop '
'checking, flash to flash only. (default: %(default)s)')
parser.add_argument(
'--test-apk',
required=True,
help='Path or name of the apk containing the tests.')
parser.add_argument(
'--test-jar',
help='Path of jar containing test java files.')
parser.add_argument(
'--timeout-scale',
type=float,
help='Factor by which timeouts should be scaled.')
parser.add_argument(
'--ui-screenshot-directory',
dest='ui_screenshot_dir', type=os.path.realpath,
help='Destination for screenshots captured by the tests')
# These arguments are suppressed from the help text because they should
# only ever be specified by an intermediate script.
parser.add_argument(
'--apk-under-test-incremental-install-json',
help=argparse.SUPPRESS)
parser.add_argument(
'--test-apk-incremental-install-json',
type=os.path.realpath,
help=argparse.SUPPRESS)
def AddJUnitTestOptions(parser):
"""Adds junit test options to |parser|."""
parser = parser.add_argument_group('junit arguments')
parser.add_argument(
'--coverage-dir',
dest='coverage_dir', type=os.path.realpath,
help='Directory to store coverage info.')
parser.add_argument(
'--package-filter',
dest='package_filter',
help='Filters tests by package.')
parser.add_argument(
'--runner-filter',
dest='runner_filter',
help='Filters tests by runner class. Must be fully qualified.')
parser.add_argument(
'-f', '--test-filter',
dest='test_filter',
help='Filters tests googletest-style.')
parser.add_argument(
'-s', '--test-suite',
dest='test_suite', required=True,
help='JUnit test suite to run.')
# These arguments are for Android Robolectric tests.
parser.add_argument(
'--android-manifest-path',
help='Path to Android Manifest to configure Robolectric.')
parser.add_argument(
'--package-name',
help='Default app package name for Robolectric tests.')
parser.add_argument(
'--resource-zip',
action='append', dest='resource_zips', default=[],
help='Path to resource zips to configure Robolectric.')
parser.add_argument(
'--robolectric-runtime-deps-dir',
help='Path to runtime deps for Robolectric.')
def AddLinkerTestOptions(parser):
parser.add_argument_group('linker arguments')
parser.add_argument(
'-f', '--gtest-filter',
dest='test_filter',
help='googletest-style filter string.')
parser.add_argument(
'--test-apk',
type=os.path.realpath,
help='Path to the linker test APK.')
def AddMonkeyTestOptions(parser):
"""Adds monkey test options to |parser|."""
parser = parser.add_argument_group('monkey arguments')
parser.add_argument(
'--browser',
required=True, choices=constants.PACKAGE_INFO.keys(),
metavar='BROWSER', help='Browser under test.')
parser.add_argument(
'--category',
nargs='*', dest='categories', default=[],
help='A list of allowed categories. Monkey will only visit activities '
'that are listed with one of the specified categories.')
parser.add_argument(
'--event-count',
default=10000, type=int,
help='Number of events to generate (default: %(default)s).')
parser.add_argument(
'--seed',
type=int,
help='Seed value for pseudo-random generator. Same seed value generates '
'the same sequence of events. Seed is randomized by default.')
parser.add_argument(
'--throttle',
default=100, type=int,
help='Delay between events (ms) (default: %(default)s). ')
def AddPerfTestOptions(parser):
"""Adds perf test options to |parser|."""
parser = parser.add_argument_group('perf arguments')
class SingleStepAction(argparse.Action):
def __call__(self, parser, namespace, values, option_string=None):
if values and not namespace.single_step:
parser.error('single step command provided, '
'but --single-step not specified.')
elif namespace.single_step and not values:
parser.error('--single-step specified, '
'but no single step command provided.')
setattr(namespace, self.dest, values)
step_group = parser.add_mutually_exclusive_group(required=True)
# TODO(jbudorick): Revise --single-step to use argparse.REMAINDER.
# This requires removing "--" from client calls.
step_group.add_argument(
'--print-step',
help='The name of a previously executed perf step to print.')
step_group.add_argument(
'--single-step',
action='store_true',
help='Execute the given command with retries, but only print the result '
'for the "most successful" round.')
step_group.add_argument(
'--steps',
help='JSON file containing the list of commands to run.')
parser.add_argument(
'--collect-chartjson-data',
action='store_true',
help='Cache the telemetry chartjson output from each step for later use.')
parser.add_argument(
'--collect-json-data',
action='store_true',
help='Cache the telemetry JSON output from each step for later use.')
parser.add_argument(
'--dry-run',
action='store_true',
help='Just print the steps without executing.')
parser.add_argument(
'--flaky-steps',
type=os.path.realpath,
help='A JSON file containing steps that are flaky '
'and will have its exit code ignored.')
# TODO(rnephew): Remove this when everything moves to new option in platform
# mode.
parser.add_argument(
'--get-output-dir-archive',
metavar='FILENAME', type=os.path.realpath,
help='Write the cached output directory archived by a step into the'
' given ZIP file.')
parser.add_argument(
'--known-devices-file',
help='Path to known device list.')
# Uses 0.1 degrees C because that's what Android does.
parser.add_argument(
'--max-battery-temp',
type=int,
help='Only start tests when the battery is at or below the given '
'temperature (0.1 C)')
parser.add_argument(
'--min-battery-level',
type=int,
help='Only starts tests when the battery is charged above '
'given level.')
parser.add_argument(
'--no-timeout',
action='store_true',
help='Do not impose a timeout. Each perf step is responsible for '
'implementing the timeout logic.')
parser.add_argument(
'--output-chartjson-data',
type=os.path.realpath,
help='Writes telemetry chartjson formatted output into the given file.')
parser.add_argument(
'--output-dir-archive-path',
metavar='FILENAME', type=os.path.realpath,
help='Write the cached output directory archived by a step into the'
' given ZIP file.')
parser.add_argument(
'--output-json-data',
type=os.path.realpath,
help='Writes telemetry JSON formatted output into the given file.')
parser.add_argument(
'--output-json-list',
type=os.path.realpath,
help='Writes a JSON list of information for each --steps into the given '
'file. Information includes runtime and device affinity for each '
'--steps.')
parser.add_argument(
'-f', '--test-filter',
help='Test filter (will match against the names listed in --steps).')
parser.add_argument(
'--write-buildbot-json',
action='store_true',
help='Whether to output buildbot json.')
parser.add_argument(
'single_step_command',
nargs='*', action=SingleStepAction,
help='If --single-step is specified, the command to run.')
def AddPythonTestOptions(parser):
parser = parser.add_argument_group('python arguments')
parser.add_argument(
'-s', '--suite',
dest='suite_name', metavar='SUITE_NAME',
choices=constants.PYTHON_UNIT_TEST_SUITES.keys(),
help='Name of the test suite to run.')
def _RunPythonTests(args):
"""Subcommand of RunTestsCommand which runs python unit tests."""
suite_vars = constants.PYTHON_UNIT_TEST_SUITES[args.suite_name]
suite_path = suite_vars['path']
suite_test_modules = suite_vars['test_modules']
sys.path = [suite_path] + sys.path
try:
suite = unittest.TestSuite()
suite.addTests(unittest.defaultTestLoader.loadTestsFromName(m)
for m in suite_test_modules)
runner = unittest.TextTestRunner(verbosity=1+args.verbose_count)
return 0 if runner.run(suite).wasSuccessful() else 1
finally:
sys.path = sys.path[1:]
_DEFAULT_PLATFORM_MODE_TESTS = ['gtest', 'instrumentation', 'junit',
'linker', 'monkey', 'perf']
def RunTestsCommand(args):
"""Checks test type and dispatches to the appropriate function.
Args:
args: argparse.Namespace object.
Returns:
Integer indicated exit code.
Raises:
Exception: Unknown command name passed in, or an exception from an
individual test runner.
"""
command = args.command
ProcessCommonOptions(args)
logging.info('command: %s', ' '.join(sys.argv))
if args.enable_platform_mode or command in _DEFAULT_PLATFORM_MODE_TESTS:
return RunTestsInPlatformMode(args)
if command == 'python':
return _RunPythonTests(args)
else:
raise Exception('Unknown test type.')
_SUPPORTED_IN_PLATFORM_MODE = [
# TODO(jbudorick): Add support for more test types.
'gtest',
'instrumentation',
'junit',
'linker',
'monkey',
'perf',
]
def RunTestsInPlatformMode(args):
def infra_error(message):
logging.fatal(message)
sys.exit(constants.INFRA_EXIT_CODE)
if args.command not in _SUPPORTED_IN_PLATFORM_MODE:
infra_error('%s is not yet supported in platform mode' % args.command)
### Set up sigterm handler.
def unexpected_sigterm(_signum, _frame):
msg = [
'Received SIGTERM. Shutting down.',
]
for live_thread in threading.enumerate():
# pylint: disable=protected-access
thread_stack = ''.join(traceback.format_stack(
sys._current_frames()[live_thread.ident]))
msg.extend([
'Thread "%s" (ident: %s) is currently running:' % (
live_thread.name, live_thread.ident),
thread_stack])
infra_error('\n'.join(msg))
signal.signal(signal.SIGTERM, unexpected_sigterm)
### Set up results handling.
# TODO(jbudorick): Rewrite results handling.
# all_raw_results is a list of lists of
# base_test_result.TestRunResults objects. Each instance of
# TestRunResults contains all test results produced by a single try,
# while each list of TestRunResults contains all tries in a single
# iteration.
all_raw_results = []
# all_iteration_results is a list of base_test_result.TestRunResults
# objects. Each instance of TestRunResults contains the last test
# result for each test run in that iteration.
all_iteration_results = []
global_results_tags = set()
@contextlib.contextmanager
def write_json_file():
try:
yield
except Exception:
global_results_tags.add('UNRELIABLE_RESULTS')
raise
finally:
json_results.GenerateJsonResultsFile(
all_raw_results, args.json_results_file,
global_tags=list(global_results_tags))
json_writer = contextlib_ext.Optional(
write_json_file(),
args.json_results_file)
@contextlib.contextmanager
def upload_logcats_file():
try:
yield
finally:
if not args.logcat_output_file:
logging.critical('Cannot upload logcat file: no file specified.')
elif not os.path.exists(args.logcat_output_file):
logging.critical("Cannot upload logcat file: file doesn't exist.")
else:
with open(args.logcat_output_file) as src:
dst = logdog_helper.open_text('unified_logcats')
if dst:
shutil.copyfileobj(src, dst)
dst.close()
logging.critical(
'Logcat: %s', logdog_helper.get_viewer_url('unified_logcats'))
logcats_uploader = contextlib_ext.Optional(
upload_logcats_file(),
'upload_logcats_file' in args and args.upload_logcats_file)
### Set up test objects.
env = environment_factory.CreateEnvironment(args, infra_error)
test_instance = test_instance_factory.CreateTestInstance(args, infra_error)
test_run = test_run_factory.CreateTestRun(
args, env, test_instance, infra_error)
### Run.
with json_writer, logcats_uploader, env, test_instance, test_run:
repetitions = (xrange(args.repeat + 1) if args.repeat >= 0
else itertools.count())
result_counts = collections.defaultdict(
lambda: collections.defaultdict(int))
iteration_count = 0
for _ in repetitions:
raw_results = test_run.RunTests()
if not raw_results:
continue
all_raw_results.append(raw_results)
iteration_results = base_test_result.TestRunResults()
for r in reversed(raw_results):
iteration_results.AddTestRunResults(r)
all_iteration_results.append(iteration_results)
iteration_count += 1
for r in iteration_results.GetAll():
result_counts[r.GetName()][r.GetType()] += 1
report_results.LogFull(
results=iteration_results,
test_type=test_instance.TestType(),
test_package=test_run.TestPackage(),
annotation=getattr(args, 'annotations', None),
flakiness_server=getattr(args, 'flakiness_dashboard_server',
None))
if args.break_on_failure and not iteration_results.DidRunPass():
break
if iteration_count > 1:
# display summary results
# only display results for a test if at least one test did not pass
all_pass = 0
tot_tests = 0
for test_name in result_counts:
tot_tests += 1
if any(result_counts[test_name][x] for x in (
base_test_result.ResultType.FAIL,
base_test_result.ResultType.CRASH,
base_test_result.ResultType.TIMEOUT,
base_test_result.ResultType.UNKNOWN)):
logging.critical(
'%s: %s',
test_name,
', '.join('%s %s' % (str(result_counts[test_name][i]), i)
for i in base_test_result.ResultType.GetTypes()))
else:
all_pass += 1
logging.critical('%s of %s tests passed in all %s runs',
str(all_pass),
str(tot_tests),
str(iteration_count))
if args.command == 'perf' and (args.steps or args.single_step):
return 0
return (0 if all(r.DidRunPass() for r in all_iteration_results)
else constants.ERROR_EXIT_CODE)
def DumpThreadStacks(_signal, _frame):
for thread in threading.enumerate():
reraiser_thread.LogThreadStack(thread)
def main():
signal.signal(signal.SIGUSR1, DumpThreadStacks)
parser = argparse.ArgumentParser()
command_parsers = parser.add_subparsers(
title='test types', dest='command')
subp = command_parsers.add_parser(
'gtest',
help='googletest-based C++ tests')
AddCommonOptions(subp)
AddDeviceOptions(subp)
AddGTestOptions(subp)
AddTracingOptions(subp)
AddCommandLineOptions(subp)
subp = command_parsers.add_parser(
'instrumentation',
help='InstrumentationTestCase-based Java tests')
AddCommonOptions(subp)
AddDeviceOptions(subp)
AddInstrumentationTestOptions(subp)
AddTracingOptions(subp)
AddCommandLineOptions(subp)
subp = command_parsers.add_parser(
'junit',
help='JUnit4-based Java tests')
AddCommonOptions(subp)
AddJUnitTestOptions(subp)
subp = command_parsers.add_parser(
'linker',
help='linker tests')
AddCommonOptions(subp)
AddDeviceOptions(subp)
AddLinkerTestOptions(subp)
subp = command_parsers.add_parser(
'monkey',
help="tests based on Android's monkey command")
AddCommonOptions(subp)
AddDeviceOptions(subp)
AddMonkeyTestOptions(subp)
subp = command_parsers.add_parser(
'perf',
help='performance tests')
AddCommonOptions(subp)
AddDeviceOptions(subp)
AddPerfTestOptions(subp)
AddTracingOptions(subp)
subp = command_parsers.add_parser(
'python',
help='python tests based on unittest.TestCase')
AddCommonOptions(subp)
AddPythonTestOptions(subp)
args, unknown_args = parser.parse_known_args()
if unknown_args:
if hasattr(args, 'allow_unknown') and args.allow_unknown:
args.command_line_flags = unknown_args
else:
parser.error('unrecognized arguments: %s' % ' '.join(unknown_args))
# --replace-system-package has the potential to cause issues if
# --enable-concurrent-adb is set, so disallow that combination
if (hasattr(args, 'replace_system_package') and
hasattr(args, 'enable_concurrent_adb') and args.replace_system_package and
args.enable_concurrent_adb):
parser.error('--replace-system-package and --enable-concurrent-adb cannot '
'be used together')
try:
return RunTestsCommand(args)
except base_error.BaseError as e:
logging.exception('Error occurred.')
if e.is_infra_error:
return constants.INFRA_EXIT_CODE
return constants.ERROR_EXIT_CODE
except: # pylint: disable=W0702
logging.exception('Unrecognized error occurred.')
return constants.ERROR_EXIT_CODE
if __name__ == '__main__':
sys.exit(main())
|
code
|
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