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Newspaper: Duke cutting off service more often this year
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From Wire Reports
Published: May 16, 2008
CHARLOTTE, N.C. (AP) — A review of state records shows Duke Energy is shutting off service to more customers this year because they can’t pay their bills.
The Charlotte Observer reviewed records on Duke, the largest utility in the Carolinas..
Duke’s Andy Thompson says the utility tries to work with customers to set up payments to allow customers to catch up. State law allows utilities to shut off service after three months of missed payments.
Duke says up to 85 percent of customers get service restored the next day, after paying a reconnection fee.
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TITLE: Confused as to why the space of all polynomials of degree at most m is a subspace of the space of polynomials
QUESTION [3 upvotes]: I am using "Linear Algebra Done Right" as a self study guide, and was confused by the following question in the text:
For $m$ a nonnegative integer, let $P_m(F)$ denote the set of all polynomials with coefficients in $F$ and degree at most $m$. You should verify that $P_m(F)$ is a subspace of $P(F)$ [where $P(F)$ is the space of all polynomials with coefficients in $F$]
The book's defines degree as:
A polynomial $p \in P(F)$ is said to have degree $m$ if there exists scalars $a_0, a_1,...a_m$ with $a_m \neq 0$ such that $p(z)=a_0+a_1z+...+a_mz^m$
For $P_m$ to be a subset of $P(F)$, $P_m$ must contain the additive inverse of $P(F)$, which is the polynomial with all 0 coefficients (according to the book). If my understanding of all this is correct, then isn't $P_0$, and subsequently any $P_m$, not a subspace of $P(F)$ because the definition of polynomials of degree $m$ requires at least one non-zero coefficient?
REPLY [2 votes]: Strictly speaking, you're correct that the zero polynomial does not have a degree. But it is conventional to think of it as having degree "$-\infty$". With this convention, then, it is in $P_m$ for each $m$.
REPLY [1 votes]: The zero vector indeed should be the polynomial with all zero coefficients. It is common to introduce the convention that this polynomial has degree $-\infty $, and thus it is included in every $P_n$.
Following the definition strictly, everything is ok. Ask yourself: is the zero polynomial excluded from $P_n$? To answer affirmatively, you will have to show that the zero polynomial has degree larger than $m$. But that is not the case!
REPLY [1 votes]: You are right: the definition of degree you quoted does not allow for $0$ to be considered a degree-zero polynomial. (In fact many authors adopt the convention that the degree of $0$ is actually $- \infty$.)
But the question is, what is the book's definition of "polynomial"? You quoted the definition of degree, but that definition takes for granted that we already know what polynomials are, and are defining what the degree of a polynomial is. Somewhere prior to that there ought to be a definition of "polynomial", and the definition ought to be such that every $a \in F$, including $a=0$, is a polynomial.
REPLY [0 votes]: The additive inverse of a polynomial is not the $0$ polynomial, but the negative of the original polynomial. The additive identity is the $0$ polynomial.
As $-p(x) + p(x) = 0$, and both $\pm p(x)$ are polynomials of the same degree, we see that $P_m$ does contain additive inverses.
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\begin{document}
\baselineskip15pt
\title[Banach algebras of generalized matrices]
{A class of Banach algebras of generalized matrices}
\author[M.M. Sadr]{Maysam Maysami Sadr}
\address{Depertment of Mathematics\\
Institute for Advanced Studies in Basic Sciences (IASBS)\\
P.O. Box 45195-1159, Zanjan 45137-66731, Iran}
\email{sadr@iasbs.ac.ir}
\subjclass[2010]{46H05; 46H35; 46H10; 47B48; 46H25.}
\keywords{Banach algebra; generalized matrix; approximate unit; ideal; derivation}
\begin{abstract}
We introduce a class of Banach algebras of generalized matrices and study the existence of approximate units, ideal structure,
and derivations of them.
\end{abstract}
\maketitle
\section{Introduction}
Let $\f{X}$ be a compact metrizable space and $\f{m}$ be a Borel probability measure on $\f{X}$. In this note we study some aspects of the
algebraic structure of a Banach algebra $\c{M}$ of generalized complex matrices whose their arrays are indexed by elements of $\f{X}^2$ and
vary continuously. The multiplication of $\c{M}$ is defined similar to the ordinary matrix multiplication and uses $\f{m}$
as the weight for arrays. See Section 2 for exact definition. In the case that $\f{m}$ has full support, $\c{M}$ is isometric isomorphic to
a subalgebra of compact operators acting on the Banach space of continuous functions on $\f{X}$. Indeed any element of $\c{M}$
defines an integral operator in a canonical way. Thus $\c{M}$ can be interpreted as a Banach algebra of integral operators or kernels (\cite{Beaver1}).
In Section 3 we investigate the existence of approximate units of $\c{M}$. In Section 4 we show that if $\f{X}$ is infinite then the center of $\c{M}$
is zero. In Section 5 we study ideal structure of $\c{M}$. In Section 6 we consider some classes of representations of $\c{M}$.
In Section 7 we show that under some mild conditions bounded derivations on $\c{M}$ are approximately inner.
\textbf{Notations.}
For a compact space $\f{X}$ and a Banach space $E$ we denote by $\b{C}(\f{X};E)$ the Banach space of continuous $E$-valued functions on $\f{X}$
with supremum norm. We also let $\b{C}(\f{X}):=\b{C}(\f{X};\mathbb{C})$. There is a canonical isometric isomorphism
$\b{C}(\f{X};E)\cong\b{C}(\f{X})\check{\ot} E$ where $\check{\ot}$ denotes the completed injective tensor product.
The phrase ``point-wise convergence topology'' is abbreviated to ``pct''.
By pct on $\b{C}(\f{X};E)$ we mean the vector topology under which a net $(f_\lambda)_\lambda\in\b{C}(\f{X};E)$ converges to $f$ if and only if
$f_\lambda(x)\to f(x)$ in the norm of $E$ for every $x\in\f{X}$. If $f$ and $f'$ are complex functions on spaces $\f{X}$ and $\f{X}'$
then $f\ot f'$ denotes the function on $\f{X}\times\f{X}'$ defined by $(x,x')\mapsto f(x)f'(x')$.
The support of a Borel measure $\f{m}$ is denoted by $\r{Sp}\f{m}$. $\c{B}_{x,\delta}$ denotes the open ball with center at $x$ and radius $\delta$.
\section{The main definitions}
Let $\f{X}$ be a compact metrizable space and $\f{m}$ be a Borel probability measure on $\f{X}$.
By analogy with matrix multiplication we let the convolution of $f,g\in\b{C}(\f{X}^2)$ be defined by
$f\star g(x,y)=\int_Xf(x,z)g(z,y)\r{d}\f{m}(z)$. Also by analogy with matrix adjoint we let $f^*\in\b{C}(\f{X}^2)$ be defined by
$f^*(x,y)=\bar{f}(y,x)$. It is easily verified that $\star$ is an associative multiplication, $*$ is an involution, and also,
$\|f\star g\|_\infty\leq\|f\|_\infty\|g\|_\infty$ and $\|f^*\|_\infty=\|f\|_\infty$; thus $\b{C}(\f{X}^2)$ becomes a Banach $*$-algebra
which we denote by $\c{M}_{\f{X},\f{m}}$. If $\f{X}$ is a finite space with $n$ distinct elements
$x_1,\cdots,x_n$ and $\r{Sp}\f{m}=\f{X}$ then the assignment $(a_{ij})\mapsto((x_i,x_j)\mapsto\frac{1}{\f{m}\{x_i\}}a_{ij})$
defines a $*$-algebra isomorphism from the algebra of $n\times n$ matrices onto $\c{M}_{\f{X},\f{m}}$.
Beside norm and pc topologies on $\c{M}_{\f{X},\f{m}}$ we need two other topologies:
Consider the canonical isometric isomorphism $f\mapsto(y\mapsto f(\cdot,y))$ from $\b{C}(\f{X}^2)$ onto $\b{C}(\f{X};\b{C}(\f{X}))$.
We define the column-wise convergence topology (cct for short)
on $\c{M}_{\f{X},\f{m}}$ to be the pull back of the pct on $\b{C}(\f{X};\b{C}(\f{X}))$ under this isomorphism.
The row-wise convergence topology (rct for short) on $\c{M}_{\f{X},\f{m}}$ is defined similarly
by using the other canonical isomorphism $f\mapsto(x\mapsto f(x,{\cdot}))$.
The pct on $\b{C}(\f{X}^2)=\c{M}_{\f{X},\f{m}}$ is contained in the intersection of cct and rct.
Column-wise and row-wise cts are \emph{adjoint} to each other in the sense that the involution
$*$ from $\c{M}_{\f{X},\f{m}}$ with cct to $\c{M}_{\f{X},\f{m}}$ with rct is a homeomorphism.
If $a_\lambda\xrightarrow{cct}a$ and $b_\lambda\xrightarrow{rct}b$ in $\c{M}_{\f{X},\f{m}}$, then
$c\star a_\lambda\xrightarrow{cct}c\star a$ and $b_\lambda\star c\xrightarrow{rct}b\star c$ for every $c$.
The assignment $(\f{X},\f{m})\mapsto\c{M}_{\f{X},\f{m}}$ can be considered as a cofunctor from the category of pairs $(\f{X},\f{m})$
to the category of Banach $*$-algebras: Suppose that $(\f{X}',\f{m}')$ is another pair of a compact metrizable space
and a Borel probability measure on it. Let $\alpha:\f{X}'\to\f{X}$
be a measure preserving continuous map. Then $\alpha$ induces a bounded $*$-algebra morphism $\c{M}\alpha$ from $\c{M}_{\f{X},\f{m}}$ into
$\c{M}_{\f{X}',\f{m}'}$ defined by $[(\c{M}\alpha)f](x',y')=f(\alpha(x'),\alpha(y'))$. By an explicit example we show that
$\c{M}$ as a functor is not full: Let $\beta:\f{X}\to\mathbb{C}$ be a continuous function with $|\beta|=1_\f{X}$ and $\beta\neq 1_\f{X}$.
Then $\hat{\beta}:\c{M}_{\f{X},\f{m}}\to\c{M}_{\f{X},\f{m}}$ defined by $(\hat{\beta}f)(x,y)=\beta(x)f(x,y)\bar{\beta}(y)$
is an isometric $*$-algebra isomorphism. It is clear that $\hat{\beta}$ is not of the form $\c{M}\alpha$ for any $\alpha:\f{X}\to\f{X}$.
Let $\f{X}_0$ be a closed subset of $\f{X}$ containing $\r{Sp}\f{m}$ and let $\iota:\f{X}_0\to\f{X}$ denote the embedding.
Then $\c{M}\iota:\c{M}_{\f{X},\f{m}}\to\c{M}_{\f{X}_0,\f{m}}$ is surjective with kernel $I:=\{f:f|_{\f{X}^2_0}=0\}$.
Thus $\c{M}_{\f{X},\f{m}}$ is an extension of $\c{M}_{\f{X}_0,\f{m}}$ by the closed self-adjoint ideal $I$. Moreover, suppose that
$\f{X}_0$ is a retract of $\f{X}$ i.e. there is a continuous map $\rho:\f{X}\to\f{X}_0$ with $\rho\iota=\r{id}_{\f{X}_0}$.
It follows from functoriality of $\c{M}$ that $(\c{M}\iota)(\c{M}\rho)$ is the identity morphism on $\c{M}_{\f{X}_0,\f{m}}$.
This shows that the mentioned extension splits strongly in the sense of \cite[Definition 1.2]{BadeDalesLykova1}.
The discussion we just had, shows that by removing the null part of $\f{m}$ from $\f{X}$
we do not lose the principal part of the structure of $\c{M}_{\f{X},\f{m}}$. We will see
that $\r{Sp}\f{m}=\f{X}$ is a crucial condition for the study of $\c{M}_{\f{X},\f{m}}$.
For any closed subset $C$ of $\f{X}$ we have $\f{m}(C)=\inf [(1_\f{X}\ot f)\star 1_{\f{X}^2}](x,y)$,
the infimum being taken over all continuous functions $f$
on $\f{X}$ with $f(\f{X})\subseteq[0,1]$ and $f(C)=\{1\}$. Using this and inner regularity of $\f{m}$ we can find the measure
of any Borel subset. Hence we can recover $\f{m}$ from $\c{M}_{\f{X},\f{m}}$.
The author does not know if the homeomorphism type of $\f{X}$ can be recovered from $\c{M}_{\f{X},\f{m}}$.
Suppose that $\f{X}$ is finite with $\r{Sp}\f{m}=\f{X}$. It is not so hard to see that if
$\phi:\c{M}_{\f{X},\f{m}}\to\c{M}_{\f{X}',\f{m}'}$ is an isometric $*$-isomorphism then
there exist a measure preserving injective and surjective map $\alpha:\f{X}'\to\f{X}$ and a function
$\beta:\f{X}\to\mathbb{C}$, with $|\beta|=1_\f{X}$, such that
$\phi=(\c{M}\alpha)\hat{\beta}$, where $\hat{\beta}$ is defined as above.
(Note that if $\phi$ is not supposed to be isometric then this assertion is wrong.)
We suggest that this conclusion is true for any arbitrary $\f{X}$ with $\r{Sp}\f{m}=\f{X}$.
In Koopman's theory, as it is well known, the operator algebras have many applications to
study of dynamical systems and ergodic theory (\cite{EisnerFarkasHaaseNagel1}).
In this direction, the study of algebraic properties of $\c{M}_{\f{X},\f{m}}$ may be useful: Let $G$ be a discrete group of measure preserving
homeomorphisms of $\f{X}$. Then $G$ acts on $\c{M}_{\f{X},\f{m}}$ by isometric automorphisms and thus it is appropriate
to consider the crossed product Banach algebra $A:=G\ltimes\c{M}_{\f{X},\f{m}}$. It is clear that any algebraic invariant of $A$ is an invariant
of the dynamical system $(\f{X},G)$. Moreover, if the suggestion stated in the preceding paragraph is true, then $(\f{X},G)$ is
completely characterized by $A$. We plan to discuss elsewhere such possible connections with ergodic theory.
\section{Approximate units of $\c{M}$}
From now on, $\f{X}$ is a fixed compact metrizable space, $\f{m}$ is a fixed Borel probability measure on $\f{X}$
with $\r{Sp}\f{m}=\f{X}$, and $\c{M}$ will denote $\c{M}_{\f{X},\f{m}}$. We also let $\f{d}$ denote a compatible metric on $\f{X}$.
A right norm- (resp. pc-, cc-, rc-) approximate unit for $\c{M}$
is a net $(u_\lambda)_\lambda$ in $\c{M}$ such that $au_\lambda\to a$
in the norm topology (resp. pct, cct, rct) for every $a\in A$. If $\sup_{\lambda}\|u_\lambda\|_\infty<\infty$ then $(u_\lambda)_\lambda$ is called bounded. (Bounded) left and two-sided norm- (resp. pc-, cc-, rc-) approximate units are defined similarly.
It is clear that every norm-approximate unit is a pc-approximate unit.
Suppose that $x\in\f{X}$ and $\delta>0$. Throughout, $\c{O}_{x,\delta}$ denotes an open set
with $\overline{\c{B}}_{x,\delta}\subseteq\c{O}_{x,\delta}\subseteq\c{B}_{x,2\delta}$ and
$\f{m}(\c{O}_{x,\delta}\setminus\c{B}_{x,\delta})<\delta\f{m}(\c{B}_{x;\delta})$;
also $\c{E}_{x,\delta}$ denotes a continuous function from $\f{X}$ to the interval $[0,1]$ such that the restriction of
$\c{E}_{x,\delta}$ to $\overline{\c{B}}_{x,\delta}$ (resp. $\f{X}\setminus\c{O}_{x,\delta}$) takes the constant value $1$ (resp. $0$).
\begin{theorem}\label{TT1}
There is a net in $\c{M}$ which is mutually a right cc-approximate unit and a left rc-approximate unit.
Thus the same net is also a two-sided pc-approximate unit.
\end{theorem}
\begin{proof}
The set of all pairs $(S,\epsilon)$, in which $S$ is a finite subset of $\f{X}$ and $\epsilon>0$, with the
ordering $((S,\epsilon)\leq(S',\epsilon'))\Leftrightarrow(S\subseteq S',\epsilon'\leq\epsilon)$, becomes a directed set.
For any pair $(S,\epsilon)$ choose $\delta>0$ such that $\delta<\epsilon$ and
$\c{B}_{y,2\delta}\cap\c{B}_{y',2\delta}=\emptyset$ for $y,y'\in S$ with $y\neq y'$, and let
$u_{S,\epsilon}=\sum_{y\in S}\frac{1}{\f{m}(\c{B}_{y,\delta})}\c{E}_{y,\delta}\ot\c{E}_{y,\delta}$.
We show that $(u_{S,\epsilon})_{(S,\epsilon)}$ is the desired net.
Let $f\in\c{M}$ and $r>0$ be arbitrary. Choose $\epsilon>0$ with $\epsilon<r$ such that for every $z,z',x$ if $\f{d}(z,z')<\epsilon$ then
$|f(x,z)-f(x,z')|<r$. If $x$ is arbitrary then for any pair $(S,\epsilon)$ with $y\in S$ we have
\begin{align*}
|f\star u_{S,\epsilon}-f|(x,y)
&=\frac{1}{\f{m}(\c{B}_{y,\delta})}|\int_{\c{B}_{y,\delta}}[f(x,z)-f(x,y)]\r{d}\f{m}(z)
+\int_{\c{O}_{x,\delta}\setminus\c{B}_{x,\delta}}f(x,z)\c{E}_{y,\delta}(z)\r{d}\f{m}(z)|\\
&\leq r+r\|f\|_\infty.
\end{align*}
This shows that $f\star u_{S,\epsilon}\to f$ in cct. Similarly it is proved that $u_{S,\epsilon}\star f\to f$ in rct.
\end{proof}
\begin{remark}\label{RR1}
The existence of a right (or left) pc-approximate unit for $\c{M}$ implies that $\r{Sp}\f{m}=\f{X}$.
An easy proof is as follows. Let $(u_\lambda)_{\lambda}$ be a right pc-approximate unit. Let $U$ be an arbitrary
nonempty open set in $\f{X}$ and let $f\in\b{C}(\f{X})$ be such that $f(\f{X}\setminus U)=\{0\}$ and $f(x)=1$ for some $x\in U$. Then we have
$1=(1_\f{X}\ot f)(x,x)=\lim_\lambda [(1_\f{X}\ot f)\star u_\lambda](x,x)=\lim_\lambda\int_Uf(z)u_\lambda(z,x)\mathrm{d}\f{m}(z)$.
This implies that $\f{m}(U)\neq0$. Hence $\r{Sp}\f{m}=\f{X}$.
\end{remark}
\begin{proposition}\label{PP1}
If $\c{M}$ has a bounded right (or left) pc-approximate unit then $\f{X}$ is finite.
\end{proposition}
\begin{proof}
Let $(u_\lambda)_{\lambda}$ be a right pc-approximate unit for $\c{M}$ bounded by $M>0$.
First of all we show that $\f{m}(\{x\})\neq0$ for every $x$. Assume, to get a contradiction, that $\f{m}(\{x\})=0$
for some $x$. Let $\epsilon>0$ be such that $\epsilon M<1/2$. There is an open neighborhood $U$ of $x$ with $\f{m}(U)<\epsilon$.
Let $f:\f{X}\to[0,1]$ be a continuous function with $f(x)=1$ and $f(\f{X}\setminus U)=\{0\}$. For every $\lambda$ we have
$|(1_\f{X}\ot f)\star u_\lambda|(x,x)\leq\int_U|f(z)u_\lambda(z,x)|\mathrm{d}\f{m}(z)\leq\epsilon M<1/2$.
But this is impossible because $[(1_\f{X}\ot f)\star u_\lambda](x,x)\to1$.
Now, since $\f{m}(\f{X})=1$, it is concluded that $\f{X}$ must be a countable space. Suppose that $\f{X}$ is not finite.
Then there is an infinite discrete subset $\{x_1,x_2,\cdots\}$ of $\f{X}$. For every $n$ let $f_n\in\c{M}$ be defined by
$f_n(z,z')=1$ if $z=z'=x_n$ and otherwise $f_n(z,z')=0$. Then we have
$1=f_n(x_n,x_n)=\lim_{\lambda}(f_n\star u_\lambda)(x_n,x_n)=\lim_\lambda\f{m}\{x_n\}u_\lambda(x_n,x_n)$.
It follows that $\f{m}\{x_n\}\geq 1/M$. But this contradicts $\lim_{n\to\infty}\f{m}\{x_n\}=0$. Hence, $\f{X}$ is finite.
\end{proof}
\begin{theorem}\label{TT2}
The following statements are equivalent.
\begin{enumerate}
\item[(a)] $\f{X}$ is finite.
\item[(b)] $\c{M}$ has a bounded right (or left) pc-approximate unit.
\item[(c)] $\c{M}$ has a unit.
\end{enumerate}
\end{theorem}
\begin{proof}
(b)$\Rightarrow$(a) is the statement of Proposition \ref{PP1}. (c)$\Rightarrow$(b) is trivial. (a)$\Rightarrow$(c) is easily verified
by analogy with ordinary matrix algebras.
\end{proof}
\begin{lemma}\label{LL1}
Let $x\in X$. The function $r\mapsto\f{m}(\c{B}_{x,r})$ is continuous at $r_0\in[0,\infty)$ if and only if $\f{m}\{y:\f{d}(x,y)=r_0\}=0$.
(Note that $\c{B}_{x,0}=\emptyset$.)
\end{lemma}
\begin{proof}
Straightforward.
\end{proof}
\begin{lemma}\label{LL2}
The function $x\mapsto\f{m}(\c{B}_{x,r})$ is continuous at $x_0$ if $\f{m}\{y:\f{d}(x_0,y)=r\}=0$.
\end{lemma}
\begin{proof}
For $\epsilon>0$ by Lemma \ref{LL1} there is $\delta>0$ such that $\f{m}(\c{B}_{x_0,r+\delta}\setminus\c{B}_{x_0,r-\delta})<\epsilon$.
Suppose that $y\in\c{B}_{x_0,\delta}$. Then
$\f{m}(\c{B}_{x_0,r-\delta})\leq\f{m}(\c{B}_{y,r})\leq\f{m}(\c{B}_{x_0,r+\delta})$. So $|\f{m}(\c{B}_{x_0,r})-\f{m}(\c{B}_{y,r})|<\epsilon$.
\end{proof}
\begin{lemma}\label{LL3}
Let $\delta>0$ be such that $\f{m}\{y:\f{d}(x,y)=\delta\}=0$ for every $x\in\f{X}$.
Then there exists $\delta'$ with $\delta<\delta'<2\delta$
such that $\f{m}(\c{B}_{x,\delta'}\setminus\c{B}_{x,\delta})<\delta\f{m}(\c{B}_{x,\delta})$ for every $x\in\f{X}$.
\end{lemma}
\begin{proof}
Assume, to reach a contradiction, that there is no $\delta'$ with the desired properties.
For sufficiently large $n$ we have $\delta+n^{-1}<2\delta$ and hence there
is a $x_n$ such that $\f{m}(\c{B}_{x_n,\delta+n^{-1}})-\f{m}(\c{B}_{x_n,\delta})\geq\delta\f{m}(\c{B}_{x_n,\delta})$.
Without lost of generality we can suppose that the sequence $(x_n)_n$ converges to an element $x$.
Let $r>0$ be arbitrary. For sufficiently large $n$ we have $\f{m}(\c{B}_{x_n,\delta+n^{-1}})\leq\f{m}(\c{B}_{x,\delta+r})$ and hence
$\delta\f{m}(\c{B}_{x_n,\delta})\leq\f{m}(\c{B}_{x,\delta+r})-\f{m}(\c{B}_{x_n,\delta})$. It follows from Lemma \ref{LL2} that
$\delta\f{m}(\c{B}_{x,\delta})\leq\f{m}(\c{B}_{x,\delta+r}\setminus\c{B}_{x,\delta})$. Letting $r\to0$ and using Lemma \ref{LL1}
we conclude that $\f{m}(\c{B}_{x,\delta})=0$, a contradiction.
\end{proof}
\begin{theorem}\label{TT3}
Suppose that the following condition is satisfied. \emph{(C1) $\f{X}$ has a compatible metric $\f{d}$ under which
there is a decreasing sequence $(\delta_n)_n$ of strictly positive numbers such that
$\inf_n\delta_n=0$ and $\f{m}\{y:\f{d}(x,y)=\delta_n\}=0$ for every $n$ and every $x\in\f{X}$.}
Then $\c{M}$ has a right (resp. left) norm-approximate unit. Moreover, that approximate unit can be chosen so as to be a sequence.
\end{theorem}
\begin{proof}
For every $n$ let $\delta'_n$ be such that the statement of Lemma \ref{LL3} is satisfied with $\delta,\delta'$ replaced by $\delta_n,\delta'_n$.
Let $K_n=\{(x,y):\f{d}(x,y)\leq\delta_n\}$ and $U_n=\{(x,y):\f{d}(x,y)<\delta'_n\}$. Choose a continuous function $E_n:\f{X}^2\to[0,1]$
such that $E_n(K_n)=\{1\}$ and $E_n(\f{X}^2\setminus U_n)=\{0\}$ and let $\c{E}_n$ (resp. $\c{E}'_n$) be defined by
$(x,y)\mapsto E_n(x,y)/\f{m}(\c{B}_{y,\delta_n})$ (resp. $(x,y)\mapsto E_n(x,y)/\f{m}(\c{B}_{x,\delta_n})$).
(Note that by Lemma \ref{LL2}, $\c{E}_n,\c{E}_n'\in\c{M}$.). Using Lemma \ref{LL3}, it is easily verified that
$(\c{E}_n)_n$ (resp. $(\c{E}_n')_n$) is a right (resp. left) norm-approximate unite for $\c{M}$.
\end{proof}
\begin{theorem}\label{TT4}
Suppose that the following condition is satisfied. \emph{(C2) $\f{X}$ has a compatible metric $\f{d}$ under which
there exists a sequence $(\delta_n)_n$ satisfying all properties stated in (C1) and,
in addition, $\f{m}(\c{B}_{x,\delta_n})=\f{m}(\c{B}_{y,\delta_n})$ for every $n$ and every $x,y\in\f{X}$.}
Then $\c{M}$ has a two-sided norm-approximate unit.
\end{theorem}
\begin{proof}
It is concluded from $\c{E}_n=\c{E}_n'$ where $\c{E}_n,\c{E}_n'$ are as in the proof of Theorem \ref{TT3}.
\end{proof}
\begin{example}\label{EE1}
If $\f{X}$ is the closure of a nonempty bounded open subset of $\mathbb{R}^n$ with the normalized $n$-dimensional Lebesgue measure and with
the Euclidean metric, then $\f{X}$ satisfies conditions of Theorem \ref{TT3}. More generally, if an open subset of a Riemannian manifold has compact
closure $\f{X}$ then $\f{X}$, with the geodesic distance $\f{d}$ and normalized Riemannian volume $\f{m}$, satisfies conditions of Theorem \ref{TT3}.
Indeed, $\f{m}\{y:\f{d}(x,y)=r\}=0$ for every $r$ and $x$.
\end{example}
\begin{example}\label{EE2}
Any closed Riemannian manifold $\f{X}$ which has constant (positive) sectional curvature
(e.g. standard spheres and tori, compact Lie groups with invariant Riemannian metrics),
with geodesic distance $\f{d}$ and normalized Riemannian volume $\f{m}$, satisfies conditions of Theorem \ref{TT4}.
Indeed, in addition to the property mentioned in Example \ref{EE1}, we have $\f{m}(\c{B}_{x,r})=\f{m}(\c{B}_{y,r})$ for every $r,x,y$.
\end{example}
\begin{example}\label{EE3}
Let $\f{X}$ be a second countable compact Hausdorff group. It is well-known that $\f{X}$ has a compatible bi-invariant metric $\f{d}$
i.e. $\f{d}(zxz',zyz')=\f{d}(x,y)$ for every $x,y,z,z'\in\f{X}$ (see \cite{Klee1} or \cite[Corollary A4.19]{HofmannMorris1}).
We show that $\f{d}$ with the normalized Haar measure $\f{m}$ satisfies (C1) and hence (because of invariant property of $\f{m}$) satisfies (C2):
Suppose, on the contrary, that there is no sequence $(\delta_n)_n$ satisfying (C1) for $\f{d}$. So there must be $\epsilon>0$ such that
$\f{m}\{y:\f{d}(e,y)=r\}\neq 0$ for every nonzero $r<\epsilon$; thus $\f{m}(\c{B}_{e,\epsilon})=\infty$, a contradiction.
\end{example}
\section{The center of $\c{M}$}
It is clear that if $\f{X}$ is finite then the center of $\c{M}$ is the one-dimensional subalgebra of scaler multiples of the unit of $\c{M}$.
But in the infinite case the situation is different:
\begin{theorem}\label{TT5}
If $\f{X}$ is infinite then the center of $\c{M}$ is zero.
\end{theorem}
\begin{proof}
Suppose that $f$ is in the center of $\c{M}$. Let $x,y$ be arbitrary in $\f{X}$ with $x\neq y$, and $\delta>0$ be such that $\f{d}(x,y)>4\delta$.
Let $g:=\frac{1}{\f{m}(\c{B}_{x,\delta})}\c{E}_{x,\delta}\ot\c{E}_{x,\delta}$ and
$h_\delta:=\frac{1}{\f{m}(\c{B}_{x,\delta})}\c{E}_{x,\delta}\ot\c{E}_{y,\delta}$.
Then we have $f\star g(x,y)=0$ and hence $g\star f(x,y)=0$. We have,
$$|f|(x,y)=|g\star f-f|(x,y)\leq\frac{1}{\f{m}(\c{B}_{x,\delta})}\int_{\c{B}_{x,\delta}}|f(z,y)-f(x,y)|\r{d}\f{m}(z)+\delta\|f\|_\infty.$$
By this inequality and continuity of $f$ we conclude that $f(x,y)=0$. Also, a simple computation shows that
$\lim_{\delta\to0} f\star h_\delta(x,y)=f(x,x)$ and $\lim_{\delta\to0} h_\delta\star f(x,y)=f(y,y)$. Thus we have $f(x,x)=f(y,y)$.
Now, suppose that $\f{X}$ is infinite. Then there is a sequence $(x_n)_{n\geq0}$ such that $x_n\to x_0$ and $x_0\neq x_n$
for every $n\geq1$. Thus $f(x_0,x_0)=\lim_{n\to\infty}f(x_0,x_n)=0$ and hence $f(x,x)=f(x_0,x_0)=0$. This completes the proof.
\end{proof}
\section{The ideal structure of $\c{M}$}
It is clear that the involution $*$ induces a one-to-one correspondence between norm- (resp. rc-, cc-, pc-) closed right ideals and
norm- (resp. cc-, rc-, pc-) closed left ideals of $\c{M}$. Also any self-adjoint right or left ideal is a two-sided ideal.
The rc-closure of any right ideal is a right ideal and the cc-closure of any left ideal is a left ideal.
For any norm-closed linear subspace ${V}$ of $\b{C}(\f{X})$ we let
$\c{R}_{V}:=\{f\in\c{M}:f({\cdot},y)\in V\}$ and $\c{L}_{V}:=\{f\in\c{M}:f(x,{\cdot})\in V\}$.
It is clear that $\c{R}_{V}^*=\c{L}_{\bar{V}}$ and $\c{L}_{V}^*=\c{R}_{\bar{V}}$ where $\bar{V}:=\{\bar{f}:f\in V\}$.
\begin{theorem}\label{TT6}
$\c{R}_{V}$ (resp. $\c{L}_{V}$) is a cc-closed right (resp. rc-closed left) ideal in $\c{M}$. Moreover, if $V$ is pc-closed
then $\c{R}_V$ (resp. $\c{L}_{V}$) is pc-closed.
\end{theorem}
\begin{proof}
It is clear that $\c{R}_{V}$ is a cc-closed linear subspace of $\c{M}$. Let $f\in\c{R}_{V}$ and $g\in\c{M}$. For every
$y$ let $h_y:\f{X}\to{V}$ be defined by $h_y(z)=f({\cdot},z)g(z,y)$. Then the Bochner integral $\int_\f{X}h_y\r{d}\f{m}$ exists and belongs to
$V$ (\cite[Proposition 1.31]{HytonenNeervenVeraarWeis1}). Since $f\star g({\cdot},y)=\int_\f{X}h_y\r{d}\f{m}$, we have $f\star g\in\c{R}_V$.
Thus $\c{R}_V$ is a right ideal. Also, $\c{L}_{V}=\c{R}_{\bar{{V}}}^*$ is a rc-closed left ideal. The second part of the theorem
is trivial.
\end{proof}
\begin{theorem}\label{TT7}
Let ${R}$ be a norm-closed right ideal of $\c{M}$ and let ${V}=\{f({\cdot},y):f\in{R}, y\in\f{X}\}$.
Then ${V}$ is a norm-closed linear subspace of $\b{C}(\f{X})$ and the cc-closure of $R$ is equal to $\c{R}_{V}$.
Moreover, if $R$ is pc-closed then $V$ is pc-closed.
\end{theorem}
\begin{proof}
Suppose that $f\in{R}$ and $y\in\f{X}$. Let $\epsilon>0$ be arbitrary and $\delta>0$ with $\delta<\epsilon$ be such that if
$\f{d}(z,z')<\delta$ then $|f(x,z)-f(x,z')|<\epsilon$ for every $x$. Then for every $x,y'$ we have
$|\frac{1}{\f{m}(\c{B}_{y,\delta})}[f\star(\c{E}_{y,\delta}\ot1)](x,y')-f(x,y)|\leq \epsilon+\epsilon\|f\|_\infty$.
This implies that there exists $F_{f,y}\in{R}$ with $F_{f,y}(x,z)=f(x,y)$ for every $x,z$.
Let $h,h'\in{V}$. Let $f,f'\in{R}$ and $y,y'\in\f{X}$ be such that $h=f({\cdot},y)$ and $h'=f'({\cdot},y')$. We have
$h+h'=[F_{f,y}+F_{f',y'}]({\cdot},z)$ for any arbitrary $z$ and thus $h+h'\in{V}$. This shows that $V$ is a linear subspace.
Suppose that $g\in\b{C}(\f{X})$ is a limit point of ${V}$.
There are sequences $(f_{n})_n\in{R}$ and $(y_n)_n\in\f{X}$ such that $f_n({\cdot},y_n)\to g$. It is clear that the sequence
$(F_{f_n,y_n})_n\in R$ converges to an element $G$ of ${R}$ with $G({\cdot},z)=g$ for every $z$. This shows that ${V}$ is norm-closed.
(A similar argument shows that if $R$ is pc-closed then $V$ is pc-closed.) To complete the proof, it is enough to show that
if $K\in\c{R}_{V}$ then there exists a net in $R$ converging to $K$ in cct. Let $K\in\c{R}_{V}$ be fixed. For every $y$ there are
$k_y\in R$ and $\alpha(y)\in\f{X}$ such that $K({\cdot},y)=k_y({\cdot},\alpha(y))$. For every $\epsilon>0$ and every finite subset $S$ of $\f{X}$
there exists $\delta>0$ with the following three properties.
\begin{enumerate}
\item[--] $\delta\|k_{y}\|_\infty<\epsilon/2$ for every $y\in S$.
\item[--] $\c{B}_{y,2\delta}\cap\c{B}_{y',2\delta}=\emptyset$ for $y,y'\in S$ with $y\neq y'$.
\item[--] If $\f{d}(z,z')<2\delta$ then $|k_{y}(x,z)-k_{y}(x,z')|<\epsilon/2$ for every $y\in S$.
\end{enumerate}
Let $K_{S,\epsilon}:=\sum_{y\in S}\frac{1}{\f{m}(\c{B}_{\alpha(y),\delta})}h_{y}\star(\c{E}_{\alpha(y),\delta}\ot\c{E}_{y,\delta})\in{R}$.
Then $\|K_{S,\epsilon}({\cdot},y)-G({\cdot},y)\|_\infty<\epsilon$ for every $y\in S$. Considering the set of all pairs
$(S,\epsilon)$ as a directed set in the obvious way, shows that $K_{S,\epsilon}\xrightarrow{cct}K$.
\end{proof}
Passing through the involution and using Theorem \ref{TT7}, we conclude that for any
norm-closed left ideal $L$ of $\c{M}$, ${V}:=\{f(x,{\cdot}):f\in{L}, x\in\f{X}\}$ is a norm-closed linear subspace and rc-closure of $L$
is equal to $\c{L}_{V}$. Moreover, if $L$ is pc-closed then $V$ is pc-closed.
\begin{corollary}\label{CC1}
The mapping ${V}\mapsto\c{R}_{V}$ (resp. ${V}\mapsto\c{L}_{V}$) establishes a 1-1 correspondence
between norm-closed linear subspaces of $\b{C}(\f{X})$ and cc-closed right (resp. rc-closed left) ideals of $\c{M}$, and also
between pc-closed linear subspaces of $\b{C}(\f{X})$ and pc-closed right (resp. left) ideals of $\c{M}$.
In particular, 1-dimensional and norm-closed 1-codimensional subspaces of $\b{C}(\f{X})$ correspond respectively to minimal and maximal
cc-closed right (resp. rc-closed left) ideals of $\c{M}$.
\end{corollary}
\begin{corollary}\label{CC2}
There is no nontrivial ideal in $\c{M}$ mutually closed under both cct and rct.
In particular, there is no nontrivial pc-closed ideal in $\c{M}$.
\end{corollary}
\begin{proof}
Let ${I}$ be a nonzero cc-closed and rc-closed ideal. There are closed linear subspaces
${V},{W}\subseteq\b{C}(\f{X})$ such that ${I}=\c{R}_{V}=\c{L}_{W}$. Since ${V}\neq0$ there are $f_0\in{V}$ and $x_0\in\f{X}$ with $f_0(x_0)=1$.
For every $g\in\b{C}(\f{X})$ we have $f_0\ot g\in\c{R}_{V}$. Thus $g=(f_0\ot g)(x_0,-)\in{W}$ and ${W}=\b{C}(\f{X})$. So, ${I}=\c{M}$.
\end{proof}
\section{Canonical representations of $\c{M}$}
For a Banach algebra $A$ a Banach space $E$ is called Banach left $A$-module if $E$ is a left $A$-module in the algebraic sense and such that
the action of $A$ on $E$ is a bounded bilinear operator. Banach right $A$-modules and Banach $A$-bimodules are defined similarly.
Let $\b{B}(E)$ denote the Banach algebra of bounded linear operators on $E$ and $\b{K}(E)\subseteq \b{B}(E)$ be the closed ideal of compact operators.
Any Banach left $A$-module structure on $E$ gives rise to a bounded representation $A\to\b{B}(E)$, $a\mapsto(\omega\mapsto a\omega)$, and vice versa.
The statements of the following theorem are standard results and can be find for instance in \cite{HalmosSunder1}.
\begin{theorem}\label{TT8}
Let $E$ denote any of the Banach spaces $\b{L}^p(\f{m})$ ($1\leq p\leq\infty$) or $\b{C}(\f{X})$. Then $\rho:\c{M}\to\b{K}(E)$, defined by
$[\rho(f)g](x)=\int_\f{X}f(x,y)g(y)\r{d}\f{m}(y)$ ($g\in E$),
is a well-defined faithful bounded representation. Moreover, the following statements hold.
\begin{enumerate}
\item[(i)] In the case that $E=\b{L}^2(\f{m})$, $\rho$ is a $*$-representation.
\item[(ii)] In the case that $E=\b{L}^\infty(\f{m})$ or $E=\b{C}(\f{X})$, $\rho$ is isometric.
\end{enumerate}
\end{theorem}
It is clear that for any Banach space $E$, $\b{C}(\f{X};E)$ is a Banach right (resp. left) $\c{M}$-module in the canonical way.
Its module action is denoted by the same symbol $\star$ and is given by $(g\star f)(y)=\int_\f{X}g(z)f(z,y)\r{d}\f{m}(z)$
(resp. $(f\star g)(x)=\int_\f{X}f(x,z)g(z)\r{d}\f{m}(z)$) for $f\in\c{M}$ and $g\in\b{C}(\f{X};E)$. Similarly, $\b{C}(\f{X}^2;E)$ becomes a Banach
$\c{M}$-bimodule.
\section{Derivations on $\c{M}$}
Let $A$ be a Banach algebra and $E$ be a Banach $A$-bimodule. A (bounded) \emph{derivation} from $A$ to $E$ is a (bounded) linear map
$D:A\to E$ satisfying $D(ab)=aD(b)+D(a)b$ ($a,b\in A$). $D$ is called \emph{inner} if there exists $\omega\in E$ such that $D(a)=a\omega-\omega a$
for every $a$. $D$ is called \emph{approximately inner} \cite{GhahramaniLoy1} if there is a net $(\omega_\lambda)_\lambda$ in $E$ such that
$D(a)=\lim_\lambda a\omega_\lambda-\omega_\lambda a$. If $(\omega_\lambda)_\lambda$ can be chosen so as to be a sequence
then $D$ is called \emph{sequentially} approximate inner.
\begin{theorem}\label{TT9}
Suppose that the condition (C2) of Theorem \ref{TT4} is satisfied, and let $E$ be a Banach $\c{M}$-bimodule
such that its module operation $\diamond:\c{M}\ot E\ot\c{M}\to E$ is continuous w.r.t. injective tensor norm, and such that for every norm approximate
unit $(\c{E}_n)_n$ of $\c{M}$ we have $\c{E}_n\diamond \omega\to \omega$ for every $\omega\in E$.
Then any bounded derivation from $\c{M}$ to $E$ is sequentially approximate inner.
\end{theorem}
\begin{proof}
Let $D:\c{M}\to E$ be a bounded derivation.
Let $\Gamma:\c{M}\check{\ot}\c{M}\to E$ be the bounded linear map defined by $f\ot g\mapsto f\diamond D(g)$.
Also let $\Lambda:\c{M}\check{\ot}\c{M}\to\c{M}$ denote the convolution product.
It is not hard to verify the following two identities
for $h\in\c{M}$ and $F\in\c{M}\check{\ot}\c{M}$.
$$\Gamma(h\star F)=h\diamond\Gamma(F),\hspace{10mm}\Gamma(F\star h)=\Lambda(F)\diamond D(h)+\Gamma(F)\diamond h.$$
Let the sequence $(\delta_n)_n$ be as in the statement of Theorem \ref{TT4} and let $\alpha_n=\f{m}(\c{B}_{x,\delta_n})$ for every $x\in\f{X}$.
By Lemma \ref{LL3} there is $r_n$ such that $\delta_n<r_n<2\delta_n$ and $\f{m}(\c{B}_{x,r_n}-\c{B}_{x,\delta_n})<\delta_n\alpha_n$.
Choose a continuous function $G_n:\f{X}^2\to[0,1]$ such that $G_n$ has constant values $1$ and $0$ respectively on
$\{(x,y):\f{d}(x,y)\leq\delta_n\}$ and $\{(x,y):\f{d}(x,y)\geq r_n\}$, and let $\c{G}_n\in\b{C}(\f{X}^4)$ be defined by
$\c{G}_n(x,z,z',y)=\frac{1}{\alpha_n}G_\delta(x,y)$. Note that we have $\Lambda(\c{G}_n)=\frac{1}{\alpha_n}G_n$.
It is not hard to verify that $(\Lambda(\c{G}_n))_n$ is a two-sided norm-approximate unit for $\c{M}$ and
$\lim_{n\to\infty}f\star\c{G}_n-\c{G}_n\star f=0$ for every $f\in\c{M}$. Let $K_n=\Gamma(\c{G}_n)\in E$. For the sequence $(K_n)_n$ and $h\in\c{M}$
we have,
\begin{align*}
\lim_{n\to\infty}h\diamond K_n-K_n\diamond h&=\lim_{n\to\infty}h\diamond \Gamma(\c{G}_n)-\Gamma(\c{G}_n)\diamond h\\
&=\lim_{n\to\infty}\Gamma(h\star\c{G}_n)-\Gamma(\c{G}_n\star h)+\Lambda(\c{G}_n)\diamond D(h)\\
&=\Gamma(\lim_{n\to\infty}h\star\c{G}_n-\c{G}_n\star h)+D(h)\\
&=D(h).
\end{align*}
This completes the proof.
\end{proof}
For any Banach space $E$, the Banach $\c{M}$-bimodule $\b{C}(\f{X}^2;E)$,
mentioned in the preceding section, satisfies the conditions of Theorem \ref{TT9}.
\bibliographystyle{amsplain}
| 12,938
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\begin{document}
\title[Difference sets, $ B_h $ sequences, and codes in $ A_n $ lattices]
{Difference sets, $ \boldsymbol{B_h} $ sequences, and codes in $ \boldsymbol{A_n} $ lattices}
\author{Mladen~Kova\v{c}evi\'{c}}
\address{Department of Electrical Engineering,
University of Novi Sad,
Trg Dositeja Obra\-dovi\'{c}a 6, 21000 Novi Sad, Serbia.}
\email{kmladen@uns.ac.rs}
\thanks{This work was supported by the Ministry of Education, Science
and Technological Development of the Republic of Serbia
(grants TR32040 and III44003), and by the EU COST action IC1104.}
\subjclass[2010]{Primary: 05B10, 11B13, 11B75, 05B45, 94B25; Secondary: 52C22, 52C07, 68P30, 68R05.}
\date{\today}
\keywords{$ B_h $ sequence, Sidon set, difference set, prime power conjecture,
$ A_n $ lattice, perfect code, tiling, group splitting, covering code.}
\begin{abstract}
The paper presents a geometric/coding theoretic interpretation of difference
sets and $ B_h $ sequences.
It is shown that an Abelian planar difference set with $ n + 1 $ elements
exists if and only if the $ A_n $ lattice admits a lattice tiling by balls
of radius $ 1 $, i.e., a linear $ 1 $-perfect code.
Similarly, $ B_{2r} $ sequences with $ n + 1 $ elements are brought into
direct correspondence with linear codes of radius $ r $ in the $ A_n $ lattice.
$ B_{2r+1} $ sequences are also represented geometrically in a slightly different
way.
Bounds on the size of $ B_{h} $ sequences that follow from this interpretation
have very simple derivations and in some cases improve upon the known bounds.
In connection to the above, existence of perfect codes of radius $ r $ is
also studied.
Several communication scenarios are described for which the obtained results
are relevant.
\end{abstract}
\maketitle
\section{Preliminaries}
\subsection{Difference sets}
Let $ G $ be a group of order $ v $ (written additively).
A set $ D \subseteq G $ of size $ k $ is said to be a $ (v, k, \lambda) $-
\emph{difference set} if every nonzero element of $ G $ can be expressed as
a difference $ d_i - d_j $ of two elements from $ D $ in exactly $ \lambda $ ways.
The parameters $ v, k, \lambda $ then necessarily satisfy the identity
$ \lambda(v-1) = k(k-1) $.
The \emph{order} of such a difference set is defined as $ k - \lambda $.
If $ G $ is Abelian, cyclic, etc., then $ D $ is also said to be Abelian, cyclic,
etc., respectively.
We shall sometimes omit the word Abelian in the sequel, as this is the only case
that will be treated.
Furthermore, we shall mostly be concerned with \emph{planar} (or \emph{simple})
difference sets -- those with $ \lambda = 1 $.
Difference sets are very well-studied, and a large body of literature is devoted
to the investigation of their properties, see \cite{designs} and references therein.
One of the most familiar problems in the area concerning the existence of these
objects for specific sets of parameters is the so-called \emph{prime power conjecture}
\cite[Conj. 7.5, p. 346]{designs}: Planar difference set of order $ n \equiv k - 1 $
exists if and only if $ n $ is a prime power (counting $ n = 1 $ as a prime power).
Existence of such sets for $ n = p^m $, $ p $ prime, $ m \in \mathbb{Z}_\sml{\geq 0} $,
was demonstrated by Singer \cite{singer}, but the necessity of this condition
remains an open problem for more than seven decades.
Difference sets have also been applied in communications and coding theory
in various settings, see for example \cite{ding, atkinson, lam+sarwate}.
\subsection{$ \boldsymbol{B_h} $ sequences}
Let $ B = \{ b_0, b_1, \ldots, b_{k-1} \} \subseteq G $, where $ G $ is an
additive Abelian group.
Then $ B $ is said to be a \emph{$ B_h $ sequence} (or \emph{$ B_h $ set}, or
\emph{Sidon set} of order $ h $) if all sums $ b_{i_1} + \cdots + b_{i_h} $,
$ 0 \leq i_1 \leq \cdots \leq i_h \leq k-1 $, are distinct.
$ B_2 $ sequences were introduced by Sidon\footnote{Though there were
some earlier appearances of the problem, see \cite{obryant}.} in \cite{sidon}
and are closely related to planar difference sets.
Namely, all the the sums $ b_i + b_j $ are distinct (up to the order of the
summands) if and only if all the differences $ b_i - b_j $ for $ i \neq j $
are distinct;
the additional requirement for difference sets is that every nonzero element
of the group can be expressed as such a difference, which is equivalent to
saying that the order of the group is $ v = k(k-1) + 1 $.
Singer's construction \cite{singer} is thus optimal for $ k - 1 $ a prime power.
This construction was later generalized by Bose and Chowla \cite{bose+chowla}
to $ B_h $ sequences in cyclic groups for arbitrary $ h \geq 1 $.
Since these pioneering papers, research in this area of combinatorial number
theory has been extensive, see \cite{obryant} for references.
It has also found many applications in coding theory, see, e.g.,
\cite{barg, derksen, graham+sloane, varshamov}.
In this note we intend to present a geometric interpretation of difference
sets and $ B_h $ sequences, offering a different view on the subject and
potentially another approach to studying them.
\subsection{$ \boldsymbol{A_n} $ lattice under $ \boldsymbol{\ell_1} $ metric}
The $ A_n $ lattice is defined as
\begin{equation}
A_n = \left\{ (x_0, x_1, \ldots, x_n) : x_i \in \mathbb{Z}, \sum_{i=0}^n x_i = 0 \right\}
\end{equation}
where $ \mathbb{Z} $ denotes the integers, as usual.
$ A_1 $ is equivalent to $ \mathbb{Z} $, $ A_2 $ to the hexagonal lattice,
and $ A_3 $ to the face-centered cubic lattice (see \cite{conway+sloane}).
The metric on $ A_n $ that we understand is essentially the $ \ell_1 $ distance
\begin{equation}
\label{eq:metricd}
d({\bf x}, {\bf y}) = \frac{1}{2} \norm{{\bf x} - {\bf y}}
= \frac{1}{2} \sum_{i = 0}^n | x_i - y_i |,
\end{equation}
where $ {\bf x} = ( x_0, x_1, \ldots, x_n ) $, $ {\bf y} = ( y_0, y_1, \ldots, y_n ) $;
the constant $ 1/2 $ is taken for convenience because $ \norm{{\bf x} - {\bf y}} $
is always even for $ {\bf x}, {\bf y} \in A_n $.
Distance $ d $ also represents the graph distance in $ A_n $.
Namely, if $ \Gamma(A_n) $ is a graph with the vertex set $ A_n $ and with
edges joining neighboring points (i.e., points at distance $ 1 $ under $ d $),
then $ d({\bf x}, {\bf y}) $ is the length of the shortest path between $ {\bf x} $
and $ \bf y $ in $ \Gamma(A_n) $.
Ball of radius $ 1 $ around $ {\bf x} \in A_n $ contains $ 2 {n+1 \choose 2} + 1 = n^2 + n + 1 $
points of the form $ {\bf x} + {\bf f}_{i,j} $, where $ {\bf f}_{i,j} $ is a
permutation of $ (1, -1, 0, \ldots, 0) $ having a $ 1 $ at the $ i $'th coordinate,
a $ -1 $ at the $ j $'th coordinate, and zeros elsewhere (with the convention
$ {\bf f}_{i,i} = {\bf 0} $).
Convex interior of the points in the ball forms a highly symmetrical
polytope having the following interesting property, among many others --
the distance of any vertex from the center is equal to the distance
between any two neighboring vertices.
\begin{figure}[h]
\centering
\includegraphics[width=0.5\columnwidth]{balls_An}
\caption{Balls of radius $ 1 $ in $ (A_2, d) $ -- hexagon, and in $ (A_3, d) $ -- cuboctahedron.}
\label{fig:balls}
\end{figure}
\noindent
Ball of radius $ r $ around $ {\bf x} \in A_n $ contains all the points with
integral coordinates in the convex interior of $ \{ {\bf x} + r {\bf f}_{i,j} \} $.
When studying packing problems, it is usually simpler to visualize $ \mathbb{Z}^n $
instead of an arbitrary lattice.
In our case there is a trivial map that makes the transition to $ \mathbb{Z}^n $
and back very easy.
For $ {\bf x} = (x_1, \ldots, x_n), {\bf y} = (y_1, \ldots, y_n) \in \mathbb{Z}^n $,
define the metric
\begin{equation}
d_a({\bf x}, {\bf y}) =
\max \left\{ \sum_{ i :\, x_i > y_i } (x_i - y_i) , \sum_{ i :\, x_i < y_i } (y_i - x_i) \right \} .
\end{equation}
This distance (restricted to $ \{0, 1, \ldots, q-1\}^n $) is frequently used
in the theory of codes for asymmetric channels \cite[Ch.\ 2.3 and 9.1]{klove}.
\begin{lemma}
\label{th:isometry}
$ (A_n, d) $ is isometric to $ (\mathbb{Z}^n, d_a) $.
\end{lemma}
\begin{proof}
For $ {\bf x} = (x_0, x_1, \ldots, x_n) $, denote $ {\bf x}' = (x_1, \ldots, x_n) $.
The map $ {\bf x} \mapsto {\bf x}' $ is the desired isometry.
Just observe that, for $ {\bf x}, {\bf y} \in A_n $,
\begin{equation}
d({\bf x}, {\bf y}) = \sum_{\substack{ i = 0 \\ x_i > y_i}}^n (x_i - y_i)
= \sum_{\substack{ i = 0 \\ x_i < y_i}}^n (y_i - x_i)
\end{equation}
because $ \sum_{i=0}^n x_i = \sum_{i=0}^n y_i = 0 $, and by considering the cases
$ x_0 \lessgtr y_0 $ it follows that
\begin{equation}
d({\bf x}, {\bf y})
= \max \left\{ \sum_{\substack{ i = 1 \\ x_i > y_i}}^n (x_i - y_i) ,
\sum_{\substack{ i = 1 \\ x_i < y_i}}^n (y_i - x_i) \right\}
= d_a({\bf x}', {\bf y}') .
\end{equation}
Furthermore, the map $ {\bf x} \mapsto {\bf x}' $ is bijective.
\end{proof}
Hence, packing and similar problems in $ (A_n, d) $ are equivalent
\footnote{This fact is mentioned in \cite{etzion} for the case $ n = 2 $, though
the interpretation via the metric $ d_a $ is not given.}
to those in $ (\mathbb{Z}^n, d_a) $.
Balls in $ (\mathbb{Z}^n, d_a) $ are ``distorted'' versions of the ones
in $ (A_n, d) $ (see Fig.\ \ref{fig:Z2tile}).
For example, ball of radius $ r $ around $ \bf 0 $ in $ (\mathbb{Z}^n, d_a) $
contains all points in $ \mathbb{Z}^n $ whose positive coordinates sum to at most
$ r $, and negative to at least $ -r $.
We shall also need the following generalization of the ball in $ (\mathbb{Z}^n, d_a) $,
\begin{equation}
\label{eq:Sn}
S_n(r^\sml{+}, r^\sml{-}) =
\left\{ {\bf x} \in \mathbb{Z}^n :
\sum_{i \, : \, x_i > 0} x_i \leq r^\sml{+} ,
\sum_{i \, : \, x_i < 0} | x_i | \leq r^\sml{-} \right\} ,
\end{equation}
where $ r^\sml{+}, r^\sml{-} \geq 0 $.
Considering the symmetry of the shape $ S_n $ in the parameters $ r^\sml{+}, r^\sml{-} $
(see Fig.\ \ref{fig:S2}), we can assume that $ r^\sml{-} \leq r^\sml{+} $.
For $ r^\sml{+} = r^\sml{-} = r $, $ S_n $ is a ball of radius $ r $ around
$ \bf 0 $ in $ (\mathbb{Z}^n, d_a) $.
\begin{figure}[h]
\centering
\includegraphics[width=0.85\columnwidth]{S2_shapes_hor4}
\caption{Shapes $ S_2(r^\sml{+},r^\sml{-}) $ for $ r^\sml{+} + r^\sml{-} = 3 $.
Black dot denotes the origin.}
\label{fig:S2}
\end{figure}
\begin{lemma}
\label{th:Anball}
The cardinality of the set $ S_n(r^\sml{+}, r^\sml{-}) $ is
\begin{equation}
\label{eq:A3ball}
| S_n(r^\sml{+}, r^\sml{-}) | =
\sum_{ m = 0 }^{ \min\{ n, r^\sml{+} \} } { n \choose m }
{ r^\sml{+} \choose m }
{ r^\sml{-} + n - m \choose n - m } .
\end{equation}
\end{lemma}
\begin{proof}
Observe the vectors in $ S_n $ having $ m $ strictly positive coordinates,
$ m \in \{ 0, \ldots, n \} $.
These coordinates can be chosen in $ n \choose m $ ways.
For each choice, the ``mass'' $ r^\sml{+} $ can be distributed over them in
$ r^\sml{+} \choose m $ ways (think of $ r^\sml{+} $ balls being placed into
$ m $ bins, where at least one ball is required in each bin).
Similarly, the mass $ -r^\sml{-} $ can be distributed over the remaining
coordinates in $ r^\sml{-} + n-m \choose n-m $ ways.
\end{proof}
\section{Difference sets as sublattices of $ A_n $}
\label{sec:diff}
In the following, when using concepts from graph theory in our setting, we have
in mind the graph representation $ \Gamma(A_n) $ of $ A_n $, as introduced above.
An $ (r, i, j) $-cover (or $ (r, i, j) $-covering code) in a graph $ \Gamma = (V, E) $
\cite{axenovich} is a set of its vertices $ S \subseteq V $ with the property that
every element of $ S $ is covered by exactly $ i $ balls of radius $ r $ centered
at elements of $ S $, while every element of $ V \setminus S $ is covered by exactly
$ j $ such balls.
Special cases of such sets, namely $ (1, i, j) $--covers, have also been studied
in the context of domination theory in graphs \cite{telle}.
In coding theory, $ (r, 1, 1) $-covers are known as $ r $-perfect codes.
An independent set in a graph $ \Gamma = (V, E) $ is a subset of its vertices
$ I \subseteq V $, no two of which are adjacent in $ \Gamma $.
\subsection{The general case}
The proof of the following theorem uses the same method that was employed to
establish the connection between lattice tilings and group splitting
\cite{stein67, hamaker, stein74, galovich, hickerson}
(see also \cite{stein+szabo}).
In fact, planar difference sets ($ \lambda = 1 $) are an instance of so-called
splitting sequences.
\begin{theorem}
\label{th:domination}
An Abelian $ (v, k, \lambda) $-difference set exists if and only if the lattice
$ A_{k-1} $ contains a $ (1, 1, \lambda) $-covering sublattice.
\end{theorem}
\begin{proof}
Suppose that $ D = \{ d_0, d_1, \ldots, d_{k-1} \} $ is a $ (v, k, \lambda) $-
difference set in an Abelian group $ G $.
Observe the sublattice
\begin{equation}
\label{eq:code}
{\mathcal L} = \left\{ {\bf x} \in A_{k-1} : \sum_{i=0}^{k-1} x_i d_i = 0 \right\}
\end{equation}
where $ x_id_i $ denotes the sum (in $ G $) of $ |x_i| $ copies of $ d_i $ if
$ x_i > 0 $, and of $ -d_i $ if $ x_i < 0 $.
$ {\mathcal L} $ is a $ (1, 1, \lambda) $-cover in $ A_{k-1} $.
To see this, consider a point $ {\bf y} = (y_0, y_1, \ldots, y_{k-1}) \notin {\mathcal L} $,
meaning that $ \sum_{i=0}^n y_i d_i = a \in G $, $ a \neq 0 $.
The neighbors of $ \bf y $ are of the form $ {\bf y} + {\bf f}_{i,j} $, $ i \neq j $.
Since $ D $ is a difference set, $ -a \in G $ can be written as a difference of
the elements from $ D $ in exactly $ \lambda $ ways, meaning that there are
$ \lambda $ different pairs $ (s,t) $ for which $ d_s - d_t = -a $, $ d_s, d_t \in D $.
For every such pair observe the point $ {\bf z}_{s,t} = {\bf y} + {\bf f}_{s,t} $.
$ {\bf z}_{s,t} \in {\mathcal L} $ because
$ \sum_{i=0}^n z_i d_i = \sum_{i=0}^n y_i d_i + d_s - d_t = a - a = 0 $.
Therefore, there are exactly $ \lambda $ points in the lattice $ \mathcal L $ that
are adjacent to $ \bf y $, i.e., such that balls of radius $ 1 $ around them cover
$ \bf y $.
To show that the elements of $ {\mathcal L} $ are covered only by the balls around
themselves (i.e., that $ \mathcal L $ is an independent set in $ \Gamma(A_n) $),
observe that if there were two points at distance $ 1 $ in $ {\mathcal L} $, then by
the same argument as above we would obtain that $ d_s - d_t = 0 $, i.e., $ d_s = d_t $
for some $ s \neq t $, which is not possible if $ | D | = k $.
\noindent
For the other direction, assume that $ {\mathcal L}' $ is a $ (1, 1, \lambda) $-covering
sublattice of $ A_{k-1} $.
Observe the quotient group $ G = A_{k-1} / {\mathcal L}' $, and take
$ D = \{ d_0, d_1, \ldots, d_{k-1} \} \subseteq G $, where
$ d_i = [{\bf f}_{i,0}] \equiv {\bf f}_{i,0} + {\mathcal L}' $ are cosets
(elements of $ G $).
Let us first assure that all the $ d_i $'s are distinct.
Suppose that $ d_s = d_t $ for some $ s \neq t $.
This implies that $ d_s - d_t = [{\bf f}_{s,t}] = [{\bf 0}] $, which means that
$ {\bf f}_{s,t} \in {\mathcal L}' $.
But since $ {\bf 0} \in {\mathcal L}' $, and $ \bf 0 $ and $ {\bf f}_{s,t} $ are
at distance $ 1 $, this would contradict the fact that $ {\mathcal L}' $ is independent.
Hence, $ | D | = k $.
Now take any nonzero element of $ G $, say $ [{\bf y}] $, $ {\bf y} \notin {\mathcal L}' $.
By assumption, $ \bf y $ is covered by exactly $ \lambda $ elements of $ {\mathcal L}' $,
i.e., $ {\bf y} + {\bf f}_{s,t} \in {\mathcal L}' $ for exactly $ \lambda $ vectors
$ {\bf f}_{s,t} $.
Since $ {\bf f}_{s,t} = {\bf f}_{s,0} - {\bf f}_{t,0} $, this means that
$ d_t - d_s = [{\bf f}_{t,0}] - [{\bf f}_{s,0}] = [{\bf y}] $ for exactly $ \lambda $
pairs $ (s,t) $.
$ D $ is therefore a $ (v,k,\lambda) $-difference set.
\end{proof}
Geometrically, the theorem states that balls of radius $ 1 $ around the points of
the sublattice $ {\mathcal L} $ overlap in such a way that every point that does
not belong to $ {\mathcal L} $ is covered by exactly $ \lambda $ balls.
(The points in $ {\mathcal L} $ -- centers of the balls -- are covered by one
ball only, and hence this notion is different than that of multitiling \cite{multitiling}.)
\begin{example}
$ D = \{ 0, 1, 2 \} $ is a $ (4, 3, 2) $-difference set
in the cyclic group $ \mathbb{Z}_4 $ (integers modulo $ 4 $).
A $ (1, 1, 2) $-covering sublattice $ {\mathcal L} \subset A_2 $ corresponding to
this difference set is illustrated in Fig.\ \ref{fig:domination}.
Points in $ {\mathcal L} $ are depicted as black, and those in $ A_2 \setminus {\mathcal L} $
as white dots.
For illustration, Fig.\ \ref{fig:dominationnotI} shows an example of a $ (1, 3, 2) $-covering
sublattice (which does not correspond to any difference set).
\myqed
\end{example}
\begin{figure}[h]
\centering
\includegraphics[width=0.75\columnwidth]{covering_code_12}
\caption{A $ (1, 1, 2) $-covering sublattice of $ A_2 $.}
\label{fig:domination}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.75\columnwidth]{covering_code_22}
\caption{A $ (1, 3, 2) $-covering sublattice of $ A_2 $.}
\label{fig:dominationnotI}
\end{figure}
\begin{remark}
We can interpret the lattice $ \mathcal L $ as an error-correcting/detecting code,
in which case the points in $ \mathcal L $ are called codewords.
When $ \lambda = 1 $, the code can correct a single error because balls of radius
one around codewords do not overlap and the minimum distance of the code is three
(here by a single error we mean the addition of a vector $ {\bf f}_{i,j} $ for some
$ i, j $, $ i \neq j $, to the ``transmitted'' codeword $ {\bf x} \in {\mathcal L} $).
For $ \lambda > 1 $, however, it can only \emph{detect} a single error reliably.
Note also that increasing $ \lambda $ increases the density of the code/lattice
$ \mathcal L $ in $ A_{k-1} $, but does not affect its error-detection capability.
The densest such lattice is therefore obtained for $ \lambda = k $ (that this is the
maximum value follows from $ \lambda(v-1) = k(k-1) $ and $ k \leq v $);
it corresponds to the trivial $ (v,v,v) $-difference set $ D = G $ in an arbitrary
group $ G $.
\myqed
\end{remark}
Note that we have not specified the order of the elements of $ D $ when
defining the corresponding lattice $ \mathcal L $ in \eqref{eq:code}
because it would affect it in an insignificant way only.
Note also that if we write $ d_i' = zd_i + g $ instead of $ d_i $ in \eqref{eq:code},
where $ z $ is a fixed integer coprime with $ v $ and $ g $ is a fixed element
of $ G $, identical lattice is obtained because
\begin{equation}
\sum_{i=0}^{k-1} x_i d_i = 0 \quad \Leftrightarrow \quad \sum_{i=0}^{k-1} x_i d_i' = 0
\end{equation}
which follows from $ \sum_{i=0}^{k-1} x_i = 0 $ and $ \operatorname{gcd}(z, v) = 1 $.
Let us recall some terminology.
Two difference sets $ D $ and $ D' $ in an Abelian group $ G $ are said to be
equivalent \cite[Rem.\ 1.11, p.\ 302]{designs} if $ D' = \{ zd + g: d \in D \} $,
for some $ z \in \mathbb{Z} $ coprime with $ v $ and some $ g \in G $.
Two codes $ \mathcal C $ and $ {\mathcal C}' $ of length $ m $ over an alphabet
$ \mathbb{A} $ are equivalent \cite[p.\ 40]{macwilliams+sloane} if there exist
$ m $ permutations of $ \mathbb{A} $, $ \pi_1, \ldots, \pi_m $, and a permutation
$ \sigma $ over $ \{ 1, \ldots, m \} $ such that
\begin{equation}
{\mathcal C}' = \big\{ \sigma( \pi_1(x_1), \ldots, \pi_m(x_m) ) : (x_1, \ldots, x_m) \in {\mathcal C} \big\} .
\end{equation}
We then have the following:
\begin{proposition}
If two difference sets $ D $ and $ D' $ are equivalent, then the corresponding
codes \emph{(}defined as in \eqref{eq:code}\emph{)} are equivalent.
\hfill \qed
\end{proposition}
\noindent
In fact, the $ \pi_i $'s are necessarily identity maps, only $ \sigma $ is
relevant here.
\subsection{Planar difference sets as perfect codes}
\label{sec:planar}
An $ r $-perfect code in a discrete metric space $ (U, d) $ is a subset
$ {\mathcal C} \subseteq U $ with the property that balls of radius $ r $
around the ``codewords'' from $ {\mathcal C} $ are disjoint and cover the
entire space $ U $.
In the above terminology $ r $-perfect codes are $ (r, 1, 1) $-covers.
Perfect codes are also studied in graph theory as one of the several variants
of dominating sets \cite{bange, domination} (see also \cite{martinez}).
When $ U $ is a vector space and $ {\mathcal C} $ its subspace, $ {\mathcal C} $
is said to be a linear code.
The same terminology is often used for lattices, namely, $ {\mathcal C} $
is called a linear code if it is a sublattice of the lattice in question.
Theorem \ref{th:domination} implies the following claim (recall that the order
of a planar difference set of size $ k $ is $ n = k - 1 $).
\begin{corollary}
There exists an Abelian planar difference set of order $ n $ if and only if
the space $ (A_n, d) $ admits a linear $ 1 $-perfect code.
\hfill \qed
\end{corollary}
Existence of such codes when $ n $ is a prime power follows from the existence
of the corresponding difference sets \cite{singer}, but the necessity of this
condition is open and is equivalent to the prime power conjecture.
\begin{conjecture}[{Prime power conjecture}]
\label{conj:ppc}
There exists a linear $ 1 $-perfect code in $ (A_n, d) $ \emph{(}or, equivalently,
in $ (\mathbb{Z}^n, d_a) $\emph{)} if and only if $ n $ is a prime power.
\myqed
\end{conjecture}
A stronger conjecture would claim the above even for nonlinear codes.
\begin{example}
Consider the $ (13, 4, 1) $-difference set $ D = \{0, 1, 3, 9\} \subset \mathbb{Z}_{13} $.
The corresponding $ 1 $-perfect code in $ (A_3, d) $ is illustrated in Fig.\ \ref{fig:A3perfect}.
The figure shows the intersection of $ A_3 $ with the plane $ x_0 = 0 $;
the intersections of a ball of radius $ 1 $ in $ A_3 $ with the planes $ x_0 = \text{const} $
are shown in Fig.\ \ref{fig:ball3color} for clarification.
\myqed
\end{example}
\begin{figure}[h]
\centering
\includegraphics[width=0.75\columnwidth]{A3perfect_w}
\caption{$ 1 $-perfect code in $ (A_3, d) $.}
\label{fig:A3perfect}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.27\columnwidth]{ball3_color_w}
\caption{Intersections of a ball in $ (A_3, d) $ with the planes $ x_0 = \text{const} $.}
\label{fig:ball3color}
\end{figure}
Another important unsolved conjecture in the field is the following:
All Abelian planar difference sets live in cyclic groups
\cite[Conj.\ 7.7, p.\ 346]{designs}.
Since the group $ G $ containing the difference set which defines the code
$ \mathcal L $ is isomorphic to $ A_n/{\mathcal L} $ (see the proof of
Theorem \ref{th:domination}), the statement that $ G $ is cyclic, i.e.,
that it has a generator, is equivalent to the following:
\begin{conjecture}[{All Abelian planar difference sets are cyclic}]
\label{conj:cyclic}
Let $ \mathcal L $ be a $ 1 $-perfect code in $ (A_n, d) $.
Then the period of $ \mathcal L $ in $ A_n $ along the direction
$ {\bf f}_{i,j} $ is equal to $ n^2 + n + 1 $ for at least one vector
$ {\bf f}_{i,j} $, $ (i,j) \in \{0, 1, \ldots, n\}^2 $.
\myqed
\end{conjecture}
\subsubsection*{The cyclic case}
In the rest of this subsection we restrict our attention to cyclic planar
difference sets of order $ n $, i.e., it is assumed that the group we are
working with is $ \mathbb{Z}_v $, $ v = n^2 + n + 1 $;
as mentioned above, this in fact might not be a restriction at all.
So let $ D = \{ d_0, d_1, \ldots, d_n \} \subset \mathbb{Z}_v $ be a
difference set and $ \mathcal L $ the corresponding code (see \eqref{eq:code}).
We shall assume that $ d_0 = 0 $, $ d_1 = 1 $.
(This is not a loss in generality because if $ D $ is a difference set,
there exist two elements, say $ d_0, d_1 \in D $, such that $ d_1 - d_0 = 1 $.
Then we can take the equivalent difference set $ D' = \{ d_i - d_0 : d_i \in D \} $
which obviously contains $ 0 $ and $ 1 $.)
In this case the generator matrix of the code (lattice) $ \mathcal L $
has the following form
\begin{equation}
B({\mathcal L}) =
\begin{pmatrix}
v & 0 & 0 & \cdots & 0 \\
-d_2 & 1 & 0 & \cdots & 0 \\
-d_3 & 0 & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
-d_n & 0 & 0 & \cdots & 1
\end{pmatrix} ,
\end{equation}
i.e., the codewords are the vectors $ {\bf x} = \boldsymbol{\xi} \cdot B({\mathcal L}) $,
$ \boldsymbol{\xi} \in \mathbb{Z}^n $ (the vectors are written as rows).
The generator matrix of the dual lattice $ {\mathcal L}^* $ is
\begin{equation}
B({\mathcal L}^*) = B({\mathcal L})^{-\sml{T}} =
\begin{pmatrix}
\frac{1}{v} & \frac{d_2}{v} & \frac{d_3}{v} & \cdots & \frac{d_n}{v} \\
0 & 1 & 0 & \cdots & 0 \\
0 & 0 & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & 1
\end{pmatrix} .
\end{equation}
We have disregarded above the $ 0 $-coordinate because $ d_0 = 0 $.
Therefore, $ B({\mathcal L}) $ is in fact a generator matrix of the
corresponding code in $ (\mathbb{Z}^n, d_a) $ (see Lemma \ref{th:isometry}).
\subsubsection*{Finite alphabet}
By taking the codewords of $ \mathcal L $ modulo $ v = n^2 + n + 1 $, one
obtains a finite code in $ \mathbb{Z}_v^n $ defined by the generator matrix
(over $ \mathbb{Z}_v $)
\begin{equation}
\begin{pmatrix}
-d_2 & 1 & 0 & \cdots & 0 \\
-d_3 & 0 & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
-d_n & 0 & 0 & \cdots & 1
\end{pmatrix} .
\end{equation}
This code is of length $ n $, has $ v^{n-1} $ codewords, and is $ 1 $-perfect
(under the obvious ``modulo $ v $ version'' of the $ d_a $ metric).
It is also systematic, i.e., the information sequence itself is a part of the
codeword.
The ``parity check'' matrix of the code is
$ H = \begin{pmatrix} 1 & d_2 & \cdots & d_n \end{pmatrix} $.
Thus, the codewords are all those vectors $ {\bf x} = (x_1, \ldots, x_n) \in \mathbb{Z}_v^n $
for which $ H \cdot {\bf x}^\sml{T} = 0 \mod v $, and the syndromes of
the correctable error vectors $ {\bf f}_{i,j} $ (with the $ 0 $-coordinate
left out) are $ H \cdot {\bf f}_{i,j}^\sml{T} = d_i - d_j $.
\section{$ B_h $ sequences as sublattices of $ A_n $}
\label{sec:Bh}
To unify notation with the one for planar difference sets, let $ k = n + 1 $
denote the size of the $ B_h $ sequence in question.
Note that if $ \{b_0, b_1, \ldots, b_n\} $ is a $ B_h $ sequence, then so is
$ \{0, b_1-b_0, \ldots, b_n-b_0\} $, and vice versa;
we shall therefore assume in the sequel that $ b_0 = 0 $.
One of the elements being zero implies that all the sums
$ b_{i_1} + \cdots + b_{i_u} $, for arbitrary $ 0 \leq u \leq h $
and $ 0 \leq i_1 \leq \cdots \leq i_u \leq n $, are distinct.
If $ S, \mathcal{L} \subseteq \mathbb{Z}^n $, $ \mathcal{L} $ a lattice,
we say that $ S $ \emph{packs} $ \mathbb{Z}^n $ with lattice $ \mathcal{L} $
if the translates $ S + {\bf x} $ and $ S + {\bf y} $ are disjoint for every
$ {\bf x}, {\bf y} \in \mathcal{L} $, $ {\bf x} \neq {\bf y} $.
In particular, we are interested in packings of the shape $ S_n(r^\sml{+}, r^\sml{-}) $
defined in \eqref{eq:Sn}.
If $ S_n(r, r) $ packs $ \mathbb{Z}^n $ with lattice $ \mathcal{L} $, we also
denominate $ \mathcal{L} $ a linear code of radius $ r $.
\begin{theorem}
\label{th:Bh}
Let $ h \geq 1 $ and $ r^\sml{+}, r^\sml{-} \geq 0 $ be integers
satisfying $ r^\sml{+} + r^\sml{-} = h $.
\begin{enumerate}[leftmargin=7mm]
\item[\emph{(}a\emph{)}]
Assume that $ B = \{ 0, b_1, \ldots, b_n \} $ is a $ B_h $ sequence in an Abelian
group $ G $ of order $ v $, and that $ B $ generates $ G $.
Then $ S_n(r^\sml{+}, r^\sml{-}) $ packs $ \mathbb{Z}^n $ with lattice
$ {\mathcal L} = \big\{ {\bf x} \in \mathbb{Z}^n : \sum_{i=1}^{n} x_i b_i = 0 \big\} $,
and $ G $ is isomorphic to $ \mathbb{Z}^n / {\mathcal L} $.
\item[\emph{(}b\emph{)}]
Conversely, if $ S_n(r^\sml{+}, r^\sml{-}) $ packs $ \mathbb{Z}^n $ with lattice
$ \mathcal{L} $, then the group $ G = \mathbb{Z}^n / \mathcal{L} $ contains a
$ B_h $ sequence of size $ n + 1 $ that generates $ G $.
\end{enumerate}
\end{theorem}
\begin{proof}
As in Theorem \ref{th:domination} the proof is an application of the
familiar group-theoretic formulation of lattice packing/tiling problems
(the formulation in \cite{stein84} is the one we used here).
We only need to observe that all the sums $ b_{i_1} + \cdots + b_{i_u} $ are
distinct, where $ u $ goes through $ \{0, 1, \ldots, h\} $ and
$ i_1 \leq \cdots \leq i_u $,
if and only if all linear combinations of the form
$ b_{i_1} + \cdots + b_{i_t} - b_{j_1} - \cdots - b_{j_s} $ are distinct,
where $ t $ goes through $ \{0, 1, \ldots, r^\sml{+}\} $, $ s $ through
$ \{0, 1, \ldots, r^\sml{-}\} $, and $ i_1 \leq \cdots \leq i_t $,
$ j_1 \leq \cdots \leq j_s $.
To paraphrase the proof in coding theoretic terms: The translates of
$ S_n(r^\sml{+}, r^\sml{-}) $ being disjoint means that the syndromes
of the correctable error-vectors (vectors from $ S_n(r^\sml{+}, r^\sml{-}) $)
are different.
Since positive coordinates of these vectors sum to $ t \leq r^\sml{+} $
and negative to $ -s \geq -r^\sml{-} $, the syndromes are precisely of
the form $ b_{i_1} + \cdots + b_{i_t} - b_{j_1} - \cdots - b_{j_s} $,
which explains the need for a $ B_h $ sequence.
\end{proof}
Hence, $ B_{2r} $ sequences correspond in a direct way to error-correcting
codes of radius $ r $ in $ (A_n, d) $.
Note also that Theorem \ref{th:Bh} implies that packings by
$ S_n(r^\sml{+}, r^\sml{-}) $ are all equivalent in the sense that the packing
lattice does not depend on the particular values of $ r^\sml{+}, r^\sml{-} $,
but only on their sum.
However, the cardinality of $ S_n(r^\sml{+}, r^\sml{-}) $ varies with $ r^\sml{+} $,
and so does the packing density $ \frac{1}{v} | S_n(r^\sml{+}, r^\sml{-}) | $.
\subsection{Bounds}
A major part of research on $ B_h $ sequences is concentrated on determining
extremal size of such sequences for given $ h, v $, or the asymptotic behavior
of $ k $ for given $ h $ and for $ v \to \infty $.
Despite a large amount of work, however, determining tight bounds remains an
open problem.
We illustrate below how nontrivial bounds can be derived in a straightforward
way using Theorem \ref{th:Bh}.
Let $ f_h(v) $ denote the size of the largest $ B_h $ sequence in \emph{all}
Abelian groups of order $ v $,
$ h_k(v) $ the largest $ h $ for which there is a $ B_h $ sequence of size
$ k $ in some Abelian group of order $ v $,
and $ \phi(h, k) $ the order of the smallest Abelian group containing some
$ B_h $ sequence of size $ k $.
This notation is from \cite{jia}, except the functions are defined to
include all finite Abelian groups, rather than just the cyclic ones.
\begin{theorem}
\label{th:bounds}
For all $ h \geq 1 $ and $ k \geq \ceil{h/2} $,
\begin{equation}
\label{eq:phik}
\phi(h, k) > \frac{ (k - \ceil{h/2})^h }{ \ceil{h/2}! \floor{h/2}! } .
\end{equation}
For all $ h \geq 1 $ and large enough $ v $,
\begin{equation}
\label{eq:fh}
f_{h}(v) < v^\frac{1}{h} \cdot \left( \floor{h/2}!\ceil{h/2}! \right)^\frac{1}{h} + \ceil{h/2} .
\end{equation}
For all $ k = n + 1 \geq 2 $ and $ h \geq 2k - 4 $,
\begin{equation}
\label{eq:phih}
\phi(h, k) > \frac{ (h - 2n + 2)^n }{ n! } \frac{ {2n \choose n} }{2^n} .
\end{equation}
For all $ k = n + 1 \geq 2 $ and large enough $ v $,
\begin{equation}
\label{eq:hk}
h_k(v) < v^\frac{1}{n} \cdot 2 \left( (n!)^3 /(2n)! \right)^\frac{1}{n} + 2n - 2 .
\end{equation}
\end{theorem}
\begin{proof}
Suppose that $ \{ 0, b_1, \ldots, b_n \} $ is a $ B_h $ sequence in a group
$ G $ of order $ v $, and $ {\mathcal L} $ the corresponding packing lattice.
By Theorem \ref{th:Bh}, $ G \cong \mathbb{Z}^n / {\mathcal L} $ and the
translations of the shape $ S_n(r^\sml{+}, r^\sml{-}) $ by vectors in
$ {\mathcal L} $ are disjoint.
Therefore, for $ n \geq r^\sml{+} $,
\begin{equation}
\begin{aligned}
v \geq | S_n(r^\sml{+}, r^\sml{-}) |
&= \sum_{ m = 0 }^{ r^\sml{+} } { n \choose m }
{ r^\sml{+} \choose m }
{ r^\sml{-} + n - m \choose n - m } \\
&\geq \sum_{ m = 0 }^{ r^\sml{+} } \frac{ (n - m + 1)^m }{ m! } { r^\sml{+} \choose m } \frac{ (n - m + 1)^{r^\sml{-}} }{ r^\sml{-}! } \\
&> \frac{ (n - r^\sml{+} + 1)^h }{ r^\sml{+}! \ r^\sml{-}! } ,
\end{aligned}
\end{equation}
where we used $ {n \choose m} \geq \frac{(n-m+1)^m}{m!} $, and the last inequality
is obtained by keeping only the summand $ m = r^\sml{+} $.
This implies \eqref{eq:phik} and \eqref{eq:fh} by taking $ r^\sml{+} = \ceil{h/2} $
(this choice minimizes the denominator $ r^\sml{+}! r^\sml{-}! $).
Similarly, if we let $ 0 \leq r^\sml{+} - n \leq r^\sml{-} $, then
\begin{equation}
\begin{aligned}
v \geq | S_n(r^\sml{+}, r^\sml{-}) |
&= \sum_{ m = 0 }^{ n } { n \choose m }
{ r^\sml{+} \choose m }
{ r^\sml{-} + n - m \choose n - m } \\
&\geq \sum_{ m = 0 }^{ n } { n \choose m } \frac{ (r^\sml{+} - m + 1)^m }{ m! } \frac{ (r^\sml{-} + 1)^{(n-m)} }{ (n-m)! } \\
&= \frac{1}{n!} \sum_{ m = 0 }^{ n } { n \choose m }^2 (r^\sml{+} - m + 1)^m (r^\sml{-} + 1)^{(n-m)} \\
&> \frac{ (r^\sml{+} - n +1)^n }{ n! } {2n \choose n} .
\end{aligned}
\end{equation}
In the last step we used the assumption $ r^\sml{-} \geq r^\sml{+} - n $ and the
identity $ \sum_{m=0}^n {n \choose m}^2 = {2n \choose n} $.
When $ r^\sml{+} = \ceil{h/2} $ we get \eqref{eq:phih} and \eqref{eq:hk}.
\end{proof}
For $ h $ fixed and $ k \to \infty $ we thus have
\begin{equation}
\label{eq:phik2}
\phi(h, k) \geq \frac{ k^h }{ \ceil{h/2}! \floor{h/2}! } + o(k^h) ,
\end{equation}
and for $ k $ fixed and $ h \to \infty $
\begin{equation}
\label{eq:phih2}
\phi(h, k) \geq \frac{ h^n }{ n! } \frac{ {2n \choose n} }{2^n} + o(h^n) .
\end{equation}
The above derivations, apart from being simple, have the advantage of
including all finite Abelian groups.
The bound in \eqref{eq:fh} is the one stated in \cite[Thm 2]{chen}, but
with an explicit error term.
It is currently the best known, and so is \eqref{eq:phik2}.
The bounds \eqref{eq:hk} and \eqref{eq:phih2} improve on the known
bounds \cite[Thm 1(v)]{jia} by a factor of $ 2^{-n}{2n \choose n} > 1 $, $ n > 1 $.
\section{$ r $-perfect codes in $ (A_n, d) $}
A natural question related to the results of Sections \ref{sec:diff} and
\ref{sec:Bh} is whether there exist $ r $-perfect codes in $ (A_n, d) $
for $ r > 1 $.
For example, such codes exist for any $ r $ in dimensions one and two:
it is easy to verify that the sublattice of $ A_1 $ spanned by the vector
$ (-2r-1, 2r+1) $ is an $ r $-perfect code in $ A_1 $, and the sublattice of
$ A_2 $ spanned by the vectors $ (-2r-1, r, r+1) $, $ (-r-1, 2r+1, r) $ is an
$ r $-perfect code in $ A_2 $
(for a study of the two-dimensional case see also \cite{costa}).
The corresponding $ B_{2r} $ sequences might therefore be called
\emph{perfect $ B_{2r} $ sequences}.
In higher dimensions, however, it does not seem possible to tile $ (A_n, d) $
by balls of radius $ r > 1 $.
We shall not be able to prove this claim here, but Theorem \ref{th:rperfect}
below is a step in this direction.
\begin{figure}[h]
\centering
\includegraphics[width=0.55\columnwidth]{Z2tile}
\caption{Bodies in $ \mathbb{R}^2 $ corresponding to a ball of radius $ 2 $ in $ \left( \mathbb{Z}^2, d_a \right) $:
The cubical tile (left) and the convex interior (right).}
\label{fig:Z2tile}
\end{figure}
Let $ S_n(r) \equiv S_n(r,r) $ be the ball of radius $ r $ around $ \bf 0 $
in $ ( \mathbb{Z}^n, d_a ) $.
Let $ D_n(r) $ be the body in $ \mathbb{R}^n $ defined as the union of unit
cubes translated to the points of $ S_n(r) $, namely,
$ D_n(r) = \bigcup_{{\bf y} \in S_n(r)} ( {\bf y} + [-1/2, 1/2]^n ) $,
and $ C_n(r) $ the body defined as the convex interior in $ \mathbb{R}^n $
of the points in $ S_n(r) $ (see Fig.\ \ref{fig:Z2tile}).
\begin{lemma}
\label{th:volumes}
The volumes of the bodies $ D_n(r) $ and $ C_n(r) $ are given by
\begin{align}
\vol(D_n(r)) &= \sum_{ m = 0 }^{ \min\{ n, r \} } { n \choose m }
{ r \choose m }
{ r + n - m \choose n - m } \\
\label{eq:volconv}
\vol(C_n(r)) &= \frac{r^n}{n!} {2n \choose n} ,
\end{align}
and they satisfy $ \lim_{r \to \infty} \vol(C_n(r)) / \vol(D_n(r)) = 1 $.
\end{lemma}
\begin{proof}
Since $ D_n(r) $ consists of unit cubes, its volume is $ \vol(D_n(r)) = |S_n(r)| $,
which gives the above expression by Lemma \ref{th:Anball}.\\
To compute the volume of $ C_n(r) $, observe its intersection with the orthant
$ x_{1}, \ldots, x_{m} > 0 $, $ x_{m+1}, \ldots, x_{n} \leq 0 $, where
$ m \in \{ 0, \ldots, n \} $.
The volume of this intersection is the product of the volumes of the $ m $-simplex
$ \big\{ (x_1, \ldots, x_m) : x_i > 0, \sum x_i \leq r \big\} $, which is known
to be $ r^m/m! $, and of the $ (n-m) $-simplex
$ \big\{ (x_{m+1}, \ldots, x_n) : x_i \leq 0, \sum x_i \geq -r \big\} $, which
is $ r^{n-m}/(n-m)! $.
This implies that
$ \vol(C_n(r)) = \sum_{m=0}^n {n \choose m} \frac{r^m}{m!} \frac{r^{n-m}}{(n-m)!} $,
which is equivalent to \eqref{eq:volconv} because $ \sum_{m=0}^n {n \choose m}^2 = {2n \choose n} $.\\
When $ r \to \infty $, $ {r \choose m} \sim \frac{r^m}{m!} $ and therefore
$ \vol(D_n(r)) \sim \frac{r^n}{n!} {2n \choose n} $.
\end{proof}
\begin{theorem}
\label{th:rperfect}
There are no $ r $-perfect codes in $ (A_n, d) $, $ n \geq 3 $, for large enough
$ r $, i.e., for $ r \geq r_0(n) $.
\end{theorem}
\begin{proof}
The proof is based on the same idea as the one for $ r $-perfect codes in
$ \mathbb{Z}^n $ under $ \ell_1 $ (also termed Manhattan or taxi) distance
\cite{golomb+welch}.
First observe that an $ r $-perfect code in $ ( \mathbb{Z}^n, d_a ) $ would
induce a tiling of $ \mathbb{R}^n $ by $ D_n(r) $, and a \emph{packing} by
$ C_n(r) $.
The relative efficiency of the latter with respect to the former is defined as
the ratio of the volumes of these bodies, $ \vol(C_n(r)) / \vol(D_n(r)) $, which
by Lemma \ref{th:volumes} tends to $ 1 $ as $ r $ grows indefinitely.
This has the following consequence: If an $ r $-perfect code exists in
$ (\mathbb{Z}^n, d_a) $ for arbitrarily large $ r $, then there exists a
tiling of $ \mathbb{R}^n $ by translates of the body $ D_n(r) $ for arbitrarily
large $ r $, which further implies that a packing of $ \mathbb{R}^n $ by translates
of the body $ C_n(r) $ exists which has efficiency arbitrarily close to $ 1 $.
But then there would also be a packing by $ C_n(r) $ of efficiency $ 1 $, i.e., a
tiling (in \cite[Appendix]{golomb+welch} it is shown that there exists a packing
whose density is the supremum of the densities of all possible packings with a
given body).
This is a contradiction.
Namely, Minkowski \cite{minkowski} (see also \cite[Thm 1]{mcmullen}) has shown
that a necessary condition for a convex body to be able to tile space is that
it be a polytope with centrally symmetric
\footnote{A polytope $ P \subset \mathbb{R}^n $ is centrally symmetric if
its translation $ \tilde{P} = P - {\bf x} $ satisfies $ \tilde{P} = -\tilde{P} $
for some $ {\bf x} \in \mathbb{R}^n $.}
facets, which $ C_n(r) $ fails to satisfy for $ n \geq 3 $.
For example, the facet which is the intersection of $ C_n(r) $ with the hyperplane
$ x_1 = -r $ is the simplex $ \big\{ (x_2, \ldots, x_n) : x_i \geq 0, \sum_{i=2}^n x_i \leq r \big\} $,
a non-centrally-symmetric body.
\end{proof}
In summary, we have shown that linear $ r $-perfect codes in $ (A_n, d) $ exist for:
\begin{itemize}
\item $ n \in \{1, 2\} $, $ r $ arbitrary,
\item $ n \geq 3 $ a prime power, $ r = 1 $.
\end{itemize}
The statement that these are the only cases (apart from the trivial one $ r = 0 $),
even if nonlinear codes are allowed, is a further strengthening of the prime power
conjecture.
It should also be contrasted with the Golomb-Welch conjecture \cite{golomb+welch}
(see also, e.g., \cite{horak2, horak}) stating that $ r $-perfect codes in
$ \mathbb{Z}^n $ under $ \ell_1 $ distance exist only in the following cases:
\begin{inparaenum}[1)]
\item $ n \in \{1, 2\} $, $ r $ arbitrary, and
\item $ r = 1 $, $ n $ arbitrary.
\end{inparaenum}
\begin{remark}
To conclude this section we note that one could also define perfect
$ B_{2r+1} $ sequences as those that give rise to tilings by the shapes
$ S_n(r+1,r) $.
In dimension $ 1 $ the problem is trivial, and in dimension $ 2 $ the
tilings exist for any $ r \geq 1 $ (see Fig.\ \ref{fig:S2_tiling}).
The tiling lattice $ {\mathcal L} $ for the shape $ S_2(r+1,r) $ is
the one spanned by the vectors $ (r+1, r+1) $ and $ (0, 3r+3) $, and
it is unique, which can be seen from the figure.
In this case, however, $ \mathbb{Z}^n / {\mathcal L} $ is not cyclic
and hence, perfect $ B_{2r+1} $ sequences of size $ 3 $ do not exist in
cyclic groups.
In higher dimensions, a statement analogous to Theorem \ref{th:rperfect} can
be proven to exclude such tilings for $ r $ large enough.
\myqed
\end{remark}
\begin{figure}[h]
\centering
\includegraphics[width=0.75\columnwidth]{S2_tiling}
\caption{Tiling of $ \mathbb{Z}^2 $ by the shape $ S_2(2,1) $.}
\label{fig:S2_tiling}
\end{figure}
\section{Applications in coding theory}
\subsection{Permutation channels}
A permutation channel \cite{kovacevic_desi, kovacevic} over an alphabet
$ \mathbb{A} $ is a communication channel that takes sequences of symbols
from $ \mathbb{A} $ as inputs, and for any input sequence outputs a random
permutation of this sequence.
This channel is intended to model packet networks based on routing in which
the receiver cannot rely on the packets being delivered in any particular order,
as well as several other communication scenarios where a similar effect occurs,
such as systems for distributed storage, data gathering in wireless sensor
networks, etc.
It was shown in \cite{kovacevic_desi} that the appropriate space for defining
error-correcting codes for such channels is
$ \left( \Delta_\ell^n, d \right) $, where
\begin{equation}
\Delta_\ell^n = \left\{ (x_0, x_1, \ldots, x_n) \in \mathbb{Z}_\sml{\geq 0}^{n+1} : \sum_{i=0}^n x_i = \ell \right\}
\end{equation}
is the discrete standard simplex, and $ d $ is the metric given by \eqref{eq:metricd}.
Notice that $ \Delta_\ell^n $ is just the translated $ A_n $ lattice restricted
to the nonnegative orthant.
This restriction is the reason that this space lacks some nice properties that
are usually exploited when studying bounds on codes, packing problems, and the
like.
In order to study the underlying geometric problem, one can disregard these
restrictions and investigate the corresponding problems in $ (A_n, d) $.
The same approach is employed for some other types of codes; for example,
studying the geometry of codes for flash memories reduces to packing problems
in $ \mathbb{Z}^n $, see, e.g., \cite{buzaglo+etzion, cassuto, schwartz}.
\subsection{Asymmetric channels}
Observe the channel with input alphabet $ \mathbb{A} = \{ 0, 1, \ldots, q-1 \} $
in which the transmitted symbols can be increased, decreased, or remain intact
(we are not concerned here with a detailed description of the channel, but
wish to make a rather general point).
In some situations, it may be desirable to treat these two types of errors
separately when designing an error-correcting code, and to impose different
bounds on the number of ``up'' and ``down'' errors that can be corrected.
This is because these types of errors can be of different nature and/or
different origin, and the probabilities of a symbol being increased/decreased
might be different.
We give below two simple examples, see also \cite{klove} for a survey of
codes for the classical $ q $-ary asymmetric channel.
\subsubsection{Particle insertion/deletion channels}
Consider the following channel model.
The transmitter sends $ x_i $ particles (or packets) in the $ i $'th time slot.
The particles are assumed identical, implying that the transmitted sequence
can be identified with a sequence of nonnegative integers
$ (x_1, \ldots, x_n) \in \mathbb{Z}^n_\sml{\geq 0} $.
In the channel, some of the particles can be lost (deletions), while new particles
can appear from the surrounding medium (insertions).
Additionally, one can also allow delays of the particles in the model \cite{kovacevic_IT}.
(Notice that if a particle is delayed, this can be thought of as it being deleted,
while another particle is being inserted several time slots later.)
Such channels are of interest in molecular communications, discrete-time queuing
systems, etc.
We want to guarantee that all patterns of $ \leq r^\sml{+} $ insertions and
$ \leq r^\sml{-} $ deletions can be corrected at the receiving side.
It is not difficult to see that this will be achieved if and only if the
translates of the set $ S_n(r^\sml{+}, r^\sml{-}) $ are disjoint.
In other words, the decoding regions are in this case defined precisely by
the set $ S_n $, and hence, packing/tiling problems in $ (\mathbb{Z}^n, d_a) $
are relevant for studying and designing good codes for channels of this type.
\subsubsection{Baseband transmission}
As a different scenario, think of the symbols in $ \mathbb{A} $ as voltage
levels (e.g., in baseband digital signal transmission, or in multi-level
flash memory \cite{cassuto,schwartz}), and suppose that we wish to assure
that the receiver will be able to recover the signal whenever the total
voltage drop (across all symbols of a particular codeword) caused by noise
is $ \leq r^\sml{-} $, and the total voltage increase $ \leq r^\sml{+} $.
This situation is essentially identical to the one in the previous example,
and it is clear that the codes in $ (\mathbb{Z}^n, d_a) $ (more precisely
their restrictions to $ \mathbb{A}^n $) are adequate constructions for
error-correction in this case.
\section*{Acknowledgment}
The author would like to thank Dr Moshe Schwartz for reading the original
version of the manuscript and providing several useful comments.
\bibliographystyle{amsplain}
| 63,749
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Title
Tetratricopeptide 39C (TTC39C) Is Upregulated During Skeletal Muscle Atrophy and is Necessary for Muscle Cell Differentiation
Year of Publication
2018. Doria Bowers, Ph.D.
Third Advisor
Dr. John Hatle, Ph.D.
Department Chair
Dr. Cliff Ross, Ph.D.
College Dean
Dr. George Rainbolt, Ph.D.
Abstract
Ttc39c has been identified as a novel gene in skeletal muscle that is upregulated in response to neurogenic atrophy in mice. Quantitative PCR and Western blot analysis confirmed that Ttc39c is expressed in both proliferating and differentiated muscle cells. Furthermore, comparison of Ttc39c expression in undifferentiated and differentiated C2C12 cells demonstrated that Ttc39c levels peak in early differentiation, but decreases as cells become fully differentiated myotubes. The transcriptional regulation of Ttc39c was examined by cloning promoter fragments of the gene and fusing it with the SEAP reporter gene. The Ttc39c reporter gene constructs were transfected into muscle cells and confirmed to have significant transcriptional activity in cultured muscle cells and were also found to be transcriptionally repressed in response to ectopic expression of myogenic regulatory factors (MRF). Furthermore, conserved E-box elements in the proximal promoter region were identified, mutated, and analyzed for their role in the transcriptional regulation of Ttc39c expression. Mutation of the conserved E-box sequences reduced the activity of the Ttc39c reporter gene, suggesting that these elements are potentially necessary for full Ttc39c expression. To determine the sub-cellular location of Ttc39c in muscle cells, the Ttc39c cDNA was fused with the green fluorescent protein (GFP), expressed in muscle cells, and visualized by confocal microscopy revealing that Tct39c is localized to the cytoplasm of proliferating myoblasts and differentiating myotubes. Furthermore, Ttc39c appears to localize to the microtubule network and differentiating muscle cells developed elongated primary cilia in response to Ttc39c ectopic expression. Additionally, Ttc39c overexpression resulted in impaired muscle cell differentiation, attenuated Hedgehog and MAP Kinase signaling, and increased expression of IFT144, a component of the intraflagellar transport complex A involved in retrograde movement in primary cilia. Interestingly, Ttc39c knockdown also resulted in abrogated muscle cell differentiation and impaired Hedgehog and MAP Kinase signaling, but did not affect IFT144 expression levels. These results suggest that muscle cell differentiation is sensitive to aberrant Ttc39c expression, that Ttc39c is necessary for proper muscle cell differentiation, and that Ttc39c may participate in retrograde transport of the primary cilia of developing muscle cells.
Suggested Citation
Hayes, Caleb, "Tetratricopeptide 39C (TTC39C) Is Upregulated During Skeletal Muscle Atrophy and is Necessary for Muscle Cell Differentiation" (2018). UNF Graduate Theses and Dissertations. 796.
| 325,258
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TITLE: product of martingales bounded in $L^2$
QUESTION [1 upvotes]: Let $(M_t)_t$ and $(M_t)_t$ be two càdlàg martingales on the same filtered probability space.
We know that $M_{\infty}$ and $N_{\infty}$ are orthogonal in $L^2$.
Is it true that $(M_t N_t)_t$ is a martingale? If so, is it uniformly integrable?
I suspect the answers are yes and no, but not sure on how to prove.
On the first statement: $MN$ is clearly adapted and càdlàg and the martingale equality should be equivalent to $$ \mathbb{E}_t [(M_{t+h}-M_t)(N_{t+h}-N_t)] = 0 \quad \forall t,h >0 $$
If possible, provide an answer that does not involve the process $[M,N]$.
REPLY [2 votes]: No, in general $(M_t \cdot N_t)_{t \geq 0}$ is not a martingale. Consider the (continous) $L^2$-martingales
$$M_t := \int_0^t f(s) \, dB_s \qquad N_t := \int_0^t g(s) \, dB_s$$
where $f,g$ are deterministic functions such that $$\int_0^{\infty} f(s)^2 \, ds < \infty \quad \int_0^{\infty} g(s)^2 \, ds < \infty$$
Then, by Itô's formula
$$\mathbb{E}(M_t \cdot N_t) = \mathbb{E} \left( \int_0^t f(s) \cdot g(s) \, ds \right) = \int_0^t f(s) \cdot g(s) \,ds$$
for any $t \in [0,\infty]$. This means that $M_{\infty}$ and $N_{\infty}$ are orthogonal if, and only if, $$\int_0^{\infty} f(s) \cdot g(s) \, ds = 0 \tag{1}$$
Obviously, we can choose $f,g$ such that $(1)$ is satisfied and
$$t \mapsto \int_0^t f(s) \cdot g(s) \, ds$$
is not constant. But this means that $\mathbb{E}(M_t \cdot N_t)$ is not constant; hence $(M_t \cdot N_t)_{t \geq 0}$ is not a martingale.
| 112,320
|
TITLE: convergence in probability confusion
QUESTION [0 upvotes]: I'm confused by the definition of convergence in probability (as discussed in Types of convergence in probability confusion?).
We have a sequence $X_1, X_2, \dots$ of random variables, and a random variable $X$. My understanding is that convergence in probability is defined as, for all $\epsilon > 0$, as $n \rightarrow \infty$,
$$ \mathbb{P} \left[ | X_n - X | \geq \epsilon \right] \rightarrow 0, $$
where the probability is defined as
$$ \mathbb{P} \left[ | X_n - X | \geq \epsilon \right] := \mathbb{P} \left( \omega \in \Omega | X_n (\omega) - X (\omega) \right). $$
So then doesn't convergence in probability depend on the sample space $\Omega$ and the probability function?
Scenario 1. Dependence on sample space: let each $X_i$ be the result of a coin flip ($1$ if heads, else $0$). Let $X$ be the result of a die roll ($1$ if even, $0$ if odd). So then the $X_i$ don't converge to $X$.
Scenario 2. Dependence on probability function: let each $X_i$ and $X$ be the result of a coin flip ($1$ if heads, else $0$). Let $X'$ be the result of a coin flip ($0$ if heads, else $1$). Then the $X_i$ converge to $X$, but not to $X'$.
If the above are true, then it seems that we can't answer questions like problem 1 of https://ocw.mit.edu/courses/18-650-statistics-for-applications-fall-2016/1aea12bd54eb19c821501f730cc247d9_MIT18_650F16_PSet1.pdf without information about the sample space and probability function.
What am I missing? Am I just overthinking?
REPLY [0 votes]: You are absolutely correct that something fishy is going on with question 1. Let us examine the definition of convergence in probability. It says that given $\varepsilon >0$, and a probability space $(\Omega,\mathcal F,\mathbb P)$, $X_n\to X$ in probability means $\mathbb P(\lvert X_n-X\rvert \geq \varepsilon)\to 0$ as $n\to \infty$. So you are correct that $X_n$ and $X$ all need to be defined on the same probability space.
However, maybe this definition is a little restrictive. Actually, what if we assume that $X_n$ is a random variable on $(\Omega_n,\mathcal F_n,\mathbb P_n)$ and that $X$ is a random variable on $(\Omega_n,\mathcal F_n,\mathbb P_n)$ for every $n$. Then we can generalize the definition as
needing $\mathbb P_n(\omega\in \Omega_n:\lvert X_n-X\rvert \geq \varepsilon)\to 0$ as $n\to \infty$. The issue is that it may not always be possible for such an $X$ to exist. But there is a special case when it does. If $X$ is almost surely constant, then we can clearly do so. In fact, if $X$ is almost surely constant, then convergence in distribution and convergence in probability are equivalent and it is common to denote both as $X_n\Rightarrow c$.
| 55,186
|
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| 298,542
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\section{Numerical experiments}\label{s_numex}
After describing the setup,
we present numerical experiments for recursive and pairwise summation (Section~\ref{s_ne1}), shifted summation (Section~\ref{s_ne2}), compensated summation (Section~\ref{s_ne3}),
and mixed-precision \texttt{FABSum} (Section~\ref{s_ne4}).
Experiments are performed in MATLAB R2022a, with unit roundoffs \cite{fp16}
\begin{itemize}
\item Half precision $u=2^{-11}\approx 4.88 \cdot 10^{-4}$.
\item Single precision $u_{\text{hi}}=2^{-24}\approx 5.96\cdot 10^{-8}$ as the high precision in \texttt{FABsum} Algorithm~\ref{alg:fabsum}.
\item Double precision $u=2^{-53}\approx 1.11\cdot 10^{-16}$ for `exact' computation.
\end{itemize}
Experiments plot errors from two rounding modes: round-to-nearest and stochastic rounding as implemented with \texttt{chop} \cite{higham2019simulating}.
The summands $x_k$ are independent uniform $[0,1]$ random variables. The plots show relative errors $|\hat{s}_n-s_n|/|s_n|$ versus $n$,
for $100\leq n\leq 10^5$.
We choose relative errors rather than absolute errors
to allow for meaningful calibration:
Relative errors $\leq u$ indicate full accuracy; while
relative errors $\geq .5$ indicate zero digits of accuracy.
For shifted summation we use the empirical mean of two extreme summands,
\begin{equation*}
c=(\min_k{x_k}+\max_k{x_k})/2.
\end{equation*}
For probabilistic bounds, the combined failure probability is $\delta+\eta=10^{-2}+10^{-3}$, hence
$\sqrt{2\ln(2/\delta)}\approx 3.26$. For $n=10^5$ and $h=n$ we get $\lambda_{n,\eta}\approx 6.2$, and in half precision $u=2^{-11}$ the higher-order errors, $1+\phi_{n,h,\eta} \approx 4.4$, have a non-negligible effect on our bounds.
\subsection{Sequential and pairwise summation}\label{s_ne1}
Figure \ref{f_ne1} shows the errors
in half precision from Algorithm \ref{alg:sum} for
sequential summation in one panel, and for pairwise summation in another
panel, along with the deterministic bounds from Theorem \ref{t_gdet},
\begin{align}
|e_n| &\leq \sum_{k=2}^n|s_k||\delta_k|\prod_{k\prec j \preceq n}|1+\delta_j|
\leq \lambda_h\,u\,\sum_{k=2}^n|s_k|\label{b_3}\\
&\leq \lambda_h\,h\,u\,\sum_{j=1}^n{|x_j|}.\label{b_4}
\end{align}
and the probabilistic bounds from Corollary~\ref{c_210},
\begin{align}
|e_n| &\leq u\sqrt{2\ln(2/\delta)}\left(1+\phi_{n,h,\eta}\right)\sqrt{\sum_{k=2}^ns_k^2}\label{b_1}\\
&\leq u \sqrt{h}\sqrt{2\ln(2/\delta)}\left(1+\phi_{n,h,\eta}\right)\sum_{k=1}^n|x_k|\label{b_2},
\end{align}
\paragraph{Sequential summation}
The bound (\ref{b_1}) remains within a factor of 2
of (\ref{b_2}). Although the higher-order error terms $1+\phi_{n,h,\eta}$ represent only a small part of the error bounds, they may
still be pessimistic, as the
bounds curve upwards for large $n$, while the actual errors
increase more slowly. The reason may be the
distribution of floating point numbers: spacing between consecutive numbers is constant within each interval $[2^t, 2^{t+1}]$, so a roundoff $\delta_k$ is affected by previous errors primarily if $\lfloor \log_2(\hat{s}_k)\rfloor \neq \lfloor \log_2(s_k)\rfloor$. Some analyses have derived deterministic error bounds for summation that do not contain second-order terms \cite{jeannerod2013improved,jeannerod2018relative,lange2019sharp,rump2012error}, and perhaps a more careful analysis will be able to do the same for probabilistic bounds. Our bounds otherwise accurately describe the behavior of stochastic rounding, but round-to-nearest suffers from stagnation for larger problem sizes.
\paragraph{Pairwise summation} The bound (\ref{b_2}) grows proportional to $\sqrt{\log_2(n)}$,
while (\ref{b_1}) remains essentially constant. The behavior
of (\ref{b_1}) may be due to the monotonically increasing partial sums for
uniform $[0, 1]$ inputs, where the final sum is likely to dominate all
previous partial sums,
$(\sum_{k=2}^ns_k^2)^{1/2} = \mathcal{O}(s_n)$. This suggests
that pairwise summation of uniform $[0, 1]$ inputs is highly accurate.
The constant bound accurately describes the behavior of the error under stochastic rounding, but not round-to-nearest. We are not sure of the exact reason for the difference in behavior between the two.
\begin{figure}
\begin{center}
\includegraphics[width = 0.49\textwidth]{rec.png}
\includegraphics[width = 0.49\textwidth]{pair.png}
\end{center}
\caption{Relative errors in half precision for recursive summation (left) and sequential summation (right) versus number of summands $n$.
The symbol (\texttt{+}) indicates round-to-nearest (RTN), and ($\mathtt{\times}$) indicates stochastic rounding (SR).
Horizontal line indicates unit roundoff $u = 2^{-11}$,
and remaining points indicate deterministic bounds (\ref{b_3}) and (\ref{b_4}) and probabilistic bounds (\ref{b_1}) and (\ref{b_2}).}
\label{f_ne1}
\end{figure}
\subsection{Shifted summation}\label{s_ne2}
Figure \ref{f_ne2} shows the errors
in half precision from Algorithm \ref{alg:shiftSum}
for shifted sequential summation and shifted pairwise summation,
along with the probabilistic bounds from Theorem~\ref{thm:shifted},
\begin{align}
|e_n| &\leq u\sqrt{2\ln(2/\delta)}\left(1+\phi_{n,h,\eta}\right)\sqrt{s_n^2 + \sum_{k=2}^nt_k^2 + \sum_{k=1}^{n+1}y_k^2}\label{b_5}\\
&\leq u\sqrt{2\ln(2/\delta)}\left(1+\phi_{n,h,\eta}\right)\left(n|c| +\sqrt{h}\sum_{k=1}^n{(|x_k-c|+|x_k|)}\right).\label{b_6}
\end{align}
A comparison with Figure \ref{f_ne1}
shows that shifting reduces both the actual errors and the bounds.
Errors are on the order of unit roundoff, in all cases:
round-to-nearest and stochastic rounding, and sequential and pairwise summation.
\begin{figure}
\begin{center}
\includegraphics[width = 0.49\textwidth]{shift.png}
\includegraphics[width = 0.49\textwidth]{shiftpair.png}
\end{center}
\caption{Relative errors in half precision for shifted sequential summation (left) and shifted pairwise summation (right) versus number of summands $n$.
The symbol (\texttt{+}) indicates round-to-nearest (RTN), and ($\mathtt{\times}$) indicates stochastic rounding (SR).
Horizontal line indicates unit roundoff $u = 2^{-11}$,
and remaining points indicate probabilistic bounds (\ref{b_5}) and (\ref{b_6}).}
\label{f_ne2}
\end{figure}
\subsection{Compensated summation}\label{s_ne3}
The first panel in Figure~\ref{f_ne3} shows the errors in half precision for Algorithm~\ref{alg:compensated}
for $10^2\leq n \leq 10^7$ summands\footnote{Our simulation of half-precision ignores the range restriction \texttt{realmax} = 65504.},
along with deterministic bounds derived from Corollary~\ref{c_cs5},
\begin{align}
|e_n| &\leq u|s_n| + 2u(1+3u)\sum_{k=2}^n|x_k| + 4u^2\sum_{k=2}^{n-1}|s_k| + \mathcal{O}(u^3) \label{b_9}\\
&\leq (3u + (4n-2)u^2)\sum_{k=1}^n|x_k| + \mathcal{O}(u^3), \label{b_10}
\end{align}
and the probabilistic bounds from Theorem~\ref{thm:comp_prob_err},
\begin{align}
|e_n|&\leq u\sqrt{2\ln(2/\delta)}
\left(|s_n| + \gamma(\sqrt{2}+\alpha u)\sqrt{\sum_{k=2}^nx_k^2}+\gamma\alpha u\sqrt{\sum_{k=2}^ns_k^2}\right) \label{b_7}\\
&\leq u\sqrt{2\ln(2/\delta)}
\left(1+\sqrt{2}+\sqrt{6}(\sqrt{n}+1) u\right)\sum_{k=1}^n|x_k| + \mathcal{O}(u^3). \label{b_8}
\end{align}
The probabilistic bounds (\ref{b_7}) and
(\ref{b_8}) track the error behavior accurately, with (\ref{b_7}) even capturing
the correct order of magnitude. This also
illustrates the higher accuracy of bounds
involving partial sums.
\begin{figure}
\begin{center}
\includegraphics[width = 0.49\textwidth]{comp.png}
\includegraphics[width = 0.49\textwidth]{FABext.png}
\end{center}
\caption{Relative errors in half precision for compensated summation (left) and mixed
precision with
\texttt{FABsum} with high precision $u_{\text{hi}}= 2^{-24}$ (right) versus number of summands $n$. The symbol (\texttt{+}) indicates round-to-nearest (RTN), and ($\mathtt{\times}$) indicates stochastic rounding (SR).
Horizontal line indicates unit roundoff $u_{\text{lo}} = 2^{-11}$,
and remaining points indicate bounds \eqref{b_9}-\eqref{b_8} (left) and \eqref{b_11}-\eqref{b_13} (right).
}
\label{f_ne3}
\end{figure}
\subsection{Mixed-precision \texttt{FABsum} summation}\label{s_ne4} The second panel of Figure~\ref{f_ne3} shows the errors for Algorithm~\ref{alg:fabsum}
with $u_{\text{lo}}=2^{-11}\approx 4.44\cdot 10^{-4}$, $u_{\texttt{hi}}=2^{-24}\approx 5.96\cdot 10^{-8}$, block
size $b=32$ and $10^2\leq n \leq 10^7$ summands, where each internal call to Algorithm~\ref{alg:sum} uses recursive summation. We also plot the deterministic first-order bound from
\cite[Eqn.~3.5]{blanchard2020class},
\begin{equation}
|e_n| \leq bu\sum_{k=1}^n|x_k| + \mathcal{O}(u^2), \label{b_11}
\end{equation}
and the probabilistic bounds derived from Theorem~\ref{t_52},
\begin{align}
|e_n| &\leq \sqrt{2\ln(2/\delta)}\left(1+\phi_{n,\tilde{h},\eta}\right)\sqrt{\sum_{k=2}^nu_k^2s_k^2} \label{b_12}\\
&\leq \sqrt{\tilde{h}}\sqrt{2\ln(2/\delta)}\left(1+\phi_{n,\tilde{h},\eta}\right)\sum_{k=1}^n|x_k|, \label{b_13}
\end{align}
where $\tilde{h} = bu^2 + (n/b)u_{\text{hi}}^2$. Errors are on the order of unit roundoff for round-to-nearest. We were surprised to observe that for stochastic rounding, errors fell to more than an order of magnitude {\em below} unit roundoff for large problem sizes. This behavior is correctly predicted by the bound in terms of the partial sums \eqref{b_12} but not the bound in terms of the inputs \eqref{b_13}, demonstrating the importance of error expressions involving the
partial sums.
| 139,284
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TITLE: Proof: $x_{1-n}$ linear dependant, w alternating multilinearform $\Rightarrow$ $w(x_1,...,x_n)=0$
QUESTION [1 upvotes]: Let $F$ be a field and $X$ a $F$-linear Space with $dim_FX=n\in\mathbb{N}$.
Let $w$ be an alternating multilinearform on $X$ and let $x_1,\cdots ,x_n\in X$ be linear dependant. Show that $w(x_1,\cdots ,x_n)=0$.
Now from wikipedia: "if the same vector is 2 times inserted in a ml form then it becomes 0"
Can someone provide me good definitions to work here?
And isn't that the reason why the determinante works in the first place?
REPLY [1 votes]: Since $\{x_1,\dots,x_n\}$ is linearly dependent there exists $a_1,\dots,a_n\in F$ not all $0$, such that $a_1x_1+\dots+a_nx_n=0$. Therefore if $a_i$ is the scalar different than $0$ we have $x_i=-\frac{a_1}{a_i}x_1-\dots-\frac{a_n}{a_i}x_n$. Evaluating $w$ and using the statement you found in wikipedia one has
$$w(x_1,\dots,x_n)=w(x_1,\dots,x_i,\dots,x_n)=w(x_1,\dots,-\frac{a_1}{a_i}x_1-\dots-\frac{a_n}{a_i}x_n,\dots,x_n)=-\frac{a_1}{a_i}w(x_1,\dots,x_1,\dots,x_n)-\dots-\frac{a_n}{a_i}w(x_1,\dots,x_n,\dots,x_n)=-0-\dots-0=0$$
| 61,399
|
Advertising and Promotions Managers Career
Career Description
Plan, direct, or coordinate advertising policies and programs or produce collateral materials, such as posters, contests, coupons, or give-aways, to create extra interest in the purchase of a product or service for a department, an entire organization, or on an account basis.
What Job Titles Advertising and Promotions Managers Might Have
- Advertising Director
- Advertising Manager
- Classified Advertising Manager
- Promotions Director
What Advertising and Promotions Managers Do
-.
What Advertising and Promotions Managers Should Be Good At
-).
- Near Vision - The ability to see details at close range (within a few feet of the observer).
- Inductive Reasoning - The ability to combine pieces of information to form general rules or conclusions (includes finding a relationship among seemingly unrelated events).
What Advertising and Promotions Managers Should Be Interested In
What Advertising and Promotions Managers.
This page includes information from O*NET OnLine by the U.S. Department of Labor, Employment and Training Administration (USDOL/ETA). Used under the CC BY 4.0 license.
| 304,352
|
TITLE: Infinitary logic doesn't have (finite) Robinson property: a counterexample?
QUESTION [3 upvotes]: Given a standard definition of an abstract logic $\mathcal{L}_A$ (cfr. Barwise & Feferman, Model-theoretic Logics, Springer-Verlag, 1985), let $E_A(\sigma)$ be the class of $\mathcal{L}_A$-sentences for a signature $\sigma$. Two $\sigma$-structures are $\mathcal{L}_A$-equivalent iff they satisfy the same sentences in $E_A(\sigma)$.
The finite Robinson property is defined as follows: given two signatures $\sigma$ and $\sigma'$, for all $\varphi \in E_A(\sigma)$, $\varphi' \in E_A(\sigma')$ and $\Psi \subseteq E_A(\sigma \cap \sigma')$, if $\Psi \cup \{\varphi\}$ and $\Psi \cup \{\varphi'\}$ are both satisfiable and every two $(\sigma \cap \sigma')$-models of $\Psi$ are $\mathcal{L}_A$-equivalent, then $\Psi \cup \{\varphi, \varphi'\}$ is satisfiable.
Let $\mathcal{L}_{\omega_1\omega}$ the abstract logic obtained from first-order logic allowing countably-long disjunctions (hence also conjunctions). An exercise in Keisler's Model Theory for Infinitary Logic: Logic with Countable Conjunctions
and Finite Quantifiers (North Holland Publishing Company, 1971) says that $\mathcal{L}_{\omega_1\omega}$ has finite Robinson property if $\Psi$ in the above definition is countable, otherwise not. So it exists an uncountable $\Psi$ which is a counterexample to the property. Any suggestion?
REPLY [3 votes]: Let $\sigma = \{P,(c_\alpha)_{\alpha<\omega_1},f\}$, and let $\sigma' = \{P,(c_\alpha)_{\alpha<\omega_1},(d_n)_{n<\omega}\}$. Here $P$ is a unary relation symbol, $f$ is a unary function symbol, and the $c_\alpha$ and $d_n$ are constant symbols. Then $\sigma\cap \sigma' = \{P,(c_\alpha)_{\alpha<\omega_1}\}$ Now define:
\begin{align*}\Psi &= \{P(c_\alpha)\land P(c_\beta)\land c_\alpha\neq c_\beta\mid \alpha<\beta<\omega_1\}\cup \{\exists x_1\dots\exists x_n(\bigwedge_{i=1}^n \lnot P(x_i)\land \bigwedge_{i\neq j} x_i\neq x_j)\mid n<\omega\}\\
\varphi &: (\forall x\, (P(x)\rightarrow \lnot P(f(x)))\land (\forall x\forall y\, (f(x) = f(y)\rightarrow x = y))\\
\varphi' &: \forall x\, (\lnot P(x)\rightarrow \bigvee_{n<\omega} (x = d_n))
\end{align*}
A model of $\Psi$ has $\omega_1$-many distinct elements named by constants and satisfying $P$ (as well as possibly other elements satisfying $P$), and infinitely many elements satisfying $\lnot P$. I claim that any two such models, say $M$ and $N$, are $\mathcal{L}_{\omega_1,\omega}(\sigma\cap \sigma')$-equivalent. Since any sentence of $\mathcal{L}_{\omega_1,\omega}$ only mentions countably many symbols, it suffices to show that for any countable signature $\sigma^*\subseteq (\sigma\cap \sigma')$, the reducts $M|_{\sigma^*}$ and $N|_{\sigma^*}$ are $\mathcal{L}_{\omega_1,\omega}(\sigma^*)$-equivalent. Now $M|_{\sigma^*}$ and $N|_{\sigma^*}$ consist of countably-many distinct elements named by constants and satisfying $P$, infinitely many other elements satisfying $P$, and infinitely many elements satisfying $\lnot P$. By the infinite Ehrenfeucht-Fraïssé game, $M|_{\sigma^*}$ and $N|_{\sigma^*}$ are $L_{\infty,\omega}(\sigma^*)$-equivalent.
A model of $\Psi\cup \{\varphi\}$ is a model $M$ of $\Psi$ together with an injective function $f\colon M\to M$ which maps $P$ into $\lnot P$. This is satisfiable, e.g. by taking $P$ to be $\omega_1$, with $c_\alpha = \alpha$, taking $\lnot P$ to be a disjoint set $X$ of size $\aleph_1$, and taking $f = g\cup g^{-1}$, where $g$ is a bijection $X\to \omega_1$.
A model of $\Psi\cup \{\varphi'\}$ is a model $M$ of $\Psi$ such that every element of $\lnot P$ is named by some constant $d_n$. This is satisfiable, e.g. by taking $P$ to be $\omega_1$, with $c_\alpha = \alpha$, taking $\lnot P$ to be a disjoint countably infinite set, each element of which is the interpretation of one of the constants $d_n$.
But $\Psi\cup \{\varphi,\varphi'\}$ is not satisfiable, because $\varphi$ forces $\lnot P$ to be uncountable, while $\varphi'$ forces $\lnot P$ to be countable.
| 3,693
|
Trekking tours
Reccomended to: travellers that are looking for an adventure, passionate about nature and exploration. The trekking tours are not all the same. Some include only very lights treks, that can be done by anybody without any kind of training or experience, and a basic accomodation, others include many walking days with an important physical effort. Some areas require a specific physical preparation to cope with long periods at elevations of more than 3000m.
Accomodation: depending on the different kinds of trekking there will be different accomodations, from small hotels to local houses and tents or huts.
Kids: the trekking tours are not reccomended to families with kids, you can find some trekking tours that are suited for guests of 14-16 years. We strongly encourage you to ask for a confirmation on the required skill level for the trekking tours.
| 356,286
|
Tell Us What You Want
Sat 23rd Feb 2002
LostCarPark has just added a new forum. This allows users to post their comments and talk to other users.
We would like to know what you want from the site. Please feel free to post your comments and suggestions. If there are sites we should like to or events we should list, please tell us.
We will do our best to act on your suggestions and improve the site for everyone!
Got some News? Submit it to our editors
| 306,371
|
TITLE: $Gal(\mathbb{Q}(\sqrt[4]{2},i)/\mathbb{Q})$, choice of automorphisms
QUESTION [2 upvotes]: I have an exercise that asks for me to find $Gal(\mathbb{Q}(\sqrt[4]{2},i)/\mathbb{Q})$. The irreducible polynomial is $(x^4-2)(x^2+1)$. Its known that the automorphisms will permute the roots of each of the irreducible polynomials. My book picks two permutations:
$$r(\sqrt[4]{2}) = i\sqrt[4]{2}, r(i) = i\\s(\sqrt[4]{2}) = \sqrt[4]{2}, s(i) = -i$$
But I know that $i\sqrt[4]{2}, -\sqrt[4]{2}, -i\sqrt[4]{2}$ are also roots of the polynomial. Why my book picks $\sqrt[4]{2}\to i\sqrt[4]{2}$ specifically?
Its clear to me that I don't need to specify the other automorphisms, because if I have $\phi(\sqrt[4]{2})$ and $\phi(i)$, then $\phi(\pm i\sqrt[4]{2}) = \phi(\pm i)\phi(\sqrt[4]{2})$, right? So I only need to choose one from each irreducible polynomial
REPLY [4 votes]: I feel like the other answer misses an important point.
The map $\sqrt[4]{2}\to -\sqrt[4]{2}$ does not generate the whole permutation group, and therefore cannot be used. This is because this is a map of order $2$. Both of the other non-trivial maps have order 4, and you can choose either of them to be your generator. $i\sqrt[4]{2}$ was probably chosen over $-i\sqrt[4]{2}$ because it was perceived to be simpler, or even just less characters, by the author. Either choice would be correct.
Identifying which roots are appropriate generators is easy. You are looking at something of the form $x^n-m$, so the roots form a cyclic group of order $n$ (to prove this!). Lay them out in $\mathbb{C}$ and connect them to form regular $n$-gon. The permutations of the roots correspond to rotations of this $n$-gon. Rotating in such as way as to move each root over by one will always generate the whole group. In your case, this would be the two permutations I mentioned as working, depending on if the rotation is clockwise ($-i\sqrt[4]{2}$) or counter clockwise ($i\sqrt[4]{2}$). To find all the generating permutations simply fix a $g$ that generates the group and compute $\{g^r|gcd(r,n)=1\}$. There is no reason to choose one element of this set over the others when writing down the generators, it's up to the author's preference.
Note that the proceeding paragraph is only correct because you are examining a cyclic group. When the Galois group of a factor of the polynomial isn't the cyclic group, modifications to what I have said need to be made. However, you're still looking for the generators of the subgroup.
| 92,161
|
TITLE: Generators of Lie groups in physics
QUESTION [1 upvotes]: I asked this question in physics SE here. But I was not satisfied.
As we solving something related to symmetry transformations, we need Lie groups. Also Lie algebra is very important to generate those transformations.
E.g. the generators of $SO(3)$:
$$\begin{pmatrix}0&i&0\\-i&0&0\\0&0&0\end{pmatrix},
\begin{pmatrix}0&0&i\\0&0&0\\-i&0&0\end{pmatrix},
\begin{pmatrix}0&0&0\\0&0&i\\0&-i&0\end{pmatrix}.
$$
Why there is an i in the front.. The thing it is not so wrong if we just consider the Lie algebra as a vector space. But when we use the Lie bracket we will also get those real ones.. But the Lie algebra is three real dimensional.. Now, it is 6.
REPLY [4 votes]: Just a reminder of facts concerning physics usage.
Because quantum physics relies on unitary operators, the symmetry group elements are unitary matrices, so then exponentials of iJ where the Js are Hermitean operators, the standard convention for operators in physics. Commutators of such Hermitean Js are antihermitean, so not in the Lie algebra: in physics structure constants are normally pure imaginary, so they can multiply hermitean generators.
This is all there is to it, as @Dietrich Burde instructs you. The highly unconventional adjoint rep generators J you wrote down conform to this convention, but, of course, as you might be familiar, physics normally uses an equivalent representation for them, this one. Multiplying them by i and exponentiating yields a unitary group element in both cases but real orthogonal group elements only for your pure imaginary (Hermitean) basis. I gather you took the real antisymmetric generators of classical rotations and physics and multiplied them by i to make them Hermitean, assuming somehow physicsists use those, which they rarely would.
| 67,460
|
I woke up this morning thinking (for some reason) about the symphonic black metal band Limbonic Art, and their wonderfully deranged song Behind the Mask Obscure.
Black metal is generally not something I’m into – neither the sound, which varies between someone being strangled on top of a pipe organ and an Airbus A380 digesting a flock of geese, or the attitude, which seems to involve burning down historic churches and stabbing people to death in “self defence”, or at least singing about such things – but symphonic metal does have some redeeming features, mostly because of the symphonic bits.
Behind the Mask Obscure is a fine example of this. It starts with xylophones, drums, bells and strings playing music that wouldn’t sound out of place at a circus parade (albeit one with particularly scary clowns). The music gets progressively heavier and heavier until the guitars, drums and synths kick in, and it sounds like the apocalypse is nigh. The music then devolves into the more typical metal sound of chainsaws dismembering live cattle while someone screeches lyrics about forests, darkness, prophecies, graveyards, being undead, wandering the land and inventing inhumanity (the kind of stuff Tom Riddle would have written in his diary if they’d had Emos in 1943). Finally it pulls itself out of the pit and goes back to the more symphonic sound, wrapping up with a flourish of strings, drums and guitars.
The middle section with its assorted moans and growls really holds no interest for me – it’s the symphonic bits I like. There is one bit of lyrics during the symphony however and it was this I spent much of my morning puzzling over. At about the 1:12 mark a male voice choir (or at least some guy with a heavy echo effect) sings…
In distance from the light, I redeem my Gloria,
In darkness I have sights, a high esteemed fantasia,
…I wonder what that could mean. Let’s take a look at it phrase by phrase shall we?
“In distance from the light” is no problem, it’s just a somewhat pretentious way of saying “In darkness”. The next phrase however is more puzzling. “I redeem my Gloria”. As far as I’m aware the verb “redeem” has two main meanings in modern English.
The first to make up for past transgressions, or make something that’s gone bad, good again. You can for instance redeem yourself by doing good things after a disgrace. You can redeem something or somebody’s reputation by making people think well of it again.
The second meaning is to exchange something for a promised reward. You can redeem a gift voucher that someone has given you. You can redeem tokens cut out of the newspaper for a chance to win a car. You can even (theoretically) redeem money for gold or silver at your nation’s central bank (although they’ll probably shoot you if you dare to try).
Neither of these meanings seems to make much sense when applied to “Gloria”, which usually means a prayer or hymn in praise of God. It seems unlikely for instance that you could hand in a prayer or hymn for the chance to win a Toyota Camry. You could perhaps redeem a poorly written Gloria with a bit of judicious editing, which is probably the meaning we’re going to have to go with.
The second line begins “In darkness I have sights”. Presumably the lyricist (who I imagine like the band is Swedish) got their inflections wrong and actually meant “sighted” – but we’ll work with what we have. There is only one possible grammatically correct interpretation of “In darkness I have sights”, which is “In darkness I possess sights”. “Sights” of course when used as a noun refers to things people see, and more specifically to things people will go out of their way to see, such people taking part in the practice of sightseeing.
So, we may ask, what sights does the singer possess? Well helpfully he explains this in the rest of the line. He has “a high esteemed fantasia”. This could be any number of things, but to me it sounds like some kind of themepark.
So, the final analysis of the lyric works out to “In darkness I edit my poorly written prayer while operating a themepark”.
I’m glad we’ve got that sorted out! 🙂
Not much else to report really. I’ve been wearing in a new pair of Docs (and as a consequence hobbling around like Torgo) and doing some work on a fairly insane post-nuclear mutant skirmish game. I’ve also been spending a fair amount of time wandering around Albany on Google Streetview (now that they’ve launched it in Australia) and avoiding as much of the Olympics as possible. Oh, and reading FreakAngels which has just started it’s second book (I’ll have to get on and update the Google Earth File).
Also, Dragons Landing is back on the air after a length hiatus. I may just send them a voicemail. Or I may not, since I’ve got a rather sore throat and any recording I make will probably sound like black metal.
Anyway, got to walk down to the village and buy some laundry detergent, otherwise I won’t have any clean clothes to wear this week. So long, farewell, auf wiedersehen, goodbye!
(I’m quoting The Sound of Music. Someone, please shoot me).
| 91,360
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\section{Error bounds for \eqref{prog:trs}} \label{sec:l2_TRS}
We now proceed to derive a $\ell_{2}$ error bound for the solution $\gest$ of \eqref{prog:trs}. Following the notation in \cite{CMT18_long}, we can represent any $x \in \mathbb{C}^n$ via $\bar{x} \in \mathbb{R}^{2n}$, where $\bar{x} = [\real(x)^T \ \imag(x)^T]^T$. Moreover, consider the matrix
\begin{eqnarray*}
H = \begin{pmatrix}
\reg L \quad & 0 \\
0 \quad & \reg L
\end{pmatrix} = \reg \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix} \otimes L
\;\; \in \matR^{2n \times 2n}
\end{eqnarray*}
where $\otimes$ denotes the Kronecker product. Then it is easy to check that \eqref{prog:trs} is equivalent to
\begin{align}
\min_{\gbar \in \mathbb{R}^{2n}: \norm{\gbar}_2^2 = n} \gbar^T H \gbar - 2 \gbar^T \zbar \label{prog:trs_real} \tag{$\text{TRS}\matR$}
\end{align}
where $\gest$ is a solution of \eqref{prog:trs} iff $\gbarest = [\real(\gest)^T \ \imag(\gest)^T]^T$ is a solution of \eqref{prog:trs_real}.
The following Lemma from \cite{CMT18_long}, which in turn is a direct consequence of \cite[Lemma 2.4, 2.8]{Sorensen82} (see also \cite[Lemma 1]{Hager01}), characterizes any solution $\gbarest$ of \eqref{prog:trs_real}.
\begin{lemma}[\cite{CMT18_long}] \label{lemma:trs_real_sol_charac}
For any given $\zbar \in \mathbb{R}^{2n}$, $\gbarest$ is a solution to \eqref{prog:trs_real} iff $\norm{\gbarest}_2^2 = n$ and $\exists \mu^{\star}$ such that
(a) $2 H + \mu^{\star} I \succeq 0$ and (b) $(2 H + \mu^{\star} I) \gbarest = 2\zbar$. Moreover, if $2 H + \mu^{\star} I \succ 0$, then the solution is unique.
\end{lemma}
Due to the equivalence of \eqref{prog:trs} and \eqref{prog:trs_real} as discussed above, this readily leads to the following characterization for any solution $\gest$ of \eqref{prog:trs}.
\begin{lemma} \label{lemma:trs_sol_charac}
For any given $z \in \mathbb{C}^n$, $\gest$ is a solution to \eqref{prog:trs} iff $\norm{\gest}_2^2 = n$ and $\exists \mu^{\star}$ such that
(a) $2 \reg L + \mu^{\star} I \succeq 0$ and (b) $(2 \reg L + \mu^{\star} I) \gest = 2z$. Moreover, if $2 \reg L + \mu^{\star} I \succ 0$, then the solution is unique.
\end{lemma}
Note that the above Lemma's do not require any sphere constraint on $\zbar, z$. We now use Lemma \ref{lemma:trs_sol_charac} to derive the following crucial Lemma which states upper and lower bounds on $\mu^{\star}$. The notation $\calN(L)$ is used to denote the null space of $L$, which is the span of $q_n$ since $G$ is connected by assumption.
\begin{lemma} \label{lemma:useful_trs_sol}
For any $z \in \calC_n$ satisfying $z \not\perp$ $\calN(L)$, we have that $\gest = 2(2 \reg L + \mu^{\star} I)^{-1} z$ is the unique solution of \eqref{prog:trs} with $\mu^{\star} \in (0,2]$. Additionally, if $\lambtil$ is an eigenvalue of $L$ satisfying
\begin{equation*}
\frac{1}{\sqrt{n}} \left(\sum_{j:\lambda_j \leq \lambtil} \abs{\dotprod{z}{q_j}}^2\right)^{1/2} > \reg \lambtil
\end{equation*}
then it holds that
\begin{equation*}
\frac{2}{\sqrt{n}} \left(\sum_{j:\lambda_j \leq \lambtil} \abs{\dotprod{z}{q_j}}^2\right)^{1/2} - 2 \reg \lambtil \leq \mu^{\star} \leq 2.
\end{equation*}
\end{lemma}
\begin{proof}
Let us denote
\begin{align} \label{eq:phimu}
\phi(\mu) := \norm{2(2\reg L + \mu I)^{-1} z}_2^2 = 4\sum_{j=1}^{n} \frac{\abs{\dotprod{z}{q_j}}^2}{(2 \reg\lambda_j + \mu)^2}.
\end{align}
If $z \not\perp$ $\calN(L)$ then $0$ is a pole of $\phi$. Hence there exists a unique $\mu^{\star} \in (0,\infty)$ such that $\gest = 2(2 \reg L + \mu^{\star} I)^{-1} z$ is a (unique) solution of \eqref{prog:trs} as it satisfies the conditions in Lemma \ref{lemma:trs_sol_charac}. Since $n = \norm{\gest}_2^2 \leq \frac{4n}{(\mu^{\star})^2}$, we obtain $\mu^{\star} \leq 2$. To obtain the lower bound on $\mu^{\star}$, note that \eqref{eq:phimu} implies
\begin{equation*}
\norm{\gest}_2^2 = n = \norm{2(2\reg L + \mu^{\star} I)^{-1} z}_2^2 \geq \frac{ \sum_{\lambda_j \leq \lambtil} \abs{\dotprod{z}{q_j}}^2}{(\reg \lambtil + \frac{\mu^{\star}}{2})^2}
\end{equation*}
which leads to the stated lower bound on $\mu^{\star}$.
\end{proof}
This is an important Lemma since localizing the value of $\mu^{\star}$ will be key for controlling the $\ell_2$ error bound for \eqref{prog:trs}. Observe that $\lambtil = 0$ always satisfies the conditions of the Lemma since $\abs{\dotprod{z}{q_n}} > 0$.
Next, we bound the the error between $\frac{\gest}{\abs{\gest}}$ and $h$ \emph{for any} $z \in \calC_n$ such that $z \not\perp \calN(L)$.
\begin{lemma} \label{lem:trs_err_bd_det}
For any $z \in \calC_n$ such that $z \not\perp$ $\calN(L)$, and $\lambbar \in [\lambmin, \lambda_1]$, the (unique) solution $\gest = 2(2 \reg L + \mu^{\star} I)^{-1} z$ of \eqref{prog:trs} satisfies the bound
\begin{align*}
\norm{\frac{\gest}{\abs{\gest}} - h}_2^2 \leq \frac{32}{(\mu^{\star})^2} (E_1 + E_2) + 8 \left(\frac{2}{\mu^{\star}} - 1 \right)^2 \left( \abs{\dotprod{h}{q_n}}^2 + \frac{\sum_{j \in \calL_{\lambbar}} \abs{\dotprod{h}{q_j}}^2}{(1+\reg \lambmin)^2}
+ \frac{\smooth_n}{\lambbar (1+\reg \lambbar)^2} \right)
\end{align*}
with $E_1, E_2$ as defined in Lemma \ref{lem:ucqp_lem_err_1}, and $\mu^{\star} \in (0,2]$.
\end{lemma}
\begin{proof}
The proof is along the lines of Lemma \ref{lem:ucqp_lem_err_1} with only few technical differences. Firstly, we can write
\begin{equation*}
\expp{2} \gest - h = \underbrace{\frac{2}{\mu^{\star}} \left(I + \frac{2\reg}{\mu^{\star}} L \right)^{-1} (\expp{2} z - h)}_{e_1} + \underbrace{\frac{2}{\mu^{\star}} \left(I + \frac{2\reg}{\mu^{\star}} L \right)^{-1}h - h}_{e_2}
\end{equation*}
which in conjunction with Proposition \ref{prop:entry_proj} leads to the bound
\begin{equation*}
\norm{\mproj{\gest} - h}_2^2 =
\norm{\mproj{\expp{2} \gest} - h}_2^2 \leq 8(\norm{e_1}_2^2 + \norm{e_2}_2^2).
\end{equation*}
Proceeding identically to the proof of Lemma \ref{lem:ucqp_lem_err_1}, it is easy to verify that $\norm{e_1}_2^2 \leq \frac{4}{(\mu^{\star})^2} E_1$.
In order to bound $\norm{e_2}_2^2$, we begin by expanding $e_2$ as
\begin{equation*}
e_2 = \frac{2}{\mu^{\star}} \left(I + \frac{2\reg}{\mu^{\star}} L \right)^{-1}h - h
= \left(\frac{2}{\mu^{\star}} - 1 \right) \dotprod{q_n}{h} q_n
+ \sum_{j=1}^{n-1} \left(\frac{\frac{2}{\mu^{\star}}}{1 + \frac{2}{\mu^{\star}}\reg \lambda_j} - 1 \right) \dotprod{q_j}{h} q_j.
\end{equation*}
Then using the orthonormality of $q_j$'s, we can bound $\norm{e_2}_2^2$ as follows.
\begin{align*}
\norm{e_2}_2^2
&\leq \left(\frac{2}{\mu^{\star}} - 1 \right)^2 \abs{\dotprod{h}{q_n}}^2 + \sum_{j=1}^{n-1} \left(\frac{(\frac{2}{\mu^{\star}} - 1)^2 + \frac{4\reg^2 \lambda_j^2}{(\mu^{\star})^2}}{(1 + \frac{2}{\mu^{\star}}\reg \lambda_j)^2} \right) \abs{\dotprod{h}{q_j}}^2 \quad (\text{using } (a-b)^2 \leq a^2 + b^2 \text{ for } a,b \geq 0 ) \\
&\leq \left(\frac{2}{\mu^{\star}} - 1 \right)^2 \abs{\dotprod{h}{q_n}}^2 + \sum_{j =1}^{n-1} \left(\frac{(\frac{2}{\mu^{\star}} - 1)^2 + \frac{4\reg^2 \lambda_j^2}{(\mu^{\star})^2}}{(1 + \reg \lambda_j)^2} \right) \abs{\dotprod{h}{q_j}}^2 \quad (\text{using } \mu^{\star} \leq 2) \\
&\leq \left(\frac{2}{\mu^{\star}} - 1 \right)^2 \left( \abs{\dotprod{h}{q_n}}^2 + \frac{ \sum_{j \in \calL_{\lambbar}} \abs{\dotprod{h}{q_j}}^2}{(1+\reg \lambmin)^2} + \frac{ \sum_{j \in \calH_{\lambbar}} \abs{\dotprod{h}{q_j}}^2}{(1+\reg \lambbar)^2} \right) + \frac{4\reg^2}{(\mu^{\star})^2 (1+\reg \lambmin)^2} \sum_{j=1}^{n-1} \lambda_j^2 \abs{\dotprod{h}{q_j}}^2 \\
&\leq \left(\frac{2}{\mu^{\star}} - 1 \right)^2 \left( \abs{\dotprod{h}{q_n}}^2 + \frac{ \sum_{j \in \calL_{\lambbar}} \abs{\dotprod{h}{q_j}}^2}{(1+\reg \lambmin)^2}
+ \frac{ \smooth_n}{\lambbar (1+\reg \lambbar)^2} \right) + \frac{4}{(\mu^{\star})^2} E_2,
\end{align*}
where in the last inequality, we used \eqref{eq:smooth_res1},\eqref{eq:smooth_res2}.
\end{proof}
When $z \in \calC_n$ is generated as in \eqref{eq:noise_mod}, the following Lemma presents a (high probability) lower bound on $\mu^{\star}$ provided $\sigma$ is small and $h$ is sufficiently smooth.
\begin{lemma} \label{lem:lowbd_mu_prob}
Let $z \in \calC_n$ be generated as in \eqref{eq:noise_mod}, then the solution of $\eqref{prog:trs}$ is unique. Moreover, suppose that for any given $k \in [n-1]$ s.t $\lambda_{n-k+1} < \lambda_{n-k}$, the following holds.
\begin{equation} \label{eq:conds_lowbd_mu}
\frac{\smooth_n}{\lambda_{n-k}} \leq \frac{n}{12}, \quad \sigma^2 \leq \frac{1}{48\pi^2}, \quad \frac{24760\log n}{\sqrt{n}} \leq \frac{1}{12} \quad \text{ and } \reg \lambda_{n-k+1} \leq \frac{1}{4}.
\end{equation}
Then with probability at least $1 - \frac{4}{n^2}$, we have that
\begin{equation*}
\frac{\mu^{\star}}{2} \geq 1 - \left(\frac{\smooth_n}{n \lambda_{n-k}} + 4\pi^2\sigma^2 + \frac{24760 \log n}{\sqrt{n}} + \reg \lambda_{n-k+1} \right).
\end{equation*}
\end{lemma}
\begin{proof}
We will lower bound the lower bound estimate of $\mu^{\star}$ from Lemma \ref{lemma:useful_trs_sol} using Proposition \ref{prop:conc_bounds}\ref{prop:conc_bds_1}. Note that $z \in \calC_n$ satisfies $z \not\perp \calN(L)$ a.s. Set $\lambtil = \lambda_{n-k+1}$ in Lemma \ref{lemma:useful_trs_sol}, and let $U$ denote the $n \times k$ matrix consisting of $q_j$'s for $n-k+1 \leq j \leq n$.
Let us first simplify the statement of Proposition \ref{prop:conc_bounds}\ref{prop:conc_bds_1} when $\sigma^2 \leq \frac{1}{8\pi^2}$. Recall from the proof of Lemma \ref{lem:lem:simp_conc} that this implies $1-\expn{8} \in [4 \pi^2 \sigma^2 , 8\pi^2 \sigma^2]$. Then the (magnitude of the) RHS of the bound in Proposition \ref{prop:conc_bounds}\ref{prop:conc_bds_1} can be upper bounded as
\begin{align}
& 4096 \log n +256 \sqrt{6} \pi^2 \sigma^2 \sqrt{k \log n}
+ 11 \log n \left(\norm{U U^T h}_{\infty} + 2\sqrt{2} \pi \sigma \underbrace{\norm{U U^T h}_2}_{\leq \sqrt{n}} \right) \nonumber \\
&\leq 6190 (\log n + \sigma^2 \sqrt{k \log n} + \log n(\norm{UU^T h}_{\infty} + \sigma \sqrt{n})) \nonumber \\
&\leq 6190 \log n (1 + \norm{U U^T h}_{\infty} + 2\sigma \sqrt{n}) \label{eq:imp_up_bound}
\end{align}
where the last inequality uses $\sigma^2 \leq \sigma$ and $k \leq n$.
Plugging \eqref{eq:imp_up_bound} in Proposition \ref{prop:conc_bounds} \ref{prop:conc_bds_1}, we conclude that with probability at least $1-\frac{4}{n^2}$,
\begin{align*}
\sum_{j:\lambda_j \leq \lambda_{n-k+1}} \abs{\dotprod{z}{q_j}}^2
&\geq 2\pi^2 \sigma^2 k + (1-4\pi^2\sigma^2) h^* UU^T h - \underbrace{6190 \log n (1 + \norm{UU^T h}_{\infty} + 2\sigma \sqrt{n})}_{\leq 24760 \sqrt{n} \log n} \\
&\geq (1-4\pi^2\sigma^2) h^* UU^T h - 24760 \sqrt{n} \log n \\
&\geq (1-4\pi^2\sigma^2) \left(n - \frac{\smooth_n}{\lambda_{n-k}} \right) - 24760 \sqrt{n} \log n \quad \text{ ($h^*UU^T h \geq n - \frac{\smooth_n}{\lambda_{n-k}}$, see \eqref{eq:smooth_res1})} \\
&\geq n - \frac{\smooth_n}{\lambda_{n-k}} - 4\pi^2\sigma^2 n - 24760 \sqrt{n} \log n \\
&= n \left(1 - \frac{\smooth_n}{n \lambda_{n-k}} - 4\pi^2\sigma^2 - \frac{24760 \log n}{\sqrt{n}} \right).
\end{align*}
Using \eqref{eq:conds_lowbd_mu}, we have $\frac{\smooth_n}{n \lambda_{n-k}} + 4\pi^2\sigma^2 + \frac{24760 \log n}{\sqrt{n}} \leq \frac{1}{4}$. This leads to the bound
\begin{equation*}
\left(\frac{1}{n} \sum_{j:\lambda_j \leq \lambda_{n-k+1}} \abs{\dotprod{z}{q_j}}^2 \right)^{1/2} \geq 1 - \frac{\smooth_n}{n \lambda_{n-k}} - 4\pi^2\sigma^2 - \frac{24760 \log n}{\sqrt{n}}
\end{equation*}
which in conjunction with Lemma \ref{lemma:useful_trs_sol} leads to the stated bound on $\mu^{\star}$. In particular, the conditions in \eqref{eq:conds_lowbd_mu} ensure $\mu^{\star} \geq 1$.
\end{proof}
Using Lemmas \ref{lem:trs_err_bd_det} and \ref{lem:lowbd_mu_prob}, we arrive at the following (high probability) bound on the error $\norm{\frac{\gest}{\abs{\gest}} - h}_2^2$ for the solution $\gest$ of $\eqref{prog:trs}$.
\begin{theorem} \label{thm:trs_err_bd_prob}
Let $z \in \calC_n$ be generated as in \eqref{eq:noise_mod}, then the solution $\gest$ of \eqref{prog:trs} is unique. For any given $k \in [n-1]$ s.t $\lambda_{n-k+1} < \lambda_{n-k}$, and any $\lambbar \in [\lambmin, \lambda_1]$ with the choice $\reg = (\frac{4\pi^2 \sigma^2 n}{\triangle \smooth_n \lambbar^2})^{1/4}$, suppose that the following conditions are satisfied.
\begin{enumerate} [label=\upshape(\roman*)]
\item\label{lem:trs_err_bd_cond1} $\smooth_n \leq \min\set{\frac{n \lambda_{n-k}}{12}, \frac{n \lambbar}{2}}$, and
\item\label{lem:trs_err_bd_cond2} $286 \left(\frac{\log n}{\sqrt{n}} \right)^{1/2} \leq \sigma \leq \min\set{\frac{1}{4\sqrt{3} \pi},\frac{\lambbar}{16 \lambda_{n-k+1}^2} \sqrt{\frac{\triangle \smooth_n}{4\pi^2 n}}}$.
\end{enumerate}
Then with probability at least $1 - \frac{8}{n^2}$, the solution $\gest \in \mathbb{C}^n$ of \eqref{prog:trs} satisfies
\begin{align} \label{eq:trs_prob_err_bd}
\norm{\frac{\gest}{\abs{\gest}} - h}_2^2
&\leq C_1 \frac{\sigma}{\lambbar} \left( \sqrt{\triangle \smooth_n n} + \frac{n^{3/2} \lambda_{n-k+1}^2}{\sqrt{\triangle \smooth_n}} \right)
+ C_2 \sigma^2(1 + \abs{\calL_{\lambbar}} + \sqrt{(1 + \abs{\calL_{\lambbar}}) \log n}) \\ \nonumber
&+ C_3 \sigma^4 n + C_4 \log n + C_5 \frac{\smooth_n^2}{n \lambda_{n-k}^2}
\end{align}
where $C_1 = 288\pi$, $C_2 = 396160$, $C_3 = 230400$, $C_4 = 262144$, $C_5 = 144$.
\end{theorem}
\begin{proof}
We simply combine Lemmas \ref{lem:trs_err_bd_det} and \ref{lem:lowbd_mu_prob}. To this end, recall that the error bound in Theorem \ref{thm:ucqp_prob_bd} is a bound on the term $8(E_1 + E_2)$. This means that if $\frac{72\log n}{\pi \sqrt{n}} \leq \sigma \leq \frac{1}{2\sqrt{2} \pi}$, then for the stated choice of $\reg$, we have with probability at least $1-\frac{4}{n^2}$,
\begin{equation} \label{eq:first_err_term_trs_hp}
32(E_1 + E_2) \leq 288 \pi \frac{\sigma}{\lambbar} \sqrt{\triangle \smooth_n n} + 396160 \sigma^2(1 + \abs{\calL_{\lambbar}} + (1 + \abs{\calL_{\lambbar}})\sqrt{\log n}) + 262144 \log n.
\end{equation}
The conditions in Lemma \ref{lem:lowbd_mu_prob} imply in particular that $\mu^{\star} \geq 1$, hence \eqref{eq:first_err_term_trs_hp} is a bound on $\frac{32(E_1 + E_2)}{\mu^{\star}}$. This takes care of the first error term in the bound in Lemma \ref{lem:trs_err_bd_det}.
In order to bound the second term therein, observe that if $\smooth_n \leq \frac{n \lambbar}{2}$, then
\begin{equation*}
\abs{\dotprod{h}{q_n}}^2 + \frac{\sum_{j \in \calL_{\lambbar}} \abs{\dotprod{h}{q_j}}^2}{(1+\reg \lambmin)^2}
+ \frac{\smooth_n}{\lambbar (1+\reg \lambbar)^2} \leq n + \frac{n}{2} = \frac{3n}{2}.
\end{equation*}
Also, condition \ref{lem:trs_err_bd_cond2} for $\sigma$ implies $24760 \frac{\log n}{\sqrt{n}} \leq \frac{3\sigma^2}{\pi^2} (\leq \frac{1}{12})$. Note that the requirement $\sigma \geq 286(\frac{\log n}{\sqrt{n}})^{1/2}$ is stricter than $\sigma \geq \frac{72 \log n}{\pi \sqrt{n}}$. Given these observations, we can bound the second term in the bound of Lemma \ref{lem:trs_err_bd_det} as follows.
\begin{align}
& 8 \left(\frac{2}{\mu^{\star}} - 1 \right)^2 \left( \abs{\dotprod{h}{q_n}}^2 + \frac{\sum_{j \in \calL_{\lambbar}} \abs{\dotprod{h}{q_j}}^2}{(1+\reg \lambmin)^2}
+ \frac{\smooth_n}{\lambbar (1+\reg \lambbar)^2} \right) \nonumber \\
&\leq 48n \left(\frac{\smooth_n}{n \lambda_{n-k}} + 4\pi^2\sigma^2 + \frac{24760 \log n}{\sqrt{n}} + \reg \lambda_{n-k+1} \right)^2 \nonumber \\
&\leq
48n \left(\frac{\smooth_n}{n \lambda_{n-k}} + 4\pi^2\sigma^2 + \frac{3\sigma^2}{\pi^2} + \reg \lambda_{n-k+1} \right)^2 \nonumber \\
&\leq 48n \left(\frac{\smooth_n}{n \lambda_{n-k}} + 40\sigma^2 + \reg \lambda_{n-k+1} \right)^2 \nonumber \\
&\leq 144 n \left( \frac{\smooth_n^2}{n^2 \lambda_{n-k}^2} + 1600 \sigma^4
+ \left(\frac{4\pi^2 \sigma^2 n}{\triangle \smooth_n \lambbar^2} \right)^{1/2} \lambda_{n-k+1}^2 \right). \label{eq:sec_err_term_trs_hp}
\end{align}
Plugging \eqref{eq:first_err_term_trs_hp} and \eqref{eq:sec_err_term_trs_hp} in Lemma \ref{lem:trs_err_bd_det} then readily yields the stated bound in the Theorem.
\end{proof}
The following Corollary provides a simplification of Theorem \ref{thm:trs_err_bd_prob} and is directly obtained by considering $k=1$ since $\lambda_n < \lambda_{n-1} = \lambmin$.
\begin{corollary} \label{cor:trs_high_prob_simp_err}
Let $z \in \calC_n$ be generated as in \eqref{eq:noise_mod}, then the solution $\gest$ of \eqref{prog:trs} is unique. For given $\lambbar \in [\lambmin, \lambda_1]$ with the choice $\reg = (\frac{4\pi^2 \sigma^2 n}{\triangle \smooth_n \lambbar^2})^{1/4}$, suppose that
\begin{equation*}
\smooth_n \leq \frac{n \lambmin}{12} \text{ and } 286 \left(\frac{\log n}{\sqrt{n}} \right)^{1/2} \leq \sigma \leq \frac{1}{4\sqrt{3} \pi}.
\end{equation*}
Then with probability at least $1 - \frac{8}{n^2}$, the solution $\gest \in \mathbb{C}^n$ of \eqref{prog:trs} satisfies
\begin{align*}
\norm{\frac{\gest}{\abs{\gest}} - h}_2^2
\leq C_1 \frac{\sigma}{\lambbar} \sqrt{\triangle \smooth_n n} +
+ C_2 \sigma^2 \left(1 + \abs{\calL_{\lambbar}} + \sqrt{(1 + \abs{\calL_{\lambbar}}) \log n} \right)
+ C_3 \sigma^4 n + C_4 \log n + C_5 \frac{\smooth_n^2}{n \lambmin^2}
\end{align*}
where the constants $C_1,\dots,C_5$ are as in Theorem \ref{thm:trs_err_bd_prob}.
\end{corollary}
We are now in a position to derive conditions under which \eqref{prog:trs} provably denoises $z$ with high probability. We begin with the following Theorem which provides these conditions in their full generality.
\begin{theorem} \label{thm:trs_prob_denoise}
Let $z \in \calC_n$ be generated as in \eqref{eq:noise_mod}, then the solution $\gest$ of \eqref{prog:trs} is unique. With constants $C_1, \dots, C_5$ as in Theorem \ref{thm:trs_err_bd_prob}, for any $\varepsilon \in (0,1)$, given $k \in [n-1]$ s.t $\lambda_{n-k+1} < \lambda_{n-k}$ and $\lambbar \in [\lambmin, \lambda_1]$ with the choice $\reg = (\frac{4\pi^2 \sigma^2 n}{\triangle \smooth_n \lambbar^2})^{1/4}$, suppose that the following conditions are satisfied.
\begin{enumerate}[label=\upshape(\roman*)]
\item\label{lem:trs_denoise_cond1} $\smooth_n \leq \min\set{\frac{n \lambda_{n-k}}{12}, \frac{n \lambbar}{2}}$, and $1 + \abs{\calL_{\lambbar}} + \sqrt{(1 + \abs{\calL_{\lambbar}}) \log n} \leq \frac{\pi^2}{5 C_2}\varepsilon n$.
\item\label{eq:trs_denoise_cond2} $\sigma \leq
\min\set{\frac{\pi \sqrt{\varepsilon}}{\sqrt{5 C_3}},\frac{\lambbar}{16 \lambda_{n-k+1}^2} \sqrt{\frac{\triangle \smooth_n}{4\pi^2 n}}}$ and
\begin{align*}
\sigma \geq \max \left\{286 \left(\frac{\log n}{\sqrt{n}} \right)^{1/2}, \frac{\sqrt{5 C_5}}{\pi} \left(\frac{\smooth_n}{n \lambda_{n-k} \sqrt{\varepsilon}} \right), \sqrt{\left(\frac{5C_4}{\varepsilon \pi^2} \right) \frac{\log n}{n}}, \frac{5C_1}{\pi^2 \varepsilon \lambbar} \left(\sqrt{\frac{\triangle \smooth_n}{n}} + \lambda_{n-k+1}^2 \sqrt{\frac{n}{\triangle \smooth_n}} \right) \right\}.
\end{align*}
\end{enumerate}
Then with probability at least $1 - \frac{10}{n^2}$, the solution $\gest \in \mathbb{C}^n$ of \eqref{prog:trs} satisfies
\begin{equation} \label{eq:trs_eps_den_bd_prob}
\norm{\frac{\gest}{\abs{\gest}} - h}_2^2 \leq \varepsilon \norm{z - h}_2^2.
\end{equation}
\end{theorem}
\begin{proof}
Recall from Lemma \ref{lem:lem:simp_conc} \ref{lem:simp_conc_item3}, that $\norm{z-h}_2^2 \geq \pi^2 \sigma^2 n$ w.p at least $1-\frac{2}{n^2}$. Conditioning on the intersection of this event and the event in Theorem \ref{thm:trs_err_bd_prob}, it suffices to ensure that
the bound in \eqref{eq:trs_prob_err_bd} is less than or equal to $\pi^2\sigma^2 n \varepsilon$. This in turn is ensured provided each term in the RHS of \eqref{eq:trs_prob_err_bd} is less than or equal to $\varepsilon \frac{\pi^2 \sigma^2 n}{5}$.
Combining the resulting conditions with those in Theorem \ref{thm:trs_err_bd_prob} yields the statement of the Theorem.
\end{proof}
The following simplification of Theorem \ref{thm:trs_prob_denoise} is obtained for $k=1$, as was done in Corollary \ref{cor:trs_high_prob_simp_err}.
\begin{corollary} \label{cor:trs_high_prob_simp_den}
Let $z \in \calC_n$ be generated as in \eqref{eq:noise_mod}, then the solution $\gest$ of \eqref{prog:trs} is unique. With constants $C_1,\dots, C_5$ as in Theorem \ref{thm:trs_err_bd_prob}, for any $\varepsilon \in (0,1)$ and $\lambbar \in [\lambmin, \lambda_1]$ with the choice $\reg = (\frac{4\pi^2 \sigma^2 n}{\triangle \smooth_n \lambbar^2})^{1/4}$, suppose that the following conditions are satisfied.
\begin{enumerate}[label=\upshape(\roman*)]
\item\label{cor:trs_denoise_simp_cond1} $\smooth_n \leq \frac{n \lambmin}{12}$ and $1 + \abs{\calL_{\lambbar}} + \sqrt{(1 + \abs{\calL_{\lambbar}}) \log n} \leq \frac{\pi^2}{5 C_2}\varepsilon n$.
\item\label{cor:trs_denoise_simp_cond2}
\begin{align*}
\max\set{286 \left(\frac{\log n}{\sqrt{n}} \right)^{1/2}, \frac{\sqrt{5 C_5}}{\pi} \left(\frac{\smooth_n}{n \lambmin \sqrt{\varepsilon}} \right), \sqrt{\left(\frac{5C_4}{\varepsilon \pi^2} \right) \frac{\log n}{n}},
\frac{5 C_1}{\pi^2 \varepsilon \lambbar} \sqrt{\frac{\triangle \smooth_n}{n}}} \leq \sigma \leq \frac{\pi \sqrt{\varepsilon}}{\sqrt{5 C_3}}.
\end{align*}
\end{enumerate}
Then with probability at least $1 - \frac{10}{n^2}$, the solution $\gest \in \mathbb{C}^n$ of \eqref{prog:trs} satisfies \eqref{eq:trs_eps_den_bd_prob}.
\end{corollary}
Finally, as done previously for \eqref{prog:ucqp}, it will be instructive to translate Theorem \ref{thm:trs_prob_denoise} for the special cases $G = K_n$, $S_n$ or $P_n$. This is stated below using the simplified version in Corollary \ref{cor:trs_high_prob_simp_den}.
\begin{corollary} \label{cor:trs_den_prob_spec_graphs}
Let $z \in \calC_n$ be generated as in \eqref{eq:noise_mod} and $\varepsilon \in (0,1)$.
\begin{enumerate}[label=\upshape(\roman*)]
\item\label{cor:trs_den_prob_eps_Kn} {($G = K_n$)} Suppose $\frac{n}{\sqrt{\log n}} \gtrsim \frac{1}{\varepsilon}$, $\smooth_n \lesssim n^2$ and $\max\set{\frac{\sqrt{\smooth_n}}{n \varepsilon} , \frac{\smooth_n}{n^2 \sqrt{\varepsilon}} , (\frac{\log n}{\sqrt{n}})^{1/2}, (\frac{\log n}{\varepsilon n})^{1/2} } \lesssim \sigma \lesssim \sqrt{\varepsilon}$. If $\reg \asymp (\frac{\sigma^2}{n^2 \smooth_n })^{1/4}$, then the (unique) solution $\gest$ of \eqref{prog:trs} satisfies \eqref{eq:trs_eps_den_bd_prob} w.h.p.
\item\label{cor:trs_den_prob_eps_Sn} {($G = S_n$)} Suppose $\frac{n}{\sqrt{\log n}} \gtrsim \frac{1}{\varepsilon}$, $\smooth_n \lesssim n$ and
$\max\set{\frac{\sqrt{\smooth_n}}{\varepsilon} , \frac{\smooth_n}{n \sqrt{\varepsilon}}, (\frac{\log n}{\sqrt{n}})^{1/2}, (\frac{\log n}{\varepsilon n})^{1/2}} \lesssim \sigma \lesssim \sqrt{\varepsilon}$. If $\reg \asymp (\frac{\sigma^2}{\smooth_n })^{1/4}$, then the (unique) solution $\gest$ of \eqref{prog:trs} satisfies \eqref{eq:trs_eps_den_bd_prob} w.h.p.
\item\label{cor:trs_den_prob eps_Pn} {($G = P_n$)} For a given $\theta \in [0,1)$, suppose $n^{\theta} + \sqrt{n^{\theta} \log n} \lesssim \varepsilon n$, $\smooth_n \lesssim \frac{1}{n}$ and
$$\max\set{\frac{n^{\frac{3 -4\theta}{2}}}{\varepsilon} \sqrt{\smooth_n}, \frac{n \smooth_n}{\sqrt{\varepsilon}}, \left(\frac{\log n}{\sqrt{n}} \right)^{1/2}, \left(\frac{\log n}{\varepsilon n} \right)^{1/2}} \lesssim \sigma \lesssim \sqrt{\varepsilon}.$$
If $\reg \asymp (\frac{\sigma^2 n^{5-4\theta}}{\smooth_n })^{1/4}$, then the (unique) solution $\gest$ of \eqref{prog:trs} satisfies \eqref{eq:trs_eps_den_bd_prob} w.h.p.
\end{enumerate}
\end{corollary}
\begin{proof}
Use Corollary \ref{cor:trs_high_prob_simp_den} with $\lambmin, \lambbar$ as in Corollary \ref{cor:ucqp_den_expec}.
\end{proof}
For $K_n$, note that only $k = 1$ meets the requirement of Theorem \ref{thm:trs_prob_denoise} since $\lambda_{n-1} = \cdots = \lambda_{1}$. For $S_n$, the only other possibility (apart from $k=1$) is to choose $k = n-1$, since $\lambda_{2} = 1 < \lambda_1 = n$. But this choice of $k$ leads to a vacuous noise regime due to the appearance of the term $\sqrt{\smooth_n} + \frac{1}{\sqrt{\smooth_n}}$ as a lower bound on $\sigma$.
\begin{remark} \label{rem:trs_exgraph_large_n}
Similarly to Remark \ref{rem:cor_ex_graph_prob_ucqp} for \eqref{prog:ucqp}, we can deduce conditions on the smoothness term $\smooth_n$ which -- when $n \rightarrow \infty$ -- lead to a non-vacuous regime for $\sigma$ of the form $o(1) \leq \sigma \lesssim \sqrt{\varepsilon}$. Here, we will only treat the case where $\varepsilon$ is fixed.
\begin{enumerate}
\item {($G = K_n$)} $\smooth_n = o(n^2)$ suffices.
\item {($G = S_n$)} $\smooth_n = o(1)$ suffices.
\item {($G = P_n$)} $\smooth_n = o(1/n)$ suffices.
\end{enumerate}
\end{remark}
\paragraph{Denoising modulo samples of a function.}
When $G = P_n$, the requirement $\smooth_n = o(1/n)$ is far from satisfactory and suggests that Corollary \ref{cor:trs_high_prob_simp_den} is weak when applied to a path graph. Indeed, when we obtain noisy modulo $1$ samples of a $M$-Lipschitz function $f:[0,1] \rightarrow \matR$, then we have $\smooth_n \asymp \frac{M^2}{n}$ as seen in \eqref{eq:func_quad_var_Bn_1}. Unfortunately, the condition on $\sigma$ in Corollary \ref{cor:trs_den_prob_spec_graphs}\ref{cor:trs_den_prob eps_Pn} becomes vacuous when $\smooth_n \asymp 1/n$. Interestingly, we can handle this smoothness regime by making use of Theorem \ref{thm:trs_prob_denoise} with a careful choice of $k$. This is made possible by the fact that the spectrum of the Laplacian of $P_n$ satisfies the condition $\lambda_{n-k+1} < \lambda_{n-k}$ for each $k \in [n-1]$. Consequently, we obtain the following Corollary of Theorems \ref{thm:trs_err_bd_prob} and \ref{thm:trs_prob_denoise}; its proof is deferred to Appendix \ref{appsec:proof_Pn_res_bet_conds}.
\begin{corollary} \label{cor:Pn_res_bet_conds}
Consider the example from Section \ref{subsec:prob_setup} where we obtain noisy modulo 1 samples of a $M$-Lipschitz function $f:[0,1] \rightarrow \matR$. If $\reg \asymp \left(\frac{\sigma^2 n^{10/3}}{M^2} \right)^{1/4}$ then the following is true for the solution $\gest$ of \eqref{prog:trs}.
\begin{enumerate}
\item If $n \gtrsim \max \set{1,M^2}$ and $(\frac{\log n}{\sqrt{n}})^{1/2} \lesssim \sigma \lesssim \min \set{1, n^{1/3} M}$ then w.h.p,
\begin{equation*}
\norm{\frac{\gest}{\abs{\gest}} - h}_2^2 \lesssim \left(\sigma \left(M + \frac{1}{M} \right) + \sigma^2 \right) n^{2/3} + \sigma^4 n + \log n + \frac{M^4}{n}.
\end{equation*}
\item For $\varepsilon \in (0,1)$,
if $n \gtrsim \max \set{(1/\varepsilon)^3, M^2}$ and
\begin{equation*}
\max\set{\frac{1}{\varepsilon n^{1/3}} \left(M + \frac{1}{M} \right) , \frac{M^2}{n \sqrt{\varepsilon}}, \left(\frac{\log n}{\sqrt{n}} \right)^{1/2}, \left(\frac{\log n}{\varepsilon n} \right)^{1/2}} \lesssim \sigma \lesssim \min\set{\sqrt{\varepsilon} , n^{1/3} M},
\end{equation*}
then $\gest$ satisfies \eqref{eq:ucqp_eps_den_bd_prob} w.h.p.
\end{enumerate}
\end{corollary}
The error bound in Corollary \ref{cor:Pn_res_bet_conds} is visibly worse than that in Corollary \ref{cor:ucqp_den_func_mod} due to the appearance of an additional $\sigma^4 n$ term. For $n$ large enough and $\varepsilon \in (0,1)$ fixed, Corollary \ref{cor:Pn_res_bet_conds} asserts that \eqref{prog:trs} succeeds in denoising in the noise regime $(\frac{\log n}{\sqrt{n}})^{1/2} \lesssim \sigma \lesssim \sqrt{\varepsilon}$. The corresponding noise regime for \eqref{prog:ucqp} is the relatively weaker requirement $\frac{M}{\varepsilon n^{1/3}} \lesssim \sigma \lesssim 1$, as seen from Corollary \ref{cor:ucqp_den_func_mod}.
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I was born the year the Soviets invaded Afghanistan—an invasion that started the international call for “jihad” to liberate the Muslim country from the Oppressors. As an Afghan American, I could see the men in my family struggle with this Islamic duty to defend their home and country. Did they have to take up arms? Was providing safety for their family enough? Should they focus on education and efforts for the next generation? Although women took up arms in small numbers, the onus was on Afghan men to fight. The “jihad” to take up arms and fight the oppressor was inculcated as men’s religious duty, reinforced by the Friday sermons; the community; and the schoolbooks for children in the refugee camps. Sayed Jamaluddin Afghani’s pan-Islamism and the concepts of Dar-ul-Harb verses Dar-ul-Islam resurfaced to evoke men’s responsibility, as the head of their household, to decide their family’s role vis-a-vis the global Ummah context.
Echoes of this religious duty of “jihadi” call to arms still reverberate in Afghanistan and the Muslim world today. And these ideas continue to strengthen and grow in force. The Taliban continue to call men, especially young men, to fulfill their manly duties by joining the “jihad” against injustice and foreign invasion in Afghanistan. The Taliban have taken up similar religious “jihadi” rhetoric of the Afghan mujahideen of earlier days to encourage joining the cause to become martyrs for the rewards of “hoories” (virgin maidens) in heaven.
ISIS/DAESH has taken this to another level. Not only does it promise martyrdom and maidens in the Hereafter, but it also promises immediate gratification in this life. By living in the “Dar ul Islam,” ISIS fighters are rewarded with money, wives, and slave women. They are promised with the glory of fighting the jihad with depictions of manliness, fulfilling the role as the male protector fighting injustice so prevalent not only in Islamic rhetoric but also in Hollywood movies of vigilantes fighting for justice.
As we look for solutions to end violent extremism in our communities, we need to take serious stock not only of the larger worldview of defining an Islamic community but also gender roles. Gender roles and masculinity are drivers in extremist violence. Solutions to violent extremism need to incorporate peaceful masculinities in order to be long term solutions. Religious actors play an important role in defining and reshaping peaceful masculinities as they have in advocating violent masculinities and must be included in the process.
The Prophet Muhammad (pbuh) did domestic chores, including childrearing, alongside his wives. He (pbuh) also took seriously his wives’ roles in politics and education. And yet the predominant conception of Islamic masculinity –the one being promoted in conflict-ridden areas—is of the man who is harsh and violent instead of loving.
Part of the project of redefining masculinities is providing emotional support to men and boys, and religious actors are best-positioned to provide this support. Challenging and reforming these structures supporting violent masculinities will be a long-term endeavor.
And most importantly, policy and programs on masculinities should be pursued in addition to, and not at the expense of, increasing resources and political will to implement commitments under the women, peace and security agenda.
Palwasha Kakar will be presenting at the Muslim Masculinity in an Age of Feminism conference:
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\begin{document}
{\begin{flushleft}\baselineskip9pt\scriptsize
\today\\
\end{flushleft}}
\begin{abstract}
The original Kelly criterion provides a strategy to maximize the
long-term growth of winnings in a sequence of simple Bernoulli bets with
an edge, that is, when the expected return on each bet is positive.
The objective of this work is to
consider more general models of returns and the continuous time,
or high frequency, limits of those models.
\end{abstract}
\maketitle
\section{Introduction}
Consider repeatedly engaging in a game of chance where one side has an edge
and seeks to optimize the betting in a way that ensures maximal long-term
growth rate of the overall wealth. This problem was posed and analyzed
by John Kelly \cite{Kelly56} at Bell Labs in 1956; the solution was
implemented and tested in a variety of setting by a
successful mathematician, gambler, and hedge fund manager Ed Thorp \cite{Thorp06}
over the period from the 60's to the early 00's.
As a motivating example, consider betting on a biased coin
toss where the return $r$ is a random variable with distribution
\begin{equation}
\label{return1}
\mathbb{P}(r=1) = p,\ \ \ \mathbb{P}(r=-1)=1- p;
\end{equation}
in what follows, we refer to this as the {\em simple Bernoulli model}.
The condition to have an edge in this setting becomes $1/2<p\leq 1$ or, equivalently,
\begin{equation}
\label{edge1}
\mathbb{E}[r] =2p-1>0.
\end{equation}
We plan on being able to make a large sequence of bets on this biased coin, resulting in an iid sequence of returns
$\{r_k\}_{k\geq 1}$ with the same distribution as $r$, and ask how much we should bet so as to maximize
long term wealth,
given that we are compounding our returns. Assume we are betting
with a fixed exposure $f$, that is, each bet involves a fixed fraction $f$ of the overall
wealth, and $f \in [0,1]$. Practically, $f \geq 0$ means {\bf no shorting} and $f \leq 1$ means {\bf no leverage}, which we refer to as the {\bf NS-NL} condition.
Then, starting with the initial amount $W_0$,
the total wealth at time $n=1,2,3,\ldots$ is the following function of $f$:
\begin{equation*}
W_n^f = W_0 \prod_{k=1}^n\big(1 + f r_k\big).
\end{equation*}
For the long-term compounder wishing to maximize their long term wealth, a natural and equivalent goal would be to find the strategy $f=f^*$ maximizing the long-term growth rate
\begin{equation}
\label{rate-dt}
g_r(f):= \lim_{n\rightarrow \infty}\frac{1}{n} \ln \frac{W_n^f}{W_0}.
\end{equation}
By direct computation,
$$
g_r(f) =
\lim_{n\rightarrow \infty}\frac{1}{n}\sum_{k=1}^n \ln(1+fr_k)
= \mathbb{E}\ln(1+fr)=p\ln(1+f) + (1-p)\ln(1-f),
$$
where the second equality follows by the law of large numbers, and therefore, after solving $g_r'(f^*)=0$,
\begin{equation}
\label{optimal1}
f^* = 2p-1,\ \
\max_{f \in [0,1]} g_r(f)= g_r(f^*)=p\ln \frac{p}{1-p}+(2-p)\ln(2-2p);
\end{equation}
note that the edge condition \eqref{edge1} ensures that $f^*$ is an admissible strategy and $g_r(f^*)>0$.
For more discussions of this result see \cite{Thorp06}.
Our objective in this paper is to derive analogues of \eqref{optimal1}
in the following situations:
\begin{enumerate}
\item the distribution of returns is a more general random variable;
\item the compounding is continuous in time;
\item the compounding is high frequency, leading to a continuous-time limit.
\end{enumerate}
In particular, we consider several scenarios when
the returns are described by L\'evy processes, which addresses
some of Thorp's questions regarding fat-tailed distributions
in finance \cite{Thorp08}.
In what follows, we write $\xi\wc\eta$ to indicate equality in distribution for
two random variables, and $X\wcL Y$ to indicate equality in law (as function-valued random elements) for two random processes. For $x>0$, $\lfloor x \rfloor$
denotes the largest integer less than or equal to $x$.
To simplify the notations, we always assume that $W_0=1$.
\section{Discrete Compounding: General Distribution of Returns}
Assume that the returns on each bet are independent random variables
$r_k,\ k\geq 1,$ with the same distribution as a given random variable $r$,
and let
\begin{equation}
\label{wealth2}
W_n^f = \prod_{k=1}^n\big(1 + f r_k\big)
\end{equation}
denote the corresponding wealth process. We also keep the NS-NL
condition on admissible strategies: $f\in [0,1]$.
For the wealth process $W^f$ to be well-defined, we need the random
variable $r$ to have the following properties:
\begin{align}
\label{r1}
&\mathbb{P}(r\geq -1)=1;\\
\label{r2}
&\mathbb{P}(r>0)>0,\ \mathbb{P}(r<0)>0;\\
\label{r3}
&\mathbb{E}|\ln(1+r)|<\infty.
\end{align}
Condition \eqref{r1} quantifies the idea that a loss in a bet should not be
more than $100\%$. Condition \eqref{r2} is basic non-degeneracy: both gains and losses are possible.
Condition \eqref{r3} is a minimal requirement to define the
long-term growth rate of the wealth process.
The key object in this section will be the function
\begin{equation}
\label{F(f)}
g_r(f)=\mathbb{E}\ln(1+fr).
\end{equation}
In particular, the following result shows that $g_r(f)$ is the long term
growth rate of the wealth process $W^f$.
\begin{prop}
\label{prop:LLN}
If \eqref{r1} and \eqref{r3} hold and $g_r(f)\not=0$,
then, for every $f\in[0,1]$, the
wealth process $W^f$ has an asymptotic representation
\begin{equation}
\label{eq:LLN-g1}
W^f_n=\exp\Big( n g_r(f)\big(1+\varepsilon_n\big)\Big),
\end{equation}
where
\begin{equation}
\label{asymp0}
\lim_{n\to \infty} \varepsilon_n=0
\end{equation}
with probability one.
\end{prop}
\begin{proof}
By \eqref{wealth2}, we have \eqref{eq:LLN-g1} with
\begin{equation}
\label{err0}
\varepsilon_n=\frac{1}{ng_r(f)}\sum_{k=1}^n\bigg( \ln(1+fr_k)-g_r(f)\bigg),
\end{equation}
and then \eqref{asymp0} follows by \eqref{r3} and the strong law of
large numbers.
\end{proof}
A stronger version of \eqref{r3} leads to a more detailed asymptotic
of $W_n^f$.
\begin{thm}
\label{th:asympt1}
Assume that \eqref{r1} holds and
\begin{equation}
\label{r3-3}
\mathbb{E}|\ln(1+r)|^2<\infty.
\end{equation}
Then
then, for every $f\in[0,1]$, the
wealth process $W^f$ has an asymptotic representation
\begin{equation}
\label{eq:LLN-g2}
W^f_n=\exp\Big( n g_r(f)+\sqrt{n}\big(\sigma_r(f)\zeta_n+
\epsilon_n\big)\Big),
\end{equation}
where $\zeta_n,\ n\geq 1, $ are standard Gaussian random variables,
\begin{equation*}
\sigma_r(f)=\Big(\mathbb{E}\big[\ln^2(1+fr)\big]-g_r^2(f)\Big)^{1/2},
\end{equation*}
and
\begin{equation*}
\lim_{n\to \infty}\epsilon_n=0
\end{equation*}
in probability.
\end{thm}
\begin{proof}
With $\varepsilon_n$ from \eqref{err0}, the result follows by the Central Limit Theorem:
$$
ng_r(f)\,\varepsilon_n= \sqrt{n}\left(\frac{1}{\sqrt{n}}
\sum_{k=1}^n\bigg(\ln(1+fr_k)-g_r(f)\bigg)\right)=
\sqrt{n}\big(\sigma_r(f)\zeta_n+
\epsilon_n\big).
$$
\end{proof}
Because the Central Limit Theorem gives convergence in distribution, the random variables
$\zeta_n$ in \eqref{eq:LLN-g2} can indeed depend on $n$. Additional assumptions about the
distribution of $r$ \cite[Theorem 1]{Zolotarev-AE}
lead to higher-order asymptotic expansions and a possibility to have
$\lim_{n\to \infty}\epsilon_n=0$ with probability one.
The following properties of the function $g_r$ are immediate
consequences of the definition and the assumptions \eqref{r1}--\eqref{r3}:
\begin{prop}
\label{prop:BasicF}
The function $f\mapsto g_r(f)$ is continuous on the closed interval
$[0,1]$ and infinitely differentiable in $(0,1)$. In particular,
\begin{equation}
\label{DF}
\frac{dg_r}{df}(f)=\mathbb{E}\left[\frac{r}{1+fr}\right],\ \
\frac{d^2g_r}{df^2}(f)=-\mathbb{E}\left[\frac{r^2}{(1+fr)^2}\right]<0.
\end{equation}
\end{prop}
\begin{cor}
\label{cor1}
The function $g_r$ achieves its maximal value on
$[0,1]$ at a point $f^*\in [0,1]$ and $g_r(f^*)\geq 0$.
If $ g_r(f^*)> 0$, then $f^*$ is unique.
\end{cor}
\begin{proof}
Note that $g_r(0)=0$ and, by \eqref{DF}, the function $g_r$ is strictly concave
(or convex up) on $[0,1]$.
\end{proof}
While concavity of $g_r$ implies that $g_r$ achieves a unique global maximal
value at a point $f^{**}$, it is possible that the domain of
the function $g_r$ is bigger than
the interval $[0,1]$ and $f^{**}\notin [0,1]$. A simple way to exclude the possibility $f^{**}<0$ is to consider returns
$r$ that are not bounded from above: $\mathbb{P}(r>c)>0$ for all $c>0$:
in this case, the function $g_r(f)=\mathbb{E}\ln(1+fr)$ is not defined for $f<0$.
Similarly, if $\mathbb{P}(r<-1+\delta)>0$ for all $\delta>0$, then the
function $g_r$ is not defined for $f>1$, excluding the possibility $f^{**}>1.$
Below are more general sufficient conditions to ensure that the point
$f^*\in [0,1]$ from Corollary \ref{cor1} is the point of global maximum of $g_r$:
$f^*=f^{**}$.
\begin{prop}
\label{prop:glob1}
If
\begin{align}
\label{global1-l}
&\lim_{f\to 0+} \mathbb{E}\left[\frac{r}{1+fr}\right]>0 \ \
\ \ \ {\rm and}\\
\label{global1-r}
&\lim_{f\to 1-} \mathbb{E}\left[\frac{r}{1+fr}\right]<0,
\end{align}
then there is a unique $f^*\in (0,1)$ such that
$$
g_r(f)< g_r(f^*)
$$
for all $f$ in the domain of $g_r$.
\end{prop}
\begin{proof}
Together with the intermediate value theorem, conditions \eqref{global1-l}
and \eqref{global1-r} imply that there is a unique $f^*\in (0,1)$ such that
$$
\frac{dg_r}{df}(f^*)=0.
$$
It remains to use strong concavity of $g_r$.
\end{proof}
Because $r\geq -1$, the expected value $\mathbb{E}[r]$ is always defined,
although $\mathbb{E}[r]=+\infty$ is a possibility. Thus, by \eqref{DF},
condition \eqref{global1-l} is equivalent to the intuitive idea of an edge:
$$
\mathbb{E}[r]>0,
$$
which, similar to \eqref{edge1}, guarantees that $g_r(f)>0$ for some $f\in (0,1)$.
Condition \eqref{global1-r} can be written as
$$
\mathbb{E}\left[\frac{r}{1+r}\right]<0,
$$
with the convention that the left-hand side can be $-\infty$. This condition
does not appear in the simple Bernoulli model, but
is necessary in general, to ensure that the edge is not too big and
leveraged gambling ($f^*>1$) does not lead to an optimal strategy.
As an example, consider the {\em general Bernoulli model} with
\begin{equation}
\label{GenBern}
\mathbb{P}(r=-a)=1-p,\ \ \mathbb{P}(r=b)=p,\ \ 0<a\leq 1,\ b>0,\ 0<p<1.
\end{equation}
The function
$$
g_r(f)=p\ln (1+fb) + (1-p)\ln(1-fa)
$$
is defined on $(-1/b, 1/a)$, achieves the global maximum at
$$
f^*=\frac{p}{a}-\frac{1-p}{b},
$$
and
$$
g_r(f^*)=p\ln p +(1-p)\ln (1-p) +\ln\frac{a+b}{a} - (p-1)\ln \frac{b}{a};
$$
we know that $g_r(f^*)\geq 0$, even though it is not at all obvious from the
above expression.
The NS-NL condition $f^*\in [0,1]$ becomes
$$
\frac{a}{a+b}\leq p \leq \min\left( \frac{ab}{a+b}\left(1+\frac{1}{b}\right), 1\right),
$$
and it is now easy to come up with a model in which $f^*>1$: for example, take
$$
a=0.1, \ b=0.5,\ p=0.5
$$
so that $f^*=4$. Given that a gain and a loss in each bet are equally likely, but the amount of a gain
is five times as much as that of a loss,
a large value of $f^*$ is not surprising, although economical and financial implications of this
type of leveraged betting are potentially very interesting and should be a subject of a
separate investigation.
Because of the logarithmic function in the definition of $g_r$, the distribution of $r$ can have a rather
heavy right tail and still satisfy \eqref{r3}. For example, consider
\begin{equation}
\label{Ch0}
r=\eta^2-1,
\end{equation}
where $\eta$ has standard Cauchy distribution with probability density function
$$
h_{\eta}(x)=\frac{1}{\pi(1+x^2)},\ \ \ -\infty< x<+\infty.
$$
Then
$$
g_r(f)=\frac{2}{\pi}\int_0^{+\infty} \frac{\ln\big((1-f)+fx^2\big)}{1+x^2}\, dx= 2\ln\big(\sqrt{f}+\sqrt{1-f}\big),
$$
where the second equality follows from \cite[(4.295.7)]{Gradshtein-Ryzhyk}.
As a result, we get a closed-form answer
$$
f^*=\frac{1}{2},\ g_r(f^*)=\ln 2.
$$
A general way to ensure \eqref{r1}--\eqref{r3} is to consider
\begin{equation}
\label{expo-model}
r=e^{\xi}-1
\end{equation}
for some random variable $\xi$ such that $\mathbb{P}(\xi>0)>0,\
\mathbb{P}(\xi<0)>0$, and $\mathbb{E}|\xi|<\infty$;
note that \eqref{Ch0} is a particular case, with $\xi=\ln\eta^2$.
Then \eqref{global1-l} and \eqref{global1-r} become, respectively,
\begin{align}
\label{global1-l-e}
&\mathbb{E}e^{\xi}>1 \ \
\ \ \ {\rm and}\\
\label{global1-r-e}
&\mathbb{E}e^{-\xi}>1.
\end{align}
For example, if $\xi$ is normal with mean $\mu\in \mathbb{R}$
and variance $\sigma^2>0$, then
$$
\mathbb{E}e^{\xi}=e^{\mu+(\sigma^2/2)},
\ \ \mathbb{E}e^{-\xi}=e^{-\mu+(\sigma^2/2)},
$$
and \eqref{global1-l-e}, \eqref{global1-r-e} are equivalent to
\begin{equation}
\label{log-nrmal}
-\frac{\sigma^2}{2}<\mu<\frac{\sigma^2}{2},
\end{equation}
which, when interpreted in terms of returns,
can indeed be considered as a ``reasonable'' edge condition: large values of $|\mu|$
do create a bias in one direction.
Note that the corresponding $f^*$ is not available in closed
form, but can be evaluated numerically.
\section{Continuous Compounding and a Case for L\'{e}vy Processes}
\label{sec:CC}
Continuous time compounding includes discrete compounding as a particular case and
makes it possible to consider more general types of return processes.
The objective of this section is to show that continuous time compounding
that leads to a non-trivial and non-random long-term growth rate of the resulting wealth
process effectively forces the return process to have independent increments.
The two main examples of such process are sums of iid random variables from the previous
section and the L\'{e}vy processes.
Writing \eqref{wealth2} as
\begin{equation}
\label{wealth1-dt}
W_{n+1}^f-W_n^f=\big(fW_n^f\big)\,r_{n+1},
\end{equation}
we see that a natural continuous time version of \eqref{wealth1-dt}
is
\begin{equation}
\label{wealth1-ct}
dW_{t}^f=fW_t^fdR_{t}
\end{equation}
for a suitable process $R=R_t,\ t\geq 0$ on a stochastic basis
\begin{equation*}
\mathbb{F}=\Big(\Omega, \mathcal{F},\ \{\mathcal{F}_t\}_{t\geq 1},
\mathbb{P}\Big)
\end{equation*}
satisfying the usual conditions \cite[Definition I.1.1]{Protter}.
We interpret \eqref{wealth1-ct} as an integral equation
\begin{equation}
\label{wealth1-ct-i}
W_{t}^f=1+f\int_0^tW_s^fdR_{s};
\end{equation}
recall that $W_0^f=1$ is the standing assumption.
Then the Bichteler-Dellacherie theorem \cite[Theorem III.22]{Protter}
implies that the process $R$ must be a semi-martingale (a sum of a martingale
and a process of bounded variation) with trajectories that, at every point,
are continuous from the
right and have limits from the left. Furthermore, if we allow the process $R$
to have discontinuities, then, by
\cite[Theorem II.36]{Protter}, we need to modify
\eqref{wealth1-ct-i} further:
\begin{equation*}
W_{t}^f=1+f\int_0^tW_{s-}^fdR_{s},
\end{equation*}
where
$$
W_{s-}=\lim_{\varepsilon\to 0, \varepsilon>0} W_{s-\varepsilon},
$$
and, assuming $R_0=0$, the process $W^f$ becomes the Dol\'{e}ans-Dade exponential
\begin{equation}
\label{DDE-1}
W^f_t=\exp\left(fR_t-\frac{f^2\langle R^c\rangle_t}{2}
\right)\prod_{0<s\leq t}
(1+f\triangle R_s)\,e^{-f\triangle R_s}.
\end{equation}
In \eqref{DDE-1}, $\langle R^c\rangle$ is the quadratic variation process of the
continuous martingale component of $R$ and $\triangle R_s=R_s-R_{s-}$.
A natural analog of \eqref{r1} is
\begin{equation}
\label{r1-ct}
\triangle R_s\geq -1,
\end{equation}
and then \eqref{DDE-1} becomes
\begin{equation}
\label{DDE-2}
W^f_t=\exp\left(fR_t-\frac{f^2\langle R^c\rangle_t}{2}+
\sum_{0<s\leq t}\Big( \ln
(1+f\triangle R_s) -f\triangle R_s\Big)\right).
\end{equation}
To proceed, let us assume that the trajectories of $R$ are continuous:
$\triangle R_s=0$ for all $s$ so that
\begin{equation*}
W^f_t=\exp\left(fR_t-f^2\langle R^c\rangle_t\right).
\end{equation*}
If, similar to \eqref{rate-dt}, we define the
long-term growth rate $g_R(f)$ by
\begin{equation}
\label{gr-ct0}
g_R(f)=\lim_{t\to \infty} \frac{\ln W^f_t}{t},
\end{equation}
then we need the limits
\begin{equation}
\label{trip0-cont}
\mu:=\lim_{t\to \infty} \frac{R_t}{t},\ \ \
\sigma^2:=\lim_{t\to \infty} \frac{\langle R^c\rangle_t}{t}
\end{equation}
to exist with probability one and with non-random numbers $\mu,\sigma^2.$
Being a semi-martingale without jumps, the process
$R$ has a representation
\begin{equation}
\label{sm-cont}
R_t=A_t+R^c_t,
\end{equation}
where $A$ is process of bounded variation;
cf. \cite[Theorem II.2.34]{LimitTheoremsforStochasticProcesses}.
Then \eqref{trip0-cont}
imply that, for large $t$,
\begin{equation}
\label{linear0}
A_t\approx \mu t,\ \
\langle R^c\rangle_t\approx \sigma^2 t,
\end{equation}
that is, a natural
way to achieve \eqref{trip0-cont} is to consider the process $R$ of the form
\begin{equation*}
R_t=\mu t+\sigma\,B_t,
\end{equation*}
where $\sigma>0$ and $B=B_t$ is a standard Brownian motion.
Then
\begin{equation}
\label{DDE-3-cont}
W^f_t=\exp\left(f\mu t+f \sigma\,B_t-\frac{f^2\sigma^2 t}{2}\right),
\end{equation}
and we come to the following conclusion:
{\em continuous time compounding with a continuous return process
effectively implies that the wealth process is a geometric Brownian motion}.
The long-term growth rate
\eqref{gr-ct0} becomes
\begin{equation}
\label{gr-ctL-cont}
g_R(f)=f\mu-\frac{f^2\sigma^2}{2},
\end{equation}
so that
\begin{equation*}
f^*=\frac{\mu}{\sigma^2},\ \ g_R(f^*)=\frac{\mu^2}{2\sigma^2},
\end{equation*}
and the NS-NL condition is
\begin{equation*}
0<\mu<\sigma^2.
\end{equation*}
Even though these results are not especially sophisticated, we will see
in the next section (Theorem \ref{prop0}) that the process
\eqref{DDE-3-cont} naturally appears as the continuous-time, or high
frequency, limit of discrete-time compounding for a large class of returns.
On the other hand, if we assume that the process $R$ is purely
discontinuous, with jumps $\triangle R_{k}=r_k$
at times $s=k\in \{1,2,3,\ldots\}$,
then
$$
R_t=0,\ t\in (0,1),\ R_t=\sum_{k=1}^{\lfloor t \rfloor} r_k,\ t\geq 1,
$$
and \eqref{DDE-1} becomes \eqref{wealth2}.
Accordingly, we will now investigate the general case \eqref{DDE-1} when the
process $R$ has both a continuous component and jumps.
To this end, we use
\cite[Proposition II.1.16]{LimitTheoremsforStochasticProcesses}
and introduce the jump measure
$\mu^R=\mu^R(dx,ds)$ of the process $R$ by putting a point mass at
every point in space-time where the process $R$ has a jump:
\begin{equation}
\label{JumpMeasure}
\mu^R(dx,ds) =
\sum_{s > 0}\delta_{(\triangle R_s,s)}(dx, ds);
\end{equation}
note that both the time $s$ and size $\triangle R_s$ of the jump can be random.
In particular, with \eqref{r1-ct} in mind,
\begin{equation}
\label{sum1}
\sum_{0<s\leq t} \Big(\ln
(1+f\triangle R_s) -f\triangle R_s\Big)=
\int_0^t\zint_{-1}^{+\infty}
\big(\ln(1+fx)-fx\big)\mu^R(dx,ds);
\end{equation}
here and below,
\begin{equation}
\label{zint}
\zint_a^b, \ \ \ a<0<b,
\end{equation}
stands for
$$
\int\limits_{(a,0)\bigcup(0,b)}.
$$
By \cite[Proposition II.2.9 and Theorem II.2.34]{LimitTheoremsforStochasticProcesses},
and keeping in mind \eqref{r1-ct}, we get the following generalization of \eqref{sm-cont}:
\begin{equation}
\label{sm-general}
\begin{split}
R_t&=A_t+R^c_t+
\int_0^t\zint_{1}^{+\infty}
x\mu^R(dx,ds)\\
&+
\int_0^t\zint_{-1}^1
x\big(\mu^R(dx,ds)-\nu(dx,s)da_s\big),
\end{split}
\end{equation}
where $a=a_t$ is a predictable non-decreasing process
and $\nu=\nu(dx,t)$ is the non-negative random time-dependent
measure on $(-1,0)\bigcup(0,+\infty)$ with the property
\begin{equation*}
\zint_{-1}^{+\infty}\min(1,x^2)\nu(dx,t)\leq 1
\end{equation*}
for all $t\geq 0$ and $\omega\in \Omega$.
Moreover
\begin{align}
\label{trip-A}
A_t&=\int_0^t \mu_s\, da_s \ {\rm \ for\ some\ predictable \ process} \
\mu=\mu_t,\\
\label{trip-B}
\langle R^c\rangle_t&=\int_0^t \sigma^2_s\,da_s\
{\rm \ for\ some\ predictable \ process} \ \sigma=\sigma_t,
\end{align}
and the process
$$
t \mapsto \int_0^t\zint_{-1}^{+\infty}
h(x)\big(\mu^R(dx,ds)-\nu(dx,s)da_s\big)
$$
is a martingale for every bounded measurable function $h=h(x)$ such that
$\limsup_{x\to 0}|h(x)|/|x|<\infty$.
To proceed, we assume that
$$
\mathbb{E} \int_0^t\zint_{-1}^{+\infty} |\ln(1+x)|\,
\nu(dx,s)\,da_s<\infty,\ t>0,
$$
which is a generalization of condition \eqref{r3}.
Then, by
\cite[Theorem II.1.8]{LimitTheoremsforStochasticProcesses}, the process
$$
t\mapsto \int_0^t\zint_{-1}^{+\infty}
\ln(1+x)\big(\mu^R(dx,ds)-\nu(dx,s)da_s\big)
$$
is a martingale.
Next, we combine \eqref{DDE-2}, \eqref{sum1}, and \eqref{sm-general},
and re-arrange the terms so that the logarithm of the wealth process
becomes
\begin{equation}
\label{logW-g}
\begin{split}
\ln W_t^f&=fA_t+fR^c_t-\frac{f^2}{2}\langle R^c_t \rangle
-f\int_0^t\zint_{-1}^1x\nu(dx,s)da_s\\
&+ \int_0^t\zint_{-1}^{+\infty} \ln(1+fx)\nu(dx,s)da_s+M^f_t,
\end{split}
\end{equation}
where
$$
M^f_t=\int_0^t\zint_{-1}^{+\infty}
\ln(1+fx)\big(\mu^R(dx,ds)-\nu(dx,s)da_s\big).
$$
In general, for equality \eqref{logW-g} to hold,
we need to make an additional assumption
\begin{equation}
\label{nu-int1}
\zint_{-1}^{1}x\nu(dx,t)<\infty
\end{equation}
for all $t\geq 0$ and $\omega\in \Omega$.
In the particular case \eqref{wealth2},
\begin{itemize}
\item $a_s=\lfloor s \rfloor$ is the step function, with unit jumps at positive integers, so that
$da_s$ is the collection of point masses at positive integers;
\item $\nu(dx,s)=F^R(dx),$ where
$F^R$ is the cumulative distribution function of the random variable $r$,
so that \eqref{nu-int1} holds automatically;
\item $\mu_t=g_f(r)+\int_{-1}^1 x\,F^R(dx)$,
$R^c_t=0$, $\sigma_t=0$;
\item $M^f_t=\sum_{0<k\leq t}\big(\ln(1+fr_k)-g_r(f)\big)$;
\item condition \eqref{LogVar-Levy} is \eqref{r3-3}.
\end{itemize}
A natural way to reconcile \eqref{linear0} with \eqref{trip-A},
\eqref{trip-B} is to take $\mu_t=\mu$, $\sigma_t=\sigma$ for some
non-random numbers $\mu\in \mathbb{R}$, $\sigma\geq 0$, and a non-random
non-decreasing function $a=a_t$ with the property
\begin{equation}
\label{limit-a}
\lim_{t\to+\infty}\frac{a_t}{t}=1.
\end{equation}
Then, to have a non-random almost-sure limit
$$
\lim_{t\to \infty} \frac{1}{t}\int_0^t\int_{-1}^{+\infty} \varphi(x)
\nu(dx,s)da_s
$$
for a sufficiently rich class of non-random test functions $\varphi$,
we have to assume that there exists a non-random non-negative
measure $F^R=F^R(dx)$ on $(-1,0)\bigcup(0,+\infty)$ such that
\begin{equation}
\label{intF1-1}
\zint_{-1}^{+\infty}\min( |x|,1)\, F^R(dx)<\infty
\end{equation}
and, for large $s$,
$$
\nu(dx,s)\approx F^R(dx).
$$
As a result, if
\begin{equation}
\label{LevyMeasure0}
\nu(dx,s)= F^R(dx)
\end{equation}
for all $s$, then
\begin{equation}
\label{triple-nr1}
A_t=\mu a_t,\ \langle R^c\rangle_t=\sigma^2a_t,\
\nu(dx,t)=F^R(dx)a_t
\end{equation}
are all non-random, and
\cite[Theorem II.4.15]{LimitTheoremsforStochasticProcesses}
implies that $R$ is a {\em process with independent increments}.
Furthermore, \eqref{triple-nr1} and the strong law of large numbers for
martingales imply
\begin{equation*}
\mathbb{P}\left(\lim_{t\to \infty} \frac{R^c_t}{t}=0\right)=1;
\end{equation*}
cf. \cite[Corollary 1 to Theorem II.6.10]{LSh-M}.
Similarly, if
\begin{equation}
\label{LogVar-Levy}
\zint_{-1}^{+\infty} \ln^2(1+x)\, F^R(dx)<\infty,
\end{equation}
then $M^f$ is a square-integrable martingale and
\begin{equation*}
\mathbb{P}\left( \lim_{t\to \infty} \frac{M^f_t}{t}=0\right)=1.
\end{equation*}
Writing
$$
\bar{\mu}=\mu-\int_{-1}^{1}x F^R(dx)
$$
the long-term growth rate \eqref{gr-ct0} becomes
\begin{equation}
\label{gr-ctL}
g_R(f)=f\bar{\mu}-\frac{f^2\sigma^2}{2}+
\zint_{-1}^{\infty}
\ln(1+fx) F^R(dx),
\end{equation}
which does include both \eqref{F(f)} and \eqref{gr-ctL-cont} as particular cases.
By direct computation, the function $f\mapsto g_R(f)$ is concave and the
domain of the function contains $[0,1]$.
Similar to Proposition \ref{prop:glob1}, we have the following result.
\begin{thm}
\label{th-LevyRate}
Consider continuous-time compounding with return process
\begin{equation}
\label{Return-Levy0}
\begin{split}
R_t&=A_t+R_t^c+
\int_0^t\zint_{1}^{+\infty}
x\mu^R(dx,ds)\\
&+
\int_0^t\zint_{-1}^1
x\big(\mu^R(dx,ds)-\nu(dx,s)da_s\big),
\end{split}
\end{equation}
where the random measure $\mu^R$ is from \eqref{JumpMeasure},
and assume that equalities \eqref{limit-a} and \eqref{triple-nr1} hold.
If $F^R$ satisfies \eqref{intF1-1}, \eqref{LogVar-Levy}, and
\begin{align*}
&\lim_{f\to 0+}
\zint_{-1}^{\infty}
\frac{x}{1+fx}\, F^R(dx)>-\bar{\mu},\\
&\lim_{f\to 1-}
\zint_{-1}^{\infty}
\frac{x}{1+fx}\, F^R(dx)<\sigma^2-\bar{\mu},
\end{align*}
then the long-term growth rate is given by \eqref{gr-ctL}, and there exists
a unique $f^*\in (0,1)$ such that
$$
g_R(f)< g_R(f^*)
$$
for all $f$ in the domain of $g_R$.
\end{thm}
By the Lebesgue decomposition theorem, the measure corresponding to the
function $a=a_t$ has a discrete, absolutely continuous, and singular components.
With \eqref{limit-a} in mind, a natural choice of the discrete component is $a_t=\lfloor t \rfloor$,
which, as we saw, corresponds to discrete compounding discussed in the previous
section. A natural choice of the absolutely continuous component is
\begin{equation*}
a_t=t.
\end{equation*}
Then
\begin{equation*}
A_t=\mu t,\ R^c_t=\sigma B_t,\ \nu(dx,t)da_t=F^R(dx)\,dt,
\end{equation*}
where $B$ is a standard Brownian motion.
By \cite[Corollary II.4.19]{LimitTheoremsforStochasticProcesses},
we conclude that the process $R$ has independent and stationary increments,
that is, {\em $R$ is a L\'{e}vy process}.
In this case, equality \eqref{Return-Levy0} is known as the L\'{e}vy-It\^{o} decomposition
of the process $R$; cf. \cite[Theorem 19.2]{Sato}.
We do not consider the singular case in this paper
and leave it for future investigation.
\section{Continuous Limit of Discrete Compounding}
\subsection{A (Simple) Random Walk Model}
Following the methodology in \cite[Section 7.1]{Thorp06},
we assume compounding a sufficiently large number $n$ of
bets in a time period $[0,T]$. The returns $r_{n,1}, r_{n,2},\ldots $
of the bets are
\begin{equation}
\label{return-nk}
r_{n,k} =
\frac{\mu}{n} + \frac{\sigma}{\sqrt{n}}\,\xi_{n,k}
\end{equation}
for some $\mu>0$, $\sigma>0$ and
independent identically distributed
random variables $\xi_{n,k},\ k=1,2,\ldots,$ with mean $0$ and variance $1$.
The classical simple random walk corresponds to
$\mathbb{P}(\xi_{n,k}=\pm 1)=1/2$ and can be considered a
{\em high frequency} version of \eqref{return1}.
Similar to \eqref{r1}, we need $r_{n,k}\geq -1$, which, in general, can only be
achieved with {\em uniform boundedness} of $\xi_{n,k}$:
\begin{equation}
\label{eq:bnd1}
|\xi_{n,k}|\leq C_0,
\end{equation}
and then, with no loss of generality, we assume that
$n$ is large enough so that
\begin{equation}
\label{small-r}
|r_{n,k}|\leq \frac{1}{2}.
\end{equation}
Similar to \eqref{edge1}, a condition to have an edge is
$$
\mathbb{E}[r_{n,k}] = \frac{\mu}{n} > 0,
$$
and, similar to \eqref{wealth2}, given $n$ bets per unit time period,
with exposure $f \in [0,1]$ in each bet, we get the following formula for the
total wealth $W_t^{n,f}$ at time $t\in (0,T]$ assuming $W_0=1$:
\begin{equation}
\label{Wntf}
W_t^{n,f} = \prod_{k=1}^{\lfloor nt \rfloor}\big(1+fr_{n,k}\big);
\end{equation}
$\lfloor nt \rfloor$ denotes the largest integer less than $nt$.
Let $B=B_t,\ t\geq 0,$ be a standard Brownian motion on
a stochastic basis $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t\geq 0},\mathbb{P})$ satisfying the usual conditions, and define the process
\begin{equation}
\label{wealth2-cont}
W^f_t= \exp\left(\left(f \mu - \frac{f^2\sigma^2}{2}\right)t + f \sigma B_t\right).
\end{equation}
Note that \eqref{wealth2-cont} is a particular case of \eqref{DDE-3-cont}.
\begin{thm}
\label{prop0}
For every $T>0$ and every $f\in [0,1]$, the sequence of processes
$ \big(W_t^{n,f},\ n\geq 1,\ t\in [0,T]\big)$ converges in law to the process $W^f=W^f_t,\ t\in [0,T]$.
\end{thm}
\begin{proof}
Writing
$$
Y^{n,f}_t= \ln W_t^{n,f},
$$
the objective is to show weak convergence, as $n\to \infty$, of $Y^{n,f}$ to the process
$$
Y^f_t=\left( f \mu - \frac{f^2\sigma^2}{2}\right)t + f \sigma B_t,\ \ t\in [0,T].
$$
The proof relies on the method of predictable characteristics
for semimartingales from \cite{LimitTheoremsforStochasticProcesses}.
More specifically, we make suitable changes in the proof of Corollary VII.3.11.
By \eqref{wealth2-cont}
$$
Y^{n,f}_t=\sum_{k=1}^{\lfloor nt \rfloor} \ln(1+fr_{n,k}).
$$
Then \eqref{return-nk} and \eqref{eq:bnd1} imply
\begin{equation*}
\mathbb{E}\bigg( Y^{n,f}_t - \mathbb{E}Y^{n,f}_t\bigg)^4
\leq \frac{C_0^4\sigma^4}{n^2}\big(nT+3nT(nT-1)\big)
\leq {3C_0^4\sigma^4T^2},
\end{equation*}
from which uniform integrability of the family
$\{Y^{n,f}_t,\ n\geq 1, \ t\in [0,T]\}$ follows.
Then, by \cite[Theorem VII.3.7]{LimitTheoremsforStochasticProcesses}, it suffices to establish the following:
\begin{align}
\label{mean0}
\lim_{n\to \infty}&
\sup_{t \leq T}
\left|\lfloor nt \rfloor \mathbb{E}\big[\ln(1+fr_{n,1})\big] - \left(f\mu-\frac{f^2\sigma^2}{2}\right)t \right| =0,\\
\label{var0}
\lim_{n\to \infty} &
\lfloor nt \rfloor \left(\mathbb{E}
\big(\ln(1+fr_{n,1})\big)^2
-
\bigg(\mathbb{E}\big[\ln(1+fr_{n,1})\big]\bigg)^2 \right)= f^2 \sigma^2 t,\ t\in [0,T],\\
\label{jump0}
\lim_{n\to \infty} &\lfloor nt \rfloor \mathbb{E}
\big[\phi\big(\ln(1+fr_{n,1})\big)\big] = 0,\ t\in [0,T].
\end{align}
Equality \eqref{jump0} must hold for all functions $\phi=\phi(x),
\ x\in \mathbb{R},$ that are
continuous and bounded on $\mathbb{R}$ and satisfy $\phi(x)=o(x^2),
\ x\to 0$, that is,
\begin{equation}
\label{phi0}
\lim_{x\to 0} \frac{\phi(x)}{x^2}=0.
\end{equation}
Equalities \eqref{mean0} and \eqref{var0} follow from
$$
r_{n,1}^2=\frac{\sigma^2}{n}\, \xi_{n,1}^2
+ \frac{2\mu\sigma\xi_{n,1}}{n^{3/2}}+\frac{\mu^2}{n^2},
$$
together with \eqref{small-r} and an elementary inequality
$$
\left| \ln(1+x)-x-\frac{x^2}{2} \right| \leq |x|^3,\ |x|\leq \frac{1}{2}.
$$
In particular,
\begin{equation*}
\mathbb{E}
\big[\big(\ln(1+fr_{n,1})\big)^2\big] = \frac{f^2\sigma^2}{n}+o(1/n),
\ n\to +\infty.
\end{equation*}
To establish \eqref{jump0}, note that \eqref{phi0} and \eqref{return-nk}
imply
$$
\phi\big(\ln (1+fr_{n,1})\big) = o(1/n),
\ n\to +\infty.
$$
\end{proof}
Similar to \eqref{rate-dt}, we define the long-term continuous time growth
rate
\begin{equation*}
g(f)=\lim_{t\to \infty} \frac{1}{t}\ln W^f_t.
\end{equation*}
Then a simple computation show that
$$
g(f)=f\mu - \frac{f^2\sigma^2}{2},
$$
and so
\begin{equation}
\label{optf-rv}
f^*=\frac{\mu}{\sigma^2}
\end{equation}
achieves the maximal long-term
continuous time growth rate
\begin{equation}
\label{optf-rv1}
g(f^*)=\frac{\mu^2}{2\sigma^2}.
\end{equation}
The NS-NL condition $f^*\in [0,1]$ holds if
$0\leq \mu\leq \sigma^2$, which, to the order $1/n$, is consistent with
\eqref{global1-l} and \eqref{global1-r}, when applied to
\eqref{return-nk}:
$$
\mathbb{E}[r_{n,k}]=\frac{\mu}{n},\ \
\mathbb{E}\left[\frac{r_{n,k}}{1+r_{n,k}}\right]
=\frac{\mu-\sigma^2}{n}+o(n^{-1}).
$$
The wealth process \eqref{wealth2-cont} is that of someone who is ``continuously" placing bets, that is, adjusts the positions instantaneously, and, for large $n$,
is a good approximation of high frequency betting \eqref{Wntf}.
In general, when the returns are given by \eqref{return-nk}, a direct optimization of
\eqref{Wntf} with respect to $f$ will not lead to a closed-form expression for the corresponding
optimal strategy $f_n^*$ , but
Theorem \ref{prop0} implies that, for sufficiently large $n$,
\eqref{optf-rv} is an approximation of $f_n^*$
and \eqref{optf-rv1} is an approximation of the corresponding long-term growth rate.
As an illustration, consider the high-frequency version
of the simple Bernoulli model \eqref{return1}:
\begin{equation}
\label{return1-hf}
\mathbb{P}\left( r_{n,k}=\frac{\mu}{n}\pm \frac{\sigma}{\sqrt{n}}\right)=\frac{1}{2},
\end{equation}
which, for fixed $n$, is
is a particular case of the general Bernoulli model
\eqref{GenBern} with $p=q=1/2$,
$$
a=\frac{\sigma}{\sqrt{n}}-\frac{\mu}{n},\ b=\frac{\sigma}{\sqrt{n}}+\frac{\mu}{n}.
$$
Then, by direct computation,
$$
f_n^*=\frac{\mu}{\sigma^2-(\mu^2/n)}\to \frac{\mu}{\sigma^2},\ n\to \infty,
$$
and
$$
\lim_{n\to \infty} g_{r_n}(f_n^*)=\frac{\mu^2}{2\sigma^2}.
$$
Even though the analysis of the proof of Theorem \ref{prop0}
shows that the convergence is uniform in $f$ on compact sub-sets of $(0,1)$,
the proof that $\lim_{n\to \infty} f^*_n=f^*$ would require
a version of Theorem \ref{prop0} with $T=+\infty$, which, for now, is not
available.
With natural modifications, Theorem \ref{prop0} extends to the setting \eqref{expo-model}.
\begin{thm}
\label{prop0-1}
Assume that
\begin{equation*}
r_{n,k}+1=\exp\left(\frac{b}{n}+\frac{\sigma}{\sqrt{n}}\,\xi_{n,k}\right),
\end{equation*}
where $b \in \mathbb{R}$, $\sigma>0$, and, for each $n\geq 1$, $k\leq n$,
the random variables
$\xi_{n,k}$ are independent and identically distributed, with zero mean, unit variance, and, for every $a>0$,
\begin{equation}
\label{moment1}
\lim_{n\to\infty}n\,
\mathbb{E}\big[|\xi_{n,1}|^2I(|\xi_{n,1}|>a\sqrt{n})\big]=0.
\end{equation}
Then the conclusion of Theorem \ref{prop0} holds with
$$
\mu=b+\frac{\sigma^2}{2}.
$$
\end{thm}
\begin{proof}
Even though a formal Taylor expansion suggests
$$
r_{n,k}=\frac{\mu}{n} + \frac{\sigma}{\sqrt{n}}\,\xi_{n,k}+o(1/n)
$$
we cannot apply Theorem \ref{prop0} directly because the
random variables $\xi_{n,k}$ are not necessarily uniformly bounded.
Still, condition \eqref{moment1}
makes it possible to verify conditions
\eqref{mean0}--\eqref{jump0}.
\end{proof}
Condition \eqref{moment1} is clearly satisfied when
$\xi_{n,k},\ k=1,2,\ldots,$ are iid standard normal, which corresponds to
\begin{equation}
\label{discr-ret-gauss}
r_{n,k}=\frac{P_{k/n} - P_{(k-1)/n}}{P_{(k-1)/n}}
\end{equation}
and
\begin{equation}
\label{GBM-returns}
P_t=e^{bt+\sigma B_t}.
\end{equation}
Thus, while the exponential model \eqref{expo-model} with log-normal
returns is not solvable in closed form, the high-frequency version leads to the
(approximately) optimal strategy
\begin{equation}
\label{st-line}
f^*=\frac{b}{\sigma^2}+\frac{1}{2},
\end{equation}
and, under \eqref{log-nrmal}, the NS-NL condition holds: $f^*\in (0,1)$.
Numerical experiments with $\sigma=1$ and $n=10$ show that the
values of the corresponding optimal $f^*_{10}$ are very close to those given
by \eqref{st-line} for all $b\in (-1/2,1/2)$.
Informally, both Theorems \ref{prop0} and \ref{prop0-1} can be considered
as particular cases of the delta method for the Donsker theorem with drift:
if the sequence of processes
$$
t\mapsto \sum_{k=1}^{\lfloor nt \rfloor} \xi_{n,k}
$$
converges, as $n\to \infty$, to the processes $t\mapsto
bt+\sigma B_t$ and $\varphi=\varphi(x)$ is a suitable
function with $\varphi(0)=0$, then one would expect the sequence of processes
$$
t\mapsto \sum_{k=1}^{\lfloor nt \rfloor} \varphi\big(\xi_{n,k}\big)
$$
to converge to the process
$$
t\mapsto \left(\varphi'(0)b+\frac{\varphi''(0)\sigma^2}{2}\right)t+|\varphi'(0)|\sigma B_t.
$$
\subsection{Beyond the Log-Normal Limit}
With the results of Section \ref{sec:CC} in mind,
we consider the following generalization of \eqref{discr-ret-gauss}:
\eqref{GBM-returns}:
$$
r_{n,k} = \frac{P_{k/n} - P_{(k-1)/n}}{P_{(k-1)/n}}, \ k=1,2,\ldots,
$$
where the process $P=P_t,\ t\geq 0$, has the form $P_t=e^{R_t}$, and
$R=R_t$ is a L\'evy process.
In other words,
\begin{equation}
\label{dctr-gen}
r_{n,k}={e}^{R_{k/n}-R_{(k-1)/n}} - 1.
\end{equation}
As in \eqref{Return-Levy0}, the process $R=R_t$ can be
decomposed into a drift, diffusion/small jump,
and large jump components according to the L\'{e}vy-It\^{o}
decomposition \cite[Theorem 19.2]{Sato}:
\begin{equation}
\label{Levy-main}
R_t={\mu}t+\sigma\,B_t+
\int_0^t\zint_{-1}^1
x\big(\mu^R(dx,ds)-F^R(dx)ds\big)+
\int_0^t\int_{|x|>1}
x\mu^R(dx,ds);
\end{equation}
we continue to use the notation $-\!\!\!\!\!\int$ first introduced in \eqref{zint}.
Now that the process $R_t$ is exponentiated,
\begin{itemize}
\item there is no need to assume that $\triangle R_t\geq -1$ ;
\item the analog of \eqref{LogVar-Levy} becomes
$\mathbb{E}|R_1|<\infty$.
\end{itemize}
Equality \eqref{Levy-main} has a natural interpretation
in terms of financial risks \cite{Thorp03}: the drift represents the
edge (``guaranteed'' return), diffusion and small jumps represent
small fluctuations of returns, and
the large jump component represents (sudden) large changes in returns.
Similar to \eqref{Wntf}, the corresponding wealth process is
\begin{equation}
\label{wp-100}
W_{t}^{n,f} = \prod_{k = 1}^{\lfloor nt \rfloor} \big(1 + f r_{n,k}\big).
\end{equation}
We have the following generalization of Theorem \ref{prop0-1}.
\begin{thm}
Consider the family of processes $W^{n,f}=W_t^{n,f}$, $t\in [0,T]$, $n\geq 1$,
$f\in [0,1],$ defined by \eqref{wp-100}. If
$r_{n,k}$ is given by \eqref{dctr-gen}, with $P_t=e^{R_t}$, and
$R=R_t$ is a L\'{e}vy process with representation \eqref{Levy-main} and $\mathbb{E}|R_1|< \infty$,
then, for every $f\in [0,1]$ and $T>0$,
$$
\lim_{n\to \infty}W^{n,f} \wcL W^f
$$
in $\mathbb{D}((0,T))$, where
\begin{equation}
\label{WP-generalLP}
\begin{split}
W^f_t &=\exp\left(f R_t + \frac{ f(1-f)\sigma^2}{2}\, t\right.\\
&+ \left. \int_0^t \zint_{\mathbb{R}} \Big[\ln\big(1+f(e^x - 1)\big) - f x\Big] \mu^R(dx, ds)\right).
\end{split}
\end{equation}
\end{thm}
\begin{proof}
By \eqref{dctr-gen} and \eqref{wp-100},
\begin{equation*}
\ln W_t^{n,f} = \sum_{k=1}^{\lfloor nt \rfloor}\ln\bigg( 1 + f \big( {e}^{R_{k/n}-R_{(k-1)/n}} - 1\big)\bigg).
\end{equation*}
\underline{Step 1:}
For $s \in \big(\frac{k-1}{n}, \frac{k}{n}\big]$, let
\begin{equation}
\label{rnks}
r_s^{n,k} = {e}^{R_s - R_{(k-1)/n}} - 1,
\end{equation}
and apply the It\^o's formula \cite[Theorem II.32]{Protter}
to the process
$$
s\mapsto \ln\big(1+f r_s^{n,k}\big),\ \ s \in \Bigg(\frac{k-1}{n}, \frac{k}{n}\Bigg].
$$
The result is
\begin{equation*}
\begin{split}
\ln\big(1+f r_s^{n,k}\big)
&= \int_{\frac{k-1}{n}}^s \frac{f (1 + r_{u-}^{n,k})}{1+f r_{u-}^{n,k}}\,
dR_u
+ \frac{\sigma^2}{2}\int_{\frac{k-1}{n}}^s \frac{f(1-f)(1+r_{u-}^{n,k})}{\big(1+fr_{u-}^{n,k}\big)^2}\, du \\
& + \int_{\frac{k-1}{n}}^s \zint_{\mathbb{R}}
\bigg[\ln\big(1-f+f{e}^{x} (r_{u-}^{n,k}+ 1)\big) \\
& - \ln(1+fr_{u-}^{n,k}) - x \frac{f(1+r_{u-}^{n,k})}{1+f r_{u-}^{n,k}}\bigg]\mu^R(dx, du).
\end{split}
\end{equation*}
\underline{Step 2:}
Putting $s = \frac{k}{n}$ in the above equality and summing over $k$,
we derive the following expression for $ \ln W_t^{n,f}$:
\begin{align}
\notag
\ln W_t^{n,f} &= \sum_{k=1}^{\lfloor nt \rfloor} \bigg(\int_{\frac{k-1}{n}}^{\frac{k}{n}} h^{(1)}_{n,k}(s) \, dR_s
+ \int_{\frac{k-1}{n}}^{\frac{k}{n}}
h^{(2)}_{n,k}(s) \, ds
+ \int_{\frac{k-1}{n}}^{\frac{k}{n}} \zint_{\mathbb{R}} h^{(3)}_{n,k}(s,x) \mu^R(dx, du)\bigg)\\
\label{main-integrals}
&= \int_0^t H^{(1)}_{n,t}(s) \, dR_s + \int_0^t H^{(2)}_{n,t}(s) \, ds + \int_0^t\zint_{\mathbb{R}} H^{(3)}_{n,t}(s,x) \mu^R(dx, ds)\,,
\end{align}
where
\begin{equation*}
\begin{split}
\displaystyle
h^{(1)}_{n,k}(s) & = \frac{f(1+r_{s-}^{n,k})}{1+fr_{s-}^{n,k}}\,,\ \
h^{(2)}_{n,k}(s) =
\frac{\sigma f(1-f)}{2}\frac{1+r_{s-}^{n,k}}{(1+fr_{s-}^{n,k})^2}\,, \\
h^{(3)}_{n,k}(s,x)&= \ln\big(1-f+fe^x(r_{s-}^{n,k}+1) \big) - \ln(1+fr_{s-}^{n,k}) - fx\frac{1+r_{s-}^{n,k}}{1+fr_{s-}^{n,k}}\,;\\
H^{(i)}_{n,t}(s) &= \sum_{k=1}^{\lfloor nt \rfloor} h^{(i)}_{n,k}(s) \mathbf{1}_{(\frac{k-1}{n}, \frac{k}{n}]}(s), \ i = 1, 2;\ \
H^{(3)}_{n,t}(s,x) = \sum_{k=1}^{\lfloor nt \rfloor} h^{(3)}_{n,k}(s,x)\mathbf{1}_{(\frac{k-1}{n}, \frac{k}{n}]}(s).
\end{split}
\end{equation*}
\underline{Step 3:}
Because
$$
\lim_{n\to \infty,\, k/n\to s} R_{(k-1)/n}=R_{s-},
$$
equality \eqref{rnks} implies
$$
\lim_{n\to +\infty,\, k/n\to s} r^{n,k}_{s-}= 0
$$
for all $s$. Consequently, we have the following
convergence in probability:
\begin{align*}
\lim_{n\to +\infty} H^{(1)}_{n,t}(s)=f,\ &
\lim_{n\to +\infty} H^{(2)}_{n,t}(s)=\frac{\sigma^2 f(1-f)}{2},\\
&\lim_{n\to +\infty} H^{(2)}_{n,t}(s,x)=\ln\big(1+f(e^x-1)\big)-fx.
\end{align*}
To pass to the corresponding limits in \eqref{main-integrals}, we need suitable bounds on the functions $H^{(i)}$, $i=1,2,3$.
Using the inequalities
$$
0<\frac{1+y}{1+ay}\leq \frac{1}{a},\ \ 0<\frac{1+y}{(1+ay)^2}
\leq \frac{1}{4a(1-a)}, \ \ \ y>-1,\ a\in (0,1),
$$
we conclude that
$$
0<h^{(1)}_{n,k}(s)\leq 1,\ 0<h^{(2)}_{n,k}(s)\leq \sigma^2,
$$
and therefore
\begin{equation}
\label{ubound1-2}
0<H^{(1)}_{n,t}(s)\leq 1,\ 0<H^{(2)}_{n,t}(s)\leq \sigma^2.
\end{equation}
Similarly, for $f\in (0,1)$ and $y > -1$,
\begin{equation}
\label{H3bnd00}
\left|\ln \frac{1-f+fe^x(y+1)}{1+fy}-fx\frac{1+y}{1+fy}\right|\leq 2\big(|x|\wedge |x|^2\big),
\end{equation}
so that
$$
|h^{(3)}_{n,k}(s,x)|\leq 2\big(|x|\wedge |x|^2\big)
$$
and
\begin{equation}
\label{ubound3}
\big|H^{(3)}_{n,t}(s)\big| \leq 2\big(|x|\wedge |x|^2\big).
\end{equation}
To verify \eqref{H3bnd00},
fix $f\in (0,1)$ and $y>-1$, and define the function
$$
z(x) = \ln \frac{1-f+fe^x(y+1)}{1+fy},\ x\in \mathbb{R}.
$$
By direct computation,
\begin{equation*}
\begin{split}
z(0) &= 0, \\
z'(x) &= \frac{fe^x(y+1)}{1-f + fe^x(y+1)}=1-\frac{1-f}{1-f + fe^x(y+1)},\\
z'(0) &= \frac{f(y+1)}{1+fy},
\end{split}
\end{equation*}
so that, using the Taylor formula,
\begin{equation}
\label{H3bnd}
\ln \frac{1-f+fe^x(y+1)}{1+fy}-fx\frac{1+y}{1+fy}
=z(x)-z(0)-xz'(0)=\int_0^x(x-u)z''(u)du.
\end{equation}
It remains to notice that
$$
0\leq z'(x)\leq 1, \ \ 0\leq z''(x)\leq 1,
$$
and then \eqref{H3bnd00} follows from \eqref{H3bnd}.
With \eqref{ubound1-2} and \eqref{ubound3} in mind, the dominated convergence theorem
\cite[Theorem IV.32]{Protter} makes it possible to pass to the limit
in probability in \eqref{main-integrals}; the convergence in the
space $\mathbb{D}$ then follows from the general
results of \cite[Section IX.5.12]{LimitTheoremsforStochasticProcesses}.
\end{proof}
The following is a representation of the
long-term growth rate of the limiting wealth process $W^f$.
\begin{thm}
\label{th:gr-rate-LP-gen}
Let $R=R_t$ be a L\'{e}vy process with representation \eqref{Levy-main}.
If $\mathbb{E}|R_1| < \infty$, then the process $W^f=W^f_t$
defined in \eqref{WP-generalLP}
satisfies
\begin{equation}
\label{GR-gen-LP}
\begin{split}
\lim_{t\to +\infty} \frac{\ln W_t^f}{t}&=f \bigg(\mu + \int_{|x|>1} x F^R(dx)\bigg)
+ \frac{ f(1-f) \sigma^2}{2}\\
&+ \int_{\mathbb{R}}\big[\ln\big(1 + f ({e}^x - 1) \big) - f x \big] F^R(dx).
\end{split}
\end{equation}
\end{thm}
\begin{proof}
By \eqref{WP-generalLP},
$$
\frac{\ln W_t^f}{t}=
f \frac{R_t}{t} + \frac{ f(1-f)\sigma^2}{2}+ \frac{1}{t}\int_0^t \int_{\mathbb{R}} \Big[\ln\big(1+f(e^x - 1)\big) - f x\Big] \mu^R(dx, ds).
$$
It remains to apply the law of large numbers
for L\'evy processes \cite[Theorem 36.5]{Sato}.
\end{proof}
If, in addition, we assume that
$$
\zint_{-1}^1 |x|F^R(dx)<\infty,
$$
that is, the small-jump component of $R$ has bounded variation, then, after a change of
variables and re-arrangement of terms, \eqref{GR-gen-LP} becomes \eqref{gr-ctL}.
On the other hand, equality \eqref{gr-ctL} is derived for a wider class of return processes
that includes L\'{e}vy processes as a particular case.
Similar to Proposition \ref{prop:glob1}, we also have the following result.
\begin{thm}
\label{th-LevyRate1}
In the setting of Theorem \ref{th:gr-rate-LP-gen}, denote the
right-hand side of \eqref{GR-gen-LP} by $g_R(f)$ and assume that
\begin{align*}
&\lim_{f\to 0+}
\int_{\mathbb{R}}
\left(\frac{e^x - 1}{1+f(e^x - 1)} - x\right)\, F^R(dx)>-\bigg(\mu
+ \frac{\sigma^2}{2} + \int_{|x|>1} x F^R(dx)\bigg),\\
&\lim_{f\to 1-}
\int_{\mathbb{R}}
\left(\frac{e^x - 1}{1+f(e^x - 1)} - x\right)\, F^R(dx)<
-\bigg(\mu + \int_{|x|>1} x F^R(dx)\bigg),
\end{align*}
Then there exists
a unique $f^*\in (0,1)$ such that
$$
g_R(f)< g_R(f^*)
$$
for all $f$ in the domain of $g_R$.
\end{thm}
\section{Continuous Limit of Random Discrete Compounding}
The objective of this section is to analyze high frequency limits for betting {\em in business time}. In other words, the number of bets is not known a priori, so that a natural model of the
corresponding wealth process is
\begin{equation}
\label{weath-mgen}
W_{t}^{n,f}=\prod_{k=1}^{\lfloor \Lambda_{n,t}\rfloor}
(1+fr_{n,k})
\end{equation}
where, for each $n$, the process $t\mapsto \Lambda_{n,t}$ is
a subordinator, that is,
a non-decreasing L\'{e}vy process, independent of all $r_{n,k}$.
To study \eqref{weath-mgen}, we will follow the methodology in
\cite{KZZ}, where convergence of processes is derived after {\em assuming}
a suitable convergence of the random variables.
The main result in this connection is as follows.
\begin{thm}
\label{th:Levy-gen0}
Consider the following objects:
\begin{itemize}
\item random variables $X_{n,k},\ n,k\geq 1$ such that
$\{X_{n,k},\ k\geq 1\}$ are iid for each $n$,
with mean zero and, for some $\beta\in [0,1]$, $m_n:=\Big(\mathbb{E}|X_{n,1}|^{\beta}\Big)^{1/\beta}<\infty$;
\item random processes $\Lambda_n=\Lambda_{n,t}$, $n\geq 1,\ t\geq 0,$
such that, for each $n$, $\Lambda_n$ is a subordinator
independent of $\{X_{n,k},\ k\geq 1\}$ with the properties
$\Lambda_{n,0}=0$, and
for some numbers $0<\delta,\delta_1\leq 1$ and $C_n>0$,
$\Big(\mathbb{E}\Lambda_{n,t}^{\delta}\Big)^{1/\delta}
\leq C_n t^{\delta_1/\delta}$.
\end{itemize}
Assume that there exist infinitely divisible random variables $Y$ and $U$
such that
$$
\lim_{n\to \infty} \sum_{k=1}^n X_{n,k} \wc \bar{Y},\
\lim_{n\to \infty} \frac{\Lambda_{n,1}}{n} \wc \bar{U}.
$$
If
\begin{equation}
\label{dens}
\sup_{n} \Big(C_n m_n^{\beta}\Big)<\infty,
\end{equation}
then, as $n\to \infty$, the sequence of processes
$$
t\mapsto \sum_{k=1}^{\lfloor \Lambda_{n,t}\rfloor} X_{n,k},\ t\in [0,T],
$$
converges, in the Skorokhod topology, to the process $Z=Z_t$
such that $Z_t=Y_{U_t}$, where $Y$ and $U$ are independent
L\'{e}vy processes satisfying $Y_1\wc\bar{Y}$ and $U_1\wc\bar{U}$.
\end{thm}
The proof is a word-for-word repetition of the arguments leading to
\cite[Theorem 1]{KZZ}:
the result of \cite{GnF}, together with the assumptions of the
theorem, implies
$$
\lim_{n\to \infty} \sum_{k=1}^{\lfloor \Lambda_{n,1}\rfloor} X_{n,k}\wc Z_1,
$$
and therefore the convergence of finite-dimensional distributions for
the corresponding processes; together
with condition \eqref{dens}, this implies the convergence in the Skorokhod space. Because we deal exclusively with L\'{e}vy processes, it is possible to
avoid the heavy machinery from \cite{LimitTheoremsforStochasticProcesses}.
We now consider the wealth process \eqref{weath-mgen}
and apply Theorem \ref{th:Levy-gen0}
with
$$
X_{n,k}=
\ln (1+fr_{n,k})-\mathbb{E}\ln (1+fr_{n,k}).
$$
On the one hand, convergence to infinitely
divisible distributions other than normal is a very diverse area,
with a variety of conditions and conclusions;
cf. \cite[Chapter XVII, Section 5]{FellerII} or a summary in
\cite[Section 16.2]{Klenke}. On the other hand, optimal strategy
\eqref{optf-rv} seems to persist.
For example, assume that
the returns $r_{n,k}$ are as in \eqref{return-nk}, and
let $\Lambda_{n,t}=S_{n^{\alpha}t}$, where $\alpha\in (0,1]$ and
$S=S_t$ is the L\'{e}vy process such that $S_1$ has the $\alpha$-stable
distribution with both scale and skewness parameters equal to $1$.
Recall that an $\alpha$-stable L\'{e}vy process $L^{\alpha}=L^{\alpha}_t$
satisfies the following equality in distribution (as processes):
\begin{equation}
\label{scale-L}
L^{\alpha}_{\gamma t}\wcL \gamma^{1/\alpha}L^{\alpha}_t, \ \gamma>0.
\end{equation}
Then
$$
\Lambda_{n,t}\wcL nS_{t}
$$
and, in the notations of Theorem \ref{th:Levy-gen0},
$\bar{Y}$ is normal with mean zero and variance $\sigma^2$.
Keeping in mind that
$$
\mathbb{E}\ln (1+fr_{n,k}) = \mathbb{E}\ln (1+fr_{n,1})=
\Big(f\mu-\frac{f^2\sigma^2}{2}\Big)n^{-1}+o(n^{-1}),
$$
we repeat the arguments from \cite[Example 1]{KZZ} to conclude that
$$
\lim_{n\to \infty} \ln W^{n,f}_t \wcL \Big(f\mu-\frac{f^2\sigma^2}{2}\Big)S_t
+ Z_t,
$$
where $Z_1$ has symmetric $2\alpha$-stable distribution.
By \eqref{scale-L},
$$
S_t\wc t^{1/\alpha}S_1,\ \lim_{t\to +\infty} t^{-1/\alpha}Z_t \wc
\lim_{t\to +\infty} t^{-1/(2\alpha)}Z_1 \wc 0,
$$
and the ``natural''
long term growth rate becomes
$$
\lim_{t\to \infty}t^{-1/\alpha}\Big(\lim_{n\to \infty} \ln W^{n,f}_t\Big) \wc
\Big(f\mu-\frac{f^2\sigma^2}{2}\Big)S_1,
$$
which is random, but, for each realization of $S$, is still maximized by
$f^*$ from \eqref{optf-rv}. Therefore, if the time with which we compound our wealth is random, then the growth rate is also random as we don't know when we will stop compounding, yet it is still maximized by a deterministic fraction. Note that, for the purpose of this
computation, the (stochastic) dependence between the processes
$S$ and $Z$ is not important.
\section{Conclusions And Further Directions}
The NS-NL condition $f^*\in [0,1]$ can fail in many situations.
Even in the simple Bernoulli model, if $p<1/2$, then the short
position $f^*=2p-1$ achieves positive long-time wealth growth:
$$
g_r(f^*)=p\ln \frac{p}{1-p}+(2-p)\ln(2-2p)=\ln 2 +p\ln p+(1-p)\ln(1-p)>0.
$$
Note that $-p\ln p-(1-p)\ln(1-p)$ is the Shannon entropy
of the Bernoulli distribution, and
the largest value of the entropy is $\ln 2$, corresponding to $p=1/2$.
When the edge is too big (cf. \eqref{GenBern}),
then $f^*>1$, that is, leveraged gambling
leads to bigger long-time wealth growth than any NS-NL strategy.
The economical and financial implications of $f^*\notin [0,1]$ are
beyond the scope of our investigation and must be studied in
a broader context of risk tolerance: even when $f^*\in (0,1)$, a certain
fraction of $f^*$ can be a smarter strategy, cf. \cite[Section 7.3]{Thorp06}.
A related observation, to be further studied in the future, is that high-frequency
betting can lead to a more aggressive strategy than the ``low
frequency'' counterpart. For example, comparing \eqref{return1} and
\eqref{return1-hf}, we see that $\mu=2p-1$ and $\sigma^2=4p(1-p) < 1$
when $p\not=1/2$.
As a result, by \eqref{optf-rv},
the optimal strategy for \eqref{return1-hf} with large $n$ is
$f^*\approx (2p-1)/(4p(1-p))> 2p-1$; recall that $f^*=2p-1$ is the optimal
strategy for the simple Bernoulli model \eqref{return1}. On the other
hand, numerical simulations suggest that, in the log-normal model
\eqref{discr-ret-gauss}, \eqref{GBM-returns}, high-frequency compounding
does not always lead to larger $f^*$.
Other problems warranting further investigation include
\begin{enumerate}
\item A dynamic strategy $f=f(t)$ with a predictable process $f$;
\item A portfolio of bets, with a vector of strategies
$\mathbf{f}=(f_1,\ldots, f_N)$.
\end{enumerate}
\bibliographystyle{amsplain}
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TITLE: What's wrong with my understanding of the scheme $\text{Isom}(E_\lambda, E_{\lambda'})$?
QUESTION [12 upvotes]: Let $\mathcal{M}_{1,1}$ be the moduli stack of elliptic curves (over the complex numbers). Define
$$\begin{eqnarray*} X &:=& \Bbb{A}^1_{\lambda} - \{0,1\}\\
X' &=& \Bbb{A}^1_{\lambda'} - \{0,1\}.\end{eqnarray*} $$
There are morphisms $X \to \mathcal{M}_{1,1}$ and $X' \to \mathcal{M}_{1,1}$ given by the families of curves
$$\begin{eqnarray*} E_\lambda := V(y^2 - x(x-1)(x-\lambda)) \\
E_{\lambda' }:= V(y^2 - x(x-1)(x-\lambda')).
\end{eqnarray*}$$
By results of Grothendieck, we know that the fiber product $\text{Isom}(E_{\lambda}, E_{\lambda'}) := X \times_{\mathcal{M}_{1,1}} X'$
is a scheme. Its $T$-points are given by
$$\text{Isom}(E_{\lambda}, E_{\lambda'})(T) = (T \to X ,T \to X', {E_{\lambda}}_T \stackrel{\simeq}{\to} {E_{\lambda'}}_T ) .$$
The isomorphism above between ${E_{\lambda}}_T$ and ${E_{\lambda'}}_T$ is a $T$-isomorphism.
My goal is to try and understand why $X \to \mathcal{M}_{1,1}$ is not étale. To do this, it is enough to show that $\text{Isom}(E_{\lambda},E_{\lambda'}) \to X$ is not \'{e}tale.
Since the automorphisms of any elliptic curve in Legendre form are given by $y \mapsto cy$ and $x \mapsto ax +b$, I can see that the scheme $\text{Isom}(E_{\lambda},E_{\lambda'})$ is given by the following conditions in $\Bbb{A}^5:$
$$\text{Isom}(E_{\lambda},E_{\lambda'}) = \operatorname{Spec} \frac{\Bbb{C}[\lambda, \frac{1}{\lambda}, \frac{1}{1-\lambda},\lambda', \frac{1}{\lambda'}, \frac{1}{1-\lambda'},a,\frac{1}{a},b,c]}{(j(\lambda) - j(\lambda'), f_1,f_2,f_3,f_4)}. $$
The polynomials $f_1,f_2,f_3,f_4$ are obtained from equating the coefficients of the relation
$$ x(x-1)(x-\lambda') = \frac{(ax+b)(ax+b-1)(ax+b-\lambda)}{c^2}.$$
Explicitly, they are given by:
$$\begin{eqnarray*}
f_1 &=& a^3 - c^2 \\
f_2 &=& 3a^2b - a^2 \lambda - a^2 + a^3(\lambda' + 1) \\
f_3 &=& 3ab^2 - 2ab\lambda - 2ab + a\lambda -a^3\lambda'\\
f_4 &=& b^3 - b^2\lambda - b^2 + b\lambda.
\end{eqnarray*} $$
Now if I compute the fiber of the map $\text{Isom}(E_{\lambda}, E_{\lambda'}) \to X$ over the $\lambda = -1$ ($j = 1728$), I get the non-reduced scheme
$$\text{Isom}(E_{\lambda}, E_{\lambda'})_{-1} = \operatorname{Spec} \frac{\Bbb{C}[\lambda', \frac{1}{\lambda'}, \frac{1}{1-\lambda'},a,\frac{1}{a}, b,c]}{ \left((2 \lambda'-1)^2 (\lambda'+1)^2 (\lambda'-2)^2,f_1,f_2',f_3',f_4'\right)}$$
where
$$\begin{eqnarray*}
f_1 &=& a^3 - c^2 \\
f_2' &=& 3b +a (\lambda'+1) \\
f_3' &=& 3b^2 - a^2\lambda' - 1\\
f_4' &=& b^3 - b.
\end{eqnarray*} $$
Hence $X \to \mathcal{M}_{1,1}$ is ramified.
However: On the other hand, I have computed the cardinality of the fiber $\text{Isom}(E_{\lambda},E_{\lambda'}) \to X$ to always be 12.
Indeed consider the $\Bbb{C}$-point of $X$ corresponding to $\lambda = -1$. There are three possibilities for $\lambda'$, namely $-1,2,1/2$. An elliptic curve with $j$-invariant 1728 has automorphism group of order 4, and so the fiber over $-1$ has cardinality $3\times 4 = 12$. The story is the same for the other values of $\lambda$.
My question is: Why am I always getting 12? I am not taking into account some non-reduced issue here? I am also confused because in my head, the fiber cardinality should jump for a ramified morphism.
Edit I was wrong previously. The fiber over $\lambda = -1$ is reduced, as Macaulay2 tells me (using the command isNormal) that the same ring with coefficients in $\Bbb{Q}$ is normal, hence reduced. Tensoring with $\Bbb{C}$ over $\Bbb{Q}$ still preserves reducedness (since $\Bbb{Q}$ is perfect). The key point is that the element $(2\lambda'-1)(\lambda'+1)(\lambda'-2)$ is already in the ideal $(f_1,f_2',f_3',f_4')$ (also confirmed by Macaulay2).
REPLY [8 votes]: Let $Leg: \mathbb P^1-\{0,1,\infty\}\to \mathcal M_{1,1}$ be the Legendre map. (This associates to $\lambda$ the elliptic curve given by $y^2 = x(x-1)(x-\lambda)$.)
The coarse moduli space map $j:\mathcal M_{1,1}\to \mathbb A^1$ is of degree $1/2$.
The morphism $\mathbb P^1-\{0,1,\infty\}\to \mathbb A^1$ is of degree $6$. (Indeed, given a $j$-invariant, there are precisely 6 possibilities for the $\lambda$-invariant of that curve: $\lambda$, $1/\lambda$, $1-\lambda$, $1/(1-\lambda)$, $\lambda/(1-\lambda)$ and $(1-\lambda)/\lambda$.) In other words, the degree of $j\circ L$ is $6$.
It follows that the degree of $L$ is the degree of $j\circ L$ divided by the degree of $j$. This gives $6\times 2 = 12$.
| 207,109
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Join us for the first annual Race Against the Odds / Rhode Island on June 7th 2015 in Providence, RI. This fun and inspiring event brings family, friends and community members together to run/walk for pediatric brain cancer awareness.
Back to All Events
Earlier Event: June 7Castle Awards Half Marathon and 5k
Later Event: June 10Brooks Mile Prediction Exhibition
| 70,097
|
Okay, I have to come clean with you: I BOUGHT something…
As in, I paid more than .99 cents for it!
I have technically broken my 2014 goal for not buying any “new” clothing on myself.
To be exact, I paid $28.
As you can see I bought a couple other items… if you’re gonna break a rule than why not break it with panache?
I love this consignment shop. I’ve shopped/sold my items there for years. And you can see from the receipt, you can get just as good a deal on gently used clothes as you can at the “other” thrift shops. Those items usually have stains, holes, and smells… ew. You don’t necessarily have to go to a “bargain” center to get a legitimate bargain is what I’m trying to say 🙂
I mean, seriously, does this most spectacular vintage winter coat LOOK like it came from a “bargain” bin?
I LOVE this thing!
Santa brought me a tripod for Christmas. I was testing it out for my pictures. Now, I don’t have to bother family or random strangers for an “after” pic. It’s a beautiful thing!
AND… now we have a way to take group shots at home. Yah for tripods. It’s truly the gift that keeps on giving.
In case you’re wondering about that bow in my hair, that’s a homemade duct tape hair clip my 8 year old niece made for me. How awesome is she? Future DIY blogger? I think so 😉
So… I broke my goal a few days out from New Year’s. What can ya do? “Nothing, so don’t worry about it,” as my friend Amy always says. Accidental Seamstress will have many new goals for 2015 (that I intend to actually keep). I hope you’ll stick around to see whether or not I live up to my own challenges.
I hope everyone had a very Merry Christmas. I hope our Jewish friends had a Happy Hanukkuh. I hope the “rest of us” that celebrate Festivus survived the “Feats of Strength” with a little dignity.
Happy Holidays
| 18,976
|
Washington officials trap state's first Asian giant hornet. Now they have until mid-September to find the nests.
BLAINE, Wash. — Washington state agriculture workers have trapped their first Asian giant hornet, and have until mid-September to destroy nests before mating season begins.
The hornet was found July 14 in a bottle trap set north of Seattle near the Canadian border, and state entomologists confirmed its identity Wednesday, according to the Washington State Department of Agriculture.
The Asian giant hornet, the world's largest at 2 inches, can decimate entire hives of honeybees and deliver a painful sting to humans. Farmers in the northwest depend on those honey bees to pollinate many crops such as apples, blueberries and cherries.
The invasive insect was first documented in the state late last year and officials have said it's not known how it arrived in North America. It normally lives in the forests and low mountains of eastern and southeast Asia.
The recently-trapped hornet in Washington is the first found in a trap rather than in the environment as the state's five previous confirmed sightings were.
"This is encouraging because it means we know that the traps work," Sven Spichiger, managing entomologist for the department said in a news release. "But it also means we have work to do."
The state now plans to search for nests using infrared cameras and place additional traps that try to capture hornets alive. If they catch live hornets, the agriculture department will try to tag and track them back to their colony so the colony can be eradicated.
Officials hope to destroy any nests by mid-September, before the colony would begin creating new reproducing queens.
More hornet coverage:Hundreds of people think they've spotted the Asian giant hornet
| 313,895
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Areaware
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| 259,869
|
Vegan Style Magnum Ice Lolly
Dairy free strawberry ice cream lollies with a chocolate coated shell from Swedish Glace. What a treat, these little lollies are soooo good. The strawberry ice cream is yummy, the chocolate coating is really good, the strawberry pearls are tasty, I really like these. Swedish Glace are one of the best vegan ice cream makers out there and this version is fantastic.
Each box of 5 lollies is dairy free, suitable for vegans. Made using soya. I bought mine from a health food shop, they had to order them in for me, which took a few days. Per 100grams is 350kcal.
Recent Comments
| 372,927
|
TITLE: Find a fraction given the repeating binary expansions
QUESTION [2 upvotes]: I can do binary expansion given a fraction just fine but the question I'm stuck on says: Find fractions for the numbers with the following binary expansion: (i) $0.00\overline{110}$ and (ii) $0.0\overline{0110}$
So I know the definition for a binary expansion is as follows: The binary expansion of a number $x_0 \in [0,1)$ is a sequence $s_0,s_1,s_2,...$ in which each $s_k$ is either $0$ or $1$ and $x_0 = \sum_{k=0}^{\infty} \frac{s_k}{2^{k+1}}$
So I tried my hand at it and I got an answer close to what I need but it's not quite there.
I figured since there is a $1$ in the $\frac{1}{8}$ and $\frac{1}{16}$ decimal place I would do $\sum_{k=1}^{\infty} \frac{1}{8^k}+\frac{1}{16^k}$ which solves to $\frac{1}{7} + \frac{1}{15} = \frac{22}{105}$
When I checked my work I got $\frac{22}{105} = 0.001101011010001...$ for its binary expansion and this is very close to what I need so I felt like I am on the right track of what to do I just messed up somewhere.
Thanks in advance for any help!
REPLY [1 votes]: The rule is the same as for decimal repeating numbers, just replace $9$ with $1$; so
$$
0.00\overline{110}=\frac{110-0}{11100}=\frac{11}{1110}
$$
and, in decimal, $3/14$. Similarly,
$$
0.0\overline{0110}=\frac{110-0}{11110}=\frac{11}{1111}
$$
which, in decimal, is $3/15=1/5$. Note that $0.00\overline{110}=0.0\overline{011}$ and $0.0\overline{0110}=0.\overline{0011}$.
With bc I get, just for rough confirmation,
bc 1.06
Copyright 1991-1994, 1997, 1998, 2000 Free Software Foundation, Inc.
This is free software with ABSOLUTELY NO WARRANTY.
For details type `warranty'.
obase=2
3/14
.0011011011011011011011011011011011011011011011011011011011011011010
1/5
.0011001100110011001100110011001100110011001100110011001100110011001
| 49,713
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TITLE: Is it possible to calculate the integral $\int_{0} ^{1} \int_{0} ^{3} \int_{4y} ^{12} \frac{5 \cos{x^{2}}}{4 \sqrt{z}} dx dy dz$?
QUESTION [0 upvotes]: I'm trying to compute the triple integral $\int_{0} ^{1} \int_{0} ^{3} \int_{4y} ^{12} \frac{5\cos{x^{2}}}{4 \sqrt{z}} dx dy dz.$ I've tried every single method I know (by substitution, by parts and changing coordinates), but I'm unable of calculating it. I'd like to know if it is possible to calculate it and, in case it is, a small hint so I can solve it myself.
REPLY [1 votes]: You need to change order of first and second integrals ($dx dy \to dy dx)$ which might be tricky, but you can do it if you sketch your domain. I might add some sort of sketch later, but for now I'll just put it here without proof.
\begin{align}
\int_0^1 \int_0^3 \int_0^{4y} \frac {5\cos x^2}{4\sqrt z} dx dy dz &= \int_0^1 \int_0^{12} \int_0^{\frac 14x} \frac {5\cos x^2}{4\sqrt z} dy dx dz = \\
&= \int_0^1 \int_0^{12} \frac {5\cos x^2}{16\sqrt z} xdx dz = \int_0^1 \int_0^{12}\frac {5 \cos x^2}{32\sqrt z} d(x^2) dz = \\
&= \int_0^1 \left .\left( \frac {5\sin x^2}{32\sqrt z} \right ) \right |_0^{144}dz = \int_0^1 \frac {5\sin 144}{32 \sqrt z} dz = \\
&= \left . \left (\frac {5\sin 144}{16} \sqrt z \right ) \right |_0^1 = \frac {5\sin 144}{16}
\end{align}
Note
By some reason WA doesn't calculate it, so here's the results from Mathematica
| 101,296
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This simple & easy to install solar light by Kingavon is manufactured with 15 bright white LEDs and a 0.65W solar panel. It has a PIR (passive infrared) detection range of 2m-6m (approx) & can be used outside as a solar security light, or inside as a solar shed light. It will light for 25 seconds once motion is detected (approx) & requires 3 x AAA rechargeable batteries which are supplied. A mounting bracket, wall plugs & screws are also supplied.. Dimensions:. ; Light Unit Size: L:13cm x W:4cm x D:7.5cm. ; Solar Panel: L:17cm x W:8.5cm. ; Solar Panel Box: W:13cm x D:10cmCheck stock and price
| 72,071
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Sotheby’s was due to begin a High Court trial with Mark Weiss Limited on April 1 but the parties have settled with Weiss paying $4.2m to Sotheby’s without any admission of liability.
Sotheby’s first filed its claim in London's High Court against Mark Weiss Limited and business partner Fairlight Art Ventures in February 2017.
The case surrounds the Frans Hals painting, Portrait of a Gentleman, supplied to Sotheby’s by Weiss and subsequently sold in 2011 to a US buyer in a $10m private deal.
In 2016 Sotheby’s reimbursed the US buyer the transaction fee and in 2017 Sotheby’s said it had completed “in-depth technical analysis which established that the work was undoubtedly a forgery” with traces of a green pigment invented in the 20th century found in the picture.
In a statement issued on April 1, Weiss”.
Ongoing court case
The trial is still going ahead between Sotheby’s and Fairlight Art Ventures, Weiss’ partner in the original acquisition in 2010 and the subsequent sale through Sotheby’s in 2011. The statement added that the court will also “deal with issues arising between Mark Weiss Limited and Fairlight in connection with the claim”.
The ‘Hals’ painting is thought to have been previously owned by Frenchman Giulano Ruffini who has been linked to a number of paintings that have had their attribution questioned.
They include a picture sold by Colnaghi as a Cranach to the Prince of Liechtenstein in 2013 and an oil attributed to Parmigianino, sold at Sotheby’s in 2012.
A statement from Sotheby’s said: “We are pleased to have resolved this litigation with Mark Weiss and we remain confident in our position against Fairlight Art Ventures. Clients transact with Sotheby’s because they know we will keep our promises if problems arise: we did so in the case of the painting Portrait of a Gentleman, which Sotheby’s concluded was a fake and not by Frans Hals.”
| 414,176
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\begin{document}
\markboth{Rozanova}{The Interplay of Regularizing Factors}
\title[Small perturbations in the Model of Upper Hybrid Oscillations]{
STUDY OF SMALL PERTURBATIONS OF A STATIONARY STATE IN A MODEL OF UPPER HYBRID PLASMA OSCILLATIONS
}
\author{Olga S. Rozanova}
\address{ Mathematics and Mechanics Department, Lomonosov Moscow State University, Leninskie Gory,
Moscow, 119991,
Russian Federation,
rozanova@mech.math.msu.su}
\subjclass{Primary 35Q60; Secondary 35L40, 34E10, 34L30}
\keywords{Quasilinear hyperbolic system,
plasma oscillations, magnetic effect, small perturbations, blow up}
\maketitle
\begin{abstract}
It is shown that a constant external magnetic field, generally speaking, is not able to prevent breaking (loss of smoothness) of relativistic plasma oscillations, even if they are arbitrarily small perturbations of the zero steady state. This result sharply differs from the non-relativistic case, for which it is possible to suppress the breaking of oscillations at any initial deviations by increasing the intensity of the magnetic field \cite {RCharx}. Nevertheless, even in the relativistic case, there are subclasses of solutions corresponding to solutions that are globally smooth in time.
\end{abstract}
\section{Introduction}
The system of equations of hydrodynamics of "cold" plasma \cite {ABR78}, \cite {GR75}
has the form
\begin{equation}
\label{base1}
\begin{array}{c}
\dfrac{\partial n }{\partial t} + \Div(n \bv)=0\,,\quad
\dfrac{\partial \bp }{\partial t} + \left( \bv \cdot \nabla \right) \bp
= e \, \left( \bE + \dfrac{1}{c} \left[\bv \times \bB\right]\right),\vspace{0.5em}\\
\gamma = \sqrt{ 1+ \dfrac{|\bp|^2}{m^2c^2} }\,,\quad
\bv = \dfrac{\bp}{m \gamma}\,,\vspace{0.5em}\\
\dfrac1{c} \frac{\partial \bE }{\partial t} = - \dfrac{4 \pi}{c} e n \bv
+ {\rm rot}\, \bB\,,\quad
\dfrac1{c} \frac{\partial \bB }{\partial t} =
- {\rm rot}\, \bE\,, \quad \Div \bB=0\,,
\end{array}
\end{equation}
where
$ e, m $ are
charge and mass of the electron (here the electron charge has a negative sign: $ e <0 $),
$c$ is the speed of light;
$ n, \bp, \bv$ are the density, momentum and velocity of
electrons;
$\gamma$ is the Lorentz factor;
$ \bE, \bB $ are vectors of electric and magnetic fields.
Plasma is considered a relativistic electron (fully ionized) liquid, neglecting recombination effects and ion motion.
The full system of equations \eqref {base1} is very difficult even for numerical simulation (monograph ~\cite {Ch_book} is devoted to these questions). It is well known that even relatively small initial perturbations can cause large-amplitude oscillations. Their evolution, as a rule, leads to the emergence of a strong singularity of the electron density ~ \cite {david72}, which is commonly called {\it breaking} of oscillations. For the rest of the solution components, the breaking process means the formation of infinite gradients.
Investigation of the dynamics of a plasma placed in an external magnetic field is a separate very complex issue. Analytical approaches, as a rule, are limited to the study of linearized models, for which one has to make assumptions about the smallness of the stationary state perturbations at all times. Even such linear waves are so complex that a special classification ~ \cite {ABR78}, \cite {GR75} is introduced for them. However, linear waves cannot help in studying the breaking phenomenon. Therefore, nonlinear models are of particular value, albeit greatly simplified in comparison with the original \eqref {base1} system, but retaining its important features and allowing to assess the possibility of delaying or completely eliminating the moment of breaking. It is they that are important for analyzing the acceleration of electrons in the wake wave of a powerful laser pulse~\cite{esarey09}.
One of such classical nonlinear models is the so-called {\it upper hybrid oscillation} model. It looks like it first appeared in ~ \cite {david72}, but has been repeatedly investigated in different contexts at the physical level of rigor (for example, \cite {Kar16}, \cite {Maity13}, \cite {Maity12}, \cite { Maity13_PRL}, \cite {verma} and literature cited there). It is assumed that the plasma is in a constant external magnetic field $ \bB = (0,0, B_0) $. The model describes the nonlinear dynamics of plasma oscillations propagating perpendicular to the external magnetic field. If we assume that the oscillations depend only on one spatial variable $ x $, then we obtain the following structure of the vectors of the momentum $ \bp $ and the electric field $\bE$~\cite{david72}:
$$
\bp(x,t) = (P_1(x,t),P_2(x,t),0), \quad \bE(x,t) = (E_1(x,t),0,0).
$$
Condition of electrostaticity
\begin{equation*}
\label{rot0}
{\rm rot}\, \bE = 0,
\end{equation*}
holds automatically.
The system of upper hybrid oscillations has the form
\begin{equation}
\begin{array}{c}
\dfrac{\partial n }{\partial t} +
\dfrac{\partial }{\partial x}
\left(n\, V_1 \right)
=0,\quad
\dfrac{\partial P_1 }{\partial t} + V_1 \dfrac{\partial P_1}{\partial x}= e\left[ E_1 + \dfrac1{c}\, V_2 B_0\right],
\quad
{V_1} = \dfrac{P_1}{m \,\gamma}, \vspace{1 ex}\\
\dfrac{\partial P_2 }{\partial t} + V_1 \dfrac{\partial P_2}{\partial x}= - \dfrac{e}{c}\,V_1 B_0,
\quad {V_2} = \dfrac{P_2}{m \,\gamma}, \quad
\gamma = \sqrt{ 1+ \dfrac{P_1^2 + P_2^2}{m^2c^2}}\,, \quad \vspace{1 ex}\\
\dfrac{\partial E_1 }{\partial t} = - 4 \,\pi \,e \,n\, V_1.
\end{array}
\label{3gl2}
\end{equation}
Note that it is not a direct consequence of (\ref {base1}) system, but retains its basic features.
We introduce the dimensionless quantities
$$
\begin{array}{c}
\rho = k_p x, \quad \theta = \omega_p t, \quad
{\hat V_1} = \dfrac{V_1}{c}, \quad
{\hat P_1} = \dfrac{P_1}{m\,c}, \quad
{\hat V_2} = \dfrac{V_2}{c}, \quad
{\hat P_2} = \dfrac{P_2}{m\,c}, \vspace{1 ex}\\
{\hat E_1} = -\,\dfrac{e\,E_1}{m\,c\,\omega_p}, \quad
{\hat N} = \dfrac{n}{n_0}, \quad
{\hat B}_0 = -\,\dfrac{e\,B_0}{m\,c\,\omega_p},
\end{array}
$$
where $ \omega_p = \left(4 \pi e^2 n_0/m \right)^{1/2}$ is the plasma frequency, $n_0$ is the unperturbed electronic
density, $k_p = \omega_p /c$.
In the new variables, system ~ (\ref {3gl2}) takes the form
\begin{equation}
\begin{array}{c}
\dfrac{\partial {\hat N} }{\partial \zt} +
\dfrac{\partial }{\partial \zr}
\left({\hat N}\, {\hat V}_1 \right)
=0,\quad
\gamma = \sqrt{ 1+ {\hat P}_1^2 + {\hat P}_2^2}\,, \quad
{{\hat V}_1} = \dfrac{{\hat P}_1}{\gamma}, \quad {{\hat V}_2} = \dfrac{{\hat P}_2}{\gamma},\vspace{1 ex}\\
\dfrac{\partial {\hat P}_1 }{\partial \zt} + {\hat V}_1 \dfrac{\partial {\hat P}_1}{\partial \zr} =
- {\hat E}_1 - {\hat V}_2 {\hat B}_0,
\quad
\dfrac{\partial {\hat P}_2 }{\partial \zt} + {\hat V}_1 \dfrac{\partial {\hat P}_2}{\partial \zr}= {\hat V}_1 {\hat B}_0,
\vspace{1 ex}\\
\dfrac{\partial {\hat E}_1 }{\partial \zt} = {\hat N}\, {\hat V}_1.
\end{array}
\label{3gl3}
\end{equation}
From the first and last equations of this system it follows
$$
\dfrac{\partial }{\partial \theta}
\left[ {\hat N} +
\dfrac{\partial }{\partial \rho} {\hat E}_1 \right] = 0.
$$
Under the traditional assumption of a constant background charge density of stationary ions, this implies a simpler expression for the electron density ${\hat N}(\rho,\theta)$:
\begin{equation}
{\hat N}(\rho,\theta) = 1 -
\dfrac{\partial {\hat E}_1(\rho,\theta) }{\partial \rho}.
\label{3gl4}
\end{equation}
Formula (\ref {3gl4}) is a special case of the Gauss theorem \cite {david72}, which in differential dimensional form has the form
$ \Div \bE = 4\,\pi\,e (n - n_0).$
Using (\ref {3gl4}) in (\ref {3gl3}), we arrive at the equations, which we will analyze further:
\begin{equation}
\begin{array}{c}
\dfrac{\partial P_1 }{\partial \zt} + V_1 \dfrac{\partial P_1}{\partial \zr} =
- E_1 - B_0\,V_2,
\quad
\dfrac{\partial P_2 }{\partial \zt} + V_1 \dfrac{\partial P_2}{\partial \zr}= B_0\, V_1,
\vspace{1 ex}\\
\gamma = \sqrt{ 1+ P_1^2 + P_2^2}\,, \quad
V_1 = \dfrac{P_1}{\gamma}, \quad {V_2} = \dfrac{P_2}{\gamma},\vspace{1 ex}\\
\dfrac{\partial E_1 }{\partial \zt} + V_1 \dfrac{\partial E_1}{\partial \zr} = V_1.
\end{array}
\label{3gl5}
\end{equation}
Here, for the sake of simplicity, we remove the upper symbol from all dimensionless components of the solution.
Consider the initial conditions
\begin{equation}\label{cd1}
P_1(\rho,0) = P_1^0(\rho), \quad P_2(\rho,0) = P_2^0(\rho), \quad
E_1(\rho,0) = E^0_1(\rho), \quad
\rho \in {\mathbb R},
\end{equation}
and we will study in the half-plane $\{(\rho,\theta)\,:\, \rho \in {\mathbb R},\; \theta
> 0\}$ the solution of the Cauchy problem \eqref{3gl5}, \eqref{cd1}.
We assume that the initial data is at least $C^2$ - smooth.
System (\ref {3gl5}) is of hyperbolic type. For such systems, there exists, locally in time, a unique solution to the Cauchy problem of the same class as the initial data, in our case it is $ C^2 $. It is also known that for such systems the loss of smoothness by the solution occurs according to one of the following scenarios: either the solution components themselves go to infinity in a finite time, or they remain bounded, but their derivatives \cite {Daf16} turn to infinity. The latter possibility is realized, for example, for homogeneous conservation laws, which include the equations of gas dynamics, where the appearance of a singularity corresponds to the formation of a shock wave.
The article is structured as follows. In Section \ref{S2}, we recall the well-known results concerning the system (\ref {3gl5}) in the special case $ B_0 = V_2 = 0 $, as well as in the non-relativistic case. In Section \ref{S3}, we find the first integrals of the solution, depending on which we classify the initial data, construct solutions in the form of a traveling wave, and also consider the extended characteristic system. In Section \ref{S4}, we consider the cases of reduction of the extended characteristic system depending on the properties of the first integrals and make changes of variables that allow linearizing it. In Section \ref{S5}, we study small deviations from the equilibrium state, estimate the effect of an external magnetic field on the time of breaking of oscillations, and make a comparison with the nonrelativistic case where possible. In Section \ref{S6}, we summarize and consider the prospects for further research.
\section{Known results}\label{S2}
{\bf 1.} Let us consider a relativistic analogue of system ~ \eqref {3gl5}, where the speed of particles much less than the speed of light. This leads to the conditions $ \gamma = 1 $ and $ \bp = \bv $, and system ~ \eqref {3gl5} takes the form
\begin{equation}
\begin{array}{c}
\dfrac{\partial V_1 }{\partial \zt} + V_1 \dfrac{\partial V_1}{\partial \zr} =
- E_1 - B_0\,V_2,
\qquad
\dfrac{\partial V_2 }{\partial \zt} + V_1 \dfrac{\partial V_2}{\partial \zr}= B_0\, V_1,
\vspace{1 ex}\\
\dfrac{\partial E_1 }{\partial \zt} + V_1 \dfrac{\partial E_1}{\partial \zr} = V_1.
\end{array}
\label{3gl5non}
\end{equation}
This system is much simpler for analysis than ~ \eqref {3gl5}, in \cite {RCharx} a criterion for the formation of singularities in terms of the initial data
\begin{equation}\label{cd2}
V_1(\rho,0) = V_1^0(\rho), \quad V_2(\rho,0) = V_2^0(\rho), \quad
E_1(\rho,0) = E_1^0(\rho), \quad
\rho \in {\mathbb R},
\end{equation}
from the class $C^2$ is obtained.
\begin{theorem} \label{T1}\cite{RCharx} For the existence of a $ C^ 1$ - smooth $ \frac {2 \pi} {\sqrt {1 + B_0^2}} $ - periodic solution $ V_1 (\theta, \rho), \,V_2 (\theta, \rho), \, E_1 (\theta, \rho) $ of the problem (\ref {3gl5non}), (\ref {cd2}) it is necessary and sufficient that at any point $ \rho \in \mathbb R $ the condition
\begin{equation} \label {crit2}
\Delta= \left( (V_1^0)' \right) ^ 2 + 2 \, (E_1^0)' +2 B_0\, (V_2^0)' - B_0^2 -1 <0
\end {equation}
holds.
If the opposite inequality holds at least at one point $ \rho_0 $, then the derivatives of the solution turn to infinity in a finite time.
\end{theorem}
This result says, in particular, that it is easy to construct initial data of a sufficiently general type corresponding to a globally smooth solution in time. For example, if we fix arbitrary initial data and increase $ | B_0 | $, then we are in just such a situation. Thus, the external magnetic field has a regularizing character.
\bigskip
{\bf 2.} Taking relativistic effects into account significantly changes the situation, and the initial data leading to a globally smooth solution must already be selected in a very special way. In the general case, the presence of a relativistic factor acts as a kind of nonlinear resonance, which leads to breaking of oscillations in a certain finite, but possibly quite long time. For the particular case of system \eqref {3gl5}, corresponding to $ B_0 = P_2 = 0 $, such a problem was solved in \cite {RChZAMP21}. The condition on the initial data, which distinguishes the class of solutions for which a globally smooth solution is possible and, in particular, a traveling wave, looks like
\begin{equation}\label{fi1}
2\sqrt{1+(P_1^0)^2}+(E_1^0)^2\equiv {\rm const}, \quad \rho_0\in \mathbb R.
\end{equation}
In \cite {RChZAMP21} (Theorem 2), a criterion for the formation of singularities in terms of the initial data for this case is obtained.
If the condition \eqref {fi1} is not met, then any small deviation of the initial data from the equilibrium $ P_1 = E_1 = 0 $ leads to the gradient catastrophe.
The aim of this work is to analyze the influence of small perturbations of the initial data for the full system \eqref {3gl5}. This is a much more complicated problem than the one that was solved for a particular case, which is associated with an increase in the dimension of the phase space of the corresponding characteristic system.
\vspace{1.5em}
\section{Analysis of the characteristic system}\label{S3}
Let us write system \eqref {3gl5} in characteristic form
\begin{eqnarray}\label{char2r}
\dfrac{dP_1}{d\theta}&=&-E_1-B_0\,V_2,\quad \dfrac{dP_2}{d\theta}=B_0 \,V_1,\quad\dfrac{dE_1}{d\theta}=V_1,\\ \dfrac{d\rho}{d\theta}&=&V_1,\quad V_i=\frac{P_i}{\sqrt{1+P_1^2+P_2^2}},\,i=1,2.\nonumber
\end{eqnarray}
First integrals of
\ \eqref {char2r} are
\begin{equation}\label{fer}
P_2-B_0 E_1 = K_1, \quad 2\sqrt{1+P_1^2+P_2^2}+E_1^2=K_2.
\end{equation}
Knowledge of the first integrals allows us to express $ E_1 $ and $ P_1 $ in terms of $ P_2 $ and from the second equation \eqref {char2r} obtain an equation for finding $ P_2 (\theta) $ along the characteristic outgoing from the point $\rho_0\in \mathbb R$:
\begin{equation}\label{P2rr}
\dfrac{dP_2}{d\theta}=\pm B_0 \frac{\sqrt{(B_0^2 K_2-(P_2-K_1)^2)^2-4 B_0^4(P_2^2+1)}}{B_0^2 K_2-(P_2-K_1)^2},
\end{equation}
where $K_i=K_i(\rho_0, 0)$, $i=1,2$. The corresponding constants can be calculated according to \eqref {fer} and, generally speaking, depend on the starting point of the characteristic.
Equation \eqref{P2rr} also allows to calculate the period of oscillation along a specific characteristic, namely:
\begin{equation*}\label{per_r}
T(\rho_0)= \frac{2}{B_0} \int\limits_{P_2^-}^{P_2^+} \frac{B_0^2 K_2-(\eta-K_1)^2}{\sqrt{(B_0^2 K_2-(\eta-K_1)^2)^2-4 B_0^4(\eta^2+1)}} \, d\eta,
\end{equation*}
where $P_2^\pm$ are smaller and larger roots of the equation $$(B_0^2 K_2-(\eta-K_1)^2)^2-4 B_0^4(\eta^2+1)=0.$$
They are chosen such that $P_2(\rho_0,0)\in (P_2^-,P_2^+)$. The specified integral can be expressed in elliptic functions. As in the case of $ B_0 = 0 $, the period of $ T (\rho_0) $ varies from point to point. However, the formula for its determination is much more complicated than the one obtained in \cite {RChZAMP21}, since to do this, you must first express $ P_2 $ in terms of $ P_1 $.
In the case under consideration, traveling waves can also be constructed. To do this, you can use equation \eqref {P2rr}, fixing in it $K_1 $ and $ K_2 $. If $P_2={\mathcal P}(\xi) $, $\xi=\rho-w \,\theta$, $w=\rm const$, then the equation for the profile of $ \mathcal P $ is
\begin{equation}\label{P2r}
\dfrac{d\mathcal P}{d\xi}=\pm B_0 \frac{\sqrt{(B_0^2 K_2-({\mathcal P}-K_1)^2)^2-4 B_0^4({\mathcal P}^2+1)}}{-(B_0^2 K_2-({\mathcal P}-K_1)^2)\,w+2 B_0^2{\mathcal P}}.
\end{equation}
We obtain an extended system describing the behavior of the derivatives of the solution along the characteristics:
\begin{equation}\label{char2dr}
\dfrac{dp_1}{d\theta}=-q_1 p_1-B_0 q_2 -e,\quad \dfrac{dp_2}{d\theta}=-q_1 p_2+B_0 q_1,
\quad\dfrac{de}{d\theta}=(1-e)q_1,
\end{equation}
where $q_1=\partial_\rho V_1
$, $q_2=\partial_\rho V_2
$, $p_1=\partial_\rho P_1
$, $p_2=\partial_\rho P_2
$,
$e=\partial_\rho E_1$,
$$q_i=\frac{p_i}{\gamma}-\frac{P_i}{\gamma^3}(p_1 P_1+p_2 P_2), i=1,2, \quad \gamma=\sqrt{1+P_1^2+P_2^2}.$$
System \eqref{char2dr} has the first integral
\begin{equation}\label{C1r}
p_2=B_0+ C_1(e-1), \quad C_1=\frac{p_2(\rho_0,0)-B_0}{e(\rho_0,0)-1},
\end{equation}
and can be reduced to two equations
\begin{eqnarray}
\dfrac{dp_1}{d\theta}= &=& -1-\frac{p_2-B_0}{C_1} -\frac{p_1^2}{\gamma}+p_1 P_1 Q- B_0\left(\frac{p_2}{\gamma}-P_2Q\right),\label{s1}\\
\dfrac{dp_2}{d\theta}= &=& (B_0-p_2)\left(\frac{p_1}{\gamma}-P_1 Q\right),\quad Q=\frac{P_1 p_1+P_2 p_2}{\gamma^3}.\label{s2}
\end{eqnarray}
If we remember that $ P_1 $ can be expressed in terms of $ P_2 $, and equation \eqref {P2rr} for $ P_2 $ is known, then it becomes clear that the analysis of the derivatives of the solution to problem \eqref {cd1} is reduced to the analysis of an autonomous system \eqref {s1}, \eqref {s2}, \eqref {P2rr}, and the last equation is decoupled. Thus, formally, the situation is the same as that studied in the relativistic case in the absence of a magnetic field \cite {RChZAMP21}. However, in the presence of an external magnetic field, the system is much more cumbersome.
\bigskip
\section{Cases of reduction and transformation of the system for derivatives}\label{S4}
\subsection{Reduction to one equation}
Let $ p_2 = B_0 $, which corresponds to $ C_1 = 0 $ in the integral \eqref{C1r}.
In this case, system \eqref {s1}, \eqref {s2} can be reduced to one equation. Indeed, \eqref {s2} holds identically, and \eqref {s1} reduces to
\begin{eqnarray}
\dfrac{dp_1}{d\theta}= &=& -1 -\frac{B_0^2}{\gamma^3}-\frac{p_1^2}{\gamma^3}-\frac{(p_1 P_2-B_0P_1)^2}{\gamma^3},\nonumber
\end{eqnarray}
whence it follows that any solution with initial data \eqref {cd1} with $ P_2 ^ 0 = B_0 \rho + \rm const $ loses smoothness in a finite time interval.
\bigskip
\subsection{The case of constant $ K_2 $.}
A more meaningful subclass of solutions are solutions for which the second of integrals \eqref {fer}, which coincides with \eqref {fi1} for the particular case $ B_0 = P_2 = 0 $, is identically constant, that is
\begin{equation}\label{K2}
2 \sqrt {1 +(P_1^0(\rho))^2+ (P_2^0(\rho))^2} +(E_1^0(\rho))^ 2 \equiv K_2={\rm const}, \, \rho\in \mathbb R.
\end{equation}
For such solutions
\begin{eqnarray} e=-\frac{p_1 P_1+p_2 P_2}{E_1(P_1,P_2)\gamma}, \quad E_1(P_1,P_2)=\pm \sqrt{K_2-2\gamma}.\label{e}\end{eqnarray}
Since \eqref{C1r} implies $e=1+\frac{p_2-B_0}{C_1}$ for $C_1\ne 0$, then
there is a relationship between $ p_1 $ and $ p_2 $. If we denote $ s = p_2-B_0 $, then \eqref {s2} reduces to system
\begin{eqnarray}
\dfrac{ds}{d\theta} &=& -L_1 s^2-L_2 s, \label{s21}\\ L_1&=&-\frac{(1+P_2^2)(E_1(P_1,P_2)\gamma+C_1 P_2)}{C_1 \gamma^3 P_1}-\frac{P_1 P_2}{\gamma^3},\nonumber \\
L_2&=&-\frac{(1+P_2^2)(E_1(P_1,P_2)\gamma+ P_2)}{\gamma^3 P_1}-\frac{B_0 P_1 P_2}{\gamma^3}.
\nonumber
\end{eqnarray}
After replacing $ y = s ^ {- 1} $ the equation \eqref {s21} becomes linear:
\begin{eqnarray}
\dfrac{dy}{d\theta} &=& L_1 +L_2 y. \label{s22}
\end{eqnarray}
\bigskip
\subsection{The case of non-constant $ K_2 $.}
Suppose that the condition \eqref {K2} is not satisfied, and the variables $ e, p_1, p_2 $ are independent. We introduce new variables $u=\frac{e}{p_1}$, $\lambda=\frac{e-1}{p_1}$ and $\sigma=\frac{p_2}{p_1}$.
They satisfy the system
\begin{eqnarray}
\dfrac{d u}{d\theta} &=& u^2+\frac{B_0\,(1+P_1^2)}{\gamma^3}u\sigma -\frac{B_0\,P_1 P_2}{\gamma^3}u -\frac{ P_1 P_2}{\gamma^3}\sigma+\frac{1+P_2^2}{\gamma^3},\label{ulam1}\\
\dfrac{d \lambda}{d\theta} &=&\lambda \left(u+ \frac{B_0\,(1+P_1^2)}{\gamma^3}\sigma -\frac{B_0\,P_1 P_2}{\gamma^3}\right),\label{ulam2}
\\
\dfrac{d\sigma}{d\theta} &=& \frac{B_0\,(1+P_1^2)}{\gamma^3}\sigma^2 +u\sigma -\frac{2 B_0\, P_1 P_2}{\gamma^3}\sigma +\frac{B_0\,(1+P_2^2)}{\gamma^3}.\label{ulam3}
\end{eqnarray}
For $ B_0 = P_2 = 0 $ this system decomposes and equations \eqref {ulam1}, \eqref {ulam2} coincide with the corresponding system from \cite{RChZAMP21}.
Notice that
$$
e=\frac{u}{u-\lambda}, \quad p_1=\frac{1}{u-\lambda}, \quad p_2=B_0+C_1 \frac{\lambda}{u-\lambda},\quad \sigma=B_0 u+(C_1-B_0)\lambda,
$$
so $ \sigma $ can be excluded from the system \eqref{ulam1} -- \eqref{ulam3} and we get two equations
\begin{eqnarray}\label{ulam4}
\dfrac{d u}{d\theta}&=& \left(1+B^2_0\,F_1\right) \,u^2+B_0(C_1-B_0)\,F_1\,u\lambda -2B_0\,F_2 u \nonumber\\&& -(C_1-B_0) F_2\lambda+F_3,\nonumber\\
\dfrac{d \lambda}{d\theta} &=&\left(1+B^2_0\,F_1\right)\,u\lambda + B_0(C_1-B_0)\,F_1\,\lambda^2 -B_0\,F_2\lambda,
\end{eqnarray}
where
\begin{eqnarray}
F_1=F_1(\theta)=\frac{1+P_1^2}{\gamma^3}, \quad F_2=F_2(\theta)=\frac{P_1 P_2}{\gamma^3}, \quad
F_3=F_3(\theta)=\frac{1+P_2^2}{\gamma^3}.\nonumber
\end{eqnarray}
\subsection{Non-constant $ K_2 $, case of constant $ K_1 $.}
Consider a subclass of solutions determined by the condition $P_2-B_0 E_1 \equiv K_1$ (see \eqref{fer}), for which $C_1=B_0 $, and the system \eqref {ulam4} takes the form
\begin{eqnarray}
\dfrac{d u}{d\theta} &=&M_1 u^2 - 2 M_2 u + M_3, \qquad
\dfrac{d \lambda}{d\theta} =\lambda (M_1 u - M_2),
\label{u}\\
M_1(\theta)&=&1+
B^2_0\,F_1\ge 1>0, \quad M_2(\theta)= B_0\,F_2,\quad M_3(\theta)=F_3.\nonumber
\end{eqnarray}
The first equation \eqref {u} is then separated from the system. Using standard variable substitutions
\begin{eqnarray}\nonumber
u(\theta)=-\frac{r'(\theta)}{M_1(\theta) r(\theta)}, \quad r(\theta)=z(\theta) \exp \left(-\int\limits_0^\theta \left(M_2(\tau)-\frac{M'_1(\tau)}{2 M_1(\tau)}\right)\, d\tau \right)
\end{eqnarray}
\eqref {u} reduces to Hill's equation
\begin{eqnarray}\label{Hill}
\dfrac{d^2 z}{d\theta^2}&+&K(\theta) z=0,\\%\quad z(0)=1,\, z'(0)=-u_0,\\
\label{Kthet}
K(\theta)&=&M_1 M_3 -M_2^2-M_2'-\frac{3}{4}\frac{(M_1')^2}{(M_1)^2}+\frac{M_1''+2 M_2 M_1'}{2 M_1}.
\end{eqnarray}
\subsection{Non-constant $ K_2 $, general case.}
Note that $ \lambda $ does not vanish for finite $ p_1 $, and we make one more change of variables: $q_1=\frac{u}{\lambda}$, $q_2=\lambda^{-1}$. With respect to the new variables, we obtain a linear inhomogeneous system of equations
\begin{eqnarray}
\dfrac{d q_1}{d\theta} &=& - B_0 F_2 \, q_1 + F_3 \, q_2-(C_1- B_0) F_2, \label{q1q2_1}\\
\dfrac{d q_2}{d\theta} &=& -(1+B_0^2 F_1) \, q_1 + B_0 F_2 \, q_2-B_0(C_1- B_0) F_1.
\label{q1q2_2}
\end{eqnarray}
This system can be reduced to a linear inhomogeneous second-order equation for $q_1(\theta)$:
\begin{eqnarray*}
\dfrac{d^2 q_1}{d\theta^2}&+&G_3(\theta) \dfrac{d q_1}{d\theta}+G_2(\theta) q_1 + G_1(\theta)=0,\\
G_1&=&(C_1-B_0) \left(F'_2-B_0 F_2^2+B_0 F_1 F_3- \frac{F_2 F_3'}{F_3}\right),\\
G_2&=&F_3(1+B_0^2 F_1)- F_2^2 B_0^2+ B_0 F_2'-\frac{ B_0 F_2 F_3'}{F_3},\\
G_3&=&-\frac{F_3'}{F_3},
\label{G}
\end{eqnarray*}
whence, in turn, using the standard change of variables
\begin{eqnarray} \label{q1w}
q_1(\theta)= e^{-\frac12\,\int\limits_0^\theta G_3(\zeta) d\zeta}\, w(\theta)= \frac{w(\theta) \sqrt{F_3(\theta)}}{\sqrt{F_3(0)}}
\end{eqnarray}
the term with the first derivative can be excluded:
\begin{eqnarray} \label{N}
\dfrac{d^2 w}{d\theta^2}+N_2(\theta) w + N_1(\theta)=0,\\
N_1=G_1 e^{\frac12\,\int\limits_0^\theta G_3(\zeta) d\zeta}=\frac{G_1 \sqrt{F_3(0)}}{\sqrt{F_3}},\quad
N_2=G_2-\frac14 G_3^2-\frac12 G_3'.\nonumber
\end{eqnarray}
It is easy to check that for constant $ K_1 $, when the equality $ C_1 = B_0 $ holds, the equation \eqref {N} is homogeneous and coincides with \eqref {Hill}.
Since $ q_1 = \dfrac {e} {e-1} $, in order to determine whether $ e $ goes to infinity (together with $ p_1 $ and $ p_2 $), it is enough to determine whether there is a time $ \theta_* $ such that $ q_1 (\theta_*) = 1 $. However, it is impossible to obtain an explicit form of the solution of system \eqref {q1q2_1}, \eqref {q1q2_2}, despite the fact that the dependence of the functions $ F_1, \, F_2, \, F_3 $ on time can be considered known. It is for this reason that we have to confine ourselves to the study of small deviations from the equilibrium state.
\section{Study of small deviations from the equilibrium state}\label{S5}
In \cite {RChZAMP21}, we proved that in the general situation a solution corresponding to an arbitrarily small deviation from the zero equilibrium state, in the relativistic case, necessarily loses its smoothness. Below investigate whether this result remains valid in the presence of an external magnetic field.
\subsection{Solutions with non-constant $ K_2 $}
\begin{theorem}\label{T2}
Let the initial data \eqref {cd1} be such that
$$ 2 \sqrt {1 +(P_1^0(\rho))^2+ (P_2^0(\rho))^2} +(E_1^0(\rho))^ 2 $$ does not equal identically to a constant. Then
derivatives of any solution to the Cauchy problem \eqref {3gl5}, \eqref {cd1} corresponding to an arbitrarily small deviation of the initial data \eqref {cd1}
from the state $ P_1 = P_2 = E_1 = 0 $ turn to infinity in a finite time.
\end{theorem}
\proof
1. Let us prove the theorem first for the case $P_2^0(\rho)-B_0 E_1^0(\rho) = K_1, \, \rho\in \mathbb R,$
when the equation \eqref {N} is homogeneous and coincides with \eqref {Hill}.
We modify the method used in \cite {RChZAMP21} and show that any solution of equation \eqref {Hill} corresponding to a small deviation from the equilibrium position is oscillating with increasing amplitude. To do this, we have to do cumbersome, but standard calculations. To distinguish a class of small perturbations, we assume that $K_2=2+\epsilon^2, $ $K_1=0$, $\epsilon\ll 1$. Then, leaving on the right-hand side of \eqref {P2rr} terms of order at most two, we obtain
\begin{equation*}
P_2(\theta)= \epsilon \frac{B_0}{\sqrt{1+B_0^2}}\,\sin(\sqrt{1+B_0^2} \theta)+O(\epsilon^2),\label{P2}
\end{equation*}
where we set the phase to be zero without loss of generality. The $ P_1 (\theta) $ function is found from the condition
\begin{equation*}
P^2_1=\frac14 \left(2+\epsilon^2 -\frac{P_2^2}{B_0^2}\right)^2-P_2^2-1.
\label{P1}
\end{equation*}
After substituting $ P_1 $ and $ P_2 $ into \eqref {Kthet} and expanding this expression in a series in $ \epsilon $ up to a power not higher than two, we obtain
\begin{eqnarray}\label{K_series}
K(\theta)=\hat a-2\hat b \cos (2\sqrt{1+B_0^2}\theta )&+& O(\epsilon^4),\\
\hat a=(1+B_0^2)-\frac{8 B_0^2(1+B_0^2)+3}{4(1+B_0^2)}\epsilon^2,&\quad &\hat b= \frac{8 B_0^2(1+B_0^2)+3}{8(1+B_0^2)}\epsilon^2.\nonumber
\end{eqnarray}
Let us make the change $\tau=\sqrt{1+B_0^2} \theta$.
If we neglect the terms higher than the second order in \eqref {K_series} and substitute the result in \eqref {Hill}, we get the well-studied Mathieu equation (e.g.,\cite {BEr67}, Chapter 16)
\begin{eqnarray}\label{Matier}
\dfrac{d^2 z}{d \tau^2}+(a-2 b \cos 2 \tau) z=0,\quad
a=\frac {\hat a}{1+B_0^2},\quad b=\frac {\hat b}{1+B_0^2}.
\end{eqnarray}
According to the Floquet theory, the boundedness or unboundedness of the solution of such an equation, together with its derivative, is completely determined by its characteristic exponent $ \mu $, defined as the solution to the equation $\cosh \mu
\pi = z_1(\pi),$ where $z_1(\theta)$ is the solution of the Mathieu equation with initial conditions
$z_1(0)=1$, $z_1'(0)=0$. Unboundedness takes place for real $ \mu $, that is, if $|\cosh \mu \pi|>1$. According to the asymptotic formula \cite {BEr67}, Section 16.3 (2), which (taking into account the misprint in the sign in the formula from Section 16.2 (15)) has the form
$$\cosh \mu \pi = \cos
\sqrt a \pi-\frac{\pi b^2}{(1-a)\sqrt a }\sin \sqrt a \pi
+O(b^4),\,b\to 0,$$ we get
\begin{equation}\label{K1B0}
\cosh \mu
\pi=-1-\frac{\pi^2}{2048}\frac{(8 B_0^2(1+B_0^2)+3)^3}{(1+B_0^2)^6}\epsilon^6+O(\epsilon^8)<-1.
\end{equation}
We see that $ K(\zt) > 0 $ (see \eqref{Hill}, \eqref{Kthet}), therefore, any solution to the Mathieu equation oscillates.
Since $ u = -\frac{r '}{M_1 r} $, then according to the second equation \eqref{Hill},
$ \frac{\lambda '}{\lambda} = - \frac{r '}{r}-M_2 $, that is $\lambda r =\lambda(0) e^{-\int\limits_0^\theta M_2(\zeta) d\zeta}$.
We set without loss of generality $ z (0) = r(0)=1 $, then $$ r '(0) = - M_1(0),\quad z'(0)= - M_1(0)+M_2(0)-\frac{M_1'(0)}{2M_1(0)}.$$ Further,
$$
\dfrac{u}{\lambda} = \dfrac{e}{e-1} = -\,\dfrac{r'}{M_1 r \lambda}=-\frac{r'}{M_1\lambda(0) e^{-\int\limits_0^\theta M_2(\zeta) d\zeta}}
=\frac{-z'+(M_2-\frac{M_1'}{2M_1})z}{\sqrt{M_1}\lambda(0)}.
$$
From these considerations it is clear that if $ e $ at some point $ \theta = \theta_* $ becomes infinity, then at this point
$ \frac {-z '+ (M_2- \frac {M_1'} {2M_1}) z} {\sqrt {M_1} \lambda (0)} $ becomes $ 1 $. If there is no such moment, then $ e $, and with it $ p_1 $ and $ p_2 $, remain bounded.
However, as we have shown, any solution of \eqref {Matier} oscillates and the amplitude of its oscillations increases. Therefore, any linear combination of a solution with its derivative has the same property (note that the expression $ M_2- \ frac {M_1 '} {2M_1} $ for all $ \theta $ remains bounded). Therefore, $ \dfrac {u} {\lambda} $ at some point in time will reach any predetermined constant, including one.
So, we have proved the theorem for the case of constant $ K_1 $.
\bigskip
2. Let us consider the general case of non-constant $ K_1 $.
The first terms of the expansion of inhomogeneity of equation \eqref {N} are of the form
\begin{equation*}
N_1(\theta)=B_0 (B_0-C_1)\left(1+\frac{5+12 B_0^2}{8(1+B_0^2)}\epsilon^2- \frac{3+10 B_0^2}{8(1+B_0^2)}\epsilon^2 \, \cos (2\sqrt{1+B_0^2}\theta ) \right)+ O(\epsilon^4).
\end{equation*}
This is a bounded continuous function for $ \theta> 0 $. We denote by $ \bar w (\theta) $ a fixed particular solution of the equation \eqref {N}, for example, with the initial conditions $ \bar w (0) = \bar w '(0) = 0 $. It is easy to see that in the zero approximation this is a bounded function. Approximations of the following orders are found from linear second-order equations with constant coefficients and do not contain resonance terms, that is, they are also bounded. The general solution \eqref {N} consists of the general solution of the corresponding homogeneous equation, which, as we proved above, is oscillating with an exponentially growing amplitude, and $ \bar w (\theta) $. Thus, for any initial conditions, $ w (\theta) $ in a finite time will reach any predetermined constant in a finite time. Therefore, $ q_1 $, which, according to \eqref {q1w}, is obtained by multiplying $ w $ by a bounded function, will reach the value 1 in a finite time.
Thus, Theorem 2 is completely proved.
\subsubsection{Influence of $ B_0 $ on the breaking time.}
As follows from Theorem 2, in the nonrelativistic case, an increase in $ | B_0 | $ always extends the class of smooth initial data, except for the case of constancy of the integral $ K_1 $ for all characteristics. In this exceptional case, $ B_0 $ participates in the selection of the initial data and its increase has the opposite effect.
In the relativistic case, in the situation where the integral $ K_1 $ is constant, a similar phenomenon also arises of a decrease in the lifetime of a smooth solution with increasing $ | B_0 | $. Indeed, as follows from \eqref {K1B0},
the correction to the quantity responsible for the exponential growth of the oscillation amplitude has the form
$$-\frac{\pi^2}{2048}\frac{(8 B_0^2(1+B_0^2)+3)^3}{(1+B_0^2)^6}\epsilon^6+o(\epsilon^6).$$
As $ | B_0 | $ increases from zero, this correction does increase in absolute value, but it has a limit at $ | B_0 | \to \infty $, which indicates that the dependence of the time of breaking of oscillations on the strength of the magnetic field ceases to be significant at large $ | B_0 | $.
When $ K_1 $ is not constant, the conclusions are not so unambiguous. The problem is that here it is necessary to take into account the influence of the particular solution of equation \eqref {N}, which is also oscillating and the oscillations can be mutually canceled out.
In addition, it should be noted that the exponential growth of the oscillation amplitude of the solution to the homogeneous equation manifests itself in a significantly larger order of smallness in $ \epsilon $, therefore, at sufficiently small times, the processes associated with the linearized equation, that is, with terms of order zero in $ \epsilon $. But this means that we are in the framework of the nonrelativistic model, for which, in the general case, the increase in $ | B_0 | $ has a regularizing character.
\bigskip
\subsection{Solutions with constant $ K_2$.}
It is clear that in this case not all solutions lose smoothness in a finite time. Indeed, a traveling wave solution \eqref {P2r} falls into the class of solutions with constant $ K_2 $. Such a solution can be constructed in the form of a small deviation from the zero equilibrium position, and its derivatives do not vanish into infinity.
\begin{theorem}\label{T4}
Let the initial data \eqref {cd1} be such that $$ 2 \sqrt {1 +(P_1^0(\rho))^2+ (P_2^0(\rho))^2} +(E_1^0(\rho))^ 2 \equiv K_2={\rm const}, \, \rho\in \mathbb R.$$ Then there is $ \epsilon_0> 0 $ such that
the solution of the Cauchy problem \eqref {3gl5}, \eqref {cd1} corresponding to a perturbation of the initial data \eqref {cd1} from the state $ P_1 = P_2 = E_1 = 0 $ of order $ \epsilon <\epsilon_0 $ keeps $ C^1 $ - smoothness if and only if at the initial moment of time the condition
\begin{equation}\label{K2c}
2e+2B_0 p_2-2B^2_0-1<0
\end{equation}
holds.
\end{theorem}
\bigskip
For the {\it proof}, as in the previous subsection, we put $K_2=2+\epsilon^2, $ $K_1=0$, $\epsilon\ll 1$, and get
$$
\begin{array}{c}
P_2(\theta)= \frac{\epsilon B_0}{\sqrt{1+B_0^2}}\,\sin(\sqrt{1+B_0^2} \theta)+O(\epsilon^2),\\ P_1(\theta)= \epsilon \,\cos(\sqrt{1+B_0^2} \theta)+O(\epsilon^2),\\
E_1(\theta)= \frac{\epsilon}{\sqrt{1+B_0^2}}\,\sin(\sqrt{1+B_0^2} \theta)+O(\epsilon^2),\quad \epsilon\to 0.
\end{array}
$$
We use equation \eqref {s22}. Its solution has the form
$$
y(\theta)= \left(y(0)+B_0+\frac{1}{C_1}\right)\,\left(\cos(\sqrt{1+B_0^2}\theta)\right) ^{\frac{1}{1+B_0^2}}-\left(B_0+\frac{1}{C_1}\right)+O(\epsilon).
$$
It is easy to calculate that the zero approximation $ y (\theta) $ does not vanish, that is, $ p_2 $, together with the rest of the derivatives, does not vanish
provided that at the initial moment inequality \eqref{K2c} holds.
\subsubsection{Comparison with the nonrelativistic case.} Let us compare this result with Theorem 1.
Considering that $ P_2 (0) = E_1 (0) $ according to the assumption that $ K_1 = 0 $ and $ p_1 (0) = 0 $
(the value of $ p_1 $ can be found from \eqref {e}), then \eqref{K2c} can be compared with the necessary and sufficient condition for the preservation of smoothness by the solution in the nonrelativistic case \eqref {crit2}, which in this situation has the form $$ 2e + 2B_0 p_2- B ^ 2_0-1 <0. $$ This comparison shows how much more stringent condition guarantees the non-overturning of sufficiently small oscillations in the relativistic case. In addition, we note that the presence of a magnetic field expands the class of initial data corresponding to globally in time smooth solutions.
\section{Conclusion}\label{S6}
We have investigated the simplest model of relativistic oscillations of a cold plasma in a constant magnetic field. It was found that even small perturbations of the trivial state of rest, as a rule, eventually lead to the formation of singularities of the solution (breaking of oscillations). However, at short times, the nature of the solution is largely determined by the leading terms of the expansion in a small parameter, which allows us to conclude that the external magnetic field has a regularizing effect. However, if in the nonrelativistic case the breaking can be completely eliminated by means of an external magnetic field, but in the relativistic case it can only be delayed.
The success of the study in this case is determined by the fact that the system of partial differential equations is nonstrictly hyperbolic, which makes it possible to write an extended system along one characteristic direction and thereby reduce it to a problem for ordinary differential equations. For the complete, unreduced, cold plasma model, this property no longer exists. However, there is every reason to believe that the technique developed in this paper can help in "splitting into processes" for the full model, where the process of breaking of oscillations is accompanied by their wave transfer.
\section*{Acknowledgments}
Supported by the Ministry of Education and Science of the Russian Federation as part of the program of the Moscow Center for Fundamental and Applied Mathematics under the agreement 075-15-2019-1621.
The authors are grateful to E.V.Chizhonkov for stimulating discussions and constant interest.
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The the Blue Zone should put materials out by 6 a.m. on Monday, April 19, and Monday, May 17, 2021.
Crews will begin picking up picking up materials on Monday of the scheduled week and work through the zone on consecutive days until all materials are collected. If materials have not been collected two days after your scheduled collection, please contact the Public Works Department at 217-403-4700 to report.
Program Details:
- The collection is available to all residential properties within the Champaign city limits.
- Materials must be in 30-gallon paper yard waste bags. NO PLASTIC BAGS OR CONTAINERS.
- Bags should be placed within 10 feet of the curbside, and five feet away from obstacles like mailboxes, trees, telephone poles, and fire hydrants.
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A Brief Overview Of THE MOST FAMOUS Casinos In The World
There are so benefits to playing online casino Korea. For one, playing is completely based on luck. However, the majority of the games are quite progressive, which means that a person’s potential for wining is really as high as their finest. However, many players say that even the virtual casinos offer actual money prizes, when in fact this isn’t the case. It all depends upon how lucky you’re.
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Along with all of this, another phrase often useful for Korean casino korea is “gambling land”. This phrase is normally found in a joking way, nonetheless it has some underlying significance. Korean culture is extremely competitive. In order to succeed, it is necessary to be constantly improving. That is why many Koreans take part in these facilities.
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We sort through the various drive modes in the 2021 Audi RS 6 Avant and 2021 Mercedes-AMG E 63 S Wagon. Wireless Phone Charger for Volvo S60 2013-2018 #1 Wireless Car Chargers Manufacturer, with over 100 styles designed for different automotive maker and model, best match your car interior, factory … These cookies …. But ads are also how we keep the garage doors open and the lights on here at Autoblog - and keep our stories free for you and for everyone. Autoblog West Coast Editor James Riswick shows what you get with the new Tremor package. Volvo low battery warning may come up for several reasons, but the most common problem is a partially discharged battery. Check that the phone supports wireless charging (Qi). The Volvo charging solution is compatible with the universal Qi standard. Excellent Charging Experience:Support charge your phones simultaneously with the maximum output of 10 watt. 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Sit Up In Class: This Brotha Gave A Thought Provoking Lesson To A Cop That Had His Gun Drawn On Him For Failure To Use A Turn Signal!
Uploaded June 03, 2020
This treatment is nothing new. Here's a statment from Randell Minott on what transpired the evening of September 6, 2017 in Kansas City, Missouri.
."
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TITLE: Find the roots of cubic equation.
QUESTION [0 upvotes]: If the function $$f(x)=x^3-9x^2+24x+c$$ has three real and distinct roots $l,m,n$, the find value of $[l]+[m]+[n]$ where $[..]$ represents greatest integer function
In my book there is no information given about $c$.
REPLY [1 votes]: The discriminant of this cubic is $-27 c^2 - 972 c - 8640 = -27(c+20)(c+16)$. To have three distinct real roots, we need the discriminant positive, so $-20 < c < -16$.
Note that the values of $f$ at $x = 1,2,3,4,5$ are $c+16, c+20, c+18,c+16,c+20$ respectively.
For $c$ between $-20$ and $-16$, the lowest root is between $1$ and $2$, the highest between $4$ and $5$. The middle root is between $2$ and $4$, is an increasing function of $c$, and is $3$ when $c=-18$. So for $-20 < c < -18$ we have $\lfloor l \rfloor + \lfloor m\rfloor + \lfloor n\rfloor = 1+2+4=7$, and for $-18 \le c < -16$ we have $1+3+4 = 8$.
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This caretaker and a breeder land tortoise (aka galapago) had a friendly thing going on.
Image taken at the Darwin Center in the Galapagos Islands.
Oh now that's cool!
Bill
who's the old man? truly a great shot!!!!!!!
a good composition and laugh!
regards, Charlie. tom
The tortoise looks a bit embarrassed in front of his mates
Cheers Ray
They sure look cosy together.
Chris
Thanks William, Fred, Tom, Ray and Chris.
The man was the caretaker for the tortoises. He was showing off a bit to our small group; he stepped over the low wall and walked up to this tortoise. Obviously, the tortoise recognized him and stretched out his neck so the man would scratch it- which he did. Then the man put his arm around the animal and gave him a hug.
Note: we were not permitted inside the containment area.
Charlie
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The night began in a similar fashion to the way it had the Saturday afternoon previous for Richmond, with Collingwood kicking the first four goals of the match. The Tigers were not to be denied this time though and showed outright desperation to secure the victory over their arch rival. The smother from Trent Cotchin in the first term, described on 3AW as ‘a game-changing smother’, wasn’t far wrong – Richmond would go on to kick 14 of the next 21 goals of the game, with Ty Vickery kicking the ice-breaker – as they went about outworking Collingwood all over the ground. The perfect skills from Collingwood in the opening 12 minutes of the game; inextricably fell away quite dramatically and Richmond used this to their advantage.
Richmond was the very definition of ineffective and uncompetitive early, but the Tigers regrouped and kicked seven of the next nine goals to take a one goal lead into the main break. It was no certainty that they would maintain the upper-hand in the second half though. They were able to expose Collingwood –through their outside pace – that turned Collingwood into a fumbling, calamity of errors. The Tigers had a 13-point lead late in the third term, but came up against a Magpie fight back. It was reminiscent of their fight back earlier in the season that was led by Scott Pendlebury. However, just when the Tiger faithful thought they would be potentially facing the same fate – Pendlebury got injured and any momentum Collingwood had, ceased to exist.
The slow possession-based game that was self sabotage for the Tigers transformed into a more urgent, corridor-based approach – the desire to win for milestone men Dustin Martin and Taylor Hunt as well as embattled coach Damien Hardwick resembled the same desire Richmond had on that famous night when they beat Sydney by a point after the siren earlier this season – they were quick, direct and looked much more dangerous than their opposition. A match between 12th and 13th on the ladder was bound to be riddled with mistakes from both teams. The winner would be who used the ball better – and as Richmond spread from the stoppages with intent and in numbers – they were the better team. Next comes the highly-fancied Geelong – the track record is poor, but Richmond has nothing to lose.
Final Score: RICH 14.8.92 to COLL 11.11.77.
TOP 3 TIGERS:
1. ALEX RANCE: The best defender in the league was immense once more. He finished with 26 disposals (10 in the first quarter), 10 intercepts, eight marks, four pressure acts and four score involvements.
2. SHAUN GRIGG: The midfielder bobbed up and did the little things, constantly creating an option for his teammates. He finished with 26 disposals, six tackles, 14 marks, 12 pressure acts, 10 score involvements and one goal.
3. JACK RIEWOLDT: The forward was extremely impressive, for the most part. He finished with 12 disposals, six marks, four pressure acts, 10 score involvements and four goals.
What were your thoughts on the game?
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\begin{document}
\begin{abstract}
A classical construction of Katz gives a purely algebraic construction of Eisenstein--Kronecker series using the Gau\ss--Manin connection on the universal elliptic curve. This approach gives a systematic way to study algebraic and $p$-adic properties of real-analytic Eisenstein series. In the first part of this paper we provide an alternative algebraic construction of Eisenstein--Kronecker series via the Poincar\'e bundle. Building on this, we give in the second part a new conceptional construction of Katz' two-variable $p$-adic Eisenstein measure through $p$-adic theta functions of the Poincar\'e bundle.
\end{abstract}
\maketitle
\section[Introduction]{Introduction}
The classical \emph{Eisenstein--Kronecker} series are defined for a lattice $\Gamma=\omega_1\ZZ+\omega_2\ZZ\subseteq \CC$, $s,t\in\frac{1}{N}\Gamma$ and integers $r-2>k\geq 0$ by the absolutely convergent series
\[
e^*_{k,r}(s,t;\Gamma):=\sum_{\gamma\in\Gamma\setminus\{-s\}} \frac{(\bar{s}+\bar{\gamma})^k}{(s+\gamma)^r} \langle \gamma,t \rangle
\]
with $\langle z,w \rangle:=\exp\left( \frac{z\bar{w}-w\bar{z}}{A(\Gamma)} \right)$ and $A(\Gamma)=\frac{\im(\omega_1\bar{\omega}_2)}{\pi}$. For arbitrary integers $k,r$ they can be defined by analytic continuation, cf.~\cite[\S 1.1]{bannai_kobayashi}. For our purposes, it is more convenient to normalize the Eisenstein--Kronecker series for integers $k,r\geq 0$, as follows:
\[
\tilde{e}_{k,r+1}(s,t;\Gamma):=\frac{(-1)^{k+r}r!}{A(\Gamma)^k}e^*_{k,r+1}(s,t;\Gamma)
\]
By varying the lattice the Eisenstein-Kronecker series define $\Ccal^\infty$-modular forms $\tilde{e}_{k,r+1}(s,t)$ of level $N$ and weight $k+r+1$. Although Eisenstein--Kronecker series are in general non-holomorphic, it turns out that they belong to the well behaved class of \emph{nearly holomorphic modular forms}.\par
A classical result of Katz gives a purely algebraic approach towards Eisenstein--Kronecker series by giving an algebraic interpretation of the Maa\ss--Shimura operator on the modular curve in terms of the Gau\ss--Manin connection. His construction has been one of the main sources for studying systematically the algebraic and $p$-adic properties of real-analytic Eisenstein series. Even today it has still influence, e.g.~it plays a key role in the proof of one of the most general known explicit reciprocity laws for Tate modules of formal $p$-divisible groups by Tsuji \cite{tsuji}. Unfortunately, the construction of the Maa\ss-Shimura operator only works in the universal situation and doesn't have good functoriality properties.\par
The study of Eisenstein--Kronecker series through the Poincar\'e bundle has been initiated by Bannai and Kobayashi \cite{bannai_kobayashi}. Bannai and Kobayashi have proven that Eisenstein--Kronecker series appear as expansion coefficients of certain translates of the \emph{Kronecker theta function}:
\begin{equation}\label{eq_KroneckerTheta}
\Theta_{s,t}(z,w)=\sum_{a,b\geq 0}\frac{\tilde{e}_{a,b+1}(s,t)}{a!b!}z^bw^a,\quad s,t\notin\Gamma
\end{equation}
The Kronecker theta function is a reduced theta function associated to the Poincar\'e bundle of an elliptic curve. For elliptic curves with complex multiplication, Bannai and Kobayashi have used this observation to study many interesting $p$-adic and algebraic properties of Eisenstein--Kronecker series at CM points \cite{bannai_kobayashi}. These results have been fruitfully applied by Bannai, Kobayashi and Tsuji for studying the algebraic de Rham and the syntomic realization of the elliptic polylogarithm for elliptic curves with complex multiplication \cite{BKT}. This approach does not immediately generalize to more general elliptic curves over arbitrary base schemes. The assumption of complex multiplication is essential to deduce the algebraicity of the involved theta function. \par
Building on the work of Bannai--Kobayashi, we provide in the first part of the paper a purely algebraic construction of Eisenstein--Kronecker series via the Poincar\'e bundle on the universal vectorial bi-extension. This construction works over arbitrary base schemes and has good functoriality properties.
In the second part of the paper, we provide a new approach to the $p$-adic interpolation of Eisenstein--Kronecker series. We will show that Katz' two-variable $p$-adic Eisenstein measure is the Amice transform of a certain $p$-adic theta function of the Poincar\'e bundle. This does not only give a conceptional approach towards the $p$-adic Eisenstein measure, it also provides a direct bridge between $p$-adic theta functions and $p$-adic modular forms.\par
Let us briefly outline the content of this paper in more detail: Theta functions provide a convenient way to describe sections of line bundles over complex abelian varieties analytically. In the first part of this paper, we will define a certain section of the Poincar\'e bundle, called \emph{the Kronecker section}. While the Kronecker section is defined for arbitrary families of elliptic curves, it can be expressed in terms of the Kronecker theta function for elliptic curves over $\CC$. The universal vectorial extension of the dual elliptic curve classifies line bundles of degree zero with an integrable connection. This gives us two integrable connections $\con{\sharp}$ and $\con{\dagger}$ on the Poincar\'e bundle over the universal vectorial bi-extension. By applying both connections iteratively to the Kronecker section and evaluating at torsion sections $s$ and $t$, we obtain a new algebraic construction of Eisenstein--Kronecker series:
\begin{thm*}
Let $\Hcal^{1}_{\mathrm{dR}}$ denote the first relative de Rham cohomology of the universal elliptic curve $\Ecal\rightarrow\Mcal$. The image of the sections
\[
(s\times t)^{*} \left( \con{\sharp}^{\circ k}\circ\con{\sharp}^{\circ r}(\scan)\right)\in\Gamma(\Mcal,\Sym^{k+r+1}\Hcal^{1}_{\mathrm{dR}})
\]
under the Hodge decomposition gives the classical Eisenstein--Kronecker series $\tilde{e}_{k,r+1}(Ds,Nt)$.
\end{thm*}
Let us refer to \cref{EP_MainThm} for more details. Although the algebraic construction of Eisenstein--Kronecker series goes back to Katz, this new point of view has several advantages. Let us remark that the construction of the sections in $\Gamma(S,\Sym^{k+r+1}\Hcal^{1}_{\mathrm{dR}} )$ applies, contrary to Katz' approach, to elliptic curves $E\rightarrow S$ over arbitrary base schemes $S$ and has good functoriality properties. Furthermore, the geometric setup is much more symmetric than the construction involving the Gau\ss--Manin connection. Indeed, the functional equation of the Eisenstein--Kronecker series is reflected by the symmetry obtained by interchanging the role of the elliptic curve and its dual. Finally, let us note that the Kronecker section provides a new construction of the logarithmic derivatives of the Kato--Siegel functions associated to a positive integer $D$. It might be worth to mention that, contrary to the classical construction of the Kato--Siegel functions, our construction even works if $D$ is not co-prime to $6$.\par
In the second part of the paper, we apply the methods of the first part to obtain a new approach to Katz' $p$-adic Eisenstein measure. In \cite{katz_padicinterpol}, Katz constructed a two-variable $p$-adic measure with values in the ring of generalized $p$-adic modular forms having certain real-analytic Eisenstein series as moments. Katz' $p$-adic Eisenstein measure is the key for studying $p$-adic congruences between real-analytic Eisenstein series. But the proof of existence of the $p$-adic Eisenstein measure uses all the predicted congruences between Eisenstein series. These congruences in turn have to be checked by hand on the $q$-expansion. We will give a more intrinsic construction of the $p$-adic Eisenstein measure via the Poincar\'e bundle. Norman's theory of $p$-adic theta functions allows us to associate a $p$-adic theta function $\pthetaD_{(a,b)}(S,T)$ to the Kronecker section for any elliptic curve with ordinary reduction over a $p$-adic base. Applying this to the universal trivialized elliptic curve gives us a two-variable power series over the ring of generalized $p$-adic modular forms. A classical result of Amice allows us to associate a two-variable $p$-adic measure $\muEisD$ to this $p$-adic theta function. It turns out that the resulting measure interpolates the Eisenstein--Kronecker series $p$-adically:
\begin{thm*}
The $p$-adic Eisenstein--Kronecker series $\EispD$ appear as moments
\[
\EispD=\int_{\Zp\times\Zp} x^k y^r \dd \muEisD(x,y)
\]
of the measure $\muEisD$ associated to the $p$-adic theta function $\pthetaD_{(a,b)}(S,T)$.
\end{thm*}
Let us refer to \cref{thm_padictheta} and \cref{cor_Eis_moments} for details. This construction does not only give a more conceptional approach towards the $p$-adic Eisenstein measure, it also provides a direct bridge between $p$-adic theta functions and $p$-adic modular forms.\par
Last but not least, one of our main leading goals was to obtain a better understanding of polylogarithmic cohomology classes. The specialization of polylogarithmic cohomology classes along torsion sections gives Eisenstein classes which play a key role in proving particular cases of deep conjectures like the Tamagawa Number Conjecture (TNC) of Bloch and Kato and its $p$-adic analogue. In particular, the syntomic Eisenstein classes play an important role for proving particular cases of the $p$-adic Beilinson conjecture\cite{bannai_kings, bannai_kings2}. We have already mentioned, that Bannai, Kobayashi and Tsuji have used the Kronecker theta function to give an explicit description of the algebraic de Rham realization and the syntomic realization of the elliptic polylogarithm for CM elliptic curves. Our aim is the generalization of these results to families of elliptic curves. In the case of the de Rham realization this goal has already been archived by Scheider in his PhD thesis by analytic methods \cite{rene}. Scheider's PhD thesis has been a great source of inspiration for us. He gave an \emph{analytic} description of the de Rham realization for the universal elliptic curve in terms of certain theta functions. But for all arithmetic applications it is indispensable to also have an explicit \emph{algebraic} description of the de Rham polylogarithm. The techniques developed in this paper allow to get rid of the analytically defined theta functions, to give a purely algebraic reinterpretation of the results of Scheider and to streamline some of his proofs \cite{deRham}. An explicit algebraic description of the de Rham realization of the elliptic polylogarithm is the cornerstone towards an explicit description of the syntomic realization which is treated in \cite{Syntomic}.\par
A naturally arising question is the question about higher dimensional generalizations. This question will be the content of future investigations.
\section*{Acknowledgement}
The results presented in this paper are part of my Ph.D. thesis at the Universit\"at Regensburg \cite{PhD}. It is a pleasure to thank my advisor Guido Kings for his guidance during the last years. Further, I would like to thank Shinichi Kobayashi for all the valuable suggestions, remarks and comments on my PhD thesis. The author would also like to thank the collaborative research centre SFB 1085 ``Higher Invariants'' by the Deutsche Forschungsgemeinschaft for its support.
\part{Real-analytic Eisenstein series via the Poincar\'e bundle}
During this work $E$ usually denotes an elliptic curve over an arbitrary base scheme $S$. The structure morphism $G\rightarrow S$ of an $S$-group scheme will be denoted by $\pi=\pi_G$ while its identity morphism will be denoted by $e=e_G$. For a commutative $S$-group scheme $G$ and some section $t\in G(S)$ let us write $T_t\colon G\rightarrow G$ for the translation morphism. Let us write $\om_{G/S}:=e^*\Omega^1_{G/S}$ for the relative co-Lie algebra of a commutative group scheme over $S$.
\section{Nearly holomorphic modular forms}
In this section we will briefly recall the theory of nearly holomorphic modular forms with an emphasize on their geometric interpretation following Urban \cite{urban}. The group of $2\times 2$ matrices with positive determinant $GL_2(\RR)^+$ acts on the upper half plane
\[
\HH:=\{\tau=x+iy\in\CC: y>0 \}
\]
by fractional linear transformations
\[
\gamma.\tau:=\frac{a\tau+b}{c\tau+d} \text{ for } \gamma=\left(\begin{matrix} a & b\\c & d \end{matrix}\right)\text{ and }\tau\in\HH.
\]
For a complex valued function $f$ on $\HH$, a non-negative integer $k$ and $\gamma\in GL_2(\RR)^+$ let us write
\[
(f|_k\gamma)(\tau):=\frac{\det(\gamma)^{k/2}}{(c\tau+d)^k}f(\gamma.\tau).
\]
Recall that holomorphic modular forms of weight $k$ and level $\Gamma\subseteq SL_2(\ZZ)$ are defined as functions on $\HH$ satisfying the following conditions:
\begin{enumerate}
\item $f$ is a holomorphic on $\HH$,
\item $f|_k\gamma=f$ for all $\gamma\in\Gamma$,
\item $f$ has a finite limit at the cusps.
\end{enumerate}
Weakening the condition $(a)$ to $f \in\Ccal^\infty(\HH)$ leads to the definition of $\Ccal^\infty$ modular forms. We have already seen in the introduction a class of $\Ccal^\infty$ modular forms of great number theoretic interest, namely the Eisenstein--Kronecker series $\tilde{e}_{a,b}(s,t;\ZZ+\tau \ZZ)$. Although the Eisenstein--Kronecker series are not holomorphic in general, they are not so far from being holomorphic. They are \emph{nearly holomorphic} in the following precise sense:
\begin{defin}\label{def_nearlyhol}
A \emph{nearly holomorphic} modular form of weight $k$ and order $\leq r$ for $\Gamma\subseteq SL_2(\RR)$ is a function on $\HH$ satisfying:
\begin{enumerate}
\item $f\in\Ccal^\infty(\HH)$,
\item $f|_k\gamma=f$ for all $\gamma\in\Gamma$,
\item There are holomorphic functions $f_0,f_1,...,f_r$ on $\HH$ such that
\[
f(\tau)=f_0(\tau)+\frac{f_1(\tau)}{y}+...+\frac{f_r(\tau)}{y^r},
\]
\item $f$ has a finite limit at the cusps.
\end{enumerate}
Let us write $\Ncal_{k}^r(\Gamma,\CC)$ for the space of nearly holomorphic modular forms of weight $k$, order $r$ and level $\Gamma$.
\end{defin}
Holomorphic modular forms of weight $k$ and level $\Gamma$ can be seen as sections of the $k$-th tensor power of the cotangent-sheaf $\om^{\otimes k}$ of the generalized universal elliptic curve $\bar{\Ecal}$ of level $\Gamma$ over the compactification $\bar{\Mcal}$ of the modular curve of level $\Gamma$. This leads in a natural way to the definition of geometric modular forms and allows to study modular forms from an algebraic perspective. For sure, $\Ccal^\infty$ modular forms allow a similar geometric description as $\Ccal^\infty$ sections of $\om^{\otimes k}$, but by passing to $\Ccal^\infty$ sections, we loose all algebraic information. For nearly holomorphic modular forms, things become much better. They allow an algebraic interpretation which we will recall in the following: The algebraic de Rham cohomology $\Hcal^1_{dR}:=R^1\bar{\pi}_*\Omega^1_{\bar{\Ecal}/\bar{\Mcal}}(\log(\bar{\Ecal}\setminus \Ecal))$ sits in a short exact sequence
\[
\begin{tikzcd}
0\ar[r] & \om\ar[r] & \Hcal^1_{dR}\ar[r] & \om^\vee\ar[r] & 0
\end{tikzcd}
\]
induced by the Hodge filtration
\[
F^0\Hcal^1_{dR}=\Hcal^1_{dR}\supseteq F^1\Hcal^1_{dR}=\om\supseteq F^2\Hcal^1_{dR}=0.
\]
This filtration does not split algebraically. But after passing to $\Ccal^\infty(\bar{\Mcal})$ sections there is a canonical splitting called the Hodge decomposition.
\[
\Hcal^1_{dR}(\Ccal^\infty)\righteq \om(\Ccal^\infty)\oplus \om^\vee(\Ccal^\infty)
\]
The Hodge filtration induces a descending filtration on $\Sym^k_{\Ocal_\Mcal}\Hcal^1_{dR}$ and by the Hodge decomposition we obtain an epimorphism
\[
\Sym^k_{\Ccal^{\infty}(\Mcal)}\Hcal^1_{dR}(\Ccal^\infty)\twoheadrightarrow \om^{\otimes k}(\Ccal^\infty).
\]
It is the Hodge decomposition which allows to pass from algebraic sections to $\Ccal^\infty$-modular forms and thereby gives us the following algebraic interpretation of nearly holomorphic modular forms:
\begin{prop}[{\cite[Prop. 2.2.3.]{urban}}]
The Hodge decomposition induces an isomorphism
\[
H^0\left(\bar{\Mcal}_{\CC},F^{k-r}\Sym^k \Hcal^1_{dR}\right)\righteq \Ncal^r_k(\Gamma,\CC).
\]
\end{prop}
Instead of working with the compactification of the modular curve one can also work with the open modular curve with a finiteness condition at the cusps. For simplicity let us restrict to the case $\Gamma_1(N)$. The modular curve of level $\Gamma_1(N)$ is the universal elliptic curve with a fixed $N$-torsion section. The finiteness condition at the cusp can be stated in terms of the Tate curve as follows: Let $A$ be an $\ZZ[ \frac{1}{N}]$-algebra. Let $\Tate(q^N)$ be the Tate curve over $A((q))$ with its canonical invariant differential $\omega_{can}$ and its canonical $\Gamma_1(N)$ level structure given by the $N$-torsion section $q$. Applying the Gau\ss-Manin connection
\[
\nabla\colon \HdR{1}{\Tate(q^N)/A((q))}\rightarrow \HdR{1}{\Tate(q^N)/A((q))}\otimes \Omega^1_{A((q))/A}
\]
to $\omega_{can}$ gives us a basis $(\omega_{can},u_{can})$ with
\[
u_{can}:=\nabla(q\frac{d}{dq})(\omega_{can}).
\]
This leads to the notion of geometric nearly holomorphic modular forms:
\begin{defin}
Let $A$ be a $\ZZ[\frac{1}{N}]$-algebra. A \emph{geometric nearly holomorphic modular form} $F$ of level $\Gamma_1(N)$, weight $k$ and order $\leq r$ defined over $A$ is a functorial assignment
\[
(E/S,t)\mapsto F_{E,t}\in\Gamma(S, F^{k-r}\Sym^k\HdR{1}{E/S} ),
\]
defined for all test objects $(E/S,t)$ consisting of an elliptic curve $E/S$ with $S$ an $\Spec{A}$-scheme and $t\in E[N](S)$ an $N$-torsion section, satisfying the following finiteness condition at the Tate curve over $A$:
\[
F_{(\Tate(q^N),q)}=\sum_{r+s=k} a_{r,s}(q)\cdot \omega_{can}^{\otimes r}\otimes u_{can}^{\otimes s}
\]
with $a_{r,s}(q)\in A\llbracket q\rrbracket\subseteq A((q))$.
\end{defin}
Finally let us remark that the two algebraic definitions of nearly holomorphic modular forms coincide: To give a geometric nearly holomorphic modular form of level $\Gamma_1(N)$ defined over $A$ is equivalent to the datum of a section
\[
H^0\left(\bar{\Mcal}_{A},F^{k-r}\Sym^k \Hcal^1_{dR}\right).
\]
If $A\subseteq \CC$ we can further pass to the analytification and apply the Hodge decomposition to relate geometric nearly holomorphic modular forms back to the $\Ccal^\infty$-definition of nearly holomorphic modular forms.
\section{The Kronecker section}\label{sec_can}
Motivated by the work of Bannai and Kobayashi \cite{bannai_kobayashi}, we would like to give an algebraic approach to Eisenstein--Kronecker series via the Poincar\'e bundle. In particular, we have to find an algebraic substitute for the Kronecker theta function appearing in \cite{bannai_kobayashi}. One way to do this is to study algebraic theta functions as done by Bannai and Kobayashi. But unfortunately the success of this approach is limited to elliptic curves with complex multiplication. Another natural approach is to study the underlying section of the Poincar\'e bundle instead of the analytically defined Kronecker theta function. In the following, we will give a precise definition of this underlying section. It will be called \emph{Kronecker section}.\par
\subsection{The Poincar\'e bundle}
In the following $E/S$ will be an elliptic curve. Let us write $e\colon S\rightarrow E$ for the unit section and $\pi_E\colon E\rightarrow S$ for the structure morphism. Let us recall the definition of the Poincar\'e bundle and thereby fix some notation. A \emph{rigidification} of a line bundle $\Lcal$ on $E$ is an isomorphism $r:e^*\Lcal \righteq \Ocal_S$. A morphism of rigidified line bundles is a morphism of line bundles respecting the rigidification. The dual elliptic curve $E^\vee$ is the $S$-scheme representing the connected component of the functor
\[
T\mapsto \mathrm{Pic}(E_T/T):=\{ \text{iso. classes of rigidified line bundles } (\Lcal,r) \text{ on } E_T/T\}
\]
on the category of $S$-schemes. The $S$-scheme $E^\vee$ is again an elliptic curve. Since a rigidified line bundle has no non-trivial automorphisms, an isomorphism class of a rigidified line bundle determines the line bundle up to unique isomorphism. This implies the existence of a universal rigidified line bundle $(\Po,r_0)$ on $E\times_S E^\vee$ with the following universal property: For any rigidified line bundle of degree zero $(\Lcal,r)$ on $E_T/T$ there is a unique morphism
\[
f:T\rightarrow E^\vee
\]
such that $(\id_E\times f)^*(\Po,r_0)\righteq (\Lcal,r)$. In particular, we obtain for any isogeny
\[
\varphi:E\rightarrow E'
\]
the \emph{dual isogeny} as the morphism $\varphi^\vee:(E')^\vee\rightarrow E^\vee$ classifying the rigidified line bundle $(\varphi\times\id)^*\Po'$ obtained as pullback of the Poincar\'e bundle $\Po'$ of $E'$. Let $\lambda:E\righteq \Ed$ be the polarization associated with the ample line bundle $\Ocal_E([e])$. More explicitly, $\lambda$ is given as
\begin{equation}\label{EP_eqAutodual}
\begin{tikzcd}[row sep=tiny]
\lambda: E \ar[r] & \Pic=:\Ed\\
P\ar[r,mapsto] & \left( \Ocal_E([-P]-[e])\otimes_{\Ocal_E} \pi^*e^*\Ocal_E([-P]-[e])^{-1}, \mathrm{can} \right).
\end{tikzcd}
\end{equation}
Here, $\mathrm{can}$ is the canonical rigidification given by the canonical isomorphism
\[
e^*\Ocal_E([-P]-[e])\otimes_{\Ocal_S} e^*\Ocal_E([-P]-[e])^{-1} \righteq \Ocal_S.
\]
We fix the identification $E\righteq \Ed$ once and for all and write again $\Po$ for the pullback of the Poincar\'e bundle along $\id\times\lambda$, i.e.
\begin{align}\label{eq_Poincare_bdl}
(\Po,r_0):&=\left( \Ocal_{E\times E}(-[e\times E]-[E\times e] + \Delta)\otimes_{\Ocal_{E\times E}} \pi_{E\times E}^* \om_{E/S}^{\otimes -1},r_0,s_0 \right)=\\
&=\left( \pr_1^* \Ocal_E([e])^{\otimes-1}\otimes\pr_2^*\Ocal_{E}([e])^{\otimes-1}\otimes \mu^*\Ocal_E([e])\otimes \pi_{E\times E}^* \om_{E/S}^{\otimes -1} ,r_0 \right).
\end{align}
Here, $\Delta=\ker \left(\mu:E\times E\rightarrow E\right)$ is the anti-diagonal and $r_0$ is the ridification induced by the canonical isomorphism
\[
e^*\Ocal_E(-[e])\righteq \om_{E/S}
\]
i.e.
\[
r_0\colon (e\times\id)^*\Po \cong \pi_E^* e^* \Ocal_E([e])^{\otimes-1}\otimes \Ocal_{E}([e])^{\otimes-1}\otimes \Ocal_E([e])\otimes \pi_{E}^* \om_{E/S}^{\otimes -1}\cong \Ocal_E.
\]
Let us remark that the Poincar\'e bundle is also rigidified along $(\id\times e)$ by symmetry, i.e. there is also a canonical isomorphism
\[
s_0\colon (\id\times e)^*\Po\cong \Ocal_E.
\]
\subsection{Definition of the Kronecker section}
We can now define the \emph{Kronecker section} which will serve as a purely algebraic substitute of the Kronecker theta function. The above explicit description of the Poincar\'{e} bundle gives the following isomorphisms of locally free $\Ocal_{E\times E}$-modules, i.\,e. all tensor products over $\Ocal_{E\times E}$:
\begin{align}\label{ch_EP_eq5}
\notag \Po\otimes \Po^{\otimes -1} &= \Po \otimes \left( \Ocal_{E\times E}(-[e\times E]-[E\times e] + \Delta)\otimes \pi_{E\times E}^* \omega_{E/S}^{\otimes-1}\right)^{\otimes-1}\cong \\
&\cong \Po \otimes \Omega^1_{E\times E/E}([e\times E]+[E\times e]) \otimes \Ocal_{E\times E}( - \Delta)
\end{align}
The line bundle $\Ocal_{E\times E}( - \Delta)$ can be identified canonically with the ideal sheaf $\Jcal_\Delta$ of the anti-diagonal $\Delta$. If we combine the inclusion
\[
\Ocal_{E\times E}( - \Delta)\cong \Jcal_\Delta\hookrightarrow \Ocal_{E\times E}
\]
with \eqref{ch_EP_eq5}, we get a morphism of $\Ocal_{E\times E}$-modules
\begin{equation}\label{ch_EP_eq6}
\Po\otimes \Po^{\otimes -1} \hookrightarrow \Po \otimes \Omega^1_{E\times \Ed/\Ed}([e\times \Ed]+[E\times e]).
\end{equation}
\begin{defin}
Let
\[
\scan\in \Gamma\left(E\times_S \Ed, \Po \otimes_{\Ocal_{E\times \Ed}} \Omega^1_{E\times \Ed/\Ed}([e\times \Ed]+[E\times e])\right)
\]
be the image of the identity element $\id_{\Po}$ under \eqref{ch_EP_eq6}. The section $\scan$ will be called \emph{Kronecker section}.
\end{defin}
\subsection{Translation operators} In the work of Bannai and Kobayashi, the Eisenstein--Kronecker series appear as expansion coefficients of certain translates of the Kronecker theta function. These translates are obtained by applying translation operators for theta functions to the Kronecker theta function. In the previous section we defined the Kronecker section, which serves as a substitute for the Kronecker theta function. This motivates to find similar translation operators for sections of the Poincar\'e bundle. In this section, we will define these translation operators: Let $\varphi\colon E\rightarrow E'$ be an isogeny of elliptic curves.
By the universal property of the Poincar\'e bundle, we get a unique isomorphism of rigidified line bundles
\begin{equation}\label{eq_Gamma_phi}
\gamma_{\id,\varphi^\vee}: (\id_{E}\times \varphi^\vee)^*\Po\righteq (\varphi\times \id_{(E')^\vee})^*\Po'.
\end{equation}
Of particular interest for us is the case $\varphi=[N]$. In this case the dual $[N]^\vee$ is just the $N$-multiplication $[N]$ on $E^\vee$. The inverse of $\gamma_{\id,\varphi^\vee}$ will be denoted by
\[
\gamma_{\varphi,\id}:(\varphi\times\id)^*\Po'\righteq (\id\times\varphi^\vee)^*\Po.
\]
For integers $N,D\geq 1$ let us define
\[
\gamma_{[N],[D]}: ([N]\times [D])^{*}\Po \righteq ([D]\times[N])^{*}\Po
\]
as the composition in the following commutative diagram
\begin{equation}\label{EP_diag1}
\begin{tikzcd}[column sep=huge]
([N]\times [D])^{*}\Po \ar[r,"({[N]}\times \id)^{*}\gamma_{\id,[D]}"]\ar[d,swap,"(\id\times {[D]})^{*}\gamma_{[N],\id}"] & ([ND]\times\id)^{*}\Po \ar[d,"({[D]}\times \id)^{*}\gamma_{[N],\id}"] \\
(\id\times[DN])^{*}\Po \ar[r,"(\id\times {[N]})^{*}\gamma_{\id,[D]}"] & ([D]\times[N])^{*}\Po.
\end{tikzcd}
\end{equation}
Indeed, this diagram is commutative since all maps are isomorphisms of rigidified line bundles and rigidified line bundles do not have any non-trivial automorphisms, i.\,e. there can be at most one isomorphism between rigidified line bundles. By the same argument we obtain the following identity for integers $N,N',D,D'\geq 1$:
\begin{align}
\label{EP_lem_gamma_c} ([D]\times[N])^*\gamma_{[N'],[D']}\circ ([N']\times [D'])^* \gamma_{[N],[D]}&=\gamma_{[NN'],[DD']}
\end{align}
Let us define the following translation operators. Later we will compare these algebraic translation operators for complex elliptic curves to the translation operators for theta functions studied in \cite{bannai_kobayashi}. It will turn out that both operators essentially agree.
\begin{defin}
For integers $N,D\geq 1$ and torsion sections $s\in E[N](S)$, $t\in \Ed[D](S)$ we have the following $\Ocal_{E\times_S \Ed}$-linear isomorphism
\[
\begin{tikzcd}
\Ucal^{[N],[D]}_{s,t}:=\gamma_{[N],[D]}\circ(T_{s}\times T_t)^*\gamma_{[D],[N]}.: (T_s\times T_t)^*([D]\times [N])^*\Po \ar[r] & ([D]\times [N])^*\Po.
\end{tikzcd}
\]
In the most important case $N=1$ we will simply write $\Ucal^{[D]}_{t}:=\Ucal^{\id,[D]}_{e,t}$.
\end{defin}
If $T$ is an $S$-scheme and $t\in E[D](T)$ is a $T$-valued torsion point of $E$, let us write $Nt$ instead of $[N](t)$. We have the following behaviour under composition.
\begin{cor}\label{ch_EP_cor_compU}
Let $D,D',N,N'\geq1$ be integers, $s\in E[N](S)$, $s'\in E[N'](S)$ and $t\in \Ed[D](S)$, $t'\in\Ed[D'](S)$. Then:
\[
\left(([D]\times[N])^*\Ucal^{[N'],[D']}_{Ds',Nt'}\right)\circ (T_{s'}\times T_{t'})^*([D']\times [N'])^*\Ucal^{[N],[D]}_{D's,N't}=\Ucal^{[NN'],[DD']}_{s+s',t+t'}
\]
\end{cor}
\begin{proof}
This follows immediately from the definition of $\Ucal^{[N],[D]}_{s,t}$ and \eqref{EP_lem_gamma_c}.
\end{proof}
Later, we would like to apply the translation operator to the Kronecker section which is a section of the sheaf
\[
\Po \otimes_{\Ocal_{E\times \Ed}} \Omega^1_{E\times \Ed/\Ed}([e\times \Ed]+[E\times e]).
\]
It is convenient to introduce the notation
\[
U^{[N],[D]}_{s,t}(s):=\left(\Ucal^{[N],[D]}_{s,t}\otimes\id_{\Omega^1_{E\times \Ed/\Ed}([e\times \Ed]+[E\times e])}\right)\Big( (T_s\times T_t)^*([D]\times [N])^*s \Big)
\]
for sections $s\in\Gamma(U,\Po \otimes_{\Ocal_{E\times \Ed}} \Omega^1_{E\times \Ed/\Ed}([e\times \Ed]+[E\times e]))$ of the Poincar\'e bundle. The resulting section $U^{[N],[D]}_{s,t}(s)$ is then a section of the sheaf
\[
([D]\times[N])^*\Big( \Po\otimes \Omega^1_{E\times_S\Ed/\Ed}([(-Ds)\times \Ed]+[E\times(-Ns)])\Big)
\]
over the open subset $(T_s\times T_t)^{-1}([D]\times [N])^{-1}U$.
\section{Real-analytic Eisenstein series via the Poincar\'e bundle}
The following section is the heart of the first part of the paper. It provides a functorial and purely algebraic construction of geometric nearly holomorphic modular forms which are proven to correspond to the analytic Eisenstein--Kronecker series under the Hodge decomposition on the universal elliptic curve. More precisely, for co-prime integers $N,D>1$ and non-zero torsion sections $s\in E[N](S),t\in \Ed[D](S)$ we will provide a section
\[
E^{k,r+1}_{s,t}\in \Gamma\left(S, \Sym^{k+r+1}_{\Ocal_S} \HdR{1}{E/S} \right).
\]
This construction is functorial in the test objects $(E/S,s,t)$. We will construct these sections by iteratively applying the universal connections of the Poincar\'e bundle on the universal bi-extension $E^\sharp\times_S E^\dagger$ and evaluating at the zero section. Let us remark that the symmetric powers of the first relative de Rham cohomology appear naturally, since the cotangent space $\omega_{E^\sharp/S}$ of the universal vectorial extension of $E^\sharp$ is canonically isomorphic to $\HdR{1}{E/S}$.
\subsection{The construction via the Poincar\'e bundle}
Let $E/S$ be an elliptic curve over some base scheme $S$. We denote by
\[
\begin{tikzcd}
E^\sharp \ar[r,"q^\sharp"] & E & \text{and} & E^\dagger \ar[r,"q^\dagger"] & \Ed
\end{tikzcd}
\]
the universal vectorial extension of $E$ and $\Ed$. Let us write $\Po^\sharp$ resp. $\Po^\dagger$ for the pullbacks of $\Po$ to $E^\sharp\times_S \Ed$ resp. $E\times_S E^\dagger$. Then, $\Po^\sharp$ resp. $\Po^\dagger$ are equipped with canonical integrable $E^\sharp$- resp. $E^\dagger$-connections $\nabla_\sharp$ resp.~$\nabla_\dagger$. Indeed, the universal vectorial extension $E^\dagger$ classifies line bundles of degree zero on $E$ equipped with an integrable connection and $(\Po^\dagger,\nabla_\dagger)$ is the universal such line bundle on $E\times_S E^\dagger$. The same construction for $E$ replaced by $E^\vee$ gives $(\Po^\sharp,\nabla_\sharp)$ on $E^\sharp\times_S E^\vee$, here note that we have a canonical isomorphism $E\righteq (E^\vee)^\vee$. Let us write $\Po^{\sharp,\dagger}$ for the pullback of $\Po$ to the universal bi-extension $E^\sharp\times_S E^\dagger$:
\[
\begin{tikzcd}
& \EN^\sharp \times_{\MN} \EN^\dagger \ar[ld,"\id\times q^\dagger",swap] \ar[rd,"q^\sharp\times \id"] & \\
\EN^\sharp \times_{\MN} E^\vee \ar[rd,"q^\sharp\times \id",swap] & & \EN \times_{\MN} \EN^\dagger \ar[ld,"\id\times q^\dagger"] \\
& \EN \times_{\MN} \EN^\vee &
\end{tikzcd}
\]
Then, $\Po^{\sharp,\dagger}$ is in a natural way equipped with both an integrable $E^\sharp$- and an integrable $E^\dagger$-connection
\begin{equation*}
\begin{tikzcd}[row sep=tiny]
\Po^{\sharp,\dagger} \ar[rr,"(\id\times q^\dagger)^*\con{\sharp}"] & & \Po^{\sharp,\dagger}\otimes_{\Ocal_{E^\sharp\times E^\dagger}} \Omega^1_{E^\sharp\times E^\dagger/E^\sharp}\\
\Po^{\sharp,\dagger} \ar[rr,"(q^\sharp\times\id)^*\con{\dagger}"] & & \Po^{\sharp,\dagger}\otimes_{\Ocal_{E^\sharp\times E^\dagger}} \Omega^1_{E^\sharp\times E^\dagger/E^\dagger}.
\end{tikzcd}
\end{equation*}
By abuse of notation we will write $\nabla_\sharp$ instead of $(\id\times q^\dagger)^*\con{\sharp}$ and $\nabla_\dagger$ instead of $(q^\sharp\times\id)^*\con{\dagger}$. The cotangent space $\om_{E^\sharp/S}$ of the universal vectorial extension of $E$ is canonically isomorphic to $\HdR{1}{E/S}$, and similarly $\om_{E^\dagger/S}\cong \HdR{1}{\Ed/S}$. It is convenient to use our chosen polarization to identify $\Hcal^1_{dR}:=\HdR{1}{E/S}\cong\HdR{1}{\Ed/S}$. With these identifications, we get
\begin{equation}\label{EP_OmegaH_eq}
\begin{tikzcd}[row sep=tiny,column sep=small]
\Omega^1_{E^\sharp\times E^\dagger/E^\sharp}\ar[r,"\sim"] & \pr_{E^\dagger}^*\Omega^1_{E^\dagger/S}\ar[r,"\sim"] & (\pi_{E^\sharp\times E^\dagger})^*\Hcal^1_{dR}\\
\Omega^1_{E^\sharp\times E^\dagger/E^\dagger}\ar[r,"\sim"] & \pr_{E^\sharp}^*\Omega^1_{E^\sharp/S}\ar[r,"\sim"] & (\pi_{E^\sharp\times E^\dagger})^*\Hcal^1_{dR}.
\end{tikzcd}
\end{equation}
Since both $\con{\sharp}$ and $\con{\dagger}$ are $(\pi_{E^\sharp\times E^\dagger})^{-1}\Ocal_S$-linear, we can define the following differential operators:
\[
\begin{tikzcd}[row sep =tiny]
\con{\sharp}\colon \Po^{\sharp,\dagger}\otimes_{\Ocal_S}\Sym^{n}_{\Ocal_S}\HdR{1}{E/S} \ar[r,"\con{\sharp}\otimes \id"] & \Po^{\sharp,\dagger}\otimes_{\Ocal_S}\Sym^{n+1}_{\Ocal_S}\Hcal^1_{dR}\\
\con{\dagger}\colon \Po^{\sharp,\dagger}\otimes_{\Ocal_S}\Sym^{n}_{\Ocal_S}\HdR{1}{E/S} \ar[r,"\con{\dagger}\otimes \id"] & \Po^{\sharp,\dagger}\otimes_{\Ocal_S}\Sym^{n+1}_{\Ocal_S}\Hcal^1_{dR}
\end{tikzcd}
\]
Applying $\con{\sharp}$ and $\con{\dagger}$ iteratively leads to
\begin{equation*}
\begin{tikzcd}[column sep=huge]
\con{\sharp,\dagger}^{k,r}\colon \Po^{\sharp,\dagger}\otimes_{\Ocal_S}\Sym^{n}_{\Ocal_S}\Hcal^1_{dR} \ar[r,"\con{\sharp}^{\circ k}\circ \con{\dagger}^{\circ r}"] & \Po^{\sharp,\dagger}\otimes_{\Ocal_S}\Sym^{n+k+r}_{\Ocal_S}\Hcal^1_{dR}.
\end{tikzcd}
\end{equation*}
\begin{rem*}
Let us remark that $\con{\sharp}$ and $\con{\dagger}$ do not commute in general, but after pullback along $(e\times e)$ the result is independent of the order of application. This explains why choosing a different order in our construction will not give any new geometric nearly holomorphic modular forms.
\end{rem*}
Next, let us consider the pullback of translates of the Kronecker section to the Poincar\'e bundle on the universal bi-extension. For co-prime integers $N,D> 1$ and non-zero torsion sections $s\in E[N](S),t\in \Ed[D](S)$ let us consider
\[
(q^\sharp\times q^\dagger)^*U^{[N],[D]}_{s,t}(\scan)\in \Gamma\left((q^\sharp\times q^\dagger)^{-1}U, ([N]\times[D])^*\left(\Po^{\sharp,\dagger} \otimes \Omega^1_{E^\sharp\times E^\dagger /E^\dagger}\right)\right).
\]
where $U:=([D]\times[N])^{-1}(T_{Ds} \times T_{Nt})^{-1}\left(E\times \Ed \setminus \{ E\times{e} \amalg e\times \Ed \}\right)$.
Via the identification in \eqref{EP_OmegaH_eq} we obtain:
\[
(q^\sharp\times q^\dagger)^*U^{N,D}_{s,t}(\scan)\in \Gamma\left((q^\sharp\times q^\dagger)^{-1}U, ([N]\times[D])^*\Po^{\sharp,\dagger} \otimes_{\Ocal_S} \Hcal^1_{dR}\right).
\]
Since we have assumed that $s$ and $t$ are non-zero and $N$ and $D$ are co-prime, the morphism
\[
(e\times e):S=S\times_S S \rightarrow E\times_S \Ed
\]
factors through the open subset $U$. By iteratively applying the universal connections to the translates of the Kronecker section, we obtain the desired geometric nearly-holomorphic modular forms:
\begin{defin}\label{def_algebraicEK}
For co-prime integers $N,D\geq 1$, non-zero torsion sections $s\in E[N](S)$, $t\in \Ed[D](S)$ and integers $k,r\geq 0$ define
\[
E^{k,r+1}_{s,t}\in \Gamma\left(S, \Sym^{k+r+1}_{\Ocal_S} \Hcal^1_{dR} \right)
\]
via
\[
E^{k,r+1}_{s,t}:=(e\times e)^*\left[ \left( ([D]\times [N])^*\con{\sharp,\dagger}^{k,r}\right)\left((q^\sharp\times q^\dagger)^*U^{[N],[D]}_{s,t}(\scan)\right) \right].
\]
We call $E^{k,r+1}_{s,t}$ \emph{algebraic Eisenstein--Kronecker series}. This construction is obviously functorial.
For later reference, let us also define
\[
\EisD:=\sum_{e\neq t\in E[D](S)} E^{k,r+1}_{s,t}.
\]
\end{defin}
\subsection{The comparison result}
Let $N,D>1$ be co-prime as above and let $\Ecal\rightarrow \Mcal$ be the universal elliptic curve over $\Spec \ZZ[\frac{1}{ND}]$ with $\Gamma_1(ND)$-level structure. Let $s\in \Ecal[N](\Mcal)$ and $t\in \Ecal[D](\Mcal)$ be the points given by the level structure of order $N$ and $D$ on $\Ecal$. Note that $N,D>1$ co-prime implies $ND>3$ and thus the moduli problem is representable. We can give an explicit description of the analytification of $\Ecal/\Mcal$ as follows: The analytification $\Ecal(\CC)/\Mcal(\CC)$ is given by
\[
\Ecal(\CC)=\CC\times \HH/\ZZ\rtimes\Gamma_1(ND) \twoheadrightarrow \Mcal(\CC)=\HH/\Gamma_1(ND)
\]
with coordinates $(z,\tau)\in \CC\times \HH$. The torsion sections $s$ and $t$ are given by $s=(\frac{1}{N},\tau)$ and $t=(\frac{1}{D},\tau)$. In particular, for each $\tau\in \HH$ we will view $s$ and $t$ as elements of $\frac{1}{N}\Gamma_\tau$ resp.~$\frac{1}{D}\Gamma_\tau$ with the varying lattice $\Gamma_\tau=\ZZ+\tau\ZZ$. By slightly abusing notation, let us write $s$ and $t$ for both, the torsion sections, as well as the associated elements in $\frac{1}{N}\Gamma_\tau$ resp.~$\frac{1}{D}\Gamma_\tau$. Let us recall that the classical Eisenstein--Kronecker series are defined for $r>k+1$ by the convergent series
\[
\tilde{e}^*_{k,r+1}(s,t;\tau):=\frac{(-1)^{k+r}r!}{A(\Gamma_\tau)^k}\sum_{\gamma\in\Gamma_\tau\setminus\{-s\}} \frac{(\bar{s}+\bar{\gamma})^k}{(s+\gamma)^{r+1}} \langle \gamma,t \rangle
\]
and for general integers $k,r$ by analytic continuation. The following result relates the algebraic Eisenstein--Kronecker series constructed in \cref{def_algebraicEK} to the classical Eisenstein--Kronecker series.
\begin{thm}\label{EP_MainThm}
Assigning to every test object $(E/S,s,t)$ the algebraic Eisenstein--Kronecker series (defined in \Cref{def_algebraicEK})
\[
E^{k,r+1}_{s,t}\in\Gamma(S,\Sym_{\Ocal_S}^{k+r+1}\HdR{1}{E/S}),
\]
gives a geometric nearly holomorphic modular form of weight $k+r+1$, order $\min(k,r)$ and level $\Gamma_1(ND)$. The classical nearly holomorphic modular forms associated to $E^{k,r+1}_{s,t} $ via the Hodge decomposition on the universal elliptic curve are the analytic Eisenstein--Kronecker series $\tilde{e}_{k,r+1}(Ds,Nt)$.
\end{thm}
\subsection{The proof of the comparison result}
The construction of $E^{k,r+1}_{s,t}$ is compatible with base change. Thus, $E^{k,r+1}_{s,t}$ is uniquely determined by its value on the universal elliptic curve $\Ecal/\Mcal$ with $\Gamma_1(ND)$-level structure. Further, the map
\[
\begin{tikzcd}[column sep=small]
\Sym^{k+r+1}_{\Ocal_{\Mcal}} \HdR{1}{\Ecal/\Mcal}\otimes_{\Ocal_{\Mcal}}\Ccal^\infty(\Mcal(\CC))\ar[r,twoheadrightarrow] & \om_{\Ecal/\Mcal}^{\otimes (k+r+1)}\left( \Ccal^\infty \right).
\end{tikzcd}
\]
induced by the Hodge decomposition on the universal elliptic curve is injective. It remains to identify the $\Cinfty$-modular form associated with $E^{k,r+1}_{s,t}$ with the Eisenstein-Kronecker series
\[
\tilde{e}_{k,r+1}(Ds,Nt)\dd z^{\otimes (k+r+1)}.
\]
We can check this identification fiber-wise and reduce the proof to the case of a single elliptic curve $E/\CC$ with $\Gamma_1(ND)$-level structure. More precisely, we fix the following setup: Let $N,D>1$ co-prime integers. Let $E/\CC$ be an elliptic curve with complex uniformization
\[
E(\CC)\righteq \CC/\Gamma,\quad \Gamma:=\ZZ+\tau\ZZ, \tau\in \HH
\]
and fixed points $s\in E(\CC)[N]$ and $t\in E(\CC)[D]$ of order $N$ resp. $D$. Again by abuse of notation, let us write $s$ and $t$ for both $s=\frac{1}{N}$ and $t=\frac{1}{D}$ and the associated $\CC$-valued points $s,t\in E(\CC)=\CC/\Gamma$.
\subsubsection{Analytification of the Poincar\'{e} bundle}\label{subsec_Analitification}
The complex uniformization and our chosen auto-duality isomorphism establish
\[
\CC\times \CC \twoheadrightarrow E(\CC)\times \Ed(\CC)
\]
as universal covering. Let us denote the coordinates on $\CC\times\CC$ by $(z,w)$. The explicit description of the Poincar\'e bundle in \eqref{eq_Poincare_bdl} allows us to trivialize its pullback $\PoC$ to this universal covering using the \emph{Kronecker theta function}
\[
\Theta(z,w):=\frac{\theta(z+w)}{\theta(z)\theta(w)},\quad \theta(z):=\exp\left(-\frac{e_2^*}{2}z^2\right)\sigma(z)
\]
as follows:
\[
\Ocal_{\CC\times\CC}\righteq \PoC,\quad 1\mapsto \trv:=\frac{1}{\Theta(z,w)}\otimes (\dd z)^\vee.
\]
The canonical isomorphism $\gamma_{\id,[D]}$ induces an isomorphism
\[
\tilde{\gamma}_{\id,[D]}: (\id\times [D])^* \PoC\righteq ( [D]\times\id)^*\PoC
\]
on the pullback $\PoC$ to the universal covering.
\begin{lem}[{\cite[Lemma 3.5.10]{rene}}]\label{EP_lemRene1}
The isomorphism $\tilde{\gamma}_{\id,[D]}$ is given by
\[
(\id\times [D])^* \PoC\righteq ( [D]\times\id)^*\PoC,\quad (\id\times [D])^*\trv \mapsto ( [D]\times\id)^*\trv.
\]
\end{lem}
\begin{proof}
The proof is an adaptation of \cite[Lemma 3.5.10]{rene}. The map
\[
(\id\times [D])^*\trv \mapsto ( [D]\times\id)^*\trv.
\]
on the universal covering descents to an isomorphism
\[
(\id\times [D])^* \Po^{an}\righteq ( [D]\times\id)^*\Po^{an}
\]
to the analytification of the Poincar\'e bundle. It is straight-forward to check that this map respects the rigidifications of the Poincar\'e bundle. Indeed, this boils down to the fact that the theta function $\theta(z)$ used in the definition of $\Theta(z,w)$ is a normalized theta function, i.e. $\theta'(z)|_{z=0}=1$. Now the claim follows from the fact that $\gamma_{\id,[D]}$ is the unique isomorphism between $(\id\times [D])^* \Po$ and $( [D]\times\id)^*\Po$ which is compatible with the rigidifications.
\end{proof}
The pullback of the Kronecker section $\scan$ to the universal covering gives a meromorphic section $\scant$ of $\PoC\otimes \Omega^1_{\CC}$. By its very construction it can be written explicitly as
\[
\scant=\Theta(z,w)\trv\otimes \dd z.
\]
The following result proves that the purely algebraic translation operators are compatible with the analytic translation operators defined in \cite[\S 1.3]{bannai_kobayashi}:
\begin{prop}\label{ch_EP_prop_Uscan_expl}
The pullback of $U^{N,D}_{s,t}(\scan)$ to the universal covering is given by the explicit formula
\[
([ D]\times[ N])^*\left( \Theta_{Ds,Nt}(z,w) \trv \otimes \dd z \right).
\]
Here we denote by $\Theta_{Ds,Nt}(z,w)$ the translates of the Kronecker theta function as defined in \cite[\S 1.3]{bannai_kobayashi}.
\end{prop}
\begin{proof}
Let us write $\tilde{U}^{N,D}_{s,t}$ for the pullback of the analytification of the translation operator. Recall, that $s$ and $t$ denote the torsion points of $E$ given by $s=\frac{1}{N}$ and $t=\frac{1}{D}$ and that $A:=A(\Gamma):=\frac{\im \tau}{\pi}$ denotes the volume of $E(\CC)$ divided by $\pi$. Before we give an explicit description of $\tilde{U}^{N,D}_{s,t}(\scant)$, let us do the following computation:
\begin{align}\label{EP_eq1}
\notag (T_{s}\times T_{t})^*\left([ N]\times [ D]\right)^*(\trv)&=\frac{1}{\Theta(Nz+1,Dw+1)} \otimes ([N]\times[D])^*(\dd z)^\vee\stackrel{(*)}{=}\\
&=\exp\left( -\frac{Nz+Dw+1}{A} \right) ([ N]\times[ D])^*(\trv)
\end{align}
Here $(*)$ follows from the transformation law of the classical theta function \cite[eq. (8)]{bannai_kobayashi}:
\[
\theta(z+\gamma)=\alpha(\gamma)\cdot \exp\left( \frac{z\bar{\gamma}}{A}+\frac{\gamma\bar{\gamma}}{2A} \right) \theta(z)
\]
By the definition of $\Theta_{Ds,Nt}(z,w)$ we have
\begin{align*}
&\Theta_{D{s},N{t}}(Dz,Nw):=\exp\left( -\frac{Nz+Dw+1}{A} \right) \Theta(Dz+D{s},Nw+N{t},\tau).
\end{align*}
The definition
\[
U^{N,D}_{s,t}(\scan):=(\gamma_{N,D}\otimes \id_\Omega)\left( (T_{s}\times T_{t})^*\left[(\gamma_{D,N}\otimes \id_\Omega)\left( ([D]\times[N])^*\scan \right) \right] \right)
\]
gives us the following explicit description of $\tilde{U}^{N,D}_{s,t}(\scant)$:\\
\resizebox{\linewidth}{!}{
\begin{minipage}{\linewidth}
\begin{align*}
&\tilde{U}^{N,D}_{s,t}(\scant)= (\tilde{\gamma}_{N,D}\otimes \id_\Omega)\left( (T_{s}\times T_{t})^*\left[(\tilde{\gamma}_{D,N}\otimes \id_\Omega)\left( ([D]\times[N])^*\scant \right) \right] \right)\stackrel{\text{Lem.}\ref{EP_lemRene1}}{=}\\
&=(\tilde{\gamma}_{N,D}\otimes \id_\Omega)\left( (T_{s}\times T_{t})^*\left[ \Theta(Dz,Nw)([N]\times[D])^*(\trv\otimes \dd z) \right] \right)=\\
&=(\tilde{\gamma}_{N,D}\otimes \id_\Omega)\big( \Theta(Dz+D{s},Nw+N{t})(T_{s}\times T_{t})^*([N]\times[D])^*(\trv\otimes \dd z) \big)\stackrel{\eqref{EP_eq1}}{=}\\
&=(\tilde{\gamma}_{N,D}\otimes \id_\Omega)\left( \Theta_{D{s},N{t}}(Dz,Nw)([N]\times[D])^*(\trv\otimes \dd z) \right)=\\
&=\Theta_{D{s},N{t}}(Dz,Nw)([D]\times[N])^*(\trv\otimes \dd z)=\\
&=([D]\times[N])^*\left( \Theta_{D{s},N{t}}(z,w) \trv\otimes \dd z\right)
\end{align*}
\end{minipage}}
\end{proof}
The analytification of the universal vectorial extension $E^\dagger$ of $\Ed$ sits in a short exact sequence (cf. \cite[Ch I, 4.4]{mazur_messing})
\[
\begin{tikzcd}
0\ar[r] & R^1(\pi_{E}^{\textit{an}})_*(2\pi i\ZZ)\ar[r] & \HdRabs{1}{E}\ar[r] & \EN^{\dagger,an}\ar[r] & 0.
\end{tikzcd}
\]
In particular, the two dimensional complex vector space $\HdRabs{1}{E}$ serves as a universal covering of $E^\dagger(\CC)$. Choosing coordinates on this universal covering is tantamount to choosing a basis of
\[
\HdRabs{1}{E}^\vee\righteq \HdRabs{1}{\Ed}.
\]
Here, this isomorphism is canonically induced by Deligne's pairing. Let us choose $[\dd w]$ and $[\dd \bar{w}]$ in $\HdRabs{1}{\Ed}$ as a basis and denote the resulting coordinates by $(w,v)$. We can summarize the resulting covering spaces in the following commutative diagram:
\[
\begin{tikzcd}
\CC^2 \ar[r,"\pr_1"]\ar[d] & \CC \ar[d]\\
E^{\dagger}(\CC)\ar[r] & \Ed(\CC).
\end{tikzcd}
\]
The pullback of the Poincar\'e bundle $\Po^\dagger$ to $E\times E^\dagger$ is equipped with a canonical integrable $E^\dagger$-connection
\[
\con{\dagger}:\Po^\dagger\rightarrow \Po^\dagger\otimes_{\Ocal_{E\times E^\dagger}} \Omega^1_{E\times E^\dagger/E^\dagger}.
\]
Let us write $\PoC^\dagger$ for the pullback of the analytification of $\PoC^\dagger$ to the universal covering. The trivializing section $\trv$ of $\PoC$ induces a trivializing section $\trv^\dagger$ of $\PoC^\dagger$.
\begin{lem}\label{eq_condagger}
\[
\con{\dagger}(\trv^\dagger)=-\frac{v}{A}\trv^\dagger\otimes \dd z
\]
\end{lem}
\begin{proof}
We use the description of the connection given by Katz \cite[Thm. C.6 (1)]{katz_eismeasure}. Katz uses different coordinates: The basis $([\eta]^\vee,[\omega]^\vee)$ of $\HdRabs{1}{E}^\vee$ gives coordinates $(w_{\mathrm{Katz}},v_{\mathrm{Katz}})$ on the universal covering of $E^\dagger(\CC)$. Comparing both bases it is straight-forward to check that these coordinates are related to our coordinates via
\begin{align*}
w_{\mathrm{Katz}}&=-w\\
v_{\mathrm{Katz}}&=-\frac{v}{A}+w\cdot\left(\frac{1}{A}+\eta(1,\tau)\right).
\end{align*}
The explicit description of the connection in \cite[Thm. C.6 (1)]{katz_eismeasure} immediately implies the following formula:
\begin{align*}
\con{\dagger}(\trv^\dagger)&=\left[-\frac{\partial_z\Theta(z,-w_{\mathrm{Katz}})}{\Theta(z,-w_{\mathrm{Katz}})} + (\zeta(z-w_{\mathrm{Katz}})-\zeta(z)+v_{\mathrm{Katz}})) \right] \trv^\dagger\otimes \dd z
\end{align*}
Using
\begin{align*}
\frac{\partial_z\Theta(z,-w_{\mathrm{Katz}})}{\Theta(z,-w_{\mathrm{Katz}})}&=\partial_z \log \Theta(z,-w_{\mathrm{Katz}})=\\
&=-w_{\mathrm{Katz}}\cdot\left(\frac{1}{A}+\eta(1,\tau)\right)+\zeta(z-w_{\mathrm{Katz}})-\zeta(z)
\end{align*}
we get
\begin{align*}
\con{\dagger}(\trv^\dagger)&=\left[ v_{\mathrm{Katz}}+ w_{\mathrm{Katz}}\cdot\left(\frac{1}{A}+\eta(1,\tau)\right) \right] \trv^\dagger\otimes \dd z
\end{align*}
and the result follows by expressing this in our coordinates.
\end{proof}
Now, let $E^\sharp$ be the universal vectorial extension of $\Ed$. Our chosen auto-duality isomorphism $E\righteq \Ed$ induces an isomorphism
\[
E^\sharp\righteq E^\dagger.
\]
The explicit description of the universal covering of $E^\dagger(\CC)$ gives us a universal covering
\[
\CC^2\twoheadrightarrow E^\sharp(\CC)
\]
of $E^\sharp(\CC)$. Let us write $(z,u)$ for the corresponding coordinates. Let us write $\PoC^\dagger$ for the pullback of the analytification of $\PoC^\dagger$ to the universal covering. The trivializing section $\trv$ of $\PoC$ induces a trivializing section $\trv^\sharp$ of $\PoC^\sharp$. By transport of structure we deduce
\begin{equation}\label{eq_consharp}
\con{\sharp}(\trv^\sharp)=-\frac{u}{A}\trv^\sharp\otimes \dd w.
\end{equation}
\begin{prop}\label{EP_prop_Ekr}
Let $E=\CC/\ZZ+\tau\ZZ$ and $s,t\in E[ND](\CC)$ as above. The algebraic Eisenstein-Kronecker series $E^{k,r+1}_{s,t}\in \Sym^{k+r+1}\HdRabs{1}{E}$ are given by the explicit formula
\[
E^{k,r+1}_{s,t}=\sum_{i=0}^{\min(k,r)} \binom{r}{i}\binom{k}{i} \frac{(-1)^i}{A^i}\tilde{e}_{k-i,r-i+1}(Ds,Nt;\tau) \cdot[\dd \bar{z}]^{\otimes i}\otimes [\dd z]^{\otimes k+r+1-i}.
\]
\end{prop}
\begin{proof}
In the following we identify the cotangent spaces of $E^\sharp$ and $E^\dagger$ with $\HdRabs{1}{E}$. More concretely, this means that we identify:
\begin{align*}
\dd z&=\dd w=[\dd z]\\
\dd u&=\dd v=[\dd \bar{z}].
\end{align*}
Let us write
\begin{align*}
\dd_{E^\dagger}&: \Oan_{E^\sharp \times E^\dagger } \rightarrow \Oan_{E^\sharp \times E^\dagger}\otimes_{\CC} \HdRabs{1}{E},\quad f\mapsto \partial_w f[\dd z]+\partial_v f[\dd \bar{z}]\\
\dd_{E^\sharp}&: \Oan_{E^\sharp \times E^\dagger } \rightarrow \Oan_{E^\sharp \times E^\dagger}\otimes_{\CC} \HdRabs{1}{E},\quad f\mapsto \partial_z f[\dd z]+\partial_u f[\dd \bar{z}]
\end{align*}
Our aim is to compute
\[
E^{k,r+1}_{s,t}=(e\times e)^*\left[ ([D]\times [N])^*\nabla^{k,r}_{\sharp,\dagger}\left( (q^\sharp\times q^\dagger)^*U^{N,D}_{s,t}(\scan) \right) \right].
\]
\cref{ch_EP_prop_Uscan_expl} gives an explicit description of the translation operators:
\[
(q^\sharp\times q^\dagger)^*U^{N,D}_{s,t}(\scan)=([D]\times[N])^*\left(\Theta_{Ds,Nt}(z,w)\trv^{\sharp,\dagger}\otimes [\dd z]\right).
\]
Using this we compute using the Leibniz rule and \cref{eq_condagger}:
\begin{align*}
&E^{k,r+1}_{s,t}=(e\times e)^*\left[ ([D]\times [N])^*\left( \nabla_{\sharp,\dagger}^{k,r}\left( \Theta_{Ds,Nt}(z,w) \trv \right)\right)\right]\otimes[\dd z]=\\
=& (e\times e)^*\left[ \nabla_{\sharp}^{\circ k}\nabla_{\dagger}^{\circ r}\left( \Theta_{Ds,Nt}(z,w) \trv \right)\right]\otimes[\dd z]=\\
=& (e\times e)^*\left[\nabla_{\sharp}^{\circ k}\left( \sum_{i=0}^r \binom{r}{i} \left( -\frac{u}{A} \right)^i \partial_z^{\circ(r-i)} \Theta_{Ds,Nt}(z,w) \trv \right)\right]\otimes[\dd z]^{\otimes (r+1)}=\\
=& \left.\sum_{j=0}^k \binom{k}{j} \sum_{i=0}^r \binom{r}{i} \dd_{E^\sharp}^{\circ (k-j)} \left[ \left(-\frac{u}{A}\right)^i \partial_z^{\circ(r-i)} \Theta_{Ds,Nt}(z,w)\right]\right|_{\substack{z=v=0\\ w=u=0}}\cdot \\
&\quad \quad\quad \cdot \left.\left( -\frac{v}{A} \right)^j\right|_{v=0} \otimes [\dd z]^{\otimes (r+j+1)}=
\end{align*}
at this point let us observe, that $\left.\left( -\frac{v}{A} \right)^j\right|_{v=0}=0$ for $j>0$. Using this we continue:
\begin{align*}
=& \left. \sum_{i=0}^r \binom{r}{i} \dd_{E^\sharp}^{\circ k} \left[ \left(-\frac{u}{A}\right)^i \partial_z^{\circ(r-i)} \Theta_{Ds,Nt}(z,w)\right]\right|_{\substack{z=v=0\\ w=u=0}} \otimes [\dd z]^{\otimes (r+1)}=\\
=& \left. \sum_{i=0}^r \binom{r}{i} \sum_{j=0}^k \binom{k}{j} \left. \dd_{E^\sharp}^{\circ k}\left[ \left(-\frac{u}{A}\right)^i\right]\right|_{w=u=0}\cdot \partial_z^{\circ(r-i)}\partial_w^{\circ(k-j)} \Theta_{Ds,Nt}(z,w)\right|_{z=w=0} \otimes [\dd z]^{\otimes (r+k-j+1)}=
\end{align*}
again observe that $\left. \dd_{E^\sharp}^{\circ k}\left[ \left(-\frac{u}{A}\right)^i\right]\right|_{w=u=0}=0$ for $i\neq j$
\begin{align*}
=& \left. \sum_{i=0}^{\min(r,k)} \binom{r}{i} \binom{k}{i} \partial_z^{\circ(r-i)}\partial_w^{\circ(k-i)} \Theta_{Ds,Nt}(z,w)\right|_{z=w=0} \left(-\frac{[\dd \bar{z}]}{A}\right)^{\otimes i} \otimes [\dd z]^{\otimes (r+k-i+1)}
\end{align*}
Now the result follows from the result of Bannai-Kobayashi, i.e. \cite[Theorem 1.17]{bannai_kobayashi}:
\[
\Theta_{s,t}(z,w)=\sum_{a,b\geq 0}\frac{\tilde{e}_{a,b+1}(s,t)}{a!b!}z^bw^a,\quad s,t\notin\Gamma
\]
\end{proof}
\begin{proof}[Proof of \cref{EP_MainThm}] The construction of $E^{k,r+1}_{s,t}$ is obviously functorial on test object $(E/S,t)$. The explicit formula in \cref{EP_prop_Ekr} proves that $E^{k,r+1}_{s,t}$ is contained in the $k+r+1-\min(k,r)$-th filtration step of the Hodge filtration. This can also be seen without using the transcendental description as follows: By the symmetry of the situation, we may assume $k\leq r$. Keeping in mind that we agreed to denote the pullback of the connection $\nabla_\dagger\colon \Po^\dagger \rightarrow \Po^\dagger\otimes\Omega^1_{E\times E^\dagger/E^\dagger}$ again by $\nabla_\dagger$, we get the formula
\begin{align*}
&\left(([D]\times[N])^*\nabla_\dagger^{\circ r}\right)(q^\sharp\times q^\dagger)^*U_{s,t}^{[N],[D]}(\scan)\\
=&(q^\sharp\times \id_{E^\dagger})^*\left[ \left(([D]\times[N])^*\nabla_\dagger^{\circ r}\right)(\id_E\times q^\dagger)^*U_{s,t}^{[N],[D]}(\scan) \right].
\end{align*}
Since $\Po^\dagger\otimes\Omega^1_{E\times E^\dagger/E^\dagger}=\Po^\dagger\otimes_{\Ocal_S}\omega_{E/S}=\Po^\dagger\otimes_{\Ocal_S}F^1\Hcal^1_{dR}$ we deduce that the section $\sigma:=\left(([D]\times[N])^*\nabla_\dagger^{\circ r}\right)(\id_E\times q^\dagger)^*U_{s,t}^{[N],[D]}(\scan)$ is contained in the filtration step
\[
([D]\times[N])^*\left( \Po^\dagger \otimes_{\Ocal_S}F^{r+1}\Sym^{r+1}\Hcal^1_{dR}\right)\subseteq ([D]\times[N])^*\left( \Po^\dagger \otimes_{\Ocal_S}\Sym^{r+1}\Hcal^1_{dR}\right).
\]
We deduce
\[
E^{k,r+1}_{s,t}=(e\times e)^*(\nabla_{\sharp}^{\circ k}(q^\sharp\times\id)^*\sigma)\in \Gamma(S,F^{r+1}\Sym^{r+1}\Hcal^1_{dR})
\]
as desired.
It remains to identify the image of $E^{k,r}_{s,t}$ under the Hodge decomposition with the Eisenstein-Kronecker series for a single elliptic curve $E/\CC$ as above. By \cref{EP_prop_Ekr} the algebraic Eisenstein-Kronecker series $E^{k,r}_{s,t}$ for $E/\CC$ is given by
\[
E^{k,r+1}_{s,t}=\sum_{i=0}^{\min(k,r)} \binom{r}{i}\binom{k}{i} \frac{(-1)^i}{A^i}\tilde{e}_{k-i,r-i+1}(Ds,Nt) \cdot[\dd \bar{z}]^{\otimes i}\otimes [\dd z]^{\otimes k+r+1-i}.
\]
The Hodge decomposition is the projection to the $[\dd z]^{\otimes k+r+1}$-part which is the $\Cinfty$ modular form
\[
\tilde{e}_{k,r+1}(Ds,Nt)\dd z^{\otimes k+r+1}.
\]
Since the analytic Eisenstein--Kronecker series are finite at the cusps, we deduce the finiteness at the Tate curve from the analytic comparison.
\end{proof}
\section{The Kronecker section and Kato-Siegel functions}
Kato--Siegel functions $\thetaD\in \Gamma(E\setminus E[D],\Ocal_E^\times)$ as introduced by Kato in \cite{kato} play an important role in modern number theory. While the values of $\thetaD$ at torsion points are closely related to elliptic units, the value of iterated derivatives of
\[
d\log \thetaD\in\Gamma(E,\Omega^1_{E/S}(E[D]))
\]
at torsion points give classical algebraic Eisenstein series. The aim of this section is to construct the logarithmic derivatives of the Kato--Siegel functions via the Kronecker section of the Poincar\'e bundle. Let us emphasize that it is not necessary that $6$ is co-prime to $D$ for the construction of the Kronecker section. This gives a new construction of the logarithmic derivatives of the Kato--Siegel functions even if the Kato--Siegel functions are not defined. This section might be skipped on the first reading. The comparison to the Kato--Siegel functions builds on some tedious computations involving the translation operators. Some results of this section are needed to prove the distribution relation of the Kronecker section which is stated in the Appendix.
\subsection{Kato-Siegel functions via the Poincar\'e bundle}
Let us slightly generalize the definition of the translation operators of \cref{sec_can}.
\begin{defin}
Let $\psi:E\rightarrow E'$ be an isogeny of elliptic curves over $S$. Let us write $\Po'$ for the Poincar\'e bundle of $E'$. For $t\in\ker\psi^\vee(S)$ define
\[
\Ucal^{\psi^\vee}_t: (\psi\times T_t)^*\Po'\righteq (\psi\times\id)^*\Po'
\]
by
\[
\Ucal^{\psi^\vee}_t:=\gamma_{\id,\psi^\vee}\circ (\id\times T_t)^*\gamma_{\psi,\id}.
\]
\end{defin}
\begin{rem}
In the case $\psi=[D]$ this coincides with our previous definition of $\Ucal^{D}_t$.
\end{rem}
For $f\in\Gamma\left(E'\times E'^\vee, \Po'\otimes \Omega^1_{E'\times E'^\vee/E'^\vee}([E'\times e]+[e\times E'^\vee]) \right)$ set
\[
U^{\psi^\vee}_t(f):=\Ucal^{\psi^\vee}_t\left( (\psi\times T_t)^*f \right)
\]
thus $U^{\psi^\vee}_t(f)$ is a section of
\[
(\psi\times\id)^*\left[\Po'\otimes \Omega^1_{E'\times E'^\vee/E'^\vee}([E'\times (-t)]+[e\times E'^\vee])\right].
\]
For $e\neq t$ the identification
\begin{align}\label{ch_EP_eq4}
&(\id\times e)^*(\psi\times\id)^*\left[\Po'\otimes \Omega^1_{E'\times E'^\vee/E'^\vee}([E'\times (-t)]+[e\times E'^\vee])\right]=\\
\notag\cong& \psi^*\left(\Omega^1_{E'/S}\left([e] \right)\right) \notag\cong \Omega^1_{E/S}(\ker \psi)
\end{align}
allows us to view $(\id\times e)^*\left( U^{\psi^\vee}_t(f) \right)$ as a global section of $\Omega^1_{E/S}(\ker\psi)$. We will implicitly use this identification in the following. Let us assume that the degree $\deg\psi$ of the isogeny $\psi$ is invertible on $S$. For $\tilde{t}\in (\ker\psi)(S)\subseteq E(S)$ let us write
\[
\Res_{\tilde{t}}:\Gamma(E,\Omega^1_{E/S}(\ker\psi))\rightarrow \Gamma(S,\Ocal_S)
\]
for the residue map along $\tilde{t}$. We keep our notation $\lambda:E\righteq E^\vee$ for the auto-duality associated to the ample line bundle of the zero section.
\begin{prop}\label{ch_EP_propomegaexists}
Let $\psi:E\rightarrow E'$ be an isogeny of elliptic curves over $S$. Let us assume that $\deg\psi$ is not a zero-divisor on $S$. For a non-zero section $t\in \ker \psi^\vee(S)\subseteq E'^\vee (S)$ the section
\[
\omega^{\psi}_{t}:=(\id\times e)^*U^{\psi^\vee}_{t}(s_{\mathrm{can},E'})\in\Gamma(E,\Omega^1_{E/S}(\ker\psi))
\]
satisfies the following properties:
\begin{enumerate}
\item\label{ch_EP_propomegaexists_a} For each finite \'{e}tale $S$-scheme $T$ with $\left|(\ker \psi) (T) \right|=\deg\psi$ we have
\[
\Res_{\tilde{t}}\omega^{\psi}_{t} = \langle \tilde{t},t\rangle
\]
for all $\tilde{t}\in \ker\psi(T)$. Here,
$$
\langle \cdot, \cdot \rangle\colon \ker\psi\times_S \ker\psi^\vee\rightarrow \mathbb{G}_m
$$
denotes Oda's pairing, cf.~\cite{oda}.
\item\label{ch_EP_propomegaexists_b} The section $\omega^{\psi}_{t}\in\Gamma\left(E,\Omega^1_{E/S}(\ker\psi)\right)$ is contained in the $\Ocal_E$-submodule
\[
\Omega^1_{E/S}\left(\psi^*([e]-[t])\right)
\]
of $\Omega^1_{E/S}(\ker\psi) $. Here, we have used our chosen auto-duality isomorphism $E'\cong E'^\vee$ to view $t$ as a section of $E'$.
\end{enumerate}
Further, $\omega^{\psi}_{t}$ is the unique section of $\Omega^1_{E/S}(\ker\psi)$ satisfying \ref{ch_EP_propomegaexists_a} and \ref{ch_EP_propomegaexists_b}.
\end{prop}
\begin{proof}
For uniqueness let $\tilde{\omega}_1$ and $\tilde{\omega}_2$ both satisfy \ref{ch_EP_propomegaexists_a} and \ref{ch_EP_propomegaexists_b}. By \ref{ch_EP_propomegaexists_a} the difference satisfies:
\[
\tilde{\omega}_1-\tilde{\omega}_2\in \Gamma(E,\ker(\bigoplus_{\tilde{t}}\Res_{\tilde{t}}))=\Gamma(E,\Omega^1_{E/S})
\]
On the other hand, \ref{ch_EP_propomegaexists_b} shows that $\tilde{\omega}_1-\tilde{\omega}_2$ vanishes along the divisor $\psi^*[t]$ and we conclude $\tilde{\omega}_1-\tilde{\omega}_2=0$.\par
Let us now prove that $\omega^{\psi}_{t}$ satisfies \ref{ch_EP_propomegaexists_b}. By its definition $s_{\mathrm{can},E'}$ is contained in the submodule
\[
\Po' \otimes \Omega^1_{E'\times E'/E'}([e\times E']+[E'\times e]) \otimes \Ocal_{E'\times E'}( - \Delta)
\]
of $\Po' \otimes \Omega^1_{E'\times E'/E'}([e\times E']+[E'\times e])$. By the definition of the translation operator the global section $(\id\times e)^*U^{\psi^\vee}_{t}(s_{\mathrm{can},E'})$ of $\Omega^1_{E/S}(\ker\psi)$ is a global section of the $\Ocal_E$-submodule
\[
\Omega^1_{E/S}\left(\psi^*([e]-[t])\right).
\]
This proves \ref{ch_EP_propomegaexists_b}.\par
The residue map is compatible with base change. Combining this with the isomorphism
\[
f^*\Omega^1_{E/S}(\ker\psi)\righteq \Omega^1_{E_T/T}(\ker\psi_T)
\]
for $f:T\rightarrow S$ finite \'{e}tale, allow us to check \ref{ch_EP_propomegaexists_a} after finite \'{e}tale base change. Thus, we may assume that $|\ker\psi(S)|=\deg\psi$. Before we do the residue computation, let us recall the definition of Oda's pairing
\[
\langle\cdot,\cdot\rangle: \ker\psi\times_S\ker\psi^\vee \rightarrow \mathbb{G}_{m,S}.
\]
Let $t\in (\ker\psi)(S)$ and $[\Lcal]\in (\ker\psi^\vee)(S)$. Since we have assumed $[\Lcal]\in (\ker\psi^\vee)(S)$, the line bundle $\psi^*\Lcal$ is trivial and we can choose an isomorphism
\[
\alpha: \psi^*\Lcal\righteq \Ocal_E.
\]
The chosen isomorphism $\alpha$ gives rise to a chain of isomorphisms
\[
\begin{tikzcd}
\Ocal_E \ar{r}{\alpha^{-1}}[swap]{\sim} & \psi^*\Lcal=T_t^*\psi^*\Lcal \ar{r}{T_t^*\alpha}[swap]{\sim} & T_t^*\Ocal_E=\Ocal_E
\end{tikzcd}
\]
and $\langle t,[\Lcal] \rangle_\psi$ is defined as the image of $1$ under this isomorphism. Our first aim is to prove
\begin{equation*}
T_{\tilde{t}}^*\omega^{\psi}_{t}=\langle \tilde{t},t \rangle_\psi\cdot \omega^{\psi}_{t}
\end{equation*}
for every $\tilde{t}\in \ker\psi(S)$. The section $t\in E'^\vee(S)$ corresponds to the isomorphism class $[(\id\times t)^* \Po']$ of line bundles. We apply Oda's pairing to $\tilde{t}$ and $[\Lcal]$ with $\Lcal:=(\id\times t)^* \Po'$. We have the following canonical choice for $\alpha$:
\[
\begin{tikzcd}[column sep=huge]
\alpha: {\psi}^* \Lcal =({\psi}\times t)^* \Po' \ar[r,"(\id\times t)^*\gamma_{\psi,\id}"] & (\id\times t)^*(\id\times {\psi^\vee})^* \Po=(\id\times e)^*\Po\cong \Ocal_E.
\end{tikzcd}
\]
Note that
\begin{align}\label{ch_EP_eq7}
\omega^{\psi}_{t}&:=(\id\times e)^* U^{\psi^\vee}_{t}(\scan)\stackrel{\mathrm{def}}{=}\notag\\
&=(\id\times e)^*\left(\left[ \left(\gamma_{\id,\psi^\vee}\circ (\id\times T_{t})^*\gamma_{\psi,\id}\right)\otimes \id_{\Omega^1} \right]\left( (\psi\times T_{t})^*(\scan) \right)\right) =\\
&=\left((\id\times t)^*\gamma_{\psi,\id} \otimes\id_{\Omega^1}\right)((\psi\times t)^*\scan)=\left( \alpha \otimes \id_{\Omega^1}\right)((\psi\times t)^*\scan).\notag
\end{align}
After tensoring
\[
\begin{tikzcd}
\Ocal_E \ar[rr,"\cdot \langle \tilde{t}{,}t \rangle_\psi",bend right]\ar[r,"\alpha^{-1}"] & \psi^*\Lcal=T_{\tilde{t}}^*\psi^*\Lcal \ar[r,"T_{\tilde{t}}^*\alpha"] & \Ocal_E
\end{tikzcd}
\]
with $\otimes_{\Ocal_E}\Omega^1_{E/S}(\ker\psi)$, we obtain
\[
\begin{tikzcd}
\Omega^1_{E/S}(\ker\psi) \ar[rr,"\cdot \langle \tilde{t}{,}t \rangle",bend right] & \psi^*\Lcal\otimes\Omega^1_{E/S}(\ker\psi)=T_{\tilde{t}}^*\psi^*\Lcal\otimes\Omega^1_{E/S}(\ker\psi)\ar[l,"\alpha\otimes\id_\Omega",swap] \ar[r,"T_{\tilde{t}}^*{\alpha\otimes\id_\Omega}"] & \Omega^1_{E/S}(\ker\psi)
\end{tikzcd}
\]
This diagram together with \eqref{ch_EP_eq7} proves
\begin{align*}
T_{\tilde{t}}^* \omega^{\psi}_{t}&= T_{\tilde{t}}^*\Big[\left( \alpha \otimes \id_{\Omega^1}\right)((\psi\times t)^*\scan)\Big]= \left( T_{\tilde{t}}^*\alpha \otimes \id_{\Omega^1}\right)\left((\psi\times t)^*\scan \right)=\\
&=\langle \tilde{t}{,}t\rangle\cdot\left( \alpha \otimes \id_{\Omega^1}\right)((\psi\times t)^*\scan)=\langle \tilde{t}{,}t \rangle\cdot\omega^{\psi}_t
\end{align*}
as desired.\par
The equation
\begin{equation*}
T_{\tilde{t}}^*\omega^{\psi}_t=\langle \tilde{t},t \rangle_\psi \cdot \omega^{\psi}_t
\end{equation*}
reduces the proof of \ref{ch_EP_propomegaexists_a} to the claim
\[
\Res_e \omega^{\psi}_t=1
\]
which can be checked by an explicit and straightforward computation locally in a neighbourhood of the zero section.
\end{proof}
The following result was obtained during the proof of the above proposition.
\begin{cor}\label{ch_EP_cortranslation} Let $\psi\colon E\rightarrow E'$ be an isogeny of elliptic curves and assume that $\deg \psi$ is not a zero-divisor on $S$. Then we have the following equality for all $\tilde{t}\in \ker\psi(S)$ and $t\in \ker\psi^\vee(S)$:
\[
T_{\tilde{t}}^*\omega^{\psi}_t=\langle \tilde{t},t \rangle_\psi \cdot \omega^{\psi}_t
\]
\end{cor}
The most important case is the case $\psi=[D]$ in this case we have produced for each $t\in\Ed[D](S)$ sections
\[
\omega^{[D]}_t\in\Gamma(E,\Omega^1_{E/S}(E[D]))
\]
via the Poincar\'e bundle. Before we can relate $\omega^{[D]}_t$ to logarithmic derivatives of Kato-Siegel functions, let us study certain compatibility relations among the $\omega^\psi_t$:
\begin{lem}\label{ch_EP_lemTrace}
Let $\psi:E\rightarrow E'$ and $\varphi:E'\rightarrow E''$ be isogenies of elliptic curves over $S$. Let us further assume that $\deg \varphi\circ\psi$ is not a zero-divisor on $S$.
\begin{enumerate}
\item\label{ch_EP_lemTrace_a} For $e\neq s\in\ker\varphi^\vee$:
\[
\omega_{s}^{\varphi\circ\psi}=\psi^*\omega_s^\varphi
\]
\item\label{ch_EP_lemTrace_b} Assume that $|\ker \varphi^\vee(S)|=\deg\varphi^\vee$. For $t\in\ker(\varphi\circ\psi)^\vee(S)$ with $\varphi^\vee(t)\neq e$ we have
\[
\sum_{s\in\ker\varphi^\vee(S)}\omega_{t+s}^{\varphi\circ\psi}=\deg\varphi \cdot \omega_{\varphi^\vee(t)}^\psi
\]
\end{enumerate}
\end{lem}
\begin{proof}\ref{ch_EP_lemTrace_b}: Both sides of the claimed equality are elements in
\[
\Gamma(E,\Omega^1_{E/S}(\ker\varphi\circ\psi)).
\]
Since $\deg \phi\circ\psi$ is not a zero-divisor on $S$, we may check the equality after inverting $\deg \phi\circ\psi$. We may further assume that there is a finite \'{e}tale map $f:T\rightarrow S$ s.t. $|\ker\varphi\circ\psi(T)|=\deg \varphi\circ\psi$. The canonical map
\[
f^*\Omega^1_{E/S}(\ker\varphi\circ\psi)\rightarrow \Omega^1_{E_T/T}(\ker\varphi_T\circ\psi_T)
\]
is an isomorphism. Since all constructions are compatible with base change, we may assume during the proof that $|\ker\varphi\circ\psi(T)|=\deg \phi\circ\psi$. In a first step we show that the difference of both sides has no residue, i.\,e.
\[
\omega_0:=\left( \sum_{s\in\ker\varphi^\vee(S)}\omega_{t+s}^{\varphi\circ\psi}-\deg\varphi \cdot \omega_{\varphi^\vee(t)}^\psi \right)\in \Gamma\Big(E,\ker\bigoplus_{\tilde{t}\in\ker \varphi\circ\psi (S) }\Res_{\tilde{t}}\Big).
\]
For $\tilde{t}\in \ker\varphi\circ \psi(S)$ we compute
\begin{align*}
\Res_{\tilde{t}}\sum_{s\in\ker\varphi^\vee(S)}\omega_{t+s}^{\varphi\circ\psi}&=\sum_{s\in\ker\varphi^\vee(S)} \langle \tilde{t},t+s\rangle_{\varphi\circ\psi}=\\
&=\langle\tilde{t}, t\rangle_{\varphi\circ\psi}\cdot\sum_{s\in\ker\varphi^\vee(S)} \langle \psi(\tilde{t}),s\rangle_{\varphi} =\\
&=\begin{cases} \deg\varphi \cdot \langle\tilde{t},\varphi^\vee(t)\rangle_{\psi} & \tilde{t}\in\ker \psi \\ 0 & \tilde{t}\notin\ker\psi. \end{cases}
\end{align*}
But this coincides with the residue of $\deg\varphi\cdot \omega_{\varphi^\vee(t)}^\psi$:
\[
\deg\varphi \Res_{\tilde{t}} \omega_{\varphi^\vee(t)}^\psi= \begin{cases} \deg\varphi\cdot \langle\tilde{t},\varphi^\vee(t)\rangle_{\psi} & \tilde{t}\in\ker \psi \\ 0 & \tilde{t}\notin\ker\psi. \end{cases}
\]
This shows $\omega_0\in\Gamma(E,\Omega^1_{E/S})$. In particular, $\omega_0$ is translation-invariant. On the other hand, we can use the behaviour of $\omega^{D}_{t}$ under translation (cf.~\cref{ch_EP_cortranslation}) to compute:
\begin{align*}
\deg \psi \cdot \omega_0&=\sum_{\tilde{s}\in \ker\psi }T^*_{\tilde{s}}\omega_0= \sum_{\tilde{s}\in \ker\psi }\left( \sum_{s\in\ker \varphi^\vee} T_{\tilde{s}}^* \omega_{t+s}^{\varphi\circ\psi}-\deg \varphi\cdot T_{\tilde{s}}^*\omega_{\varphi^\vee(t)}^\psi \right)=\\
&=\sum_{\tilde{s}\in \ker\psi }\left( \sum_{s\in\ker \varphi^\vee} \langle \tilde{s},t+s\rangle_{\varphi\circ\psi}\cdot \omega_{t+s}^{\varphi\circ\psi}-\deg \varphi\cdot \langle \tilde{s},\varphi^\vee(t)\rangle_{\psi}\cdot \omega_{\varphi^\vee(t)}^\psi \right)=\\
&=\underbrace{\left(\sum_{\tilde{s}\in \ker\psi } \langle \tilde{s},\varphi^\vee(t)\rangle_{\psi} \right)}_{=0}\cdot\left( \sum_{s\in\ker \varphi^\vee} \omega_{t+s}^{\varphi\circ\psi}-\deg \varphi\cdot \omega_{\varphi^\vee(t)}^\psi \right)=0
\end{align*}
Since $\deg\psi$ is not a zero-divisor on $S$, we conclude $\omega_0=0$ as desired.\par
\ref{ch_EP_lemTrace_a} can be proven along the same lines.
\end{proof}
\begin{lem}\label{ch_EP_lemTrace2}
Let
\[
\begin{tikzcd}
E\ar[r,"\varphi"]\ar[d,"\psi"] & E' \ar[d,"\psi'"] \\
E'\ar[r,"\varphi'"] & E
\end{tikzcd}
\]
be a commutative diagram of isogenies of elliptic curves over $S$. Let us further assume that $\deg\varphi'\circ\psi$ is not a zero-divisor on $S$. Let $t\in\ker(\varphi')^\vee(S)$ with $(\psi')^\vee(t)\neq e$.
\begin{enumerate}
\item\label{ch_EP_lemTrace2_a} For $s\in(\ker\psi'^\vee)(S)$ we have
\[
\omega^{\varphi'\circ\psi}_{s+t}=\sum_{\tilde{s}\in \ker\psi(S)} \langle \tilde{s},\varphi'^\vee (s) \rangle_\psi \cdot (T_{-\tilde{s}})^*\omega^{\varphi}_{\psi'^\vee(t)}
\]
\item\label{ch_EP_lemTrace2_b} We have
\[
\sum_{\tilde{s}\in \ker\psi(S)} (T_{\tilde{s}})^*\omega^{\varphi}_{\psi'^\vee(t)}=\psi^* \omega^{\varphi'}_{t}.
\]
\end{enumerate}
\end{lem}
\begin{proof}\ref{ch_EP_lemTrace2_a}: Both sides of the claimed equality are elements in
\[
\Gamma(E,\Omega^1_{E/S}(\ker\varphi'\circ\psi)).
\]
After inverting $\deg \phi'\circ\psi$ there is a finite \'{e}tale map $f:T\rightarrow S$ s.t. $|\ker\varphi'\circ\psi(T)|=\deg \varphi'\circ\psi$. The canonical map
\[
f^*\Omega^1_{E/S}(\ker\varphi'\circ\psi)\rightarrow \Omega^1_{E_T/T}(\ker\varphi'_T\circ\psi_T)
\]
is an isomorphism and since all constructions are compatible with base change, we may assume during the proof that $|\ker\varphi'\circ\psi(T)|=\deg \phi\circ\psi$. In a first step we show that the difference of both sides has no residue, i.\,e.
\[
\omega_0:=\left(\omega^{\varphi'\circ\psi}_{s+t}-\sum_{\tilde{s}\in \ker\psi(S)} \langle \tilde{s},\varphi'^\vee (s) \rangle_\psi \cdot (T_{-\tilde{s}})^*\omega^{\varphi}_{\psi'^\vee(t)} \right)\in \Gamma\Big(E,\Omega^1_{E/S}\Big).
\]
For $\tilde{t}\in (\ker\varphi'\circ \psi)(S)$ we compute
\begin{align*}
& \Res_{\tilde{t}}\sum_{\tilde{s}\in \ker\psi(S)} \langle \tilde{s},\varphi'^\vee (s) \rangle_\psi \cdot (T_{-\tilde{s}})^*\omega^{\varphi}_{\psi'^\vee(t)}\\
=& \begin{cases} \langle \tilde{s},\varphi'^\vee (s) \rangle_\psi \cdot \langle \tilde{t}-\tilde{s},\psi'^\vee (t) \rangle_\varphi & \tilde{t}-\tilde{s}\in\ker\varphi \\ 0 & \tilde{t}-\tilde{s}\notin\ker\varphi \end{cases}\\
=& \begin{cases} \langle \varphi(\tilde{s}),s \rangle_{\psi'} \cdot \langle \psi(\tilde{t}),t \rangle_{\varphi'} & \varphi(\tilde{t})=\varphi(\tilde{s}) \\ 0 & \varphi(\tilde{t})\neq\varphi(\tilde{s})\end{cases}\\
=& \langle \varphi(\tilde{t}),s \rangle_{\psi'} \cdot \langle \psi(\tilde{t}),t \rangle_{\varphi'}=\langle \tilde{t},s+t \rangle_{\varphi'\circ\psi}
\end{align*}
But this coincides with the residue of $\omega^{\varphi'\circ\psi}_{s+t}$. This shows $\omega_0\in\Gamma(E,\Omega^1_{E/S})$. In particular, $\omega_0$ is translation-invariant.
\begin{align*}
\deg\varphi\cdot \omega_0&=\sum_{\tilde{t}\in\ker\varphi(S)}T_{\tilde{t}}^*\omega_0=\\
&=\sum_{\tilde{t}\in\ker\varphi(S)}\left( T_{\tilde{t}}^*\omega_{s+t}^{\varphi'\circ\psi} -\sum_{\tilde{s}\in\ker\psi(S)} \langle \tilde{s},\varphi'^\vee(s)\rangle_{\psi}T_{-\tilde{s}}^*T_{\tilde{t}}^*\omega^\varphi_{\psi'^\vee(t)}\right)=\\
&=\underbrace{\left(\sum_{\tilde{t}\in\ker\varphi(S)}\langle \tilde{t},\psi'^\vee(t)\rangle_{\varphi}\right)}_{=0}\cdot\omega_0
\end{align*}
Since $\deg\varphi$ is not a zero-divisor on $S$, we conclude $\omega_0=0$ as desired.\par
\ref{ch_EP_lemTrace2_b}: Follows by setting $s=e$ in \ref{ch_EP_lemTrace2_a} and using \cref{ch_EP_lemTrace} \ref{ch_EP_lemTrace_b}.
\end{proof}
The classical Kato--Siegel functions are only defined if $D$ is co-prime to $6$, while the definition of $\omega_D$ (defined below) makes sense without this hypothesis:
\begin{cor}\label{PE_propKatoSiegel}
For $D>1$ invertible on $S$, let $T\rightarrow S$ be finite \'etale with $|E[D](T)|=D^2$. Let us define
\[
\omega^D:=\sum_{e\neq t\in \Ed_T[D](T)} \omega^{[D]}_t \in\Gamma(E,\Omega_{E/S}(E[D])).
\]
If $D$ is furthermore co-prime to $6$, then $\omega^D$ coincides with the logarithmic derivative of the Kato--Siegel function $\thetaD$, i.e.
\[
\omega^D=\dlog \thetaD.
\]
\end{cor}
\begin{proof}
The logarithmic derivative $\dlog \thetaD\in\Gamma(E,\Omega^1_{E/S}(E[D]))$ is uniquely determined by the following two properties:
\begin{enumerate}
\item Its residue is
\[
\Res (\dlog \thetaD)=D^2\mathds{1}_e-\mathds{1}_{E[D]}
\]
where
\[
\Res: \Omega^1_{E/S}(\log E[D])\rightarrow (i_{E[D]})_*\Ocal_{E[D]}
\]
is the residue map and $\mathds{1}_e$ resp. $\mathds{1}_{E[D]}$ are the functions in $(i_{E[D]})_*\Ocal_{E[D]}$ which have the constant value one along $e$ resp. $E[D]$.
\item It is trace compatible, i.\,e. for each $N$ coprime to $D$ we have
\[\Tr_{[N]}\dlog \thetaD=\dlog \thetaD.\]
\end{enumerate}
The residue condition for $\omega^D$ follows from \cref{ch_EP_propomegaexists}. The trace compatibility follows by applying \cref{ch_EP_lemTrace2} \ref{ch_EP_lemTrace2_b} with $\psi$ and $\psi'$ equal to $[N]$ and $\varphi$ and $\varphi'$ equal to $[D]$.
\end{proof}
\part{p-adic interpolation of real-analytic Eisenstein series}
\section[$p$-adic theta functions for sections of the Poincar\'e bundle]{p-adic theta functions for sections of the Poincar\'e bundle}
In \cite{norman_padictheta} P.~Norman discussed constructions of $p$-adic theta functions associated to sections of line bundles on Abelian varieties over algebraically closed $p$-adic fields. A similar method can be used to construct $p$-adic theta functions for sections of the Poincar\'e bundle for Abelian schemes with ordinary reduction over more general $p$-adic base schemes. We will discuss the construction of $p$-adic theta functions for elliptic curves but it immediately generalizes to the higher dimensional case.\par
Let $p$ be a fixed prime. Let $R$ be a $p$-adic ring, i.\,e. $R$ is complete and separated in its $p$-adic topology, and set $S:=\Spec R$. An elliptic curve $E/S$ will be said to have ordinary reduction if $E\times_S \Spec R/pR$ is fiber-wise an ordinary elliptic curve. For the moment let $n\geq 1$ be a fixed integer. Let $C:=C_n$ resp.~$D:=D_n$ be the connected components of $E[p^n]$ resp.~$E^\vee[p^n]$. Let us write $i: C\hookrightarrow E$ and $j:D\hookrightarrow \Ed$ for the inclusions. We define
\[
\varphi:E\twoheadrightarrow E/C=:E'
\]
and note that its dual $\varphi^\vee\colon (E')^\vee\rightarrow E^\vee$ is \'{e}tale since we assumed $E/S$ to have ordinary reduction. Let us further write $j'\colon D'\hookrightarrow (E')^\vee$ for the inclusion of the connected component of $(E')^\vee[p^n]$. Since $\varphi^\vee$ is \'etale it induces an isomorphism on connected components of $p^n$-torsion groups, i.e.:
\[
\begin{tikzcd}
D'\ar[r,hook,"j'"] \ar[d,"\cong"] & (E')^\vee \ar[d,"\varphi^\vee"]\\
D \ar[r,hook,"j"] & E^\vee
\end{tikzcd}
\]
Let us write $\Phi:\Ocal_D\righteq \Ocal_{D'}$ for the induced isomorphism of structure sheaves. In particular by pulling back the Poincar\'e bundle along this diagram we obtain an $\id\times \Phi$-linear isomorphism
\[
(\id\times j)^*\Po\righteq (\id\times j')^*(\id\times\varphi^\vee)^*\Po.
\]
Let us write $\Po'$ for the Poincar\'e bundle on $E'\times_S(E'^\vee)$. Restricting along $(i\times\id)$ and composing with $(i\times\id)^*\gamma_{\id,\varphi^\vee}$ gives an $\id\times\Phi$-linear isomorphism
\begin{equation}\label{eq_Triv1}
(i \times j)^*\Po\righteq (i\times j')^*(\id\times\varphi^\vee)^*\Po\righteq (i\times j')^*(\varphi\times\id)^*\Po'.
\end{equation}
Since $\varphi\circ i$ factors through the zero section, we have the identity
\[
\varphi\circ i=\varphi \circ e\circ \pi_C.
\]
Using this, we obtain an $\id\times\Phi^{-1}$-linear isomorphism
\begin{equation}\label{eq_Triv2}
(i\times j')^*(\varphi\times\id_{E'^\vee})^*\Po'=(\pi_C\times \id_{D'})^*(e\times\id_{D'})^*(\varphi\times j')^*\Po\stackrel{(a)}{\cong} \pi_C^*\Ocal_{D'}\stackrel{(b)}{\cong} \Ocal_C\otimes_{\Ocal_S}\Ocal_D.
\end{equation}
Here, we have used the rigidification of the Poincar\'e bundle in $(a)$ and $\Phi^{-1}$ in $(b)$. Finally, the composition of \eqref{eq_Triv1} and \eqref{eq_Triv2} gives an $\Ocal_{C_n}\otimes_{\Ocal_S}\Ocal_{D_n}$-linear isomorphism
\[
\triv_n: \Po|_{C_n\times D_n}=(i\times j)^*\Po \righteq \Ocal_{C_n}\otimes_{\Ocal_S}\Ocal_{D_n}.
\]
It is straightforward to check that this isomorphism is compatible with restriction along $C_n\hookrightarrow C_m$ for $n\leq m$. Let us write $\hat{E}$ and $ \Edf$ for the formal groups obtained by completion of $E$ and $\Ed$ along the zero section. By passing to the limit over $n$ we obtain
\begin{equation}\label{eq_Triv3}
\triv:\Po|_{\hat{E}\times_S \Edf}\righteq \Ocal_{\hat{E}}\hat{\otimes}_{\Ocal_S}\Ocal_{\Edf}.
\end{equation}
Let us introduce the notation $\Pof:=\Po|_{\hat{E}\times_S \Edf}$ for the restriction of the Poincar\'e bundle to $\hat{E}\times_S \Edf$.
\begin{defin}
For a section $s\in\Gamma(U,\Po)$ with $U\subseteq E\times_S\Ed$ an open subset containing the zero section $e\times_S e$, let us define the \emph{$p$-adic theta function associated to $s$} by
\[
\vartheta_s:=\triv(s)\in \Gamma(\hat{E}\times_S\Edf, \Ocal_{\hat{E}}\hat{\otimes}_{\Ocal_S}\Ocal_{\Edf}).
\]
\end{defin}
\section[$p$-adic Eisenstein-Kronecker series]{p-adic Eisenstein-Kronecker series}
In the first part of this paper we have given a construction of real-analytic Eisenstein series via the Poincar\'e bundle. More precisely we constructed geometric nearly holomorphic modular forms
\[
E^{k,r}_{s,t} \in \Gamma(S,\Sym^{k+r}\HdR{1}{E/S})
\]
which give rise to the classical Eisenstein-Kronecker series after applying the Hodge-decomposition on the modular curve. It was first observed by Katz in \cite{katz_padicinterpol} that one can get $p$-adic modular forms associated to geometric nearly holomorphic modular forms by applying the unit root decomposition on the universal trivialized elliptic curve instead of the Hodge decomposition. Let us recall this construction. For more details we refer to Katz' paper \cite{katz_padicinterpol}.\par
Let $R$ be a $p$-adic ring and let us write $S=\Spec R$. A trivialization of an elliptic curve $E/S$ is an isomorphism
\[
\beta: \Ef\righteq \Gmf{S}
\]
of formal groups over $S$. For a natural number $N\geq 1$ coprime to $p$, a trivialized elliptic curve with $\Gamma(N)$-level structure is a triple $(E,\beta,\alpha_N)$ consisting of an elliptic curve $E/S$, a trivialization $\beta$ and a level structure $\alpha_N:(\ZZ/N\ZZ)^2_S\righteq E[N]$. Let $(\Etriv,\beta,\alpha_N)$ be the universal trivialized elliptic curve over the moduli scheme $\Mtriv$ of trivialized elliptic curves of level $\Gamma(N)$. The scheme $\Mtriv$ is affine. Let us write $V_p(\Gamma(N))$ or sometimes just $\VpN$ for the ring of global sections of $\Mtriv$. Following Katz, the ring $\VpN$ will be called ring of generalized $p$-adic modular forms. For more details we refer to \cite[Ch. V]{katz_padicinterpol}. Let us recall the definition of the unit root decomposition. Dividing $\Etriv$ by its canonical subgroup $C$, again gives a trivialized elliptic curve
\[
(E'=\Etriv/C,\beta',\alpha_N')
\]
with $\Gamma(N)$-level structure over $\Spec \VpN$. The corresponding morphism
\[
\Frob: \VpN\rightarrow \VpN
\]
classifying this quotient will be called \emph{Frobenius morphism} of $\Mtriv=\Spec \VpN$. In particular, the quotient map $\Etriv\rightarrow \Etriv'=\Etriv/C$ induces a $\Frob$-linear map
\[
F: \Frob^*\HdR{1}{\Etriv/\Mtriv}=\HdR{1}{\Etriv'/\Mtriv}\rightarrow \HdR{1}{\Etriv/\Mtriv}
\]
which is easily seen to respect the Hodge filtration
\[
\begin{tikzcd}
0 \ar[r] & \om_{\Etriv/\Mtriv}\ar[r] & \HdR{1}{\Etriv/\Mtriv}\ar[r] & \om_{\Etriv^\vee/\Mtriv}^\vee\ar[r] & 0.
\end{tikzcd}
\]
Further, the induced $\Frob$-linear endomorphism of $\om_{\Etriv^\vee/\Mtriv}^\vee$ is bijective while the induced $\Frob$-linear map on $\om_{\Etriv/\Mtriv}$ is divisible by $p$. This induces a decomposition
\begin{equation}\label{IP_eq1}
\HdR{1}{\Etriv/\Mtriv}=\om_{\Etriv/\Mtriv}\oplus \UR
\end{equation}
where $\UR\subseteq \HdR{1}{\Etriv/\Mtriv}$ is the unique $F$-invariant $\Ocal_\Mtriv$-submodule on which $F$ is invertible. $\UR$ is called the unit root space and \eqref{IP_eq1} is called unit root decomposition. Let us write
\[
u\colon \HdR{1}{\Etriv/\Mtriv}\rightarrow \om_{\Etriv/\Mtriv}
\]
for the projection induced by the unit root decomposition. Let us further observe, that the isomorphism
\[
\beta: \Eftriv\righteq \Gmf{S}
\]
gives a canonical generator $\omega:=\beta^{-1}(\frac{d T}{1+T})$ of $\om_{\Etriv/\Mtriv}$. This generator gives an isomorphism
\[
\om_{\Etriv/\Mtriv}\righteq \Ocal_{\Mtriv}, \quad \omega\mapsto 1.
\]
Let us consider the following variant of the Eisenstein--Kronecker series $E^{k,r+1}_{s,t}$: For a positive integer $D$ let us define
\[
\EisD:=\sum_{e\neq t\in \Etriv[D]} E^{k,r+1}_{s,t}
\]
One could equally well work with the Eisenstein--Kronecker series $E^{k,r+1}_{s,t}$. The main reason to concentrate on $\EisD$ is, that we want to compare the Eisenstein--Kronecker series with the real-analytic Eisenstein series studied by Katz. For this comparison it is convenient to work with the variant $\EisD$.
\begin{defin}
Let $\Etriv/\Mtriv$ be the universal trivialized elliptic curve with $\Gamma(N)$-level structure. Let $D>0$, $(0,0)\neq (a,b)\in\ZZ/N\ZZ$ and $s\in \Etriv[N]$ the associated $N$-torsion section. The $p$-adic Eisenstein-Kronecker series
\[
\EispD\in\VpN
\]
are defined as the image of $\EisD$ under the unit root decomposition
\[
\Sym^{k+r+1}\HdR{1}{\Etriv/\Mtriv} \twoheadrightarrow \om_{\Etriv/\Mtriv}^{k+r+1} \righteq \Ocal_{\Mtriv}.
\]
\end{defin}
Katz defines generalized $p$-adic modular forms $2\Phi_{k,r,f}\in\VpN$ for $k,r\geq 1$ and $f:(\ZZ/N\ZZ)^2\rightarrow \Zp$. For the precise definition we refer to \cite[\S 5.11]{katz_padicinterpol}. Essentially, he applies the differential operator
\[
\Theta:\Sym^k \HdR{1}{\Etriv/\Mtriv}\rightarrow \Sym^k \HdR{1}{\Etriv/\Mtriv}\otimes_{\Ocal_\Mtriv} \Omega^1_{\Mtriv/\Zp} \hookrightarrow \Sym^{k+2} \HdR{1}{\Etriv/\Mtriv}
\]
obtained by Gau\ss--Manin connection and Kodaira--Spencer isomorphism to classical Eisenstein series and finally uses the unit root decomposition in order to obtain $p$-adic modular forms. We have the following comparison result.
\begin{prop}\label{PI_propKatz} We have the following equality of $p$-adic modular forms:
\[
\EispD=2N^{-k}\left[ D^{k-r+1} \Phi_{r,k,\delta_{(a,b)}}- \Phi_{r,k,\delta_{(Da,Db)}} \right]
\]
where $\delta_{(a,b)}$ is the function on $(\ZZ/N\ZZ)^2$ with $\delta_{(a,b)}(a,b)=1$ and zero else.
\end{prop}
\begin{proof}
Since both sides of the equation are given by applying the unit root decomposition to classes in $\Sym^{k+r+1}\HdR{1}{\Etriv/\Mtriv}$ it suffices to compare these classes. This can be done on the universal elliptic curve of level $\Gamma(N)$. It is further enough to compare the associated $\Ccal^\infty$-modular forms obtained by applying the Hodge decomposition on the universal elliptic curve. The $\Ccal^\infty$-modular form associated with $2\phi_{k,r,f}$ is according to Katz \cite[3.6.5, 3.0.5]{katz_padicinterpol} given by
\[
(2\phi_{k,r,f})^{an}=(2G_{k+r+1,-r,f})^{an}=(-1)^{k+r+1}k!\left( \left(\frac{N}{A(\tau)}\right)^r \zeta_{k+r+1}\left( \frac{k-r+1}{2},1,\tau,f \right) \right)
\]
where $\zeta_{k+r+1}$ is the Epstein zeta function obtained by analytic continuation of
\begin{equation}\label{PI_eq4}
\zeta_k(s,1,\tau,f)=N^{2s}\sum_{(0,0)\neq (n,m)}\frac{f(n,m)}{(m\tau+n)^k|m\tau+n|^{2s-k}},\quad \mathrm{Re}(s)>1.
\end{equation}
On the other hand the $\Ccal^\infty$-modular form corresponding to $\EisD$ is according to \cref{EP_MainThm} given by
\begin{align*}
&\sum_{(0,0)\neq (c,d)\in (\ZZ/D\ZZ)^2} \tilde{e}_{k,r+1}(\frac{Da}{N}\tau+\frac{Db}{N},\frac{Nc}{D}\tau+\frac{Nd}{D})=\\
=&\frac{(-1)^{k+r}r!}{A^k}\sum_{(0,0)\neq (c,d)\in (\ZZ/D\ZZ)^2}\sum_{(m,n)\in\ZZ^2} \frac{(\frac{Da}{N}\bar{\tau}+\frac{Db}{N}+m\bar{\tau}+n)^k}{(\frac{Da}{N}\tau+\frac{Db}{N}+m\tau+n)^{r+1}}\langle\gamma,\frac{Nc}{D}\tau+\frac{Nd}{D}\rangle=\\
=&\frac{(-1)^{k+r}r!}{A^k}\left(\sum_{(m,n)\in\ZZ^2} \frac{(\frac{Da}{N}\bar{\tau}+\frac{Db}{N}+m\bar{\tau}+n)^k}{(\frac{Da}{N}\tau+\frac{Db}{N}+m\tau+n)^{r+1}}\underbrace{\sum_{(c,d)\in (\ZZ/D\ZZ)^2}\exp\left(\frac{2\pi i}{D}N(dm-cn)\right)}_{=0\text{ if }(m,n)\notin (D\ZZ)^2} \right)-\\
&-\tilde{e}_{k,r+1}(\frac{Da}{N}\tau+\frac{Db}{N},0)=\\
=&\frac{(-1)^{k+r}r!}{A^k}\left(\sum_{\gamma\in\Gamma} \frac{(\frac{Da}{N}\bar{\tau}+\frac{Db}{N}+Dm\bar{\tau}+Dn)^k}{(\frac{Da}{N}\tau+\frac{Db}{N}+Dm\tau+Dn)^{r+1}}\cdot D^2\right)-\tilde{e}_{k,r+1}(\frac{Da}{N}\tau+\frac{Db}{N},0)=\\
=&D^{k-r+1}\tilde{e}_{k,r+1}(\frac{a}{N}\tau+\frac{b}{N},0)-\tilde{e}_{k,r+1}(\frac{Da}{N}\tau+\frac{Db}{N},0)
\end{align*}
The analytic Eisenstein--Kronecker series $\tilde{e}_{k,r+1}(\frac{a}{N}\tau+\frac{b}{N},0)$ appearing in the description of the $\Ccal^\infty$-modular form $\EisD$ are defined by
\begin{align*}
\tilde{e}_{k,r+1}(\frac{a}{N}\tau+\frac{b}{N},0)&:=(-1)^{k+r+1}r!\frac{K^*_{k+r+1}(\frac{a}{N}\tau+\frac{b}{N},0,r+1,\tau)}{A(\tau)^k}\stackrel{\text{\cite[Prop. 1.3]{bannai_kobayashi}}}{=}\\
&=(-1)^{k+r+1}k!\frac{K^*_{k+r+1}(0,\frac{a}{N}\tau+\frac{b}{N},k+1,\tau)}{A(\tau)^r}
\end{align*}
with the Eisenstein--Kronecker--Lerch $K^*_{k}(0,\frac{a}{N}\tau+\frac{b}{N},s,\tau)$ series which is given by analytic continuation of
\begin{equation}\label{PI_eq5}
K^*_{k}\left(0,\frac{a}{N}\tau+\frac{b}{N},s,\tau\right):=\sum_{(0,0)\neq(m,n)} \frac{(m\bar{\tau}+n)^k}{|m\tau+n|^{2s}}\exp\left(2\pi i\frac{ma-nb}{N} \right).
\end{equation}
Comparing \eqref{PI_eq4} and \eqref{PI_eq5} shows
\begin{equation}\label{PI_eq6}
K^*_{k}\left(0,\frac{a}{N}\tau+\frac{b}{N},s+\frac{k}{2},\tau\right)=N^{1-2s}\zeta_k(s,1,\tau,\hat{\delta}_{(a,b)}).
\end{equation}
Using this, we compute
\begin{align*}
N^{-k}(2\phi_{k,r,\hat{\delta}_{a,b}})^{an}&=N^{-k}(-1)^{k+r+1}k!\left( \frac{N}{A(\tau)}\right)^r\zeta_{k+r+1}(\frac{k-r+1}{2},1,\tau,\hat{\delta}_{a,b})=\\
&=N^{-k}(-1)^{k+r+1}k!\left(\frac{N}{A(\tau)}\right)^r N^{k-r}K^*_{k+r+1}(0,s,k+1;\tau)=\\
&=\tilde{e}_{k,r+1}(\frac{a}{N}\tau+\frac{b}{N},0)
\end{align*}
Finally, let us recall from \cite{katz_padicinterpol} the identity $\phi_{k,r,f}=\phi_{r,k,\hat{f}}$. Now, the analytic identity
\begin{align*}
N^{-k}&\left[ D^{k-r+1} \left(2 \phi_{r,k,\delta_{a,b}}\right)^{an}-\left(2\phi_{r,k,\delta_{Da,Db}}\right)^{an} \right]=\\
=&D^{k-r+1}\tilde{e}_{k,r+1}(\frac{a}{N}\tau+\frac{b}{N},0)-\tilde{e}_{k,r+1}(D\frac{a}{N}\tau+D\frac{b}{N},0)
\end{align*}
proves the desired algebraic identity on the universal elliptic curve and thereby the proposition.
\end{proof}
\section[$p$-adic Eisenstein--Kronecker series and $p$-adic theta functions]{p-adic Eisenstein--Kronecker series and p-adic theta functions}
Let $N,D$ be positive integers co-prime to $p$. Let us again write $\Etriv/\Mtriv$ for the universal trivialized elliptic curve of level $\Gamma(N)$. Let $s$ be the $N$-torsion section given by $(0,0)\neq(a,b)\in(\ZZ/N\ZZ)^2$. Let us write $\pthetaD_{(a,b)}\in \Gamma(\Eftriv\times_\Mtriv \Edftriv,\Ocal_{\Eftriv\times \Edftriv})$ for the $p$-adic theta function associated to the section
\begin{equation}\label{eq_scanD}
\sum_{e\neq t \in \Eftriv[D]}U_{s,t}^{N,D}(\scan).
\end{equation}
More precisely: The trivialization $\beta:\Eftriv \righteq \Gmf{\Mtriv}$ gives us a canonical invariant differential $\omega:=\beta^* \frac{d S}{1+S}\in\Gamma(\Mtriv,\om_{\Etriv/\Mtriv})$. Since $\Gamma(\Etriv,\Omega^1_{\Etriv/\Mtriv})=\Gamma(\Mtriv,\om_{\Etriv/\Mtriv})$ we obtain an isomorphism
\[
\Omega^1_{\Etriv/\Mtriv}\righteq \Ocal_{\Mtriv}.
\]
Using this isomorphism allows us to view
\[
\sum_{e\neq t \in \Eftriv[D]}U_{s,t}^{N,D}(\scan)\Big|_{\Eftriv\times\Edftriv}
\]
as a section of
\[
\Gamma(\Eftriv\times\Edftriv, ([D]\times[N])^*\Pof)\cong \Gamma(\Eftriv\times\Edftriv,\Pof).
\]
The last isomorphism is induced by $N$ resp. $D$ multiplication on the formal groups. Finally, we define $\pthetaD_{(a,b)}\in \Gamma(\Eftriv\times_\Mtriv \Edftriv,\Ocal_{\Eftriv\times \Edftriv})$ as the image of \eqref{eq_scanD} under the trivialization map
\[
\Gamma(\Eftriv\times_\Mtriv \Edftriv,\Pof)\righteq \Gamma(\Eftriv\times_\Mtriv \Edftriv,\Ocal_{\Eftriv\times \Edftriv}).
\]
Let us write
\[
\partial_{\Eftriv}\colon \Ocal_{\Eftriv} \rightarrow \Ocal_{\Eftriv},\quad \partial_{\Edftriv}\colon \Ocal_{\Edftriv} \rightarrow \Ocal_{\Edftriv}
\]
for the invariant derivations associated to the invariant differential $\omega$. The following result relates invariant derivatives of our $p$-adic theta function $\pthetaD_{(a,b)}$ to the $p$-adic Eisenstein--Kronecker series, thus it can be seen as a $p$-adic version of the Laurent expansion
\begin{equation*}
\Theta_{s,t}(z,w)=\sum_{k,r\geq 0}\frac{\tilde{e}_{k,r+1}(s,t)}{k!r!}z^rw^k,\quad s,t\notin\Gamma
\end{equation*}
of the Kronecker theta function due to Bannai and Kobayashi.
\begin{thm}\label{thm_padictheta}
Let $\Etriv/\Mtriv$ be the universal trivialized elliptic curve of level $\Gamma(N)$. We have the following equality of generalized $p$-adic modular forms
\[
\EispD=(e\times e)^*\left(\partial_{\Edftriv}^{\circ k} \partial_{\Eftriv}^{\circ r} \pthetaD_{(a,b)}\right).
\]
\end{thm}
We will present the proof in \cref{sec_proofpadic}. As an immediate consequence of this result we get a new construction of Katz' two-variable $p$-adic Eisenstein measure: Let $R$ be a $p$-adic ring. A $R$-valued $p$-adic measure on a pro-finite Abelian group $G$ is an $R$-linear map $C(G,R)\rightarrow R$, where $C(G,R)$ denotes the $R$-module of $R$-valued continuous functions on $G$. Let us write $\Meas(\Zp^2,R)$ for the set of all $R$-valued measures on $\Zp^2$. According to a theorem of Y. Amice there is an isomorphism of $R$-algebras:
\[
R\llbracket S,T \rrbracket \righteq \Meas(\Zp^2,R), \quad f\mapsto \mu_f
\]
which is uniquely characterized by
\[
\int_{\Zp^2} x^k y^l\dd \mu_f(x,y)=\left.\partial_S^{\circ k}\partial_T^{\circ l} f \right|_{S=T=0}
\]
where $\partial_T=(1+T)\frac{\partial}{\partial T}$ and $\partial_S=(1+S)\frac{\partial}{\partial S}$ are the invariant derivations on the two copies of $\Gmf{R}$. Let us call $\mu_f$ the \emph{Amice transform} associated to $f$.\par
The trivialization $\beta: \Eftriv\righteq \Gmf{\Mtriv}$ together with the autoduality $\Eftriv\righteq \Edftriv$ allows us to view $\pthetaD_{(a,b)}$ as a two-variable power series with coefficients in the ring $R:=\VpN$ of generalized $p$-adic modular forms:
\[
\pthetaD_{(a,b)}(S,T)\in R\llbracket S,T\rrbracket.
\]
Here, $S$ is the variable of $\Gmf{\Mtriv}$ corresponding to $\Etriv$ and the variable $T$ corresponds to the dual elliptic curve. The Amice transform of the $p$-adic theta function $\pthetaD_{(a,b)}$ gives us a $p$-adic measure on $\Zp\times\Zp$ which will be called $\muEisD$. As an immediate corollary of the above result we get the $p$-adic Eisenstein--Kronecker series as moments of the $p$-adic measure $\muEisD$:
\begin{cor}\label{cor_Eis_moments} The $p$-adic Eisenstein-Kronecker series $\EispD$ appear as moments
\[
\EispD=\int_{\Zp\times\Zp} x^k y^r \dd \muEisD(x,y)
\]
of the measure $\muEisD$ associated to the $p$-adic theta function $\pthetaD_{(a,b)}(S,T)$.
\end{cor}
This corollary gives a more concise construction of the $p$-adic Eisenstein measure then the original one in \cite{katz_padicinterpol}: In \cite{katz_padicinterpol} the existence of the $p$-adic Eisenstein measure has been proven by checking all predicted $p$-adic congruences between the corresponding $p$-adic modular on the $q$-expansion. In our construction, the $p$-adic Eisenstein measure appears more naturally as the Amice transform of a $p$-adic theta function. We obtain the $p$-adic congruences for free as a by product without checking them in the first instance.
\section{The geometric logarithm sheaves}
For the proof of \cref{thm_padictheta} it is necessary to study the structure of the restrictions of the Poincar\'e bundle to $\Ef\times_S \Edf$ more carefully. In his PhD thesis \cite{rene} Scheider has proven that the de Rham logarithm sheaves appear naturally by restricting the Poincar\'e bundle $\Po^\dagger$ to infinitesimal thickenings of the elliptic curve. At this place it is not necessary to develop the theory of the de Rham logarithm sheaves, but keeping this relation in mind motivates many of the properties of $\hat{\Po}:=\Po|_{\Ef\times \Edf}$.
\subsection{Basic properties} Let $E/S$ be an elliptic curve over a $p$-adic ring $S=\Spec R$ with fiber-wise ordinary reduction. As before, let us write $E^\dagger$ for the universal vectorial extension of the dual of the elliptic curve $E$. The pullback of the Poincar\'e bundle $\Po$ to $E\times_{S} E^\dagger$ is denoted by $\Po^\dagger$ and carries the universal integrable connection $\nabla_{\dagger}$. Motivated by Scheider's results, let us define
\[
\Lnf:=(\pr_{\Ef})_* \left(\Po|_{\Ef\times_S \Inf^n_e \Ed}\right)
\]
and
\[
\Lnf^\dagger:=(\pr_{\Ef})_* \left(\Po^\dagger|_{\Ef\times_S \Inf^n_e E^\dagger}\right).
\]
The connection on $\Po^\dagger$ induces an $\Ocal_S$-linear connection $\nabla_\dagger^{(n)}$ on the $\Ocal_{\Ef}$-module $\Lnf^\dagger$. Let us write $\Hcal_{\Ef}:=\HdR{1}{\Ed/S}\otimes\Ocal_{\Ef}$ and $\om_{\Ef}:=\om_{\Ed/S}\otimes\Ocal_{\Ef}$. Since $\Lnf$ is obtained by restriction of $\Po$ to $\Ef\times_S\Inf^n_e\Ed$, we obtain transition maps
\[
\Lnf\twoheadrightarrow \hat{\Lcal}_{n-1}
\]
by further restriction along $\Ef\times_S\Inf^{n-1}_e \Ed\hookrightarrow \Ef\times_S\Inf^n_E\Ed$. Similarly, we obtain
\[
\Lnf^\dagger\twoheadrightarrow \hat{\Lcal}_{n-1}^\dagger.
\]
The fact that $\Po^\dagger$ is the pullback of $\Po$ along $E\times_S \Ed\twoheadrightarrow E\times_S E^\dagger$ gives inclusions
\[
\Lnf\hookrightarrow \Lnf^\dagger.
\]
The decompositions $\Ocal_{\Inf^1_e \Ed}=\Ocal_S\oplus \om_{\Ed/S}$ and $\Ocal_{\Inf^1_e E^\dagger}=\Ocal_S\oplus \Hcal$ induce short exact sequences
\[
\begin{tikzcd}
0\ar[r] & \om_{\Ef}\ar[r] & \widehat{\Lcal}_1\ar[r] & \Ocal_{\Ef}\ar[r] & 0
\end{tikzcd}
\]
and
\begin{equation}\label{eq_ses1}
\begin{tikzcd}
0\ar[r] & \Hcal_{\Ef}\ar[r] & \widehat{\Lcal}^\dagger_1\ar[r] & \Ocal_{\Ef}\ar[r] & 0.
\end{tikzcd}
\end{equation}
The maps in the last exact sequence are horizontal if we equip $\Ocal_{\Ef}$ and $\Hcal_{\Ef}$ with the trivial pullback connections relative $S$, i.e.
\[
\Hcal_{\Ef}=\Ocal_{\Ef}\otimes_{\Ocal_S}\Hcal\rightarrow \Omega^1_{\Ef/S}\otimes_{\Ocal_S} \Hcal,\quad (f\otimes h)\mapsto df\otimes h
\]
and similarly for $\Ocal_{\Ef}$. In \eqref{eq_Triv3} we have defined a trivialization isomorphism
\[
\triv\colon \Pof\righteq \Ocal_{\Ef}\hat{\otimes}\Ocal_{\Edftriv}.
\]
This isomorphism induces a $\Ocal_{\Ef}$-linear trivialization map
\[
\triv^{(n)}\colon \Lnf\righteq \Ocal_{\Ef}\otimes_{\Ocal_S} \Ocal_{\Inf^n_e \Ed}.
\]
Since $\Po^\dagger$ is the pullback of $\Po$ we also obtain
\[
\triv^{(n)}\colon \Lnf^\dagger\righteq \Ocal_{\Ef}\otimes_{\Ocal_S} \Ocal_{\Inf^n_e E^\dagger}.
\]
Let us observe that in the special case $n=1$ the trivialization map splits the above short exact sequences, i.e. we get
\[
\triv^{(1)}\colon \widehat{\Lcal}_1\righteq \Ocal_{\Ef}\oplus \om_{\Ef}
\]
and
\[
\triv^{(1)}\colon \widehat{\Lcal}^\dagger_1\righteq \Ocal_{\Ef}\oplus \Hcal_{\Ef}.
\]
The last map is not horizontal if we equip the right hand side with the trivial $S$-connections.
\subsection{Comultiplication maps} In this subsection let us introduce certain canonical comultiplication maps on the infinitesimal geometric logarithm sheaves. Using the $\Gm{S}$-biextension structure of the Poincar\'e bundle let us construct certain natural comultiplication maps on $\Lnf$ and $\Lnf^\dagger$. Such a construction already appeared in the PhD thesis of Ren\'e Scheider \cite[\S 2.4.2]{rene}. As before, let $E/S$ be an elliptic curve over a $p$-adic ring $S=\Spec R$ with fiber-wise ordinary reduction and let us write $E^\dagger$ for the universal vectorial extension of $\Ed$. Let
\[
\iota^{\dagger}_n:E_n^\dagger:=\Inf^n_e E^{\dagger}\hookrightarrow E^{\dagger}
\]
denote the inclusion of the $n$-th infinitesimal neighbourhood $E_n^\dagger$ of $e$ in $E^\dagger$. For the time being we will use the convention to denote by $\times$ and $\otimes$ the product and tensor product over $S$. Recall that the Poincar\'{e} bundle $\Po^{\dagger}$ is equipped with a natural $\Gm{S}$-biextension structure, i.\,e. isomorphisms
\begin{alignat}{3}\label{GL_eq1}
(\mu_E\times \id_{E^{\dagger}})^* \Po^{\dagger} &\righteq \pr_{1,3}^*\Po^\dagger \otimes \pr_{2,3}^*\Po^{\dagger} &&\quad\text{ on }\quad E\times E\times E^{\dagger} \\
(\id_E\times \mu_{E^{\dagger}})^* \Po^{\dagger} &\righteq \pr_{1,2}^*\Po^\dagger \otimes \pr_{1,3}^*\Po^{\dagger} &&\quad\text{ on }\quad E\times E^{\dagger}\times E^{\dagger}\nonumber
\end{alignat}
satisfying certain compatibilities, cf. \cite[exp. VII]{SGA7_I}. Here, $\mu$ denotes the multiplication and $\pr_{i,j}$ the projection on the $i$-th and $j$-th component of the product. Now, fix some integers $n,m\geq 1$ and define $\Po^{\dagger}_n:=(\id\times \iota^{\dagger}_n)^*\Po^{\dagger}$. Restricting
\[
\mu_{E^{\dagger}}: E^{\dagger}\times E^{\dagger}\rightarrow E^{\dagger}
\]
to $E_n^{\dagger}\times E^{\dagger}_m$ gives
\[
\mu_{n,m}: E_n^{\dagger}\times E^{\dagger}_m\rightarrow E^{\dagger}_{n+m}.
\]
Restricting \eqref{GL_eq1} along
\[
E\times E_n^{\dagger}\times E_m^{\dagger} \hookrightarrow E\times E^{\dagger}\times E^{\dagger}
\]
results in
\[
\Po^{\dagger}_{n+m}\rightarrow (\pr_{12})^*\Po^{\dagger}_n \otimes_{\Ocal_{E\times E_n^{\dagger}\times E_m^{\dagger} }} (\pr_{13})^*\Po^{\dagger}_m.
\]
Using the unit of the adjunction between $(\id\times \mu_{n,m})_*$ and $(\id\times \mu_{n,m})^*$, we obtain
\[
\Po^{\dagger}_{n+m}\rightarrow (\id\times \mu_{n,m})_*\left[ (\pr_{12})^*\Po^{\dagger}_n\otimes (\pr_{13})^*\Po^{\dagger}_m \right].
\]
Taking the pushforward along $\pr_E$ gives:
\[
\xi_{n,m}: \widehat{\Lcal}_{n+m}^\dagger\rightarrow \widehat{\Lcal}_{n}^\dagger\otimes_{\Ocal_E}\widehat{\Lcal}_{m}^\dagger
\]
Since the $\Gm{S}$-biextension structure is compatible with the connection, we get that $\xi_{n,m}$ is horizontal. Using the compatibilities of the $\Gm{S}$-biextension structure, one deduces the following commutative diagrams:
\begin{equation}\label{GL_eq2}
\begin{tikzcd}
\widehat{\Lcal}_{n+m}^\dagger \ar[r,"\xi_{n,m}"]\ar[rd,"\xi_{m,n}",swap] & \widehat{\Lcal}_{n}^\dagger \otimes_{\Ocal_E} \widehat{\Lcal}_{m}^\dagger \ar[d,"can"] \\
& \widehat{\Lcal}_{m}^\dagger \otimes_{\Ocal_E} \widehat{\Lcal}_{n}^\dagger
\end{tikzcd}
\end{equation}
and
\begin{equation}\label{GL_eq3}
\begin{tikzcd}
\widehat{\Lcal}_{n+m+l}^\dagger \ar[r,"\xi_{n+m,l}"]\ar[d,"\xi_{n,m+l}",swap] & \widehat{\Lcal}_{n+m}^\dagger \otimes_{\Ocal_E} \widehat{\Lcal}_{l}^\dagger \ar[d,"\xi_{n,m}\otimes \id"] \\
\widehat{\Lcal}_{n}^\dagger \otimes_{\Ocal_E} \widehat{\Lcal}_{l+m}^\dagger \ar[r,"\id\otimes \xi_{m,l}"] & \widehat{\Lcal}_{n}^\dagger \otimes_{\Ocal_E} \widehat{\Lcal}_{m}^\dagger \otimes_{\Ocal_E} \widehat{\Lcal}_{l}^\dagger.
\end{tikzcd}
\end{equation}
Thus, we obtain well-defined maps
\[
\widehat{\Lcal}_{n}^\dagger \rightarrow \underbrace{\widehat{\Lcal}_{1}^\dagger\otimes_{\Ocal_E} ... \otimes_{\Ocal_E} \widehat{\Lcal}_{1}^\dagger }_{n \text{ times}}.
\]
The diagram \eqref{GL_eq2} shows that this map is invariant under transposing any of the $n$ factors on the right hand side. Thus, letting the symmetric group $S_n$ act by permuting the factors we see that $\Lnf^\dagger \rightarrow \left( \widehat{\Lcal}_1^\dagger \right)^{\otimes n}$ factors through the invariants of the $S_n$ action. We denote the resulting map by
\begin{equation}\label{GLdagger_eq4}
\widehat{\Lcal}_{n}^\dagger \hookrightarrow \TSym^n_{\Ocal_{\Ef}} \widehat{\Lcal}_{1}^\dagger:=\left[ \left( \widehat{\Lcal}_{1}^\dagger\right)^{\otimes n} \right]^{S_n}.
\end{equation}
This map is horizontal, if we equip the right hand side with the tensor product connection induced by $\nabla^{(1)}_\dagger$. Similarly, we get $\Ocal_{\Ef}$-linear maps
\begin{equation}\label{GL_eq4}
\widehat{\Lcal}_{n} \hookrightarrow \TSym^n_{\Ocal_{\Ef}} \widehat{\Lcal}_{1}.
\end{equation}
A similar construction applies to the multiplication of the formal groups $\Ef^\vee$:
\[
\Ocal_{\Edftriv}\rightarrow \Ocal_{\Edftriv}\hat{\otimes} \Ocal_{\Edftriv}.
\]
This multiplication induces comultiplication maps
\begin{equation}\label{eq_formal_comult}
\Ocal_{\Inf^n_e \Ed}\rightarrow \TSym^n_{\Ocal_S} \Ocal_{\Inf^1_e\Ed}
\end{equation}
which are compatible with the comultiplication maps on $\widehat{\Lcal}_{n}$:
\begin{equation}\label{eq_comult}
\begin{tikzcd}
\widehat{\Lcal}_{n}\ar[d,"\cong"]\ar[r] & \TSym^n_{\Ocal_{\Ef}} \widehat{\Lcal}_{1} \ar[d,"\cong"] \\
\Ocal_{\Ef}\otimes_{\Ocal_S}\Ocal_{\Inf^n_e E^\vee}\ar[r] & \TSym^n_{\Ocal_{\Ef}}\left( \Ocal_{\Ef}\otimes_{\Ocal_S}\Ocal_{\Inf^1_e E^\vee}\right).
\end{tikzcd}
\end{equation}
In this diagram the lower horizontal map is induced by tensoring \eqref{eq_formal_comult} with $\Ocal_{\Ef}$. The comultiplication maps for $\Ef^\vee$ can be identified with taking iterated invariant derivatives. More precisely, the map
\[
\Ocal_{\Inf^n_e \Ed}\rightarrow \TSym^n_{\Ocal_S} \Ocal_{\Inf^1_e\Ed}= \TSym^n_{\Ocal_S} (\Ocal_S\oplus \om_{\Ed/S})\cong \bigoplus_{k=0}^n \TSym^k_{\Ocal_S}\om_{\Ed/S}
\]
coincides with the map $f\mapsto \left(e^*(\partial^{\circ k} f\right)_{k=0}^n$ where $\partial$ is the map induced by the invariant derivative $\partial\colon \Ocal_{\Edf}\rightarrow \om_{\Ed/S}\otimes\Ocal_{\Edf}$.
\begin{lem}\label{lem_TrivializeLn} Let us assume that $S=\Spec R$ is flat over $\Zp$. Under this assumption, the comultiplication maps
\[
\widehat{\Lcal}_{n} \hookrightarrow \TSym^n \widehat{\Lcal}_{1},\quad \widehat{\Lcal}^\dagger_{n} \hookrightarrow \TSym^n \widehat{\Lcal}^\dagger_{1}
\]
are injective and isomorphisms on the generic fiber $E_{\Qp}:=E\times_S\Spec \Qp$
\[
\widehat{\Lcal}_{n,E_{\Qp}} \righteq \TSym^n \widehat{\Lcal}_{1,E_{\Qp}},\quad \widehat{\Lcal}^\dagger_{n,E_{\Qp}} \righteq \TSym^n \widehat{\Lcal}^\dagger_{1,E_{\Qp}}.
\]
\end{lem}
\begin{proof}
We give the proof for $\widehat{\Lcal}_{n}$. The proof for $\widehat{\Lcal}^\dagger_{n}$ is completely analogous. By \eqref{eq_comult} it is enough to prove that the map
\[
\Ocal_{\Inf^n_e E^\vee}\rightarrow \TSym^n_{\Ocal_{S}}\left( \Ocal_{\Inf^1_e E^\vee}\right)
\]
is injective and an isomorphism after inverting $p$. The isomorphism $\Ocal_{\Inf^1_e E^\vee}\cong \Ocal_{S}\oplus \om_{\Ed/S}$ gives
\[
\Ocal_{\Inf^n_e E^\vee}\rightarrow \TSym^n_{\Ocal_{S}}\left( \Ocal_{\Inf^1_e E^\vee}\right)=\bigoplus_{k=0}^n \TSym^k_{\Ocal_S}\om_{\Ed/S}.
\]
But this map is just the map sending $f\in\Ocal_{\Inf^n_e E^\vee}$ to $(e^*(\partial^{\circ k} f) )_{k=0}^n$ which is injective if $R$ is flat over $\Zp$ and an isomorphism if $p$ is invertible.
\end{proof}
By combining the comultiplication with the trivialization $\triv^{(1)}\colon \widehat{\Lcal}_{1}\righteq \Ocal_{\Eftriv}\oplus \om_{\Eftriv}$ we obtain
\begin{equation}\label{eq_Lntriv}
\widehat{\Lcal}_{n}\rightarrow \TSym^n_{\Ocal_{\Ef}} \widehat{\Lcal}_{1}=\TSym^n_{\Ocal_{\Ef}} (\Ocal_{\Ef}\oplus \om_{\Ef})\righteq \bigoplus_{k=0}^n \TSym^k_{\Ocal_{\Ef}} \om_{\Ef}.
\end{equation}
and
\begin{equation}\label{eq_Lndtriv}
\widehat{\Lcal}^\dagger_{n}\rightarrow \TSym^n_{\Ocal_{\Ef}} \widehat{\Lcal}^\dagger_{1}=\TSym^n_{\Ocal_{\Ef}}( \Ocal_{\Ef}\oplus \Hcal_{\Ef})\righteq \bigoplus_{k=0}^n \TSym^k_{\Ocal_{\Ef}} \Hcal_{\Ef}.
\end{equation}
In particular, whenever we are in a situation where $\Hcal$ and $\om_{\Ed/S}$ can be generated by global sections, such generators and \cref{lem_TrivializeLn} give us an explicit $\Ocal_{\Ef}$-basis of $\Lnf$ and $\Lnf^\dagger$ on the generic fiber.
\subsection{The Frobenius structure}
Let $\Etriv/\Mtriv$ be the universal trivialized elliptic curve with $\Gamma(N)$-level structure. Our next aim is to define a \emph{Frobenius structure} on $\Lnf$, i.e. an isomorphism
\[
\Lnf\righteq \phi_{\Eftriv}^* \Lnf
\]
where $\phi_{\Eftriv}$ is a Frobenius lift. Let $\varphi:\Etriv\rightarrow \Etriv'=\Etriv/C$ be the quotient of the universal trivialized elliptic curve by its canonical subgroup. Since $\Etriv'$ is again a trivialized elliptic curve with $\Gamma(N)$-level structure, it is the pullback of $E/M$ along a unique map $\Frob\colon M\rightarrow M$. This gives us a diagram
\begin{equation}\label{PI_eq7}
\begin{tikzcd}
\Etriv \ar[r,"\varphi"]\ar[rd,"\pi"] & \Etriv'\ar[r]\ar[r,"\widetilde{\mathrm{Frob}}"]\ar[d,"\pi_{\Etriv'}"] & \Etriv\ar[d,"\pi"]\\
& \Mtriv \ar[r,"\Frob"] & \Mtriv
\end{tikzcd}
\end{equation}
with the square being Cartesian. Let us define
\[
\phi_{\Eftriv}:=\left(\widetilde{\mathrm{Frob}}\circ \varphi\right)\Big|_{\Eftriv}:\Eftriv\rightarrow \Eftriv
\]
as the restriction of the upper horizontal composition in the above diagram to the formal group $\Eftriv$. The map $\phi_{\Eftriv}$ gives us a Frobenius lift on the formal group $\Eftriv$. We want to construct an $\Ocal_{\Eftriv}$-linear isomorphism
\[
(\pr_{\Eftriv})_*\Pof \righteq (\pr_{\Eftriv})_*(\phi_{\Eftriv}\times \id_{\Edftriv})^*\Pof.
\]
We will do this in two steps: Let us write $\Pof'$ for the Poincar\'e bundle of $\Etriv'$ restricted to the formal scheme $\widehat{\Etriv'}\times_\Mtriv \widehat{\Etriv'}^\vee$. Restricting the map $\gamma_{\id,\varphi^\vee}$ to $\widehat{\Etriv'}\times_\Mtriv \widehat{\Etriv'}^\vee$ gives the isomorphism
\[
(\id\times\varphi^\vee|_{\widehat{\Etriv'}^\vee})^*\Pof\righteq (\varphi|_{\Eftriv}\times\id)^*\Pof'
\]
The dual isogeny $\varphi^\vee$ is \'etale, hence it induces an isomorphism of formal groups over $\Mtriv$:
\[
\varphi^\vee|_{(\widehat{\Etriv'})^\vee}\colon (\widehat{\Etriv'})^\vee\righteq \Edftriv.
\]
We get
\begin{equation}\label{eq_Phi1}
(\pr_{\Eftriv})_*\Po\Big|_{\Eftriv\times\Edftriv}\righteq (\pr_{\Eftriv})_* (\varphi|_{\Eftriv}\times\id)^*\left(\Po'\Big|_{\widehat{\Etriv'}\times \widehat{\Etriv'}^\vee}\right).
\end{equation}
On the other hand, by the compatibility of the Poincar\'e bundle with base change along the Cartesian diagram
\begin{equation*}
\begin{tikzcd}
\Etriv'\ar[r]\ar[r,"\widetilde{\mathrm{Frob}}"]\ar[d,"\pi_{\Etriv'}"] & \Etriv\ar[d,"\pi"]\\
\Mtriv'=\Mtriv \ar[r,"\Frob"] & \Mtriv,
\end{tikzcd}
\end{equation*}
and using the identification $\Etriv'\times_{\Mtriv'} \Etriv'^\vee=\Etriv'\times_\Mtriv \Etriv^\vee$ we get an isomorphism
\begin{equation}\label{eq_Phi2}
(\widetilde{\mathrm{Frob}}\times_\Mtriv \id_{\Edtriv})^*\Po\righteq \Po'.
\end{equation}
Composing \eqref{eq_Phi1} with \eqref{eq_Phi2} gives the desired isomorphism of $\Ocal_{\Eftriv}$-modules:
\[
(\pr_{\Eftriv})_*\Pof \righteq (\pr_{\Eftriv})_*(\phi_{\Eftriv}\times \id_{\Edftriv})^*\Pof.
\]
Replacing $(\pr_{\Eftriv})_*\Pof$ by $\Lnf=(\pr_{\Eftriv})_*(\Pof|_{\Eftriv\times \Inf^n_e \Edtriv})$ in the above construction gives an $\Ocal_{\Eftriv}$-linear isomorphism
\[
\Psi\colon \Lnf\righteq \phi_{\Eftriv}^* \Lnf.
\]
Let us write $\Po^{\dagger}$ for the pullback of $\Po$ along $\Etriv\times_\Mtriv\Etriv^\dagger\rightarrow \Etriv\times_\Mtriv \Edtriv$, where $\Etriv^\dagger$ is the universal vectorial extension of $\Edtriv$. Let us write $\Pof^\dagger$ for the restriction along the formal completion $\Eftriv\times_\Mtriv\Eftriv^\dagger$. Along the same lines as above, we obtain $\Ocal_{\Eftriv}$-linear morphisms
\begin{equation}\label{eq_Phidagger}
(\pr_{\Eftriv})_*\Pof^\dagger\rightarrow (\pr_{\Eftriv})_*(\phi_{\Eftriv}\times \id_{\Eftriv^\dagger})^*\Pof^\dagger.
\end{equation}
and
\begin{equation}\label{eq_Phidagger_Ln}
\Psi\colon\Lnf^\dagger\rightarrow \phi_{\Eftriv}^*\Lnf^\dagger.
\end{equation}
This map is horizontal if both sides are equipped with the canonical $\Mtriv$-connections.\par
\subsection{A basis for the geometric logarithm sheaves} Let $\Etriv/\Mtriv$ be the universal trivialized elliptic curve with $\Gamma(N)$-level structure. Now let us construct a canonical basis for the infinitesimal geometric logarithm sheaves on the universal trivialized elliptic curve. In \eqref{eq_Lntriv} and \eqref{eq_Lndtriv} we have defined maps
\[
\widehat{\Lcal}_{n}\rightarrow \bigoplus_{k=0}^n \TSym^k_{\Ocal_{\Eftriv}} \om_{\Eftriv}
\]
and
\[
\widehat{\Lcal}^\dagger_{n}\rightarrow \bigoplus_{k=0}^n \TSym^k_{\Ocal_{\Eftriv}} \Hcal_{\Eftriv}.
\]
The rigidification $\Edftriv\righteq \Eftriv\righteq \Gmf{\Mtriv}$ gives a canonical generator $\omega\in\Gamma(\Mtriv,\om_{\Edtriv/\Mtriv})$. Now $\omega$ generates $\om_{\Eftriv}$ as $\Ocal_{\Eftriv}$-module. The tensor symmetric algebra $\TSym^\bullet \om_{\Eftriv}$ is a graded ring with divided powers given by
\[
(\cdot)^{[k]}\colon \om_{\Eftriv}\rightarrow \TSym^{k} \om_{\Eftriv}, \quad x\mapsto x^{[k]}:=\underbrace{x\otimes...\otimes x}_{k\text{-times}}.
\]
for any positive integer $k$. Using the isomorphism $\widehat{\Lcal}_1\cong \Ocal_{\Eftriv}\oplus \om_{\Eftriv}$, we may view $\omega$ as a section of $\widehat{\Lcal}_1$. The divided power structure give a canonical $\Ocal_{\Eftriv}$-basis $(\hat{\omega}^{[k]})_{k=0}^n$ of $\TSym^n \widehat{\Lcal}_{1}$ defined by
\[
\hat{\omega}^{[k]}:=\omega^{[k]}\in \Gamma(\Eftriv,\TSym^n \widehat{\Lcal}_{1}).
\]
Similarly, let us write $[\omega]$ for the image of $\omega$ under the inclusion of the Hodge filtration:
\[
\om_{\Edtriv/\Mtriv}\hookrightarrow \Hcal.
\]
There is a unique section $[u]\in\Gamma(\Mtriv,\UR)$ in the unit root part of $\Hcal$ with $\langle [u],[\omega]\rangle=1$. This gives a basis $([\omega],[u])$ of $\Hcal$. Let us write
\[
\hat{\omega}^{[k,l]}:=[\omega]^{[k]}\cdot [u]^{[l]}\in \Gamma(\Eftriv,\TSym^n \widehat{\Lcal}^\dagger_{1})
\]
for the basis induced by $([\omega],[u])$ using the divided power structure on the tensor symmetric algebra $\TSym^{\bullet} \widehat{\Lcal}^\dagger_{1}$.
\begin{lem}
We have canonical $\Ocal_{\Eftriv}$-linear decompositions:
\[
\widehat{\Lcal}_{n,\Etriv_{\Qp}}\righteq \bigoplus_{k=0}^n \hat{\omega}^{[k]}\cdot \Ocal_{\Eftriv_{\Qp}},\quad \widehat{\Lcal}^\dagger_{n,\Etriv_{\Qp}}\righteq \bigoplus_{k+l\leq n} \hat{\omega}^{[k,l]}\cdot \Ocal_{\Eftriv_{\Qp}}.
\]
These decompositions are compatible with the transition maps
\[
\widehat{\Lcal}_{n,\Etriv_{\Qp}}\twoheadrightarrow \widehat{\Lcal}_{n-1,\Etriv_{\Qp}},\quad \widehat{\Lcal}^\dagger_{n,\Etriv_{\Qp}}\twoheadrightarrow \widehat{\Lcal}^\dagger_{n-1,\Etriv_{\Qp}}
\]
and the inclusion $\widehat{\Lcal}_{n,\Etriv_{\Qp}}\hookrightarrow \widehat{\Lcal}^\dagger_{n,\Etriv_{\Qp}}$.
\end{lem}
\begin{proof}
The decomposition is an immediate consequence of the bases $(\hat{\omega}^{[k,l]})_{k+l\leq n}$ and $(\hat{\omega}^{[k]})_{k=0}^n$ of $\TSym^{n} \widehat{\Lcal}^\dagger_{1}$ and $\TSym^{n} \widehat{\Lcal}_{1}$ together with \cref{lem_TrivializeLn}. The compatibility with transition maps and the canonical inclusion follow by tracing back the definitions.
\end{proof}
\begin{lem}
The Frobenius structure
\[
\Psi\colon\Lnf^\dagger\rightarrow \phi_{\Eftriv}^*\Lnf^\dagger
\]
is explicitly given by the formula
\[
\Psi(\hat{\omega}^{[k,l]})=p^l\cdot \phi_{\Eftriv}^*\hat{\omega}^{[k,l]}.
\]
\end{lem}
\begin{proof}
By the commutativity of the diagram
\[
\begin{tikzcd}
\Lnf^\dagger\ar[r]\ar[d,"\Psi"] & \TSym^n\hat{\Lcal}_1^\dagger\ar[d,"\TSym^n\Psi"]\\
\phi_{\Eftriv}^*\Lnf^\dagger\ar[r] & \TSym^n\phi_{\Eftriv}^*\hat{\Lcal}_1^\dagger
\end{tikzcd}
\]
we are reduced to prove the formulas in the case $n=1$, i.e.
\begin{align*}
\Psi(\hat{\omega}^{[0,0]})&=\phi_{\Eftriv}^*\hat{\omega}^{[0,0]}\\
\Psi(\hat{\omega}^{[1,0]})&=\phi_{\Eftriv}^*\hat{\omega}^{[1,0]}\\
\Psi(\hat{\omega}^{[0,1]})&=p\cdot\phi_{\Eftriv}^*\hat{\omega}^{[0,1]}.
\end{align*}
Since $\Psi$ is compatible with the decomposition $\hat{\Lcal}_1^\dagger=\Ocal_{\Eftriv}\oplus \Hcal_{\Eftriv}$, the Frobenius structure on
$\Ocal_{\Eftriv}$ is the canonical isomorphism $\Ocal_{\Eftriv}\cong \phi_{\Eftriv}^*\Ocal_{\Eftriv}$ induced by $\phi_{\Eftriv}\colon\Eftriv\rightarrow \Eftriv$. Since $\hat{\omega}^{[0,0]}$ corresponds to $1\in \Ocal_{\Eftriv}$, we get $\Psi(\hat{\omega}^{[0,0]})=\phi_{\Eftriv}^*\hat{\omega}^{[0,0]}$. The restriction of $\Psi$ to $\Hcal_{\Eftriv}$ is by construction the $\Ocal_{\Eftriv}$-linear extension of the map
\[
\Hcal=\HdR{1}{\Edtriv/\Mtriv}\rightarrow \HdR{1}{(\Etriv')^\vee/\Mtriv}\cong \Frob^* \HdR{1}{\Edtriv/\Mtriv}
\]
induced by the diagram
\begin{equation*}
\begin{tikzcd}
\Etriv \ar[r,"\varphi"]\ar[rd,"\pi"] & \Etriv'\ar[r]\ar[r,"\widetilde{\mathrm{Frob}}"]\ar[d,"\pi_{\Etriv'}"] & \Etriv\ar[d,"\pi"]\\
& \Mtriv \ar[r,"\Frob"] & \Mtriv.
\end{tikzcd}
\end{equation*}
Since $\hat{\omega}^{[0,1]}$ corresponds to $[u]$ and $\hat{\omega}^{[1,0]}$ corresponds to $[\omega]$, we have to prove that $[u]$ maps to $p\cdot \Frob^*[u]$ and $[\omega]$ maps to $\Frob^*[\omega]$. But this is known to be true, see for example \cite[p. 23]{bannai_kings}.
\end{proof}
\begin{lem}
The $\Mtriv$-connection on $\widehat{\Lcal}^\dagger_{1}$ induces an $\Mtriv$-connection $\nabla_{(n)}$ on $\TSym^n \widehat{\Lcal}^\dagger_{1}$ which is explicitly given by the formula
\[
\con{(n)}(\hat{\omega}^{[k,l]})=c\cdot (l+1)\hat{\omega}^{[k,l+1]}\otimes \omega
\]
for some $c\in \Zp$.
\end{lem}
\begin{proof}
The connection on $\TSym^n \widehat{\Lcal}^\dagger_{1} $ is the tensor product connection induced from $\widehat{\Lcal}^\dagger_{1}$. Thus, it is enough to prove the formula in the case $n=1$. Here, $\TSym^1\widehat{\Lcal}^\dagger_{1}=\widehat{\Lcal}^\dagger_{1} $ and $\con{(1)}=\nabla_{\dagger}^{(1)}$. By the horizontality of \eqref{eq_ses1}, the $\Mtriv$-connection is trivial on $\Hcal_{\Eftriv}$, i.e. $\nabla^{(1)}_\dagger(\hat{\omega}^{[1,0]})=\nabla^{(1)}_\dagger(\hat{\omega}^{[0,1]})=0$. It remains to determine $\nabla^{(1)}_\dagger(\hat{\omega}^{[0,0]})$. We will use the horizontality of
\[
\Psi\colon \widehat{\Lcal}^\dagger_{1} \rightarrow \phi_{\Eftriv}^* \widehat{\Lcal}^\dagger_{1}
\]
to determine $\nabla^{(1)}_\dagger(\hat{\omega}^{[0,0]})$: In terms of the basis $\hat{\omega}^{[*,*]}$ we have
\begin{align*}
\Psi(\hat{\omega}^{[0,0]})&=\phi_{\Eftriv}^*\hat{\omega}^{[0,0]}\\
\Psi(\hat{\omega}^{[1,0]})&=\phi_{\Eftriv}^*\hat{\omega}^{[1,0]}\\
\Psi(\hat{\omega}^{[0,1]})&=p\cdot\phi_{\Eftriv}^*\hat{\omega}^{[0,1]}.
\end{align*}
We already know that $$\nabla^{(1)}_\dagger(\hat{\omega}^{[0,0]})=f_{1,0}\cdot \hat{\omega}^{[1,0]} + f_{0,1}\cdot \hat{\omega}^{[0,1]}$$ for suitable $f_{0,1},f_{1,0}\in\Gamma(\Eftriv,\Ocal_{\Eftriv})$. Now, the horizontality of \eqref{eq_Phidagger} expresses as the explicit formula
\[
f_{1,0}\cdot \phi_{\Eftriv}^*\hat{\omega}^{[1,0]} + p\cdot f_{0,1}\cdot \phi_{\Eftriv}^*\hat{\omega}^{[0,1]}=p\cdot \phi_{\Eftriv}^*\left(f_{1,0}\hat{\omega}^{[1,0]}\right)+p\cdot \phi_{\Eftriv}^*\left(f_{0,1}\hat{\omega}^{[0,1]}\right).
\]
i.e. $f_{1,0}=p\cdot \phi_{\Eftriv}^*f_{1,0}$ and $f_{0,1}=\phi_{\Eftriv}^*f_{0,1}$. In particular, we get $f_{1,0}=0$ and $f_{0,1}\in\Zp$ which proves the claim.
\end{proof}
\begin{rem}
With a little bit more effort, one can indeed prove that $c=1$ in the above lemma. Since the proof needs a more careful study of the (formal) logarithm sheaves, we will not present it here. The full proof will be given in the upcoming work \cite{Syntomic}.
\end{rem}
By restricting sections of the Poincar\'e bundle $\Po$ to $\Ef\times_S\Edf$ we obtain sections of $\hat{\Lcal}_n:=\pr_*(\Po|_{\Ef\times_S\Edf})$. We would like to give an explicit description of such elements in terms of the basis $(\hat{\omega}^{[k]})_{k=0}^n$ of $\TSym^n\hat{\Lcal}_n$.
\begin{lem}
Let $s\in\Gamma(\Eftriv\times\Edftriv,\Pof)$ and write $\vartheta\in\Gamma(\Eftriv\times\Edftriv,\Ocal_{\Eftriv\times\Edftriv})$ for the associated section obtained by the trivialization. Then
\[
s\Big|_{\Eftriv\times \Inf^n_e \Edtriv}\mapsto \left( (\id\times e)^*\left(\partial_{\Edftriv}^{\circ k}\vartheta\right)\cdot \omega^{[k]} \right)_{k=0}^n
\]
under
\[
\Gamma(\Eftriv\times\Inf^n_e \Edtriv,\Pof)=\Gamma(\Eftriv,\widehat{\Lcal}^\dagger_{n})\rightarrow \bigoplus_{k=0}^n \Gamma(\Eftriv,\TSym^k\om_{\Eftriv}).
\]
Here, $\partial_{\Edftriv}\colon\Ocal_{\Edftriv}\rightarrow \Ocal_{\Edftriv} $ denotes the canonical invariant derivation.
\end{lem}
\begin{proof}
This follows from \eqref{eq_comult} and the fact that
\[
\Ocal_{\Inf^n_e \Etriv^\vee}\rightarrow \TSym^n_{\Ocal_{\Mtriv}}\left( \Ocal_{\Inf^1_e \Etriv^\vee}\right)=\bigoplus_{k=0}^n \TSym^k \om_{\Edtriv/\Etriv}.
\]
maps $f\in\Ocal_{\Inf^n_e \Etriv^\vee}$ to $(e^*\partial_{\Edftriv}^{\circ k} f \cdot\omega^{[k]})_{k=0}^n$.
\end{proof}
\section[Proof of the Theorem]{Proof of the Theorem}\label{sec_proofpadic}
\begin{proof}[Proof of \cref{thm_padictheta}]
The Poincar\'e bundle $\Po^{\sharp,\dagger}$ is equipped with two integrable connections $\con{\sharp}$ and $\con{\dagger}$. Let us write $\Pof^{\sharp,\dagger}$ for the restriction of $\Po^{\sharp,\dagger}$ to the formal completion of $\Etriv^\sharp\times_\Mtriv \Etriv^\dagger$ along the the zero section. The unit root decomposition and the canonical generator of $\om_{\Edtriv/\Mtriv}$ induce a projection:
\[
\HdR{1}{\Edtriv/\Mtriv}\twoheadrightarrow \om_{\Edtriv/\Mtriv}\righteq \Ocal_\Mtriv.
\]
With this identification we obtain differential operators
\[
\con{\sharp}: \Pof^{\sharp,\dagger}\rightarrow \Pof^{\sharp,\dagger}\otimes_{\Ocal_\Mtriv} \HdR{1}{\Edtriv/\Mtriv} \twoheadrightarrow \Pof^{\sharp,\dagger}
\]
and similarly
\[
\con{\dagger}: \Pof^{\sharp,\dagger}\rightarrow \Pof^{\sharp,\dagger}\otimes_{\Ocal_\Mtriv} \HdR{1}{\Etriv/\Mtriv} \twoheadrightarrow \Pof^{\sharp,\dagger}.
\]
Tensoring the trivialization $\Pof\righteq \Ocal_{\Eftriv\times\Edftriv}$ with $\Ocal_{\Eftriv^\sharp\times\Eftriv^\dagger}$ gives a trivialization $\Pof^{\sharp,\dagger}\righteq \Ocal_{\Eftriv^\sharp\times\Eftriv^\dagger}$. Further, the unit root decomposition induces canonical projections $\Ocal_{\Eftriv^\sharp}\twoheadrightarrow \Ocal_{\Eftriv}$ and $\Ocal_{\Eftriv^\dagger}\twoheadrightarrow \Ocal_{\Eftriv^\vee}$. By the definition of the geometric nearly holomorphic modular forms $E^{k,r+1}_{s,t}$ it suffices to prove the commutativity of the following diagrams
\[
\begin{tikzcd}
\Pof^{\sharp,\dagger}\ar[d,"\cong"] \ar[r,"\con{\sharp}"] & \Pof^{\sharp,\dagger}\ar[d,"\cong"]\\
\Ocal_{\Eftriv^\sharp}\hat{\otimes}_{\Ocal_\Mtriv}\Ocal_{\Eftriv^\dagger}\ar[d,two heads] & \Ocal_{\Eftriv^\sharp}\hat{\otimes}_{\Ocal_\Mtriv}\Ocal_{\Eftriv^\dagger}\ar[d,two heads]\\
\Ocal_{\Eftriv}\hat{\otimes}_{\Ocal_\Mtriv}\Ocal_{\Eftriv^\vee} \ar[r,"\id\otimes \partial_{\Eftriv^\vee}"] & \Ocal_{\Eftriv}\hat{\otimes}_{\Ocal_\Mtriv}\Ocal_{\Eftriv^\vee}
\end{tikzcd}
\quad
\begin{tikzcd}
\Pof^{\sharp,\dagger}\ar[d,"\cong"] \ar[r,"\con{\dagger}"] & \Pof^{\sharp,\dagger}\ar[d,"\cong"]\\
\Ocal_{\Eftriv^\sharp}\hat{\otimes}_{\Ocal_\Mtriv}\Ocal_{\Eftriv^\dagger}\ar[d,two heads] & \Ocal_{\Eftriv^\sharp}\hat{\otimes}_{\Ocal_\Mtriv}\Ocal_{\Eftriv^\dagger}\ar[d,two heads]\\
\Ocal_{\Eftriv}\hat{\otimes}_{\Ocal_\Mtriv}\Ocal_{\Eftriv^\vee} \ar[r,"\partial_{\Eftriv}\otimes \id"] & \Ocal_{\Eftriv}\hat{\otimes}_{\Ocal_\Mtriv}\Ocal_{\Eftriv^\vee}
\end{tikzcd}
\]
We prove the commutativity of the right diagram, the other case is completely symmetric. Recall that $\Po^{\sharp,\dagger}$ with the connection $\con{\dagger}$ is obtained by pullback of the universal connection $\con{\Po^\dagger}$ on the Poincar\'e bundle $\Po^\dagger$ along the projection $\Etriv^\sharp\times_\Mtriv \Etriv^\dagger\rightarrow \Etriv\times_\Mtriv \Etriv^\dagger$. Thus we can deduce the commutativity of the right diagram from the commutativity of
\begin{equation*}
\begin{tikzcd}
\Pof^{\dagger}\ar[d,"\cong"] \ar[r,"\con{\dagger}"] & \Pof^{\dagger}\ar[d,"\cong"]\\
\Ocal_{\Eftriv}\hat{\otimes}_{\Ocal_\Mtriv}\Ocal_{\Eftriv^\dagger}\ar[d,two heads] & \Ocal_{\Eftriv}\hat{\otimes}_{\Ocal_\Mtriv}\Ocal_{\Eftriv^\dagger}\ar[d,two heads]\\
\Ocal_{\Eftriv}\hat{\otimes}_{\Ocal_\Mtriv}\Ocal_{\Eftriv^\vee} \ar[r,"\partial_{\Eftriv}\otimes \id"] & \Ocal_{\Eftriv}\hat{\otimes}_{\Ocal_\Mtriv}\Ocal_{\Eftriv^\vee}.
\end{tikzcd}
\end{equation*}
It is enough to prove the commutativity of this diagram restricted to $\Eftriv\times_\Mtriv \Inf^n_e \Eftriv^\dagger$ for all $n\geq 1$, i.e. we have to prove for all $n\geq 1$ the commutativity of
\begin{equation}\label{diag_Pof_n}
\begin{tikzcd}
\Lnf^\dagger\ar[d,"\cong"] \ar[r,"\nabla^{(n)}_\dagger"] & \Lnf^\dagger\ar[d,"\cong"]\\
\Ocal_{\Eftriv}\hat{\otimes}_{\Ocal_\Mtriv}\Ocal_{\Inf^n_e \Etriv^\dagger}\ar[d,two heads] & \Ocal_{\Eftriv}\hat{\otimes}_{\Ocal_\Mtriv}\Ocal_{\Inf^n_e \Etriv^\dagger}\ar[d,two heads]\\
\Ocal_{\Eftriv}\hat{\otimes}_{\Ocal_\Mtriv}\Ocal_{\Inf^n_e \Etriv^\vee} \ar[r,"\partial_{\Eftriv}\otimes \id"] & \Ocal_{\Eftriv}\hat{\otimes}_{\Ocal_\Mtriv}\Ocal_{\Inf^n_e \Etriv^\vee}.
\end{tikzcd}
\end{equation}
For the commutativity of the diagram \eqref{diag_Pof_n} let us consider the following diagram:
\[
\begin{tikzcd}[row sep=tiny, column sep=tiny]
\Lnf^\dagger\ar[rr,"\nabla_{\dagger}^{(n)}"]\ar[dd,"\triv^{(n)}"]\ar[rd,hook] & & \Lnf^\dagger\ar[rd,hook]\ar[dd] & \\
& \TSym^n_{\Ocal_{\Eftriv}} \widehat{\Lcal}_1^\dagger \ar[crossing over]{rr}[near start]{\con{(n)}}\ar[dd,"\triv^{(1)}"] & & \TSym^n_{\Ocal_{\Eftriv}} \widehat{\Lcal}_1^\dagger\ar[dd] \\
\Ocal_{\Eftriv}\otimes_{\Ocal_\Mtriv}\Ocal_{\Inf^n_e \Etriv^\dagger}\ar[dd] & & \Ocal_{\Eftriv}\otimes_{\Ocal_\Mtriv}\Ocal_{\Inf^n_e \Etriv^\dagger}\ar[dd] & \\
& \TSym^n_{\Ocal_{\Eftriv}} \left(\Ocal_{\Eftriv}\otimes_{\Ocal_\Mtriv}\Ocal_{\Inf^1_e \Etriv^\dagger}\right) && \TSym^n_{\Ocal_{\Eftriv}} \left(\Ocal_{\Eftriv}\otimes_{\Ocal_\Mtriv}\Ocal_{\Inf^1_e \Etriv^\dagger}\right)\ar[dd]\\
\Ocal_{\Eftriv}\otimes_{\Ocal_\Mtriv}\Ocal_{\Inf^n_e \Etriv^\vee}\ar[rr]\ar[rd,hook] & & \Ocal_{\Eftriv}\otimes_{\Ocal_\Mtriv}\Ocal_{\Inf^n_e \Etriv^\vee}\ar[rd,hook]& \\
& \TSym^n_{\Ocal_{\Eftriv}}\left( \Ocal_{\Eftriv}\otimes_{\Ocal_\Mtriv}\Ocal_{\Inf^1_e \Etriv^\vee}\right)\ar[crossing over, leftarrow, near start]{uu}{}\ar[rr] & & \TSym^n_{\Ocal_{\Eftriv}}\left( \Ocal_{\Eftriv}\otimes_{\Ocal_\Mtriv}\Ocal_{\Inf^1_e \Etriv^\vee}\right)
\end{tikzcd}
\]
In this diagram, the commutativity of the left and the right face is just the compatibility of the co-multiplication maps. The commutativity of the lower face is obvious. The upper face commutes by the horizontality of the co-multiplication maps. The commutativity of the front face follows from the explicit formula of $\con{(n)}$. Now, the commutativity of the back face is deduced from the commutativity of the other faces and the injectivity of the comultiplication maps.
\end{proof}
\section[$p$-adic interpolation of $p$-adic Eisenstein--Kronecker series]{p-adic interpolation of p-adic Eisenstein--Kronecker series}
As always when one has a $p$-adic measure $\mu$ on $\Zp$ it is only possible to define the moment function
\[
\Zp\ni s\mapsto \int_{\Zp^\times} \langle x\rangle^s \dd \mu(x)
\]
with $\langle\cdot\rangle\colon \Zp^\times\twoheadrightarrow (1+p\Zp)$ after restriction to $\Zp^\times$. Let us consider again the universal trivialized elliptic curve $(\Etriv/\Mtriv,\beta,\alpha_N)$ with $\Gamma(N)$-level structure. For $e\neq s\in \Etriv[N](\Mtriv)$ corresponding to $(a,b)$ via the level structure we have defined the $p$-adic measure $\muEisD$. Let us write $\muEisDp:=\muEisD|_{\Zp^\times\times \Zp} $ for the restriction of the measure $\muEisDp$ to $\Zp^\times\times \Zp$. It will be convenient to view $\muEisDp$ as a measure on $\Zp\times\Zp$ by extending by zero. We can easily deduce the following statement from Katz \cite[\S 6.3]{katz_padicinterpol} and \cref{PI_propKatz}. Katz deduces this result by comparing the $q$-expansions of the moments. In this section we sketch a different proof using the geometry of the Poincar\'e bundle. \par
\begin{thm}
\[
\int_{\Zp^\times\times \Zp} f(x,y)\dd \muEisD=\int_{\Zp\times \Zp} f(x,y)\dd \muEisD-\Frob \int_{\Zp\times \Zp} f(p\cdot x,y)\dd \muEisD.
\]
\end{thm}
\begin{proof}[Sketch of the proof:] In the \cref{appendix_dist} we prove a distribution relation for the Kronecker section. The distribution relation for the isogeny $\varphi\colon E\rightarrow E'$ implies the formula
\[
p\cdot\Frob(\pthetaD([p](S),T))=\sum_{\zeta\in \Gmf{S}[p]} \pthetaD(S+_{\Gmf{S}}\zeta,T).
\]
Thus we get
\begin{equation}\label{eq_Theta1}
\pthetaD(S,T)-\frac{1}{p}\sum_{\zeta\in \Gmf{S}[p]} \pthetaD(S+_{\Gmf{S}}\zeta,T)=\pthetaD(S,T)-\Frob(\pthetaD([p](S),T)).
\end{equation}
It is well known, that the Amice transform of $f(S)-\frac{1}{p}\sum_{\zeta\in \hat{\mathbb{G}}_m[p]} f(S+_{\hat{\mathbb{G}}_m}\zeta)$ is the measure $\mu_f|_{\Zp^\times}$. In particular, the left hand side is the Amice transform of $\muEisDp$. Now the statement follows from \eqref{eq_Theta1}, since the inverse of the Amice transform can be computed by
\[
f(T)=\int_{\Zp}(1+T)^x\dd \mu_f(x).
\]
\end{proof}
\appendix
\section{The distribution relation}\label{appendix_dist}
The Kronecker theta function is known to satisfy a distribution relation \cite{bannai_kobayashi}. The aim of this appendix is to prove a similar distribution relation for the underlying Kronecker section. In order to state the general distribution relation we need a further generalization of the translation operators. Let us consider the following commutative diagram
\begin{equation}\label{diag1}
\begin{tikzcd}
E\ar[r,"{[D']}"]\ar[d,"\psi"] & E \ar[d,"\psi"] \\
E'\ar[r,"{[D']}"] & E'
\end{tikzcd}
\end{equation}
of isogenies of elliptic curves. The most important case is the case $E=E'$. Let us write $\gamma_{[D'],\psi^\vee}$ for the diagonal of the commutative diagram
\[
\begin{tikzcd}[column sep=huge, row sep=huge]
([D']\times\psi^\vee)^*\Po \ar[rd,"\gamma_{[D'],\psi^\vee}"]\ar[r,"({[D']}\times\id)^*\gamma_{\id,\psi^\vee}"]\ar[d,swap,"(\id\times\psi^\vee)^*\gamma_{[D'],\id}"] & (\psi\circ[D']\times\id)^*\Po' \ar[d,"(\psi\times\id)^*\gamma_{[D'],\id}"] \\
(\id\times (\psi\circ[D'])^\vee)^*\Po' \ar[r,swap,"(\id\times{[D']})^*\gamma_{\id,\psi^\vee}"] & (\psi\times [D'])^*\Po'.
\end{tikzcd}
\]
and $\gamma_{\psi,[D']}$ for its inverse.
\begin{defin}
Let $s\in (\ker\psi)(S)$ and $t\in(E^{\prime,\vee}[D'])(S)$. The translation operator
\[
\Ucal_{s,t}^{\psi,[D']}\colon (T_s\times T_t)^*([D']\times\psi^\vee)^*\Po \rightarrow ([D']\times\psi^\vee)^*\Po
\]
is defined as the composition $\Ucal_{s,t}^{\psi,[D']}:=\gamma_{\psi,[D']}\circ(T_s\times T_t)^*\gamma_{[D'],\psi^\vee}$.
\end{defin}
As before let us write
\[
U_{s,t}^{\psi,[D']}(f):=(\Ucal_{s,t}^{\psi,[D']}\otimes\id)\left( (T_s\times T_t)^*([D']\times\psi^\vee)^*f \right)
\]
for sections $f\in\Gamma\left(E\times E^\vee, \Po\otimes \Omega^1_{E\times E^\vee/E^\vee}([E\times e]+[e\times E^\vee]) \right)$.
Now, we can state the distribution relation which is motivated from the theta function distribution relation given in \cite[Proposition 1.16]{bannai_kobayashi}.
\begin{thm}\label{ch_EP_thmdist}Let $E$ and $E'$ be elliptic curves fitting into a commutative diagram as \eqref{diag1}. Let $N,D$ and $D'$ be integers and let us assume that $N,D,D'$ and $\ker\psi$ are non-zero-divisors on $S$. Let us further assume that all torsion sections of the finite group schemes $\ker\psi$ and $E'^\vee[D']$ are already defined over $S$, i.e.
\[
|(\ker\psi)(S)|=\deg\psi,\quad |(E'^\vee[D'])(S)|=(D')^2.
\]
Then, for $t\in E'^\vee[D](S)$, $s\in E[N](S)$:
\begin{align*}
&\sum_{\substack{\alpha\in (\ker\psi)(S),\\ \beta\in (E'^\vee[D'])(S)}} U^{[N]\circ \psi,~[D]\circ [D']}_{s+\alpha,~t+\beta}(\scan )=\\
=&((D')^2)\cdot \left( ([D]\times[N])^*\gamma_{\psi,[D']} \right)\left((\psi\times [D'])^*U^{N,D}_{\psi(s),[D'](t)}(s_{\mathrm{can},E'})\right)
\end{align*}
\end{thm}
In its simplest but still interesting form the distribution relation specializes to the following equality:
\begin{cor}\label{ch_EP_cor_dist1}
For $E/S$ with $\tilde{D}$ invertible on $S$ and $|E[\tilde{D}](S)|=\tilde{D}^2$:
\[
\sum_{e\neq t\in \Ed[\tilde{D}](S)} U^{\tilde{D}}_{t}(\scan)=\tilde{D}^2\cdot \gamma_{1,\tilde{D}}\left( (\id \times [\tilde{D}])^*(\scan)\right)-([\tilde{D}]\times \id)^*(\scan)
\]
\end{cor}
\begin{proof}
This is the special case $D=N=1$, $\psi=\psi=\id$ and $[D']=[D']=[\tilde{D}]$ of \cref{ch_EP_thmdist}.
\end{proof}
Let us define
\[
\scan^D:=D^2\cdot \gamma_{1,D}\left( (\id \times [D])^*(\scan)\right)-([D]\times \id)^*(\scan).
\]
the above Corollary states that
\[
\sum_{e\neq t\in \Ed[D](S)} U^{D}_{t}(\scan)=\scan^D.
\]
If $\varphi\colon E\rightarrow E'$ is an isogeny with $\deg\varphi$ being a non-zero-divisor on $S$, we obtain by summing over all $t\in\Ed$ and all $s\in \ker\varphi$ the following special case of the distribution relation:
\begin{cor}\label{ch_EP_cor_dist2}
Let $\varphi\colon E\rightarrow E'$ be an isogeny of elliptic curves over $S$ with $\deg \varphi$ a non-zero-divisor on $S$. For $D$ a non-zero-divisor on $S$ we have
\[
\sum_{\tau\in \ker\varphi(S)} ([D]\times\id)^*\Ucal_{\tau,e}^{\varphi,\id}\left( (T_\tau\times \varphi^\vee)^*\scan^D \right) =([D]\times\id)^* \gamma_{\varphi,\id}\left( (\varphi \times \id)^*(s_{\mathrm{can},E'}^D)\right).
\]
\end{cor}
\begin{proof}
After inverting $D$ and making a finite \'etale base change, we may assume that $|E[D](S)|=D^2$. The general distribution relation gives us the identity
\begin{align}\label{eq_dist1}
\sum_{\substack{\tau\in (\ker\varphi)(S),\\ t\in (E'^\vee[D])(S)}} U^{\varphi,D}_{\tau,t}(\scan )=(D^2)\cdot \gamma_{\varphi,[D]}\left((\varphi\times[D])^*s_{\text{can},E'}\right).
\end{align}
On the other hand, again by the general distribution relation we have
\begin{align}\label{eq_dist2}
\sum_{\substack{\tau\in (\ker\varphi)(S)}} U^{\varphi,D}_{\tau,e}(\scan )&=([D]\times\id)^*\gamma_{\varphi,\id}\left((\varphi\times\id)^*U_{e,e}^{\id,[D]}(s_{\text{can},E'})\right)=\\
&=([D]\times\id)^*\Big[\gamma_{\varphi,\id}\left((\varphi\times\id)^*s_{\text{can},E'})\right)\Big].
\end{align}
By \cref{ch_EP_cor_dist1} and a straight forward computation, we can identify the following two sums
\[
\sum_{\substack{\tau\in (\ker\varphi)(S),\\ e\neq t\in (E'^\vee[D])(S)}} U^{\varphi,D}_{\tau,t}(\scan )=\sum_{\tau\in \ker\varphi(S)} ([D]\times\id)^*\Ucal_{\tau,e}^{\varphi,\id}\left( (T_\tau\times \varphi^\vee)^*\scan^D \right).
\]
Subtracting \eqref{eq_dist2} from \eqref{eq_dist1} and using the last identification proves the corollary.
\end{proof}
\begin{rem}
For a CM elliptic curve $E$ over $\CC$, one can use the methods from section \ref{subsec_Analitification} to describe the analytification of translations of the Kronecker section in terms of translations of the Kronecker theta function. In this case, the distribution relation specializes to the analytic distribution relation in \cite{bannai_kobayashi}.
\end{rem}
\subsection{Density of torsion sections}
Before we prove the distribution relation let us recall the density of torsion sections for elliptic curves:
\begin{lem}
Let $N>1$ and $E/S$ be an elliptic curve with $N$ invertible on $S$. For $\Fcal$ a locally free $\Ocal_E$-module of finite rank, $U\subseteq E$ open and $s\in\Gamma(U,\Fcal)$ we have: The section $s$ is zero, if and only if $t^*s=0$ for all $T\rightarrow S$ finite \'{e}tale, $n\geq 0$ and $t\in E[N^n](T)$.
\end{lem}
\begin{proof}
By the sheaf property we may prove this locally and reduce to the case $\Fcal=\Ocal_E^r$, $r\geq 0$. By \cite[Thm. 11.10.9]{EGA4_3} we are further reduced to prove the result in the case $S=\Spec k$ for a field $k$. In this case the result is well-known, cf.~\cite[(5.30) Thm, and the remark (2) afterwards]{moonen_vdgeer}.
\end{proof}
\begin{rem}
If we take all torsion points different from zero, we still get a universally schematically dense family. Indeed, a priori the family is then only universally schematically dense in the open subscheme $U=E\setminus\{e(S)\}$, but the inclusion $U\hookrightarrow E$ is also universally schematically dense, since it is the complement of a divisor \cite[cf. the remark after Lemma 11.33]{goertz}.
\end{rem}
\subsection{Proof of the distribution relation}
For the proof of the distribution relation we will need the following lemma:
\begin{lem}\label{ch_EP_lem_Ustomega}
Let $\psi\colon E\rightarrow E'$ be an isogeny of elliptic curves and $D,\tilde{D}$ be positive integers.
\begin{enumerate}
\item\label{ch_EP_lem_Ustomega_b} For $\tilde{t}\in E'^\vee[\tilde{D}](S)$ we have:
\begin{align*}
&([\tilde{D}]\times\id)^*\gamma_{\psi,[D']}\circ(\psi\times [D'])^*\Ucal^{[\tilde{D}]}_{D'\tilde{t}}=\\
=&([D']\times \psi^\vee)^*\Ucal^{[\tilde{D}]}_{\psi^\vee(\tilde{t})}\circ ([\tilde{D}]\times T_{\tilde{t}})^*\gamma_{\psi,[D']}
\end{align*}
\item\label{ch_EP_lem_Ustomega_a} For $\alpha \in (\ker\psi)(S)$ and $\beta\in (E'^\vee[D'])(S)$ with $\psi^\vee(\beta)\neq e$ we have
\[
(\id\times e)^* U^{\psi,[D']}_{\alpha,\beta}(\scan)=T_\alpha^*\omega^{[D']}_{\psi^\vee(\beta)}\,.
\]
\item\label{ch_EP_lem_Ustomega_c} For $\tilde{t}\in E'^\vee[\tilde{D}](S)$ we have:
\[
([D']\times\psi^\vee)^*\Ucal^{[\tilde{D}]}_{\psi^\vee(\tilde{t})}\circ ([\tilde{D}]\times T_{\tilde{t}})^*\Ucal^{\psi,[D']}_{\tilde{D}s,t}=\Ucal^{\psi,[D'\cdot\tilde{D}]}_{s,t+\tilde{t}}
\]
\end{enumerate}
\end{lem}
\begin{proof}
\ref{ch_EP_lem_Ustomega_b}: We have the following commutative diagrams:
\[
\begin{tikzcd}[column sep=1.5in]
(\id \times T_{\tilde{t}})^*(\psi\circ[\tilde{D}] \times [D'])^*\Po'
\ar[r,"{(\id \times T_{\tilde{t}})^*(\psi \times [D'])^*\gamma_{[\tilde{D}],\id}}"]
\ar[d,equal]
& (\id \times T_{\tilde{t}})^*(\psi \times [D'\cdot\tilde{D}])^*\Po'
\ar[d,"{(\id \times T_{\tilde{t}})^*\gamma_{\psi,[D'\cdot\tilde{D}]}}"]\\
(\id \times T_{\tilde{t}})^*(\psi\circ[\tilde{D}] \times [D'])^*\Po'
\ar[r,"{(\id \times T_{\tilde{t}})^*([\tilde{D}] \times \id )^*\gamma_{\psi,[D']}}"]
& (\id \times T_{\tilde{t}})^*([D'\cdot\tilde{D}] \times \psi^\vee)^*\Po
\end{tikzcd}
\]
\[
\begin{tikzcd}[column sep=1.5in]
(\psi \times [D'\cdot\tilde{D}])^*\Po'
\ar[r,"{(\psi \times [D'])^*\gamma_{\id,[\tilde{D}]}}"]
\ar[d,"{(\id \times T_{\tilde{t}})^*\gamma_{\psi,[D'\cdot\tilde{D}]}}"]
& (\psi\circ[\tilde{D}] \times [D'])^*\Po'
\ar[d,"{\gamma_{\psi\circ[\tilde{D}],[D']}}"]\\
(\id \times T_{\tilde{t}})^*([D'\cdot\tilde{D}] \times \psi^\vee)^*\Po
\ar[r,"{(\id \times T_{\tilde{t}})^*([D'] \times \psi^\vee)^*\gamma_{[\tilde{D}],\id}}"]
& ([D'] \times \psi^\vee\circ[\tilde{D}])^*\Po
\end{tikzcd}
\]
\[
\begin{tikzcd}[column sep=1.5in]
(\psi\circ[\tilde{D}] \times [D'])^*\Po'
\ar[r,"{([\tilde{D}] \times \id)^*\gamma_{\psi,[D']}}"]
\ar[d,"{\gamma_{[\tilde{D}]\circ\psi,[D']}}"]
& ([D'\cdot\tilde{D}] \times \psi^\vee)^*\Po
\ar[d,equal]\\
([D'] \times \psi^\vee\circ[\tilde{D}])^*\Po
\ar[r,"{([D'] \times \psi^\vee)^*\gamma_{\id,[\tilde{D}]}}"]
& ([D'\cdot\tilde{D}] \times \psi^\vee)^*\Po
\end{tikzcd}
\]
The composition of the upper horizontal arrows in the three diagrams is
\begin{align*}
([\tilde{D}]\times \id)^*\gamma_{\psi,[D']}\circ (\psi\times[D'])^*\gamma_{\id,[\tilde{D}]}\circ (\id\times T_{\tilde{t}})^*(\psi\times[D'])^*\gamma_{[\tilde{D}],\id}
\end{align*}
while the composition of the lower horizontal arrows is:
\[
([D']\times\psi^\vee)^*\gamma_{\id,[\tilde{D}]}\circ (\id\times T_{\tilde{t}})^*([D']\times\psi^\vee)^*\gamma_{[\tilde{D}],\id}\circ ([\tilde{D}]\times T_{\tilde{t}})^*\gamma_{\psi,[D']}
\]
The commutativity shows that both compositions are equal, i.\,e. it gives the middle equality in
\begin{align*}
&([\tilde{D}]\times\id)^*\gamma_{\psi,[D']}\circ(\psi\times [D'])^*\Ucal^{[\tilde{D}]}_{D'\tilde{t}}=\\
=&([\tilde{D}]\times \id)^*\gamma_{\psi,[D']}\circ (\psi\times[D'])^*\gamma_{\id,[\tilde{D}]}\circ (\id\times T_{\tilde{t}})^*(\psi\times[D'])^*\gamma_{[\tilde{D}],\id}=\\
=&([D']\times\psi^\vee)^*\gamma_{\id,[\tilde{D}]}\circ (\id\times T_{\tilde{t}})^*([D']\times\psi^\vee)^*\gamma_{[\tilde{D}],\id}\circ ([\tilde{D}]\times T_{\tilde{t}})^*\gamma_{\psi,[D']}=\\
=&([D']\times \psi^\vee)^*\Ucal^{[\tilde{D}]}_{\psi^\vee(\tilde{t})}\circ ([\tilde{D}]\times T_{\tilde{t}})^*\gamma_{\psi,[D']}.
\end{align*}
\ref{ch_EP_lem_Ustomega_a}: Let us first prove the following equality
\begin{equation}\label{eq2}
\Ucal^{\psi,[D']}_{\alpha,\beta}=([D']\times\id)^*\left[\gamma_{\psi,\id}\circ(T_{D'\alpha}\times\id)^*\gamma_{\id,\psi^\vee}\right]\circ(T_\alpha\times\psi^\vee)^*\Ucal_{\psi^\vee(\beta)}^{[D']}
\end{equation}
The proof of this equality follows the same lines as the proof of \ref{ch_EP_lem_Ustomega_b}. So let us only write the equations instead of all commutative diagrams:
\begin{align*}
\tag{I} &(\psi\times\id)^*\gamma_{\id,[D']}=([D']\times\id)^*\gamma_{\id,\psi^\vee}\circ\gamma_{\psi,[D']}\\
\tag{II} &(T_\alpha\times\psi^\vee)^*\gamma_{\id,[D']}\circ(T_\alpha\times T_\beta)^*\gamma_{[D']\circ\psi,\id}=\\
=&(T_\alpha\times\id)^*\gamma_{\psi,[D']}\circ(\psi\times T_\beta)^*\gamma_{[D'],\id}\\
\tag{III} &(\id\times\psi^\vee)^*\gamma_{[D'],\id}=\gamma_{[D']\circ\psi,\id}\circ([D']\times\id)^*\gamma_{\id,\psi^\vee}
\end{align*}
These identities follow easily from the fact that $\gamma_{[D'],\id}$ is induced from the universal property of the Poincar\'e bundle. Using these identities we compute
\begin{align*}
&(\psi\times\id)^*\gamma_{\id,[D']}\circ (T_\alpha\times T_\beta)^*\left[ (\psi\times\id)^*\gamma_{[D'],\id}\circ([D']\times\id)^*\gamma_{\id,\psi^\vee} \right]=\\
=&(T_\alpha\times\id)^*(\psi\times\id)^*\gamma_{\id,[D']}\circ (\psi\times T_\beta)^*\gamma_{[D'],\id}\circ(T_\alpha\times T_\beta)^*([D']\times\id)^*\gamma_{\id,\psi^\vee} \stackrel{(I)}{=}\\
=&([D']\circ T_\alpha\times\id)^*\gamma_{\id,\psi^\vee}\circ(T_\alpha\times\id)^*\gamma_{\psi,[D']}\circ (\psi\times T_\beta)^*\gamma_{[D'],\id}\circ([D']\circ T_\alpha\times T_\beta)^*\gamma_{\id,\psi^\vee} \stackrel{(II)}{=}\\
=&([D']\circ T_\alpha\times\id)^*\gamma_{\id,\psi^\vee}\circ(T_\alpha\times\psi^\vee)^*\gamma_{\id,[D']}\circ(T_\alpha\times T_\beta)^*\gamma_{[D']\circ\psi,\id}\circ([D']\circ T_\alpha\times T_\beta)^*\gamma_{\id,\psi^\vee} \stackrel{(III)}{=}\\
=&([D']\times\id)^*(T_{[D'](\alpha)}\times\id)^*\gamma_{\id,\psi^\vee}\circ(T_\alpha\times\psi^\vee)^*\gamma_{\id,[D']}\circ(T_\alpha\times T_\beta)^*(\id\times\psi^\vee)^*\gamma_{[D'],\id}.
\end{align*}
Now equation \eqref{eq2} follows by precomposing this computation with $([D']\times\id)^*\gamma_{\psi,\id}\circ$. Observing that
\[
(\id\times e)^*\left[\gamma_{\psi,\id}\circ(T_{D'\alpha}\times\id)^*\gamma_{\id,\psi^\vee}\right]
\]
is just the canonical isomorphism $T_{D'\alpha}^*\Ocal_{E}\righteq \Ocal_{E}$, we deduce $(a)$ from \eqref{eq2} by applying $(\id\times e)^*$.\\
\ref{ch_EP_lem_Ustomega_c}: Follows along the same lines as \ref{ch_EP_lem_Ustomega_a} and \ref{ch_EP_lem_Ustomega_b}.
\end{proof}
\begin{proof}[Proof of the distribution relation]
Since $M:=N\cdot D\cdot D'\cdot \deg\psi$ is a non-zero-divisor, we may prove the equality after inverting $M$. Thus, let us assume that $M$ is invertible on $S$. Let us write
\begin{align*}
A&:=\sum_{\substack{\alpha\in (\ker\psi)(S),\\ \beta\in (E'^\vee[D'])(S)}} U^{[N]\circ \psi,~[D\cdot D']}_{s+\alpha,~t+\beta}(\scan )\\
B&:=(D')^2\cdot \left( ([D]\times[N])^*\gamma_{\psi,[D']} \right)\left((\psi\times [D'])^*U^{[N],[D]}_{\psi(s),[D'](t)}(s_{\text{can},E'})\right).
\end{align*}
Our aim is to prove $A=B$. Using the Zariski covering $(S[\frac{1}{\tilde{D}}])_{\tilde{D}>1, (\tilde{D},M)=1}$ together with a density of torsion section argument we reduce the proof of the equality $(\id\times\tilde{t})^*A=(\id\times\tilde{t})^*B$ for all $\tilde{D}$ torsion points $\tilde{t}\in E'^\vee[\tilde{D}](T)$ with $\tilde{D}$ invertible on $T$. Further, since $([D\cdot D']\times e)^*\Ucal^{[\tilde{D}]}_{([N]\circ\psi^\vee)(\tilde{t})}$ is an isomorphism, we are reduced to prove the following: For all $\tilde{D}>1$ coprime to $M$ we have the following equality:
\begin{enumerate}
\item[$(*)_{\tilde{D}}$] For all pairs $(T,\tilde{t})$ with $T$ an $S$-scheme, $\tilde{D}$ invertible on $T$ and $e\neq \tilde{t}\in E'^\vee[\tilde{D}](T)$ we have
\begin{align}\label{ch_EP_thmdist_eq2}
\notag&\left(([D\cdot D']\times e)^*\Ucal^{[\tilde{D}]}_{([N]\circ\psi^\vee)(\tilde{t})} \right)\left[([\tilde{D}]\times \tilde{t})^*A\right]=\\
=&\left(([D\cdot D']\times e)^*\Ucal^{[\tilde{D}]}_{([N]\circ\psi^\vee)(\tilde{t})} \right)\left[([\tilde{D}]\times \tilde{t})^*B\right]=
\end{align}
\end{enumerate}
We compute the left hand side for arbitrary $(T,\tilde{t})$:
\begin{align}\label{ch_EP_thmdist_eq3}
&\left(([D\cdot D']\times e)^*\Ucal^{[\tilde{D}]}_{([N]\circ\psi^\vee)(\tilde{t})}\right)\left(([\tilde{D}]\times \tilde{t})^*A\right)\stackrel{\text{Lem.} \ref{ch_EP_lem_Ustomega} (c)}{=}\\
\notag =&(\id \times e)^*\left(\sum_{\substack{\alpha\in (\ker\psi)(S),\\ \beta\in (E'^\vee[D'])(S)}} U^{[N]\circ\psi,~[D\tilde{D} D']}_{(\tilde{D})^{-1}(s+\alpha),~\tilde{t}+t+\beta}(\scan) \right)\stackrel{\text{Lem.} \ref{ch_EP_lem_Ustomega}(b)}{=}\\
\notag =&\sum_{\substack{\alpha\in (\ker\psi)(S),\\ \beta\in (E'^\vee[D'])(S)}} (T_{(\tilde{D})^{-1}(s+\alpha)})^* \omega^{[D\tilde{D} D']}_{([N]\circ\psi^\vee)(\tilde{t}+t+\beta)}=\\
\notag =&(T_{(\tilde{D})^{-1}(s)})^*\sum_{\substack{\alpha\in (\ker\psi)(S),\\ \beta\in (E'^\vee[D'])(S)}} T_{\alpha}^* \omega^{[D\tilde{D} D']}_{([N]\circ\psi^\vee)(\tilde{t}+t+\beta)}\stackrel{\text{Lem.} \ref{ch_EP_lemTrace2}\ref{ch_EP_lemTrace2_b}}{=}\\
\notag =&(T_{(\tilde{D})^{-1}(s)})^*\sum_{\beta\in (E'^\vee[D'])(S)} \psi^* \omega^{[D\tilde{D} D']}_{N(\tilde{t}+t+\beta)}\stackrel{\text{Lem.} \ref{ch_EP_lemTrace}\ref{ch_EP_lemTrace_b}}{=}\\
\notag =&(D')^2\cdot(T_{(\tilde{D})^{-1}(s)})^* \psi^* \omega^{[D\tilde{D}]}_{(D'\cdot N)(\tilde{t}+t)}
\end{align}
Before we simplify the right hand side of the above equation, we use \cref{ch_EP_lem_Ustomega} to simplify the following expression:
\begin{align*}
&([D\cdot D']\times[N]\circ\psi^\vee)^*\Ucal^{[\tilde{D}]}_{([N]\circ\psi^\vee)(\tilde{t})}\circ([\tilde{D}]\times T_{\tilde{t}})^*([D]\times[N])^*\gamma_{\psi,[D']}\circ\\
&\circ([\tilde{D}]\times T_{\tilde{t}})^*(\psi\times[D'])^*\Ucal^{[N],[D]}_{\psi(s),D't}=\\
=&([D]\times[N])^*\left[ ([D']\times\psi^\vee)^*\Ucal^{[\tilde{D}]}_{([N]\circ\psi^\vee)(\tilde{t})}\circ([\tilde{D}]\times T_{\tilde{Nt}})^*\gamma_{\psi,[D']}\right]\circ\\
&\circ([\tilde{D}]\times T_{\tilde{t}})^*(\psi\times[D'])^*\Ucal^{[N],[D]}_{\psi(s),D't}\stackrel{\text{Lem.}\ref{ch_EP_lem_Ustomega}\ref{ch_EP_lem_Ustomega_b}}{=}\\
=&([D\tilde{D}]\times[N])^*\gamma_{\psi,[D']}\circ(\psi\times[D'])^*\Big[([D]\times[N])^*\Ucal_{ND'\tilde{t}}^{[\tilde{D}]}\circ\\
&\circ([\tilde{D}]\times T_{D'\tilde{t}})^*\Ucal^{[N],[D]}_{\psi(s),D't}\Big]\stackrel{\text{Cor.}\ref{ch_EP_cor_compU}}{=}\\
=&([D\tilde{D}]\times[N])^*\gamma_{\psi,[D']}\circ(\psi\times[D'])^*\Ucal^{[N],[D\tilde{D}]}_{\tilde{D}^{-1}(\psi(s)),D'(t+\tilde{t})}
\end{align*}
Using this and again \cref{ch_EP_lem_Ustomega}, the right hand side of \eqref{ch_EP_thmdist_eq2} is:
\begin{align*}
&\left(([D\cdot D']\times e)^*\Ucal^{\tilde{D}}_{([N]\circ\psi^\vee)(\tilde{t})} \right)\left[([\tilde{D}]\times \tilde{t})^*B\right]=\\
=&(D')^2\cdot(\id\times e)^*\left( (\psi\times[D'])^*U^{[N],[D\tilde{D}]}_{\tilde{D}^{-1}\psi(s),[D'](t+\tilde{t})}(s_{\mathrm{can},E'}) \right)\stackrel{\text{Lem.} \ref{ch_EP_lem_Ustomega} (a)}{=}\\
=& (D')^2\cdot \psi^* T_{\tilde{D}^{-1}\psi(s)}^*\omega^{[D\tilde{D}]}_{ND'(t+\tilde{t})}=(D')^2\cdot T_{\tilde{D}^{-1}s}^*\psi^*\omega^{[D\tilde{D}]}_{ND'(t+\tilde{t})}
\end{align*}
Comparing this last equation to \eqref{ch_EP_thmdist_eq3} shows that the equation in $(*)_{\tilde{D}}$ holds for all pairs $(T,\tilde{t})$.
\end{proof}
\bibliographystyle{amsalpha}
\bibliography{PaperEisensteinPoincare}
\end{document}
| 51,426
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Carl Jung
"Psychotherapy involves the therapist listening to your experiences, exploring connections between your present feelings and actions and the past events. It aims to help you understand more about yourself and your relationships. Therapists have different approaches and styles of working... some will take the lead with questions, while others will support your train of thought." The Department of Health, 'Choosing Talking Therapies'
| 221,429
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TITLE: Is there a parametric equation for a right triangle?
QUESTION [1 upvotes]: I've been trying to find a parametric equation for a triangle because I was inspired by J.M. isn't a mathematician's answer where they provided this parametric equation for a rectangle:
$x=p(|\cos t|\cos t+ |\sin t|\sin t)$
$y=q(|\cos t|\cos t- |\sin t|\sin t)$
For the right triangle, I think we can still use parts of this equation. For example, $|\cos t|\cos t+ |\sin t|\sin t$ remains constant in quadrants 1 and 3. So here, we'd need a function constant in 1 and 2, one for 2 and 3, and then 1 diagonal side with 2 changing functions.
I think we can use $\cos^2 t+|\sin t|\sin t$ as the equation constant in quadrants 1 and 2. And then from there, I think we can put that as the y and the other one as the x like so:
$x=|\cos t|\cos t- |\sin t|\sin t$
$y=\cos^2 t+|\sin t|\sin t$
Where does one go next? How can we do this with an equation for a triangle building off of the parametric equation for a rectangle that we already have? Thanks!
REPLY [1 votes]: Well... It seems I answered my own question with my edit. Because putting in
$x=p(|\cos t|\cos t+ |\sin t|\sin t)$
$y=q(|\cos t|\cos t- |\sin t|\sin t)$
yields
Thanks all! And thank you to another user for a deleted answer they gave with this cool way to create any triangle from its vertices.
| 52,762
|
\begin{document}
\title{Boundary Layer Problems for the Two-dimensional Inhomogeneous
Incompressible Magnetohydrodynamics Equations\hspace{-4mm}}
\author{Jincheng Gao$^{\dag}$ \quad Daiwen Huang$^{\ddag}$ \quad Zheng-an Yao$^{\dag}$\\[10pt]
\small {$^\dag $School of Mathematics, Sun Yat-sen University,}\\
\small {Guangzhou, 510275, P.R.China}\\[5pt]
\small {$^\ddag $Institute of Applied Physics and Computational Mathematics,}\\
\small {Beijing, 100088, P.R.China}\\[5pt]
}
\footnotetext{Email: \it gaojch5@mail.sysu.edu.cn(J.C. Gao),
\it hdw55@tom.com(D.W.Huang)
\it mcsyao@mail.sysu.edu.cn(Z.A.Yao).}
\date{}
\maketitle
\begin{abstract}
In this paper, we study the well-posedness of boundary layer problems for the
inhomogeneous incompressible magnetohydrodynamics(MHD) equations,
which are derived from the two-dimensional density-dependent incompressible MHD equations.
Under the assumption that initial tangential magnetic field is not zero
and density is a small perturbation of the outer constant flow in supernorm,
the local-in-time existence and uniqueness of inhomogeneous incompressible MHD boundary layer equation
are established in weighted Conormal Sobolev spaces by energy method.
As a byproduct, the local-in-time well-posedness of homogeneous incompressible MHD boundary layer
equations with any large initial data can be obtained.
\end{abstract}
\tableofcontents
\section{Introduction and Main Result}
In this paper, we consider the boundary layer problems in the small viscosity and resistivity
limit for the two-dimensional inhomogeneous incompressible Magnetohydrodynamics(MHD) equation
in a period domain $\Omega =: \left\{(x, y): x\in \mathbb{T}, y\in \mathbb{R}^+\right\}$:
\begin{equation}\label{eq1}
\left\{
\begin{aligned}
&\p_t \rho^\ep+{\rm div}(\rho^\ep \mathbf{u}^\ep)=0,\\
&\rho^\ep \p_t \mathbf{u}^\ep+\rho^\ep(\mathbf{u}^\ep \cdot \nabla) \mathbf{u}^\ep
-\mu \ep \Delta \mathbf{u}^\ep+\nabla p^\ep=(\mathbf{h}^\ep \cdot \nabla)\mathbf{h}^\ep,\\
&\p_t \mathbf{h}^\ep-\nabla \times (\mathbf{u}^\ep \times \mathbf{h}^\ep)
-\k \ep \Delta \mathbf{h}^\ep=0,\\
&{\rm div}\mathbf{u}^\ep=0,\quad {\rm div}\mathbf{h}^\ep=0.
\end{aligned}
\right.
\end{equation}
Here, we assume the viscosity and resistivity coefficients have the same order of a small parameter $\ep$.
The unknown functions $\rho^\ep$ denotes the density of fluid,
$\mathbf{u}^\ep=(u_1^\ep, u_2^\ep)$ denotes the velocity vector,
$\mathbf{h}^\ep=(h_1^\ep, h_2^\ep)$ denotes the magnetic field,
and $p^\ep=\widetilde{p}^\ep+\frac{|\mathbf{h}^\ep|^2}{2}$
represents the total pressure with $\widetilde{p}^\ep$ the pressure of fluid.
This system \eqref{eq1} can be used as model to descritbe a viscous fluid that
is incompressible but has nonconstant density,
and hence, it is much more complex than the classical incompressible MHD equation with constant density.
To complete the system \eqref{eq1}, the boundary conditions are given by
\begin{equation}\label{bc1}
u_1^\ep|_{y=0}=u_2^\ep|_{y=0}=0,\quad \p_y h_1^\ep|_{y=0}=h_2^\ep|_{y=0}=0.
\end{equation}
As the parameter $\varepsilon$ tends to zero in the system \eqref{eq1},
we obtain the following system formally
\begin{equation*}\label{ideal}
\left\{
\begin{aligned}
&\p_t \rho^0+{\rm div}(\rho^0 \mathbf{u}^0)=0,\\
&\rho^0 \p_t \mathbf{u}^0+\rho^0(\mathbf{u}^0 \cdot \nabla) \mathbf{u}^0
+\nabla p^0=(\mathbf{h}^0 \cdot \nabla)\mathbf{h}^0,\\
&\p_t \mathbf{h}^0-\nabla \times (\mathbf{u}^0 \times \mathbf{h}^0)=0,\\
&{\rm div}\mathbf{u}^0=0,\quad {\rm div}\mathbf{h}^0=0.
\end{aligned}
\right.
\end{equation*}
which is the inhomogeneous incomrpessible ideal MHD system
with the unknown function $(\rho^0, \mathbf{u}^0, \mathbf{h}^0)$.
To find out the terms in \eqref{eq1} whose contributions are essential
for the boundary layer, we use the same scaling as the one used in
\cite{{Oleinik2},{Liu-Xie-Yang}}
$$
t=t,\quad x=x,\quad \widetilde{y}=\varepsilon^{-\frac{1}{2}}y,
$$
and set
$$
\begin{aligned}
&\rho(t, x, \widetilde{y})=\rho^\ep (t, x, y),\quad
p(t, x, \widetilde{y})=p^\ep (t, x, y),\\
&u_1(t, x, \widetilde{y})=u_1^\ep (t, x, y),\quad
u_2(t, x, \widetilde{y})=\ep^{-\frac{1}{2}}u_2^\ep (t, x, y),\\
&h_1(t, x, \widetilde{y})=h_1^\ep (t, x, y),\quad
h_2(t, x, \widetilde{y})=\ep^{-\frac{1}{2}}h_2^\ep(t, x, y),
\end{aligned}
$$
then the system \eqref{eq1}, after taking the leading order, is reduced to
\begin{equation}\label{eq2}
\left\{
\begin{aligned}
&\p_t \rho +u_1 \p_x \rho+u_2 \p_y \rho=0,\\
&\rho \p_t u_1+\rho u_1 \p_x u_1+\rho u_2 \p_y u_1-\mu \p_y^2 u_1+\p_x p=h_1 \p_x h_1+h_2 \p_y h_1,\\
&\partial_y p=0,\\
&\p_t h_1+\p_y(u_2 h_1-u_1 h_2)=\k \p_y^2 h_1,\\
&\p_t h_2-\p_x(u_2 h_1-u_1 h_2)=\k \p_y^2 h_2,\\
&\p_x u_1+\p_y u_2=0,\quad \p_x h_1+\p_y h_2=0,
\end{aligned}
\right.
\end{equation}
where $(t, x, y)\in [0, T]\times \Omega$, here we have replaced
$\widetilde{y}$ by $y$ for simplicity of notations.
Indeed, the nonlinear boundary layer system \eqref{eq2}
becomes the classical well-known unsteady boundary layer system
if the density becomes constant and magnetic field vanishes(cf.\cite{Schlitchting}).
The third equation of system \eqref{eq2} implies that the leading order of
boundary layers for the total pressure $p^\varepsilon(t, x, y)$ is
invariant across the boundary layer, and should be matched to the outflow pressure
$P(t, x)$ on top of boundary layer, that is, the trace of pressure of idea MHD flow.
Hence, we obtain
$$
p(t, x, y)\equiv P(t, x).
$$
Furthermore, the density $\rho(t, x, y)$, tangential component $u_1(t, x, y)$ of velocity flied,
$h_1(t, x, y)$ of magnetic field, should match the outflow density $\theta(t, x)$,
tangential velocity $U(t, x)$ and tangential magnetic field $H(t, x)$, on the top of boundary layer, that is
\begin{equation*}
\rho(t, x, y)\rightarrow \theta(t, x), \quad
u_1(t, x, y)\rightarrow U(t, x), \quad
h_1(t, x, y) \rightarrow H(t, x),
~{\rm as}~y \rightarrow +\infty,
\end{equation*}
where $\theta(t, x), U(t, x)$ and $H(t, x)$ are the trace of density,
tangential velocity and tangential magnetic field respectively.
Then, we have the following matching conditions:
\begin{equation}\label{ma-co}
\p_t \theta+U\partial_x \theta=0,\quad
\theta \p_t U+\theta U\p_x U+\p_x P=H\p_x H,\quad
\p_t H+U\partial_x H-H\partial_x U=0.
\end{equation}
Moreover, by virtue of the boundary condition \eqref{bc1}, one attains the following boundary condition
\begin{equation}\label{bc2}
\left.u_1 \right|_{y=0}=
\left.u_2 \right|_{y=0}=
\left.\partial_y h_1\right|_{y=0}=
\left. h_2\right|_{y=0}=0.
\end{equation}
In this paper, we consider the outer flow $(\theta, U, H)=(1, 1, 1)$,
which implies the pressure $p$ being a constant.
On the other hand, it is noted that the fifth equation of \eqref{eq2}
is a direct consequences of the fourth equation of \eqref{eq2}.
Hence, we only need to study the following initial boundary value problem for
the inhomogeneous incompressible MHD boundary layer equation
\begin{equation}\label{eq3}
\left\{
\begin{aligned}
&\p_t \rho+u_1 \p_x \rho +u_2\p_y \rho=0,\\
&\rho \p_t u_1+\rho u_1\p_x u_1+ \rho u_2 \p_y u_1-\mu \partial_y^2 u_1
=h_1 \p_x h_1+h_2\p_y h_1,\\
&\p_t h_1+\p_y(u_2 h_1-u_1 h_2)-\k \p_y^2 h_1=0,\\
&\p_x u_1+\p_y u_2=0, \quad \p_x h_1+\p_y h_2=0,
\end{aligned}
\right.
\end{equation}
where the density $\rho:=\rho(t, x, y)$, velocity field $(u_1, u_2):=(u_1(t,x,y), u_2(t,x,y))$,
the magnetic field $(h_1, h_2):=(h_1(t,x,y),h_2(t,x,y))$ are unknown functions.
The boundary conditions for equation \eqref{eq3} are given by
\begin{equation}\label{bc3}
\left\{
\begin{aligned}
&\left.u_1\right|_{y=0}=\left. u_2\right|_{y=0}
=\left.\p_y h_1 \right|_{y=0}=\left. h_2\right|_{y=0}=0,\\
&\lim_{y\rightarrow +\infty }\rho(t,x,y)
=\lim_{y\rightarrow +\infty }u_1(t,x,y)
=\lim_{y\rightarrow +\infty }h_1(t,x,y)=1.
\end{aligned}
\right.
\end{equation}
Let us first introduce some weighted Sobolev spaces for later use.
For any $l \in \mathbb{R}$, denote by $L^2_l(\Omega)$ the weighted Lebesgue space
with respect to the spatial variables:
$$
L_l^2(\Omega):= \{f(x, y):\Omega \rightarrow \mathbb{R},\
\|f\|_{L_l^2(\Omega)}^2 := \int_\Omega \la y \ra^{2l} |f(x,y)|^2 dxdy<+\infty, \
\la y \ra\triangleq 1+y\},
$$
and denote the weighted $L^\infty_l(\Omega)$ Lebesgue space by
$$
L^\infty_l(\O):=\{f(x, y):\Omega \rightarrow \mathbb{R},\
\|f\|_{L_l^\infty(\Omega)}:=
\underset{(x, y)\in \O}{\rm esssup}|\la y \ra^{l}f(x, y)|<+\infty,\
\la y \ra\triangleq 1+y\}.
$$
To define the conormal Sobolev spaces, we will use the notation:
$
Z_1=\p_x, Z_2=\varphi(y)\p_y,
$
where the function $\varphi(y)\triangleq \frac{y}{1+y}$.
Then, we can define the conormal Sobolev spaces as follows:
$$
H^{m,l}_{co}\triangleq \{f\in L^2_l(\Omega)|\
Z^I f \in L_l^2(\Omega),\ |I|\le m\},
$$
where $I=(I_1, I_2)$ and $Z^I=Z_1^{I_1} Z_2^{I_2}$. We also use the notation
\begin{equation*}\label{ndef1}
\|u\|_{m,l}^2=\sum_{|\a|\le m}\|Z^\a u\|_{L^2_l(\O)}^2,\quad
\|u\|_{m,l,\infty}^2=\sum_{|\a|\le m}\|Z^\a u\|_{L_l^\infty(\O)}^2.
\end{equation*}
It is easy to check that
$$
Z_i Z_j=Z_j Z_i, \quad j,k=1,2,
$$
and
$$
\p_y Z_1 =Z_1 \p_y,\quad \p_y Z_2 \neq Z_2 \p_y.
$$
For later use and notational convenience, set
$Z_\tau=(\p_t, Z_1)$ and
$
\z^\a =Z_\tau^{\a_1} Z_2^{\a_2}=\p_t^{\a_{11}}Z_1^{\a_{12}} Z_2^{\a_2},
$
where $\a, \a_1, \a_2$ are the differential multi-indices with
$\a=(\a_1, \a_2), \a_1=(\a_{11}, \a_{12})$,
and we also use the notation
\begin{equation*}\label{ndef2}
\|f(t)\|_{\H^m_l}^2=\sum_{|\a|\le m}\|\z^\a f(t)\|_{L^2_l(\Omega)}^2,
\quad \|f(t)\|_{\H^{m,\infty}_l}^2=\sum_{|\a|\le m}\| \z^{\a} f(t)\|_{L_l^\infty(\O)}^2
\end{equation*}
for smooth space-time function $f(x,t)$. We also use
\begin{equation*}\label{ndef3}
\|f(t)\|_{\H^{m}_{l, tan}}^2
=\sum_{|\a_1|\le m}\| Z_\t^{\a_1}f(t)\|_{L_l^2(\O)}^2, \quad
\|f(t)\|_{\H^{m, \infty}_{l, tan}}^2
=\sum_{|\a_1|\le m}\| Z_\t^{\a_1}f(t)\|_{L_l^\infty(\O)}^2.
\end{equation*}
Finally, we define the functional space $\mathcal{B}_l^m(T)$ for a pair of function
$(\rho, u_1, h_1)=(\rho, u_1, h_1)(x, y, t)$ as follows:
\begin{equation}
\mathcal{B}^m_l(T)=\{(\rho-1, u_1-1, h_1-1)\in L^\infty([0, T]; L^2_l(\O)):
\underset{0\le t \le T}{\rm esssup}
\|(\rho, u_1, h_1)(t)\|_{\mathcal{B}^m_l}<+\infty\},
\end{equation}
where the norm $\|\cdot\|_{\mathcal{B}^m_l}=\|\cdot\|_{\overline{\mathcal{B}}^m_l}
+\|\cdot\|_{\widehat{\mathcal{B}}^m_l}$ is given by
\begin{equation}\label{norm-BX}
\|(\rho, u_1, h_1)(t)\|_{\overline{\mathcal{B}}^m_l}:=
\|(\rho-1, u_1-1, h_1-1)(t)\|_{\H^m_l}^2
+\|\p_y(\rho, u_1, h_1)(t)\|_{\H^{m-1}_l}^2
+\| \p_y \rho(t)\|_{\H^{1,\infty}_1}^2,
\end{equation}
and
\begin{equation}\label{norm-BY}
\begin{aligned}
\|(\rho, u_1, h_1)(t)\|_{\widehat{\mathcal{B}}^m_l}:=
&\sum_{i=0}^{m-1} \|\p_t^i (\p_x \rho, \p_y \rho, \p_x u_1, \p_x h_1)(t)\|_{\H^m_l}^2
+\sum_{i=0}^{m-1} \|\p_t^i \p_y(\p_x^2 \rho, \p_y^2 \rho)(t)\|_{\H^{1,\infty}_0}^2\\
&+\sum_{i=0}^{m-1} \|\p_t^i \p_y(\p_x \rho, \p_y \rho, \p_x u_1, \p_x h_1)(t)\|_{\H^{m-1}_l}^2.
\end{aligned}
\end{equation}
In the present article, we supplement the MHD boundary layer equation \eqref{eq3}
with the initial data
\begin{equation}
(\rho, u_1, h_1)(0, x, y)=(\rho_0, u_{10}, h_{10})(x, y),
\end{equation}
satisfying
\begin{equation}\label{id0}
0<m_0 \le \rho_0 \le M_0 <+\infty,
\end{equation}
and
\begin{equation}\label{id1}
\|(\rho_0, u_{10}, h_{10})\|_{\mathcal{B}^m_l}\le C_0<+\infty,
\end{equation}
where $m_0, M_0, C_0>0$ are positive constants and
\textbf{ the time derivatives of initial data in \eqref{id1}
are defined through the MHD boundary layer equations \eqref{eq3}}.
Hence, we set
\begin{equation}\label{id2}
\mathcal{B}^{m, l}_{BL, ap}=\{(\rho-1, u_1-1, h_1-1)\in H_l^{4m}|\p_t(\rho, u_1,h_1), k=1,...,m
~{\rm are~ defined~ throgh~Eq.}~\eqref{eq3}\}
\end{equation}
and
\begin{equation}\label{id3}
\mathcal{B}^{m,l}_{BL}={\rm the ~closure~of}~\mathcal{B}^{m, l}_{BL, ap}~{\rm in~the~norm~}\|\cdot\|_{\mathcal{B}^m_l}.
\end{equation}
Now, we can state the main results with respect to the well-posedness theory for the inhomogeneous
incompressible MHD boundary layer equations \eqref{eq3}-\eqref{bc3} in this paper as follows.
\begin{theorem}[Main Thoerem]\label{local}
Let $m \ge 5$ be an integer and $l \ge 2$ be a real number.
Assume the initial data $(\rho_0, u_{10}, h_{10})\in \mathcal{B}^{m,l}_{BL}$
given in \eqref{id3} and satisfying \eqref{id0} and \eqref{id1}.
Moreover, there exists a small constant $\delta_0>0$ such that
\begin{equation}\label{condition1}
h_{10}(x,y)\ge 2\delta_0,~{\rm for~all}~(x, y)\in \Omega,
\end{equation}
and
\begin{equation}\label{condition2}
\|\rho_0-1\|_{L^\infty_0(\O)}\le \frac{2l-1}{16}\delta^2_0, \quad
\|\p_y u_{10}\|_{L^\infty_1(\Omega)}\le (2\delta_0)^{-1}.
\end{equation}
Then, there exist a positive time $0<T_0=T_0(\mu, \k,m, l, \d_0,
\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l},
\|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l})$ and a unique solution
$(\rho, u_1, u_2, h_1, h_2)$ to the initial boundary value problem \eqref{eq3}-\eqref{bc3}, such that
\begin{equation}\label{main-estimate}
\begin{aligned}
&\sup_{0\le t \le T_0}\{\|(\rho-1, u_1-1, h_1-1)(t)\|_{\H^m_l}^2
+\|\p_y(\rho, u_1, h_1)(t)\|_{\H^{m-1}_l}^2
+\| \p_y \rho(t)\|_{\H^{1,\infty}_1}^2\}\\
&\quad +\int_0^{T_0} \|\p_y(\sqrt{\mu} u_1, \sqrt{\k}h_1)(t)\|_{\H^m_l}^2 dt
+\int_0^{T_0} \|\p_y^2(\sqrt{\mu} u_1, \sqrt{\k}h_1)(t)\|_{\H^{m-1}_l}^2 dt
\le \widehat{C}_0<+\infty,
\end{aligned}
\end{equation}
where $\widehat{C}_0$ depends only on $l, \d_0$, and $\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}$.
\end{theorem}
\begin{remark}
Note that we choose the initial data with higher regularity and Conormal Sobolev space
as our basic space since we construct the approximation system \eqref{eq4} to
establish the well-posedness for the MHD boundary layer system \eqref{eq3}.
\end{remark}
\begin{remark}
Note that the approach for the well-posedness result in Theorem \ref{local} can be generalized to
study the nonlinear problem \eqref{eq3} with a non-trivial Euler outflow $(1, U, H)$
satisfying the equation \eqref{ma-co}.
\end{remark}
\begin{remark}
We should point out that the initial condition \eqref{condition2} is not required when
the incompressible magnetohydrodynamics flows are the case of homogeneous
(cf. Remark \ref{remark-condition}).
In other words, the local-in-time well-posedness of boundary layer system \eqref{eq3}
with any large initial data can be obtained only under the condition \eqref{condition1}
when the density is constant. This will improve the recent interesting result \cite{Liu-Xie-Yang}.
\end{remark}
We now review some related works to the problem studied in this paper.
The MHD system \eqref{eq1} is a combination of the inhomogeneous incompressible
Navier-Stokes equations of fluid dynamics and Maxwell's equations of electromagnetism.
Since the study for MHD system has been along with that for Navier-Stokes one,
let us recall some results about the inhomogeneous incompressible Navier-Stokes equations.
If the initial density is bounded away from zero, Kazhikov \cite{Kazhikov} proved that:
the inhomogeneous incompressible Navier-Stokes equations have at least one global weak solutions in the energy space.
This result can be generalized to case of initial data with vacuum(cf.\cite{{Simon},{Lions}}).
Choe and Kim \cite{Choe-Kim} proved the existence and uniqueness of local
strong solutions to the initial value problem or the initial boundary
value problem even though the initial vacuum exists.
Recently, the large-time decay and stability to any given global
smooth solutions of the 3D incompressible inhomogeneous Navier-Stokes equations were obtained \cite{Abidi}.
Let's go back to the MHD system \eqref{eq1}, it is known that Gerbeau and Le Bris\cite{GL}
(see also Desjardins and Le Bris \cite{BD}) established the global existence of weak solutions
of finite energy in the whole space or in the torus.
The global existence of strong solution with small initial data in some Besov spaces was
considered by Abidi and Paicu \cite{Abidi-Paicu}.
Recently, Gui \cite{Gui} has shown that the 2D incompressible inhomogeneous magnetohydrodynamics system
with a constant viscosity is globally well-posed for a generic family of the variations of the
initial data and an inhomogeneous electrical conductivity.
When magnetic field vanishes, the MHD system \eqref{eq1} turns to be
the classical well-known incompressible Navier-Stokes equations if the density being constant.
As the viscosity $\varepsilon$ tends to zero, the Navier-Stokes equations
will become the Euler equations.
There are lots of literatures on the uniform bounds and the vanishing viscosity
limit for the Navier-Stokes equations without boundaries \cite{{Constantin}, {Constantin-Foias},{Kato}, {Masmoudi1}}.
The time of existence $T^\varepsilon$ always depend on the viscosity coefficient
when the boundary appears. It is difficult to prove that
the existence of time stays bounded away from zero.
However, for the domain with some special types of Navier-slip boundary conditions,
some uniform $H^3$ (or $W^{2,p}$, with $p$ large enough) estimates and
a uniform time of existence have recently been established \cite{{Beira1},{Beira2},{Xiao-Xin1}}.
This uniform control in some limited regularity Sobolev spaces
can be obtained because these special boundary conditions gives arise to
the main part of the boundary layer vanishes.
For the three dimensional domain with smooth boundary,
Masmoudi and Rousset \cite{Masmoudi-Rousset} recently
obtained conormal uniform estimates for the incompressible
Navier-Stokes equations with Naiver-slip type boundary condition.
Furthermore, they also applied the compact argument to establish
the convergence of the viscous solution to the inviscid ones.
This result was generalized to the compressible flow \cite{Wang-Xin-Yong},
which also shown that the boundary layers for density must be weaker than the one for the velocity.
The vanishing viscosity limit of the incompressible Navier-Stokes equations that,
in a bounded domain with Dirichlet boundary condition, is an important problem in
both physics and mathematics.
This is due to the formation of a boundary layer, where the solution undergoes a sharp transition
from a solution of the Euler system to the zero non-slip boundary condition on boundary of the
Navier-Stokes system. This boundary layer satisfies the Prandtl system formally.
Indeed, Prandtl \cite{Prandtl} derived the Prandtl equations for boundary layer
from the incompressible Navier-Stokes equations with non-slip boundary condition.
The first systematic work in rigorous mathematics was obtained by Oleinik \cite{{Oleinik3},{Oleinik4}},
in which she established the local in time well-posedness of the Prandtl equations
in dimension two by applying the Crocco transformation under the monotonicity condition
on the tangential velocity field in the normal direction to the boundary.
For more extensional mathematical results, the interested readers can refer to
the classical book finished by Oleinik and Samokhin \cite{Oleinik2}.
By taking care of the cancelation in the convection term to overcome the
loss of derivative in the tangential direction of velocity, the researchers
in \cite{Xu-Yang-Xu} and \cite{Masmoudi} independently used the simply
energy method to establish well-posedness theory for the
two-dimensional Prandtl equations in the framework of Sobolev spaces.
For more results in this direction, the interested readers can refer to \cite{{Grenier-Guo-Nguyen1},{Grenier-Guo-Nguyen2},{Gie-Temam}}
and references therein.
Under the influence of electro-magnetic field, the system of
magnetohydrodynamics(denoted by MHD) is a fundamental system
to describe the movement of electrically conducting fluid,
for example plasmas and liquid metals(cf.\cite{Alfven}).
On one hand, G\'{e}rard-Varet and Prestipino \cite{G-V-P}
provided a systematic derivation of boundary layer models in magnetohydrodynamics,
through an asymptotic analysis of the incompressible MHD system.
Furthermore, they also performed some stability analysis for the boundary layer
system, and emphasized the stabilizing effect of the magnetic field.
On the other hand, if both the hydrodynamic Reynolds numbers and magnetic Reynolds numbers
tend to infinity at the same rate, the local in time well-posedness of
the boundary layer system was obtained \cite{Liu-Xie-Yang} if
there exists a small constant $\d_0$ such that
\begin{equation}\label{Yang-C}
|\la y \ra^{l+1}\p_y^i(u_{10}, h_{10})(x, y)|\le (2\d_0)^{-1},~{\rm for~}i=1,2,~(x, y)\in \O,
\end{equation}
and \eqref{condition1} hold on. In other words, the local in time well-posedness of MHD boundary layer
system holds on under the condition on the initial tangential magnetic field is not zero
instead of the monotonicity condition on the tangential velocity field.
Finally, we point out that it is an outstanding open problem to rigorously justify
the validity of expansion in the inviscid limit.
On one hand, Sammartino and Caflisch \cite{{Sammartino-Caflisch1},{Sammartino-Caflisch2}}
obtained the well-posedness in the framework of analytic functions without the
monotonicity condition on the velocity field and justified the boundary layer expansion
for the unsteady incompressible Navier-Stokes equations.
Furthermore, Guo and Nguyen \cite{Guo-Nguyen1} concerned nonlinear ill-posedness of the Prandtl equation
and an invalidity of asymptotic boundary layer expansion of incompressible fluid flow
near a solid boundary. Furthermore, they also shown that the asymptotic boundary layer expansion was not
valid for nonmonotonic shear layer flow in Sobolev spaces and verified that Oleinik's monotonic solutions were well-posed.
For the incompressible steady Navier-Stokes equations, Guo and Nguyen\cite{Guo-Nguyen2} justified the boundary
layer expansion for the flow with a non-slip boundary condition on a moving plate.
As the magnetic field appears, the Prandtl ansatz boundary layer expansion for the unsteady MHD system
was justified \cite{Liu-Xie-Yang2} when no-slip boundary and
perfect conducting boundary conditions are imposed on velocity field and magnetic field respectively.
The rest of this paper is organized as follows.
In section \ref{sa}, we explain the main difficulty and our approach to establish the
local-in-time well-posedness theory for the Prandtl type equation \eqref{eq3}.
In Section \ref{A-P-Est}, one establishes the a priori estimates for the nonlinear problem \eqref{eq5}.
The local-in-time existence and uniqueness of equation \eqref{eq3}
in Weighted Conormal Sobolev space are given in Section \ref{local-in-times}.
Finally, some useful inequalities and important equivalent relations\
will be stated in Appendixs \ref{appendixA} and \ref{appendixB}.
Before we proceed, let us comment on our notation.
Throughout this paper, all constants $C$ may be different in different lines.
Subscript(s) of a constant illustrates the dependence of the constant, for example,
$C_s$ is a constant depending on $s$ only.
Denote by $\p_y^{-1}$ the inverse of the derivative $\p_y$, i.e.,
$(\p_y^{-1}f)(y):=\int_0^y f(z)dz$.
Moreover, we also use the notation $[A, B]=AB-BA$, to denote the commutator between $A$ and $B$.
Finally, $\mathcal{P}_i(\cdot, \cdot)$ stands for a polynomial function independent of $\es$,
and the index $i$ denote it changing from line to line.
\section{Difficulties and Outline of Our Approach}\label{sa}
The main of this section is to explain main difficulties of proving Theorem \ref{local}
as well as our strategies for overcoming them.
In order to solve the Prandtl type equation \eqref{eq3} in certain $H^m$ Sobolev space,
the main difficulty comes from the vertical velocity $u_2=-\p_y^{-1}\p_x u_1$
(and vertical magnetic field $h_2=-\p_y^{-1}\p_x h_1$) creates
a loss of $x-$derivative, so the standard energy estimates can not apply directly.
The main idea of establishing the well-posedness of inhomogeneous incompressible
MHD boundary layer equations \eqref{eq3}-\eqref{bc3} is to apply the so-called
\emph{vanishing viscosity and nonlinear cancelation methods}.
To this end, we consider the following approximate problem:
\begin{equation}\label{eq4}
\left\{
\begin{aligned}
&\p_t \r+u_1^\es \p_x \r+ u_2^\es \p_y \r-\es\p_x^2 \r-\es \p_y^2 \r
=-\es \p_x r_1-\es \p_y r_2,\\
&\r \p_t u_1^\es+\r u_1^\es \p_x u_1^\es+ \r u_2^\es \p_y u_1^\es
-\es \p_x^2 u_1^\es-\mu \p_y^2 u_1^\es=h_1^\es \p_x h_1^\es+h_2^\es \p_y h_1^\es-\es \p_x r_u,\\
&\p_t h_1^\es+\p_y(u_2^\es h_1^\es-u_1^\es h_2^\es)
-\es \p_x^2 h_1^\es-\k \p_y^2 h_1^\es=-\es \p_x r_h,\\
&\p_x u_1^\es+\p_y u_2^\es=0, \quad \p_x h_1^\es+\p_y h_2^\es=0,
\end{aligned}
\right.
\end{equation}
for any parameter $\es>0$. Here the functions $r_1, r_2, r_u$ and $r_h$ are defined by
\begin{equation}\label{rdef}
(r_1, r_2, r_u, r_h)(t, x, y)=\sum_{i=0}^{m-1} \frac{t^i}{i!}\partial_t^i
(\p_x \rho, \p_y \rho, \p_x u_{1}, \p_x h_{1})(0,x,y),
\end{equation}
which gives that by direct calculation
\begin{equation}\label{ID-def}
\p_t^i(\r, u_1^\es, h_1^\es)(0, x, y)=\p_t^i(\rho, u_1, h_1)(0, x, y),\quad 0\le i \le m.
\end{equation}
To complete the system \eqref{eq4}, the boundary conditions are given by
\begin{equation}\label{bc4}
\left\{
\begin{aligned}
&\left.\p_y \r|_{y=0}=u^{\es}_1\right|_{y=0}=\left. u^{\es}_2\right|_{y=0}
=\left.\p_y h^{\es}_1 \right|_{y=0}=\left. h^{\es}_2\right|_{y=0}=0,\\
&\lim_{y\rightarrow +\infty }\r(t,x,y)
=\lim_{y\rightarrow +\infty }u^{\es}_1(t,x,y)
=\lim_{y\rightarrow +\infty }h^{\es}_1(t,x,y)=1.
\end{aligned}
\right.
\end{equation}
Since the local-in-time existence and uniqueness of regularized Eqs.\eqref{eq4}-\eqref{bc4}
can be obtained easily in $H^m$ Sobolev space for any $\es>0$, we hope that
the solution $(\rho^\es, u_1^{\es}, u_2^\es, h_1^\es, h_2^\es)$ of regularized equation \eqref{eq4}
will converge to the solution $(\rho, u_1, u_2, h_1, h_2)$ of original Prandtl type
equation \eqref{eq3} as $\es$ tends to zero.
To this end, we need to get the uniform \emph{a priori} estimates of solution
$(\rho^\es, u_1^{\es}, u_2^\es, h_1^\es, h_2^\es)$
in an existence time independent of $\es$.
Although the idea of local-in-times well-posedness of MHD boundary layer equation,
which only needs that the background tangential magnetic field has a lower positive bound instead of monotonicity
assumption on the tangential velocity, comes from the recent result \cite{Liu-Xie-Yang},
we have to overcome some essential difficulties when the density of fluid changes from a constant to unknown quantity.
First of all, we should work on the Conormal Sobolev space to obtain some energy estimates
independent of small coefficient $\es$ since there is boundary condition for the
first equation of \eqref{eq4}.
To control the vertical velocity $u_2^\es=-\p_y^{-1} \p_x u_1^\es$ by the horizontal velocity $u_1^\es$,
we need to apply the Hardy type inequality by adding a weight $(1+y)^1$.
Since the conormal derivative $Z_2=\varphi(y) \p_y$ does not communicate with the normal
derivative $\p_y$, we need to choose the Sobolev space with suitable weight(actually taking $l \ge 2$)
to close the energy estimate, which is the first novelty in our paper.
Similar to the Prandtl equation, the main difficulty in the analysis on the system \eqref{eq4}
in the Sobolev framework is the loss of $x-$derivative in the vertical components
$u_2^\es$ and $h_2^\es$ appearing in the terms
$u_2^\es \p_y \r, \r u_2^\es \p_y u_1^\es-h_2^\es \p_y h_1^\es$ and
$u_2^\es \p_y h_1^\es-h_2^\es \p_y u_1^\es$ in the first, second and third equations of \eqref{eq4}, respectively.
Motivated by the recent result \cite{Liu-Xie-Yang}, we construct some quantities $(\vr_m, \u_m, \h_m)$
(see the definitions \eqref{equi-r}, \eqref{eqi-u}, \eqref{equi-h} respectively) to avoid the loss of $x$ derivative
and obtain the estimate for $(\vr_m, \u_m, \h_m)$ in $L^2_l-$norm independent of $\es$ by the energy method.
To establish the relation between the quantities $(\vr_m, \u_m, \h_m)$ and $\p_x^m(\rho^\es, u_1^\es, h_1^\es)$,
we need to control the quantity $\p_y(\rho^\es, u_1^\es, h_1^\es)$ in $L^\infty_1-$norm.
To this end, we apply the low order tangential derivative estimate $\mathcal{E}_{m,l}(t)$(see the definition \eqref{eml})
to control the quantity $\p_y(u_1^\es, h_1^\es)$ in $L^\infty_1-$norm, which can be achieved by the
Sobolev embedding inequality. Then, it is easy to get the almost equivalent relation $X_{m, l}(t) \sim Y_{m,l}(t)$
(see the definitions in \eqref{ydef} and \eqref{xdef} respectively).
This is the second novelty in our paper, and avoid the important condition \eqref{Yang-C} required
in \cite{Liu-Xie-Yang} for the MHD boundary layer equation with constant density .
\section{A Priori Estimates}\label{A-P-Est}
In this section, we will establish a priori estimates(independent of $\es$),
which are crucial to prove the Theorem \ref{local}.
First of all, let us define
\begin{equation}\label{transf}
\vr:=\r-1,\
\u:=u_1^\es-1+e^{-y},\ \v=u_2^\es,\quad
\h:=h_1^\es-1, \ \g=h_2^\es,
\end{equation}
then it follows from equation \eqref{eq4} that
\begin{equation}\label{eq5}
\left\{
\begin{aligned}
&\p_t \vr+(u^\es+1-e^{-y})\p_x \vr +v^\es \p_y \vr
-\es\p_x^2 \vr-\es \p_y^2 \vr=-\es \p_x r_1-\es \p_y r_2,\\
&\rho^\es \partial_t u^\es+\rho^\es(u^\es+1-e^{-y})\partial_x u^\es+\rho^\es v^\es\partial_y u^\es
+\rho^\es v^\es e^{-y}\\
&\quad =\es \partial_x^2 u^\es+\mu \partial_y^2 u^\es
+(h^\es+1)\partial_x h^\es+g^\es \partial_y h^\es-\es \p_x r_u-\mu e^{-y},\\
&\partial_t h^\es+(u^\es+1-e^{-y})\partial_x h^\es+v^\es \partial_y h^\es
-\es \partial_x^2 h^\es-\k \partial_y^2 h^\es
=(h^\es+1)\partial_x u^\es+g^\es\partial_y(u^\es-e^{-y})-\es \p_x r_h,\\
&\p_x \u+\p_y \v=0, \quad \p_x \h+\p_y \g=0,\\
&(\vr,\u,\h)|_{t=0}:=(\vr_0,\u_0,\h_0),
\end{aligned}
\right.
\end{equation}
with the boundary conditions
\begin{equation}\label{bc5}
\left\{
\begin{aligned}
&\partial_y \varrho^\es|_{y=0}=u^\es|_{y=0}=v^\es|_{y=0}=
\partial_y h^\es|_{y=0}=g^\es|_{y=0}=0,\\
&\underset{y\rightarrow +\infty}{\lim}\varrho^\es
=\underset{y\rightarrow +\infty}{\lim}u^\es
=\underset{y\rightarrow +\infty}{\lim}h^\es=0,\\
\end{aligned}
\right.
\end{equation}
Due to the relation \eqref{transf}, we can get the relation between two initial data as follow:
\begin{equation}\label{id-relation}
\vr_0=\rho_0-1,\quad \u_0=u_{10}-1+e^{-y},\quad \h_0=h_{10}-1,
\end{equation}
and hence, we have the estimates:
\begin{equation*}
\|(\vr_0, \u_0, \h_0)\|_{\H^m_l}^2
+\|\p_y(\vr_0, \u_0, \h_0)\|_{\H^{m-1}_l}^2
+\| \p_y \vr_0\|_{\H^{1,\infty}_1}^2
\le C(1+\|(\rho_0, u_{10}, h_{10})\|_{\overline{{\mathcal{B}}}^m_l}),
\end{equation*}
and
\begin{equation*}
\|(r_1, r_2, r_u, r_h)(t)\|_{\H^m_l}^2
+\|\p_y(r_1, r_2, r_u, r_h)(t)\|_{\H^{m-1}_l}^2
+\|\p_y(\p_x r_1, \p_y r_2)(t)\|_{\H^{1,\infty}_0}^2
\le C\|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}.
\end{equation*}
Here the norms $\|\cdot \|_{\overline{{\mathcal{B}}}^m_l}$ and
$\|\cdot \|_{\widehat{\mathcal{B}}^m_l}$
are defined by \eqref{norm-BX} and \eqref{norm-BY} respectively,
and the time derivatives of initial data
are defined through the MHD boundary layer equation \eqref{eq3}.
Let us define
\begin{equation}
\begin{aligned}
\ta_{m,l}(\vr, \u, \h)(t):=
&\sup_{0\le s \le t}\{1+\|(\vr, \u, \h)(s)\|_{\H^m_l}^2+\|\p_y(\vr, \u, \h)(s)\|_{\H^{m-1}_l}^2
+\| \p_y \vr (s)\|_{\H^{1,\infty}_1}^2\}\\
&+\es \int_0^t\|\p_x(\vr, \u, \h)\|_{\H^m_l}^2d\t
+\int_0^t \|\p_y(\sqrt{\es}\vr, \sqrt{\mu}\u, \sqrt{\k}\h)\|_{\H^m_l}^2d\t\\
&+\es \int_0^t\|\p_{xy}(\vr, \u, \h)\|_{\H^{m-1}_l}^2d\t
+\int_0^t \|\p_y^2(\sqrt{\es}\vr, \sqrt{\mu}\u, \sqrt{\k}\h)\|_{\H^{m-1}_l}^2d\t.
\end{aligned}
\end{equation}
Next, we will prove the following a priori estimates independent of $\es$
for the regularized MHD boundary layer equations \eqref{eq5}-\eqref{bc5}.
\begin{theorem}[\emph{a priori estimates}]\label{theo a priori}
Let $m \ge 5$ be an integer, $l \ge 2$ be a real number
and $\epsilon \in (0, 1]$, and $(\vr, \u, \v, \h, \g)$ be sufficiently smooth solution,
defined on $[0, T^\es]$, to the regularized MHD boundary layer equations \eqref{eq5}-\eqref{bc5}.
The initial data $(\vr_0,\u_0,\h_0)$ is defined by $(\rho_0, u_{10}, h_{10})$
given in Theorem \ref{local} through the relation \eqref{id-relation}.
Then, there exists a time $T_a=T_a(\mu, \k,m, l, \d_0,
\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l},
\|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l})>0$
independent of $\es$ such the following a priori estimates hold
for all $t \in [0, \min(T_a, T^\es)]$:
\begin{equation}\label{3a1}
\ta_{m,l}(\vr, \u, \h)(t) \le 2 C_l \mathcal{P}_0(\d^{-1}_0, \|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}),
\end{equation}
and
\begin{equation}\label{3a2}
\|\p_y(\u-e^{-y})(t)\|_{L^\infty_1(\O)} \le \d_0^{-1},\quad
\|\vr(t)\|_{L^\infty_0(\O)}\le \frac{3(2l-1)}{32}\delta^2_0,\quad
\h(t,x, y)+1 \ge \d_0,
\end{equation}
for all $(t, x, y) \in [0, \min(T_a, T^\es)] \times \O$.
\end{theorem}
\begin{remark}
When the parameter $\es=0$, the regularized Prandtl type equations \eqref{eq5} become
the original Prandtl type equation \eqref{eq3}, and hence Theorem \ref{theo a priori}
also provides a priori estimates for the original MHD boundary layer equation \eqref{eq3}.
\end{remark}
Throughout this section, for any small constant $\d$,
we assume that the following a priori assumptions:
\begin{equation}\label{a2}
h^\es(t,x,y)+1\ge \d,
\end{equation}
and
\begin{equation}\label{a1}
\|\vr(t)\|_{L^\infty_0(\O)}\le \frac{2l-1}{2}\d^2,\quad
\|\p_y (\u-e^{-y})(t)\|_{L^\infty_1(\O)}\le \d^{-1},
\end{equation}
hold on for any $(t, x, y) \in [0, T^\es] \times \O$.
Thanks to the smallness of $\d$, we find
\begin{equation}\label{low-bou}
\frac{1}{2}\le \rho^\es(t, x, y)\le \frac{3}{2}
\end{equation}
for $(t, x, y)\in [0, T^\es] \times \O$.
\subsection{Weighted $\H^m_l-$Estimates with Conormal Derivative}
In this subsection, we will derive the weighted estimates for the quantities
$Z_\t^{\a_1} Z_2^{\a_2}(\vr, \u, \h)$ with $|\a_1|+|\a_2|=m, |\a_1|\le m-1$.
This goal is easy to reach by the standard energy method because one order
tangential derivative loss is allowed.
For notational convenience, we denote
\begin{equation}\label{eml}
\mathcal{E}_{m,l}(t):=\sum_{\substack{ |\alpha| \le m \\
|\a_1| \le m-1}}
\|\z^\a(\vr, \u, \h)(t)\|_{L^2_l(\O)}^2,
\end{equation}
and
\begin{equation}\label{Q}
\begin{aligned}
Q(t):=\underset{0\le s \le t}{\sup}\{
&\|Z_\t \vr(s)\|_{L^\infty_0(\O)}^2+\|(\u, \h)(s)\|_{\H^{1,\infty}_{0, tan}}^2
+\|(\v, \g)(s)\|_{\H^{1, \infty}_{1, tan}}^2\\
&+\|(\p_y \vr, \p_y \u, \p_y \h, \frac{\v}{\varphi})(s)\|_{\H^{1,\infty}_1}^2\}.
\end{aligned}
\end{equation}
\begin{proposition}\label{Lower-estimate}
Let $(\vr, \u, \v, \h, \g)$ be sufficiently smooth solution, defined on $[0, T^\es]$,
to the equations \eqref{eq5}-\eqref{bc5}. Then, it holds on
\begin{equation*}
\begin{aligned}
&\sup_{0\le \t \le t}\e_{m,l}(\t)
+\sum_{\substack{0 \le |\alpha| \le m \\ |\a_1| \le m-1}}
\es \int_0^t \|\p_x \z^\a(\vr, \u, \h)\|_{L^2_l(\O)}^2 d\tau\\
& + \sum_{\substack{0 \le |\alpha| \le m \\ |\a_1| \le m-1}}
\int_0^t \|(\sqrt{\es} \p_y \z^\a \vr, \sqrt{\mu} \p_y \z^\a \u, \sqrt{\k} \p_y \z^\a\h)\|_{L^2_l(\O)}^2 d\tau\\
\le
& C\|(\vr_0, \u_0, \h_0)\|_{\H^m_l}^2
+C t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}\\
&+C_{\mu, \k, m, l}(1+Q^2(t))\int_0^t (1+\|(\vr, \u, \h)\|_{\H^{m}_l}^2+\|\p_y(\vr, \u, \h)\|_{\H^{m-1}_l}^2)d\tau.
\end{aligned}
\end{equation*}
\end{proposition}
The Proposition \ref{Lower-estimate} will be proved in Lemma \ref{lemma32}.
Now, we give the proof for the case $m=0$.
\begin{lemma}\label{lemma31}
For smooth solution $(\vr, \u, \v, \h,\g)$ of the equations \eqref{eq5}-\eqref{bc5}, then it holds on
\begin{equation}\label{31}
\begin{aligned}
&\sup_{\tau\in [0, t]} \|(\vr, \u, \h)(\t)\|_{L^2_l(\Omega)}^2
+\es \int_0^t \|\p_x(\vr, \u, \h)\|_{L^2_l(\Omega)}^2 d\tau
+\int_0^t \|\p_y(\sqrt{\es} \vr, \sqrt{\mu} \u, \sqrt{\k} \h)\|_{L^2_l(\Omega)}^2 d\tau\\
&\le C\|(\vr_0, \u_0, \h_0)\|_{L^2_l(\Omega)}^2
+\!C\!\!\int_0^t\!\! \|(r_1, r_2, r_u, r_h)\|_{L^2_l(\Omega)}^2d\t
+\!C_{\mu, \k, l}(1+Q(t))\!\int_0^t \!\!(1+ \|(\vr,\u,\h)\|_{\H^1_l}^2)d\tau.
\end{aligned}
\end{equation}
\end{lemma}
\begin{proof}
First of all, multiplying \eqref{eq5}$_2$ by $\y \u$, integrating over $\O $
and integrating by parts, we find
\begin{equation}\label{311}
\begin{aligned}
&\frac{d}{dt}\frac{1}{2}\int_\O \ya \r |\u |^2 dxdy
+\es \int_\O \y |\p_x \u|^2dxdy
+\mu \int_\O \ya |\p_y \u |^2dxdy\\
&=\frac{1}{2} \int_\O \ya |\u |^2(\p_t \r+(\u+1-e^{-y})\p_x \r
+\v \p_y \r )dxdy\\
&\quad +l\int_\O \yb \r \v |\u |^2 dxdy
-2\l \mu \int_\O \yb \u \cdot \p_y \u dxdy\\
&\quad
-\int_\O \rho^\es v^\es e^{-y}\cdot \la y\ra^{2l}u^\es dxdy
+\int_\O [(h^\es+1)\partial_x h^\es+g^\es \partial_y h^\es]\cdot \la y\ra^{2l}u^\es dxdy\\
&\quad +\int_\O (-\es \p_x r_u-\mu e^{-y})\cdot \ya \u dxdy,
\end{aligned}
\end{equation}
where we have used the boundary condition \eqref{bc5} and the divergence free condition \eqref{eq5}$_4$.
By routine checking, we have, after using the definition of $Q(t)$ in \eqref{Q},
\begin{equation*}\label{312}
\begin{aligned}
|\int_\O \ya |\u |^2(\p_t \r+(\u+1-e^{-y})\p_x \r+\v \p_y \r )dxdy|
\le C(1+Q(t))\|\u\|_{L^2_l(\Omega)}^2,
\end{aligned}
\end{equation*}
By virtue of estimate for density \eqref{low-bou}, we get
\begin{equation*}\label{313}
|\int_\O \yb \r \v |\u|^2 dxdy|\le C\|\la y \ra^{-1} \v\|_{L^\infty_0(\O)}\|\u\|_{L^2_l(\Omega)}^2.
\end{equation*}
Using the H\"{o}lder and Cauchy-Schartz inequalities, it follows
\begin{equation*}\label{314}
|\mu \int_\O \yl \u \cdot \p_y \u dxdy|
\le \frac{\mu}{4}\int_\O \y |\p_y \u|^2 dxdy
+C_\mu \int_\O \la y \ra^{2(l-1)}|\u|^2 dxdy.
\end{equation*}
and
\begin{equation*}\label{315}
|\int_\O (-\es \p_x r_u-\mu e^{-y})\cdot \y \u dxdy|
\le \frac{1}{2}\es \|\p_x \u\|_{L^2_l(\O)}^2+ C(1+\|r_u\|_{L^2_l(\Omega)}^2+\|\u\|_{L^2_l(\Omega)}^2).
\end{equation*}
Applying the divergence-free condition \eqref{eq5}$_4$, H\"{o}lder and Hardy inequalities, we get
\begin{equation*}\label{316}
|\int_\O \r \v e^{-y}\cdot \y \u dxdy|
\le C\|\v\|_{L^2_{l-1}(\Omega)}\|\u\|_{L^2_{l}(\O )}
\le C_l\|\p_y \v\|_{L^2_{l}(\O )}\|\u\|_{L^2_{l}(\O )}
\le C_l\|\u\|_{\H^1_l}^2.
\end{equation*}
Integrating by part and applying the divergence-free condition \eqref{eq5}$_4$, we find
\begin{equation*}\label{317}
\begin{aligned}
&\int_\O [(\h+1)\p_x \h+\g \p_y \h]\cdot \y \u dxdy\\
=
&-\int_\O \y \p_x \h \u \h dxdy-\int \y \h (\h+1) \p_x \u dxdy\\
&-\int_\O \y \p_y \g \u \h dxdy-2l\int \yl \g \u \h dxdy\\
&-\int_\O \y \h \g \p_y \u dxdy\\
\le
&-\int_\O \y \h (\h+1) \p_x \u dxdy-\int \y \h \g \p_y \u dxdy\\
&+C\|\la y \ra^{-1} \g\|_{L^\infty_0(\O)}\|\u\|_{L^2_l(\Omega)}\|\h\|_{L^2_l(\Omega)}.
\end{aligned}
\end{equation*}
Substituting the above estimates into \eqref{311},
and integrating the resulting inequality over $[0, t]$, we obtain
\begin{equation*}
\begin{aligned}
&\|\sqrt{\r} \u(t)\|_{L^2_l(\Omega)}^2
+\es \int_0^t\|\p_x \u\|_{L^2_l(\Omega)}^2 d\tau
+\mu \int_0^t\|\p_y \u\|_{L^2_l(\Omega)}^2 d\tau\\
&+\int_0^t \int_\O \y \h\cdot ( (\h+1) \p_x \u + \g \p_y \u )dxdyd\tau\\
&\le \|\sqrt{\r_0} \u_0\|_{L^2_l(\Omega)}^2+C\int_0^t\|r_u\|_{L^2_l(\Omega)}^2d\t
+C_{\mu, l}(1+Q(t))\int_0^t (1+\|\u\|_{\H^1_l}^2+\|\h\|_{L^2_l(\Omega)}^2) d\tau.
\end{aligned}
\end{equation*}
Similarly, based on the equations \eqref{eq5}$_3$ and \eqref{eq5}$_1$, it follows directly
\begin{equation*}
\begin{aligned}
&\|\h(t)\|_{L^2_l(\Omega)}^2
+\es \int_0^t \|\p_x \h\|_{L^2_l(\O)}^2 d\tau
+\k \int_0^t \|\p_y \h\|_{L^2_l(\O)}^2 d\tau\\
&-\int_0^t \int_\O \y \h \cdot [(\h+1) \p_x \u + \g \p_y \u ]dxdyd\tau\\
&\le \|\h_0\|_{L^2_l(\Omega)}^2+C\int_0^t \|r_h\|_{L^2_l(\O)}^2d\t
+C_{\k, l}(1+Q(t))\int_0^t (1+\|\h\|_{\H^1_l}^2) d\tau.
\end{aligned}
\end{equation*}
and
\begin{equation*}
\|\vr\|_{L^2_l(\Omega)}^2
\!+\es \!\int_0^t\! \|(\p_x \vr, \p_y \vr)\|_{L^2_l(\Omega)}^2 d\tau
\le \|\vr_0\|_{L^2_l(\Omega)}^2+C\!\int_0^t \!\|(r_1, r_2)\|_{L^2_l(\O)}^2d\t
\!+C(1+Q(t))\!\int_0^t\! \|\vr\|_{L^2_l(\Omega)}^2 d\tau.
\end{equation*}
Therefore, we collect above estimates to complete the proof of Lemma \ref{lemma31}.
\end{proof}
Now, we establish the following estiamte:
\begin{lemma}\label{lemma32}
For smooth solution $(\vr, \u, \v, \h,\g)$ of the equations \eqref{eq5}-\eqref{bc5}, then it holds on
\begin{equation}\label{321}
\begin{aligned}
&\sup_{0\le \t \le t}\e_{m,l}(\t)
+\sum_{\substack{|\alpha| \le m \\ |\a_1| \le m-1}}
\es \int_0^t \|\p_x \z^\a(\vr, \u, \h)\|_{L^2_l(\O)}^2 d\tau\\
& + \sum_{\substack{|\alpha| \le m \\ |\a_1| \le m-1}}
\int_0^t \|(\sqrt{\es} \p_y \z^\a \vr, \sqrt{\mu} \p_y \z^\a \u, \sqrt{\k} \p_y \z^\a\h)\|_{L^2_l(\O)}^2 d\tau\\
\le
& C\|(\vr_0, \u_0, \h_0)\|_{\H^m_l}^2
+C\int_0^t \|(r_1, r_2, r_u, r_h)\|_{\H^m_l}^2 d\t\\
&+C_{\mu, \k, m, l}(1+Q^2(t))\int_0^t (1+\|(\vr, \u, \h)\|_{\H^{m}_l}^2+\|\p_y(\vr, \u, \h)\|_{\H^{m-1}_l}^2)d\tau.
\end{aligned}
\end{equation}
\end{lemma}
\begin{proof}
The case $m=0$ has been proved in the Lemma \ref{lemma31}. Then, we give the proof for the case $m \ge 1$.
Step 1: Applying the differential operator $\z^\alpha(1 \le |\a|\le m, |\a_{1}| \le m-1)$
to the equation \eqref{eq5}$_2$, we can obtain the evolution equation for $\z^\a u^\es$:
\begin{equation}\label{322}
\begin{aligned}
&\r \p_t \z^\a u^\es+\r(\u+1-e^{-y})\p_x \z^\a \u
+\r \v \p_y \z^\a \u-\es \p_x^2 \z^\a \u-\mu \z^\a \p_y^2 \u\\
=&(\h+1)\p_x \z^\a \h+\g \p_y \z^\a \h
+\z^\a(-\es \p_x r_u-\mu e^{-y})+C_{11}^\a+\C_{12}^\a+\C_{13}^\a+\C_{14}^\a+\C_{15}^\a+\C_{16}^\a.
\end{aligned}
\end{equation}
where $\C_{1i}^\a(i=1,...,6)$ are defined by
\begin{equation*}\label{323}
\begin{aligned}
&\C_{11}^\a=-[\z^\a, \r \p_t]\u,\quad
\C_{12}^\a=-[\z^\a, \r(\u+1-e^{-y})\p_x]\u, \quad
\C_{13}^\a=-[\z^\a, \r \v \p_y]\u,\\
&\C_{14}^\a=[\z^\a, (\h+1)\p_x]\h,\quad
\C_{15}^\a=[\z^\a, \g \p_y]\h,\quad
\C_{16}^\a=-\z^\a(\r \v e^{-y}).
\end{aligned}
\end{equation*}
Multiplying the equation \eqref{322} by $\ya \z^\a \u$, integrating over $\Omega$ and
applying the boundary condition \eqref{bc5}, we have
\begin{equation}\label{324}
\begin{aligned}
&\frac{d}{dt}\frac{1}{2}\int_\O \ya \r |\z^\a \u|^2 dxdy
+\es \int_\O \ya |\p_x \z^\a \u|^2 dxdy\\
=
&\frac{1}{2}\int_\O \ya |\z^\a \u|^2(\partial_t \rho^\es+(u^\es+1-e^{-y})\partial_x \rho^\es+v^\es\partial_y \rho^\es)dxdy\\
&+l\int_\O \yb \r \v |\z^\a \u|^2 dxdy+\mu \int_\O \z^\a \p_y^2 \u \cdot \ya \z^\a \u dxdy\\
&+\int_\O \{(\h+1)\p_x \z^\a \h+\g \p_y \z^\a \h\}\cdot \ya \z^\a \u dxdy\\
&+\int_\O \z^\a(-\es \p_x r_u-\mu e^{-y})\cdot \ya \z^\a \u dxdy
+\sum_{i=1}^6\int_\O \C_{1i}^\a \cdot \ya \z^\a \u dxdy.
\end{aligned}
\end{equation}
By routine checking, it follows that
\begin{equation}\label{325}
|\int_\O \ya |\z^\a \u|^2(\partial_t \rho^\es+(u^\es+1-e^{-y})\partial_x \rho^\es+v^\es\partial_y \rho^\es)dxdy|
\le C(1+Q(t))\|\z^\a \u\|_{L^2_{l }(\Omega)}^2,
\end{equation}
and
\begin{equation}\label{326}
|\int_\O \yb \r \v |\z^\a \u|^2 dxdy|
\le C\|\la y \ra^{-1} \v \|_{L^\infty_0(\Omega)}\|\z^\a \u\|_{L^2_{l }(\Omega)}^2.
\end{equation}
The integration by part with respect to $y$ variable yields directly
\begin{equation}\label{327}
\begin{aligned}
&\int_\O \z^\a \p_y^2 \u \cdot \ya \z^\a \u dxdy\\
=
&\int_\T \z^\a \p_y \u \cdot \z^\a \u|_{y=0} dx
-2l\int_\O \z^\a \p_y \u \cdot \yb \z^\a \u dxdy\\
&-\int_\O \z^\a \p_y \u \cdot \ya \p_y \z^\a \u dxdy
+\int_\O [\z^\a, \p_y] \p_y \u \cdot \ya \z^\a \u dxdy\\
=
&\int_\T \z^\a \p_y \u \cdot \z^\a \u|_{y=0} dx
-2l\int_\O \z^\a \p_y \u \cdot \yb \z^\a \u dxdy\\
&-\int_\O \ya |\p_y \z^\a \u|^2 dxdy
-\int_\O [\z^\a, \p_y] \u \cdot \ya \p_y \z^\a \u dxdy\\
&
+\int_\O [\z^\a, \p_y] \p_y \u \cdot \ya \z^\a \u dxdy.
\end{aligned}
\end{equation}
If $\a_2=0$, the boundary condition \eqref{bc5} implies $\z^\a \u|_{y=0}=0$.
If $\a_2\neq 0$, we apply the property of $\varphi$, which vanishes on the boundary,
to get $\z^\a \u|_{y=0}$, and hence
\begin{equation}\label{328}
\int_\T \z^\a \p_y \u \cdot \z^\a \u|_{y=0} dx=0.
\end{equation}
Next, we deal with term involving $[\z^\a, \p_y]$.
It is worth noting that the operator $Z_\tau=(\p_t, \p_x)$ communicates with $\p_y$,
we obtain $[\z^\a, \p_y]\u=0$ for $\a_2=0$.
By direct computation, we find for $\a_2 \neq 0$
\begin{equation*}\label{329}
[Z_2^{\a_2}, \p_y]\u
=-\sum_{1\le k \le \a_2}C_{\a_2, k}\p_y Z_2^{k-1} \varphi \cdot Z_2^{\a_2-k}\p_y \u.
\end{equation*}
This and the H\"{o}lder inequality yield directly
\begin{equation}\label{3210}
|\int [\z^\a, \p_y] \u \cdot \ya \p_y \z^\a \u dxdy|
\le
\frac{1}{4}\|\p_y \z^\a \u\|_{L^2_{l }(\Omega)}^2
+C_m \|\p_y \u\|_{\H^{m-1}_{l}}^2.
\end{equation}
Using the relation \eqref{329} and Cauchy-Schwartz inequality, we obtain
\begin{equation}\label{3211}
\begin{aligned}
&2l|\int_\O \z^\a \p_y \u \cdot \yb \z^\a \u dxdy|\\
=
& 2l|\int_\O ([\z^\a, \p_y]\u+\p_y \z^\a \u) \cdot \yb \z^\a \u dxdy|\\
\le
&\frac{1}{4}\|\p_y \z^\a \u\|_{L^2_{l}(\Omega)}^2
+C_{m,l}(\|[\z^\a, \p_y]\u\|_{L^2_{l}(\Omega)}^2
+\|\z^\a \u\|_{L^2_{l}(\Omega)}^2)\\
\le
&
\frac{1}{4}\|\p_y \z^\a \u\|_{L^2_{l}(\Omega)}^2
+C_{m,l}(\| \u\|_{\H^{m}_{l}}^2+\|\p_y \u\|_{\H^{m-1}_{l}}^2).
\end{aligned}
\end{equation}
Next, we deal with the term $\int_\O [\z^\a, \p_y] \p_y \u \cdot \ya \z^\a \u dxdy$.
For a smooth function $f$, it follows
\begin{equation}\label{3212}
[\z^\a, \p_y]f=\sum_{\b_2 \neq 0, \b_2+\ga_2=\a_2}
C_{\b_2, \ga_2}\varphi Z_2^{\b_2}(\frac{1}{\varphi})\p_y Z_2^{\ga_2}Z_\tau^{\a_1}f.
\end{equation}
By virtue of the definition of $\varphi$, it holds on by computating directly
\begin{equation}\label{3213}
|\varphi Z_2^{\b_2}(\frac{1}{\varphi})|\le C, \quad
|\p_y\{\varphi Z_2^{\b_2}(\frac{1}{\varphi})\}| \le C.
\end{equation}
For $\b_2 \neq 0$ and $\b_2+\ga_2=\a_2$, the integration by part yields immediately
\begin{equation}\label{3214}
\begin{aligned}
&\int_\O \varphi Z_2^{\b_2}(\frac{1}{\varphi})\p_y Z_2^{\ga_2}Z_\tau^{\a_1} \p_y \u \cdot \ya \z^\a \u dxdy\\
=
&
-\int_\O \varphi Z_2^{\b_2}(\frac{1}{\varphi}) Z_2^{\ga_2}Z_\tau^{\a_1} \p_y \u \cdot \ya \p_y \z^\a \u dxdy\\
&
-2l\int_\O \varphi Z_2^{\b_2}(\frac{1}{\varphi}) Z_2^{\ga_2}Z_\tau^{\a_1} \p_y \u \cdot \yb \z^\a \u dxdy\\
&
-\int_\O \p_y\{\varphi Z_2^{\b_2}(\frac{1}{\varphi})\} Z_2^{\ga_2}Z_\tau^{\a_1} \p_y \u \cdot \ya \z^\a \u dxdy,
\end{aligned}
\end{equation}
where the boundary term in the above equality vanishes since the quantity $\z^{\a} \u|_{y=0}=0$.
Then, applying the relation \eqref{3212}, estimate \eqref{3213}, H\"{o}lder and Cauchy inequalities, we obtain
\begin{equation*}
\begin{aligned}
\int_\O \varphi Z_2^{\b_2}(\frac{1}{\varphi})\p_y Z_2^{\ga_2}Z_\tau^{\a_1} \p_y \u \cdot \ya \z^\a \u dxdy
\le
\frac{1}{4}\int \ya |\p_y \z^\a \u|^2 dxdy+C_{m,l}(\|\u \|_{\H^{m}_l}^2+\|\p_y \u\|_{\H^{m-1}_l}^2),
\end{aligned}
\end{equation*}
and hence
\begin{equation}\label{3215}
\begin{aligned}
|\int_\O [\z^\a, \p_y] \p_y \u \cdot \ya \z^\a \u dxdy|
\le \frac{1}{4}\int \ya |\p_y \z^\a \u|^2 dxdy+
C_{m, l}(\| \u\|_{\H^{m}_{l}}^2+\|\p_y\u\|_{\H^{m-1}_{l-1}}^2).
\end{aligned}
\end{equation}
Plugging the estimates \eqref{328}, \eqref{3210},
\eqref{3211} and \eqref{3215} into \eqref{327}, we conclude
\begin{equation}\label{3216}
\mu\int_\O \z^\a \p_y^2 \u \cdot \ya \z^\a \u dxdy
\le -\frac{1}{2}\mu\|\p_y \z^\a \u\|_{L^2_{l}(\Omega)}^2
+C_{\mu, m, l}(\| \u\|_{\H^{m}_{l}}^2+\|\p_y \u\|_{\H^{m-1}_{l}}^2).
\end{equation}
Integrating by part, applying the boundary condition \eqref{bc5}
and divergence-free condition \eqref{eq5}$_4$, we find
\begin{equation}\label{3217}
\begin{aligned}
&\int_\O \{(\h+1)\p_x \z^\a \h+\g \p_y \z^\a \h\}\cdot \ya \z^\a \u dxdy\\
=
&-\int_\O \ya (\p_x \h+\p_y \g)\z^\a \u \cdot \z^\a \h dxdy\\
&-\int_\O ((\h+1)\p_x \z^\a \u+\g \p_y \z^\a \u)\cdot \ya \z^\a \h dxdy\\
&-2l\int_\O \yb \g \z^\a \u \cdot \z^\a \h dxdy\\
\le
&-\int_\O ((\h+1)\p_x \z^\a \u+\g \p_y \z^\a \u)\cdot \ya \z^\a \h dxdy\\
&+C_l \|\la y \ra^{-1}\g\|_{L^\infty_0(\O)}
\|\z^\a \u\|_{L^2_{l}(\Omega)}\|\z^\a \h\|_{L^2_{l}(\Omega)}.
\end{aligned}
\end{equation}
Using the H\"{o}lder and Cauchy inequalities, it follows
\begin{equation}\label{3218}
\begin{aligned}
&|\int_\O \z^\a(-\es \p_x r_u-\mu e^{-y})\cdot \ya \z^\a \u dxdy|\\
&\le \frac{1}{2}\es \|\p_x \z^\a \u\|_{L^2_l(\O)}^2
+C(1+\|\z^\a r_u\|_{L^2_{l}(\Omega)}^2+\|\z^\a \u\|_{L^2_{l}(\Omega)}^2).
\end{aligned}
\end{equation}
Substituting the estimates \eqref{325}, \eqref{326}, \eqref{327},
\eqref{3216}, \eqref{3217} and \eqref{3218} into the equality \eqref{324},
and integrating over $[0, t]$, we conclude
\begin{equation}\label{3219}
\begin{aligned}
&\int_\O \ya \r |\z^\a \u|^2 dxdy
+\int_0^t \int_\O \ya (\es |\p_x \z^\a \u|^2+\mu|\p_y \z^\a \u|^2) dxdy\\
&+\int_0^t \int_\O ((\h+1)\p_x \z^\a \u+\g \p_y \z^\a \u)\cdot \ya \z^\a \h dxdyd\tau\\
\le
& \int_\O \ya \r_0 |\z^\a \u_0|^2 dxdy
+C\int_0^t \|\z^{\a} r_u\|_{L^2_l(\Omega)}^2 d\t
+\sum_{i=1}^6\int_0^t \|\C_{1i}^\a\|_{L^2_l(\Omega)}^2d\tau\\
& +C_{\mu, m, l}(1+Q(t))\int_0^t (1+\|(\u, \h) \|_{\H^m_l}^2+\|\p_y \u \|_{\H^m_l}^2)d\tau.
\end{aligned}
\end{equation}
Now, we claim the following estimate, which will be shown later:
\begin{equation}\label{ulc1}
\sum_{i=1}^6 \int_0^t \|\C_{1i}^\a\|_{L^2_{l}(\Omega)}^2 d \tau
\le C_{m,l}(1+Q^2(t))\int_0^t (\|(\vr, \u, \h)\|_{\H^m_l}^2+\|(\p_y \u, \p_y \h)\|_{\H^{m-1}_l}^2) d\tau.
\end{equation}
For the moment we can substitute the estimate \eqref{ulc1} into inequality \eqref{3219}, and hence
\begin{equation}\label{3220}
\begin{aligned}
&\int_\O \ya \r |\z^\a \u|^2 dxdy
+\int_0^t \int_\O \ya (\es |\p_x \z^\a \u|^2+\mu|\p_y \z^\a \u|^2) dxdy\\
&+\int_0^t \int_\O ((\h+1)\p_x \z^\a \u+\g \p_y \z^\a \u)\cdot \ya \z^\a \h dxdyd\tau\\
\le
& \int_\O \ya \r_0 |\z^\a \u_0|^2 dxdy
+C\int_0^t \|\z^{\a} r_u\|_{L^2_l(\Omega)}^2 d\t\\
& +C_{\mu, m, l}(1+Q^2(t))\int_0^t (1+\|(\vr, \u, \h)\|_{\H^m_l}^2+\|(\p_y \u, \p_y \h)\|_{\H^{m-1}_l}^2) d\tau.
\end{aligned}
\end{equation}
Step 2: Applying operator $\z^\a$ to the first equation of \eqref{eq5},
multiplying by $\ya \z^\a \vr$ and integrating over $\Omega$, we find
\begin{equation}\label{3221}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\int_\O \ya |\z^\a \vr|^2 dxdy
+\es \int_\O \ya |\p_x \z^\a \vr|^2 dxdy\\
=
&\es\int_\O \z^\a \p_y^2 \vr \cdot \ya \z^\a \vr dxdy
+l \int_\O \yb \v |\z^\a \vr|^2 dxdy\\
&+\int_\O (\C_{21}^\a+\C_{22}^\a)\cdot \ya \z^\a \vr dxdy
-\es \int_\O \z^\a (\p_x r_1+\p_y r_2) \cdot \ya \z^\a \vr dxdy,
\end{aligned}
\end{equation}
where $\C_{21}^\a$ and $\C_{22}^\a$ are defined by
\begin{equation*}\label{3222}
\C_{21}^\a=-[\z^\a, (\u+1-e^{-y})\p_x]\vr, \quad
\C_{22}^\a=-[\z^\a, \v \p_y]\vr.
\end{equation*}
Similar to the equality \eqref{327}, the integration by part gives
\begin{equation}\label{3223}
\begin{aligned}
&\es\int_\O \z^\a \p_y^2 \vr \cdot \ya \z^\a \vr dxdy\\
=
&\es\int_{\mathbb{T}} \z^\a \p_y \vr \cdot \z^\a \vr|_{y=0} dx
-\es\int_\O \ya [\z^\a, \p_y] \vr \cdot \p_y \z^\a \vr dxdy\\
&-\es\int_\O \ya |\p_y \z^\a \vr|^2 dxdy
-2l \es\int_\O \yb \z^\a \p_y \vr \cdot \z^\a \vr dxdy\\
&
+\es\int_\O [\z^\a, \p_y] \p_y \vr \cdot \ya \z^\a \vr dxdy.
\end{aligned}
\end{equation}
If $\a_2=0$, the boundary condition $\p_y \vr|_{y=0}=0$
implies $\z^\a \p_y \vr|_{y=0}=0$. If $\a_2\neq 0$, we get from the
definition of $Z_2=\varphi(y)\p_y=\frac{y}{y+1}\p_y$ that $\z^\a \p_y \vr|_{y=0}=0$,
and hence
\begin{equation*}\label{3224}
\int_{\mathbb{T}} \z^\a \p_y \vr \cdot \z^\a \vr|_{y=0} dx=0.
\end{equation*}
The other terms on the right hand side of \eqref{3223}
can take the idea as estimate \eqref{3215}, we conclude
\begin{equation}\label{3225}
\es\int_\O \z^\a \p_y^2 \vr \cdot \ya \z^\a \vr dxdy
\le
-\frac{3}{4}\es\int_\O \ya |\p_y \z^\a \vr|^2 dxdy+C_{m, l}(\|\vr\|_{\H^m_l}^2+\|\p_y \vr\|_{\H^{m-1}_l}^2).
\end{equation}
By routine checking, it follows
\begin{equation}\label{3226}
\begin{aligned}
| \int_{\O} \yb \v |\z^\a \vr|^2 dxdy|
\le C \|\la y \ra^{-1} \v\|_{L^\infty_0(\O)}\|\z^\a \vr\|_{L^2_l(\Omega)}^2.
\end{aligned}
\end{equation}
Applying the integration by part and Cauchy inequality, we obtain
\begin{equation}\label{3226a}
\begin{aligned}
&|\es\int_{\O} \z^\a(\p_x r_1+\p_y r_2)\cdot \ya \z^\a \vr dxdy|\\
\le
&\frac{\es}{2}\|(\p_x \z^\a \vr, \p_y \z^\a \vr)\|_{L^2_l(\Omega)}^2
+C(\|\z^\a r_1\|_{L^2_l(\Omega)}^2+\|r_2\|_{\H^m_l}^2)+C_l\|\z^\a \vr\|_{L^2_l(\Omega)}^2.
\end{aligned}
\end{equation}
Substituting estimates \eqref{3225} and \eqref{3226} into \eqref{3221},
integrating the resulting inequality over $[0, t]$, we get
\begin{equation}\label{3227}
\begin{aligned}
&\int_\O \ya |\z^\a \vr|^2 dxdy
+\es \int_0^t \int_\O \ya (|\p_x \z^\a \vr|^2+|\p_y \z^\a \vr|^2 )dxdy d\tau\\
\le
&\int_\O \ya |\z^\a \vr_0|^2 dxdy
+C\int_0^t \|(r_1, r_2)\|_{\H^m_l}^2 d\t
+\int_0^t \|(\C_{21}^\a, \C_{22}^\a)\|_{L^2_l(\Omega)}^2d\tau\\
&+C_{m, l}(1+\|\la y \ra^{-1} \v\|_{L^\infty_0(\O)})
\int_0^t(\|\vr\|_{\H^m_l}^2+\|\p_y \vr\|_{\H^{m-1}_l}^2)d\tau.
\end{aligned}
\end{equation}
Similar to the claim estimates \eqref{ulc1}, we can justify the estimate
\begin{equation*}\label{3228}
|\int_0^t \|(\C_{21}^\a, \C_{22}^\a)\|_{L^2_l(\Omega)}^2d\tau|
\le C_{m, l}(1+Q(t))\int_0^t (\|(\vr, \u)\|_{\H^m_l}^2+\|\p_y \vr\|_{\H^{m-1}_l}^2)d\tau.
\end{equation*}
This and inequality \eqref{3227} yield directly
\begin{equation}\label{3229}
\begin{aligned}
&\int_\O \ya |\z^\a \vr|^2 dxdy
+\es \int_0^t \int_\O \ya (|\p_x \z^\a \vr|^2+|\p_y \z^\a \vr|^2 )dxdy d\tau\\
\le
&\!\int_\O \!\ya |\z^\a \vr_0|^2 dxdy
+C\!\int_0^t\!\! \| (r_1, r_2)\|_{\H^m_l}^2 d\t
+\!C_{m, l}(1+Q(t))\!\int_0^t\!\! (\|(\vr, \u)\|_{\H^m_l}^2+\|\p_y \vr\|_{\H^{m-1}_l}^2)d\tau.
\end{aligned}
\end{equation}
Step 3: Applying operator $\z^\a$ to the third equation of \eqref{eq5}
multiplying by $\ya \z^\a \h$ and integrating over $\Omega$, it follows
\begin{equation*}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\int_\O \ya |\z^\a \h|^2 dxdy
+\es \int \ya |\p_x \z^\a \h|^2 dxdy
-\k \int_\O \z^\a \p_y^2 \h \cdot \ya \z^\a \h dxdy\\
=
&l \int_\O \yb \v |\z^\a \h|^2 dxdy
+\int_\O \{(\h+1)\p_x \z^\a \u+\g \p_y \z^\a \u\} \cdot \ya \z^\a \h dxdy\\
&+\int_\O \z^\a(\g e^{-y}-\es \p_x r_h)\cdot \ya \z^\a \h dxdy
+\int_\O (\C_{31}^\a+\C_{32}^\a+\C_{33}^\a+\C_{34}^\a)\cdot \ya \z^\a \h dxdy,
\end{aligned}
\end{equation*}
where $\C_{3i}^\a(i=1,2,3,4)$ are defined by
$$
\begin{aligned}
&\C_{31}^\a=-[\z^\a, (\u+1-e^{-y})\p_x]\h, \quad
\C_{32}^\a=-[\z^\a, \v \p_y]\h,\\
&\C_{33}^\a=[\z^\a, (\h+1)\p_x]\u,\quad
\C_{34}^\a=[\z^\a, \g \p_y](\u-e^{-y}).
\end{aligned}
$$
Following the idea as the estimate \eqref{3229}, we can verify the following estimate
\begin{equation*}\label{3230}
\begin{aligned}
&\int_\O \ya |\z^\a \h|^2 dxdy
+\int_0^t \int_\O \ya (\es|\p_x \z^\a \h|^2+\k |\p_y \z^\a \h|^2) dxdyd\tau\\
&-\int_0^t \int_\O \{(\h+1)\p_x \z^\a \u+\g \p_y \z^\a \u\}
\cdot \ya \z^\a \h dxdyd\tau\\
\le
&\!\int_\O \!\ya |\z^\a \h_0|^2 dxdy+C\!\!\int_0^t\! \!\|\z^\a r_h\|_{L^2_l(\Omega)}^2 d\t
+C_{\k, m, l}(1+Q(t))\!\!\int_0^t \!(\|(\u, \h)\|_{\H^{m}_l}^2+\|(\p_y \u, \p_y \h)\|_{\H^{m}_l}^2)d\tau,
\end{aligned}
\end{equation*}
which, together with the estimates \eqref{3220} and \eqref{3229} completes the proof of lemma
after taking the summation over all $|\a|\le m$ and $|\a_1|\le m-1$ .
\end{proof}
\underline{\textbf{Proof of claim estimate \eqref{ulc1}}}
Now we give the estimate for $\int_0^t \|\C_{1i}^\a\|_{L^2_l(\Omega)}^2d\tau(i=1,...,6)$.
By virtue of the Moser type inequality \eqref{ineq-moser}, we find
\begin{equation*}
\begin{aligned}
\int_0^t \|\C_{11}^\a\|_{L^2_{l}(\Omega)}^2 d \tau
&\le C_m \sum_{|\b|\ge 1, \b +\ga =\a} \int_0^t \|\z^\b \vr \cdot \z^{\ga} \p_t \u \|_{L^2_l(\Omega)}^2 d\tau\\
&\le C_m\|\z^{E_i} \vr \|_{L^\infty_0(\O)}^2 \int_0^t \|\p_t \u \|_{\H^{m-1}_{l}}^2 d \tau
+ C_m\|\p_t \u\|_{L^\infty_0(\O)}^2 \int_0^t \|\z^{E_i} \vr\|_{\H^{m-1}_{l}}^2 d \tau\\
&\le C_m\|(\z^{E_i}\vr, \p_t \u)\|_{L^\infty_0(\O)}^2 \int_0^t \|(\vr, \u)\|_{\H^{m}_{l}}^2 d \tau.
\end{aligned}
\end{equation*}
Similarly, we conclude the following estimate
\begin{equation*}
\int_0^t \|\C_{12}^\a\|_{L^2_{l}(\Omega)}^2 d \tau
\le C_m(1+\|(\u, \z^{E_i} \vr, \z^{E_i} \u)\|_{L^\infty_0(\O)}^4)\int_0^t(1+\|(\vr, \u)\|_{\H^m_l}^2)d\tau,
\end{equation*}
and
\begin{equation*}
\int_0^t \|\C_{14}^\a\|_{L^2_{l}(\Omega)}^2 d \tau
\le C_m \|\z^{E_i} \h\|_{L^\infty_0(\O)}^2 \int_0^t \|\h\|_{\H^m_l}^2 d\tau.
\end{equation*}
Finally, we deal with the term $\int_0^t \|\C_{13}^\a\|_{L^2_{l}(\Omega)}^2 d \tau$.
By direct computation, it is easy to check that
\begin{equation*}
\C_{13}^\a=\underset{|\b|\ge1,\ \b+\ga=\a}{\sum}C_{\b,\ga}
\z^\b(\r \v)\z^\ga \p_y \u+\r \v [\z^\a, \p_y]\u.
\end{equation*}
Using the estimate \eqref{low-bou}, we get
\begin{equation}\label{u3}
\int_0^t \|\r \v [\z^\a, \p_y]\u\|_{L^2_l(\Omega)}^2 d\tau
\le C_m\| \v\|_{L^\infty_0(\O)}^2 \int_0^t \|\p_y \u\|_{\H^{m-1}_l}^2 d\tau.
\end{equation}
In order to control the velocity $\v$, the idea is to apply the Hardy inequality
and divergence-free condition to transform into the velocity $\p_x \u$ in some weighted Sobolev norm.
By virtue of $|\a_1|\le m-1$ and $|\a_1|+\a_2=m$, it follows $\a_2 \ge 1$, and hence
we can get $\ga_2 \ge 1$ if $\b_2=0$. Thus it follows from the Moser type inequality \eqref{ineq-moser} that
\begin{equation}\label{u1}
\begin{aligned}
&\underset{|\b|\ge1,\ \b+\ga=\a}{\sum}\int_0^t
\|\z^\b(\r \v)\z^\ga \p_y \u\|_{L^2_l(\Omega)}^2 d\tau\\
\le
& C_m\|Z_\tau^{e_i} (\r \v)\|_{L^\infty_1(\O)}^2 \int_0^t \|Z_2 \p_y \u\|_{\H^{m-2}_{l-1}}^2 d\tau
+ C_m\|Z_2 \p_y \u\|_{L^\infty_1(\O)}^2 \int_0^t \| Z_\tau^{e_i} (\r \v)\|_{\H^{m-2}_{l-1}}^2 d\tau.
\end{aligned}
\end{equation}
Using the divergence-free condition, Hardy and Moser type inequalities \eqref{ineq-moser}, we get
\begin{equation*}
\begin{aligned}
\int_0^t \| Z_\tau^{e_i} (\r \v)\|_{\H^{m-2}_{l-1}}^2 d\tau
\le
&C_m\|\r\|_{L^\infty_0(\O)}^2 \int_0^t \| \v\|_{\H^{m-1}_{l-1}}^2 d\tau
+C_m\|\v\|_{L^\infty_0(\O)}^2 \int_0^t \| \vr \|_{\H^{m-1}_{l-1}}^2 d\tau\\
\le
&C_{m,l}\int_0^t \| \p_y \v\|_{\H^{m-1}_{l}}^2 d\tau
+C_m\|\v\|_{L^\infty_0(\O)}^2 \int_0^t \| \vr \|_{\H^{m-1}_{l-1}}^2 d\tau\\
\le
&C_{m,l}(1+\|\v\|_{L^\infty_0(\O)}^2)\int_0^t \|(\vr, \u)\|_{\H^{m}_{l}}^2 d\tau.
\end{aligned}
\end{equation*}
This and the inequality \eqref{u1} yield directly
\begin{equation}\label{u4}
\begin{aligned}
\underset{|\b|\ge1,\ \b+\ga=\a}{\sum}\int_0^t
\|\z^\b(\r \v)\z^\ga \p_y \u\|_{L^2_l(\Omega)}^2 d\t
\le C_{m,l}(1+Q^2(t))\int_0^t (\|(\vr, \u)\|_{\H^{m}_{l}}^2+\|\p_y \u\|_{\H^{m-1}_{l}}^2) d\t.
\end{aligned}
\end{equation}
If $\b_2 \ge 1$, we get after using the Moser type inequality \eqref{ineq-moser} that
\begin{equation}\label{u2}
\begin{aligned}
&\underset{|\b|\ge1,\ \b+\ga=\a}{\sum}\int_0^t
\|\z^\b(\r \v)\z^\ga \p_y \u\|_{L^2_l(\Omega)}^2 d\t\\
\le
&C_m \|Z_2 (\r \v)\|_{L^\infty_1(\O)}^2 \int_0^t \| \p_y \u\|_{\H^{m-1}_{l-1}}^2 d\t
+C_m \|\p_y \u\|_{L^\infty_1(\O)}^2 \int_0^t \| Z_2 (\r \v)\|_{\H^{m-1}_{l-1}}^2 d\t.
\end{aligned}
\end{equation}
In view of the fact $Z_2 (\r \v)=Z_2 \r \v+\r Z_2 \v$, we apply divergence-free condition,
Hardy and Moser type inequalities to get
\begin{equation*}
\begin{aligned}
\int_0^t \| Z_2 (\r \v)\|_{\H^{m-1}_{l-1}}^2 d\tau
\le
&C_m\|(Z_2 \r , \v)\|_{L^\infty_0(\O)}^2
\int_0^t (\| \p_y \v\|_{\H^{m-1}_{l}}^2+\| \vr \|_{\H^{m}_{l-1}}^2) d\tau\\
&+C_m(1+\|Z_2 \v\|_{L^\infty_0(\O)}^2)
\int_0^t (\| \u \|_{\H^{m}_{l-1}}^2+ \| \vr \|_{\H^{m-1}_{l-1}}^2)d\tau\\
\le
&C_{m,l}(1+\|(\v, \p_x \u, Z_2 \r)\|_{L^\infty_0(\O)}^2)\int_0^t \|(\vr, \u)\|_{\H^{m}_{l}}^2 d\tau.
\end{aligned}
\end{equation*}
which, along with \eqref{u2}, gives directly
\begin{equation}\label{u5}
\underset{|\b|\ge1,\ \b+\ga=\a}{\sum}\int_0^t
\|\z^\b(\r \v)\z^\ga \p_y \u\|_{L^2_l(\Omega)}^2 d\tau
\le C_{m,l}(1+Q^2(t))\int_0^t (\|(\vr, \u)\|_{\H^{m}_{l}}^2+\|\p_y \u\|_{\H^{m-1}_{l}}^2) d\tau.
\end{equation}
The combination of the estimates \eqref{u3}, \eqref{u4} and \eqref{u5} yields directly
\begin{equation*}
\int_0^t \|\C_{13}^\a\|_{L^2_{l}(\Omega)}^2 d \tau
\le C_{m,l}(1+Q^2(t))\int_0^t (\|(\vr, \u)\|_{\H^m_l}^2+\|\p_y \u\|_{\H^{m-1}_l}^2)d\tau.
\end{equation*}
Similarly, by routine checking, we may conclude that
\begin{equation*}
\int_0^t (\|\C_{14}^\a\|_{L^2_{l}(\Omega)}^2+\|\C_{15}^\a\|_{L^2_{l}(\Omega)}^2+ \|\C_{16}^\a\|_{L^2_{l}(\Omega)}^2) d \tau
\le C_{m,l}(1+Q(t))\int_0^t (\|(\vr, \u, \h)\|_{\H^m_l}^2+\|\p_y \h\|_{\H^{m-1}_l}^2)d\tau.
\end{equation*}
Therefore, we complete the proof of the claim estimate \eqref{ulc1}.
\subsection{Weighted $\H^m_l-$Estimates only on Tangential Derivative}
In this subsection, we hope to establish the estimate for the quantity
$Z_\t^{\a_1}(\vr, \u, \h)$ with $|\a_1|=m$.
However, there is an essential difficulty to achieve this goal since
the terms $\v \p_y \vr$, $\r \v \p_y(\u-e^{-y})-\g \p_y \h$ and
$\v \p_y \h-\g \p_y (\u-e^{-y})$ will create the loss of one derivative in the tangential variable $x$.
In other words, $\v=-\p_y^{-1}\p_x u$ and $\g=-\p_y^{-1}\p_x \h$, by the divergence-free condition \eqref{eq5}$_4$,
create a loss of $x-$derivative that prevents us to apply the standard energy estimates.
To overcome this essential difficulty, we take the strategy of the recent interesting result
\cite{Liu-Xie-Yang} that only needs that the background tangential magnetic field
has a lower positive bound instead of monotonicity assumption on the tangential velocity.
However, due to the density being a unknown function instead of a constant, we need to take some
new ideas to deal with the terms $\v \p_y \vr$ and $\r \v \p_y(\u-e^{-y})$.
First of all, applying $Z_\t^{\a_1}(|\a_1|= m)$ differential operator to the equation \eqref{eq5}$_3$,
we find
\begin{equation}\label{3b1}
\begin{aligned}
&\{\p_t+(\u+1-e^{-y})\p_x +\v \p_y-\es \p_x^2 -\k \p_y^2\}(Z_\t^{\a_1} \h)
+Z_\t^{\a_1} \v \p_y \h\\
&=(\h+1)\p_x Z_\t^{\a_1} \u+\g \p_y Z_\t^{\a_1} \u
+Z_\t^{\a_1} \g \p_y(\u-e^{-y})-\es Z_\t^{\a_1} \p_x r_h+f_h,
\end{aligned}
\end{equation}
where the function $f_h$ is defined by
\begin{equation*}\label{3b2}
\begin{aligned}
f_h=
&-[Z_\t^{\a_1}, (\u+1-e^{-y})\p_x]\h
+[Z_\t^{\a_1}, (\h+1)\p_x]\u\\
&-\sum_{\substack{\b_1+\ga_1=\a_1 \\ \b_1 \neq 0, \b_1 \neq \a_1}}
C_{\b_1, \ga_1}Z_\t^{\b_1} \v Z_\t^{\ga_1} \p_y \h
+\sum_{\substack{\b_1+\ga_1=\a_1 \\ \b_1 \neq 0, \b_1 \neq \a_1}}
C_{\b_1, \ga_1} Z_\t^{\b_1} \g Z_\t^{\ga_1} \p_y \u.
\end{aligned}
\end{equation*}
To eliminate to difficult term $Z_\t^{\a_1} \v \p_y \h$, following the idea as in \cite{Liu-Xie-Yang},
we introduce the stream function $\ps$ satisfying
\begin{equation*}\label{3b3}
\p_y \ps=\h, \quad \p_x \ps = -\g, \quad \ps|_{y=0}=0.
\end{equation*}
Then, we can deduce from the equation \eqref{eq5}$_3$
and boundary condition \eqref{bc5} that
\begin{equation*}\label{3b4}
\p_t \ps+(\u+1-e^{-y})\p_x \ps+\v (\p_y \ps+1)
-\es \p_x^2 \ps-\k \p_y^2 \ps=-\es \p_y^{-1}\p_x r_h.
\end{equation*}
Applying $Z_\t^{\a_1}(|\a_1|= m)$ differential operator to above equation, it follows
\begin{equation}\label{3b5}
\{\p_t+(\u+1-e^{-y})\p_x+\v \p_y -\es \p_x^2 -\k \p_y^2 \}Z_\t^{\a_1}\ps
+Z_\t^{\a_1} \v(\h+1)=-\es \p_y^{-1} Z_\t^{\a_1}\p_x r_h+f_{\psi},
\end{equation}
where the function $f_{\psi}$ is defined by
\begin{equation*}\label{3b6}
f_{\psi}=
-[Z_\t^{\a_1}, (\u+1-e^{-y})\p_x] \ps
-\sum_{\substack{\b_1+\ga_1=\a_1 \\ \b_1 \neq 0, \b_1 \neq \a_1}}
C_{\b_1, \ga_1} Z_\t^{\b_1} \v Z_\t^{\ga_1} \p_y \ps.
\end{equation*}
Set $\eta_h :=\frac{\p_y \h}{\h+1}$ and define the quantity
\begin{equation}\label{equi-h}
\h_m :=Z_\t^{\a_1} \h-\eta_h Z_\t^{\a_1} \ps,
\end{equation}
then multiplying the equation \eqref{3b5} by $\eta_h$ and substituting the equation \eqref{3b1},
the difficult term $Z_\t^{\a_1} \v \p_y \h$ in \eqref{3b1} can be eliminated.
Hence, we get the evolution equation for the quantity $\h_m$ as
\begin{equation}\label{eqhm}
\begin{aligned}
&\p_t \h_m+(\u+1-e^{-y})\p_x \h_m +\v \p_y \h_m-\es \p_x^2 \h_m-\k \p_y^2 \h_m\\
&-(\h+1)\p_x Z_\t^{\a_1}\u-\g \p_y Z_\t^{\a_1} \u-Z_\t^{\a_1} \g \p_y(\u-e^{-y})\\
=
& -\es Z_\t^{\a_1} \p_x r_h+\es \eta_h \p_y^{-1} Z_\t^{\a_1} \p_x r_h
+f_h-\eta_h f_\psi+2 \es \p_x \eta_h \p_x Z_\t^{\a_1} \ps
+2\k\p_y \eta_h \p_y Z_\t^{\a_1} \ps\\
&+Z_\t^{\a_1} \ps (\p_t+(\u+1-e^{-y})\p_x+\v \p_y-\es \p_x^2 -\k \p_y^2)\eta_h.
\end{aligned}
\end{equation}
Similarly, after applying $Z_\t^{\a_1}(|\a_1|= m)$ operator to the first equation of \eqref{eq5}, we get
\begin{equation}\label{3b7}
\{\p_t+(\u+1-e^{-y})\p_x +\v \p_y-\es \p_x^2 -\es \p_y^2\}(Z_\t^{\a_1} \vr)
+Z_\t^{\a_1} \v \p_y \vr
=-\es Z_\t^{\a_1}\p_x r_1-\es Z_\t^{\a_1}\p_y r_2+f_\rho,
\end{equation}
where the function $f_\rho$ is defined by
\begin{equation*}\label{3b8}
\begin{aligned}
f_\rho=
-[Z_\t^{\a_1}, (\u+1-e^{-y})\p_x]\vr
-\sum_{\substack{\b_1+\ga_1=\a_1 \\ \b_1 \neq 0, \b_1 \neq \a_1}}
C_{\b_1, \ga_1}Z_\t^{\b_1} \v Z_\t^{\ga_1} \p_y \vr.
\end{aligned}
\end{equation*}
Set $\eta_\rho=\frac{\p_y \vr}{\h+1}$ and define
\begin{equation}\label{equi-r}
\vr_m:=Z_\t^{\a_1} \vr-\eta_\rho Z_\t^{\a_1} \ps,
\end{equation}
we multiply the equation \eqref{3b5}
by $\eta_\rho$ and substitute to the equation \eqref{3b7},
and hence, the evolution for the quantity $\vr_m$ as follows
\begin{equation}\label{eqrm}
\begin{aligned}
&\p_t \vr_m+(\u+1-e^{-y})\p_x \vr_m +\v \p_y \vr_m-\es \p_x^2 \vr_m-\es \p_y^2 \vr_m\\
=
&f_\rho-\eta_\rho f_\psi+2 \es \p_x \eta_\rho \p_x Z_\t^{\a_1} \ps
-\k \eta_\rho \p_y^2 Z_\t^{\a_1} \ps+\es \p_y^2 (\eta_\rho Z_\t^{\a_1} \ps)\\
&+Z_\t^{\a_1} \ps (\p_t+(\u+1-e^{-y})\p_x+\v \p_y-\es \p_x^2 )\eta_\rho
-\es Z_\t^{\a_1} (\p_x r_1+\p_y r_2)+\es \eta_\rho \p_y^{-1} Z_\t^{\a_1} \p_x r_h.
\end{aligned}
\end{equation}
Finally, applying $Z_\t^{\a_1}(|\a_1|= m)$ differential operator to the equation \eqref{eq5}$_2$, we get
\begin{equation}\label{3b9}
\begin{aligned}
&\{\r \p_t+\r (\u+1-e^{-y})\p_x +\r \v \p_y-\es \p_x^2 -\mu \p_y^2\}(Z_\t^{\a_1} \u)
+\r Z_\t^{\a_1} \v \p_y (\u-e^{-y})\\
&=(\h+1)\p_x Z_\t^{\a_1} \h+\g \p_y Z_\t^{\a_1} \h
+Z_\t^{\a_1} \g \p_y \h-\es Z_\t^{\a_1}\p_x r_u+f_u,
\end{aligned}
\end{equation}
where the function $f_u$ is defined by
\begin{equation*}\label{3b10}
\begin{aligned}
f_u=
&-[Z_\t^{\a_1}, \r \p_t]\u-[Z_\t^{\a_1}, \r(\u+1-e^{-y})\p_x]\u+[Z_\t^{\a_1}, (\h+1)\p_x]\h\\
&-\sum_{\substack{\b_1+\ga_1=\a_1 \\ \b_1 \neq 0 }}
C_{\b_1, \ga_1}Z_\t^{\b_1} \r Z_\t^{\ga_1} \v \p_y (\u-e^{-y})
-\sum_{\substack{\b_1+\ga_1=\a_1 \\ \b_1 \neq 0, \b_1 \neq \a_1}}
C_{\b_1, \ga_1}Z_\t^{\b_1} (\r \v) Z_\t^{\ga_1} \p_y \u\\
&+\sum_{\substack{\b_1+\ga_1=\a_1 \\ \b_1 \neq 0, \b_1 \neq \a_1}}
C_{\b_1, \ga_1} Z_\t^{\b_1} \g Z_\t^{\ga_1} \p_y \h.
\end{aligned}
\end{equation*}
Set $\eta_u=\frac{\p_y(\u-e^{-y})}{\h+1}$, multiplying the equation \eqref{3b5}
by $\r \eta_u$ and substituting to the equation \eqref{3b9}, we find for the quantity
\begin{equation}\label{eqi-u}
\u_m:=Z_\t^{\a_1} \u-\eta_u Z_\t^{\a_1} \ps
\end{equation}
satisfying the evolution as follows
\begin{equation}\label{equm}
\begin{aligned}
&\r \p_t \u_m+\r(\u+1-e^{-y})\p_x \u_m +\r \v \p_y \u_m-\es \p_x^2 \u_m-\mu \p_y^2 \u_m\\
&-(\h+1)\p_x Z_\t^{\a_1}\h-\g \p_y Z_\t^{\a_1} \h-Z_\t^{\a_1} \g \p_y \h\\
=\
&f_u-\r \eta_u f_\psi-Z_\t^{\a_1} \ps (\r \p_t+\r(\u+1-e^{-y})\p_x+\r\v \p_y)\eta_u\\
&-\es(\r-1)\eta_u \p_x^2 Z_\t^{\a_1}\ps-\k \r \eta_u \p_y^2 Z_\t^{\a_1} \ps
+2\es \p_x \eta_u \p_x Z_\t^{\a_1} \ps\\
&+\es \p_x^2 \eta_u Z_\t^{\a_1} \ps+\mu \p_y^2( \eta_u Z_\t^{\a_1} \ps )
-\es Z_\t^{\a_1} \p_x r_u+\es \rho^\es \eta_u \p_y^{-1} Z_\t^{\a_1} \p_x r_h.
\end{aligned}
\end{equation}
Let us define the functional:
\begin{equation}\label{ydef}
X_{m,l}(t):=1+\e_{m,l}(t)+\|(\vr_m,\u_m, \h_m)(t)\|_{L^2_l(\O)}^2+\|\p_y(\vr, \u, \h)(t)\|_{\H^{m-1}_l}^2
+\| \p_y \vr (t)\|_{\H^{1,\infty}_1}^2,
\end{equation}
where $\e_{m,l}(\t)$ is defined by \eqref{eml}.
Then, we will establish the following estimate in this subsection.
\begin{proposition}\label{Tanential-estimate}
Let $(\vr, \u, \v, \h, \g)$ be sufficiently smooth solution, defined on $[0, T^\es]$,
to the equations \eqref{eq5}-\eqref{bc5}.
Under the assumptions of conditions \eqref{a2} and \eqref{a1}, it holds on
\begin{equation*}
\begin{aligned}
&\sup_{\t \in [0, t]}\|(\vr_m,\u_m, \h_m)(\t)\|_{L^2_l(\O)}^2
+\es\int_0^t \|(\p_x \vr_m, \p_x \u_m, \p_x \h_m)(\t)\|_{L^2_l(\O)}^2 d\t\\
&+\int_0^t(\es\|\p_y \vr_m(\t)\|_{L^2_l(\O)}^2+\mu\|\p_y \u_m(\t)\|_{L^2_l(\O)}^2+\k\|\p_y \h_m(\t)\|_{L^2_l(\O)}^2) d\t\\
\le
&C\|(\vr_m, \u_m , \h_m)(0)\|_{L^2_l(\O)}^2
+C t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}^2
+C_{\mu, \k, m, l} \d^{-6}(1+Q^3(t))\int_0^t X_{m,l}(\t)d\t.
\end{aligned}
\end{equation*}
\end{proposition}
First of all, we establish the weighted $L^2-$estimate for the quantity $\vr_m$.
\begin{lemma}\label{lemma33}
Let $(\vr, \u, \v, \h, \g)$ be sufficiently smooth solution, defined on $[0, T^\es]$,
to the equations \eqref{eq5}.
Under the assumption of condition \eqref{a2}, then we have the following estimate for $0<\d_1<1$:
\begin{equation*}\label{331}
\begin{aligned}
&\sup_{\t \in [0, t]}\|\vr_m(\t)\|_{L^2_l(\O)}^2
+\es \int_0^t (\|\p_x \vr_m\|_{L^2_l(\O)}^2+\|\p_y \vr_m\|_{L^2_l(\O)}^2)d\tau\\
\le
&C\|\vr_{m}(0)\|_{L^2_l(\O)}^2
+C\!\int_0^t \|(r_1, r_2, r_h)\|_{\H^{m}_{l, tan}}^4 d\t
+C\d_1 \int_0^t ( \es \|\p_x \h_m\|_{L^2_l(\O)}^2+\!\k \|\p_y \h_m\|_{L^2_l(\O)}^2)d\t\\
&+C_{\k, m, l}\d^{-4}(1+Q^2(t))\int_0^t (\mathcal{E}_{m,l}(\t)
+\|(\vr_m, \u_m, \h_m)(\t)\|_{L^2_l(\O)}^2+\|\p_y \vr(\t)\|_{\H^{m-1}_l}^2)d\t.
\end{aligned}
\end{equation*}
\end{lemma}
\begin{proof}
Due to the fact $\p_y \vr|_{y=0}$ and $\ps|_{y=0}=0$, it follows $\p_y \eta_\rho|_{y=0}=0$.
Then, multiplying the equation \eqref{eqrm} by $\ya \vr_m$, integrating over $[0, t]\times \O$
and integrating by part, we get
\begin{equation}\label{333}
\begin{aligned}
&\frac{d}{dt}\int_\O \ya |\vr_m|^2 dxdy+\es \int_\O \ya |\p_x \vr_m|^2 dxdyd\t
+\es \int_\O \ya |\p_y \vr_m|^2 dxdyd\t \\
=
&l\int_\O \yb \v |\vr_m|^2 dxdy-2l\es \int_\O \ya \p_y \vr_m \cdot \vr_m dxdy
+2 \es \int_\O \p_x \eta_\rho \p_x Z_\t^{\a_1} \ps \cdot \ya \vr_m dxdy\\
&+\int_\O[-\k \eta_\rho \p_y^2 Z_\t^{\a_1} \ps+\es \p_y^2 (\eta_\rho Z_\t^{\a_1} \ps)
-\es \p_x^2 \eta_\rho Z_\t^{\a_1} \ps]\cdot \ya \vr_m dxdy \\
&+\int_\O Z_\t^{\a_1} \ps (\p_t+(\u+1-e^{-y})\p_x+\v \p_y )\eta_\rho
\cdot \ya \vr_m dxdy\\
&+\int_\O(f_\rho-\eta_\rho f_\psi-\es Z_\t^{\a_1} (\p_x r_1+\p_y r_2)+\es \eta_\rho \p_y^{-1} Z_\t^{\a_1} \p_x r_h)\cdot \ya \vr_m dxdy.
\end{aligned}
\end{equation}
Using the H\"{o}lder and Cauchy inequalities, it follows
\begin{equation*}\label{334}
\begin{aligned}
&|l\int_\O \yb \v |\vr_m|^2 dxdy-2l\es \int_\O \ya \p_y \vr_m \cdot \vr_m dxdy|\\
\le
&C_l \|\v\|_{L^\infty_{-1}(\O)}\|\vr_m\|_{L^2_l(\O)}^2+2l \es \|\p_y \vr_m\|_{L^2_l(\O)}\|\vr_m\|_{L^2_l(\O)}\\
\le
&\frac{1}{8} \es \|\p_y \vr_m\|_{L^2_l(\O)}^2+C_l(1+\|\v\|_{L^\infty_{-1}(\O)}^2)\|\vr_m\|_{L^2_l(\O)}^2,
\end{aligned}
\end{equation*}
and
\begin{equation}\label{335}
|\es \int_\O \p_x \eta_\rho \p_x Z_\t^{\a_1} \ps\cdot \ya \vr_m dxdy|
\le \es \|\p_x \eta_\rho \p_x Z_\t^{\a_1} \ps\|_{L^2_l(\O)}\|\vr_m\|_{L^2_l(\O)}.
\end{equation}
By virtue of the fact $\p_x \eta_\rho=\frac{1}{\h+1}\{{\p_{xy}^2 \vr} -\frac{\p_y \vr \p_x \h}{\h+1 }\}$, we get
\begin{equation*}\label{336}
\|\p_x \eta_\rho \p_x Z_\t^{\a_1} \ps\|_{L^2_l(\O)}
\le (\|\p_{xy}^2 \vr\|_{L^\infty_1(\O)}+\d^{-1}\| \p_{y} \vr\|_{L^\infty_1}\| \p_{x} \h\|_{L^\infty_0(\O)})
\|\frac{\p_x Z_\t^{\a_1} \ps}{\h+1}\|_{L^2_{l-1}(\O)},
\end{equation*}
where we have used the fact $\h+1\ge \d$.
This and inequalities \eqref{b13} and \eqref{335} give
\begin{equation*}\label{337}
|\es \int_\O \p_x \eta_\rho \p_x Z_\t^{\a_1} \ps\cdot \ya \vr_m dxdy|
\le \d_1 \es \|\p_x \h_m\|_{L^2_l(\O)}^2+C_l \d^{-4}(1+Q^2(t))\|(\vr_m, \h_m)\|_{L^2_l(\O)}^2.
\end{equation*}
Using the H\"{o}lder and Cauchy inequalities, it follows
\begin{equation*}
\begin{aligned}
|\k \int_\O \eta_\rho \p_y^2 Z_\t^{\a_1} \ps \cdot \ya \vr_m dxdy|
\le \d_1 \k \|\p_y Z_\t^{\a_1} \h\|_{L^2_l(\O)}^2+C_\k \|\eta_\rho\|_{L^\infty_0(\O)}^2 \|\vr_m\|_{L^2_l(\O)}^2,
\end{aligned}
\end{equation*}
which, together with the estimate \eqref{b14}, yields directly
\begin{equation*}\label{338}
|\k \int_\O \eta_\rho \p_y^2 Z_\t^{\a_1} \ps \cdot \ya \vr_m dxdy|
\le \d_1 \k \|\p_y \h_m\|_{L^2_l(\O)}^2+C_{\k, l} \d^{-2} Q(t)\|(\vr_m, \h_m)\|_{L^2_l(\O)}^2.
\end{equation*}
In view of $\p_y \vr|_{y=0}=0$ and $\ps|_{y=0}=0$, it is easy
to justify the fact $\p_y (\eta_\rho Z_\t^{\a_1} \ps)|_{y=0}=0$,
and hence, we integrating by part to get directly
\begin{equation*}\label{339}
\begin{aligned}
&\es \int_\O \p_y^2 (\eta_\rho Z_\t^{\a_1} \ps)\cdot \ya \vr_m dxdy\\
=
&-\es \int_\O \p_y \eta_\rho Z_\t^{\a_1} \ps \cdot (2l \yb \vr_m+\ya \p_y \vr_m) dxdy\\
&-\es \int_\O \eta_\rho Z_\t^{\a_1} \h \cdot(2l \yb \vr_m+\ya \p_y \vr_m) dxdy.
\end{aligned}
\end{equation*}
It follows from the H\"{o}lder inequality that
\begin{equation}\label{3310}
\begin{aligned}
&|\es \int_\O \p_y \eta_\rho Z_\t^{\a_1} \ps \cdot (2l \yb \vr_m+\ya \p_y \vr_m) dxdy|\\
\le
&\|\p_y \eta_\rho Z_\t^{\a_1} \ps\|_{L^2_{l-1}(\O)}\| \vr_m\|_{L^2_{l}(\O)}
+\|\p_y \eta_\rho Z_\t^{\a_1} \ps\|_{L^2_{l}(\O)}\|\p_y \vr_m\|_{L^2_{l}(\O)}.
\end{aligned}
\end{equation}
Thanks to the relation $\p_y \eta_\rho=\frac{1}{\h+1}\{\p_y^2 \vr-\frac{\p_y \vr \p_y \h}{\h+1}\}$, we get
\begin{equation}\label{3311}
\begin{aligned}
\|\p_y \eta_\rho Z_\t^{\a_1} \ps\|_{L^2_{l}(\O)}
\le
& (\|Z_2 \p_y \vr\|_{L^\infty_1(\O)}+\d^{-1}\|Z_2 \vr\|_{L^\infty_1(\O)}\| \p_y \h \|_{L^\infty_0(\O) })
\|\frac{1}{\varphi(y)} \frac{Z_\t^{\a_1} \ps}{\h+1}\|_{L^2_{l-1}(\O)} \\
\le
& C_l \d^{-2}(\|Z_2 \p_y \vr\|_{L^\infty_1(\O)}+\|Z_2 \vr\|_{L^\infty_1(\O)}\| \p_y \h \|_{L^\infty_0(\O)})
\|\h_m\|_{L^2_{l}(\O)},
\end{aligned}
\end{equation}
where, in the last inequality, we have used the following estimate
\begin{equation*}\label{3312}
\|\frac{1}{\varphi(y)} \frac{Z_\t^{\a_1} \ps}{\h+1}\|_{L^2_{l-1}(\O)}
\le C_l \d^{-1}\|\h_m\|_{L^2_l(\O)}.
\end{equation*}
Combining \eqref{3310} with \eqref{3311}, we conclude that
\begin{equation}\label{3313}
|\es \int_\O \p_y \eta_\rho Z_\t^{\a_1} \ps \cdot (2l \yb \vr_m+\ya \p_y \vr_m) dxdy|
\le \frac{1}{8} \es \|\p_y \vr_m\|_{L^2_l(\O)}^2
+C_l \d^{-4}Q^2(t) \|(\vr_m, \h_m)\|_{L^2_{l}(\O)}^2.
\end{equation}
Using the Cauchy inequality and the estimate \eqref{b12}, we show
\begin{equation*}\label{3314}
\begin{aligned}
&|\es \int_\O \eta_\rho Z_\t^{\a_1} \h \cdot(2l \yb \vr_m+\ya \p_y \vr_m) dxdy\\
\le
& \frac{1}{8} \es \|\p_y \vr_m\|_{L^2_l(\O)}^2
+C_l \d^{-4}(1+\|\p_y \vr \|_{L^\infty_0(\O)}^4
+\|\p_y \h \|_{L^\infty_1(\O)}^4)\|(\vr_m, \h_m)\|_{L^2_{l}(\O)}^2.
\end{aligned}
\end{equation*}
This and the inequality \eqref{3313} give
\begin{equation}\label{3315}
|\es \int_\O \p_y^2 (\eta_\rho Z_\t^{\a_1} \ps)\cdot \ya \vr_m dxdy|
\le
\frac{1}{4} \es \|\p_y \vr_m\|_{L^2_l(\O)}^2
+C_l \d^{-4}(1+Q^2(t))\|(\vr_m,\h_m)\|_{L^2_l(\O)}^2.
\end{equation}
The integration by part with respect to $x$ variable yields immediately
\begin{equation*}\label{3316}
\es \int_\O Z_\t^{\a_1} \ps \p_x^2 \eta_\rho \cdot \ya \vr_m dxdy
=
\es \int_\O \ya \p_x \eta_\rho (Z_\t^{\a_1} \ps \p_x \vr_m+\p_x Z_\t^{\a_1} \ps \vr_m)dxdy.
\end{equation*}
By virtue of the Holder inequality, we get
\begin{equation}\label{3317}
|\es \int_\O \ya \p_x \eta_\rho Z_\t^{\a_1} \ps \p_x \vr_m dxdy|
\le \es \|\p_x \eta_\rho Z_\t^{\a_1} \ps\|_{L^2_{l}(\O)}\|\p_x \vr_m\|_{L^2_{l}(\O)}.
\end{equation}
Due to the fact $\p_x \eta_\rho=\frac{1}{\h+1}\{\p_{xy}^2 \vr-\frac{\p_y \vr \p_x \h}{\h+1}\}$, it follows
\begin{equation*}\label{3318}
\begin{aligned}
\|\p_x \eta_\rho Z_\t^{\a_1} \ps\|_{L^2_{l}(\O)}
&\le(\|\p_{xy}^2 \vr\|_{L^\infty_1(\O)}+\d^{-1}\|\p_y \vr\|_{L^\infty_1(\O)}\|\p_x \h\|_{L^\infty_0(\O)})
\|\frac{Z_\t^{\a_1} \ps}{\h+1}\|_{L^2_{l-1}(\O)}\\
&\le C_l \d^{-2}(\|\p_{xy}^2 \vr\|_{L^\infty_1(\O)}+\|\p_y \vr\|_{L^\infty_1(\O)}\|\p_x \h\|_{L^\infty_0(\O)})
\|\h_m\|_{L^2_{l}(\O)},
\end{aligned}
\end{equation*}
where we have used the estimate \eqref{b11} in the last inequality.
This and inequality \eqref{3317} yield directly
\begin{equation}\label{3319}
\begin{aligned}
|\es \int_\O \ya \p_x \eta_\rho Z_\t^{\a_1} \ps \p_x \vr_m dxdy|
\le
\frac{1}{8}\es \|\p_x \vr_m\|_{L^2_{l}(\O)}^2+C_l \d^{-4}(1+Q^2(t))\| \h_m\|_{L^2_{l}(\O)}^2.
\end{aligned}
\end{equation}
Similarly, it is easy to justify that
\begin{equation*}\label{3319a}
|\es \int_\O \ya \p_x \eta_\rho \p_x Z_\t^{\a_1} \ps \vr_m dxdy|
\le \d_1 \es \|\p_x \h_m\|_{L^2_{l}(\O)}^2+C_l \d^{-4}(1+Q^2(t))\|(\vr_m, \h_m)\|_{L^2_{l}(\O)}^2.
\end{equation*}
which, along with \eqref{3319}, yields directly
\begin{equation}\label{3320}
\begin{aligned}
&|\es \int_\O Z_\t^{\a_1} \ps \p_x^2 \eta_\rho \cdot \ya \vr_m dxdy|\\
&\le
\es (\frac{1}{8}\|\p_x \vr_m\|_{L^2_l(\O)}^2+\d_1 \|\p_x \h_m\|_{L^2_l(\O)}^2)
+C_l \d^{-4}(1+Q^2(t))\|(\vr_m, \h_m)\|_{L^2_l(\O)}^2.
\end{aligned}
\end{equation}
Using the H\"{o}lder inequality and estimate \eqref{b11}, we get
\begin{equation*}\label{3321}
|\int_\O Z_\t^{\a_1} \ps (\p_t+(\u+1-e^{-y})\p_x+\v \p_y)\eta_\rho \cdot \ya \vr_m dxdy|
\le C_l \d^{-2}(1+Q(t)) \|(\vr_m, \h_m)\|_{L^2_l(\O)}^2.
\end{equation*}
The application of H\"{o}lder and Cauchy inequalities gives directly
\begin{equation*}\label{3322}
|\int_\O (f_\rho-\eta_\rho f_\psi) \cdot \ya \vr_m dxdy|
\le C(\|f_\rho\|_{L^2_l(\O)}^2+ \| f_\psi\|_{L^2_{l-1}(\O)}^2)
+C(1+\|\eta_\rho\|_{L^\infty_1(\O)}^2)\|\vr_m\|_{L^2_l(\O)}^2.
\end{equation*}
Integrating by part and applying the Cauchy inequality, it follows
\begin{equation*}
\begin{aligned}
&|\int_\O(-\es Z_\t^{\a_1} (\p_x r_1+\p_y r_2)+\es \eta_\rho \p_y^{-1} Z_\t^{\a_1} \p_x r_h)\cdot \ya \vr_m dxdy|\\
\le
&\frac{\es}{8}\|(\p_x \vr_m, \p_y \vr_m)\|_{L^2_l(\O)}^2
+\|Z_\t^{\a_1}(r_1, r_2, r_h)\|_{L^2_l(\O)}^4
+C_l\d^{-4}(1+Q^2(t))(1+\|\vr_m\|_{L^2_l(\O)}^2).
\end{aligned}
\end{equation*}
Combining the above estimates of terms for the righthand side of \eqref{333}, and
integrating the resulting inequality over $[0, t]$, we get
\begin{equation*}\label{3323}
\begin{aligned}
&\|\vr_m(t)\|_{L^2_l(\O)}^2+\frac{1}{2}\es \int_0^t (\|\p_x \vr_m\|_{L^2_l(\O)}^2+\|\p_y \vr_m\|_{L^2_l(\O)}^2)d\tau\\
\le
&\|\vr_m(0)\|_{L^2_l(\O)}^2+\d_1 \int_0^t(\es \|\p_x \h_m\|_{L^2_l(\O)}^2+\k \|\p_y \h_m\|_{L^2_l(\O)}^2)d\t
+\int_0^t \|Z_\t^{\a_1}(r_1, r_2, r_h)\|_{L^2_l(\O)}^4 d\d\\
&+C\int_0^t (\|f_\rho\|_{L^2_l(\O)}^2+\|f_\psi\|_{L^2_{l-1}(\O)}^2) d\t
+C_{\k,l}\d^{-4}(1+Q^2(t))\int_0^t(1+ \|(\vr_m, \h_m)\|_{L^2_l(\O)}^2)d\t.
\end{aligned}
\end{equation*}
On the other hand, we applying the Moser type inequality \eqref{ineq-moser} to get
\begin{equation*}\label{3324}
\int_0^t(\|f_\rho\|_{L^2_l(\O)}^2+\| f_\psi\|_{L^2_{l-1}(\O)}^2) d\tau
\le C_{m,l} Q(t)\int_0^t (\|(\vr, \u, \h)\|_{\H^{m}_l}^2+\|\p_y \vr\|_{\H^{m-1}_l}^2) d\tau,
\end{equation*}
which along with the estimate \eqref{b22} completes the proof of lemma.
\end{proof}
Next, we establish the estimate for the quantities $\u_m$ and $\h_m$.
Indeed, it is easy to check that
\begin{equation*}\label{3b11}
\begin{aligned}
&-(\h+1)\p_x Z_\t^{\a_1} \h-\g \p_y Z_\t^{\a_1} \h- Z_\t^{\a_1} \g \p_y \h\\
=
&-(\h+1)\p_x \h_m-\g \p_y \h_m-(\h+1)\p_x \eta_h Z_\t^{\a_1} \ps
-\g \p_y \eta_h Z_\t^{\a_1} \ps-\g \eta_h Z_\t^{\a_1} \h,
\end{aligned}
\end{equation*}
and
\begin{equation*}\label{3b12}
\begin{aligned}
&-(\h+1)\p_x Z_\t^{\a_1}\u-\g \p_y Z_\t^{\a_1} \u-Z_\t^{\a_1}\g \p_y(\u-e^{-y})\\
=
&-(\h+1)\p_x \u_m-\g \p_y \u_m-(\h+1)\p_x \eta_u Z_\t^{\a_1} \ps
-\g \p_y \eta_u Z_\t^{\a_1} \ps-\g \eta_u Z_\t^{\a_1} \h,
\end{aligned}
\end{equation*}
which were first observed in \cite{Liu-Xie-Yang}.
Then, we will have the following estimates:
\begin{lemma}\label{lemma34}
Let $(\vr, \u, \v, \h, \g)$ be sufficiently smooth solution, defined on $[0, T^\es]$,
to the equations \eqref{eq5}.
Under the assumption of conditions \eqref{a2} and \eqref{a1}, it holds on
\begin{equation*}\label{341}
\begin{aligned}
&\underset{0\le \t \le t}{\sup}\|(\u_m, \h_m)(\t)\|_{L^2_l(\O)}^2
+\int_0^t \|\sqrt{\es}\p_x (\u_m, \h_m)\|_{L^2_l(\O)}^2 d\t
+\int_0^t \|\p_y (\sqrt{\mu} \u_m, \sqrt{\k} \h_m)\|_{L^2_l(\O)}^2 d\t\\
\le
&C\|(\u_m, \h_m)(0)\|_{L^2_l(\O)}^2
+C\int_0^t \|(r_u, r_h)\|_{\H^{m}_{l, tan}}^4d\t
+C_{\mu, \k, m, l} \d^{-6}(1+Q^3(t))\int_0^t \mathcal{E}_{m,l}(\t) d\t.\\
& +C_{\mu, \k, m, l}
\d^{-6}(1+Q^3(t))\int_0^t (\|(\vr_m, \u_m, \h_m)\|_{L^2_l(\O)}^2+\|(\p_y \u, \p_y \h)\|_{\H^{m-1}_l}^2)d\t.
\end{aligned}
\end{equation*}
\end{lemma}
\begin{proof}
Multiplying the equation \eqref{equm} by $\ya \u_m$, integrating over $\O$ and
integrating by part, we find
\begin{equation}\label{342}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\int_\O \ya \r |\u_m|^2 dxdy
+\es \int_\O \ya |\p_x \u_m|^2 dxdy+\mu \int_\O \ya |\p_y \u_m|^2 dxdy\\
&+\int_\O \ya (\h+1)\p_x \u_m \cdot \h_m dxdy
+\int_\O \ya \g \p_y \u_m \cdot \h_m dxdy=\sum_{i=1}^9 I_i,
\end{aligned}
\end{equation}
where $I_i(i=1,...,9)$ are defined by
\begin{equation*}
\begin{aligned}
&I_1=\frac{1}{2}\int_\O \ya |\u_m|^2(\p_t \r+(\u+1-e^{-y})\p_x \r+\v \p_y \r)dxdy,\\
&I_2=l \int_\O \yb \r \v |\u_m|^2 dxdy, \quad I_3=-2 l \int_\O \yb (\mu\p_y \u_m +\g \h_m)\cdot \u_m dxdy,\\
&I_4=\int_\O (-(\h+1)\p_x \eta_h Z_\t^{\a_1} \ps
-\g \p_y \eta_h Z_\t^{\a_1} \ps-\g \eta_h Z_\t^{\a_1} \h )\cdot \ya \u_m dxdy,\\
&I_5=\int_\O[-Z_\t^{\a_1} \ps (\r \p_t+\r(\u+1-e^{-y})\p_x+\r\v \p_y)\eta_u]\cdot \ya \u_m dxdy,\\
&I_6=\int_\O(f_u-\r \eta_u f_\psi
-\k \r \eta_u \p_y^2 Z_\t^{\a_1} \ps+2\es \p_x \eta_u \p_x Z_\t^{\a_1} \ps)\cdot \ya \u_m dxdy,\\
&I_7=\int_\O(\es \p_x^2 \eta_u Z_\t^{\a_1} \ps
+\mu \p_y^2( \eta_u Z_\t^{\a_1} \ps )\cdot \ya \u_m dxdy, \quad
I_8=\es\int_\O (1-\r )\eta_u \p_x^2 Z_\t^{\a_1}\ps \cdot \ya \u_m dxdy,\\
&I_9=\int_\O(-\es Z_\t^{\a_1} \p_x r_u+\es \rho^\es \eta_\rho \p_y^{-1} Z_\t^{\a_1} \p_x r_h)\cdot \ya \u_m dxdy.
\end{aligned}
\end{equation*}
By routine checking, we may show that
\begin{equation*}\label{343}
|I_1|+|I_2|\le C_l (1+\|(\u, \v, \p_t \vr, \p_x \vr, \p_y \vr) \|_{L^\infty_0(\O)}^2)\|\u_m\|_{L^2_l(\O)}^2.
\end{equation*}
By virtue of the Holder and Cauchy inequalities, we find
\begin{equation*}\label{344}
|I_3|\le \frac{1}{8} \mu \|\p_y \u_m\|_{L^2_l(\O)}^2
+C_{\mu, l}(1+\|\g\|_{L^\infty_{-1}(\O)}^2)(\| \u_m\|_{L^2_l(\O)}^2+\|\h_m\|_{L^2_l(\O)}^2).
\end{equation*}
Deal with the term $I_4$.
By virtue of $\h+1\ge \d$ and estimate \eqref{b12}, we apply the H\"{o}lder inequality to get
\begin{equation}\label{345}
\begin{aligned}
&|\int_\O \g \eta_h Z_\t^{\a_1} \h \cdot \ya \u_m dxdy|\\
\le
&\d^{-1}\|\g\|_{L^\infty_{-1}(\O)}\|\p_y \h\|_{L^\infty_1(\O)}
\|Z_\t^{\a_1} \h\|_{L^2_l(\O)}\|\u_m\|_{L^2_l(\O)}\\
\le
&C_l\d^{-2}(\|\g\|_{L^\infty_{-1}(\O)}^2+\|\p_y \h\|_{L^\infty_1(\O)}^2)
(\|\u_m\|_{L^2_l(\O)}^2+\|\h_m\|_{L^2_l(\O)}^2).
\end{aligned}
\end{equation}
Due to the fact
$(\h+1)\p_x \eta_h=\p_{xy}\h-\frac{\p_y \h \p_x \h}{\h+1}$,
we apply the estimate \eqref{b11} and H\"{o}lder inequality to get
\begin{equation}\label{346}
\begin{aligned}
&|\int_\O (\h+1)\p_x \eta_h Z_{\t}^{\a_1}\ps\cdot \ya \u_m dxdy|\\
\le
&(1+\|\h\|_{L^\infty_0(\O)})\|(\h+1)\p_x \eta_h\|_{L^\infty_1(\O)}
\|\frac{ Z_{\t}^{\a_1}\ps}{\h+1}\|_{L^2_{l-1}(\O)}\|\u_m\|_{L^2_l(\O)}\\
\le
&C_l \d^{-2}(1+Q^2(t))\|(\u_m, \h_m)\|_{L^2_l(\O)}^2.
\end{aligned}
\end{equation}
Similarly, we also have
\begin{equation*}\label{347}
|\int_\O \g \p_y \eta_h Z_\t^{\a_1} \ps \cdot \ya \u_m dxdy|
\le C_l \d^{-2}(1+Q^2(t))\|(\u_m, \h_m)\|_{L^2_l(\O)}^2.
\end{equation*}
This, along with inequalities \eqref{345} and \eqref{346}, yields directly
\begin{equation*}\label{348}
|I_4|\le C_l \d^{-2}(1+Q^2(t)) \|(\u_m, \h_m)\|_{L^2_l(\O)}^2,
\end{equation*}
Similarly, we also get that
\begin{equation*}\label{349}
|I_5|\le C_l \d^{-2}(1+Q^2(t)) \|(\u_m, \h_m)\|_{L^2_l(\O)}^2.
\end{equation*}
Deal with the term $I_6$.
By virtue of the H\"{o}lder and Cauchy inequalities, we get
\begin{equation}\label{3410}
|\int_\O (f_u-\rho^\es \eta_u f_\psi) \cdot \ya \u_m dxdy|
\le C(\|f_u\|_{L^2_l(\O)}^2+ \| f_\psi\|_{L^2_{l-1}(\O)}^2)
+C(1+\|\eta_u\|_{L^\infty_1(\O)}^2)\|\u_m\|_{L^2_l(\O)}^2.
\end{equation}
Similar to the estimate \eqref{345}, it is easy to check that
\begin{equation}\label{3411}
\begin{aligned}
&|\k \int_\O \r \eta_u \p_y^2 Z_\t^{\a_1} \ps \cdot \ya \u_m dxdy|\\
\le
&C\k\d^{-1}\|\p_y \u\|_{L^\infty_0(\O)}\|\p_y Z_\t^{\a_1} \h\|_{L^2_l(\O)}\|\u_m\|_{L^2_l(\O)}\\
\le
&\d_2 \k \|\p_y \h_m\|_{L^2_l(\O)}^2
+C_{\k, l}\d^{-2}(\| Z_2 \p_y \h \|_{L^\infty_1(\O)}^2+\|(\p_y \u, \p_y \h)\|_{L^\infty_0(\O)}^2)
\|(\u_m, \h_m)\|_{L^2_l(\O)}^2,
\end{aligned}
\end{equation}
where we have used the estimate \eqref{b14} in the last inequality.
Similarly, we also have
\begin{equation*}\label{3412}
\begin{aligned}
&|2\es \int_\O \p_x \eta_u \p_x Z_\t^{\a_1} \ps \cdot \ya \u_m dxdy|\\
\le
&2\es (\|\p_{xy}^2 \u\|_{L^\infty_1(\O)}+\|\p_y \u\|_{L^\infty_1(\O)}\|\p_x \h\|_{L^\infty_0(\O) })
\|\frac{\p_x Z_\t^{\a_1} \ps}{\h+1}\|_{L^2_{l-1}(\O)}\|\u_m\|_{L^2_l(\O)}\\
\le
&\d_1 \es \|\p_x \h_m\|_{L^2_l(\O)}^2
+C_l \d^{-4}(1+\|(\p_{xy}\u, \p_y \u)\|_{L^\infty_1(\O)}^4+\|\p_x \h\|_{L^\infty_0(\O)}^4)\|(\u_m, \h_m)\|_{L^2_l(\O)}^2.
\end{aligned}
\end{equation*}
This, along with inequalities \eqref{3410} and \eqref{3411}, yields directly
\begin{equation*}\label{3413}
\begin{aligned}
|I_6|
\le
&\d_2 \k \|\p_x \h_m\|_{L^2_l(\O)}^2+\d_2 \es \|\p_y \h_m\|_{L^2_l(\O)}^2
+ C(\|f_u\|_{L^2_l(\O)}^2+ \| f_\psi\|_{L^2_{l-1}(\O)}^2)\\
&+C_{\k, l}\d^{-4}(1+Q^2(t))\|(\u_m, \h_m)\|_{L^2_l(\O)}^2.
\end{aligned}
\end{equation*}
Now, we give the estimate for the term $I_7$.
Similar to the estimates \eqref{3315} and \eqref{3320}, we can obtain
\begin{equation*}\label{3414}
|\es \int_\O \p_x^2 \eta_u Z_\t^{\a_1} \ps \cdot \ya \u_m dxdy|
\le \frac{1}{8} \es \|\p_x \u_m\|_{L^2_l(\O)}^2
+C_l \d^{-4}(1+Q^2(t))\|(\u_m, \h_m)\|_{L^2_{l}(\O)}^2,
\end{equation*}
and
\begin{equation*}\label{3415}
|\es \int_\O \p_y^2 (\eta_u Z_\t^{\a_1} \ps)\cdot \ya \u_m dxdy|
\le \frac{1}{8} \mu \|\p_y \u_m\|_{L^2_l(\O)}^2+C_{\mu, l}\d^{-4}(1+Q^2(t))\|(\u_m, \h_m)\|_{L^2_{l}(\O)}^2,
\end{equation*}
and hence, it follows
\begin{equation*}\label{3416}
|I_7|\le
\frac{1}{8}(\es\|\p_x \u_m\|_{L^2_l(\O)}^2+\mu\|\p_y \u_m\|_{L^2_l(\O)}^2)
+C_{\mu, l}\d^{-4} (1+Q^2(t))\|(\u_m, \h_m)\|_{L^2_l(\O)}^2.
\end{equation*}
Finally, we deal with the term $I_8$. Indeed, the integration by part with respect to $x$ yields directly
\begin{equation}\label{3417}
\begin{aligned}
I_8
=&\es \int_\O \ya \p_x \vr \eta_u \u_m \cdot \p_x Z_\t^{\a_1}\ps dxdy\\
&+\es \int_\O \ya \vr \eta_u \p_x \u_m \cdot \p_x Z_\t^{\a_1}\ps dxdy\\
&+\es \int_\O \ya \vr \p_x \eta_u \u_m \cdot \p_x Z_\t^{\a_1}\ps dxdy\\
=&I_{81}+I_{82}+I_{83}.
\end{aligned}
\end{equation}
By virtue of the estimate \eqref{b13}, Holder and Cauchy equalities, we find
\begin{equation}\label{3418}
\begin{aligned}
I_{81}
&\le \es \| \p_x \vr \|_{L^\infty_0(\O)}\| \p_y (\u-e^{-y}) \|_{L^\infty_1(\O)}
\|\u_m\|_{L^2_l(\O)} \|\frac{\p_x Z^{\a_1}_\t \ps}{h+1}\|_{L^2_{l-1}(\O)}\\
&\le \d_2 \es \|\p_x \h_m\|_{L^2_l(\O)}^2
+C_l\d^{-4}(1+\|\p_y \u\|_{L^\infty_1(\O)}^4+\|(\p_x \vr, \p_x \h)\|_{L^\infty_0(\O)}^4)
\|(\u_m, \h_m)\|_{L^2_l(\O)}^2.
\end{aligned}
\end{equation}
Similar to the estimate \eqref{3412}, it is easy to justify
\begin{equation}\label{3419}
I_{83}\le
\d_2 \es \|\p_x \h_m\|_{L^2_l(\O)}^2+C_l \d^{-4}(1+\|(\p_{xy}^2 \u, \p_y \u)\|_{L^\infty_1(\O)}^4
+\|\p_x \h\|_{L^\infty_0(\O)}^4)\|(\u_m,\h_m)\|_{L^2_l(\O)}^2.
\end{equation}
Using the H\"{o}lder inequality and the estimate \eqref{b13}, it follows
\begin{equation*}\label{3420}
\begin{aligned}
I_{82}
\le
& \es \|\vr \|_{L^\infty_0(\O)}\|\p_y (\u-e^{-y})\|_{L^\infty_1(\O)}
\|\p_x \u_m\|_{L^2_l(\O)} \|\frac{\p_x Z^{\a_1}_\t \ps}{\h+1}\|_{L^2_{l-1}(\O)}\\
\le
& \frac{4\es\d^{-1}}{2l-1} \|\vr \|_{L^\infty_0(\O)}\| \p_x \h\|_{L^\infty_0(\O)} \|\p_y (\u-e^{-y})\|_{L^\infty_1(\O)}
\|\p_x \u_m\|_{L^2_l(\O)} \|\h_m\|_{L^2_l(\O)}^2\\
&+ \frac{2\es\d^{-1}}{2l-1} \|\vr \|_{L^\infty_0(\O)}\|\p_y (\u-e^{-y})\|_{L^\infty_1(\O)}
\|\p_x \u_m\|_{L^2_l(\O)} \|\p_x \h_m\|_{L^2_l(\O)}\\
\le
& 2\es\| \p_x \h\|_{L^\infty_0(\O)}
\|\p_x \u_m\|_{L^2_l(\O)} \|\h_m\|_{L^2_l(\O)}+ \es\|\p_x \u_m\|_{L^2_l(\O)} \|\p_x \h_m\|_{L^2_l(\O)},
\end{aligned}
\end{equation*}
where we have used the condition {$\|\p_y (\u-e^{-y})\|_{L^\infty_1(\O)}\le \d^{-1}$
and $\|\vr \|_{L^\infty_0(\O)} \le (l-\frac{1}{2})\d^2$} in the last inequality.
and hence, it follows
\begin{equation}\label{3421}
I_{82}\le
(\frac{1}{2}+\d_2)\es\|\p_x \h_m\|_{L^2_l(\O)}^2
+\frac{1}{2}\es \|\p_x \u_m\|_{L^2_l(\O)}^2
+C \| \p_x \h\|_{L^\infty_0(\O)}^2 \|\h_m\|_{L^2_l(\O)}^2,
\end{equation}
Then, substituting the estimates \eqref{3418}, \eqref{3419} and \eqref{3421} into \eqref{3417}, we get
\begin{equation*}\label{3422}
|I_{8}|\le
(\frac{1}{2}+3\d_2)\es\|\p_x \h_m\|_{L^2_l(\O)}^2
+\frac{1}{2}\es \|\p_x \u_m\|_{L^2_l(\O)}^2
+C_l \d^{-4}(1+Q^2(t))\|(\u_m, \h_m)\|_{L^2_l(\O)}^2.
\end{equation*}
Finally, integrating by part and applying the Cauchy inequality, we get
\begin{equation*}
|I_9|\le \frac{\es}{8}\|\p_x \u_m\|_{L^2_l(\O)}^2
+\|Z_\t^{\a_1}(r_u, r_h)\|_{L^2_l(\O)}^4+C_l \d^{-4}(1+Q^2(t))\|\u_m\|_{L^2_l(\O)}^2.
\end{equation*}
Then, substituting the estimates of $I_1$ through $I_9$
into the equality \eqref{342}, and integrating the resulting inequality over $[0, t]$, we get that
\begin{equation}\label{3423}
\begin{aligned}
& \|\sqrt{\vr}\u_m(t)\|_{L^2_l(\O)}^2 +\frac{\es}{2} \int_0^t \|\p_x \u_m\|_{L^2_l(\O)}^2 d\t
+\frac{\mu}{2} \int_0^t \| \p_y \u_m\|_{L^2_l(\O)}^2 d\t\\
&+\int_0^t \int_\O \ya [(\h+1)\p_x \u_m \cdot \h_m+ \g \p_y \u_m \cdot \h_m] dxdy d\t\\
\le
&\|(\sqrt{\vr}\u_m)(0)\|_{L^2_l(\O)}^2
+(\frac{1}{2}+3\d_2) \es \int_0^t \|\p_x \h_m\|_{L^2_l(\O)}^2 d\t
+\d_2 \k\int_0^t \|\p_y \h_m\|_{L^2_l(\O)}^2 d\t\\
&+C\int_0^t \|Z_\t^{\a_1}(r_u, r_h)\|_{L^2_l(\O)}^4d\t
+C\int_0^t (\|f_u\|_{L^2_l(\O)}^2+ \| f_\psi\|_{L^2_{l-1}(\O)}^2) d\t\\
&
+C_{\mu, \k, l} \d^{-4}(1+Q^2(t))\int_0^t \|(\u_m, \h_m)\|_{L^2_l(\O)}^2 d\t.
\end{aligned}
\end{equation}
Applying the Moser type inequality \eqref{ineq-moser}, it is easy to justify
\begin{equation}\label{3424}
\int_0^t\! \|f_u\|_{L^2_l(\O)}^2 d\t
\le C_{m, l}(1+Q^2(t))\! \int_0^t \!(\|(\vr, \u, \h)\|_{\H^m_l}^2\!+\|(\p_y \u, \p_y \h)\|_{\H^{m-1}_l}^2)d\t.
\end{equation}
Similarly, thanks to the equation \eqref{eqhm}, it is easy to obtain the estimate
\begin{equation}\label{3425}
\begin{aligned}
&\|\h_m (t)\|_{L^2_l(\O)}^2+\frac{3}{4}\es \int_0^t \|\p_x \h_m\|_{L^2_l(\O)}^2 d\t
+\frac{3}{4} \k\int_0^t \|\p_y \h_m\|_{L^2_l(\O)}^2 d\t\\
&-\int_0^t \int_\O \ya [(\h+1)\p_x \u_m \cdot \h_m+ \g \p_y \u_m \cdot \h_m] dxdy d\t\\
\le
&\|\h_m(0)\|_{L^2_l(\O)}^2
+C\int_0^t \| Z_\t^{\a_1}r_h\|_{L^2_l(\O)}^4 d\t
+C_{\k, m, l} \d^{-6}(1+Q^3(t))\int_0^t \mathcal{E}_{m,l}(\t) d\t\\
&+C_{\k, m, l} \d^{-6}(1+Q^3(t))\int_0^t (\|(\u_m, \h_m)\|_{L^2_l(\O)}^2+\|(\p_y \u, \p_y \h)\|_{\H^{m-1}_l}^2)d\t.
\end{aligned}
\end{equation}
Therefore, combining the estimates \eqref{3423}-\eqref{3425} with \eqref{b22},
and choosing $\d_2$ small enough, we complete the proof of lemma.
\end{proof}
Therefore, combining the estimates in Lemmas \ref{lemma33} and \ref{lemma34}
and choosing the constant $\d_1$ small enough, then we complete the proof of Proposition \ref{Tanential-estimate}.
\begin{remark}\label{remark-condition}
To deal with the term $\es\int_\O (1-\r )\eta_u \p_x^2 Z_\t^{\a_1}\ps \cdot \ya \u_m dxdy$
(i.e., the term $I_8$ on the right handside of equality \eqref{342}),
we require the assumption of condition \eqref{a1}.
In other words, the condition \eqref{a1} is not needed for the homogeneous flow($\r \equiv 1$)
since this difficult term will disappear.
\end{remark}
\subsection{Weighted $\H^{m-1}_l-$Estimates for Normal Derivative}
In this subsection, we shall provide an estimate for
$\|(\p_y \vr, \p_y \u, \p_y \h)\|_{\H^{m-1}_l}$,
which will be given as follows:
\begin{proposition}\label{Normal-estimate}
Let $(\vr, \u, \v, \h, \g)$ be sufficiently smooth solution, defined on $[0, T^\es]$,
to the equations \eqref{eq5}-\eqref{bc5}. Under the assumption of condition \eqref{a2}, it holds on
\begin{equation*}
\begin{aligned}
&\sup_{0\le \t \le t}\|(\p_y \vr, \p_y \u, \p_y \h)(\t)\|_{\H^{m-1}_l}^2
+\es \int_0^t \|\p_x(\p_y \vr, \p_y \u, \p_y \h) \|_{\H^{m-1}_l}^2 d\t\\
&+\int_0^t (\es \|\p_y^2 \vr\|_{\H^{m-1}_l}^2+\mu \|\p_y^2 \u\|_{\H^{m-1}_l}^2
+\k \|\p_y^2 \h\|_{\H^{m-1}_l}^2) d\t\\
\le
&C\|(\p_y \vr_0, \p_y \u_0, \p_y \h_0)\|_{\H^{m-1}_l}^2
+C_{\mu, \k} t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}
+C_{\mu, \k, m, l}\d^{-2}(1+Q^3(t))\int_0^t X_{m,l}(\t) d\t.
\end{aligned}
\end{equation*}
\end{proposition}
First of all, we establish the estimate for the quantity $\p_y \vr$ in $\H^{m-1}_l$ norm.
To this end, differentiating the density equation \eqref{eq5}$_1$ with respect to $y$ variable,
we get the evolution equation for $\p_y \vr$:
\begin{equation}\label{3c1}
(\p_t +(\u+1-e^{-y})\p_x +\v \p_y -\es \p_x^2-\es \p_y^2)\p_y \vr
=f_1,
\end{equation}
where the function $f_1$ is defined by
\begin{equation*}\label{3c2}
f_1:=-\es \p_y (\p_x r_1+\p_y r_2)-(\p_y \u+e^{-y})\p_x \vr+\p_x \u \p_y \vr.
\end{equation*}
\begin{lemma}\label{lemma37}
For smooth solution $(\vr, \u, \v, \h,\g)$ of the equations \eqref{eq5}-\eqref{bc5}, then it holds on
\begin{equation*}\label{371}
\begin{aligned}
&\sup_{\tau \in [0 ,t]}\|\p_y \vr(\t)\|_{\H^{m-1}_l}^2
+\es\int_0^t (\|\p_{xy} \vr\|_{\H^{m-1}_l}^2+\|\p_y^2 \vr\|_{\H^{m-1}_l}^2) d\tau\\
\le
&\|\p_y \vr_0\|_{\H^{m-1}_l}^2\!+\!\!\int_0^t\!\! \|\p_y (r_1, r_2)\|_{\H^{m-1}_l}^2 d\tau
+C_{m,l}(1+Q(t))\!\!\int_0^t\!\!(1+\|(\vr, \u)\|_{\H^m_l}^2+\|\p_y(\vr, \u)\|_{\H^{m-1}_l}^2)d\tau.
\end{aligned}
\end{equation*}
\end{lemma}
\begin{proof}
We will give the proof of the estimate \eqref{371} by induction.
First of all, multiplying \eqref{3c1} by $\ya \p_y \vr$, integrating over $\O$ and
integrating by part with respect to $x$ variable, we find
\begin{equation*}\label{373}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\int_{\O} \ya |\p_y \vr|^2 dxdy
+\es \int_{\O} \ya |\p_x \p_y \vr|^2 dxdy\\
=
&\es \int_{\O} \p_y^3 \vr \cdot \ya \p_y \vr dxdy
+l \int_{\O} \yb \v |\p_y \vr|^2 dxdy
+\int_{\O} f_1 \cdot \ya \p_y \vr dxdy.
\end{aligned}
\end{equation*}
Integrating by part and applying the boundary condition $\p_y \vr|_{y=0}=0$, we get
\begin{equation*}\label{374}
\begin{aligned}
&\es \int_\O \p_y^3 \vr \cdot \ya \p_y \vr dxdy\\
=
&\es \int_\T \p_y^2 \vr \cdot \p_y \vr|_{y=0} dx
-\es \int \p_y (\ya \p_y \vr) \cdot \p_y^2 \vr dxdy\\
=
&-\es \int_\O \ya |\p_y^2 \vr|^2 dxdy
-2l \es \int_\O \yb \p_y \vr \cdot \p_y^2 \vr dxdy\\
\le
&-\frac{1}{2}\es \int \ya |\p_y^2 \vr|^2 dxdy
+C_l \|\p_y \vr\|_{L^2_{l-1}(\Omega)}^2,
\end{aligned}
\end{equation*}
where we have used the H\"{o}lder and Cauchy inequalities in the last inequality.
Thus we get after integrating the resulting inequality over $[0, t]$ with time variable
\begin{equation*}
\begin{aligned}
& \int_{\O} \ya |\p_y \vr|^2 dxdy
+\es \int_0^t\int_{\O} \ya (|\p_x \p_y \vr|^2+|\p_y^2 \vr|^2) dxdyd\t\\
\le
&\int_{\O} \ya |\p_y \vr_0|^2 dxdy
+2l \int_0^t \int_{\O} \yb |\v| |\p_y \vr|^2 dxdyd\t\\
&
+2\int_0^t \int_{\O} \ya |f_1| |\p_y \vr| dxdyd\t
+C_l \int_0^t \|\p_y \vr\|_{L^2_{l-1}(\Omega)}^2 d\t,
\end{aligned}
\end{equation*}
which implies directly
\begin{equation*}\label{375}
\begin{aligned}
& \int_{\O} \ya |\p_y \vr|^2 dxdy
+\es \int_0^t(\|\p_x \p_y \vr\|_{L^2_l(\O)}^2+\|\p_y^2 \vr\|_{L^2_l(\O)}^2) d\t\\
\le
&\int_{\O} \ya |\p_y \vr_0|^2 dxdy
+\int_0^t \|\p_y(r_1, r_2)\|_{L^2_l(\O)}^2 d\t
+C_l(1+Q(t)) \int_0^t (\|\vr\|_{\H^1_l}^2+\|\p_y \vr\|_{L^2_l(\O)}^2)d\t.
\end{aligned}
\end{equation*}
Obviously, this inequality implies the estimate \eqref{371} holds on
for $m=1$. To prove the general case, assume that \eqref{371}
is proven for $k \le m-2$, we need to prove it holds on also for $k=m-1$.
Applying the operator $\z^\a (|\a|= m-1)$ to the equation \eqref{3c1}, we get
\begin{equation}\label{376}
(\p_t+(\u+1-e^{-y})\p_x +\v \p_y)\z^\a \p_y \vr
-\es \z^\a \p_x^2 \p_y \vr-\es \z^\a \p_y^3 \vr
=\z^\a f_1+\C_{41}+\C_{42},
\end{equation}
where $\C_{4i}(i=1,2)$ are defined by
$$
\C_{41}=-[\z^\a, (\u+1-e^{-y})\p_x]\p_y \vr, \quad
\C_{42}=-[\z^\a, \v \p_y]\p_y \vr.
$$
Multiplying the equation \eqref{376} by $\ya \z^\a \p_y \vr$,
integrating over $\O \times [0, t]$,
and integrating by part with respect to $x$ variable, we find
\begin{equation}\label{377}
\begin{aligned}
& \frac{1}{2}\int_\O \ya |\z^\a \p_y \vr|^2 dxdy
+\es \int_0^t \int_\O \ya |\p_x \z^\a \p_y \vr|^2 dxdy d\tau\\
=
&\frac{1}{2}\int_\O \ya |\z^\a \p_y \vr_0|^2 dxdy+I_{21}+I_{22}+I_{23}+I_{24}+I_{25},
\end{aligned}
\end{equation}
where the term $I_{2i}(i=1,...,5)$ are defined by
\begin{equation*}
\begin{aligned}
&I_{21}=\es \int_0^t\int_\O \z^\a \p_y^3 \vr \cdot \ya \z^\a \p_y \vr dxdyd\tau,\quad
I_{22}=\int_0^t \int_\O \z^\a f_1 \cdot \ya \z^\a \p_y \vr dxdyd\tau,\\
&I_{23}=l \int_0^t \int_\O \yb \v |\z^\a \p_y \vr|^2 dxdyd\tau,\quad
I_{24}=\int_0^t \int_\O \C_{41} \cdot \ya \z^\a \p_y \vr dxdyd\tau,\\
&I_{25}=\int_0^t \int_\O \C_{42} \cdot \ya \z^\a \p_y \vr dxdyd\tau.
\end{aligned}
\end{equation*}
Similar to the estimate \eqref{3225}, we can get
\begin{equation*}\label{378}
I_{21}
\le -\frac{1}{2}\es \int_0^t \int_\O \ya |\p_y \z^\a \p_y \vr|^2 dxdyd\tau
+C_{m,l}\int_0^t (\es\|\p_y \vr\|_{\H^{m-1}_l}^2+\es\|\p_y^2 \vr\|_{\H^{m-2}_l}^2)d\tau.
\end{equation*}
It is easy to justify
\begin{equation*}\label{379}
|I_{23}|\le C_l \|\v \|_{L^\infty_0(\O)}^2 \int_0^t \|\p_y \vr\|_{\H^{m-1}_l}^2 d\tau
\end{equation*}
Applying the Moser type inequality \eqref{ineq-moser}, we conclude
\begin{equation*}\label{3710}
\begin{aligned}
|I_{22}|
\le
&\frac{1}{4}\es \int_0^t(\|\p_x \z^\a \p_y \vr\|_{L^2_l(\O)}^2+\|\p_y \z^\a \p_y \vr\|_{L^2_l(\O)}^2)d\t
+\int_0^t \|\p_y (r_1, r_2)\|_{\H^{m-1}_l}^2 d\tau\\
&+C_{m,l}(1+Q(t))\int_0^t (1+\|(\vr, \u)\|_{\H^m_l}^2+\|(\p_y \vr, \p_y \u)\|_{\H^{m-1}_l}^2) d\tau,
\end{aligned}
\end{equation*}
and
\begin{equation*}\label{3710}
|I_{24}| \le C_m(1+Q(t))\int_0^t (1+\|\u \|_{\H^m_l}^2+\|\p_y \vr\|_{\H^{m-1}_l}^2)d\tau.
\end{equation*}
Finally, we deal with the term $I_{25}$.
It follows from the H\"{o}lder inequality that
\begin{equation}\label{3711}
|I_{25}|\le \int_0^t \|\C_{42} \|_{L^2_l(\Omega)} \|\z^\a \p_y \vr\|_{L^2_l(\Omega)} d\tau.
\end{equation}
It is easy to check that
$$
[\z^\a, \v \p_y]\vr=[\z^\a, \v]\p_y^2 \vr+\v [\z^\a, \p_y]\p_y \vr.
$$
Since the coefficient $\es$
of the quantity $\p_y^2 \vr$ in \eqref{eq5}$_1$ is sufficiently small,
it is not expected to establish a estimate which is uniform in $\es$
for $\|\p_y^2 \vr\|_{L^\infty_0(\O)}$ or $\|\p_y^2 \vr\|_{\H^{m-1}_l}$.
Hence, we first write
$$
[Z_2^{\a_2}, \p_y]\p_y \vr=\sum_{\b_2 \neq 0, \b_2+\ga_2=\a_2}
C_{\b_2, \ga_2}Z_2^{\b_2}(\frac{1}{\varphi})Z_2^{\ga_2+1}\p_y \vr,
$$
and get
\begin{equation}\label{3712}
\begin{aligned}
\int_0^t \|\v Z_2^{\b_2}(\frac{1}{\varphi})Z_2^{\ga_2+1}Z_\tau^{\a_1 }\p_y \vr\|_{L^2_l(\O)}^2 d\tau
\le
C\|\frac{\v}{\varphi}\|_{L^\infty_0(\O)}^2 \int_0^t \|\p_y \vr\|_{\H^{m-1}_l}^2 d\tau,
\end{aligned}
\end{equation}
where we have used the estimate \eqref{3213} in the last inequality.
Similarly, we have
$$
[\z^\a, \p_y] \p_y^2 \vr=\sum_{|\b+\ga|\le m-1, |\ga|\le m-2}
C_{\b, \ga, \varphi}\z^\b (\frac{\v}{\varphi})\z^\ga (Z_2 \p_y \vr)
$$
If $\b =0$, it is easy to verify
\begin{equation*}\label{3713}
\int_0^t \|\frac{\v}{\varphi}\z^\ga (Z_2 \p_y \vr)\|_{L^2_l}^2 d\tau
\le \|\frac{\v}{\varphi}\|_{L^\infty_0(\O)}^2\int_0^t \|\p_y \vr\|_{\H^{m-1}_l}^2 d\tau.
\end{equation*}
If $\b \neq 0$, the application of Moser type inequality \eqref{ineq-moser} yields directly
\begin{equation}\label{3714}
\begin{aligned}
&\int_0^t \|\z^\b (\frac{\v}{\varphi})\z^\ga (Z_2 \p_y \vr)\|_{L^2_l(\O)}^2 d\tau\\
\le
&C\|\z^{E_i} (\frac{\v}{\varphi})\|_{L^\infty_1(\Omega)}^2
\int_0^t \|Z_2 \p_y \vr\|_{\H^{m-2}_{l-1}}^2 d\tau\\
& +C\|Z_2 \p_y \vr\|_{L^\infty_1(\Omega)}^2
\int_0^t \|\z^{E_i} (\frac{\v}{\varphi}) \|_{\H^{m-2}_{l-1}}^2 d\tau.
\end{aligned}
\end{equation}
Using the Hardy inequality and divergence-free condition
of velocity in \eqref{eq5}$_4$, we get
\begin{equation*}\label{3715}
\begin{aligned}
\|\z^{E_i} (\frac{\v}{\varphi}) \|_{\H^{m-2}_{l-1}}^2
\le \|\frac{\v}{y} \|_{\H^{m-1}_{l}}^2
\le C_l\|\p_x \u \|_{\H^{m-1}_{l}}^2
\le C_l\| \u \|_{\H^{m}_{l}}^2,
\end{aligned}
\end{equation*}
which, together with \eqref{3714}, yields directly
\begin{equation*}\label{3716}
\begin{aligned}
\int_0^t \|\z^\b (\frac{\v}{\varphi})\z^\ga (Z_2 \p_y \vr)\|_{L^2_l(\O)}^2 d\tau
\le
C_{l} \|(\z^{E_i} (\frac{\v}{\varphi}),Z_2 \p_y \vr\|_{L^\infty_1(\Omega)}^2
\int_0^t (\| \u \|_{\H^{m}_{l}}^2+\|\p_y \vr\|_{\H^{m-1}_l}^2) d\tau.
\end{aligned}
\end{equation*}
This and inequality \eqref{3712} give directly
\begin{equation*}\label{3717}
|I_{25}|\le C_{m,l} Q(t)\int_0^t (\| \u \|_{\H^{m}_{l}}^2+\|\p_y \vr\|_{\H^{m-1}_l}^2) d\tau.
\end{equation*}
Therefore, substituting the estimate of $I_{21}$ through $I_{25}$ into \eqref{377}
and using the induction assumption to eliminate the term $\es \int_0^t \|\p_y^2 \vr\|_{\H^{m-2}_l}^2d\tau$,
then the proof of this lemma is completed.
\end{proof}
Next, we establish the estimate for $\|\p_y \u\|_{\H^{m-1}_l}$.
Although $\p_y \u$ does not vanish on the boundary, we can take $-\p_y^2 \u$
as the text function thanks to the coefficient $\mu>0$ in \eqref{eq5}$_2$.
\begin{lemma}\label{lemma38}
For smooth solution $(\vr, \u, \v, \h,\g)$ of the equations \eqref{eq5}-\eqref{bc5}, then it holds on
\begin{equation*}\label{381}
\begin{aligned}
&\sup_{0\le \tau \le t}\|\p_y \u(\t)\|_{\H^{m-1}_l}^2
+\int_0^t( \es\|\p_{xy} \u\|_{\H^{m-1}_l}^2+\mu \|\p_y^2 \u\|_{\H^{m-1}_l}^2) d\tau\\
&\le \|\p_y \u_0\|_{\H^{m-1}_l}^2
\!+C_\mu \!\!\int_0^t\!\! \|\p_y r_u\|_{\H^{m-1}_l}^2 d\tau
\!+\!C_{\mu, m, l}(1+Q^2(t))\!\!\int_0^t\!\! (1\!+\|(\vr, \u ,\h)\|_{\H^m_l}^2\!+\|(\p_y \u, \p_y \h)\|_{\H^{m-1}_l}^2)d\tau.
\end{aligned}
\end{equation*}
\end{lemma}
\begin{proof}
First of all, multiplying the equation \eqref{eq5}$_2$ by $-\ya \p_y^2 \u$
and integrating over $\Omega$, we find
\begin{equation*}\label{383}
\begin{aligned}
&\int_\O (-\r \p_t \u +\es \p_x^2 \u+\mu \p_y^2 \u)\cdot \ya \p_y^2 \u dxdy\\
=
&\int_\O (\es \p_x r_u+\mu e^{-y})\cdot \ya \p_y^2 \u dxdy-\int_\O f_2 \cdot \ya \p_y^2 \u dxdy,
\end{aligned}
\end{equation*}
where $f_2 $ is defined by
\begin{equation*}\label{384}
f_2:=-\r (\u+1-e^{-y})\p_x \u-\r \v \p_y \u-\r \v e^{-y}+(\h+1)\p_x \h+\g \p_y \h.
\end{equation*}
Integrating by part and applying the boundary condition $\u|_{y=0}=0$, we get
\begin{equation*}\label{385}
\begin{aligned}
&\es \int_\O \p_x^2 \u \cdot \ya \p_y^2 \u dxdy\\
=
&\es \int_{\T} \p_x^2 \u \cdot \p_y \u|_{y=0} dx
-\es \int_\O \p_x^2 \p_y \u \cdot \ya \p_y \u dxdy\\
&-2l\es \int_\O \p_x^2 \u \cdot \yb \p_y \u dxdy\\
=
&\es \int_\O \ya |\p_x\p_y \u|^2 dxdy
+2l\es \int_\O \yb \p_x \u \cdot \p_x \p_y \u dxdy.
\end{aligned}
\end{equation*}
Similarly, we get that
\begin{equation*}\label{386}
\begin{aligned}
-\int \r \p_t \u \cdot \ya \p_y^2 \u dxdy
=
&\frac{1}{2}\frac{d}{dt}\int \ya \r |\p_y \u|^2 dxdy
-\frac{1}{2}\int \ya \p_t \r |\p_y \u|^2 dxdy\\
&+\int \ya \p_y \r \p_t \u \cdot \p_y \u dxdy
+2l \int \yb \r \p_t \u \cdot \p_y \u dxdy.
\end{aligned}
\end{equation*}
Based on the above estimates, we can conclude that
\begin{equation*}\label{387}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\int_\O \ya \r |\p_y \u|^2 dxdy
+\es \int_\O \ya |\p_x\p_y \u|^2 dxdy
+\mu \int_\O \ya |\p_y^2 \u |^2 dxdy\\
=
&-\int_\O f_2 \cdot \ya \p_y^2 \u dxdy
+\frac{1}{2}\int_\O \ya \p_t \r |\p_y \u|^2 dxdy
-\int_\O \ya \p_y \r \p_t \u \cdot \p_y \u dxdy\\
&-2l \int_\O \yb \r \p_t \u \cdot \p_y \u dxdy
-2l\es \int_\O \yb \p_x \u \cdot \p_x \p_y \u dxdy,
\end{aligned}
\end{equation*}
which, integrating over $[0, t]$, yields directly
\begin{equation*}\label{388}
\begin{aligned}
&\sup_{\t \in [0, t]}\|\sqrt{\vr}\p_y \u(\t)\|_{L^2_l(\O)}^2
+\es \int_0^t \|\p_{xy}\u\|_{L^2_l(\O)}^2 d\t
+\mu \int_0^t \|\p_{y}^2 \u\|_{L^2_l(\O)}^2 d\t\\
\le
&\|\sqrt{\vr_0}\p_y \u_0\|_{L^2_l(\O)}^2
\!+C_\mu \!\int_0^t \!\|\p_y r_u\|_{L^2_l(\O)}^2 d\t
\!+\!C_{\mu, l}(1+Q(t))\!\int_0^t\!(1+\|(\u, \h)\|_{\H^1_l(\O)}^2\!+\!\|(\p_y \u, \p_y \h)\|_{L^2_l(\O)}^2)d\t.
\end{aligned}
\end{equation*}
This implies the estimate \eqref{381} holds on for $m=1$. To prove the general case, let us assume that \eqref{381}
is proven for $k \le m-2$, we need to prove it also holds on for $k=m-1$.
Applying $\z^\a(|\a|=m-1)$ operator to the second equation of \eqref{eq5}, multiplying the resulting equation
by $-\ya \p_y \z^\a \p_y\u $ and integrating over $\Omega$, we find
\begin{equation}\label{389}
\begin{aligned}
&\int_\O(- \r \p_t \z^\a \u +\es \p_x^2 \z^\a \u+\mu \z^\a \p_y^2 \u)\cdot \ya \p_y \z^\a \p_y \u dxdy\\
&=\int_\O [\z^\a, \rho]\p_t\u \cdot \ya \p_y \z^\a \p_y \u dxdy
+\int_\O \z^\a (-f_2+\es \p_x r_u+\mu e^{-y})\cdot \ya \p_y \z^\a \p_y \u dxdy.
\end{aligned}
\end{equation}
In view of the boundary condition $\u|_{y=0}=0$ and the definition of $\varphi(y)$,
we can justify that $\z^\a \u|_{y=0}=0$.
Then, integrating by part and applying the fact
$\z^\a \u|_{y=0}=0$, one arrives at
\begin{equation*}\label{3810}
\begin{aligned}
&\es \int_\O \p_x^2 \z^\a \u \cdot \ya \p_y \z^\a \p_y \u dxdy\\
=
&\es \int_{\mathbb{T}} \p_x^2 \z^\a \u \cdot \z^\a \p_y \u|_{y=0} dx
-\es \int_\O \ya \p_y \p_x^2 \z^\a \u \cdot \z^\a \p_y \u dxdy\\
&-2l \es \int_\O \yb \p_x^2 \z^\a \u \cdot \z^\a \p_y \u dxdy\\
=
&
\es \int_\O \ya |\p_x \z^\a \p_y \u|^2dxdy
+\es \int_\O \ya [\z^\a, \p_y]\p_x^2 \u \cdot \z^\a \p_y \u dxdy\\
&+2l \es \int_\O \yb \p_x \z^\a \u \cdot \p_x \z^\a \p_y \u dxdy\\
\ge
&\frac{1}{2}\es \int_\O \ya |\p_x \z^\a \p_y \u|^2dxdy
-C(\es \| \p_x \p_y \u\|_{\H^{m-2}_l}^2+ \| \z^\a \p_x \u\|_{L^2_l(\Omega)}^2).
\end{aligned}
\end{equation*}
By virtue of the Cauchy-Schwarz inequality, it is easy to justify that
\begin{equation*}\label{3811}
\begin{aligned}
&\mu \int_\O \z^\a \p_y^2 \u \cdot \ya \p_y \z^\a \p_y \u dxdy\\
=
&\mu \int_\O \ya |\p_y \z^\a \p_y \u|^2 dxdy
+\mu \int_\O \ya [\z^\a, \p_y]\p_y \u \cdot \p_y \z^\a \p_y \u dxdy\\
\ge
&\frac{1}{2}\mu \int_\O \ya |\p_y \z^\a \p_y \u|^2 dxdy
-C\|[\z^\a, \p_y]\p_y \u\|_{L^2_l(\Omega)}^2\\
\ge
&\frac{1}{2}\mu \int_\O \ya |\p_y \z^\a \p_y \u|^2 dxdy
-C\|\p_y^2 \u\|_{\H^{m-2}_l}^2.
\end{aligned}
\end{equation*}
Integrating by part and applying the boundary condition $\z^\a \u|_{y=0}=0$, we get
\begin{equation*}\label{3812}
-\int_\O \r \p_t \z^\a \u \cdot \ya \p_y \z^\a \p_y \u dxdy
=
\frac{1}{2}\frac{d}{dt}\int_\O \ya \r |\z^\a \p_y \u|^2 dxdy
+II.
\end{equation*}
where $II$ is defined by
\begin{equation*}\label{3813}
\begin{aligned}
II=
&-\frac{1}{2} \int_\O \ya \p_t \r |\z^\a \p_y \u|^2 dxdy
-\int_\O \ya \r [\z^\a, \p_y]\p_t \u \cdot \z^\a \p_y \u dxdy\\
&+\int_\O \ya \p_y \r \p_t \z^\a \u \cdot \z^\a \p_y \u dx dy
+2l \int_\O \yb \r \p_t \z^\a \u \cdot \z^\a \p_y \u dx dy,
\end{aligned}
\end{equation*}
which can be estimated as follows
\begin{equation*}\label{3814}
|II|\le C_l \|(\p_t \r, \p_y \r)\|_{L^\infty_0(\O)}(\|\u\|_{\H^{m}_l}^2+\|\p_y \u\|_{\H^{m-1}_l}^2).
\end{equation*}
Using the H\"{o}lder and Cauchy inequalities, it follows
\begin{equation*}\label{3815}
\begin{aligned}
&| \int_\O [\z^\a, \r ]\p_t\u \cdot \ya \p_y \z^\a \p_y \u dxdy|\\
&\le
\frac{\mu}{4}\|\ya \p_y \z^\a \p_y \u \|_{L^2_l(\Omega)}^2
+C_\mu\|[\z^\a, \r ]\p_t\u\|_{L^2_l(\Omega)}^2,
\end{aligned}
\end{equation*}
and
\begin{equation*}\label{3816}
\begin{aligned}
&|\int_\O \z^\a (f_2+\es \p_x r_u+\mu e^{-y})\cdot \ya \p_y \z^\a \p_y \u dxdy|\\
\le
&\frac{\mu}{4}\|\ya \p_y \z^\a \p_y \u \|_{L^2_l(\Omega)}^2
+C_\mu( 1+\|\z^\a r_u\|_{L^2_l(\Omega)}^2+\|\z^\a f_2\|_{L^2_l(\Omega)}^2).
\end{aligned}
\end{equation*}
Substituting the above estimates into \eqref{389},
and integrating the inequality over $[0, t]$, we get
\begin{equation*}\label{3817}
\begin{aligned}
&\int_\O \ya \r |\z^\a \p_y \u|^2 dxdy
+\int_0^t \int_\O \ya (\es|\p_x \z^\a \p_y \u|^2+\mu |\p_y \z^\a \p_y \u|^2)dxdy d\tau\\
&\le \int_\O \ya \r_0 |\z^\a \p_y \u_0|^2 dxdy
+C \es \int_0^t \| \p_x \p_y \u\|_{\H^{m-2}_l}^2 d\tau
+C \mu \int_0^t \| \p_y^2 \u\|_{\H^{m-2}_l}^2 d\tau\\
&
+C_\mu \int_0^t \|[\z^\a, \r ]\p_t\u \|_{L^2_l(\Omega)}^2 d\tau
+C_\mu \int_0^t ( 1+\|\z^\a f_2\|_{L^2_l(\Omega)}^2+\|\z^\a r_u\|_{L^2_l(\Omega)}^2)d\tau\\
&+C_l (1+\|(\p_t \r, \p_y \r)\|_{L^\infty_0(\O)})
\int_0^t (\|\u\|_{\H^{m}_l}^2+\|\partial_y \u\|_{\H^{m-1}_l}^2)d\tau.
\end{aligned}
\end{equation*}
By virtue of the Cauchy and Morse type inequality \eqref{ineq-moser}, we get
\begin{equation*}\label{3818}
\begin{aligned}
&|\int_0^t \int_\O [\z^\a, \r ]\p_t\u \cdot \ya \p_y \z^\a \p_y \u dxdy d\tau|\\
\le
& \frac{\mu}{4}\int_0^t \|\p_y \z^\a \p_y \u\|_{L^2_l(\Omega)}^2d\tau
+C_\mu \int_0^t \|[\z^\a, \r ]\p_t\u\|_{L^2_l(\Omega)}^2d\tau\\
\le
& \frac{\mu}{4}\int_0^t \|\p_y \z^\a \p_y \u\|_{L^2_l(\Omega)}^2d\tau
+C_\mu \|(\z^{E_i}\vr, \p_t \u)\|_{L^\infty_0(\O)}^2
\int_0^t \|(\vr, \u)\|_{\H^m_l}^2 d\tau.
\end{aligned}
\end{equation*}
Similarly, by routine checking, we may show
\begin{equation*}\label{3819}
\int_0^t \|\z^\a f_2\|_{L^2_l(\Omega)}^2 d\tau
\le C(1+Q^2(t))\int_0^t (\|(\vr, \u ,\h)\|_{\H^m_l}^2
+\|(\p_y \u, \p_y \h)\|_{\H^{m-1}_l}^2)d\tau.
\end{equation*}
Thus, it is easy to justify the following estimate for $|\a|=m-1$
\begin{equation}\label{3820}
\begin{aligned}
&\sup_{0\le \tau \le t}\int_\O \ya \r |\z^\a \p_y \u|^2 dxdy
+\int_0^t \int_\O \ya (\es|\p_x \z^\a \p_y \u|^2+\mu |\p_y \z^\a \p_y \u|^2)dxdy d\tau\\
&\le \int_\O \ya \r_0 |\z^\a \p_y \u_0|^2 dxdy
+C \es \int_0^t \| \p_x \p_y \u\|_{\H^{m-2}_l}^2 d\tau
+C \mu \int_0^t \| \p_y^2 \u\|_{\H^{m-2}_l}^2 d\tau\\
&
+C_\mu \int_0^t \|\p_y r_u\|_{\H^{m-1}_l}^2 d\tau
+C_{\mu, m, l}(1+Q^2(t))\int_0^t (\|(\vr, \u ,\h)\|_{\H^m_l}^2
+\|(\p_y \u, \p_y \h)\|_{\H^{m-1}_l}^2)d\tau.
\end{aligned}
\end{equation}
Since the terms $\es \int_0^t \| \p_x \p_y \u\|_{\H^{m-2}_l}^2 d\tau$
and $\mu \int_0^t \| \p_y^2 \u\|_{\H^{m-2}_l}^2 d\tau$ in \eqref{3820}
can be obtained by induction, we complete the proof of this lemma.
\end{proof}
Similarly, we can obtain the following estimates for the quantity $\|\p_y \h\|_{\H^{m-1}_l}$.
\begin{lemma}\label{lemma39}
For smooth solution $(\vr, \u, \v, \h,\g)$ of the equations \eqref{eq5}-\eqref{bc5}, then it holds on
\begin{equation*}\label{391}
\begin{aligned}
&\sup_{0 \le \tau \le t}\|\p_y \h\|_{\H^{m-1}_l}^2
+\es \int_0^t \|\p_{xy} \h\|_{\H^{m-1}_l}^2 d\tau
+\k \int_0^t \|\p_y^2 \h\|_{\H^{m-1}_l}^2 d\tau\\
&\le \|\p_y \h_0\|_{\H^{m-1}_l}^2
+C_\k \int_0^t \|\p_y r_h\|_{\H^{m-1}_l}^2 d\tau
+C_{\k, m, l}(1+Q(t))\int_0^t (\|(\u, \h)\|_{\H^m_l}^2
+\|(\p_y \u, \p_y \h)\|_{\H^{m-1}_l}^2)d\tau.
\end{aligned}
\end{equation*}
\end{lemma}
Finally, we give the proof for the estimate in Proposition \ref{Normal-estimate}.
Indeed, we recall the estimate(see \eqref{b22} in appendix \ref{appendixB}) as follows
\begin{equation*}
\|Z_\t^{\a_1}(\vr, \u, \h)\|_{L^2_l(\O)}^2
\le C_l \d^{-2}(1+\|\p_y (\vr, \u, \h)\|_{L^\infty_1(\O)}^2)\|(\vr_m, \u_m, \h_m)\|_{L^2_l(\O)}^2,
\end{equation*}
which, together with the estimates in Lemmas \ref{lemma37}, \ref{lemma38} and \ref{lemma39},
completes the proof of estimate in Proposition \ref{Normal-estimate}.
\subsection{$L^\infty-$Estimates}
To close the estimate, we need to control the $L^\infty-$norm of $(\r, \u, \v, \h, \g)$ in $Q(t)$.
Then, we have
\begin{proposition}\label{Infinity-estimate}
Let $(\vr, \u, \v, \h, \g)$ be sufficiently smooth solution, defined on $[0, T^\es]$,
to the equations \eqref{eq5}-\eqref{bc5}, then we have the following estimates:
\begin{equation*}
Q(t)\le C(1+\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}
+t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}+X_{m,l}^3(t)),
\end{equation*}
and
\begin{equation*}
\begin{aligned}
\|\p_y \vr(t)\|_{\H^{1,\infty}_1}^2
\le
&C(\|\p_y \vr_0\|_{\H^{1,\infty}_1}^2+\|(\vr_0, \u_0, \h_0)\|_{\H^3_0}^2)
+\!C t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}^2
+\! C(1+Q(t))\!\int_0^t\! X_{m,l}^6(\t) d\t,
\end{aligned}
\end{equation*}
for $m \ge 5, l \ge 2$.
\end{proposition}
We point out that the Proposition will be proved in Lemmas \ref{Lemma310}
and \ref{Lemma311}.
First of all, due to the coefficients $\mu>0$ and $\k>0$,
we can apply the Sobolev inequality, and equations \eqref{eq5} to establish
the estimates as follows.\\
\begin{lemma}\label{Lemma310}
Let $(\vr, \u, \v, \h, \g)$ be sufficiently smooth solution, defined on $[0, T^\es]$,
to the equations \eqref{eq5}-\eqref{bc5}, then we have the following estimates:
\begin{equation}\label{3101}
\|Z_\t \vr(t)\|_{L^\infty_0(\O)} +\|(\u, \h)(t)\|_{\H^{1,\infty}_{0, tan}}
\le C(\e_{3,0}^{\frac{1}{2}}(t)+\|\p_y(\vr, \u, \h)(t)\|_{\H^2_0}),
\end{equation}
\begin{equation}\label{3102}
\|(\v, \g)(t)\|_{\H^{1, \infty}_{1, tan}} \le C \e_{4,2}^\frac{1}{2}(t),
\end{equation}
\begin{equation}\label{3104}
\|(\frac{\v}{\varphi})(t)\|_{\H^{1,\infty}_1}
\le C(\e_{4,2}^{\frac{1}{2}}(t)+\|\p_y \u(t)\|_{\H^3_2}),
\end{equation}
\begin{equation}\label{3103}
\begin{aligned}
\|(\p_y \u, \p_y \h)(t)\|_{\H^{1,\infty}_1}
\le
&C(1+\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}^{\frac{1}{2}}
+t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}^{\frac{1}{2}})\\
& +C(\e_{5,1}^{\frac{3}{2}}(t)+\|\p_y(\vr, \u, \h)(t)\|_{\H^3_1}^3), \quad m \ge 5,\ l \ge 1.
\end{aligned}
\end{equation}
\end{lemma}
\begin{proof}
By virtue of the Sobolev inequality \eqref{sobolev}
and the definition of $\mathcal{E}_{m, l}(t)$(see \eqref{eml}), then we get
\begin{equation*}\label{3105}
\|\u\|_{L^\infty_0(\O)}
\le C(\|\u \|_{L^2_0(\O)} +\|\p_x \u \|_{L^2_0(\O)}
+\|\p_y \u \|_{L^2_0(\O)} +\|\p_{xy} \u\|_{L^2_0(\O)})
\le C(\e_{2,0}^{\frac{1}{2}}(t) +\|\p_y \u(t)\|_{\H^1_0}).
\end{equation*}
Similarly, it is easy to justify
\begin{equation*}\label{3106}
\|\h\|_{L^\infty_0(\O)} \le C(\e_{2,0}^{\frac{1}{2}}(t)+\|\p_y \h\|_{\H^1_0}),
\quad
\|Z_\t^{e_i}(\vr, \u, \h)\|_{L^\infty_0(\O)}
\le C(\e_{3,0}^{\frac{1}{2}}(t)+\|\p_y(\vr, \u, \h)\|_{\H^2_0}).
\end{equation*}
Thus we obtain the estimate \eqref{3101}.
Using the Hardy inequality and Sobolev inequality \eqref{sobolev}, we get
\begin{equation}\label{3107}
\begin{aligned}
\|\v\|_{L^\infty_1(\O)}
&\le C(\|\v \|_{L^2_1(\O)}+\|\p_x \v \|_{L^2_1(\O)}
+\|\p_y \v \|_{L^2_1(\O)}+\|\p_{xy} \v\|_{L^2_1(\O)})\\
&\le C(\|\p_y \v \|_{L^2_2(\O)}+\|\p_{xy} \v \|_{L^2_2(\O)}
+\|\p_{x} \u \|_{L^2_1(\O)}+\|\p_{xx} \u\|_{L^2_1(\O)})\\
&\le C\e_{3,2}^\frac{1}{2}(t),
\end{aligned}
\end{equation}
where we have used the divergence-free condition in the last inequality.
Similarly, we obtain
\begin{equation}\label{3108}
\|\g\|_{L^\infty_1(\O)}^2 \le C \e_{3,2}^\frac{1}{2}(t),
\quad
\|Z_\t^{e_i}\v\|_{L^\infty_1(\O)}
+\|Z_\t^{e_i}\g\|_{L^\infty_1(\O)}
\le C\e_{4,2}^\frac{1}{2}(t).
\end{equation}
Then, the combination of estimates \eqref{3107} and \eqref{3108} yields the estimate \eqref{3102}.
By virtue of the Sobolev inequality \eqref{sobolev}, we get
\begin{equation*}\label{3109}
\|\p_y \u\|_{L^\infty_1(\O)}
\le C(\|\p_y \u\|_{L^2_1(\O)}+\|\p_{xy} \u\|_{L^2_1(\O)}
+\|\p_{yy} \u\|_{L^2_1(\O)}+\|\p_{xyy} \u\|_{L^2_1(\O)}).
\end{equation*}
In view of the equation \eqref{eq5}$_2$ and the estimate \eqref{3108}
with the weight $0$ instead of $1$, we find
\begin{equation*}\label{31010}
\begin{aligned}
\|\p_y^2 \u\|_{L^2_1(\O)} \le
& C(1+\|\p_x r_u\|_{L^2_1(\O)} +\|\es \p_x^2 \u\|_{L^2_1(\O)})
+C(1+\|(\u, \h)\|_{L^\infty_0(\O)} )\|(\p_x \u, \p_x \h)\|_{L^2_1(\O)} \\
&+C(\|\p_t \u\|_{L^2_1(\O)} +\|(\v, \g)\|_{L^\infty_0(\O)}) (1+\|(\p_y \u, \p_y \h)\|_{L^2_1(\O)})\\
\le
&C(1+ \|\p_x r_u\|_{L^2_1(\O)} +\e_{3,1}(t)+\|\p_y(\u, \h)\|_{\H^1_1}^2).
\end{aligned}
\end{equation*}
Similarly, it is easy to justify
\begin{equation*}\label{31011}
\|\p_{xyy} \u\|_{L^2_1(\O)}
\le C(1+ \|\p_x^2 r_u\|_{L^2_1(\O)}+\e_{4,1}^{\frac{3}{2}}(t)+\|\p_y(\vr, \u, \h)\|_{\H^2_1}^3).
\end{equation*}
Thus we can conclude the estimate
\begin{equation}\label{31012-1}
\|\p_y \u\|_{L^\infty_1(\O)} \le C(1+\|(\p_x r_u, \p_x^2 r_u)\|_{L^2_1(\O)}
+\e_{4,1}^{\frac{3}{2}}(t)+\|\p_y(\vr, \u, \h)\|_{\H^2_1}^3).
\end{equation}
By virtue of the definition of $r_u$ in \eqref{rdef}, we get for $m\ge 4, l \ge 1$ that
\begin{equation*}
\|(\p_x r_u, \p_x^2 r_u)\|_{L^2_1(\O)}
\le \|(\rho_0, u_{10}, h_{10})\|_{\mathcal{\mathcal{B}}^m_l}^{\frac{1}{2}}
+C t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}^{\frac{1}{2}},
\end{equation*}
which, together with the estimate \eqref{31012-1}, yields directly
\begin{equation}\label{31012a}
\|\p_y \u\|_{L^\infty_1(\O)}
\le C(1+\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}^{\frac{1}{2}}
+C t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}^{\frac{1}{2}}
+\e_{4,1}^{\frac{3}{2}}(t)+\|\p_y(\vr, \u, \h)\|_{\H^2_1}^3).
\end{equation}
Similarly, we get for $m \ge 4, l \ge 1$
\begin{equation}\label{31012-a}
\begin{aligned}
\|\p_y \h\|_{L^\infty_1(\O)} \le
C(1+\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}^{\frac{1}{2}}
+C t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}^{\frac{1}{2}}
+\e_{4,1}(t)+\|\p_y(\u, \h)\|_{\H^2_1}^2),
\end{aligned}
\end{equation}
and for $m \ge 5, l \ge 1$
\begin{equation}\label{31012}
\begin{aligned}
&\!\|\z^{E_i} \p_y \u\|_{L^\infty_1(\O)}
\le\! C(1+\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}^{\frac{1}{2}}
\!+C t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}^{\frac{1}{2}}
\!+\e_{5,1}^{\frac{3}{2}}(t)\!+\!\|\p_y(\vr, \u, \h)\|_{\H^3_1}^3),\\
& \!\|\z^{E_i} \p_y \h\|_{L^\infty_1(\O)}
\!\le C(1+\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}^{\frac{1}{2}}
\!+C t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}^{\frac{1}{2}}
\!+\e_{5,1}(t)\!+\|\p_y(\u, \h)\|_{\H^3_1}^2).
\end{aligned}
\end{equation}
Then, the combination of estimates \eqref{31012a}, \eqref{31012-a} and \eqref{31012} yields \eqref{3103}.
Finally, we give the estimate for the quantity $\|\frac{\v}{\varphi}\|_{\H^{1,\infty}_1}$.
Since vertical velocity $\v$ vanishes on the boundary(i.e., $\v|_{y=0}=0$), we get
$
\la y \ra^2|\v|\le C y(\|\v\|_{L^\infty_1(\O)}+\|\p_y \v\|_{L^\infty_2(\O)}).
$
Using Sobolev inequality \eqref{sobolev} and divergence-free condition \eqref{eq5}$_4$, it follows
\begin{equation}\label{31013}
\|\frac{\v}{\varphi}\|_{L^\infty_1(\O)}
\le C(\|\v\|_{L^\infty_1(\O)} +\|\p_y \v\|_{L^\infty_2(\O)})
\le C(\e_{3,2}^{\frac{1}{2}}(t)+\|\p_y \u\|_{\H^2_2}).
\end{equation}
Similarly, we also get that
\begin{equation}\label{31014}
\|Z_\t^{e_i}(\frac{\v}{\varphi})\|_{L^\infty_1(\O)}^2 \le C(\e_{4,2}^{\frac{1}{2}}(t)+\|\p_y \u\|_{\H^3_2}).
\end{equation}
By virtue of the fact $\p_y(\frac{1}{\varphi})=-\frac{1}{y^2}$,
we get after using the divergence-free condition \eqref{eq5}$_4$
\begin{equation*}\label{31015}
\begin{aligned}
\|Z_2 (\frac{\v}{\varphi})\|_{L^\infty_1(\O)}
&\le C(\|y \p_y(\frac{1}{\varphi})\v\|_{L^\infty_0(\O)}+\|\p_y \v\|_{L^\infty_1(\O)})\\
&\le C(\|\frac{\v}{y}\|_{L^\infty_0(\O)}+\|\p_x \u\|_{L^\infty_1(\O)})
\le C\|\p_x \u\|_{L^\infty_1(\O)},
\end{aligned}
\end{equation*}
where we have used the fact $|\v|\le y \|\p_y \v\|_{L^\infty_0(\O)}$ in the last inequality.
Using the above inequality and Sobolev inequality \eqref{sobolev}, we conclude
\begin{equation*}
\|\la y \ra Z_2 (\frac{\v}{\varphi})\|_{L^\infty_0(\O)} \le C(\e_{3, 1}^{\frac{1}{2}}(t)+\|\p_y \u\|_{\H^2_1}),
\end{equation*}
which, together with the estimates \eqref{31013} and \eqref{31014}, yields directly
\begin{equation*}
\|\frac{\v}{\varphi}\|_{\H^{1,\infty}_1}\le C(\e_{4,2}^{\frac{1}{2}}(t)+\|\p_y \u\|_{\H^3_2}).
\end{equation*}
Therefore, we complete the proof of Lemma \ref{Lemma310}.
\end{proof}
By virtue of the estimates \eqref{3101}-\eqref{3104} in Lemma \ref{Lemma310},
then $Q(t)$ can be controlled as follows:
\begin{equation*}
Q(t)\le C(1+\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}
+ t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}+X_{m,l}^3(t))
\end{equation*}
for $m \ge 5, l \ge 1$.
To close the estimate, we still need to establish the estimate for the quantity $\|\p_y \vr(t)\|_{\H^{1,\infty}_1}^2$.
Since the quantity $y\p_y^2 \vr$ does not communicate with the diffusive term, this prevents
us to apply the maximum principle of transport-diffusion equation.
We should point out that Masmoudi and Rousset \cite{Masmoudi-Rousset} have applied
some estimates of one dimensional Fokker-Planck type equation to achieve this target.
However, we can only apply the $L^\infty-$estimate of heat equation(cf. \ref{eheat} in Lemma \ref{A-heat})
to achieve this goal since $\p_y \v$ vanishes on the boundary due to $\u|_{y=0}=0$ and divergence-free condition.
\begin{lemma}\label{Lemma311}
Let $(\vr, \u, \v, \h, \g)$ be sufficiently smooth solution, defined on $[0, T^\es]$,
to the equations \eqref{eq5}. Then, it holds on
\begin{equation*}\label{31101}
\begin{aligned}
\|\p_y \vr(t)\|_{\H^{1,\infty}_1}^2
\le
&C(\|\p_y \vr_0\|_{\H^{1,\infty}_1}^2+\|(\vr_0, \u_0, \h_0)\|_{\H^3_0}^2)
+C t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}^2
+C(1+Q(t))\int_0^t X_{m,l}^6(\t) d\t,
\end{aligned}
\end{equation*}
for $m \ge 5, l \ge 2$.
\end{lemma}
\begin{proof}
By virtue of the Sobolev inequality \eqref{sobolev}, it is easy to justify that
\begin{equation}\label{31102}
\begin{aligned}
\|\p_y \vr\|_{\H^{1,\infty}_1}^2
\le
&\|\p_y \vr\|_{L^\infty_0(\O)}^2+\|Z_2 \vr\|_{L^\infty_1(\O)}^2
+\|Z_\t^{e_i}\p_y \vr\|_{L^\infty_0(\O)}^2+\|y\p_{yy}\vr\|_{L^\infty_0(\O)}^2\\
\le
&C(\|(\p_y \vr, Z_\t^{e_i}\p_y \vr, y\p_{yy}\vr)\|_{L^\infty_0(\O)}^2
+\e_{3,1}(t)+\|\p_y \vr(t)\|_{\H^3_1}^2).
\end{aligned}
\end{equation}
First of all, since the quantity $\p_y \vr$ satisfies the evolution equation
\eqref{3c1}, we may apply the maximum principle of transport-diffusion equation
\eqref{3c1} to get
\begin{equation*}\label{31103}
\|\p_y \vr\|_{L^\infty_0(\O)}
\le \|\p_y \vr_0\|_{L^\infty_0(\O)}+\int_0^t \|f_1\|_{L^\infty_0(\O)}d\t,
\end{equation*}
where $f_1$ is defined in \eqref{3c2}.
Thus we apply the Cauchy-Schwarz inequality to get
\begin{equation}\label{31104}
\begin{aligned}
\|\p_y \vr\|_{L^\infty_0(\O)}^2
\le
& 2\|\p_y \vr_0\|_{L^\infty_0(\O)}^2+2t \int_0^t \|f_1\|_{L^\infty_0(\O)}^2d\t\\
\le
&2\|\p_y \vr_0\|_{L^\infty_0(\O)}^2+2 t\int_0^t \|\p_y(\p_x r_1, \p_y r_2)\|_{L^\infty_0(\O)}^2 d\t\\
&+Ct\int_0^t (1+\e_{4,0}^2(\t)+\|\p_y(\vr, \u, \h)\|_{\H^2_0}^4+\|\p_y \vr\|_{L^\infty_0(\O)}^4) d\t.
\end{aligned}
\end{equation}
Next, Applying the tangential differential derivatives $Z_\t^{e_i}(i=1,2, e_1=(1, 0), e_2=(0, 1))$
on the evolution equation \eqref{3c1}, we get the evolution for $Z_\t^{e_i}\p_y \vr$:
\begin{equation*}\label{31105}
(\p_t +(\u+1-e^{-y})\p_x +\v \p_y -\es \p_x^2-\es \p_y^2)Z_\t^{e_i}\p_y \vr
=Z_\t^{e_i}f_1-Z_\t^{e_i}\u \p_{xy}\vr-Z_\t^{e_i} \v \p_y^2 \vr,
\end{equation*}
and hence it follows from the maximum principle and Cauchy-Schwarz inequality that
\begin{equation*}\label{31106}
\|Z_\t^{e_i}\p_y \vr\|_{L^\infty_0(\O)}^2 \le 2\|Z_\t^{e_i}\p_y \vr_0\|_{L^\infty_0(\O)}^2
+2t\int_0^t \|(Z_\t^{e_i} f_1, Z_\t^{e_i}\u \p_{xy}\vr, Z_\t^{e_i} \v \p_y^2 \vr)\|_{L^\infty_0(\O)}^2 d\t.
\end{equation*}
Since the vertical velocity $\v$ vanishes on the boundary(i.e., $\v|_{y=0}=0$), we conclude
\begin{equation*}\label{31107}
\begin{aligned}
&\|Z_\t^{e_i}\u \p_{xy}\vr\|_{L^\infty_0(\O)}^2+\|Z_\t^{e_i} \v \p_y^2 \vr\|_{L^\infty_0(\O)}^2\\
\le
&\|Z_\t^{e_i}\u\|_{L^\infty_0(\O)}^2 \|\p_{xy}\vr\|_{L^\infty_0(\O)}^2
+\|\frac{Z_\t^{e_i} \v}{\varphi}\|_{L^\infty_0(\O)}^2\|Z_2 \p_y \vr\|_{L^\infty_0(\O)}^2\\
\le
&C(\e_{4,1}^2(t)+\|\p_y \u\|_{\H^3_1}^4+\|\p_y \vr\|_{\H^{1,\infty}_0}^4).
\end{aligned}
\end{equation*}
Thus we obtain the estimate
\begin{equation}\label{31108}
\begin{aligned}
\|Z_\t^{e_i}\p_y \vr\|_{L^\infty_0(\O)}^2
\le
&2\|Z_\t^{e_i}\p_y \vr_0\|_{L^\infty_0(\O)}^2
+2t\int_0^t(\|Z_\t^{e_i} \p_y(\p_x r_1, \p_y r_2)\|_{L^\infty_0(\O)}^2+\|r_u\|_{\H^3_0}^4)d\t\\
&+C t\int_0^t (1+\e_{5,1}^2(\t)+\|\p_y(\vr, \u, \h)\|_{\H^3_1}^4+\|\p_y \vr\|_{\H^{1,\infty}_0}^4)d\t.
\end{aligned}
\end{equation}
Finally, we deal with the term $\|y \p_y^2 \vr\|_{L^\infty_0(\O)}$.
The main difficulty is the estimate of $y \p_y^2 \vr$,
since the communicator of this quantity with the Laplacian involves
two derivatives in the normal variable.
Let $\chi(y)$ be a smooth compactly supported function which takes
the value one in the vicinity of $0$ and is supported in $[0, 1]$,
and hence, we get
\begin{equation*}
\p_y \vr=\chi(y)\p_y \vr+(1-\chi(y))\p_y \vr \triangleq \varrho^b+\varrho^{int},
\end{equation*}
where $\varrho^b$ is compactly supported in $y$ and $\varrho^{int}$ is supported away from the boundary.
Since $H^m_{co}$ norm is equivalent to the usual $H^m$ norm if
the function is support away from the boundary, we apply
the Sobolev inequality \eqref{sobolev} to get
\begin{equation}\label{31109}
\|y \p_y \varrho^{int}\|_{L^\infty_0(\O)}\le C\|\p_y \vr\|_{\H^3_1}.
\end{equation}
On the other hand, due to the equation \eqref{3c1}, we can get the evolution equation for $\varrho^b$:
\begin{equation}\label{311010}
(\p_t-\es \p_y^2)\varrho^b=\es \chi \p_x^2\vr +\chi f_1+R_1+R_2
\end{equation}
where $R_i(i=1,2)$ are defined by
\begin{equation*}
R_1=-\es \chi'' \p_y \vr-2\es \chi' \p_y^2 \vr,
\quad
R_2=-\chi(\u+1-e^{-y})\p_{xy} \vr-\chi \v \p_y^2 \vr.
\end{equation*}
Applying the estimate \eqref{eheat} to the equation \eqref{311010}, it follows
\begin{equation}\label{311011}
\begin{aligned}
\|y\p_y \varrho^b\|_{L^\infty_0(\O)}^2
\le
& C(\|\varrho^b_0\|_{L^\infty_0(\O)}^2+\|y\p_y \varrho^b_0\|_{L^\infty_0(\O)}^2)
+C \es^2\int_0^t \|(\chi \p_x^2\vr, y \p_y(\chi\p_x^2\vr))\|_{L^\infty_0(\O)}^2 d\t\\
& +C\int_0^t \|(\chi f_1, y\p_y(\chi f_1), R_1, y\p_y R_1, R_2, y\p_y R_2) \|_{L^\infty_0(\O)}^2 d\t.
\end{aligned}
\end{equation}
In view of the definition of $\chi$ and the Sobolev inequality, we find
\begin{equation}\label{311012}
|\es^2\int_0^t \|(\chi \p_x^2\vr, y \p_y(\chi\p_x^2\vr))\|_{L^\infty_0(\O)}^2 d\t|
\le C\es^2 \int_0^t \|\p_y \vr\|_{\H^4_0}^2 d\t+C\es^2 \int_0^t \|\vr\|_{\H^4_0}^2 d\t,
\end{equation}
and
\begin{equation}\label{311014}
\|R_1\|_{L^\infty_0(\O)}^2+\|y\p_y R_1\|_{L^\infty_0(\O)}^2\le C \|\vr\|_{\H^5_0}^2.
\end{equation}
Using the $L^\infty-$estimates in Lemma \ref{Lemma310}, we conclude
\begin{equation}\label{311013}
\begin{aligned}
\|(\chi f_1,y\p_y(\chi f_1))\|_{L^\infty_0(\O)}^2
\le
&\|(\p_{xy} r_1, \p_{y}^2 r_2, Z_2 \p_{xy} r_1, Z_2 \p_{y}^2 r_2)\|_{L^\infty_0(\O)}^2
+\|r_u\|_{\H^3_0}^4\\
&+C(1+\e_{5,0}^2+\|\p_y(\vr, \u, \h)\|_{\H^3_0}^4+\|\p_y \vr\|_{\H^{1,\infty}_0}^4).
\end{aligned}
\end{equation}
By virtue of $\v|_{y=0}=0$, we may apply the Taylor formula
and divergence-free condition \eqref{eq5}$_4$ to get
\begin{equation*}\label{311015}
\|\chi \v \p_{y}^2 \vr\|_{L^\infty_0(\O)}
\le \|\p_y \v\|_{L^\infty_0(\O)} \|\chi y \p_{y}^2 \vr\|_{L^\infty_0(\O)}
\le C \|\p_x \u\|_{L^\infty_0(\O)} \|Z_2 \p_y \vr\|_{L^\infty_0(\O)},
\end{equation*}
and along with the Sobolev inequality \eqref{sobolev} yields directly
\begin{equation}\label{311016}
\begin{aligned}
\|R_2\|_{L^\infty_0(\O)}^2
&\le C(1+\|\u\|_{L^\infty_0(\O)}^2)\|\p_{xy} \vr\|_{L^\infty_0(\O)}^2
+ C \|\p_x \u\|_{L^\infty_0(\O)}^2 \|Z_2 \p_y \vr\|_{L^\infty_0(\O)}^2\\
&\le C(1+\e_{3,0}^2(t)+\|\p_y \u\|_{\H^2_0}^4+\|\p_y \vr\|_{\H^{1,\infty}_0}^4).
\end{aligned}
\end{equation}
By routine checking, we may check that
\begin{equation*}\label{311017}
\begin{aligned}
y\p_y R_2
&=\chi' y [(\u+1-e^{-y})\p_{xy} \vr+ \v \p_y^2 \vr]
+\chi y[\p_y(\u+1-e^{-y})\p_{xy} \vr+ \p_y \v \p_y^2 \vr]\\
&\quad+\chi (\u+1-e^{-y})y\p_{xyy} \vr+ \chi \v y \p_y^3 \vr.
\end{aligned}
\end{equation*}
By virtue of the definition $\chi$, it follows
\begin{equation}\label{311018}
\begin{aligned}
\|\chi' y [(\u+1-e^{-y})\p_{xy} \vr+ \v \p_y^2 \vr]\|_{L^\infty_0(\O)}^2
\le
C(1+\e_{3,1}^2(t)+\|\p_y \u\|_{\H^1_0}^4+\|\p_y \vr\|_{\H^{1,\infty}_0}^4),
\end{aligned}
\end{equation}
and
\begin{equation}\label{311019}
\begin{aligned}
&\|\chi y[\p_y(\u+1-e^{-y})\p_{xy} \vr+ \p_y \v \p_y^2 \vr]\|_{L^\infty_0(\O)}^2\\
&\le C(1+\es^4\|r_u\|_{\H^{2}_0}^4+\e^6_{4,0}(t)+\|\p_y(\vr, \u, \h)\|_{\H^2_0}^{12}
+\|\p_y \vr\|_{\H^{1,\infty}_0}^4).
\end{aligned}
\end{equation}
Since the velocity $\u$ vanishes on the boundary, the application of Taylor formula yields immediately
\begin{equation}\label{311020}
\begin{aligned}
\|\chi (\u+1-e^{-y})y\p_{xyy} \vr\|_{L^\infty_0(\O)}^2
&\le (1+\|\p_y \u\|_{L^\infty_0(\O)}^2)\|\chi y^2 \p_{xyy} \vr\|_{L^\infty_0(\O)}^2\\
&\le C(1+\|\p_y \u\|_{L^\infty_0(\O)}^2)\|\varphi (y)Z_2 \p_{xy} \vr\|_{L^\infty_0(\O)}^2,
\end{aligned}
\end{equation}
where we have used the fact that $y$ is equivalent to $\frac{y}{1+y}$
if $y\in [0, c_0]$.
Using the fact $\u|_{y=0}=0$ and $\p_x \u+\p_y \v=0$, we have $\p_y \v|_{y=0}=0$,
and hence, the Taylor formula implies for $\xi \in [0, y]$
\begin{equation*}\label{311021}
\v(t,x,y)=\v(t,x,0)+y\p_y \v(t,x,0)+\frac{1}{2}y^2 \p_y^2 \v(t,x, \xi)
=\frac{1}{2}y^2 \p_y^2 \v(t,x, \xi),
\end{equation*}
where we have used the fact $\v|_{y=0}=\p_y\v|_{y=0}=0$. Thus, it follows
\begin{equation}\label{311022}
\|\chi \v y \p_y^3 \vr\|_{L^\infty_0(\O)}^2
\le C\|\p_y^2 \v\|_{L^\infty_0(\O)}^2\|\chi y^3 \p_y^3 \vr\|_{L^\infty_0(\O)}^2
\le C\|\p_{xy}\u\|_{L^\infty_0(\O)}^2\|\varphi(y)Z_2^2 \p_y \vr\|_{L^\infty_0(\O)}^2.
\end{equation}
Then the combination of estimates \eqref{311020}, \eqref{311022}
and Sobolev inequality \eqref{sobolev} yields directly
\begin{equation*}
\|\chi (\u+1-e^{-y})y\p_{xyy} \vr+ \chi \v y \p_y^3 \vr\|_{L^\infty_0(\O)}^2
\le C(1+\|r_u\|_{\H^2_0}^4+\e_{5,0}^6(t)+\|\p_y(\vr, \u, \h)\|_{\H^4_0}^{12}),
\end{equation*}
which, together with the estimates \eqref{311018} and \eqref{311019}, yields directly
\begin{equation}\label{311023}
\|y\p_y R_2\|_{L^\infty_0(\O)}^2
\le C(1+\|r_u\|_{\H^3_0}^4+\e_{5,1}^6(t)+\|\p_y(\vr, \u, \h)\|_{\H^4_0}^{12}+\|\p_y \vr\|_{\H^{1,\infty}_0}^4).
\end{equation}
Then, we can get from the estimates \eqref{311012}, \eqref{311013}, \eqref{311014}, \eqref{311016}
and \eqref{311023} that
\begin{equation*}
\begin{aligned}
\|y\p_y \varrho^b\|_{L^\infty_0(\O)}^2
\le
&C(\|\varrho^b_0\|_{L^\infty_0(\O)}^2+\|y\p_y \varrho^b_0\|_{L^\infty_0(\O)}^2)
+C\int_0^t \|(\p_{xy} r_1, \p_{y}^2 r_2, Z_2 \p_{xy} r_1, Z_2 \p_{y}^2 r_2)\|_{L^\infty_0(\O)}^2 d\t\\
&+C\int_0^t \|r_u\|_{\H^3_0}^4 d\t
+C\int_0^t(1+\e_{5,1}^6(\t)+\|\p_y(\vr, \u, \h)\|_{\H^4_0}^{12}+\|\p_y \vr\|_{\H^{1,\infty}_0}^4)d\t,
\end{aligned}
\end{equation*}
which, together with the estimate \eqref{31109}, yields directly
\begin{equation}\label{311024}
\begin{aligned}
\|y\p_y^2 \rho\|_{L^\infty_0(\O)}^2
\le
&C(\|\p_y \vr_0\|_{L^\infty_0(\O)}^2+ \|Z_2 \p_y \vr_0\|_{L^\infty_0(\O)}^2)+C\|\p_y \vr\|_{\H^3_1}^2\\
&+C\int_0^t (\|r_u\|_{\H^3_0}^4+
\|(\p_{xy} r_1, \p_{y}^2 r_2, Z_2 \p_{xy} r_1, Z_2 \p_{y}^2 r_2)\|_{L^\infty_0(\O)}^2)d\t\\
&+C\int_0^t(1+\e_{5,1}^6(\t)+\|\p_y(\vr, \u, \h)\|_{\H^4_0}^{12}+\|\p_y \vr\|_{\H^{1,\infty}_0}^4)d\t.
\end{aligned}
\end{equation}
Therefore, substituting the estimates \eqref{31104}, \eqref{31108} and \eqref{311024} into \eqref{31102},
we complete the proof of lemma.
\end{proof}
\subsection{Proof of Theorem \ref{theo a priori}}
Based on the estimates obtained so far, we can complete the proof of Theorem \ref{theo a priori} in this subsection.
First of all, we give the proof for the estimate \eqref{3a1}.
For two parameters $R$ and $\d$, which will be defined later, we define
\begin{equation*}\label{3d1}
\begin{aligned}
T^\es_* :=&\sup\left\{T\in [0, 1]\ |\ \Theta_{m,l}(t)\le R, \
\|\p_y (\u-e^{-y})(t)\|_{L^\infty_1(\O)}\le \d^{-1}, \right.\\
&\quad \quad \quad \quad \left.
\|\vr(t)\|_{L^\infty_0(\O)}\le \frac{2l-1}{2}\d^2, \
h^\es(t,x,y)+1\ge \d, \ \forall t \ \in [0, T], \ (x, y)\in \O\right\}.
\end{aligned}
\end{equation*}
Now, we write
\begin{equation}\label{3d2}
\begin{aligned}
&\n_{m,l}(t)
:=\sup_{0\le s \le t}\{1+\e_{m,l}(s)+\|(\vr_m,\u_m, \h_m)(s)\|_{L^2_l(\O)}^2
+\|(\p_y \vr, \p_y \u, \p_y \h)(s)\|_{\H^{m-1}_l}^2+\|\p_y \vr(s)\|_{\H^{1,\infty}_1}^2\}\\
&\quad \quad \
+\!\int_0^t \!\!\!\|\p_y^2(\sqrt{\es} \vr, \sqrt{\mu} \u, \sqrt{\k} \h)\|_{\H^{m-1}_l}^2d\t
\!+\es \!\int_0^t\!\!\! \|\p_{xy}(\vr,\u,\h)\|_{\H^{m-1}_l}^2 d\t\!+\!\int_0^t\! (\D_x^{m,l}\!+\!\D_y^{m,l})(\t)d\t,
\end{aligned}
\end{equation}
where $D_x^{m,l}(t)$ and $D_y^{m,l}(t)$ are defined by
\begin{equation}\label{3d3}
\D_x^{m,l}(t)=
\sum_{\substack{0 \le |\alpha| \le m \\ |\a_1| \le m-1}}\es \|\p_x \z^\a(\vr, \u, \h)(t)\|_{L^2_l(\O)}^2
+\es\|\p_x (\vr_m, \u_m, \h_m)(t)\|_{L^2_l}^2
\end{equation}
and
\begin{equation}\label{3d3-0}
\begin{aligned}
\D_y^{m,l}(t)=
&\sum_{\substack{0 \le |\alpha| \le m \\ |\a_1| \le m-1}}
\|\p_y(\sqrt{\es}\z^\a \vr,\sqrt{\mu}\z^\a \u,\sqrt{\k}\z^\a\h)(t)\|_{L^2_l(\O)}^2\\
&+\|\p_y(\sqrt{\es}\vr_m, \sqrt{\mu}\u_m, \sqrt{\k} \h_m)(t)\|_{L^2_l(\O)}^2.
\end{aligned}
\end{equation}
From the estimates in Propositions \ref{Lower-estimate}, \ref{Tanential-estimate},\ref{Normal-estimate},\ref{Infinity-estimate},
we may conclude for $T_1 \le T^\es_*$ that
\begin{equation}\label{3d4}
\n_{m,l}(t) \le C\d^{-2}(1+\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}^6)
+C_{\mu, \k}t\|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}^6
+C_{\mu, \k, m, l}\d^{-12} t \n_{m,l}^{12}(t), t \in [0, T_1].
\end{equation}
On the other hand, recall the almost equivalently relations
(see Lemma \ref{equi-control})
\begin{equation}\label{3d5}
\Theta_{m,l}(t)
\le C\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}^4
+C t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}^4
+C_l \d^{-8}\n_{m,l}^{12}(t),
\quad \forall t \in [0, T_1],
\end{equation}
and
\begin{equation}\label{3d6}
\n_{m,l}(t)
\le C\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}^4
+C t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}^4
+C\d^{-8} \Theta_{m,l}^{12}(t), \quad \forall t \in [0, T_1],
\end{equation}
and hence, we may deduce from the estimates \eqref{3d4}, \eqref{3d5} and \eqref{3d6} that
\begin{equation*}\label{3d7}
\begin{aligned}
\Theta_{m,l}(T_1)
\le
&C_{l} \mathcal{P}_0(\d^{-1}, \|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l})
+C_{\mu, \k, m, l} T_1 \mathcal{P}_1(\d^{-1}, \|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l})\\
&+C_{\mu, \k, m, l}T_1 \mathcal{P}_2(\d^{-1}, \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l})
+C_{\mu, \k, m, l} T_1 \mathcal{P}_3(\d^{-1}, R).
\end{aligned}
\end{equation*}
Choose constant $\d=\frac{\d_0}{2}$ and
$R=4 C_l \mathcal{P}_0(\d_0^{-1}, \|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l})$, we obtain
\begin{equation*}\label{3d8}
\begin{aligned}
\Theta_{m,l}(T_1)
\le
&C_l \mathcal{P}_0(\d^{-1}_0, \|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l})
+C_{\mu, \k, m, l}T_1 \mathcal{P}_4(\d^{-1}_0, \|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l})\\
&+C_{\mu, \k, m, l}T_1\mathcal{P}_5(\d^{-1}_0, \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l})
+C_{\mu, \k, m, l}T_1\mathcal{P}_6(\d^{-1}_0, \|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}).
\end{aligned}
\end{equation*}
Choose the time
$T_1=\min\{
\frac{\overline{C}_{0}}
{\mathcal{P}_4(\d^{-1}_0, \|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l})},
\frac{\overline{C}_{0}}
{\mathcal{P}_5(\d^{-1}_0, \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l})},
\frac{\overline{C}_{0}}
{\mathcal{P}_6(\d^{-1}_0, \|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l})},\}$,
and hence, it follows
\begin{equation*}\label{3d9}
\Theta_{m,l}(T_1)\le 2 C_l \mathcal{P}_0(\d^{-1}_0, \|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l})=\frac{R}{2}.
\end{equation*}
Here the constant $\overline{C}_{0}:=\frac{\mathcal{P}_0(\d^{-1}_0, \|(\rho_0, u_{10},h_{10})\|_{\overline{\mathcal{B}}^m_l})}{3C_{\mu, \k, m, l}}$.
For any smooth function $W(t, x, y)$, it is easy to justify
\begin{equation}\label{relation}
W(t, x, y)=W(0, x, y)+\int_0^t \p_s W(s, x, y)ds,
\end{equation}
Using the relation \eqref{relation} and the Sobolev inequality \eqref{sobolev}, we get
\begin{equation*}
\begin{aligned}
\h(t, x, y)+1
&\ge \h_0(x, y)+1 -Ct\sup_{0\le s \le t}\|(\h, \p_y \h)\|_{\H^2_0} \\
&\ge 2\d_0-2C_l t \sqrt{\mathcal{P}_0(\d_0^{-1}, \|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l})}.
\end{aligned}
\end{equation*}
Choose $T_2=\min\{T_1, \frac{\d_0}
{2 C_l \sqrt{\mathcal{P}_0(\d_0^{-1}, \|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l})}}\}$,
it follows
\begin{equation*}\label{3d10}
\h(t,x,y)+1 \ge \d_0=2\d,\quad \text{for~all}~(t, x, y)\in [0, T_2]\times \O.
\end{equation*}
Similarly, we get from the relation \eqref{relation} and the estimate \eqref{31012}$_2$ that
\begin{equation*}
\begin{aligned}
\|\p_y(\u-e^{-y})(t)\|_{L^\infty_1(\O)}
&\le \|\p_y(\u_0-e^{-y})\|_{L^\infty_1(\O)}
+t\sup_{0\le s \le t}\|\p_y \p_s \u(s)\|_{L^\infty_1(\O)}\\
&\le (2\d_0)^{-1}
+Ct(1+\|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}
+(\mathcal{P}_0(\d_0^{-1}, \|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}))^{\frac{3}{2}}).
\end{aligned}
\end{equation*}
Choosing $T_3=\min\{T_2, \frac{1}{2 \d_0 C(1+\|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}
+(\mathcal{P}_0(\d_0^{-1}, \|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}))^{\frac{3}{2}})} \}$,
and hence
\begin{equation*}\label{3d11}
\begin{aligned}
\|\p_y(\u-e^{-y})(t)\|_{L^\infty_1(\O)}
\le \d_0^{-1}=(2\d)^{-1},\quad \text{for~all}~t\in [0, T_3].
\end{aligned}
\end{equation*}
Finally, from the relation \eqref{relation} and the Sobolev inequality \eqref{sobolev}, we find
\begin{equation*}
\begin{aligned}
\|\vr(t)\|_{L^\infty_0(\O)}
&\le \|\vr_0\|_{L^\infty_0(\O)}+Ct\sup_{0\le s \le t}\|(\vr, \p_y \vr)(s)\|_{\H^2_0}\\
&\le \frac{2l-1}{16}\delta^2_0+2 C_l t\sqrt{\mathcal{P}_0(\d_0^{-1}, \|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l})}.
\end{aligned}
\end{equation*}
Choose
$T_4=\min\{T_3, \frac{2l-1}{64C_l \sqrt{\mathcal{P}_0(\d_0^{-1},
\|(\rho_0, u_{10},h_{10})\|_{\overline{\mathcal{B}}^m_l})}}\d_0^2\}$, we obtain
\begin{equation*}\label{3d12}
\|\vr(t)\|_{L^\infty_0(\O)}\le \frac{3(2l-1)}{32}\delta^2_0=\frac{3(2l-1)}{8}\delta^2,
\end{equation*}
for all $t\in [0, T_4]$.
Obviously, we conclude that there exists $T_4>0$ depending only on
$\mu, \k, m, l, \d_0$ and the initial data
(hence independent of parameter $\es$) such that for $T\le \min\{T_4, T^\es\}$,
the estimates \eqref{3a1} and \eqref{3a2} hold on. Of course, it holds that $T_4\le T^\es_*$.
Indeed otherwise, our criterion about the continuation of
the solution would contradict the definition of $T^\es_*$.
Then, taking $T_a=T_4$, we obtain the estimate \eqref{3a2} and closes the a priori assumptions
\eqref{a2} and \eqref{a1}. Therefore, the proof of Theorem \ref{theo a priori} is completed.
\section{Local-in-time Existence and Uniqueness}\label{local-in-times}
In this section, we will establish the local-in-times existence and uniqueness of solutions
to the inhomogeneous incompressible MHD boundary layer equations \eqref{eq3}-\eqref{bc3}.
\subsection{Existence for the MHD Boundary Layer System}
We shall use the a priori estimates obtained thus far to prove local in time
existence result. For $m \ge 5$ and $l \ge 2$, consider initial data
such that $ \|(\rho_0, u_{10}, h_{10})\|_{\mathcal{B}^m_l}\le C_0<+\infty$.
For such initial data, we are not aware of a local well-posedness result for
the equations \eqref{eq5}-\eqref{bc5}. Since $(\rho_0, u_{10}, h_{10}) \in \mathcal{B}^{m,l}_{BL}$, there
exists a sequence of smooth approximate initial data
$(\rho_0^{\sigma}, u_{10}^{\sigma}, h_{10}^{\sigma}) \in \mathcal{B}^{m,l}_{BL, ap}$
($\sigma$ being a regularization parameter), which have enough spatial regularity
so that the time derivatives at the initial time can be defined by the equation \eqref{eq3}
and boundary compatibility condition are satisfied.
Then, it follows to get a positive time $T^{\es, \sigma}>0$($T^{\es, \sigma}$
depends on $\es, \sigma$, and the initial data) for which a solution
$(\varrho^{\es, \sigma}, u^{\es, \sigma}, h^{\es, \sigma})$ exists in Sobolev spaces $H^{4m}_l(\O)$
and $(v^{\es, \sigma}, g^{\es, \sigma})$ exists in Sobolev spaces $H^{4m}_{l-1}(\O)$ respectively.
Applying the a priori estimates given in Theorem \ref{theo a priori} to
$(\varrho^{\es, \sigma}, u^{\es, \sigma}, h^{\es, \sigma})$,
we obtain a uniform time $T_a>0$ and a constant $C_1$(independent of $\es$ and $\sigma$),
such that it holds on
\begin{equation}\label{ext1}
\ta_{m,l}(\varrho^{\es, \sigma}, u^{\es, \sigma}, h^{\es, \sigma})(t) \le C_1, \quad
\|\varrho^{\es, \sigma}(t)\|_{L^\infty_0(\O)}\le \frac{2l-1}{2}\delta^2_0,\quad
\| \p_y (u^{\es, \sigma}-e^{-y})(t)\|_{L^\infty_1(\O)}\le \delta_0^{-1},
\end{equation}
and
\begin{equation}\label{ext2}
h^{\es, \sigma}(t,x,y)+1\ge \delta_0,
\end{equation}
where $t \in [0, T_0], T_0:=\min(T_a, T^{\es, \sigma})$.
Based on the uniform estimates for $(\varrho^{\es, \sigma}, u^{\es, \sigma}, h^{\es, \sigma})$, one can pass the limit
$\es \rightarrow 0^+$ and $\sigma \rightarrow 0^+$
to get a strong solution $(\varrho, u, h)$ satisfying \eqref{eq3}
by using a strong compactness arguments.
Indeed, it follows from \eqref{ext1} that $(\varrho^{\es, \sigma}, u^{\es, \sigma}, h^{\es, \sigma})$
is bounded uniformly in $L^\infty([0, T_2]; H^m_{co})$,
while $\p_y(\varrho^{\es, \sigma}, u^{\es, \sigma}, h^{\es, \sigma})$
is bounded uniformly in $L^\infty([0, T_0]; H^{m-1}_{co})$,
and $\p_t (\varrho^{\es, \sigma}, u^{\es, \sigma}, h^{\es, \sigma})$
is bounded uniformly in $L^\infty([0, T_0]; H^{m-1}_{co})$.
Then, it follows from a strong compactness argument that $(\varrho^{\es, \sigma}, u^{\es, \sigma}, h^{\es, \sigma})$
is compact in $\mathcal{C}([0, T_0]; H^{m-1}_{co,loc})$.
Due to $\kappa>0$, it is easy to check that
$h^{\es, \sigma}$ is compact in $\mathcal{C}([0, T_0]; H^{2}_{loc})$.
In particular, there exists a sequence $\es_n, \sigma_n \rightarrow 0^+$
and $(\varrho, u, h)\in \mathcal{C}([0, T_0]; H^{m-1}_{co,loc})$ such that
$$
(\varrho^{\es_n, \sigma_n}, u^{\es_n, \sigma_n}, h^{\es_n, \sigma_n}) \rightarrow (\varrho, u, h)~~ {\rm in}~ ~
\mathcal{C}([0, T_0]; H^{m-1}_{co,loc})~ ~{\rm as} ~~\es^n, \sigma^n \rightarrow 0^+,
$$
and
$$
h^{\es_n, \sigma_n} \rightarrow h ~~ {\rm in}~ ~
\mathcal{C}([0, T_0]; H^{2}_{loc})~ ~{\rm as} ~~\es^n, \sigma^n \rightarrow 0^+,
$$
Furthermore, we apply the Sobolev inequality to get
\begin{equation*}
\begin{aligned}
&\underset{0\le \t \le t}{\sup}\|(\p_y^{-1} \p_x u^{\es_n, \sigma_n}-\p_y^{-1} \p_x u)(\t)\|_{L^\infty_{0, co}(\O)}\\
&\le C\underset{0\le \t \le t}{\sup} \|\p_y^{-1}\p_x (u^{\es_n, \sigma_n}-u)(\t)\|_{H^{1}_{co, loc}}^{\frac{1}{2}}
\|\p_x (u^{\es_n, \sigma_n}-u)(\t)\|_{H^{1}_{co, loc}}^{\frac{1}{2}}\\
&\le C\underset{0\le \t \le t}{\sup} \|(u^{\es_n, \sigma_n}-u)(\t)\|_{H^{2}_{co, loc}}
\rightarrow 0, ~~{\rm as}~~\es^n, \sigma^n \rightarrow 0^+.
\end{aligned}
\end{equation*}
Hence we denote $v(t, x, y)=-\int_0^y \p_x u(t, x, \xi)d\xi$,
which satisfies the divergence-free condition $\p_x u+\p_y v=0$.
Similarly, we denote $g(t, x, y)=-\int_0^y h(t, x, \xi)d\xi$, which satisfies
\begin{equation*}
\underset{0\le \t \le t}{\sup}\|(g^{\es_n, \sigma_n}- g)(\t)\|_{L^\infty_{0, co}(\O)}
\le C\underset{0\le \t \le t}{\sup} \|(h^{\es_n, \sigma_n}-h)(\t)\|_{H^{1}_{co, loc}}
\rightarrow 0, ~~{\rm as}~~\es^n, \sigma^n \rightarrow 0^+.
\end{equation*}
By routine checking, we may show that $(\rho, u_1, u_2, h_1, h_2):=(\varrho+1, u+1-e^{-y}, v, h+1, g)$
is a solution of the original MHD boundary layer system \eqref{eq3}.
Finally, applying the lower semicontinuity of norms to the bound \eqref{ext1},
one obtains the estimate \eqref{main-estimate} for the solution $(\rho, u_1, h_1)$.
Since $h^{\es_n, \sigma_n}$ converges uniformly to $h$, then we can get $h_1\ge \d$ from \eqref{ext2}.
\subsection{Uniqueness for the MHD Boundary Layer System}
In this subsection, we will show the uniqueness of solution to the MHD boundary
layer equations \eqref{eq3}-\eqref{bc3}.
Let $(\rho_1, u_1, v_1, h_1, g_1)$ and $(\rho_2, u_2, v_2, h_2, g_2)$ be two solutions
in the existence time $[0, T_a]$, constructed in the previous subsection,
with respect to the initial data $(\rho_0^1, u_0^1, h_0^1)$ and $(\rho_0^2, u_0^2, h_0^2)$ respectively.
Let us set
$$
(\overline{\rho}, \overline{u}, \overline{v}, \overline{h}, \overline{g})
=(\rho_1-\rho_2, u_1-u_2, v_1-v_2, h_1-h_2, g_1-g_2),
$$
then they satisfy the following evolution
\begin{equation}\label{eq7}
\left\{
\begin{aligned}
&\p_t \orho+u_1 \p_x \orho+v_1 \p_y \orho+\ou \p_x \rho_2+ \ov \p_y \rho_2=0,\\
&\rho_1\p_t \ou+\rho_1 u_1 \p_x \ou+\rho_1 v_1 \p_y \ou+\rho_1 \ou \p_x u_2+\rho_1 \ov \p_y u_2-\mu \p_y^2 \ou \\
&\quad =-\orho \p_t u_2-\orho u_2 \p_x u_2-\orho v_2 \p_y u_2
+\oh \p_x h_1+h_2 \p_x \oh+g_1 \p_y \oh+\og \p_y h_2,\\
&\p_t \oh+u_1 \p_x \oh+v_1 \p_y \oh+\ou \p_x h_2+\ov \p_y h_2-\k \p_y^2 \oh
=\oh \p_x u_1+h_2 \p_x \ou+g_1 \p_y \ou+\og \p_y u_2,\\
&\p_x \ou+\p_y \ov=0,\quad \p_x \oh+\p_y \og=0,\\
\end{aligned}
\right.
\end{equation}
with the boundary condition and initial data
\begin{equation*}
(\ou, \ov, \p_y \oh, \og)|_{y=0}=\mathbf{0},
\quad \lim_{y \rightarrow +\infty}(\orho, \ou, \oh)=\mathbf{0}, \quad
(\orho, \ou, \oh)|_{t=0}=\mathbf{0}.
\end{equation*}
Here we assume the two solutions $(\rho_1, u_1, v_1, h_1, g_1)$ and $(\rho_2, u_2, v_2, h_2, g_2)$
have the same initial data $(\rho_0^1, u_0^1, h_0^1)=(\rho_0^2, u_0^2, h_0^2)$.
Denote by $\ophi:=\p_y^{-1}\oh=\p_y^{-1}(h_1-h_2)$, it follows
\begin{equation*}
\p_t \ophi+u_1 \p_x \ophi+ v_1 \p_y \ophi-\ou g_2+\ov h_2-\k \p_y^2 \ophi=0.
\end{equation*}
Define
$
\eta_1:=\frac{\p_y \rho_2}{h_2}, \
\eta_2:=\frac{\p_y u_2}{h_2}, \
\eta_3:=\frac{\p_y h_2}{h_2},
$
and introduce the new quantities:
\begin{equation}\label{eqi-quantity}
\irho:=\orho-\eta_1 \ophi,\quad \iu:=\ou-\eta_2 \ophi,\quad \ih:=\oh-\eta_3 \ophi.
\end{equation}
Next, we can obtain that through direct calculation, $(\irho, \iu, \ih)$ satisfies the following
initial boundary value problem:
\begin{equation}\label{uniq-eq}
\left\{
\begin{aligned}
&\p_t \irho+u_1 \p_x \irho+v_1 \p_y \irho=-\k \eta_1 \p_y \ih
-a_{11}\ou- a_{12}\oh-a_{13} \ophi,\\
&\rho_1\p_t \iu+\rho_1 u_1 \p_x \iu+\rho_1 v_1 \p_y \iu
-h_2 \p_x \ih-g_1 \p_y \ih-\mu \p_y^2 \iu,\\
&\quad =-\k \rho_1 \eta_2 \p_y \ih+\mu \p_y(\p_y \eta_2 \ophi+\eta_2 \oh)
-a_{21}\orho-a_{22}\ou- a_{23}\oh-a_{24} \ophi,\\
&\p_t \ih+u_1 \p_x \ih+v_1 \p_y \ih-h_2 \p_x \iu-g_1 \p_y \iu-\k \p_y^2 \ih
=-\k \eta_3 \p_y \ih-a_{31}\ou- a_{32}\oh-a_{33} \ophi,
\end{aligned}
\right.
\end{equation}
where
\begin{equation*}
\left\{
\begin{aligned}
&a_{11}=\p_x \rho_2+\eta_1 g_2, \quad a_{12}=\k \eta_1 \eta_3,
\quad a_{13}=\p_t \eta_1+u_1 \p_x \eta_1+v_1 \p_y \eta_1+\k \eta_1 \p_y \eta_3,\\
&a_{21}=\p_t u_2 +u_2 \p_x u_2+ v_2 \p_y u_2,\quad
a_{22}=\rho_1 \p_x u_2+\rho_1 \eta_2 g_2,\quad
a_{23}=\k \rho_1 \eta_2 \eta_3-\p_x h_1-g_1 \eta_3,\\
&
a_{24}=\rho_1 \p_t \eta_2+\rho_1 u_1 \p_x \eta_2+\rho_1 v_1 \p_y \eta_2
+\k \rho_1 \eta_2 \p_y \eta_3-h_2 \p_x \eta_3-g_1 \p_y \eta_3,\\
&a_{31}=\eta_3 g_2+\p_x h_2,\quad
a_{32}=\k \eta_3^2+2\k \p_y \eta_3-g_1 \eta_2-\p_x u_1,\\
& a_{33}=\p_t \eta_3+u_1 \p_x \eta_3+v_1 \p_y \eta_3+\k \eta_3 \p_y \eta_3
-h_2 \p_x \eta_2-g_1 \p_y \eta_2.
\end{aligned}
\right.
\end{equation*}
By virtue of the relation
$\ih:=\oh-\frac{\p_y h_2}{h_2}\ophi =h_2\p_y(\frac{\ophi}{h_2})$, it is easy to justify
$\frac{\ophi}{h_2}=\p_y^{-1}(\frac{\ih}{h_2})$,
and hence, we can apply the Hardy inequality to obtain the estimate:
$$
\|\frac{\ophi}{h_2}\|_{L^2_{-1}(\O)}\le C\|\frac{1}{h_2}\|_{L^\infty_0(\O)}\|\ih\|_{L^2_0(\O)}.
$$
Thus we can apply the standard energy method for the equation \eqref{uniq-eq}
to establish the following estimate, which we omit the proof for brevity of presentation.
\begin{proposition}\label{uniqueness-energy}
Let $(\rho_1, u_1, v_1, h_1, g_1)$ and $(\rho_2, u_2, v_2, h_2, g_2)$
be two solutions of MHD boundary layer equations \eqref{eq3}-\eqref{bc3} with the same initial data,
and satisfying the estimate \eqref{main-estimate} respectively.
Then, there exists a positive constant
$$
C=C(T_a, \d_0, \|(\rho_1, u_1, h_1)(t)\|_{\overline{\mathcal{B}}^m_l},
\|(\rho_2, u_2, h_2)(t)\|_{\overline{\mathcal{B}}^m_l})>0,
$$
such that the quantity $(\irho, \iu, \ih)$ given by \eqref{eqi-quantity} satisfies
\begin{equation}\label{unique-estimate}
\|(\irho, \iu, \ih)(t)\|_{L^2}^2
+\int_0^t \|\p_y(\sqrt{\mu}\ \iu, \sqrt{\k}\ \ih)(\t)\|_{L^2}^2 d\t
\le C\int_0^t \|(\irho, \iu, \ih)(\t)\|_{L^2}^2 d\t.
\end{equation}
\end{proposition}
Then, we can prove the uniqueness of the solutions to \eqref{eq3}-\eqref{bc3} as follows.
\begin{proof}[\textbf{Proof of Uniqueness.}]
Applying Gronwall's lemma to the estimate \eqref{unique-estimate}, we obtain $ (\irho, \iu, \ih)\equiv 0 $.
Then, we substitute $\ih \equiv 0$ into equality $\frac{\ophi}{h_2}=\p_y^{-1}(\frac{\ih}{h_2})$ to get $\ophi \equiv 0$.
From the definition \eqref{eqi-quantity} we get $(\rho_1, u_1, h_1)\equiv (\rho_2, u_2, h_2)$ due to the fact
$ (\irho, \iu, \ih)\equiv 0 $ and $\ophi \equiv 0$.
Finally, it follows from the divergence-free condition and the boundary condition $\ov|_{y=0}=0$ that
$\ov=-\p_y^{-1} \p_x \ou=0$, which implies the fact $v_1 \equiv v_2$.
Similarly, it holds on $g_1 \equiv g_2$.
Therefore, we complete the proof of uniqueness of the solution for the
MHD boundary layer equations \eqref{eq3}-\eqref{bc3} completely.
\end{proof}
\appendix
\section{Calculus Inequalities}\label{appendixA}
In this appendix, we will introduce some basic inequalities that be used frequently in this paper.
First of all, we introduce the following Hardy type inequality, which can refer to \cite{Masmoudi}.
\begin{lemma}\label{Hardy}
Let the proper function $f:\mathbb{T}\times \mathbb{R}^+ \rightarrow \mathbb{R}$,
and satisfies $f(x, y)|_{y=0}=0$ and $ \underset{{y \to +\infty}}{\lim} f(x,y) = 0$.
If $\lambda > - \frac{1}{2}$, then it holds on
\begin{equation} \label{Hardy}
\|f\|_{L^2_\lambda (\mathbb{T}\times \mathbb{R}^+)} \le \frac{2}{2\lambda +1}
\|\partial_y f\|_{L^2_{\lambda+1} (\mathbb{T}\times \mathbb{R}^+)}.
\end{equation}
\end{lemma}
Next, we will state the following Sobolev-type inequality.
\begin{lemma}
Let the proper function $f:\mathbb{T}\times \mathbb{R}^+ \rightarrow \mathbb{R}$,
and satisfies $ \underset{{y \to +\infty}}{\lim} f(x,y) = 0$.
Then there exists a universal constant $C>0$ such that
\begin{equation}\label{sobolev-1}
\|f\|_{L^\infty_0(\mathbb{T}\times \mathbb{R}^+)}
\le C(\|\p_y f\|_{L^2_0(\mathbb{T}\times \mathbb{R}^+)}+\|\p_{xy}^2 f\|_{L^2_0(\mathbb{T}\times \mathbb{R}^+)})^{\frac{1}{2}}
(\|f\|_{L^2_0(\mathbb{T}\times \mathbb{R}^+)}+\|\p_x f\|_{L^2_0(\mathbb{T}\times \mathbb{R}^+)})^{\frac{1}{2}},
\end{equation}
or equivalently
\begin{equation}\label{sobolev}
\|f\|_{L^\infty_0(\mathbb{T}\times \mathbb{R}^+)}
\le C(\|f\|_{L^2_0(\mathbb{T}\times \mathbb{R}^+)}+\|\partial_x f\|_{L^2_0(\mathbb{T}\times \mathbb{R}^+)}
+\|\partial_y f\|_{L^2_0(\mathbb{T}\times \mathbb{R}^+)}+\|\partial_{xy}^2 f\|_{L^2_0(\mathbb{T}\times \mathbb{R}^+)}).
\end{equation}
\end{lemma}
\begin{proof}
Indeed, the estimate \eqref{sobolev} follows directly from estimate
\eqref{sobolev-1} and the Cauchy-Schwartz inequality. Hence, we only give the proof
for the estimate \eqref{sobolev-1}.
On one hand, thanks to the one-dimensional Sobolev
inequality for the $y-$variable, we get
\begin{equation}\label{a21}
|f(x, y)|^2 \le C(\int_0^\infty |\p_\xi f(x, \xi)|^2d\xi)^{\frac{1}{2}}
(\int_0^\infty |f(x, \xi)|^2d\xi)^{\frac{1}{2}}.
\end{equation}
On the other hand, we apply the following one-dimensional Sobolev
inequality for $x-$variable to get
\begin{equation}\label{a22}
|f(x, y)|^2\le C(\|f(y)\|_{L^2(\mathbb{T})}^2+\|\p_x f(y)\|_{L^2(\mathbb{T})}^2),
\quad
|\p_y f(x, y)|^2\le C(\|\p_y f(y)\|_{L^2(\mathbb{T})}^2+\|\p_{xy} f(y)\|_{L^2(\mathbb{T})}^2).
\end{equation}
Therefore, substituting the estimate \eqref{a22} into \eqref{a21}, we complete
the proof of estimate \eqref{sobolev-1}.
\end{proof}
Now we will state the Moser type inequality as follow:
\begin{lemma}\label{moser}
Denote $\O:=\mathbb{T}\times \mathbb{R}^+$, let the proper
functions $f(t, x, y): \mathbb{R}^+\times \O \rightarrow \mathbb{R}$
and $g(t, x, y): \mathbb{R}^+ \times \O \rightarrow \mathbb{R}$.
Then, there exists a constant $C_m>0$ such that
\begin{equation}\label{ineq-moser}
\int_0^t \|(\z^{\b} f \z^{\ga}g)(\t)\|_{L^2_l(\O)}^2d\t
\le C_m(\|\la y \ra^{l_1} f\|_{L^\infty_{x,y,t}}^2\int_0^t \|g\|_{\H^{m}_{l_2}}^2 d\t
+\|\la y \ra^{l_1} g\|_{L^\infty_{x,y,t}}^2\int_0^t \|f\|_{\H^{m}_{l_2}}^2 d\t),
\end{equation}
where $|\b+\ga|=m$ and $l_1+l_2=l$.
\end{lemma}
\begin{proof}
For any $p\ge 2$, due to the relation $|Z_2 f|^p=Z_2(f Z_2 f |Z_2 f|^{p-2})-(p-1)f Z_2^2 f|Z_2 f|^{p-2}$,
we find
\begin{equation*}
\int_{\mathbb{R}^+} \la y \ra^{\theta l p}|Z_2 f|^p dy
=\int_{\mathbb{R}^+}\la y \ra^{\theta l p} Z_2(f Z_2 f |Z_2 f|^{p-2}) dy
-(p-1) \int_{\mathbb{R}^+}\la y \ra^{\theta l p}f Z_2^2 f|Z_2 f|^{p-2}dy.
\end{equation*}
Integrating by part and applying the H\"{o}lder inequality, we find for $0 \le \theta \le 1$
and $0\le \theta_1 \le \frac{\theta}{2}$ that
\begin{equation*}
\begin{aligned}
\|\la y \ra^{\theta l} Z_2 f\|_{L_0^p(\mathbb{R}^+)}^p
&\le C_p \int_{\mathbb{R}^+}\la y \ra^{\theta l p-1} |f|(|Z_2 f|+|Z_2^2 f|)|Z_2 f|^{p-2} dy\\
&\le C_p\|\la y \ra^{\theta l} Z_2 f\|_{L_0^p(\mathbb{R}^+)}^{p-2}
\|\la y \ra^{\theta_1 l} f\|_{L_0^q(\mathbb{R}^+)}
\|\la y \ra^{(2\theta-\theta_1) l}(|Z_2 f|,|Z_2^2 f|)\|_{L_0^r(\mathbb{R}^+)},
\end{aligned}
\end{equation*}
and hence, it follows
\begin{equation*}
\|\la y \ra^{\theta l} Z_2 f\|_{L_0^p(\mathbb{R}^+)}^2
\le C_p \|\la y \ra^{\theta_1 l} f\|_{L_0^q(\mathbb{R}^+)}
\sum_{1\le k \le 2}\|\la y \ra^{(2\theta-\theta_1) l}Z_2^k f\|_{L_0^r(\mathbb{R}^+)}.
\end{equation*}
Here $\frac{1}{q}+\frac{1}{r}=\frac{2}{p}$.
Integrating with $t$ and $x$ variables, and applying H\"{o}lder inequality, we get
\begin{equation*}
\|\la y \ra^{\theta l} Z_2 f\|_{L^p(Q_T)}^2
\le C_p \|\la y \ra^{\theta_1 l} f\|_{L^q(Q_T)}
\sum_{1\le k \le 2}\|\la y \ra^{(2\theta-\theta_1) l}Z_2^k f\|_{L^r(Q_T)}.
\end{equation*}
Here $Q_T=\O \times [0, T]$.
Similarly, it is easy to justify for $i=0,1$,
\begin{equation*}
\|\la y \ra^{\theta l} Z_i f\|_{L^p_0(Q_T)}^2
\le C_p \|\la y \ra^{\theta_1 l} f\|_{L^q_0(Q_T)}
\sum_{1\le k \le 2}\|\la y \ra^{(2\theta-\theta_1) l}Z_i^k f\|_{L^r_0(Q_T)}.
\end{equation*}
Here $\frac{1}{q}+\frac{1}{r}=\frac{2}{p}$.
By multiple application of the above inequality, we get(proof by induction)
\begin{equation*}
\|\la y \ra^{(|\a|\theta-(|\a|-1)\theta_1)l} \z^\a f\|_{L^{p_1}_0(Q_T)}
\le C_p \|\la y \ra^{\theta_1 l} f\|_{L^{q_1}_0(Q_T)}^{1-\frac{|\a|}{m}}
\sum_{1\le |\b| \le m}\|\la y \ra^{(m\theta-(m-1)\theta_1) l} \z^{\b} f\|_{L^{r_1}_0(Q_T)}^{{\frac{|\a|}{m}}},
\end{equation*}
where $\frac{1}{p_1}=\frac{1}{q_1}(1-\frac{|\a|}{m})+\frac{|\a|}{r_1 m}$ and $1\le |\a| \le m-1$.
Then, we get for $|\beta|+|\gamma|=m$ that
\begin{equation*}
\begin{aligned}
\|\la y \ra^{l}\z^{\b}f \z^{\ga} g\|_{L^2_0(Q_T)}^2
&\le C \|\la y \ra^{\frac{|\b|}{m}l+(1-\frac{2|\b|}{m})l_1} \z^{\b}f \|_{L^{\frac{2m}{|\b|}}_0(Q_T)}^2
\|\la y \ra^{\frac{|\ga|}{m}l+(\frac{2|\b|}{m}-1)l_1}\z^{\ga} f\|_{L^{\frac{2m}{|\ga|}}_0(Q_T)}^2\\
&\le C_m \|\la y \ra^{l_1}f\|_{L^\infty_0(Q_T)}^{2(1-\frac{|\b|}{m})}
\sum_{1\le |\b| \le m}\|\la y \ra^{l-l_1}\z^{\b}f\|_{L^2_0(Q_T)}^{\frac{2|\b|}{m}}\\
&\quad \quad \times
\|\la y \ra^{l_1} g\|_{L^\infty_0(Q_T)}^{2(1- \frac{|\ga|}{m})}
\sum_{1\le |\ga| \le m}\|\la y \ra^{l-l_1}\z^{\ga} g\|_{L^2_0(Q_T)}^{\frac{2|\ga|}{m}}\\
&\le C_m \|f\|_{L^\infty_{l_1}(Q_T)}^2\sum_{1\le |\b| \le m}\|\z^\b g\|_{L^2_{l-l_1}(Q_T)}^2
+C_m \|g\|_{L^\infty_{l_1}(Q_T)}^2\sum_{1\le |\b| \le m}\|\z^\b f\|_{L^2_{l-l_1}(Q_T)}^2.
\end{aligned}
\end{equation*}
Therefore, we complete the proof of this lemma.
\end{proof}
Finally, we establish the following $L^\infty-$estimate with weight for the heat equation.
\begin{lemma}\label{A-heat}
For the heat equation $\p_t F(t, x)-\es \p_x^2 F(t, x)=G(t, x), \ (t, x) \in \mathbb{R}^+\times \mathbb{R}^+$;
with the boundary condition $F(t, x)|_{x=0}=0$ and initial data $F(t, x)|_{t=0}=F_0$.
Then, it holds on
\begin{equation}\label{eheat}
\|x\p_x F\|_{L^\infty_0(\mathbb{R}^+)}\le C(\|F_0\|_{L^\infty_0(\mathbb{R}^+)}+\|x\p_x F_0\|_{L^\infty_0(\mathbb{R}^+)})
+C\int_0^t (\|G\|_{L^\infty_0(\mathbb{R}^+)}+\|x\p_x G\|_{L^\infty_0(\mathbb{R}^+)}) d\t,
\end{equation}
where $C$ is a constant independent of the parameter $\es$.
\end{lemma}
\begin{proof}
First of all, let us consider the heat equation
\begin{equation}\label{a41}
\p_t H(t, x)-\es \p_x^2 H(t, x)=0,\quad (t, x) \in \mathbb{R}^+\times \mathbb{R}^+;
\end{equation}
with the initial data and boundary condition
\begin{equation*}
H(t, x)|_{t=0}=H_0(x),\quad x \in \mathbb{R}^+;
\quad H(t, x)|_{x=0}=0,\quad t \in \mathbb{R}^+.
\end{equation*}
In order to transform the problem \eqref{a41} into a problem in the whole space,
let us define $\widetilde{H}(t, x)$ by
\begin{equation*}\label{a42}
\widetilde{H}(t, x)= H(t, x), \ x>0;\quad
\widetilde{H}(t, x)= -H(t, -x), \ x<0,
\end{equation*}
and define the initial data $\widetilde{H}_0(x)$ by
\begin{equation*}
\widetilde{H}_0(x)= H_0(x), \ x>0;\quad
\widetilde{H}_0(x)= -H_0(-x), \ x<0.
\end{equation*}
It is easy to justify that the function $\widetilde{H}(t, x)$ solves the following evolution equation
\begin{equation}\label{a43}
\p_t \widetilde{H}(t, x)-\es \p_x^2 \widetilde{H}(t, x)=0,\quad (t, x) \in \mathbb{R}^+\times \mathbb{R};
\quad \widetilde{H}(t, x)|_{t=0}=\widetilde{H}_0(x), \quad x \in \mathbb{R}.
\end{equation}
Define $S(t, x)=\frac{1}{\sqrt{4\pi \es t}}e^{-\frac{|x|^2}{\sqrt{4\es t}}}$,
then the solution of evolution \eqref{a43} can be expressed as
\begin{equation}\label{seq}
\widetilde{H}(t, x)=\int_{\mathbb{R}} \widetilde{H}_0(\xi)S(t, x-\xi)d\xi,
\end{equation}
which implies directly
\begin{equation*}\label{a44}
x \p_x \widetilde{H}(t, x)=\int_{\mathbb{R}} \widetilde{H}_0(\xi)x \p_x S(t, x-\xi)d\xi.
\end{equation*}
In view of the relation
$x \p_x S(t, x-\xi)=(x-\xi)\p_x S(t, x-\xi)+\xi\p_x S(t, x-\xi)$, we get
\begin{equation*}\label{a45}
x \p_x \widetilde{H}(t, x)=\int_{\mathbb{R}} \widetilde{H}_0(\xi)(x-\xi) \p_x S(t, x-\xi)d\xi
+\int_{\mathbb{R}} \widetilde{H}_0(\xi)\xi \p_x S(t, x-\xi)d\xi.
\end{equation*}
Due to $\int_{\mathbb{R}}|(x-\xi) \p_x S(t, x-\xi)|d\xi \le C$, it follows
\begin{equation*}\label{a46}
|\int_{\mathbb{R}} \widetilde{H}_0(\xi)(x-\xi) \p_x S(t, x-\xi)d\xi|
\le C\|\widetilde{H}_0\|_{L^\infty_0(\mathbb{R})}.
\end{equation*}
Using the equality $\p_x S(t, x-\xi)=-\p_{\xi} S(t, x-\xi)$, the integration by part yields directly
\begin{equation*}\label{a47}
|\int_{\mathbb{R}}\widetilde{H}_0(\xi)\xi \p_x S(t, x-\xi)d\xi|
\le C(\|\widetilde{H}_0\|_{L^\infty_0(\mathbb{R})}
+\|x \p_x \widetilde{H}_0\|_{L^\infty_0(\mathbb{R})}),
\end{equation*}
and hence, we get
\begin{equation*}\label{a48}
\begin{aligned}
\|x \p_x H (t, x)\|_{L^\infty_0(\mathbb{R}^+)} \le
\|x \p_x \widetilde{H}(t, x)\|_{L^\infty_0(\mathbb{R})}
\le C \|(\widetilde{H}_0, x \p_x \widetilde{H}_0)\|_{L^\infty_0(\mathbb{R})})
\le C (\|{H}_0\|_{L^\infty_0(\mathbb{R}^+)}
+\|x \p_x {H}_0\|_{L^\infty_0(\mathbb{R}^+)}).
\end{aligned}
\end{equation*}
This, along with representation \eqref{seq} and the
well-known Duhamel formula, we complete the proof of this lemma.
\end{proof}
\section{Almost Equivalence of Weighted Norms}\label{appendixB}
In this subsection we will use the quantity $\h_m$ in weighted norm,
and $\h$ and its derivatives in $L^\infty$ norm to control the quantities $Z_\t^{\a_1}\h$ and $Z_\t^{\a_1}\ps$
in weighted norm. To derive these estimates, we shall apply the Lemma \ref{Hardy},
which was introduced previously in Section \ref{appendixA}.
\begin{lemma}\label{equivalent}
Let the stream function $\ps(t, x, y)$ satisfies
$\p_y \ps=\h, \p_x \ps = -\g, \ps|_{y=0}=0$.
There exists a constant $\d\in (0, 1)$, such that $\h(t, x, y)+1\ge \d, \forall(t, x, y)\in [0, T]\times \O$.
Then, for $l \ge 1$ and $|\a_1|=m$, we have the following estimates:
\begin{equation}\label{b11}
\|\frac{Z^{\a_1}_\t \ps}{\h+1}\|_{L^2_{l-1}(\O)}\le \frac{2 \d^{-1}}{2l-1} \| \h_m \|_{L^2_l(\O)},
\end{equation}
\begin{equation}\label{b12}
\|Z^{\a_1}_\t \h\|_{L^2_l}(\O) \le \|\h_m\|_{L^2_l(\O)}
+\frac{2 \d^{-1}}{2l-1} \|\p_y \h\|_{L^\infty_1(\O)} \| \h_m \|_{L^2_l(\O)},
\end{equation}
\begin{equation}\label{b13}
\|\frac{\p_x Z^{\a_1}_\t \ps}{\h+1}\|_{L^2_{l-1}(\O)}
\le \frac{2\d^{-1}}{2l-1} \|\p_x \h_m\|_{L^2_{l}(\O)}
+\frac{4\d^{-2}}{2l-1} \|\p_x \h\|_{L^\infty_0(\O)}\|\h_m\|_{L^2_{l}(\O)},
\end{equation}
\begin{equation}\label{b14}
\|\p_y Z^{\a_1}_\t \h\|_{L^2_l(\O)}
\le \|\p_y \h_m \|_{L^2_l(\O)}+C_l \d^{-1}(\|\p_y \h \|_{L^\infty_0(\O)}
+\|Z_2 \p_y \h\|_{L^\infty_1(\O)})\| \h_m\|_{L^2_l(\O)},
\end{equation}
where the constant $C_l$ depends only on $l$.
\end{lemma}
\begin{proof}
(i) By virtue of the definition $\h_m=Z_\t^{\a_1}\h-\frac{\p_y \h}{\h+1} Z_\t^{\a_1} \ps$,
it is easy to obtain $\h_m=(\h+1)\p_y(\frac{Z_\t^{\a_1} \ps}{\h+1})$.
Integrating over $[0, y]$ and applying the boundary condition $\ps|_{y=0}=0$, we have
\begin{equation}\label{b15}
\frac{Z_\t^{\a_1} \ps}{\h+1}= \int_0^y \frac{\h_m}{\h+1}d\xi,
\end{equation}
and along with the Hardy inequality \eqref{Hardy}, yields directly
\begin{equation}\label{b16}
\| \frac{Z_\t^{\a_1} \ps}{\h+1} \|_{L^2_{l-1}(\O)}
\le \| \int_0^y \frac{\h_m}{\h+1}d\xi\|_{L^2_{l-1}(\O)}
\le \frac{2}{2l-1}\|\frac{\h_m}{\h+1} \|_{L^2_l(\O)}
\le \frac{2 \d^{-1}}{2l-1} \| \h_m \|_{L^2_l(\O)},
\end{equation}
where we have used the fact $\h+1\ge \d$ in the last inequality.
(ii)In view of the relation $Z_\t^{\a_1}\h=\h_m+\frac{\p_y \h}{\h+1} Z_\t^{\a_1} \ps$, we get
\begin{equation*}
\|Z^{\a_1}_\t \h\|_{L^2_l(\O)}
\le \|\h_m\|_{L^2_l(\O)}+\|\p_y \h\|_{L^\infty_1(\O)}
\|\frac{ Z^{\a_1}_\t \ps}{\h+1}\|_{L^2_{l-1}(\O)},
\end{equation*}
which, together with estimate \eqref{b16}, yields directly
\begin{equation}\label{b17}
\|Z^{\a_1}_\t \h\|_{L^2_l(\O)}
\le \|\h_m\|_{L^2_l(\O)}+\frac{2 \d^{-1}}{2l-1} \|\p_y \h\|_{L^\infty_1(\O)} \| \h_m \|_{L^2_l(\O)}.
\end{equation}
(iii)Differentiating the equality ${Z_\t^{\a_1} \ps}=(\h+1)\int_0^y \frac{\h_m}{\h+1}d\xi$
with $x$ variable, we find
\begin{equation*}
\p_x Z_\t^{\a_1} \ps
=\p_x \h \int_0^y \frac{\h_m}{\h+1} d\xi
+(\h+1)\int_0^y \frac{\p_x \h_m}{\h+1}d\xi
-(\h+1)\int_0^y \frac{\h_m \p_x \h}{(\h+1)^2}d\xi,
\end{equation*}
which, implies that
\begin{equation*}
\frac{\p_x Z_\t^{\a_1} \ps}{\h+1}
=\frac{\p_x \h}{\h+1}\int_0^y \frac{\h_m}{\h+1} d\xi
+\int_0^y \frac{\p_x \h_m}{\h+1}d\xi
-\int_0^y \frac{\h_m \p_x \h}{(\h+1)^2}d\xi,
\end{equation*}
and hence, we apply the Hardy inequality \eqref{Hardy} and $\h+1 \ge \d$ to get
\begin{equation}\label{b18}
\begin{aligned}
\|\frac{\p_x Z^{\a_1}_\t \ps}{\h+1}\|_{L^2_{l-1}(\O)}
&\le \frac{4}{2l-1}\|\frac{\p_x \h}{\h+1}\|_{L^\infty_0(\O)}\|\frac{\h_m}{\h+1}\|_{L^2_{l}(\O)}
+\frac{2}{2l-1} \| \frac{\p_x \h_m}{\h+1} \|_{L^2_{l}(\O)}\\
&\le \frac{2\d^{-1}}{2l-1} \|\p_x \h_m\|_{L^2_{l}(\O)}
+\frac{4\d^{-2}}{2l-1} \|\p_x \h\|_{L^\infty_0(\O)}\|\h_m\|_{L^2_{l}(\O)}.
\end{aligned}
\end{equation}
(iv)Differentiating the equality $Z_\t^{\a_1}\h=\h_m+\frac{\p_y \h}{\h+1} Z_\t^{\a_1} \ps$
with the $y$ variable, it follows
\begin{equation*}
\p_y Z^{\a_1}_\t \h
=\p_y \h_m +\p_y^2 \h \frac{Z^{\a_1}_\t \ps}{\h+1}+\p_y \h \p_y ( \frac{Z^{\a_1}_\t \ps}{\h+1} ),
\end{equation*}
which, together with the relation \eqref{b15}, yields
\begin{equation*}
\begin{aligned}
\p_y Z^{\a_1}_\t \h
&=\p_y \h_m +\p_y^2 \h \int_0^y \frac{\h_m}{\h+1}d\xi+\eta_h \h_m\\
&=\p_y \h_m +Z_2 \p_y \h \frac{1}{\varphi(y)}\int_0^y \frac{\h_m}{\h+1}d\xi+\eta_h \h_m,
\end{aligned}
\end{equation*}
where $\varphi(y)=\frac{y}{1+y}$ and $\eta_h=\frac{\p_y \h}{\h+1}$.
The application of Hardy inequality \eqref{Hardy} and $\h+1\ge \d$ yields
\begin{equation}\label{b19}
\begin{aligned}
\|\p_y Z^{\a_1}_\t \h\|_{L^2_l(\O)}
\le& \|\p_y \h_m \|_{L^2_l(\O)}+\frac{2\d^{-1}}{2l-1}\|Z_2 \p_y \h\|_{L^\infty_1(\O)}
\| \h_m \|_{L^2_{l}(\O)}\\
& +\d^{-1}\|\p_y \h \|_{L^\infty_0(\O)}\| \h_m\|_{L^2_l(\O)}\\
\le& \|\p_y \h_m \|_{L^2_l(\O)}
+\frac{(2l+1) \d^{-1}}{2l-1}(\|\p_y \h \|_{L^\infty_0(\O)}
+\|Z_2 \p_y \h\|_{L^\infty_1(\O)})\| \h_m\|_{L^2_l(\O)}.
\end{aligned}
\end{equation}
Therefore, the estimates \eqref{b16}-\eqref{b19} imply the estimates \eqref{b11}-\eqref{b14}.
\end{proof}
Let us define
\begin{equation}\label{xdef}
Y_{m,l}(t):=1+{{\|(\vr, \u, \h)(t)\|_{\H^m_l}^2}}+\|\p_y(\vr, \u, \h)(t)\|_{\H^{m-1}_l}^2.
+\| \p_y \vr (t)\|_{\H^{1,\infty}_1}^2,
\end{equation}
and hence we will establish the following almost equivalent relation.
\begin{lemma}\label{equi-control}
Let $(\vr, \u, \v, \h, \g)$ be sufficiently smooth solution, defined on $[0, T^\es]$,
to the regularized MHD boundary layer equations \eqref{eq5}-\eqref{bc5}.
There exists a constant $\d\in (0, 1)$, such that $\h(t, x, y)+1\ge \d, \forall(t, x, y)\in [0, T]\times \O$.
Then, for $m \ge 4$ and $l \ge 1$, it holds on
\begin{equation}\label{ab1}
\Theta_{m,l}(t)
\le C\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}^4
+C t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}^4
+C_l \d^{-8}\n_{m,l}^{12}(t),
\end{equation}
and
\begin{equation}\label{ab2}
\n_{m,l}(t)
\le C\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}^4
+C t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}^4
+C\d^{-8} \Theta_{m,l}^{12}(t),
\end{equation}
where $\Theta_{m,l}(t)$ and $\n_{m,l}(t)$ are defined in \eqref{3a1} and \eqref{3d2} respectively.
\end{lemma}
\begin{proof}
By virtue of the definition $\vr_m=Z_\t^{\a_1} \vr-\frac{\p_y \vr}{\h+1}Z_\t^{\a_1}\ps$
and the estimate \eqref{b11}, we find
\begin{equation*}\label{b21}
\begin{aligned}
\|Z_\t^{\a_1} \vr\|_{L^2_l(\O)}^2
&\le \|\vr_m\|_{L^2_l(\O)}^2+\|\p_y \vr\|_{L^\infty_1(\O)}^2\|\frac{Z_\t^{\a_1 }\ps}{\h+1}\|_{L^2_{l-1}(\O)}^2\\
&\le \|\vr_m\|_{L^2_l(\O)}^2+C_l \d^{-2}\|\p_y \vr\|_{L^\infty_1(\O)}^2\|\h_m\|_{L^2_{l}(\O)}^2.
\end{aligned}
\end{equation*}
Similarly, we can obtain for $|\a_1|=m$ that
\begin{equation*}\label{b22}
\|Z_\t^{\a_1}(\u, \h)\|_{L^2_l(\O)}^2
\le \|(\u_m, \h_m)\|_{L^2_l(\O)}^2+C_l \d^{-2}(1+\|\p_y (\u, \h)\|_{L^\infty_1(\O)}^2)\|\h_m\|_{L^2_{l}(\O)}^2.
\end{equation*}
The combination of the above two estimates yields directly
\begin{equation}\label{b22}
\|Z_\t^{\a_1}(\vr, \u, \h)\|_{L^2_l(\O)}^2
\le C_l \d^{-2}(1+\|\p_y (\vr, \u, \h)\|_{L^\infty_1(\O)}^2)\|(\vr_m, \u_m, \h_m)\|_{L^2_l(\O)}^2,
\end{equation}
and hence, we have for $m \ge 4, l \ge 1$
\begin{equation}\label{b23}
\begin{aligned}
\|Z_\t^{\a_1}(\vr, \u, \h)\|_{L^2_l(\O)}^2
&\le C\|\p_y(\u, \h)\|_{L^\infty_1(\O)}^4
+C\d^{-4}(1+\|(\vr_m, \u_m, \h_m)\|_{L^2_l(\O)}^4+\|\p_y \vr\|_{L^\infty_1(\O)}^4)\\
&\le C\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}^2
+C t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}^2+C_l \d^{-4}(1+X_{m,l}^6(t)),
\end{aligned}
\end{equation}
Due to the definition of
$X_{m,l}(t)$ and $Y_{m,l}(t)$ in \eqref{ydef} and \eqref{xdef} respectively, we get from \eqref{b23} that
\begin{equation}\label{requi-E1}
Y_{m,l}(t)\le C\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}^2
+C t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}^2+C_l \d^{-4}(1+X_{m,l}^6(t)).
\end{equation}
On the other hand, by virtue of the definition of $\vr_m(t)$ and the estimate \eqref{b11}, we find
\begin{equation*}\label{b24}
\begin{aligned}
\|\vr_m(t)\|_{L^2_l(\O)}^2
&\le \|Z_\t^{\a_1} \vr(t)\|_{L^2_l(\O)}^2
+\|\p_y \vr(t)\|_{L^\infty_1(\O)}^2\|\frac{Z_\t^{\a_1 }\ps}{\h+1}(t)\|_{L^2_{l-1}(\O)}^2\\
&\le \|Z_\t^{\a_1} \vr(t)\|_{L^2_l(\O)}^2+C_l\d^{-2}\|\p_y \vr(t)\|_{L^\infty_1(\O)}^2\|Z_\t^{\a_1}\h(t)\|_{L^2_{l}(\O)}^2,
\end{aligned}
\end{equation*}
and hence, we also have
\begin{equation*}\label{b25}
\begin{aligned}
\|(\u_m, \h_m)(t)\|_{L^2_l(\O)}^2
\le \|Z_\t^{\a_1}(\u, \h)(t)\|_{L^2_l(\O)}^2
+C\d^{-2}(1+\|\p_y (\u, \h)(t)\|_{L^\infty_1(\O)}^2)\|Z_\t^{\a_1}\h(t)\|_{L^2_{l}(\O)}^2.
\end{aligned}
\end{equation*}
Then, the combination of the above estimates yields directly
\begin{equation} \label{b26}
\|(\vr_m, \u_m, \h_m)(t)\|_{L^2_l}^2
\le C\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}^2
+C t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}^2
+C_l \d^{-4}(1+Y_{m,l}^6(t)),
\end{equation}
where $m \ge 4, l \ge 1$. According to the definition of $X_{m,l}(t)$ and $Y_{m,l}(t)$,
we get from \eqref{b26} that
\begin{equation}\label{requi-E2}
X_{m,l}(t)\le C\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}^2
+C t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}^2+C_l \d^{-4}(1+Y_{m,l}^6(t)).
\end{equation}
Next, by virtue of the definition $\vr_m(t)$ and estimate \eqref{3103}, we find
\begin{equation*}\label{b27}
\begin{aligned}
\|\p_x Z_\t^{\a_1} \vr\|_{L^2_l(\O)}^2
\le
&\|\p_x \vr_m\|_{L^2_l(\O)}^2+C_l \d^{-2}\|\p_y \vr\|_{L^\infty_1(\O)}^2\|\p_x \h_m\|_{L^2_l(\O)}^2\\
&+C_l \d^{-4}(\|\p_{xy} \vr\|_{L^\infty_1(\O)}^2
+\|\p_{y} \vr\|_{L^\infty_1(\O)}^2\|\p_{x} \h\|_{L^\infty_1(\O)}^2)\|\h_m\|_{L^2_l(\O)}^2\\
\le
&\|\p_x \vr_m\|_{L^2_l(\O)}^2+C_l \d^{-2} X_{m,l}(t) \|\p_x \h_m\|_{L^2_l(\O)}^2
+C_l \d^{-4}(1+ X_{m,l}^3(t)).
\end{aligned}
\end{equation*}
Similarly, by routine checking, we may conclude that
\begin{equation*}\label{b28}
\begin{aligned}
\|\p_x Z_\t^{\a_1} (\u, \h)\|_{L^2_l(\O)}^2
\le
&C(\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}^2
+t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}^2)
+C_l\d^{-8}(1+X_{m,l}^6(t))\\
&\!+\!C_l \d^{-2}(1+\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}
+t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}+X_{m,l}^3(t) )\|\p_x \h_m\|_{L^2_l(\O)}^2\\
&+\|\p_x (\u_m, \h_m)\|_{L^2_l(\O)}^2.
\end{aligned}
\end{equation*}
for $m\ge 5, l \ge 1$, and hence it follows
\begin{equation}\label{b29}
\begin{aligned}
\es\|\p_x(\vr, \u, \h)\|_{\H^m_l}^2
&\le C_l \d^{-2}(1+\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}
+t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}+X_{m,l}^3(t) )D_x^{m,l}(t)\\
&+C(\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}^2
+t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}^2)+C_l \d^{-8}(1+X_{m,l}^6(t)),
\end{aligned}
\end{equation}
where $D_x^{m,l}(t)$ is defined in \eqref{3d3}.
By virtue of the definition $\vr_m(t)$ and estimate \eqref{3101}, we get
\begin{equation*}\label{b210}
\begin{aligned}
\|\p_x \vr_m\|_{L^2_l(\O)}^2
\le
&\|\p_x Z_\t^{\a_1} \vr\|_{L^2_l(\O)}^2
+C\d^{-2}\|\p_y \vr\|_{L^\infty_1(\O)}^2\|\p_x Z_\t^{\a_1}\h\|_{L^2_{l}(\O)}^2\\
&+C\d^{-4}(\|\p_{xy}\vr\|_{L^\infty_1(\O)}^2
+\|\p_y \vr\|_{L^\infty_1(\O)}^2\|\p_x \h\|_{L^\infty_0(\O)}^2)\|Z_\t^{\a_1}\h\|_{L^2_l(\O)}\\
\le
&\|\p_x Z_\t^{\a_1} \vr\|_{L^2_l(\O)}^2
+C\d^{-2} Y_{m,l}(t) \|\p_x Z_\t^{\a_1}\h\|_{L^2_{l}}^2+C\d^{-4}(1+ Y_{m,l}^3(t)).
\end{aligned}
\end{equation*}
Similarly, by routine checking, we may conclude that
\begin{equation*}\label{b211}
\begin{aligned}
\|\p_x (\u_m, \h_m)\|_{L^2_l(\O)}^2
\le
&C\d^{-2}(1+\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}
+t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}+Y_{m,l}^3(t) )
\|\p_x Z_\t^{\a_1}\h\|_{L^2_{l}(\O)}^2\\
&+C(\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}^2
+t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}^2)+C\d^{-8}(1+Y_{m,l}^6(t))\\
&+\|\p_x Z_\t^{\a_1} (\u, \h)\|_{L^2_l(\O)}^2,
\end{aligned}
\end{equation*}
and hence, it follows
\begin{equation}\label{b212}
\begin{aligned}
D_x^{m,l}(t)
\le
&C\d^{-2}(1+\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}
+t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}+ Y_{m,l}^3(t) )\es\|\p_x(\vr, \u, \h)\|_{\H^m_l}^2\\
&+C(\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}^2
+t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}^2)+C\d^{-8}(1+Y_{m,l}^6(t)).
\end{aligned}
\end{equation}
where $D_x^{m,l}(t)$ is defined in \eqref{3d3}.
Similarly, we can justify the estimates
\begin{equation}\label{b213}
\begin{aligned}
\|\p_y(\sqrt{\es} \vr, \sqrt{\mu} \u, \sqrt{\k} \h)\|_{\H^m_l}^2
\le &C(\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}^4
+t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}^4)\\
&+D_y^{m,l}(t)+C_l \d^{-8}(1+X_{m,l}^{12}(t)),
\end{aligned}
\end{equation}
and
\begin{equation}\label{b214}
\begin{aligned}
D_y^{m,l}(t)
\le
&C(\|(\rho_0, u_{10}, h_{10})\|_{\overline{\mathcal{B}}^m_l}^4
+t \|(\rho_0, u_{10}, h_{10})\|_{\widehat{\mathcal{B}}^m_l}^4)\\
&+\|\p_y(\sqrt{\es} \vr, \sqrt{\mu} \u, \sqrt{\k} \h)\|_{\H^m_l}^2+C\d^{-8}(1+Y_{m,l}^{12}(t)).
\end{aligned}
\end{equation}
Therefore, the combination of estimates \eqref{requi-E1}, \eqref{requi-E2},
\eqref{b29}, \eqref{b212}, \eqref{b213} and \eqref{b214} can establish the
estimates \eqref{ab1} and \eqref{ab2}.
\end{proof}
\section*{Acknowledgements}
Jincheng Gao's research was partially supported by
Fundamental Research Funds for the Central Universities(Grants No.18lgpy66)
and NNSF of China(Grants No.11801586).
Daiwen Huang's research was partially supported by the NNSF of China(Grants No.11631008)
and National Basic Research Program of China 973 Program(Grants No.2007CB814800).
Zheng-an Yao's research was partially supported by NNSF of China(Grant No.11431015).
| 36,584
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New Zealand V8s in progress!
March 27
The NZV8s 2008/09 mod will soon be available as an add-on pack for the Template Mod.
Read more about the NZV8s 2008/09 Mod or check the forum.
The Template Mod!
March 26
More info soon about the new Template Mod. The basic idea is to allow people to paint textures for the included cars - lower formula, f1, touring car, sports car and maybe more.
DTM 2006 Mod & Trackpack released!
November 25
The DTM 2006 mod is now available.
Read more about the DTM 2006 Mod or check the forum and download the mod at GP4DB.com.
Le Mans Bugatti released!
November 25
As well as the DTM 2006 mod, the trackpack is also available. The star of the pack is the Bugatti track which is available at GP4DB.com.
Read more about Le Mans Bugatti
DTM 2006 Mod!
November 1
A new DTM 2006 mod has been announced. You'll soon be driving all the Audi and Mercedes on all the tracks - including the new Le Mans Bugatti!
1988 F1 Mod news!
October 29
Some screenshots of new Rial ARC-01 and Dallara BMS188 for the 1988 F1 mod are now online
WSC 1989 Mod news!
October 27
Some final renders of the Aston Martin AMR1 from the upcoming WSC 1989 mod are now online
1994 Mod news!
June 9
Some renders of the cars from the upcoming 1994 Formula 1 mod are online now.
This great looking mod has been worked on for a few years by Konstantin (mckey415), Diego (Öggo), Alex (Duffer) & others, and is now closer to release than it ever has been!
Formula Renault 2.0 Mod released!
June 3
The mod and trackpack are now available for download.
Includes over 30 of the drivers that took part in 2007, along with all the extras any good mod should have
Find more information about the FR 2.0 NEC 2007 mod and download it from GP4db.com
Dijon in progress and some screenshots
February 12
A new track is in progress - Dijon Prenois and there are some new screenshots of Mexico City, Le Mans, and the WSC 1989 project.
1988 Mod 1.25 update released!
December 3
The 1988 update is now available.
Read more about the 1988 Mod or check the forum and download the mod at GP4DB.com.
Anderstorp & new WSC '89 screenshots!
December 1
We're very happy to announce our Anderstorp 1978 circuit, and welcome Kaid to the WSC 1989 project. Some new screenshots of his Toyota and Nissan, as well as Eriks new Porsche are online.
DTM 2005 Mod & Trackpack released!
November 25
After a long wait the mod and trackpack including all nine tracks is now available!
Read more about the DTM 2005 Mod or check the forum and download the mod at GP4DB.com.
Sporstcars, Mexico, Le Mans and an update for 1988!
November 16
Welcome to the latest version of the site. There is now alot more info about our existing mods and tracks, and this is a good time to announce a few new projects:
1989 World Sportscar Championship - a brand new mod for GP4, featuring some amazing cars, top drivers and legendary track!
Mexico City - the first of eight or more curcuits being built as part of the WSC '89 mod!
Le Mans Bugatti - the shorter version of the classic track being worked on as part of an, as yet, unannounced mod!
1988 Mod update - the last big announcement for now is an update to an old mod. With Philippe (filou16) working on some nice updates to the 1988 cars, its time to update the mod as well. Along with a more modern mod structure, all the missing teams - Euro Brun, Coloni, Rial, Larrousse, AGS, Osella & Dallara - and drivers will be added!
In the next few weeks we'll have some more news - a mod or two to release, another new mod (or two!) to announce, and maybe even another couple of circuits to show off!
In the meantime take a look around - there is some new info and screenshots of most of the mods and circuits, especially the upcoming DTM 2005 mod!
The forum is still online, and is in the (slow and painful!) process of being updated.
All content © grandprixX.com 19xx-2007 | 379958 Visits |
| 205,494
|
I’m HERE!
Posted in Uncategorized on April 4, 2008 | 6 Comments »
It was a good trip!
I was able to connect with Steve and we had a fantastic lunch together. I learned all about his family and his MBA adventures and his upcoming trip to Vietnam; we really hit it off and I’m so glad that he was willing to spend part of his afternoon in [...]
| 63,647
|
TITLE: Evaluating the integral $\int e^{tx - 2|x|} dx$
QUESTION [0 upvotes]: I need to evaluate the following integral:
$$\int_{-\infty}^{\infty} e^{tx - 2|x|} dx \tag*{for $t \in (-2, 2)$}$$
To attempt to evaluate this, I being by splitting the integral in two:
$$
\int_{-\infty}^{\infty} e^{tx - 2|x|} dx \\
= \int_{-\infty}^{0} e^{tx + 2x} dx + \int_{0}^{\infty} e^{tx - 2x} dx \\
\text{Let }u = tx + 2x\text{, }du = t + 2 dx\\
\text{Let }v = tx - 2x\text{, }dv = t - 2 dx\\
= \int_{-\infty}^{0} \frac{e^{u}}{t + 2} du + \int_{0}^{\infty} \frac{e^{v}}{t - 2} dv \\
= \frac{1}{t + 2}\int_{-\infty}^{0} e^{u} du + \frac{1}{t - 2}\int_{0}^{\infty} e^{v}dv \\
= \frac{1}{t + 2}(1) + \frac{1}{t - 2}\int_{0}^{\infty} e^{tx - 2x} dx \\
$$
At this point I was unsure how to proceed, so I went to Wolfram Alpha to see how it evaluted the integral. Wolfram Alpha gives the following for the indefinite integral:
$$
\int e^{tx - 2|x|} dx
= \begin{cases}
\frac{e^{tx + 2x}}{t + 2} & x < 0 \\
\frac{(t+2)e^{tx - 2x}-4}{t^2 - 4} & x > 0 \\
0 & x = 0
\end{cases}
$$
How do I proceed with evaluating this integral so no integral terms remain, and how does what I've found for the improper integral (if correct) reconcile with Wolfram Alpha's indefinite integral results?
REPLY [1 votes]: $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\ic}{\mathrm{i}}
\newcommand{\mc}[1]{\mathcal{#1}}
\newcommand{\mrm}[1]{\mathrm{#1}}
\newcommand{\pars}[1]{\left(\,{#1}\,\right)}
\newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
\newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
\newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
\newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$
\begin{align}
\left.\int_{-\infty}^{\infty}\expo{tx - 2\verts{x}}\,\dd x
\,\right\vert_{\ t\ \in\ \pars{-2,2}} & =
\int_{0}^{\infty}
\bracks{\expo{tx - 2\verts{x}} + \expo{t\pars{-x} - 2\verts{-x}}}\,\dd x
\\[5mm] &=
\int_{0}^{\infty}
\bracks{\expo{-\pars{2 - t}x} + \expo{-\pars{2 + t}x}}\,\dd x =
\left.{\expo{-\pars{2 - t}x} \over t - 2} +
{\expo{-\pars{2 + t}x} \over -t - 2}\right\vert_{\ 0}^{\ \infty}
\\[5mm] & =
-\,{1 \over t - 2} - {1 \over -t - 2} =
\bbx{\ds{4 \over 4 - t^{2}}}
\end{align}
Note that
$\ds{\int_{-\infty}^{\infty}\mrm{f}\pars{x}\,\dd x =
\int_{0}^{\infty}\bracks{\mrm{f}\pars{x}+ \mrm{f}\pars{-x}}\,\dd x}$.
| 65,801
|
Our weekend accommodation in Staffordshire, deep in the Manifold Valley, was a National Trust cottage called Darfar. The National Trust hamlet of Wetton Mill consists of two holiday cottages, a farmhouse and a privately owned tea room.
Beautiful setting of Wetton Mill
Wetton Mill Cottages
Wetton Mill Tea Shop
After a stormy night Sunday morning began to brighten up and a walk up and down the valley was in order. First we headed down river for about a mile along the former track of the Leek and Manifold Light Railway (1904-1934) to Thor’s Cave 250 feet up from the valley bottom.
Manifold River almost flooded
Manifold River and Wetton Mill Bridge
Thor’s Cave information board
Thor’s Cave
Approaching Thor’s Cave
The Cave (but I didn’t go inside)
The View from the mouth of Thor’s Cave
After a quick lunch at the tea room we headed up the valley and through the Swainsley Tunnel. An attempt to find a suitable track over the tunnel failed and we returned along the route we had originally taken. As dusk began to fall we arrived back at our cosy Darfar cottage.
Darfar
We had one would-be visitor
| 272,735
|
\begin{document}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\title{Asymptotic bounds for the number of closed and privileged words}
\author{Daniel Gabric \\
School of Computer Science \\
University of Waterloo \\
Waterloo, ON N2L 3G1 \\
Canada\\
\href{mailto:dgabric@uwaterloo.ca}{\tt dgabric@uwaterloo.ca} }
\date{}
\maketitle
\begin{abstract}
A word $w$ has a border $u$ if $u$ is a non-empty proper prefix and suffix of $u$. A word $w$ is said to be \emph{closed} if $|w| \leq 1$ or if $w$ has a border that occurs exactly twice in $w$. A word $w$ is said to be \emph{privileged} if $|w| \leq 1$ or if $w$ has a privileged border that occurs exactly twice in $w$. Let $C_k(n)$ (resp.~$P_k(n)$) be the number of length-$n$ closed (resp. privileged) words over a $k$-letter alphabet. In this paper we improve existing upper and lower bounds on $C_k(n)$ and $P_k(n)$. We prove that $C_k(n) \in \Theta(k^n/n)$. Let $\log_k^{\circ 0}(n) = n$ and $\log_k^{\circ j}(n) = \log_k(\log_k^{\circ j-1}(n))$ for $j\geq 1$. We also prove that for all $j\geq 0$ there exist constants $N_j$, $c_j$, and $c_j'$ such that
\[c_j\frac{k^n}{n\log_k^{\circ j}(n)\prod_{i=1}^j\log_k^{\circ i}(n)}\leq P_k(n) \leq c_j'\frac{k^n}{n\prod_{i=1}^j\log_k^{\circ i}(n)}\] for all $n>N_j$.
\end{abstract}
\section{Introduction}
Let $\Sigma_k$ denote the $k$-letter alphabet $\{0,1,\ldots, k-1\}$. A word $w$ has a \emph{border} $u$ if $u$ is a non-empty proper prefix and suffix of $w$. A word that has a border is said to be \emph{bordered}. Otherwise, it is said to be \emph{unbordered}. A word $w$ is said to be \emph{closed} if $|w|\leq 1$ or if $w$ has a border that occurs exactly twice in $w$. If $u$ is a border $w$ and $u$ occurs in $w$ exactly twice, then we say $w$ is \emph{closed by} $u$. It is easy to see that if a word $w$ is closed by a word $u$, then $u$ must be the largest border in $w$. Otherwise $u$ would occur more than two times in $w$. A word $w$ is said to be \emph{privileged} if $|w| \leq 1$ or if $w$ is closed by a privileged word.
\begin{example}
The English word {\tt bonobo} has the border {\tt bo} and only contains two occurrences of {\tt bo}. Thus {\tt bonobo} is a closed word, closed by {\tt bo}. Since $|{\tt bo}| >1$ and {\tt bo} is unbordered and therefore not privileged, we have that {\tt bonobo} is not privileged.
The French word {\tt entente} is closed by {\tt ente}. Furthermore {\tt ente} is closed by {\tt e}. But $|{\tt e}|\leq 1$, so {\tt ente} is privileged and therefore so is {\tt entente}.
\end{example}
Both closed words~\cite{Fici:2011} and privileged words~\cite{Kellendonk&Lenz&Savinien:2013} have been introduced relatively recently, although some equivalent formulations of closed words that have been defined previously. Closed words are equivalent to periodic-like words~\cite{Carpi&deLuca:2001} and codewords in prefix-synchronized codes~\cite{Gilbert:1960,Guibas&Odlyzko:1978}. Since their introduction, there has been much research into the properties of closed and privileged words~\cite{Peltomaki:2013,Bucci:2013,DeLuca&Fici&Karhumaki&Lepisto&Zamboni:2013, Badkobeh&Fici&Liptak:2015, Schaeffer&Shallit:2016, Fici:2017, Jahannia:2022}. One problem that has received some interest lately~\cite{Forsyth2016, Nicholson2018, Rukavicka2020, Rukavicka:2022} is to find good upper and lower bounds for the number of closed and privileged words.
Let $C_k(n)$ denote the number of length-$n$ closed words over $\Sigma_k$. Let $C_k(n,t)$ denote the number of length-$n$ closed words over $\Sigma_k$ that are closed by a length-$t$ word. Let $P_k(n)$ denote the number of length-$n$ privileged words over $\Sigma_k$. Let $P_k(n,t)$ denote the number of length-$n$ privileged words over $\Sigma_k$ that are closed by a length-$t$ privileged word. See Tables~\ref{table:ClosedTable},~\ref{table:PrivilegedTable}, and~\ref{table:PrivilegedClosedTable} for sample values of $C_2(n)$, $C_2(n,t)$, $P_2(n)$, and $P_2(n,t)$ for small $n$, $t$.
Every privileged word is a closed word, so any upper bound on $C_k(n)$ is also an upper bound on $P_k(n)$. Furthermore, any lower bound on $P_k(n)$ is also a lower bound on $C_k(n)$.
\begin{itemize}
\item Forsyth et al.~\cite{Forsyth2016} showed that $P_2(n) \geq 2^{n-5}/n^2$ for all $n>0$. \item Nicholson and Rampersad~\cite{Nicholson2018} improved and generalized this bound by showing that there are constants $c$ and $n_0$ such that $P_k(n) \geq c\frac{k^n}{n(\log_k(n))^2}$ for all $n \geq n_0$.
\item Rukavicka~\cite{Rukavicka2020} showed that there is a constant $c$ such that $C_k(n) \leq c\ln n\frac{k^n}{\sqrt{n}}$ all $n>1$.
\item Rukavicka~\cite{Rukavicka:2022} also showed that for every $j\geq 3$, there exist constants $\alpha_j$ and $n_j$ such that $P_k(n) \leq \alpha_j\frac{k^n\sqrt{\ln n}}{\sqrt{n}}\ln^{\circ j}(n)\prod\limits_{i=2}^{j-1}\sqrt{\ln^{\circ i}(n)}$ length-$n$ privileged words for all $n\geq n_j$ where $\ln^{\circ 0}(n) = n$ and $\ln^{\circ j}(n) = \ln(\ln^{\circ j-1}(n))$.
\end{itemize}
So the best upper and lower bounds for both $C_k(n)$ and $P_k(n)$ are widely separated, and can be much improved. In this paper we improve the existing upper and lower bounds on $C_k(n)$ and $P_k(n)$. We prove the following two theorems.
\begin{theorem}\label{theorem:mainC}
Let $k\geq 2$ be an integer.
\begin{enumerate}[label=(\alph*)]
\item There exist constants $N$ and $c$ such that $C_k(n)\geq c\frac{k^n}{n}$ for all $n>N$. \label{theorem:mainCLower}
\item There exist constants $N'$ and $c'$ such that $C_k(n) \leq c'\frac{k^n}{n}$ for all $n>N'$. \label{theorem:mainCUpper}
\end{enumerate}
\end{theorem}
\begin{theorem}\label{theorem:mainP}
Let $k\geq 2$ be an integer. Let $\log_k^{\circ 0}(n)=n$ and $\log_k^{\circ j}(n) = \log_k(\log_k^{\circ j-1}(n))$ for $j\geq 1$.
\begin{enumerate}[label=(\alph*)]
\item For all $j\geq 0$ there exist constants $N_j$ and $c_j$ such that \[P_k(n)\geq c_j\frac{k^n}{n\log_k^{\circ j}(n)\prod_{i=1}^j\log_k^{\circ i}(n)}\] for all $n>N_j$. \label{theorem:mainPLower}
\item For all $j\geq 1$ there exist constants $N_j'$ and $c_j'$ such that \[P_k(n) \leq c_j'\frac{k^n}{n\prod_{i=1}^j\log_k^{\circ i}(n)}\]
for all $n>N_j'$.\label{theorem:mainPUpper}
\end{enumerate}
\end{theorem}
Before we proceed, we give a heuristic argument as to why $C_k(n)$ is in $\Theta(\frac{k^n}{n})$. Consider a ``random" length-$n$ word $w$. Let $\ell = \log_k(n) + c$. The probability that $w$ has a length-$\ell$ border $u$ is around $k^{n-\ell}/k^n = \frac{1}{k^cn}$. Suppose $w$ has a length-$\ell$ border. Now suppose we drop the first and last character of $w$ to get $w'$. If $w'$ were randomly chosen (which it is not), then we could use the linearity of expectation to get that the expected number of occurrences of $u$ in $w'$ is approximately $(n-2)k^{-\ell} \approx k^{-c}$. Thus for $c$ large enough we have that $u$ does not occur in $w$ with high probability, and so $w$ is closed. Therefore there are approximately $k^{n-\ell} \in \Theta(\frac{k^n}{n})$ length-$n$ closed words.
\begin{table}[H]
\centering
\begin{tabular}{|c|cccccccccc|}
\hline
\backslashbox{$n$}{$t$} & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10$ \\
\hline
10 & 2 & 30 & 70 & 50 & 30 & 12 & 6 & 2 & 2 & 0 \\
11 & 2 & 42 & 118 & 96 & 54 & 30 & 13 & 6 & 2 & 2 \\
12 & 2 & 60 & 200 & 182 & 114 & 54 & 30 & 12 & 6 & 2 \\
13 & 2 & 88 & 338 & 346 & 214 & 126 & 54 & 30 & 12 & 6 \\
14 & 2 & 132 & 570 & 640 & 432 & 232 & 126 & 54 & 30 & 12 \\
15 & 2 & 202 & 962 & 1192 & 828 & 474 & 240 & 126 & 54 & 30 \\
16 & 2 & 314 & 1626 & 2220 & 1612 & 908 & 492 & 240 & 126 & 54 \\
17 & 2 & 494 & 2754 & 4128 & 3112 & 1822 & 956 & 504 & 240 & 126 \\
18 & 2 & 784 & 4676 & 7670 & 6024 & 3596 & 1934 & 982 & 504 & 240 \\
19 & 2 & 1252 & 7960 & 14264 & 11636 & 7084 & 3828 & 1992 & 990 & 504 \\
20 & 2 & 2008 & 13588 & 26524 & 22512 & 13928 & 7632 & 3946 & 2026 & 990 \\
\hline
\end{tabular}
\captionsetup{justification=centering}
\caption{Some values of $C_2(n,t)$ for $n$, $t$ where $10 \leq n \leq 20$ and $1\leq t \leq 10$.}
\label{table:ClosedTable}
\end{table}
\begin{table}[H]
\centering
\begin{tabular}{|c|cccccccccc|}
\hline
\backslashbox{$n$}{$t$} & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ & $9$ & $10$ \\
\hline
10 & 2 & 16 & 22 & 8 & 6 & 2 & 2 & 0 & 2 & 0 \\
11 & 2 & 26 & 38 & 16 & 10 & 6 & 4 & 2 & 2 & 2 \\
12 & 2 & 42 & 68 & 30 & 18 & 4 & 6 & 2 & 2 & 0 \\
13 & 2 & 68 & 122 & 58 & 38 & 14 & 10 & 6 & 4 & 2 \\
14 & 2 & 110 & 218 & 108 & 76 & 20 & 14 & 8 & 6 & 2 \\
15 & 2 & 178 & 390 & 204 & 148 & 46 & 24 & 18 & 14 & 6 \\
16 & 2 & 288 & 698 & 384 & 288 & 86 & 48 & 16 & 18 & 8 \\
17 & 2 & 466 & 1250 & 724 & 556 & 178 & 92 & 36 & 32 & 26 \\
18 & 2 & 754 & 2240 & 1364 & 1076 & 344 & 190 & 64 & 36 & 28 \\
19 & 2 & 1220 & 4016 & 2572 & 2092 & 688 & 388 & 136 & 70 & 56 \\
20 & 2 & 1974 & 7204 & 4850 & 4068 & 1342 & 772 & 268 & 138 & 52 \\
\hline
\end{tabular}
\captionsetup{justification=centering}
\caption{Some values of $P_2(n,t)$ for $n$, $t$ where $10 \leq n \leq 20$ and $1\leq t \leq 10$.}
\label{table:PrivilegedTable}
\end{table}
\begin{table}[H]
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$n$ & $P_2(n)$ & $C_2(n)$ & $n$ & $P_2(n)$ & $C_2(n)$ \\
\hline
0 & 1 & 1 & 13 & 328 & 1220 \\
1 & 2 & 2 & 14 & 568 & 2240 \\
2 & 2 & 2 & 15 & 1040 & 4132 \\
3 & 4 & 4 & 16 & 1848 & 7646 \\
4 & 4 & 6 & 17 & 3388 & 14244 \\
5 & 8 & 12 & 18 & 6132 & 26644 \\
6 & 8 & 20 & 19 & 11332 & 49984 \\
7 & 16 & 36 & 20 & 20788 & 94132 \\
8 & 20 & 62 & 21 & 38576 & 177788 \\
9 & 40 & 116 & 22 & 71444 & 336756 \\
10 & 60 & 204 & 23 & 133256 & 639720 \\
11 & 108 & 364 & 24 & 248676 & 1218228 \\
12 & 176 & 664 & 25 & 466264 & 2325048 \\
\hline
\end{tabular}
\captionsetup{justification=centering}
\caption{Some values of $P_2(n)$ and $C_2(n)$ for $n\leq 25$.}
\label{table:PrivilegedClosedTable}
\end{table}
\section{Preliminary results}
In this section we give some necessary results and definitions in order to prove our main results. Also throughout this paper, we use $c$'s, $d$'s, and $N$'s to denote positive real constants (dependent on $k$).
Let $w$ be a length-$n$ word. Suppose $w$ is closed by a length-$t$ word $u$. Since $u$ is also the largest border of $w$, it follows that $w$ cannot be closed by another word. This implies that \[C_k(n) = \sum_{i=1}^{n-1}C_k(n,t)\text{ and }P_k(n) = \sum_{i=1}^{n-1}P_k(n,t)\] for $n>1$.
Let $B_k(n,u)$ denote the number of length-$n$ words over $\Sigma_k$ that are closed by the word $u$. Let $A_k(n,u)$ denote the number of length-$n$ words over $\Sigma_k$ that do not contain the word $u$ as a factor.
The \emph{auto-correlation}~\cite{Guibas&Odlyzko:1978,Guibas&Odlyzko:1980, Guibas&Odlyzko:1981} of a length-$t$ word $u$ is a length-$t$ binary word $a(u)=a_1a_2\cdots a_t$ where $a_i=1$ if and only if $u$ has a border of length $t-i+1$. The \emph{auto-correlation polynomial} $f_{a(u)}(z)$ of $a(u)$ is defined as
\[f_{a(u)}(z) = \sum_{i=0}^{t-1}a_{t-i}z^i\]
For example, the word $u={\tt alfalfa}$ has auto-correlation $a(u) = 1001001$ and auto-correlation polynomial $f_{a(u)}(z) = z^6+z^3+1$.
\begin{comment}
\begin{theorem}[Section 7~\cite{Guibas&Odlyzko:1981}]
Let $k\geq 2$ and $t\geq 4$ be integers. Then
\[A_k(n,u) = C_u\theta_u^n + O(1.7^n)\]
where
\[\theta_u = k-\frac{1}{f_{a(u)}(k)} - \frac{f_{a(u)}'(k)}{(f_{a(u)}(k))^3}+O\bigg(\frac{t^2}{k^{3t}}\bigg)\text{ and }C_u = \frac{1}{1-(k-\theta_u)^2f_{a(u)}'(\theta_u)}.\]
\end{theorem}
\begin{theorem}[Section 7~\cite{Guibas&Odlyzko:1981}]
Let $k\geq 2$ be an integer. If $f_{a(u)}(2) > f_{a(v)}(2)$ for words $u$, $v$, then $\theta_{u} \geq \theta_v$ and $A_k(n,u) \geq A_k(n,v)$ for all $n\geq 1$.
\end{theorem}
\begin{corollary}
Let $k\geq 2$ and $t\geq 1$ be integers. Let $u=0^t$. Let $v$ be a length-$t$ word. Then $\theta_{u} \geq \theta_v$ and $A_k(n,u)\geq A_k(n,v)$ for all $n\geq 1$.
\end{corollary}
\begin{theorem}[Section 4~\cite{Rubinchik:2016}]
Let $k\geq 2$ and $t\geq 4$ be integers. Let $u=0^t$. Then
\[\theta_{u} = k-\frac{k-1}{k^t}-\frac{t(k-1)^2}{k^{2t+1}}+O\bigg(\frac{kt+t^2}{k^{3t}}\bigg)\text{ and }C_{u} = 1+O\bigg(\frac{t}{k^t}\bigg).\]
\end{theorem}
\end{comment}
We now prove two technical lemmas that will be used in the proofs of Theorem~\ref{theorem:mainC}~\ref{theorem:mainCUpper} and Theorem~\ref{theorem:mainP}~\ref{theorem:mainPUpper}.
\begin{lemma}
\label{lemma:binomial}
Let $k,t\geq 2$ be integers, and let $\gamma$ be a real number such that $0< \gamma \leq \frac{6}{t}$. Then
\[k^t - \gamma tk^{t-1} \leq (k-\gamma)^t \leq k^t - \gamma t k^{t-1} + \frac{1}{2}\gamma^2 t(t-1) k^{t-2}.\]
\end{lemma}
\begin{proof}
The case when $k=2$ was proved in a paper by Forsyth et al.~\cite[Lemma 9]{Forsyth2016}. We generalize their proof to $k\geq 3$.
When $t=2$, we have $k^2 - 2k\gamma \leq (k-\gamma)^2 \leq k^2 - 2k\gamma+\gamma^2 $. So suppose $t\geq 3$. By the binomial theorem, we have
\begin{align}
(k-\gamma)^t &= \sum_{i=0}^t k^{t-i}(-\gamma)^i{t \choose i} = k^t - \gamma t k^{t-1} + \sum_{i=2}^t k^{t-i}(-\gamma)^i{t \choose i} \nonumber \\
&\geq k^t - \gamma t k^{t-1} + \sum_{j=1}^{\lfloor(t-1)/2\rfloor} \bigg(k^{t-2j}\gamma^{2j}{t \choose 2j} - k^{t-2j-1}\gamma^{2j+1}{t\choose 2j+1}\bigg). \nonumber
\end{align}
So to show that $k^t - \gamma tk^{t-1} \leq (k-\gamma)^t$, it is sufficient to show that
\begin{equation}
k^{t-2j}\gamma^{2j}{t \choose 2j} \geq k^{t-2j-1}\gamma^{2j+1}{t\choose 2j+1}
\label{foo3}
\end{equation}
for $1\leq j \leq \lfloor (t-1)/2\rfloor \leq (t-1)/2$.
By assumption we have that $\gamma \leq \frac{6}{t}$, so $\gamma \leq \frac{6}{t-2}$ and thus $\gamma t - 2\gamma \leq 6$. Adding $2\gamma -2$ to both sides we get $\gamma t -2 \leq 4+2\gamma$, and so $\frac{\gamma t - 2 }{\gamma +2}\leq 2$. If $i\geq 2 \geq \frac{\gamma t - 2 }{\gamma +2}$, then $(\gamma +2)i \geq \gamma t - 2$. This implies that $2(i+1) \geq \gamma(t-i)$, and \[\frac{k}{\gamma} \geq \frac{2}{\gamma} \geq \frac{t-i}{i+1} = \frac{{t \choose i+1}}{{t \choose i}}.\]
Therefore letting $i=2j$, we have that $k{t\choose 2j} \geq \gamma {t\choose 2j+1}$. Multiplying both sides by $k^{t-2j-1}\gamma^{2j}$ we get $k^{t-2j}\gamma^{2j}{t\choose 2j} \geq k^{t-2j-1}\gamma^{2j+1}{t\choose 2j+1}$, which proves
\eqref{foo3}.
Now we prove that $(k-\gamma)^t \leq k^t - \gamma t k^{t-1} + \frac{1}{2}\gamma^2 t(t-1) k^{t-2}$. Going back to the binomial expansion of $(k-\gamma)^t$, we have
\begin{align}
(k-\gamma)^t &= k^t - \gamma t k^{t-1} +\frac{1}{2}\gamma^2 t(t-1) k^{t-2}+\sum_{i=3}^t k^{t-i}(-\gamma)^i{t \choose i}\nonumber \\
&\leq k^t - \gamma t k^{t-1} + \frac{1}{2}\gamma^2 t(t-1) k^{t-2} \nonumber \\
&- \sum_{j=1}^{\lfloor(t-2)/2\rfloor} \bigg(k^{t-2j-1}\gamma^{2j+1}{t \choose 2j+1} - k^{t-2j-2}\gamma^{2j+2}{t\choose 2j+2}\bigg).\nonumber
\end{align}
So to show that $(k-\gamma)^t \leq k^t - \gamma t k^{t-1} + \frac{1}{2}\gamma^2 t(t-1) k^{t-2}$, it is sufficient to show that \[k^{t-2j-1}\gamma^{2j+1}{t \choose 2j+1} \geq k^{t-2j-2}\gamma^{2j+2}{t\choose 2j+1}\] for $1\leq j \leq \lfloor (t-2)/2\rfloor$. But we already have already proved that $k{t \choose i} \geq \gamma {t\choose i+1}$. Letting $i=2j$, we have that $k{t \choose 2j+1} \geq \gamma {t\choose 2j+2}$. Multiplying both sides by $k^{t-2j-2}\gamma^{2j+1}$ we get $k^{t-2j-1}\gamma^{2j+1}{t \choose 2j+1} \geq k^{t-2j-2}\gamma^{2j+2}{t\choose 2j+2}$.
\end{proof}
Let $\log_k^{\circ 0}(n) = n$ and $\log_k^{\circ j}(n) = \log_k(\log_k^{\circ j-1}(n))$ for $j\geq 1$.
\begin{lemma}\label{lemma:logLimit}
Let $i\geq 1$ and $k\geq 2$ be integers. Then for any constant $\gamma > 0$, we have \[\lim_{n\to \infty} \frac{\log_k^{\circ i}( n^\gamma)}{\log_k^{\circ i}(n)}=\begin{cases}
\gamma , & \text{if $i=1$;} \\
1, & \text{if $i>1$.}
\end{cases}.\]
\end{lemma}
\begin{proof}
When $i=1$ we have $\lim\limits_{n\to \infty} \frac{\log_k^{}( n^\gamma)}{\log_k^{}( n)} =\gamma \lim\limits_{n\to \infty} \frac{\log_k^{}( n)}{\log_k^{}( n)}=\gamma$.
The proof is by induction on $i$. Since we will use L'H\^{o}pital's rule to evaluate the limit, we first compute the derivative of $\log_k^{\circ i}( n^\lambda)$ with respect to $n$ for any constant $\lambda >0$. We have \[\frac{d}{dn}\log_k^{\circ i}( n^\lambda) = \frac{\lambda}{n\prod\limits_{j=1}^{i-1}\log_k^{\circ j}(n^\lambda )}.\]
In the base case, when $i=2$, we have
\begin{align*}
\lim_{n\to \infty} \frac{\log_k^{\circ 2}( n^\gamma)}{\log_k^{\circ 2}(n)} &= \lim_{n\to \infty} \frac{\frac{\gamma}{n \log_k(n^\gamma )}}{\frac{1}{n \log_k(n )}} = 1.
\end{align*}
Suppose $i>2$. Then we have
\begin{align*}
\lim_{n\to \infty} \frac{\log_k^{\circ i}( n^\gamma)}{\log_k^{\circ i}(n)} &= \lim_{n\to \infty} \frac{\frac{\gamma}{n\prod\limits_{j=1}^{i-1}\log_k^{\circ j}(n^\gamma )}}{\frac{1}{n\prod\limits_{j=1}^{i-1}\log_k^{\circ j}(n)}} =\lim_{n\to \infty} \frac{ {\prod\limits_{j=2}^{i-1}\log_k^{\circ j}(n)}}{\prod\limits_{j=2}^{i-1}\log_k^{\circ j}(n^\gamma )} = 1.
\end{align*}
\end{proof}
\section{Closed words}
\subsection{Lower bound}
We first state a useful lemma from a paper of Nicholson and Rampersad~\cite{Nicholson2018}.
\begin{lemma}[Nicholson and Rampersad~\cite{Nicholson2018}]\label{lemma:nich&ramp}
Let $k\geq 2$ be an integer. Let $t$ be the unique integer such that \[\frac{\ln k}{k-1}k^t\leq n-t< \frac{\ln k}{k-1}k^{t+1}.\] Let $u$ be a length-$t$ word. There exist constants $N_0$ and $d$ such that for $n-t> N_0$ we have \[B_k(n,u) \geq d\frac{k^n}{n^2}.\]
\end{lemma}
We now use the previous lemma to prove Theorem~\ref{theorem:mainC}~\ref{theorem:mainCLower}.
\begin{proof}[Proof of Theorem~\ref{theorem:mainC}~\ref{theorem:mainCLower}]
The number $C_k(n,t)$ of length-$n$ words closed by a length-$t$ word is clearly equal to the sum, over all length-$t$ words $u$, of the number $B_k(n,u)$ of length-$n$ words closed by $u$. Thus we have that \[C_k(n,t) = \sum_{|u| = t}B_k(n,u).\] Let $t= \lfloor \log_k(n-t) + \log_k(k-1) - \log_k(\ln k )\rfloor$. By Lemma~\ref{lemma:nich&ramp} there exist constants $N_0$ and $d$ such that for $n-t > N_0$ we have $B_k(n,u) \geq d k^n/n^2$. Clearly $t \leq \log_k(n)+1$ for all $n\geq 1$. Since $t$ is asymptotically much smaller than $n$, there exists a constant $N> N_0$ such that $n-t > N_0$ for all $n>N$. Thus for $n>N$ we have
\begin{align*}
C_k(n) &\geq C_k(n,t) = \sum_{|u| = t}B_k(n,u) \geq \sum_{|u|=t}d\frac{k^n}{n^2} = k^{t}\bigg(d \frac{k^n}{n^2}\bigg) \\
&=d k^{\lfloor \log_k(n-t) + \log_k(k-1) - \log_k(\ln k )\rfloor} \frac{k^n}{n^2} \geq d_0k^{\log_k(n-t) + \log_k(k-1) - \log_k(\ln k )} \frac{k^n}{n^2} \\
&\geq d_1(n-t)\frac{k^n}{n^2} \geq d_1(n-\log_k(n)-1)\frac{k^n}{n^2}\geq c\frac{k^n}{n}
\end{align*}
for some constant $c>0$.
\end{proof}
\begin{comment}
Let $\mathbf{u}_n$ denote the set of all length-$n$ unbordered words over $\Sigma_k$.
Let $u_n$ denote the number of length-$n$ unbordered words over $\Sigma_k$.
Let $U_k(n)$ denote the number of length-$n$ closed words $w$ over $\Sigma_k$ that are closed by an unbordered word.
Let $U_k(n,t)$ denote the number of length-$n$ closed words over $\Sigma_k$ that are closed by a length-$t$ unbordered word.
\begin{proposition}
For all $n > 1$ we have,
\[U_k(n) = \sum_{t=1}^{\lfloor n/2 \rfloor} U_k(n,t).\]
\end{proposition}
\begin{proof}
Let $t$ be a positive integer in the range $1\leq t < n$. Suppose a length-$n$ word $w$ is closed by a length-$t$ unbordered word $u$. If $t > n/2$, then $u$ overlaps itself in $w$. But this is impossible, since $u$ is unbordered. So $U_k(n,t)=0$ for $t>n/2$. From here it is sufficient to show that $w$ cannot be closed by two different unbordered words. Suppose $w$ is also closed by another unbordered word $v$. Without loss of generality, let $|v| < t$. Then $v$ is both a prefix and suffix of $u$. So $u$ is bordered, a contradiction. Thus we have
\[U_k(n) = \sum_{t=1}^{\lfloor n/2 \rfloor} U_k(n,t).\]
\end{proof}
\begin{lemma}\label{lemma:countUnb}
Let $n$, $t$, and $k$ be integers such that $n\geq 2t\geq 1$ and $k\geq 2$. Let $u$ be a length-$t$ unbordered word. Then the number of length-$n$ words that are closed by $u$ is $A_k(n-2t,u)$.
\end{lemma}
\begin{proof}
Let $w$ be a length-$n$ word. Suppose $u$ is a border of $w$. Since $|w| = n \geq 2t = 2|u|$, we can write $w = uvu$ where $|v| = n-2t$. It is clear that if $w$ is closed by $u$, then $v$ cannot contain any occurrence of $u$. Therefore it suffices to show that if $v$ does not contain $u$ as a factor, then $w$ is closed by $u$. Suppose $v$ does not contain $u$ as a factor. Furthermore suppose that $w$ is not closed by $u$. Since $v$ does not contain $u$ as a factor, we have that $w=uvu$ must contain a third occurrence of $u$ that overlaps the first occurrence of $u$ or the last occurrence of $u$. But this implies that $u$ is bordered, a contradiction. Therefore the number of length-$n$ words that are closed by $u$ is the number of words of length $|v|=n-2t$ that do not contain $u$ as a factor.
\end{proof}
\begin{lemma}
\label{lemma:shortUnbRec}
Let $n,t\geq 1$ and $k\geq 2$ be integers. Let $u$ be a length-$t$ unbordered word. Then
\[ A_{k}(n,u) = \begin{cases}
k^n, & \text{if $n < t$;} \\
kA_k(n-1,u) - A_k(n-t,u), & \text{if $n\geq t$.}
\end{cases}
\]
\end{lemma}
\begin{proof}
Suppose $n < t$. Then any length-$n$ word is shorter than $u$, and thus cannot contain $u$ as a factor. So $A_k(n,u) = k^n$.
Suppose $n\geq t$. Let $w$ be a length-$(n-1)$ word. We prove that $w$ does not contain $u$ as a factor and there exists an $a\in \Sigma_k$ such that $wa$ has $u$ as a suffix if and only if $w = w' v$ where $v$ is a length-$(t-1)$ prefix of $u$ and $w'$ is a length-$(n-t)$ word that does not contain $u$ as a factor. The forward direction is obvious. Suppose $w=w'v$ where $w'$ and $v$ are defined as above. Let $a$ be the last symbol of $u$. Then $wa=w'va = w'u$. So there exists an $a\in \Sigma_k$ such that $wa$ ends in $u$. Now suppose that $w$ contains $u$ as a factor. Since $w'$ does not contain $u$ as a factor, the first occurrence of $u$ in $w$ must overlap the suffix $v$. But since $v$ is a prefix of $u$, this implies $u$ is bordered, a contradiction. So $w$ does not contain $u$ as a factor.
Now we are ready to prove the recurrence. Suppose $w$ does not contain $u$ as a factor. Consider the word $wa$ where $a\in \Sigma_k$. There are $kA_k(n-1,u)$ choices for $wa$. But $wa$ can possibly contain $u$ as a suffix. However, we proved that $wa$ has $u$ as a suffix if and only if $w=w'v$ where $u=va$ and $w'$ is a length-$(n-t)$ word that does not contain $u$ as a factor. Therefore
\[A_k(n,u) = kA_k(n-1,u) - A_k(n-t,u).\]
\end{proof}
\begin{corollary}
\label{corollary:shortUnbRec}
Let $n,t\geq 1$ and $k\geq 2$ be integers. Let $u$ be a length-$t$ unbordered word. Then
\[ A_{k}(n,u) = \begin{cases}
k^n, & \text{if $n < t$;} \\
k^{n-t}(k^t-(n-t)-1), & \text{if $t\leq n < 2t$;}\\
kA_k(n-1,u) - A_k(n-t,u), & \text{if $n\geq 2t$.}
\end{cases}
\]
\end{corollary}
\begin{corollary}
Let $n,t\geq 1$ and $k\geq 2$ be integers. Let $u$ be a length-$t$ unbordered word. The expression $A_k(n,u)$ satisfies
\[ A_{k}(n,u) = \begin{cases}
k^n, & \text{if $n < t$;} \\
1+(k-1)\sum\limits_{i=1}^{t-1}A_k(n-i,u) + (k-2)\sum\limits_{j=0}^{n-t}A_k(j,u), & \text{if $n\geq t$.}
\end{cases}
\]
\end{corollary}
\begin{proof}
Let \[R_k(n,u) = 1+(k-1)\sum_{i=1}^{t-1}A_k(n-i,u) + (k-2)\sum_{j=0}^{n-t}A_k(j,u).\] We prove by induction on $n$ that $R_k(n,u)=A_k(n,u)$ for all $n\geq t$.
By Corollary~\ref{corollary:shortUnbRec} we have that $A_k(n,u) = k^{n-t}(k^t-(n-t)-1)$ for $t \leq n < 2t$. We will now show, by induction on $n$, that $R_k(n,u) = k^{n-t}(k^t-(n-t)-1)$ for $t\leq n < 2t$. In the base case when $n=t$ we have
\begin{align*}
R_k(n,u) &= 1+(k-1)\sum_{i=1}^{t-1}A_k(t-i,u) + (k-2)\sum_{j=0}^{0}A_k(j,u)\\
&= (k-1)\sum_{i=1}^{t-1}k^{t-i} + (k-1)\\
&= (k-1)\sum_{i=1}^{t}k^{t-i} = (k-1)\frac{k^t - 1}{k-1} = k^t-1\\
&= k^{n-t}(k^t-(n-t)-1)
\end{align*}
Now we show that $R_k(n,u) = k^{n-t}(k^t-(n-t)-1)$ for all $t <n<2t$. Let $n = t +p$ where $p$ is an integer with $1\leq p < t$. Then
\begin{align}
R_k(t+p,u) &= 1+(k-1)\sum_{i=1}^{t-1}A_k(t+p-i,u) + (k-2)\sum_{j=0}^{p}A_k(j,u) \nonumber\\
&= 1 + (k-1)\bigg(\sum_{i=1}^{p}A_k(t+p-i,u) + \sum_{i=p+1}^{t-1}A_k(t+p-i)\bigg) +(k-2)\sum_{j=0}^{p}k^j \nonumber\\
&= 1+(k-1)\bigg(\sum_{i=0}^{p-1}A_k(t+i,u) + \sum_{i=p+1}^{t-1}A_k(i)\bigg) +(k-2)\frac{k^{p+1}-1}{k-1} \nonumber \\
&= 1+(k-1)\bigg(\sum_{i=0}^{p-1}k^i(k^t-i-1) + \frac{k^t-k^{p+1}}{k-1}\bigg) +(k-2)\frac{k^{p+1}-1}{k-1}\label{mark}
\end{align}
We now evaluate the remaining sum. We have
\begin{align*}
\sum_{i=0}^{p-1}k^i(k^t-i-1) &=(k^t-1)\sum_{i=0}^{p-1}k^i - \sum_{i=0}^{p-1}ik^i \\
&= (k^t-1)\frac{k^p-1}{k-1} - \frac{(p-1)k^{p+1} - pk^p + k}{(k-1)^2}
\end{align*}
Continuing from (\ref{mark}), we have
\begin{align*}
R_k(t+p,u)&= 1+(k^t-1)(k^p-1) - \frac{(p-1)k^{p+1} - pk^p + k}{k-1} \\
&+ k^t-k^{p+1} +((k-1)-1)\frac{k^{p+1}-1}{k-1} \\
&= 1+k^{t+p}-k^p- \frac{pk^{p+1} - pk^p + (k-1)}{k-1} \\
&= 1+ k^{t+p}-k^p-(k-1)\frac{pk^p+1}{k-1} \\
&=k^p(k^t-p-1)
\end{align*}
So $R_k(n,u) = k^{n-t}(k^t-(n-t)-1)=A_k(n,u)$ for all $t\leq n < 2t$.
Suppose $n\geq 2t$. Then
\begin{align*}
A_k(n,u) &= kA_k(n-1,u) - A_k(n-t,u)
&= (k-1)A_k(n-1,u) + R_k(n-1,u) - A_k(n-t,u) \\
&= (k-1)A_k(n-1,u) + 1+(k-1)\sum_{i=1}^{t-1}A_k(n-1-i,u) \\&+ (k-2)\sum_{j=0}^{n-1-t}A_k(j,u)- A_k(n-t,u) \\
&= 1+(k-1)\sum_{i=0}^{t-1}A_k(n-1-i,u)+ (k-2)\sum_{j=0}^{n-1-t}A_k(j,u)- A_k(n-t,u)\\
&= 1+(k-1)\sum_{i=1}^{t}A_k(n-i,u)+ (k-2)\sum_{j=0}^{n-1-t}A_k(j,u)- A_k(n-t,u)\\
&= 1+(k-1)\sum_{i=1}^{t-1}A_k(n-i,u)+ (k-2)\sum_{j=0}^{n-t}A_k(j,u)\\
&= R_k(n,u).
\end{align*}
\end{proof}
\begin{corollary}\label{corollary:sameUnb}
Let $n,t\geq 1$ and $k\geq 2$ be integers. Let $u$ and $v$ be length-$t$ unbordered words. Then $A_k(n,u) = A_k(n,v)$.
\end{corollary}
\begin{lemma}
Let $n$, $t$, and $k$ be integers such that $n\geq 2t\geq 1$ and $k\geq 2$. Then the number $U_k(n,t)$ of length-$n$ closed words that are closed by a length-$t$ unbordered word is $U_k(n,t) = u_t A_k(n-2t,0^{t-1}1)$.
\end{lemma}
\begin{proof}
We have that $A_k(n,u) = A_k(n,v)$ for length-$t$ unbordered words $u$, $v$ by Corollary~\ref{corollary:sameUnb}. Clearly $0^{t-1}1$ is unbordered, so $A_k(n,0^{t-1}1)=A_k(n,u)$ for all length-$t$ unbordered words $u$. Lemma~\ref{lemma:countUnb} says that the number of length-$n$ words closed by a length-$t$ unbordered word $u$ is $A_k(n-2t,u)$. Thus
\[U_k(n,t) =\sum_{u{\scaleobj{0.75}\in} \mathbf{u}_t}A_k(n-2t,u)=\sum_{u{\scaleobj{0.75}\in}{\mathbf{u}_t}}A_k(n-2t,0^{t-1}1)= u_t A_k(n-2t,0^{t-1}1).\]
\end{proof}
Since $(A_k(n,0^{t-1}1))_n$ satisfies a linear recurrence, we can easily find good estimates on the zeroes of its characteristic polynomial.
\begin{theorem}\label{theorem:alphaBound}
Let $k,t \geq 2$ be integers. Let \[\alpha(t) = k-\frac{1}{k^{t-1}-\frac{t-1}{k}-\frac{(t-1)^2}{k^{t-1}}}.\]
Then $\alpha(t+1) \leq k - \alpha(t+1)^{-t}$.
\end{theorem}
\begin{proof}
The case when $k=2$ was proved in~\cite[Theorem 10]{Forsyth2016}. We follow their proof to generalize to $k\geq 3$.
It can be easily proved that \[\frac{k^2-1}{k^2}t^2 \geq \frac{2t^3}{k^{t+1}}+\frac{t^4}{k^{2t}}\]
for all $k, t\geq 2$. Adding $tk^{t-1}$ to both sides and subtracting the two terms on the right-hand-side from both sides we get
\begin{align}
tk^{t-1} &\leq tk^{t-1} +\frac{k^2-1}{k^2}t^2 -\frac{2t^3}{k^{t+1}}-\frac{t^4}{k^{2t}}\nonumber \\
&= \bigg(\frac{t}{k}+\frac{t^2}{k^t}\bigg)\bigg(k^t- \frac{t}{k} - \frac{t^2}{k^t}\bigg).\nonumber
\end{align}
Let $\gamma(t) = \Big(k^t - \frac{t}{k}-\frac{t^2}{k^{t-1}}\Big)^{-1}$. Then
\[\gamma(t) t k^{t-1} \leq \frac{t}{k} + \frac{t^2}{k^t}.\]
Multiplying by $-1$ and adding $k^t$ to both sides we get
\[k^t - \gamma(t) t k^{t-1} \geq k^t - \frac{t}{k} - \frac{t^2}{k^t}.\]
It is easy to see that $\gamma(t) \leq 6/t$ for $t\geq 2$. Thus by Lemma~\ref{lemma:binomial} we have that \[(k-\gamma(t))^t \geq k^t - \gamma(t) t k^{t-1} \geq k^t - \frac{t}{k} - \frac{t^2}{k^t}.\]
So \[\gamma(t) \geq (k-\gamma(t))^{-t}.\] Therefore \[\alpha(t+1) \leq k - \alpha(t+1)^{-t}\] since $\alpha(t+1) = k-\gamma(t)$.
\end{proof}
\begin{lemma}\label{lemma:UnborderedBound}
Let $n\geq 0$, $t\geq 2$, and $k\geq 2$ be integers. Then $\alpha(t)^{n}\leq A_k(n,0^{t-1}1)$.
\end{lemma}
\begin{proof}
The proof is by induction on $n$. Let $\gamma(t) = \Big(k^{t-1} - \frac{t-1}{k}-\frac{(t-1)^2}{k^{t-1}}\Big)^{-1}$. Suppose $n<t$. Then \[\alpha(t)^{n} =(k-\gamma(t))^{n} \leq k^{n} = A_k(n,0^{t-1}1).\]
Suppose $n \geq t$. Firstly, notice that for $t\geq 2$ we have that $\alpha(t) -1 =k- \gamma(t) - 1\geq k-2$ since $\gamma(t) \leq 1$. Then
\begin{align}
A_k(n,0^{t-1}1) &= 1+(k-1)\sum_{i=1}^{t-1}A_k(n-i,0^{t-1}1) + (k-2)\sum_{j=0}^{n-t}A_k(j,0^{t-1}1) \nonumber \\
&\geq 1+ (k-1)\sum_{i=1}^{t-1}\alpha(t)^{n-i}+ (k-2)\sum_{j=0}^{n-t}\alpha(t)^j\nonumber \\
&= 1+ (k-1)\frac{\alpha(t)^n - \alpha(t)^{n-t+1}}{\alpha(t)-1} + (k-2)\frac{\alpha(t)^{n-t+1} - 1}{\alpha(t)-1}\nonumber \\
&= \frac{(k-1)\alpha(t)^n - \alpha(t)^{n-t+1}+(\alpha(t)-1) -(k-2)}{\alpha(t)-1}\nonumber \\
&\geq \frac{(k-1)\alpha(t)^n - \alpha(t)^{n-t+1}}{\alpha(t)-1}.\nonumber
\end{align}
By Theorem~\ref{theorem:alphaBound}, we have that $\alpha(t) -1 \leq (k-1) - \alpha(t)^{-t+1}$. Therefore
\[A_k(n,0^{t-1}1) \geq \frac{(k-1)\alpha(t)^n - \alpha(t)^{n-t+1}}{\alpha(t)-1} = \alpha(t)^n\frac{(k-1) - \alpha(t)^{-t+1}}{\alpha(t)-1} \geq \alpha(t)^n.\]
\end{proof}
\begin{theorem}
Let $k\geq 2$ be an integer. Then there exist constants $N>0$ and $c''>0$ such that \[C_k(n) \geq c''\frac{k^n}{n}\]
for all $n > N$.
\end{theorem}
\begin{proof}
Let $N'>0$ be a constant such that $\lfloor \log_k n \rfloor \geq 2$ for all $n>N'$.
From Lemma~\ref{lemma:limitBound}, we have that there exists a constant $N''>0$ such that \[\bigg(\frac{\alpha(\lfloor \log_k n \rfloor)}{k}\bigg)^{n-2\lfloor \log_k n \rfloor}\geq \frac{e^{-k}}{2}\] for all $n > N''$. Let $N = \max(N',N'')$. Let $c_0,c_1,c_2$ be positive real constants such that the following inequalities hold for all $n>N$.
\begin{align}
C_k(n)\geq U_k(n) &= \sum_{t=1}^{\lfloor n/2\rfloor} U_k(n,t)= \sum_{t=1}^{\lfloor n/2\rfloor} u_tA_k(n-2t,0^{t-1}1) \nonumber \\
&\geq u_{\lfloor \log_k n\rfloor} A_k(n-2\lfloor \log_k n\rfloor, \lfloor \log_k n\rfloor)\nonumber \\[.1in]
&\geq c_0 k^{\lfloor \log_k n \rfloor }\alpha(\lfloor \log_k n\rfloor)^{n-2\lfloor \log_k n\rfloor}\quad \text{ (by Lemma~\ref{lemma:UnborderedBound})}\nonumber \\[.1in]
&= c_0 \frac{k^n}{k^{\lfloor \log_k n\rfloor}}\bigg(\frac{\alpha(\lfloor \log_k n\rfloor)}{k}\bigg)^{n-2\lfloor \log_k n\rfloor}\nonumber \\[.1in]
&\geq c_1 \frac{k^n}{n} \frac{e^{-k}}{2} \quad \text{ (by Lemma~\ref{lemma:limitBound})}\nonumber \\[.1in]
&= c_2 \frac{k^n}{n}. \nonumber
\end{align}
\end{proof}
\end{comment}
\subsection{Upper bound}
Before we proceed with upper bounding $C_k(n)$, we briefly outline the direction of the proof. First, we begin by bounding $C_k(n,t)$ for $t< n/2$ and $t\geq n/2$. We show that for $t< n/2$, the number of length-$n$ words closed by a particular length-$t$ word $u$ is bounded by the number of words of length $n-2t$ that do not have $0^t$ as a factor. For $t\geq n/2$ we prove that $C_k(n,t)$ is negligibly small. Next, we prove upper bounds on the number of words that do not have $0^t$ as a factor, allowing us to finally bound $C_k(n)$.
\begin{lemma}\label{lemma:BboundA}
Let $n$, $t$, and $k$ be integers such that $n\geq 2t\geq 2$ and $k\geq 2$. Let $u$ be a length-$t$ word. Then \[B_k(n,u) \leq A_k(n-2t,0^t).\]
\end{lemma}
\begin{proof}
Recall that $B_k(n,u)$ is the number of length-$n$ words that are closed by the word $u$.
Let $w$ be a length-$n$ word closed by $u$ where $|w| = n \geq 2t = 2|u|$. Then we can write $w=uvu$ where $v$ does not contain $u$ as a factor. This immediately implies that $B_k(n,u) \leq A_k(n-2t,u)$. But from a result of Guibas and Odlyzko~\cite[Section 7]{Guibas&Odlyzko:1981}, we have that if $f_{a(u)}(2) > f_{a(v)}(2)$ for words $u$, $v$, then $A_k(m,u) \geq A_k(m,v)$ for all $m\geq 1$. The auto-correlation polynomial only has $0$ or $1$ as coefficients, depending on the $1$'s and $0$'s in the auto-correlation. Thus the auto-correlation $p$ that maximizes $f_p(2)$ is clearly $p=1^t$. The words that achieve this auto-correlation are words of the form $a^t$ where $a\in \Sigma_k$. Therefore we have
\[B_k(n,u) \leq A_k(n-2t,u) \leq A_k(n-2t,0^t).\]
\end{proof}
\begin{lemma}\label{lemma:Cntbound}
Let $n$, $t$, and $k$ be integers such that $n\geq 2t\geq 2$ and $k\geq 2$. Then
\[C_k(n,t) \leq k^t A_k(n-2t,0^t).\]
\end{lemma}
\begin{proof}
The number $C_k(n,t)$ of length-$n$ words closed by a length-$t$ word is equal to the sum, over all length-$t$ words $u$, of the number $B_k(n,u)$ of length-$n$ words closed by $u$. Thus we have that \[C_k(n,t) = \sum_{|u| = t}B_k(n,u).\]
By Lemma~\ref{lemma:BboundA} we have that $B_k(n,v) \leq A_k(n-2t,0^t)$ for all length-$t$ words $v$. Therefore \[C_k(n,t)=\sum_{|u| = t}B_k(n,u) \leq \sum_{|u|=t} A_k(n-2t, 0^t) \leq k^tA_k(n-2t,0^t).\]
\end{proof}
\begin{corollary}\label{corollary:Cbound}
Let $n\geq 1$ and $k\geq 2$ integers. Then
\[C_k(n) \leq \sum_{t=1}^{\lfloor n/2\rfloor} k^t A_k(n-2t,0^t) + nk^{\lceil n/2\rceil}.\]
\end{corollary}
\begin{proof}
It follows from Lemma~\ref{lemma:Cntbound} that \[C_k(n)=\sum_{t=1}^{n-1}C_k(n,t) \leq \sum_{t=1}^{\lfloor n/2\rfloor} k^t A_k(n-2t,0^t) + \sum_{t=\lfloor n/2\rfloor +1}^{n-1} C_k(n,t).\]
Now we show that \[\sum_{t=\lfloor n/2\rfloor +1}^{n-1} C_k(n,t)\leq nk^{\lceil n/2\rceil}.\]
Let $w=w_0w_1\cdots w_{n-1}$ be a word of length $n$ that is closed by a word $u$ of length $t>\lfloor n/2\rfloor $. Then $w = ux = yu$ for some words $x$, $y$. So $w_i = w_{i+(n-t)}$ for all $i$, $0\leq i < t$. This implies that $w= v^i v'$ where $v$ is the length-$(n-t)$ prefix of $w$, $i=\lfloor n/|v|\rfloor$, and $v'$ is the length-$(n-i|v|)$ prefix of $v$. Since $t > \lfloor n/2 \rfloor$, we have that $n-t < \lceil n/2\rceil$. We see that $w$ is fully determined by the word $v$. So since $|v| < \lceil n/2\rceil$, we have $C_k(n,t) \leq k^{\lceil n/2\rceil}$. Thus
\[\sum_{t=\lfloor n/2\rfloor +1}^{n-1} C_k(n,t)\leq \sum_{t=\lfloor n/2\rfloor +1}^{n-1} k^{\lceil n/2\rceil} \leq nk^{\lceil n/2\rceil}.\]
\end{proof}
\begin{lemma}\label{lemma:A0t}
Let $n\geq 0$, $t\geq 1$, and $k\geq 2$ be integers. Then
\[ A_k(n,0^t) = \begin{cases}
k^n, & \text{if $n < t$;} \\
(k-1)\sum\limits_{i=1}^t A_k(n-i,0^t), & \text{if $n\geq t$.}
\end{cases}
\]
\end{lemma}
\begin{proof}
If $n<t$, then any length-$n$ word is shorter than $0^t$, and thus cannot contain $0^t$ as a factor. So $A_k(n,0^t)=k^n$.
Suppose $n\geq t$. Let $w$ be a length-$n$ word that does not contain $0^t$ as a factor. Let us look at the symbols that $w$ ends in. Since $w$ does not contain $0^t$, we have that $w$ ends in anywhere from $0$ to $t-1$ zeroes. So $w$ is of the form $w = w' b 0^i$ where $i$ is an integer with $0\leq i \leq t-1$, $b\in \Sigma_k-\{0\}$, and $w'$ is a length-$(n-i-1)$ word that does not contain $0^t$ as a factor. There are $k-1$ choices for $b$, and $A_k(n-i-1,0^t)$ choices for $w'$. So there are $(k-1)A_k(n-i-1,0^t)$ words of the form $w'b0^i$. Summing over all possible $i$ gives
\[A_k(n,0^t)=(k-1)\sum_{i=1}^t A_k(n-i,0^t).\]
\end{proof}
\begin{corollary}
Let $n\geq 0$, $t\geq 1$, and $k\geq 2$ be integers. Then
\[ A_k(n,0^t) = \begin{cases}
k^n, & \text{if $n < t$;} \\
k^n-1, & \text{if $n=t$;}\\
kA_k(n-1,0^t) - (k-1)A_k(n-t-1,0^t), & \text{if $n> t$.}
\end{cases}
\]
\end{corollary}
\begin{proof}
Compute $A_k(n,0^t)-A_k(n-1,0^t)$ with the recurrence from Lemma~\ref{lemma:A0t} and the result follows.
\end{proof}
\begin{comment}
\begin{proof}
Suppose $n=t$. Then \[A_k(n,0^t) = (k-1)\sum_{i=1}^t A_k(t-i,0^t) = (k-1)\sum_{i=1}^tk^{t-i} = k^n-1.\]
Suppose $n>t$. Then
\begin{align*}
A_k(n,0^t) - A_k(n-1,0^t) &= (k-1)\sum_{i=1}^t A_k(n-i,0^t) - (k-1)\sum_{i=1}^t A_k(n-1-i,0^t)\\ &= (k-1)A_k(n-1,0^t)-(k-1)A_k(n-t-1,0^t).
\end{align*}
Therefore \[A_k(n,0^t) =kA_k(n-1,0^t) - (k-1)A_k(n-t-1,0^t).\]
\end{proof}
\end{comment}
\begin{corollary}\label{corollary:recUpp}
Let $n\geq 0$, $t\geq 1$, and $k\geq 2$ be integers. Then
\[ A_k(n,0^t) = \begin{cases}
k^n, & \text{if $n < t$;} \\
k^{n-t}(k^t -1)-(n-t)k^{n-t-1}(k-1), & \text{if $t \leq n \leq 2t$;}\\
kA_k(n-1,0^t) - (k-1)A_k(n-t-1,0^t), & \text{if $n> 2t$.}
\end{cases}
\]
\end{corollary}
Since $(A_k(n,0^{t}))_n$ satisfies a linear recurrence, we know that the asymptotic behaviour of $A_k(n,0^t)$ is determined by the real roots of the polynomial $x^{t+1}-kx^t+k-1=0$. We use this fact to find an upper bound for $A_k(n,0^t)$.
\begin{theorem}
\label{theorem:betaUpper}
Let $t\geq 1$ and $k\geq 2$ be integers. Let \[\beta_k(t) = k-(k-1)k^{-t-1}.\] Then $\beta_k(t) \geq k-(k-1)\beta_k(t)^{-t}$.
\end{theorem}
\begin{proof}
Since $\beta_k(t) \leq k$, we have that $\beta_k(t)^{-t} \geq k^{-t}\geq {k^{-t-1}}$. This implies that
\[\beta_k(t) = k-(k-1)k^{-t-1} \geq k-(k-1)\beta_k(t)^{-t}.\]
\end{proof}
\begin{lemma}
\label{lemma:baseCaseUpper}
Let $k,t\geq 2$ be integers. Let $n$ be an integer such that $2t \leq n\leq 3t$. Then $A_k(n,0^t) \leq \beta_k(t)^n$.
\end{lemma}
\begin{proof}
The proof is by induction on $n$. By Corollary~\ref{corollary:recUpp} we have that $A_k(n,0^t) = k^{n-t}(k^t -1)-(n-t)k^{n-t-1}(k-1)$ for $t \leq n \leq 2t$. Let $\gamma(t) = (k-1)k^{-t-1}$.
Suppose, for the base case, that $n=2t$. Then
\begin{align*}
A_k(2t,0^t) &= k^{t}(k^t -1)-tk^{t-1}(k-1) = k^{2t} - k^{t-2}(k^2 + tk(k-1))\\
&= k^{2t} - \gamma(t) k^{2t-1}\frac{(k^2 + tk(k-1))}{k-1} \\
&\leq k^{2t} - \gamma(t) t k^{2t-1}.
\end{align*}
Clearly $\gamma(t) \leq 6/t$ for all $t\geq 2$, so $A_k(2t)\leq k^{2t} - \gamma(t) t k^{2t-1} \leq (k- \gamma(t))^{2t} = \beta_k(t)^{2t}$.
Suppose that $2t < n \leq 3t$. Furthermore let $n = 2t + i + 1$ where $i$ is an integer such that $0\leq i < t$. Notice that $A_k(n-t-1,0^t) = A_k(t+i,0^t) = k^{i}(k^t-1) - ik^{i-1}(k-1)$. Then
\begin{align*}
A_k(2t+i+1,0^t) &= kA_k(2t+i,0^t) - (k-1)A_k(t+i,0^t)\\
&\leq k(k-\gamma(t))^{2t+i} - (k-1)(k^{i}(k^t-1) - ik^{i-1}(k-1)) \\
&= (k-\gamma(t))^{2t+i+1} + \gamma(t)(k-\gamma(t))^{2t+i}- (k-1)(k^{i}(k^t-1) - ik^{i-1}(k-1))\\
&= \beta_k(t)^{2t+i+1} + \gamma(t)\beta_k(t)^{2t+i}- (k-1)(k^{i}(k^t-1) - ik^{i-1}(k-1))
\end{align*}
To prove the desired bound, namely that $A_k(2t+i+1,0^t) \leq \beta_k(t)^{2t+i+1}$, it is sufficient to show that $\beta_k(t)^{2t+i} \leq \gamma(t)^{-1}(k-1)(k^{i}(k^t-1) - ik^{i-1}(k-1))$. We begin by upper bounding $\beta_k(t)^{2t+i}$ with Lemma~\ref{lemma:binomial}. We have
\begin{align}
\beta_k(t)^{2t+i} &\leq k^{2t+i} - \gamma(t)(2t+i)k^{2t+i-1}+\frac{1}{2}\gamma(t)^2(2t+i)(2t+i-1)k^{2t+i-2} \nonumber \\
&\leq k^{2t+i} - 2(k-1)tk^{t+i-2}+\frac{9}{2}(k-1)^2t^2k^{i-4} \nonumber\\
&\leq k^{2t+i+1} -(k-1)k^{2t+i} - 2(k-1)tk^{t+i-2}+\frac{9}{2}(k-1)^2t^2k^{i-4} \nonumber\\
&= k^{2t+i+1} -k^{t+i}\Big((k-1)k^t + 2(k-1)tk^{-2} -\frac{9}{2}(k-1)^2t^2k^{-t-4} \Big) . \label{lastLine}
\end{align}
It is easy to verify that $(k-1)k^t \geq k+t(k-1)$ and $2(k-1)tk^{-2} -\frac{9}{2}(k-1)^2t^2k^{-t-4} \geq 0$ for all $t\geq 2$. Thus, continuing from (\ref{lastLine}), we have
\begin{align*}
\beta_k(t)^{2t+i} &\leq k^{2t+i+1} -k^{t+i}(k + t(k-1) ) \leq k^{2t+i+1} -k^{t+i}(k + i(k-1) ) \\
&= \frac{k^{t+1}}{k-1}(k-1)(k^{t+i} - k^{i} - ik^{i-1}(k-1))\\
&= \gamma(t)^{-1} (k-1)(k^i(k^t-1) - ik^{i-1}(k-1)).
\end{align*}
\end{proof}
\begin{lemma}
\label{lemma:upperBoundA0}
Let $n$, $t$, and $k$ be integers such that $n\geq 2t \geq 4$ and $k\geq 2$. Then $A_k(n,0^t) \leq \beta_k(t)^{n}$.
\end{lemma}
\begin{proof}
The proof is by induction on $n$. The base case, when $2t\leq n \leq 3t$, is taken care of in Lemma~\ref{lemma:baseCaseUpper}.
Suppose $n>3t$. Then
\[
A_k(n,0^t) = (k-1)\sum_{i=1}^t A_k(n-i,0^t)\leq (k-1)\sum_{i=1}^t \beta_k(t)^{n-i}= (k-1)\frac{\beta_k(t)^n - \beta_k(t)^{n-t}}{\beta_k(t) - 1}.\]
By Theorem~\ref{theorem:betaUpper}, we have that $\beta_k(t) -1 \geq (k-1)-(k-1)\beta_k(t)^{-t}$. Therefore
\[A_k(n,0^t) \leq (k-1)\frac{\beta_k(t)^n - \beta_k(t)^{n-t}}{\beta_k(t) - 1}= \beta_k(t)^n \frac{(k-1) - (k-1)\beta_k(t)^{-t}}{\beta_k(t)-1} \leq \beta_k(t)^n.\]
\end{proof}
\begin{proof}[Proof of Theorem~\ref{theorem:mainC}~\ref{theorem:mainCUpper}]
First notice that $A_k(n,0) = (k-1)^n$, since $A_k(n,0)$ is just the number of length-$n$ words that do not contain $0$.
Let $N'$ be a positive integer such that the following inequalities hold for all $n>N'$.
\begin{align}
C_k(n) &\leq \sum_{t=2}^{\lfloor n/2\rfloor} k^t A_k(n-2t,0^t) + kA_k(n-2,0) + nk^{\lceil n/2\rceil} \nonumber \\
&\leq \sum_{t=2}^{\lfloor n/2\rfloor} k^t \beta_k(t)^{n-2t} + k(k-1)^{n-2} + nk^{\lceil n/2\rceil}\nonumber \\
&\leq \sum_{t=2}^{\lfloor n/2\rfloor} k^t \bigg(k-\frac{k-1}{k^{t+1}}\bigg)^{n-2t} + d_2\frac{k^n}{n}= k^n\sum_{t=2}^{\lfloor n/2\rfloor} \frac{1}{k^{t}} \bigg(1-\frac{k-1}{k^{t+2}}\bigg)^{n-2t} + d_2\frac{k^n}{n}\nonumber \\
&\leq k^n\Bigg(\sum_{t=2}^{\lfloor \log_k n\rfloor} \frac{1}{k^{t}} \bigg(1-\frac{k-1}{k^{t+2}}\bigg)^{n-2t} +\sum_{t=\lfloor \log_k n\rfloor+1}^{\lfloor n/2\rfloor} \frac{1}{k^{t}}\Bigg) + d_2\frac{k^n}{n}\nonumber \\
&\leq k^n\Bigg(\sum_{t=2}^{\lfloor \log_k n\rfloor } \frac{1}{k^{t}} \bigg(1-\frac{k-1}{k^{t+2}}\bigg)^{n-2\lfloor \log_k n\rfloor}+ \frac{d_3}{n}\Bigg) + d_2\frac{k^n}{n}\nonumber\\
&\leq k^n\sum_{t=2}^{\lfloor \log_k n\rfloor } \frac{1}{k^{t}} \bigg(1-\frac{k-1}{k^{t+2}}\bigg)^{n/2}+ d_4\frac{k^n}{n}. \label{firstSum}
\end{align}
Now we bound the sum in~(\ref{firstSum}).
Let $h(x) = (1-(k-1)k^{-2}x)^{n/2}$. Notice that $h(x)$ is monotonically decreasing on the interval $x\in (0,1)$. So for $k^{-t-1} \leq x \leq k^{-t}$ we have that $h(x) \geq h(k^{-t})$. Thus
\[\frac{1}{k^{t}} \bigg(1-\frac{k-1}{k^{t+2}}\bigg)^{n/2}\leq \frac{k-1}{k^{t}} \bigg(1-\frac{k-1}{k^{t+2}}\bigg)^{n/2} \leq k\bigg(\bigg(\frac{1}{k^{t}}-\frac{1}{k^{t+1}}\bigg)h(k^{-t})\bigg) \leq k\int_{k^{-t-1}}^{k^{-t}} h(x) \, dx.\] Going back to (\ref{firstSum}) we have
\[C_k(n) \leq k^n\sum_{t=2}^{\lfloor \log_k n\rfloor } k\int_{k^{-t-1}}^{k^{-t}} h(x) \, dx+ d_4\frac{k^n}{n} \leq k^{n+1}\int_0^1 h(x) \, dx+ d_4\frac{k^n}{n}.\]
Evaluating and bounding the definite integral, we have
\begin{align*}
\int_0^1 h(x) \, dx &= -\frac{k^2}{k-1}\bigg[\frac{(1-(k-1)k^{-2}x)^{n/2+1}}{n/2+1}\bigg]_{x=0}^{x=1}\\
&= -\frac{k^2}{k-1}\bigg(\frac{(1-(k-1)k^{-2})^{n/2+1} -1}{n/2+1}\bigg) \\
&\leq d_5\bigg(\frac{1-(1-(k-1)k^{-2})^{n/2+1}}{n/2+1}\bigg) \leq d_5\frac{1}{n/2+1} \leq \frac{d_6}{n}. \\
\end{align*}
Putting everything together, we have that
\[C_k(n) \leq k^{n+1}\int_0^1 h(x) \, dx+ d_4\frac{k^n}{n} \leq d_6\frac{k^{n+1}}{n} + d_4\frac{k^n}{n} \leq c'\frac{k^n}{n}\]
for some constant $c'>0$.
\end{proof}
\section{Privileged words}
\subsection{Lower bound}
In this section we provide a family of lower bounds for the number of length-$n$ privileged words. We use induction to prove these bounds. The basic idea is that we start with the lower bound by Nicholson and Rampersad, and then use it to bootstrap ourselves to better and better lower bounds.
\begin{proof}[Proof of Theorem~\ref{theorem:mainP}~\ref{theorem:mainPLower}]
The proof is by induction on $j$. Let $t= \lfloor \log_k(n-t) + \log_k(k-1) - \log_k(\ln k )\rfloor$. We clearly have $0\leq t \leq \log_k(n) + 1$ for all $n\geq 1$. Let $u$ be a length-$t$ privileged word. By Lemma~\ref{lemma:nich&ramp} we have that there exist constants $N_0$ and $c_0$ such that $P_k(n)\geq B_k(n,u) \geq c_0\frac{k^n}{n^2}$ for all $n > N_0$. So the base case, when $j=0$, is taken care of.
Suppose $j>0$. By induction we have that there exist constants $N_{j-1}$ and $c_{j-1}$ such that \[P_k(n)\geq c_{j-1}\frac{k^n}{n\log_k^{\circ j-1}(n)\prod_{i=1}^{j-1}\log_k^{\circ i}(n)}\] for all $n>N_{j-1}$. By Lemma~\ref{lemma:nich&ramp} we have
\begin{align*}
P_k(n) \geq P_k(n,t) \geq \sum_{\substack{|u|=t\\u\text{ privileged}}}B_k(n,u) \geq \sum_{\substack{|u|=t\\u\text{ privileged}}}d\frac{k^n}{n^2} =dP_k(t) \frac{k^n}{n^2}.
\end{align*}
for $n>N_0$. Since $t \leq \log_k(n) +1$, we have that $\frac{1}{\log_k^{\circ i}(t)}\geq \frac{1}{\log_k^{\circ i}(\log_k(n)+1)}$ for all $i\geq 0$. Thus continuing from above we have
\begin{align*}
P_k(n) &\geq dc_{j-1}\frac{k^t}{t\log_k^{\circ j-1}(t)\prod_{i=1}^{j-1}\log_k^{\circ i}(t)} \frac{k^n}{n^2}\geq d_7\frac{k^{\log_k(n-t) + \log_k(k-1) - \log_k(\ln k )}}{t\log_k^{\circ j-1}(t)\prod_{i=1}^{j-1}\log_k^{\circ i}(t)}\frac{k^n}{n^2} \\
&\geq d_8\frac{1}{t\log_k^{\circ j-1}(t)\prod_{i=1}^{j-1}\log_k^{\circ i}(t)}\frac{k^n}{n}\\
&\geq d_9\frac{1}{(\log_k(n)+1)\log_k^{\circ j-1}(\log_k(n)+1)\prod_{i=1}^{j-1}\log_k^{\circ i}(\log_k(n)+1)}\frac{k^n}{n} \\
&\geq c_j\frac{k^n}{n\log_k^{\circ j}(n)\prod_{i=1}^{j}\log_k^{\circ i}(n)}
\end{align*}
for all $n>N_j$ where $N_j > \max(N_0, N_{j-1})$.
\end{proof}
\subsection{Upper bound}
In Theorem~\ref{theorem:mainC}~\ref{theorem:mainCUpper} we proved that $C_k(n) \in O(\frac{k^n}{n})$. Since every privileged word is also a closed word, this is also shows that $P_k(n) \in O(\frac{k^n}{n})$. This bound improves on the existing bound on privileged words but it does not show that $P_k(n)$ and $C_k(n)$ behave differently asymptotically. We show that $P_k(n)$ is much smaller than $C_k(n)$ asymptotically by proving upper bounds on $P_k(n)$ that show $P_k(n) \in o(\frac{k^n}{n})$.
\begin{lemma}\label{lemma:Pntbound}
Let $n$, $t$, and $k$ be integers such that $n\geq 2t\geq 2$ and $k\geq 2$. Then
\[P_k(n,t) \leq P_k(t)A_k(n-2t,0^t).\]
\end{lemma}
\begin{proof}
The number of length-$n$ privileged words closed by a length-$t$ privileged word is equal to the sum, over all length-$t$ privileged words $u$, of the number $B_k(n,u)$ of length-$n$ words closed by $u$. Thus we have that \[P_k(n,t) = \sum_{\substack{|u|=t\\u\text{ privileged}}}B_k(n,u).\]
By Lemma~\ref{lemma:BboundA} we have that $B_k(n,v) \leq A_k(n-2t,0^t)$ for all length-$t$ words $v$. Therefore
\[ P_k(n,t) = \sum_{\substack{|u|=t\\u\text{ privileged}}}B_k(n,u) \leq \sum_{\substack{|u|=t\\u\text{ privileged}}}A_k(n-2t,0^t) \leq P_k(t) A_k(n-2t,0^t).\]
\end{proof}
\begin{proof}[Proof of Theorem~\ref{theorem:mainP}~\ref{theorem:mainPUpper}]
For $n\geq 2t$ we can use Lemma~\ref{lemma:Pntbound} to bound $P_k(n,t)$. But for $n < 2t$, we can use Corollary~\ref{corollary:Cbound} and the fact that $P_k(n,t) \leq C_k(n,t)$. We get
\[P_k(n) = \sum_{t=1}^{n-1} P_k(n,t) \leq \sum_{t=1}^{\lfloor n/2\rfloor} P_k(t) A_k(n-2t,0^t) + nk^{\lceil n/2\rceil}.\]
The proof is by induction on $j$. The base case, when $j=0$, is taken care of by Theorem~\ref{theorem:mainC}~\ref{theorem:mainCUpper}.
Suppose $j>0$. Then there exist constants $N_{j-1}'$ and $c_{j-1}'$ such that \[P_k(n) \leq c_{j-1}'\frac{k^n}{n\prod_{i=1}^{j-1}\log_k^{\circ i}(n)}\] for all $n>N_{j-1}'$.
We now bound $P_k(n)$. First we let $N_j' > N_{j-1}'$ be a constant such that the following inequalities hold for all $n>N_j'$. We have
\begin{align}
P_k(n) &\leq \sum_{t=1}^{\lfloor n/2\rfloor} P_k(t) A_k(n-2t,0^t) + nk^{\lceil n/2\rceil} \nonumber \\
&\leq \sum_{t=N_{j}'+1}^{\lfloor n/2\rfloor} c_{j-1}'\frac{k^t}{t\prod_{i=1}^{j-1}\log_k^{\circ i}(t)} \beta_k(t)^{n-2t} + \sum_{t=1}^{N_{j}'} P_k(t) A_k(n-2t,0^t) + nk^{\lceil n/2\rceil} \nonumber \\
&\leq \sum_{t=N_{j}'+1}^{\lfloor n/2\rfloor} c_{j-1}'\frac{k^t}{t\prod_{i=1}^{j-1}\log_k^{\circ i}(t)}\bigg(k-\frac{k-1}{k^{t+1}}\bigg)^{n-2t} + d_{10}\sum_{t=2}^{N_{j}'} \bigg(k-\frac{k-1}{k^{t+1}}\bigg)^{n-2t} + d_{11}\frac{k^n}{n^2} \nonumber \\
&\leq c_{j-1}'k^n\sum_{t=N_{j}'+1}^{\lfloor n/2\rfloor} \frac{1}{k^tt\prod_{i=1}^{j-1}\log_k^{\circ i}(t)} \bigg(1-\frac{k-1}{k^{t+2}}\bigg)^{n-2t} + d_{12}\frac{k^n}{n^2} \nonumber\\
&\leq c_{j-1}'k^n\bigg(d_{13}\sum_{t=N_{j}'+1}^{\lfloor \log_k(n)\rfloor} \frac{1}{k^tt\prod_{i=1}^{j-1}\log_k^{\circ i}(t)} \bigg(1-\frac{k-1}{k^{t+2}}\bigg)^{n/2}+\sum_{t=\lfloor \log_k(n)\rfloor+1}^{\lfloor n/2\rfloor} \frac{1}{k^tt\prod_{i=1}^{j-1}\log_k^{\circ i}(t)} \bigg) + d_{12}\frac{k^n}{n^2} \nonumber \\
&\leq c_{j-1}'k^n\bigg(d_{13}\sum_{t=N_{j}'+1}^{\lfloor \log_k(n)\rfloor} \frac{1}{k^tt\prod_{i=1}^{j-1}\log_k^{\circ i}(t)} \exp{\bigg(\frac{n}{2}\ln\bigg(1-\frac{k-1}{k^{t+2}}\bigg)\bigg)}\nonumber \\
&\hspace*{2cm}+\sum_{t=\lfloor \log_k(n)\rfloor+1}^{\infty} \frac{1}{k^tt\prod_{i=1}^{j-1}\log_k^{\circ i}(t)} \bigg) + d_{12}\frac{k^n}{n^2} \label{Pline}
\end{align}
The sum on line~(\ref{Pline}) is clearly convergent. We have
\begin{align*}
\sum_{t=\lfloor \log_k(n)\rfloor+1}^{\infty}\frac{1}{k^tt\prod_{i=1}^{j-1}\log_k^{\circ i}(t)} &\leq \frac{1}{(\lfloor \log_k(n)\rfloor+1)\prod_{i=1}^{j-1}\log_k^{\circ i}(\lfloor \log_k(n)\rfloor+1)}\sum_{t=\lfloor \log_k(n)\rfloor+1}^{\infty}\frac{1}{k^t} \\
&\leq d_{14}\frac{1}{\log_k(n)\prod_{i=1}^{j-1}\log_k^{\circ i}(\log_k(n))}\frac{1}{n} \leq d_{14}\frac{1}{n\prod_{i=1}^{j}\log_k^{\circ i}(n)}.
\end{align*}
Now we upper bound the sum \[D_n= \sum_{t=N_{j}'+1}^{\lfloor \log_k(n)\rfloor} \frac{1}{k^tt\prod_{i=1}^{j-1}\log_k^{\circ i}(t)} \exp{\bigg(\frac{n}{2}\ln\bigg(1-\frac{k-1}{k^{t+2}}\bigg)\bigg)}.\] It is well-known that $\ln(1-x) \leq -x$ for $|x| < 1$. Thus, letting $\alpha= \frac{k-1}{2k^2}$, we have \[\exp{\bigg(\frac{n}{2}\ln\bigg(1-\frac{k-1}{k^{t+2}}\bigg)\bigg)} \leq \exp{\Big(-\alpha\frac{n}{k^{t}}\Big)}.\]
We reverse the order of the series, by letting $t=\lfloor \log_k(n)\rfloor -t+N_{j}'+1$. We also shift the index of the series down by $N_{j}'+1$. We have
\begin{align}
D_n &= \sum_{t=0}^{\lfloor \log_k(n)\rfloor-N_{j}' - 1} \frac{1}{k^{\lfloor \log_k(n)\rfloor -t}(\lfloor \log_k(n)\rfloor -t)\prod_{i=1}^{j-1}\log_k^{\circ i}(\lfloor \log_k(n)\rfloor -t)}\exp{\Big(-\alpha\frac{n}{k^{\lfloor \log_k(n)\rfloor -t}}\Big)} \nonumber \\
&\leq d_{15}\sum_{t=0}^{\lfloor \log_k(n)\rfloor-N_{j}' - 1} \frac{k^{t}}{n(\log_k(n)-t)\prod_{i=1}^{j-1}\log_k^{\circ i}(\log_k(n) -t)} \exp{(-\alpha k^t)} \nonumber\\
&\leq d_{15}\frac{1}{n\prod_{i=1}^{j}\log_k^{\circ i}(n)}\sum_{t=0}^{\lfloor \log_k(n)\rfloor-N_{j}' - 1} \frac{k^{t}}{\prod\limits_{i=0}^{j-1}\frac{\log_k^{\circ i}(\log_k(n) -t)}{\log_k^{\circ i+1}(n)}} \exp{(-\alpha k^t)}\label{Dline}\\
\intertext{
Suppose $\beta$ is a positive constant strictly between $0$ and $1$ such that $\beta\log_k(n)$ is an integer and $\beta \log_k(n) < \lfloor \log_k(n)\rfloor-N_{j}' - 1$. If $t \leq \beta \log_k(n)$, then $\frac{\log_k^{\circ i}(\log_k(n) -t)}{\log_k^{\circ i+1}(n)} \geq\frac{\log_k^{\circ i+1}(n^{1-\beta})}{\log_k^{\circ i+1}(n)} \geq d_i'$ for some $d_i'>0$ by Lemma~\ref{lemma:logLimit}. If $t> \beta\log_k(n)$, then $\frac{\log_k^{\circ i}(\log_k(n) -t)}{\log_k^{\circ i+1}(n)} \geq \frac{\log_k^{\circ i}(N_j'+1)}{\log_k^{\circ i+1}(n)}$. We split up the sum in $D_n$ in two parts. One sum with $t\leq \beta\log_k(n)$ and one with $t> \beta\log_k(n)$. Continuing from~(\ref{Dline}) we get}
&\leq d_{15}\frac{1}{n\prod\limits_{i=1}^{j}\log_k^{\circ i}(n)}\bigg( \sum_{t=1}^{\beta \log_k(n)} \frac{k^{t}}{\prod\limits_{i=0}^{j-1}d_i'} \exp{(-\alpha k^t)}+\prod\limits_{i=0}^{j-1}\bigg(\frac{\log_k^{\circ i+1}(n)}{\log_k^{\circ i}(N_{j}'+1)}\bigg)\sum_{t=\beta \log_k(n)+1}^{\lfloor \log_k(n)\rfloor-N_{j}' - 1} k^{t} \exp{(-\alpha k^t)}\bigg)\nonumber \\
&\leq d_{15}\frac{1}{n\prod\limits_{i=1}^{j}\log_k^{\circ i}(n)}\bigg( d_{16}\sum_{t=1}^{\infty } t\exp{(-\alpha t)}+d_{17}\prod\limits_{i=1}^{j}\log_k^{\circ i}(n)\sum_{t=kn^\beta}^{\infty} t \exp{(-\alpha t)}\bigg)\nonumber
\end{align}
The first and second sum are both clearly convergent. It is also easy to show that both of them can be bounded by a constant multiplied by the first term. Thus we have that \[D_n \leq d_{15}\frac{1}{n\prod_{i=1}^{j}\log_k^{\circ i}(n)}\bigg( d_{18}+d_{19}\prod\limits_{i=1}^{j}\log_k^{\circ i}(n)\frac{kn^\beta}{\exp{(\alpha k n^\beta)}}\bigg)\leq d_{20}\frac{1}{n\prod_{i=1}^{j}\log_k^{\circ i}(n)}.\]
Putting everything together and continuing from line~(\ref{Pline}), we get
\[ P_k(n) \leq c'k^n\bigg(d_{13}D_n+d_{14}\frac{1}{n\prod_{i=1}^{j}\log_k^{\circ i}(n)}\bigg) + d_{12}\frac{k^n}{n^2} \leq c_j'\frac{k^n}{n\prod_{i=1}^{j}\log_k^{\circ i}(n)}\]
for some constant $c_j'>0$.
\end{proof}
\section{Open problems}
We conclude by posing some open problems.
\begin{enumerate}
\item In this paper we showed that $C_k(n) \in \Theta(\frac{k^n}{n})$. In other words, we showed that $C_k(n)$ can be bounded above and below by a constant times $k^n/n$ for $n$ sufficiently large. Can we do better than this? Does the limit \[\lim_{n\to \infty} \frac{C_k(n)}{k^n/n}\] exist? If it does exist, what does the limit evaluate to? Can one find good bounds on the limit?
\item In this paper we also gave a family of upper and lower bounds for $P_k(n)$. But for every $j\geq 0$, the upper and lower bounds on $P_k(n)$ are asymptotically separated by a factor of $1/\log_k^{\circ j}(n)$. Does there exist a $g(n)$ such that $P_k(n)\in \Theta(\frac{k^n}{g(n)})$? If such a function $g(n)$ exists, then does the limit \[\lim_{n\to \infty} \frac{P_k(n)}{k^n/g(n)}\] exist?
\end{enumerate}
\section{Acknowledgements}
Thanks to Jeffrey Shallit for introducing me to this problem and for helpful discussions and suggestions.
\bibliographystyle{new2}
\bibliography{abbrevs,simplest}
\end{document}
| 192,942
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BRAIN IMAGINGPrefrontal functional dissociation in the semantic network of patients with schizophreniaPark, Il Hoa c; Park, Hae-Jeonga b; Chun, Ji-Wonc; Kim, Eung Yeopb; Kim, Jae-Jina b cAuthor Information Departments of aPsychiatry bDiagnostic Radiology cInstitute of Behavioral Science in Medicine, Yonsei University College of Medicine, Seoul, Korea Correspondence to Dr Jae-Jin Kim, MD, PhD, Yongdong Severance Hospital, 612 Eonjuro, Gangnam-gu, Seoul 135-720, Korea Tel: +82 2 2019 3341; fax: +82 2 3462 4304; e-mail: [email protected], [email protected] Received 29 May 2008; accepted 12 June 2008 NeuroReport: September 17, 2008 - Volume 19 - Issue 14 - p 1391-1395 doi: 10.1097/WNR.0b013e32830cebff Buy Metrics Abstract We examined whether deficient prefrontal control over the semantic network exists in patients with schizophrenia. Fourteen patients with schizophrenia and 14 healthy controls performed a comparison task, judging semantic congruity according to an abstract category in an event-related functional MRI paradigm. In the control group, prefrontal–temporal networks consisting of the left inferior frontal gyrus and right inferior frontal sulcus converging at the left posterior superior temporal sulcus were identified as activated during semantic demand of incongruence. In the patients with schizophrenia, we observed a loss of the recruitment of the right inferior frontal sulcus and the prefrontal–temporal network. These findings indicate that cognitive modulation of semantic processing may be dysfunctional in patients with schizophrenia. © 2008 Lippincott Williams & Wilkins, Inc.
| 138,672
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Special Sessions
Propositions for Special Sessions must be sent to the Organizing Committee, including the title of the special session, and a short abstract (max 10 lines):
Jérôme Vicente: vicente@ut-capitole.fr
Olivier Brossard: olivier.brossard@ut-capitole.fr
Choice of Themes
Abstracts
For regular sessions, abstracts will be reviewed by two members of the Scientific Board. For special sessions, abstracts will be reviewed by the Session Manager and the Organizing Committee. The results will be individually sent to each author by email.
Final Papers
Regular Sessions
Special Sessions
| 384,053
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chapter \<open>Abstract Formulation of Tarski's Theorems\<close>
text \<open>We prove Tarski's proof-theoretic and semantic theorems about the
non-definability and respectively non-expressiveness (in the standard model) of truth\<close>
(*<*)
theory Tarski
imports Goedel_Formula Standard_Model_More
begin
(*>*)
section \<open>Non-Definability of Truth\<close>
context Goedel_Form
begin
context
fixes T :: 'fmla
assumes T[simp,intro!]: "T \<in> fmla"
and Fvars_T[simp]: "Fvars T = {xx}"
and prv_T: "\<And>\<phi>. \<phi> \<in> fmla \<Longrightarrow> Fvars \<phi> = {} \<Longrightarrow> prv (eqv (subst T \<langle>\<phi>\<rangle> xx) \<phi>)"
begin
definition \<phi>T :: 'fmla where "\<phi>T \<equiv> diag (neg T)"
lemma \<phi>T[simp,intro!]: "\<phi>T \<in> fmla" and
Fvars_\<phi>T[simp]: "Fvars \<phi>T = {}"
unfolding \<phi>T_def PP_def by auto
lemma bprv_\<phi>T_eqv:
"bprv (eqv \<phi>T (neg (subst T \<langle>\<phi>T\<rangle> xx)))"
unfolding \<phi>T_def using bprv_diag_eqv[of "neg T"] by simp
lemma prv_\<phi>T_eqv:
"prv (eqv \<phi>T (neg (subst T \<langle>\<phi>T\<rangle> xx)))"
using d_dwf.bprv_prv'[OF _ bprv_\<phi>T_eqv, simplified] .
lemma \<phi>T_prv_fls: "prv fls"
using prv_eqv_eqv_neg_prv_fls2[OF _ _ prv_T[OF \<phi>T Fvars_\<phi>T] prv_\<phi>T_eqv] by auto
end \<comment> \<open>context\<close>
theorem Tarski_proof_theoretic:
assumes "T \<in> fmla" "Fvars T = {xx}"
and "\<And>\<phi>. \<phi> \<in> fmla \<Longrightarrow> Fvars \<phi> = {} \<Longrightarrow> prv (eqv (subst T \<langle>\<phi>\<rangle> xx) \<phi>)"
shows "\<not> consistent"
using \<phi>T_prv_fls[OF assms] consistent_def by auto
end \<comment> \<open>context @{locale Goedel_Form}\<close>
section \<open>Non-Expressiveness of Truth\<close>
text \<open>This follows as a corollary of the syntactic version, after taking prv
to be isTrue on sentences.Indeed, this is a virtue of our abstract treatment
of provability: We don't work with a particular predicate, but with any predicate
that is closed under some rules --- which could as well be a semantic notion of truth (for sentences).\<close>
locale Goedel_Form_prv_eq_isTrue =
Goedel_Form
var trm fmla Var num FvarsT substT Fvars subst
eql cnj imp all exi
fls
prv bprv
enc
P
S
for
var :: "'var set" and trm :: "'trm set" and fmla :: "'fmla set"
and Var num FvarsT substT Fvars subst
and eql cnj imp all exi
and fls
and prv bprv
and enc ("\<langle>_\<rangle>")
and S
and P
+
fixes isTrue :: "'fmla \<Rightarrow> bool"
assumes prv_eq_isTrue: "\<And> \<phi>. \<phi> \<in> fmla \<Longrightarrow> Fvars \<phi> = {} \<Longrightarrow> prv \<phi> = isTrue \<phi>"
begin
theorem Tarski_semantic:
assumes 0: "T \<in> fmla" "Fvars T = {xx}"
and 1: "\<And>\<phi>. \<phi> \<in> fmla \<Longrightarrow> Fvars \<phi> = {} \<Longrightarrow> isTrue (eqv (subst T \<langle>\<phi>\<rangle> xx) \<phi>)"
shows "\<not> consistent"
using assms prv_eq_isTrue[of "eqv (subst T \<langle>_\<rangle> xx) _"]
by (intro Tarski_proof_theoretic[OF 0]) auto
text \<open>NB: To instantiate the semantic version of Tarski's theorem for a truth predicate
isTruth on sentences, one needs to extend it to a predicate "prv" on formulas and verify
that "prv" satisfies the rules of intuitionistic logic.\<close>
end \<comment> \<open>context @{locale Goedel_Form_prv_eq_isTrue}\<close>
(*<*)
end
(*>*)
| 94,643
|
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| 209,748
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TITLE: Recommend books for this syllabus
QUESTION [4 upvotes]: (Note: Mods, this is not a maths question per se, so please delete immediately if this is not in line with what is expected here. To all those gearing up to vote down - save it, vote to delete instead.)
I'm planning to buy a few books (Elementary to Intermediate) for the following syllabus. Can you folks recommend a number of books to cover this effectively? I'm not looking for books that will have excessive literature - I'm looking for concice, practice oriented books - something like the Schaum's series.
Numbers of books should be low
Books should introductory to intermediate, not very advanced
Amazon availability would be a huge help
Thanks! The syllabus is below:
TOPICS
1.
Solution of Quadratic equations with real coefficients.
2.
Arithmetic, geometric and harmonic progressions,
arithmetic, geometric and harmonic means,
sums of finite arithmetic and geometric progressions,
infinite geometric series,
sums of squares and cubes of the first n natural numbers.
3.
Permutations and combinations,
Binomial theorem for a positive integral index,
properties of binomial coefficients.
4.
Matrices as a rectangular array of real numbers,
equality of matrices,
addition, multiplication by a scalar and product of matrices,
transpose of a matrix,
determinant of a square matrix,
inverse of a square matrix,
properties of these matrix operations, diagonal,
symmetric and skew-symmetric matrices and their properties,
solutions of simultaneous linear equations using matrices.
Gauss-Jordan Method of Solution of simultaneous linear equations.
5.
Linear Algebra:
Dependence & independence of vectors,
bases and dimensions, spanning,
properties of quadratic forms.
7.
Two dimensions: Cartesian coordinates,
distance between two points, shift of origin.
Equation of a straight line in various forms,
distance of a point from a line;
Lines through the point of intersection of two given lines,
equation of the bisector of the angle between two lines;
Equation of a circle,
equations of tangent, normal and chord.
8.
Differential calculus: Real valued functions of a real variable,
into, onto and one-to-one functions,
sum, difference, product and quotient of two functions,
composite functions,
absolute value, polynomial, rational, exponential and logarithmic functions.
Limit and continuity of a function,
limit and continuity of the sum, difference, product and quotient of two functions,
Even and odd functions, inverse of a function, continuity of composite functions,
intermediate value property of continuous functions. Derivative of a function, derivative of the sum, difference, product and quotient of two functions, chain rule, derivatives of polynomial, rational, exponential and logarithmic functions. Derivatives of implicit functions; increasing and decreasing functions, maximum and minimum values of a function; partial derivatives; Lagrange’s Mean Value Theorem;
Applications: maxima and minima, optimization.
9.
Integral calculus: Integration as the inverse process of differentiation, indefinite integrals of standard functions, definite integrals and their properties. Integration by parts, integration by the methods of substitution and partial fractions, application of definite integrals to the determination of areas involving simple curves, continuous compounding, average value of functions.
10.
Formation of ordinary differential equations, solution of homogeneous differential equations, separation of variables method, linear first order differential equations.
11.
Numerical Analysis: solution of polynomial & transcendental equations using numerical methods such as Bisection, Newton-Raphson methods, Lagrange’s and Newton’s Interpolating polynomials.
REPLY [0 votes]: Stewart Calculus: Early Transcendentals and
Lay's Linear Algebra and It's Applications
Will cover 2,4,5,8,9 and get you started on 10
If you're concerned with price just buy an older edition there are plenty on Amazon.
| 31,169
|
\vspace{-2mm}
\renewcommand{\thesectiondis}[2]{\Alph{section}:}
\appendices
\section{Proof of Thm. \ref{thm:bound_memory_order}}\label{app:proof_memoryorder}
The proof is motivated from the derivation presented in \cite[Thm. 4]{POLK} and is presented here for the proposed algorithm. Consider the function iterates $f_{i,t}$ and $f_{i,t+1}$ of agent $i$ generated from Algorithm \ref{alg:soldd} at $t$-th and $(t+1)$-th instant. The function iterates $f_{i,t}$ and $f_{i,t+1}$ are parametrized by dictionary $\bbD_{i,t}$ and $\bbD_{i,t+1}$ and weights $\bbw_{i,t}$ and $\bbw_{i,t+1}$, respectively. The dictionary size corresponding to $f_{i,t}$ and $f_{i,t+1}$ in dictionary $\bbD_{i,t}$ and $\bbD_{i,t+1}$ are denoted by $M_{i,t}$ and $M_{i,t+1}$, respectively. The kernel dictionary $\bbD_{i,t+1}$ is formed from $\tbD_{i,t+1}= [\bbD_{i,t},\;\;\bbx_{i,t}]$ by selecting a subset of $M_{i,t+1}$ columns from $\tilde{M}_{i,t+1}=M_{i,t}+1$ number of columns of $\tbD_{i,t+1}$ that best approximate $\tilde{f}_{i,t+1}$ in terms of Hilbert norm error, i.e., $\|f_{i,t+1} - \tilde{f}_{i,t+1} \|_{\ccalH} \leq \eps $, where $\eps$ is the error tolerance.
Suppose the model order of function $f_{i,t+1}$ is less than equal to that of $f_{i,t}$, i.e., $M_{i,t+1}\le M_{i,t}$, which holds when the stopping criteria of KOMP is violated for dictionary $\tbD_{i,t+1}$:
\begin{align}\label{eq:dist_eq2}
\min_{j=1,\dots,M_{i,t}+1} \gamma_j\le \eps,
\end{align}
where $\gamma_j$ is the minimal approximation error with dictionary element $\bbd_{i,j}$ removed from dictionary $\tbD_{i,t+1}$ defined as
\begin{align}\label{eq:dist_eq3}
\gamma_j=\min_{\bbw\in \reals^{\tilde{M}_{i,t+1}-1}} \|\tilde{f}_{i,t+1}(\cdot)-\sum_{k\in \ccalI \setminus \{j\}} w_k\kappa(\bbd_{i,k}, \cdot)\|_\ccalH,
\end{align}
where $\ccalI=\{1,\dots,M_{i,t}+1\}$.
Observe that \eqref{eq:dist_eq2} lower bounds the approximation error $\gamma_{M_{i,t}+1}$ of removing the most recently added feature vector $\bbx_{i,t}$. Thus if $\gamma_{M_{i,t}+1}\le \eps$, then \eqref{eq:dist_eq2} is satisfied and the relation $M_{i,t+1}\le M_{i,t}$ holds, implying the model order does not grow. Hence it is adequate to consider $\gamma_{M_{i,t}+1}$.
Using the definition of $\tilde{f}_{i,t+1}$ and denoting $\ccalI'\coloneqq \ccalI \setminus \{M_{i,t}+1\}$, we write $\gamma_{M_{i,t}+1}$ as
\begin{align}\label{eq:dist_eq4}
&\gamma_{M_{i,t}+1}\\
&=\min_{\bbu\in \reals^{M_{i,t}}} \|{f}_{i,t} -\eta \nabla_{f_i}\hat{\ccalL}_{t}(\bbf_t,\bbmu_t)-\sum_{k\in \ccalI'} u_k\kappa(\bbd_{i,k}, \cdot)\|_\ccalH. \nonumber
\end{align}
The minimizer of \eqref{eq:dist_eq4} is obtained for $\bbu^*$ is obtained via a least-squares computation, and takes the form:
\begin{align}\label{eq:dist_eq6}
\bbu^*&=(1-\eta\lambda)\bbw_{i,t}-\eta\Big[\ell_i'(f_{i,t}(\bbx_{i,t}),y_{i,t}) \\
&+ \sum_{j\in n_i}\mu_{ij,t}h'_{ij}(f_{i,t}(\bbx_{i,t}),f_{j,t}(\bbx_{i,t}))\Big]\bbK_{\bbD_{i,t},\bbD_{i,t}}^{-1}\boldsymbol{\kappa}_{\bbD_{i,t}}(\bbx_{i,t}).\nonumber
\end{align}
Further, we define $\ell_i'(f_{i,t})$ as
\begin{align}\label{eq:l_prime}
\ell_i'(f_{i,t}):=&\ell_i'(f_{i,t}(\bbx_{i,t}),y_{i,t}) \nonumber\\
&+ \sum_{j\in n_i}\mu_{ij,t}h'_{ij}(f_{i,t}(\bbx_{i,t}),f_{j,t}(\bbx_{i,t}))\big],
\end{align}
Substituting $\bbu^*$ from \eqref{eq:dist_eq6} into \eqref{eq:dist_eq4} and applying Cauchy-Schwartz with \eqref{eq:l_prime}, we obtain $\gamma_{M_{i,t}+1}$ as
\begin{align}\label{eq:dist_eq8}
\gamma_{M_{i,t}+1}
&\le |{\eta\ell_i'(f_{i,t})}| \\
&\times \Big\lVert \kappa(\bbx_{i,t},\cdot)-\Big[\bbK_{\bbD_{i,t},\bbD_{i,t}}^{-1}\boldsymbol{\kappa}_{\bbD_{i,t}}(\bbx_{i,t})\Big]^T\boldsymbol{\kappa}_{\bbD_{i,t}}(\cdot)\Big\rVert_\ccalH.\nonumber
\end{align}
It can be observed from the right hand side of \eqref{eq:dist_eq8} that the Hilbert-norm term can be replaced by using the definition of subspace distance from \eqref{eq:dist1}. Thus, we get
\begin{align}\label{eq:dist_eq9}
\gamma_{M_{i,t}+1} \le |{\eta\ell_i'(f_{i,t})}| ~\text{dist}(\kappa(\bbx_{i,t}, \cdot), \ccalH_{\bbD_{i,t}}).
\end{align}
Now for $\gamma_{M_{i,t}+1}\le \eps$, the right hand side of \eqref{eq:dist_eq9} should also be upper bounded by $\eps$ and thus can be written as
\begin{align}\label{eq:dist_eq10}
\text{dist}(\kappa(\bbx_{i,t}, \cdot), \ccalH_{\bbD_{i,t}}) \le \frac{\eps}{\eta|{\ell_i'(f_{i,t})}|},
\end{align}
where we have divided both the sides by $|{\ell_i'(f_{i,t})}|$. Note that if \eqref{eq:dist_eq10} holds, then $\gamma_{M_{i,t}+1}\le \eps$ and since $\gamma_{M_{i,t}+1}\geq \min_j \gamma_j$, we may conclude that \eqref{eq:dist_eq2} is satisfied. Implying the model order at the subsequent steps does not grow, i.e., $M_{i,t+1}\le M_{i,t}$.
Now, let's take the contrapositive of the expression in \eqref{eq:dist_eq10} to observe that growth in the model order ($M_{i,t+1}= M_{i,t}+1$) implies that the condition
\begin{align}\label{eq:dist_eq11}
\text{dist}(\kappa(\bbx_{i,t}, \cdot), \ccalH_{\bbD_{i,t}}) > \frac{\eps}{\eta|{\ell_i'(f_{i,t})}|},
\end{align}
holds. Therefore, every time a new point is added to the model, the corresponding kernel function, i.e., $\kappa(\bbx_{i,t}, \cdot)$ is at least $\frac{\eps}{\eta|{\ell_i'(f_{i,t})}|}$ distance far away from every other kernel
function in the current model defined by dictionary $\bbD_{i,t}$.
Now, to have a bound on the right-hand side term of \eqref{eq:dist_eq11}, we bound the denominator of the right-hand side of \eqref{eq:dist_eq11}. Thus, we upper bound $|{\ell_i'(f_{i,t})}|$ as
\begin{align}\label{eq:dist_eq12_temp}
&|{\ell_i'(f_{i,t})}|\nonumber\\
&=|{\big[\ell_i'(f_{i,t}(\bbx_{i,t}),y_{i,t}) + \sum_{j\in n_i}\mu_{ij,t}h'_{ij}(f_{i,t}(\bbx_{i,t}),f_{j,t}(\bbx_{i,t}))\big]}|\nonumber\\
&\le |\ell_i'(f_{i,t}(\bbx_{i,t}),y_{i,t})| + | \sum_{j\in n_i}\mu_{ij,t}h'_{ij}(f_{i,t}(\bbx_{i,t}),f_{j,t}(\bbx_{i,t}))|\nonumber\\
& \le C + L_h E R_{i,t},
\end{align}
where in the first equality we have used the definition of $\ell_i'(f_{i,t})$ from \eqref{eq:l_prime} and the second inequality is obtained by using triangle inequality. To obtain the last inequality in \eqref{eq:dist_eq12_temp}, we use Assumption \ref{as:second} and \ref{as:third}, with $R_{i,t}=\max_{j\in n_i}|\mu_{ij,t}|$ and upper-bounded $|n_i|$ by the total number of edges $E$. Subsequently, we denote the right hand side of \eqref{eq:dist_eq12_temp} as ${R_{M_{i,t}}}\coloneqq C + L_h E R_{i,t}$. Substituting into \eqref{eq:dist_eq11} yields
\begin{align}\label{eq:dist_eq13}
\text{dist}(\kappa(\bbx_{i,t}, \cdot), \ccalH_{\bbD_{i,t}}) > \frac{\eps }{\eta {R_{M_{i,t}}}}.
\end{align}
Hence, the stopping criterion for the newest point is violated whenever it satisfies the condition, $\|\phi(\bbx_{i,t})-\phi(\bbd_{i,k})\|_2\le \frac{\eps }{\eta {R_{M_{i,t}}}}$ for $k\in\{1,\dots,M_{i,t}\}$ meaning $\phi(\bbx_{i,t})$ can be well approximated by $\phi(\bbd_{i,k})$ with already existing point $\bbd_{i,k}$ in the dictionary. Now for the finite model order proof, we proceed in a manner similar to the proof of \cite[Thm. 3.1]{1315946}. Since feature space $\ccalX$ is compact and $\phi(\bbx)=\kappa(\bbx,\cdot)$ is continuous (as $\kappa$ is continuous), we can deduce that $\phi(\ccalX)$ is compact. Therefore, the covering number (number of balls with radius $\varrho=\frac{\eps }{\eta {R_{M_{i,t}}}}$ to cover the set $\phi(\ccalX)$) of set $\phi(\ccalX)$ is finite \cite{zhou2002covering}. The covering number of a set is finite if and only if its packing number (maximum number of points in $\phi(\ccalX)$ separated by distance larger than $\varrho$) is finite. This means that the number of points in $\phi(\ccalX)$ separated by distance $\varrho$ is finite. Note that KOMP retains points which satisfy $\|\phi(\bbd_{i,j})-\phi(\bbd_{i,k})\|_2 >\varrho$, i.e., the dictionary points are $\varrho$ separated. Thus, when the packing number of $\phi(\ccalX)$ with scale $\varrho$ is finite, the number of dictionary points is also finite.
From \cite[Proposition 2.2]{1315946}, we know that for a Lipschitz continuous Mercer
kernel $\kappa$ on a compact set $\ccalX \subset \reals^p$, there exists a constant $\beta$ depending upon the $\ccalX$ and the kernel function such that for any training set $\{\bbx_{i,t}\}_{t=1}^\infty$ and any $\vartheta>0$, the number of elements in the dictionary satisfies
\begin{align}\label{eq:dist_eq14}
M_{i,t}\le \beta \Big(\frac{1}{\vartheta}\Big)^{2p}.
\end{align}
From \eqref{eq:dist_eq13}, observe $\vartheta=\frac{\eps }{\eta {R_{M_{i,t}}}}$. With \eqref{eq:dist_eq14} we may write
$M_{i,t}\le \beta ({\eta {R_{M_{i,t}}}}/{\eps})^{2p}$,
which is the required result stated in \eqref{eq:agent_mo} of Thm. \ref{thm:bound_memory_order}. To obtain the model order $M_t$ of the multi-agent system, we sum the model order of individual nodes across network, i.e.,
$M_t=\sum_{i=1}^N M_{i,t}$
as stated in Thm. \ref{thm:bound_memory_order}.\hfill $\blacksquare$
\section{Proof of Thm. \ref{thm:convergence}}
\label{app:proof_of_theorem}
Before discussing the proof, we introduce the following compact notations to make the analysis clear and compact.
We further use the following short-hand notations to denote the expressions involving $h_{ij}(\cdot,\cdot)$ as
\begin{align}\label{notation}
g_{ij}(f_t(\bbx_t)):=& h_{ij}(f_{i,t}(\bbx_{i,t}),f_{j,t}(\bbx_{i,t}))-\gamma_{ij}
\end{align}
Moreover, the $E$-fold stacking of the constraints across all the edges is denoted as $\bbG^t(\bbf)\coloneqq \text{vec}[g_{ij}(f(\bbx_t))]$ and $\bbG(\bbf)\coloneqq \mbE_{\bbx_t}[\bbG^t(\bbf)]$. The intermediate results required for the proof are stated in Lemma \ref{thm:bound_gap}-\ref{lemma:inst_lagrang_diff} (detailed in the supplementary material). Consider the statement of Lemma \ref{lemma:inst_lagrang_diff} (c.f. \eqref{eq:inst_lagrang_diff}), expand the left hand side of \eqref{eq:inst_lagrang_diff} using the definition of \eqref{eq:stochastic_approx}, further utilizing the notation of $S^t(\bbf_t)$ stated in \eqref{eq:main_prob} and $\bbG^t(\cdot)$ in \eqref{notation}, we can write
\begin{align}
&S^t(\bbf_t) + \ip{\bbmu,\bbG^t(\bbf_t) + \nu\one} - \frac{\delta\nu}{2}\norm{\bbmu}^2\nonumber\\
&\qquad\quad-S^t(\bbf)-\ip{\bbmu_t,\bbG^t(\bbf) + \nu\one} + \frac{\delta\nu}{2}\norm{\bbmu_t}^2\nonumber
\end{align}
\begin{align}\label{eq:converproof1}
&\leq \frac{1}{2\eta}\Delta_t + \frac{\eta}{2}\big( 2\|{\nabla}_{\bbf}\hat{\ccalL}_{t}(\bbf_{t},\bbmu_{t})\|_{\ccalH}^2+\|\nabla_{\bbmu}\hat{\ccalL}_{t}(\bbf_t,\mathbf{\bbmu}_t)\|^2\big)\nonumber \\
&\quad+\frac{\sqrt{V}\eps}{\eta}\|\bbf_t-\bbf\|_{\ccalH}+\frac{V\eps^2}{\eta}.
\end{align}
where $\Delta_t=(\|\bbf_t-\bbf\|_{\ccalH}^2-\|\bbf_{t+1}-\bbf\|_{\ccalH}^2+\|\bbmu_{t}-\bbmu\|^2-\|\bbmu_{t+1}-\bbmu\|^2)$. Next, let us take the total expectation on both sides of \eqref{eq:converproof1}. From Lemma
\ref{lemma:bound_primal_dual_grad}, substituting the upper bounds of $\|{\nabla}_{\bbf}\hat{\ccalL}_{t}(\bbf_{t},\bbmu_{t})\|_{\ccalH}^2$ and $\|\nabla_{\bbmu}\hat{\ccalL}_{t}(\bbf_t,\mathbf{\bbmu}_t)\|^2$ to \eqref{eq:converproof1}, the right hand side of \eqref{eq:converproof1} can be written as
\begin{align}\label{inter}
&\mbE\Big[\frac{1}{2\eta}\Delta_t +\frac{\sqrt{V}\eps}{\eta}\|\bbf_t-\bbf\|_{\ccalH}+\frac{V\eps^2}{\eta}\Big]\nonumber
\\
&\quad+\mbE\Big[\frac{\eta}{2}\big(2(4V X^2 C^2 + 4V X^2 L_h^2 E \|\bbmu_t\|^2+2V\lambda^2 \cdot R_{\ccalB}^2)\nonumber
\\
&\quad\quad\quad + E\Big((2K_1+2L_h^2X^2\cdot R_{\ccalB}^2)+ 2\delta^2\eta^2\|\bbmu_t\|^2 \Big)\Big].
\end{align}
Since each individual $f_{i,t}$ and $f_i$ for $\in\{i,\dots,V\}$ in the ball $\ccalB$ have finite Hilbert norm and is bounded by $R_{\ccalB}$, the term $\|\bbf_t-\bbf\|_{\ccalH}$ can be upper bounded by $2\sqrt{V}R_{\ccalB}$. Next, we define $K\coloneqq 8V X^2 C^2 +4V\lambda^2 \cdot R_{\ccalB}^2+2EK_1+2EL_h^2X^2\cdot R_{\ccalB}^2$. Now using the the bound of $\|\bbf_t-\bbf\|_{\ccalH}$ and the definition of $K$, we can upper bound the expression in \eqref{inter}, and then collectively writing the left and right hand side terms together, we get
\begin{align}\label{eq:converproof3}
&\mbE\Big[{S(\bbf_t)}-{S(\bbf)}+ \ip{\bbmu,{\bbG(\bbf_t)} + \nu\one} - \frac{\delta\eta}{2}\norm{\bbmu}^2 \\
&\qquad -\ip{\bbmu_t,{\bbG(\bbf)} + \nu\one}\Big] \nonumber\\
& \leq \mbE\Big[\frac{1}{2\eta}\Delta_t +{\frac{{2}V\eps}{\eta}}R_{\ccalB}+\frac{V\eps^2}{\eta}\Big]+\mbE\Big[\frac{\eta}{2}\big(K+C(\delta)\|\bbmu_t\|^2\big)\Big]. \nonumber
\end{align}
where $C(\delta):=8V X^2 L_h^2 E + 2E\delta^2\eta^2 -\delta$. Next, we select the constant parameter $\delta$ such that $C(\delta)\leq 0$, which then allows us to drop the term involving $\|\bbmu_t\|^2$ from the second expected term of right-hand side of \eqref{eq:converproof3}. Further, take the sum of the expression in \eqref{eq:converproof3} over times $t=1,\dots,T$, assume the initialization $\bbf_1=0\in\ccalH^V$ and $\bbmu_1=0\in\mbR_+^E$, we obtain
\begin{align}
\sum_{t=1}^T&\mbE\Big[{S(\bbf_t)}-{S(\bbf)}+ \ip{\bbmu,{\bbG(\bbf_t)} + \nu\one} - \frac{\delta\eta}{2}\norm{\bbmu}^2 \label{eq:converproof5}\\
&\qquad -\ip{\bbmu_t,{\bbG(\bbf)} + \nu\one}\Big]\nonumber\\
&\leq \frac{1}{2\eta}\big[\|\bbf\|_{\ccalH}^2+\|\bbmu\|^2\big]+ \frac{{2}V\eps T R_{\ccalB}+V\eps^2 T}{\eta}+\frac{\eta K T}{2}.\nonumber
\end{align}
where we drop the negative terms remaining after the telescopic sum since $\|\bbf_{T+1}-\bbf\|_{\ccalH}^2$ and $\|\bbmu_{T+1}-\bbmu\|^2$ are always positive. It can be observed from \eqref{eq:converproof5} that the right-hand side of this inequality is deterministic. We now take $\bbf$ to be the solution $\bbf_\nu^\star$ of \eqref{eq:prob_zero_cons}, which in turn implies $\bbf_\nu^\star$ must satisfy the inequality constraint of \eqref{eq:prob_zero_cons}. This means that $\bbf_\nu^\star$ is a feasible point, such that $\sum_{t=1}^T\sum_{(i,j)\in\ccalE}\mu_{ij,t}(g_{ij}(f^\star_{i}(\bbx_{i,t}),f^\star_{j}(\bbx_{i,t}))+\nu)\leq 0$ holds. Thus we can simply drop this term in \eqref{eq:converproof5} and collecting the terms containing $\|\bbmu\|^2$ together, we obtain
\begin{align}\label{eq:converproof6}
&\sum_{t=1}^T\mbE\Big[{S(\bbf_t)}-{S(\bbf_\nu^\star)}+\ip{\bbmu,{\bbG(\bbf_t)}+\nu\one}\Big] - z(\eta,T)\|\bbmu\|^{2}\nonumber\\
&\qquad\leq \frac{1}{2\eta}\|\bbf_\nu^\star\|_{\ccalH}^2+ {\frac{V\eps T}{\eta}}({2}R_{\ccalB}+\eps)+\frac{\eta K T}{2}.
\end{align}
where $z(\eta,T):=\frac{\delta\eta T}{2}+\frac{1}{2\eta}$. Next, we maximize the left-hand side of \eqref{eq:converproof6} over $\bbmu$ to obtain the optimal Lagrange multiplier which controls the growth of the long-term constraint violation, whose closed-form expression is given by
\begin{align}\label{eq:bar_mu}
\bar{\mu}_{ij}= \frac{1}{2(\delta\eta T+1/ \eta)}\big[\sum_{t=1}^T\mbE [g_{ij}(f_t(\bbx_t))]+\nu\big]_+.
\end{align}
Therefore, selecting $\bbmu=\bar{\bbmu}$ in \eqref{eq:converproof6}, we obtain
\begin{align}\label{eq:converproof8}
&\sum_{t=1}^T\mbE\big[{S(\bbf_t)}-{S(\bbf_\nu^\star)}\big]+\!\!\!\!\sum_{(i,j)\in\ccalE}\!\!\!\!\frac{\Big[\sum_{t=1}^T\Big(\mbE[g_{ij}(f_t(\bbx_t))]+\nu\Big)\Big]_+^2}{2(\delta\eta T+1/ \eta)}\nonumber\\
&\qquad\qquad\leq \frac{1}{2\eta}\|\bbf_\nu^\star\|_{\ccalH}^2+ {\frac{V\eps T}{\eta}}({2}R_{\ccalB}+\eps)+\frac{\eta K T}{2}.
\end{align}
Firstly, consider the objective error sequence $\mbE\big[{S(\bbf_t)}-{S(\bbf_\nu^\star)}\big]$, we observe from \eqref{eq:converproof8} that the second term present on the left-side of the inequality can be dropped without affecting the inequality owing to the fact that it is positive. So we obtain
\begin{align}
\sum_{t=1}^T\mbE\big[{S(\bbf_t)}-{S(\bbf_\nu^\star)}\big]
\leq \frac{\|\bbf_\nu^\star\|_{\ccalH}^2}{2\eta}+ {\frac{V\eps T}{\eta}}({2}R_{\ccalB}+\eps)+\frac{\eta K T}{2}.\nonumber
\end{align}
Using Lemma \ref{thm:bound_gap} and summing over $t=1, \dots, T$, we have
\begin{align}\label{eq:converproof9_1}
\sum_{t=1}^T \big[{S(\bbf_\nu^*)}-{S(\bbf^*)}\big]\le \frac{{4}VR_{\ccalB}(C X+\lambda R_{\ccalB})}{\xi}\nu T.
\end{align}
Adding these inequalities, and then taking the average over $T$, we obtain
\begin{align}\label{eq:converproof9_1_temp}
\frac{1}{T}\sum_{t=1}^T\mbE\big[{S(\bbf_t)}-{S(\bbf^*)}\big]&\leq \frac{1}{2\eta T}\|\bbf_\nu^\star\|_{\ccalH}^2+ {\frac{V\eps}{\eta}}({2}R_{\ccalB}+\eps)\nonumber\\
&\hspace{-1cm} +\frac{\eta K}{2} + \frac{{4}VR_{\ccalB}(C X+\lambda R_{\ccalB})}{\xi}\nu.
\end{align}
Since $\eps\leq {2}R_{\ccalB}$, we upper bound $\eps$ present in the second term of the left hand side of \eqref{eq:converproof9_1_temp} by ${2}R_{\ccalB}$. Next, using the definition $\alpha=\eps/\eta$ from Thm. \ref{thm:bound_memory_order}, and setting step size $\eta=1/\sqrt{T}$ and $\nu=\zeta T^{-1/2} + \Lambda \alpha$ for some $(\zeta, \Lambda)>0$, we obtain
\begin{align}\label{eq:converproof9_1_final}
\frac{1}{T}\sum_{t=1}^T\mbE\big[{S(\bbf_t)}-{S(\bbf^*)}\big]\leq\mathcal{O}(T^{-1/2}+\alpha)
\end{align}
which is the required result in \eqref{eq:func_order}.
\begin{comment}
{\color{red} The rest of the proof has many errors. Specifically, Notice the distinction between
\begin{align}
S^t(\bbf) &= \sum_{i\in\ccalV} \ell_i(f_i(\bbx_{i,t}),y_{i,t}) + \frac{\lambda}{2}\norm{f_i}_{\ccalH}^2 \\
S(\bbf) & = \mbE_{\bbx_{i,t},y_{i,t}} [S^t(\bbf)] \\
S^t(\bbf_t) &= \sum_{i\in\ccalV} \ell_i(f_{i,t}(\bbx_{i,t}),y_{i,t}) + \frac{\lambda}{2}\norm{f_{i,t}}_{\ccalH}^2
\end{align}
where the second equation expectation is conditioned on $\bbf$. This distinction must be maintained throughout the rest of the proof. Note that $\bbG$ and $\bbG^t$ have also been defined similarly. You must take care when defining the Lagrangian and standard Lagrangian: are they with respect to the expectations or empirical sums? In online setting they are with respect to expectations. Therefore the standard Lagragian should be defined as
\begin{align}
\ccalL^s(\bbf_t,\bbmu) = \mbE\left[S^t(\bbf_t) + \ip{\bbmu,\bbG^t(\bbf_t)+\nu\one}\right]
\end{align}}
\end{comment}
Next, we establish the bound on the growth of the constraint violation. For this, we first denote $\ccalL^s$ as the standard Lagrangian for \eqref{eq:prob_zero_cons} and write it for ${\bbf}$ and $\bbmu$ as, $\ccalL^s({\bbf},\bbmu) = {S(\bbf)} + \ip{\bbmu,{\bbG(\bbf)}+\nu\one}$. The standard Lagrangian for ${(\bbf_t,\bbmu)}$ and $(\bbf,\bbmu_t)$ are defined similarly. Now using the expressions of $\ccalL^s(\bbf_t,\bbmu)$ and $\ccalL^s(\bbf,\bbmu_t)$, we rewrite \eqref{eq:converproof5} as
\begin{align}\label{eq:converproof10_4}
&\sum_{t=1}^T \mbE\big[\ccalL^s(\bbf_t,\bbmu)- \ccalL^s(\bbf,\bbmu_t)\big]\le \frac{1}{2\eta}\mbE\big[\|\bbf\|_{\ccalH}^2+\|\bbmu\|^2\big]\nonumber\\
&\quad\quad+ \frac{{2}V\eps T R_{\ccalB}+V\eps^2 T}{\eta}+\frac{\eta K T}{2} + \frac{\delta\eta T}{2}\mbE\|\bbmu\|^{2}.
\end{align}
{Since $(\bbf_\nu^*,\bbmu_\nu^*)$ is the optimal pair for standard Lagrangian $\ccalL^s(\bbf,\bbmu)$ of \eqref{eq:prob_zero_cons} and assuming $\mathbf{1}_i$ to be a vector of all zeros except the $i$-th entry which is unity, write for $\bbmu=\mathbf{1}_i+\bbmu_\nu^*$:
\begin{align}\label{eq:converproof10_5}
&\mbE\big[\ccalL^s(\bbf_t,\mathbf{1}_i+\bbmu_\nu^*)\big]-\mbE\big[\ccalL^s(\bbf_\nu^*,\bbmu_\nu^*)\big]\nonumber\\
&=\mbE\big[{S(\bbf_t)}+ \langle \mathbf{1}_i+\bbmu_\nu^*, \bbG(\bbf_t)+ \nu\mathbf{1}\rangle\big]-\mbE\big[\ccalL^s(\bbf_\nu^*,\bbmu_\nu^*)\big]\nonumber\\
&=\mbE\big[{S(\bbf_t)}+ \langle \bbmu_\nu^*, \bbG(\bbf_t)+ \nu\mathbf{1}\rangle\big]-\mbE\big[\ccalL^s(\bbf_\nu^*,\bbmu_\nu^*)\big]\nonumber\\
&\quad + \mbE\big[\langle \mathbf{1}_i, \bbG(\bbf_t)+ \nu\mathbf{1}\rangle\big]\nonumber\\
&=\mbE\big[\ccalL^s(\bbf_t,\bbmu_\nu^*)-\ccalL^s(\bbf_\nu^*,\bbmu_\nu^*)\big] + \mbE\big[G_i(\bbf_t)+\nu\big].
\end{align}
The first equality in \eqref{eq:converproof10_5} is written by using the definition of standard Lagrangian $\ccalL^s$ and $\bbG$ denotes the stacking of the constraints of all edges as defined in the paragraph following \eqref{notation}. In the second equality, we split $\langle \mathbf{1}_i+\bbmu_\nu^*, \bbG(\bbf_t)+ \nu\mathbf{1}\rangle$ into $\langle \bbmu_\nu^*, \bbG(\bbf_t)+ \nu\mathbf{1}\rangle$ and $\langle \mathbf{1}_i, \bbG(\bbf_t)+ \nu\mathbf{1}\rangle$ using additivity of the inner product. Via the definition of $\ccalL^s$, we rewrite ${S}(\bbf_t)+ \langle \bbmu_\nu^*, \bbG(\bbf_t)+ \nu\mathbf{1}\rangle$ in the second equality as $\ccalL^s(\bbf_t,\bbmu_\nu^*)$ in the third equality. The last term in the third equality is written using the definition of $\mathbf{1}_i$ and $G_i$ denotes the $i$-th constraint of the stacked constraint vector $\bbG$.
Since $(\bbf_\nu^*,\bbmu_\nu^*)$ is a saddle point of $\ccalL^s$, it holds that
\begin{align}\label{eq:converproof10_6}
\mbE\big[\ccalL^s(\bbf_\nu^*,\bbmu_t)\big]\le \mbE\big[\ccalL^s(\bbf_\nu^*,\bbmu_\nu^*)\big]\le \mbE\big[\ccalL^s(\bbf_t,\bbmu_\nu^*)\big].
\end{align}
From the relation of \eqref{eq:converproof10_6}, we know the first term in the third equality of \eqref{eq:converproof10_5} is positive, and thus drop it to write:
\begin{align}\label{eq:converproof10_7}
&\mbE\big[G_i(\bbf_t)+\nu\big]
\le \mbE\big[\ccalL^s(\bbf_t,\mathbf{1}_i+\bbmu_\nu^*)\big]-\mbE\big[\ccalL^s(\bbf_\nu^*,\bbmu_\nu^*)\big]\nonumber\\
&\quad =\mbE\big[\ccalL^s(\bbf_t,\mathbf{1}_i+\bbmu_\nu^*)\big]-\mbE\big[\ccalL^s(\bbf_\nu^*,\bbmu_t)\big]+\mbE\big[\ccalL^s(\bbf_\nu^*,\bbmu_t)\big]\nonumber\\
&\quad\quad-\mbE\big[\ccalL^s(\bbf_\nu^*,\bbmu_\nu^*)\big]\nonumber\\
& \quad \le \mbE\big[\ccalL^s(\bbf_t,\mathbf{1}_i+\bbmu_\nu^*)\big]-\mbE\big[\ccalL^s(\bbf_\nu^*,\bbmu_t)\big]
\end{align}
where in the second equality we have added and subtracted $\mbE\big[\ccalL^s(\bbf_\nu^*,\bbmu_t)\big]$ and the last inequality comes from the fact that $\mbE\big[\ccalL^s(\bbf_\nu^*,\bbmu_t)\big]-\mbE\big[\ccalL^s(\bbf_\nu^*,\bbmu_\nu^*)\big]\le 0$ from the relation \eqref{eq:converproof10_6}.
Now, summing \eqref{eq:converproof10_7} over $t=1, \dots, T$ and using \eqref{eq:converproof10_4} we get:
\begin{align}\label{eq:converproof10_7_temp}
&\sum_{t=1}^T\mbE\big[G_i(\bbf_t)+\nu\big] \le \sum_{t=1}^T\big(\mbE\big[\ccalL^s(\bbf_t,\mathbf{1}_i+\bbmu_\nu^*)\big]-\mbE\big[\ccalL^s(\bbf_\nu^*,\bbmu_t)\big]\Big)\nonumber\\
& \le \frac{1}{2\eta}\Big[\|\bbf_\nu^*\|_{\ccalH}^2+\|\mathbf{1}_i+\bbmu_\nu^*\|^2\Big] + V\alpha T (2R_{\ccalB} + \eps)+\frac{ \eta K T}{2} \nonumber\\
&\qquad + \frac{\delta\eta {T}}{2}\|\mathbf{1}_i+\bbmu_\nu^*\|^{2},
\end{align}
where $\alpha=\eps/\eta$ defined in Thm. \ref{thm:bound_memory_order}.
Next, using Assumption \ref{as:fifth},
the first term in \eqref{eq:converproof10_7_temp} by $R_{\ccalB}^2$, and the second term on the right hand side of \eqref{eq:converproof10_7_temp} is bounded as
\begin{align}\label{eq:converproof10_8_1}
\|\mathbf{1}_i+\bbmu_\nu^*\|^2&\le 2 \|\mathbf{1}_i\|^2 + 2 \|\bbmu_\nu^*\|^2
\le 2 + 2\Big(\sum_{i=1}^E \mu_{i,\nu}^*\Big)^2\nonumber\\
&\qquad\quad\le 2 + 2 \Big(\frac{{4}VR_{\ccalB}(C X+\lambda R_{\ccalB})}{\xi}\Big)^2.
\end{align}
In the second inequality of \eqref{eq:converproof10_8_1}, we have used the fact that each $\mu_{i,\nu}^*$ is positive thus allowing us to use the inequality $(a^2+b^2+c^2)\le (a+b+c)^2$ where $a,b$ and $c$ are positive. In the last inequality of \eqref{eq:converproof10_8_1} we have used the upper bound of \eqref{eq:boundgap_9} (in the supplementary) to bound the $\big(\sum_{i=1}^E \mu_{i,\nu}^*\big)^2$ term.
Now setting step size $\eta=1/\sqrt{T}$ and using Assumption \ref{as:fifth} and the upper bound of \eqref{eq:converproof10_8_1}, we get
\begin{align}\label{eq:converproof10_8_1_1}
&\sum_{t=1}^T\mbE\big[G_i(\bbf_t)\big] \leq \sqrt{T} \Gamma + V\alpha T (2R_{\ccalB} + \eps)-\nu T.
\end{align}
where $\Gamma \coloneqq \frac{1}{2}\bigg[R_{\ccalB}^2 + (1+\delta)\bigg(2 + 2 \Big(\frac{{4}VR_{\ccalB}(C X+\lambda R_{\ccalB})}{\xi}\Big)^{2}\bigg) + K\bigg]$. If $\eps\leq 2 R_{\ccalB}$, then it follows from \eqref{eq:converproof10_8_1_1} that
\begin{align}\label{eq:converproof10_8_1_2}
\frac{1}{T}\sum_{t=1}^T\mbE\big[G_i(\bbf_t)\big] \leq \frac{1}{\sqrt{T}}\Gamma + 4VR_{\ccalB}\alpha -\nu.
\end{align}
Setting $\nu=\zeta T^{-1/2} + \Lambda \alpha$, where $\zeta\geq \Gamma$ and $\Lambda \geq 4VR_{\ccalB}$ ensures the aggregation of constraints gets satisfied on long run, i.e., $\sum_{t=1}^T\mbE\Big[G_i(\bbf_t)\Big]\le 0$, as stated in \eqref{eq:constr_order}. Analogous logic applies for all edges $i\in \ccalE$.\hfill $\blacksquare$
\begin{comment}
\red{To be modified: Select $\eps=P\eta^2=P/T$ in \eqref{eq:converproof10_8_1_1} to write
\begin{align}
\sum_{t=1}^T\mbE\big[G_i(\bbf_t)+\nu\big]& \le \frac{\sqrt{T}}{2}\Gamma + \frac{VP^2}{\sqrt{T}}\label{eq:converproof10_8_2}
\end{align}
where $\Gamma\coloneqq R_{\ccalB}^2+(1+\delta)[2 + 2 (\frac{{4}VR_{\ccalB}(C X+\lambda R_{\ccalB})}{\xi})^2] + 4VPR_{\ccalB} + K $. For $T$ sufficiently large such that $T\geq \frac{2VP^2}{\Gamma}$. Thus $\frac{VP^2}{\sqrt{T}}\le \frac{\Gamma}{2}\sqrt{T}$. With the upper bound of $\frac{VP^2}{\sqrt{T}}$ in \eqref{eq:converproof10_8_2}, we get
\begin{align}\label{eq:converproof10_9}
\sum_{t=1}^T\mbE\Big[G_i(\bbf_t)\Big]& \le \Gamma\sqrt{T}-\nu T .
\end{align}
Setting $\nu=\zeta T^{-1/2}$, where $\zeta\geq \Gamma$ ensures the aggregation of constraints gets satisfied on long run, i.e.,
$\sum_{t=1}^T\mbE\Big[G_i(\bbf_t)\Big]\le 0$,
signifying aggregate constraint violation is null, as stated in \eqref{eq:cons_order_temp}. Analogous logic applies for all edges $i\in \ccalE$.} \hfill $\blacksquare$
\end{comment}
}
\begin{comment}
\section{Proof of Thm. \ref{thm:order}}
\label{app:proof_of_theorem}
\begin{proof}
The proof depends on the result of Lemma \ref{lemma:inst_lagrang_diff} defined in \eqref{eq:inst_lagrang_diff}. We expand the left-hand side of \eqref{eq:inst_lagrang_diff} using \eqref{eq:stochastic_approx} and use the definition of $S^t(\bbf_t)$ stated in the Thm. \ref{thm:order} and obtain the following expression,
\begin{align}
&\sum_{i\in\ccalV}\bigg[\ell_i(f_{i,t}\big(\bbx_{i,t}), y_{i,t}\big)+\frac{\lambda}{2}\|f_{i,t} \|^2_{\ccalH} \bigg]\nonumber\\
&\quad+\sum_{(i,j)\in\ccalE} \left\{ \bigg[\mu_{ij}(g_{ij}(f_{i,t}(\bbx_{i,t}),f_{j,t}(\bbx_{i,t})))\bigg] - \frac{\delta\eta}{2}\mu_{ij}^{2}\right\}\nonumber
\\
&\quad-\sum_{i\in\ccalV}\bigg[\ell_i(f_{i}\big(\bbx_{i,t}), y_{i,t}\big)+\frac{\lambda}{2}\|f_{i} \|^2_{\ccalH} \bigg]\nonumber
\end{align}
\begin{align}
&\quad+\sum_{(i,j)\in\ccalE} \left\{\bigg[\mu_{ij,t}(g_{ij}(f_{i}(\bbx_{i,t}),f_{j}(\bbx_{i,t})))\bigg]-\frac{\delta\eta}{2}\mu_{ij,t}^{2}\right\}\nonumber\\
&\leq \frac{1}{2\eta}\bigg(\|\bbf_t-\bbf\|_{\ccalH}^2-\|\bbf_{t+1}-\bbf\|_{\ccalH}^2+\|\bbmu_{t}-\bbmu\|^2-\|\bbmu_{t+1}-\bbmu\|^2\bigg)\nonumber\\
&\quad+ \frac{\eta}{2}\bigg( 2\|{\nabla}_{\bbf}\hat{\ccalL}_{t}(\bbf_{t},\bbmu_{t})\|_{\ccalH}^2+\|\nabla_{\bbmu}\hat{\ccalL}_{t}(\bbf_t,\mathbf{\bbmu}_t)\|^2\bigg)\nonumber\\
&\quad+\frac{\sqrt{V}\eps_t}{\eta}\|\bbf_t-\bbf\|_{\ccalH}+\frac{V\eps_t^2}{\eta}.\label{eq:converproof1}
\end{align}
Before discussing the analysis, we introduce the following compant notations to make the analysis clear and compact. We will denote the expressions involving $g_{ij}(\cdot,\cdot)$ as
\begin{align}\label{notation}
g_{ij}(f_t(\bbx_t)):=& g_{ij}(f_{i,t}(\bbx_{i,t}),f_{j,t}(\bbx_{i,t}))\nonumber\\
g_{ij}(f(\bbx_t)):=& g_{ij}(f_{i}(\bbx_{i,t}),f_{j}(\bbx_{i,t})).
\end{align}
Next, we compute the expectation not only on the random pair $(\bbx,\bby)$ but also on the entire algorithm history, i.e., on sigma algebra $\ccalF_t$ which measures the algorithm history for times $u\leq t$, i.e., $\ccalF_t \supseteq \{\bbx_u, \bby_u, \bbf_u, \bbmu_u\}_{u =0}^{t-1} $ on both sides of \eqref{eq:converproof1}. The left hand side of \eqref{eq:converproof1} becomes
\begin{align}\label{eq:converproof02}
&\mbE\Big[S^t(\bbf_t)-S^t(\bbf)+\sum_{(i,j)\in\ccalE} [\mu_{ij}(g_{ij}(f_t(\bbx_t)))-\mu_{ij,t}(g_{ij}(f(\bbx_t)))]\nonumber\\
&\quad- \frac{\delta\eta}{2}\|\bbmu\|^{2}+\frac{\delta\eta}{2}\|\bbmu_t\|^{2}\Big]
\end{align}
where we utilize the notation defined in \eqref{notation}. Further, by substituting the bounds of $\|{\nabla}_{\bbf}\hat{\ccalL}_{t}(\bbf_{t},\bbmu_{t})\|_{\ccalH}^2$ and $\|\nabla_{\bbmu}\hat{\ccalL}_{t}(\bbf_t,\mathbf{\bbmu}_t)\|^2$ given in \eqref{eq:lemma1_final} and \eqref{eq:lemma1_2_final} to \eqref{eq:converproof1}, the right hand side of \eqref{eq:converproof1} can be written as
\begin{align}\label{inter}
&\mbE\Bigg[\frac{1}{2\eta}\bigg(\|\bbf_t-\bbf\|_{\ccalH}^2-\|\bbf_{t+1}-\bbf\|_{\ccalH}^2+\|\bbmu_{t}-\bbmu\|^2-\|\bbmu_{t+1}-\bbmu\|^2\bigg)\nonumber\\
&\quad+\frac{\sqrt{V}\eps_t}{\eta}\|\bbf_t-\bbf\|_{\ccalH}+\frac{V\eps_t^2}{\eta}\Bigg]\nonumber\\
&\quad+\mbE\Bigg[\frac{\eta}{2}\bigg(2(4V' C^2 + 4V' L_h^2 M \|\bbmu_t\|^2+2V\lambda^2 \cdot R_{\ccalB}^2)\nonumber\\
&\quad\quad\quad + M\Big((2K_1+2L_h^2X^2\cdot R_{\ccalB}^2)+ 2\delta^2\eta^2\|\bbmu_t\|^2 \Big)\bigg)\Bigg].
\end{align}
Since each individual $f_{t,i}$ and $f_i$ for $\in\{i,\dots,V\}$ in the ball $\ccalB$ have finite Hilbert norm and is bounded by $R_{\ccalB}$, the term $\|\bbf_t-\bbf\|_{\ccalH}$ can be upper bounded by $2\sqrt{V}R_{\ccalB}$. Next, we define $K\coloneqq 8V' C^2 +4V\lambda^2 \cdot R_{\ccalB}^2+2MK_1+2ML_h^2X^2\cdot R_{\ccalB}^2)$. Now using the the bound of $\|\bbf_t-\bbf\|_{\ccalH}$ and the definition of $K$, we can upper bound the expression in \eqref{inter}, {and then collectively writing the left and right hand side terms together, we get}
{\begin{align}\label{eq:converproof3}
&\mbE\Big[S^t(\bbf_t)-S^t(\bbf)+\sum_{(i,j)\in\ccalE} [\mu_{ij}(g_{ij}(f_t(\bbx_t)))\nonumber\\
&\quad-\mu_{ij,t}(g_{ij}(f(\bbx_t)))] - \frac{\delta\eta}{2}\|\bbmu\|^{2}\Big]\nonumber\\
& \leq \mbE\Bigg[\frac{1}{2\eta}\bigg(\|\bbf_t-\bbf\|_{\ccalH}^2-\|\bbf_{t+1}-\bbf\|_{\ccalH}^2+\|\bbmu_{t}-\bbmu\|^2-\|\bbmu_{t+1}-\bbmu\|^2\bigg)\nonumber\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+{\frac{{2}V\eps_t}{\eta}}.R_{\ccalB}+\frac{V\eps_t^2}{\eta}\Bigg]\nonumber\\
&\quad+\mbE\Bigg[\frac{\eta}{2}\bigg(K+\Big(8V' L_h^2 M+ 2M\delta^2\eta^2-\delta\Big)\|\bbmu_t\|^2\bigg)\Bigg].
\end{align}}
Now, we select the constant parameter $\delta$ such that $8V' L_h^2 M + 2M\delta^2\eta^2 -\delta\leq 0$, which then allows us to drop the term involving $\|\bbmu_t\|^2$ from the second expected term of right-hand side of \eqref{eq:converproof3}. Further, we set the approximation budget $\eps_t=\eps$, take the sum of the expression {in \eqref{eq:converproof3} over times $t=1,\dots,T$}, assume the initialization $\bbf_1=0\in\ccalH^V$ and $\bbmu_1=0\in\mbR_+^M$, we get
\begin{align}\label{eq:converproof5}
\sum_{t=1}^T&\mbE\bigg[S^t(\bbf_t)-S^t(\bbf)+\sum_{(i,j)\in\ccalE} [\mu_{ij}(g_{ij}(f_t(\bbx_t)))\nonumber\\
&\quad\quad\quad\quad-\mu_{ij,t}(g_{ij}(f(\bbx_t)))]\bigg] - \frac{\delta\eta T}{2}\|\bbmu\|^{2}\nonumber\\
&\leq \frac{1}{2\eta}\big(\|\bbf\|_{\ccalH}^2+\|\bbmu\|^2\big)+ \frac{{2}V\eps T R_{\ccalB}+V\eps^2 T}{\eta}+\frac{\eta K T}{2}.
\end{align}
where we drop the negative terms remaining after the telescopic sum since $\|\bbf_{T+1}-\bbf\|_{\ccalH}^2$ and $\|\bbmu_{T+1}-\bbmu\|^2$ are always positive.
It can be observed from \eqref{eq:converproof5} that the right-hand side of this inequality is deterministic. We now take $\bbf$ to be the solution $\bbf^\star$ of \eqref{eq:main_prob}, which in turn implies $\bbf^\star$ must satisfy the inequality constraint of \eqref{eq:main_prob}. This means that $\bbf^\star$ is a feasible point, such that $\sum_{t=1}^T\sum_{(i,j)\in\ccalE}\mu_{ij,t}\Big(g_{ij}(f^\star_{i}(\bbx_{i,t}),f^\star_{j}(\bbx_{i,t}))\Big)\leq 0$ holds. Thus we can simply drop this term
and collecting the terms containing $\|\bbmu\|^2$ together, we further obtain
{\begin{align}\label{eq:converproof6}
&\sum_{t=1}^T\mbE\Bigg[\big[S^t(\bbf_t)-S^t(\bbf^\star)\big]+\sum_{(i,j)\in\ccalE} [\mu_{ij}(g_{ij}(f_t(\bbx_t)))]\Bigg] - z(\eta,T)\|\bbmu\|^{2}\nonumber\\
&\leq \frac{1}{2\eta}\|\bbf^\star\|_{\ccalH}^2+ {\frac{V\eps T}{\eta}}\Big({2}R_{\ccalB}+\eps\Big)+\frac{\eta K T}{2}.
\end{align}}
where $z(\eta,T):=\Big(\frac{\delta\eta T}{2}+\frac{1}{2\eta}\Big)$. Next, we maximize the left-hand side of \eqref{eq:converproof6} over $\bbmu$ to obtain the optimal Lagrange multiplier which controls the growth of the long-term constraint violation. Specifically, the function of $\bbmu$ has a minimizer $\bar{\bbmu}\in\mbR_+^M$. Thus for any $i=1,\dots,V$ and $j=1,\dots,M$, the optimal value of $\bbmu_{ij}$ is given by
\begin{align}
\bar{\bbmu}_{ij}=\mbE\Bigg[\frac{1}{2(\delta\eta T+1/ \eta)}\sum_{t=1}^T\bigg[\Big(g_{ij}(f_t(\bbx_t))\Big)\bigg]_+\Bigg].\nonumber
\end{align}
Now, substituting $\bar{\bbmu}$ in place of $\bbmu$ in the left hand side of \eqref{eq:converproof6}, it becomes
\begin{align}\label{eq:converproof7}
&\mbE\Bigg[\sum_{t=1}^T\big[S^t(\bbf_t)-S^t(\bbf^\star)\big]+\sum_{(i,j)\in\ccalE}\frac{\Big[\sum_{t=1}^Tg_{ij}(f_t(\bbx_t))\Big]_+^2}{2(\delta\eta T+1/ \eta)}\Bigg].
\end{align}
The first term in \eqref{eq:converproof7} denotes the primal optimality gap and the second term denotes the commutative constraint violation.
We consider step-size $\eta=1/\sqrt{T}$ and approximation budget $\eps=P\eta^2=P/T$, where $P>0$ is a fixed constant. Substituting these in \eqref{eq:converproof7} and then considering the upper bound in \eqref{eq:converproof6}, we get
\begin{align}\label{eq:converproof8}
&\mbE\Bigg[\sum_{t=1}^T\big[S^t(\bbf_t)-S^t(\bbf^\star)\big]+\sum_{(i,j)\in\ccalE}\frac{\Big[\sum_{t=1}^Tg_{ij}(f_t(\bbx_t))\Big]_+^2}{2\sqrt{T}(\delta+1)}\Bigg]\nonumber\\
&\leq \frac{\sqrt{T}}{2}K_2.
\end{align}
where $K_2\coloneqq \Big(\|\bbf^\star\|_{\ccalH}^2+ 4VPR_{\ccalB}+\frac{2VP^2}{T}+K\Big)$. This expression serves as the basis which allows us to derive convergence result of both the objective function and the feasibility of the proposed iterates. Considering first the objective error sequence $\mbE\big[S^t(\bbf_t)-S^t(\bbf^\star)\big]$, we observe from \eqref{eq:converproof8} that the second term present on the left-side of the inequality can be dropped without affecting the inequality owing to the fact that it is positive. So we obtain
\begin{align}\label{eq:converproof9}
\sum_{t=1}^T\mbE\big[S^t(\bbf_t)-S^t(\bbf^\star)\big]
\leq \mathcal{O}(\sqrt{T})
\end{align}
which is as stated in \eqref{eq:func_order} of Thm. \ref{thm:order}.
Next, we establish the sublinear growth of the constraint violation in $T$. For this consider the objective error sequence,
\begin{align}\label{eq:converproof10}
S^t(\bbf_t)-S^t(\bbf^\star)&=\mbE\sum_{i\in\ccalV}\big[\ell_i(f_{i,t}\big(\bbx_{i,t}), y_{i,t}\big)-\ell_i(f_i^\star\big(\bbx_{i,t}), y_{i,t}\big)\big]\nonumber\\
&\qquad+\frac{\lambda}{2}\sum_{i\in\ccalV}\Big(\|f_{i,t} \|^2_{\ccalH}- \|f_i^\star \|^2_{\ccalH}\Big).
\end{align}
Next, we bound the objective error sequence as
\begin{align}\label{eq:converproof11}
|S^t(\bbf_t)-S^t(\bbf^\star)|&\leq \mbE\sum_{i\in\ccalV}\big[|\ell_i(f_{i,t}\big(\bbx_{i,t}), y_{i,t}\big)-\ell_i(f_i^\star\big(\bbx_{i,t}), y_{i,t}\big)|\big]\nonumber\\
&\quad+\frac{\lambda}{2}\sum_{i\in\ccalV}|\|f_{i,t} \|^2_{\ccalH}- \|f_i \|^2_{\ccalH}|\\
&\hspace{-2cm}\leq \mbE\sum_{i\in\ccalV} C|f_{i,t}\big(\bbx_{i,t})-f_i^\star\big(\bbx_{i,t})|+\frac{\lambda}{2}\sum_{i\in\ccalV}|\|f_{i,t} \|^2_{\ccalH}- \|f_i \|^2_{\ccalH}|,\nonumber
\end{align}
where using triangle inequality we write the first inequality and then using Assumption \eqref{as:second} of Lipschitz-continuity condition we write the second inequality. Further, using reproducing property of $\kappa$ and Cauchy-Schwartz inequality, we simplify $|f_{i,t}\big(\bbx_{i,t})-f_i^\star\big(\bbx_{i,t})|$ in \eqref{eq:converproof11} as
\begin{align}\label{eq:converproof12}
|f_{i,t}\big(\bbx_{i,t})-f_i^\star\big(\bbx_{i,t})|&=|\langle f_{i,t}-f_i^\star,\kappa(\bbx_{i,t},\cdot)\rangle|\\
&\leq \|f_{i,t}-f_i^\star\|_{\ccalH}\cdot \|\kappa(\bbx_{i,t},\cdot)\|_{\ccalH}\leq {2}R_\ccalB X\nonumber
\end{align}
where the last inequality comes from Assumption \ref{as:first} and \ref{as:fourth}.
Now, we consider the $|\|f_{t,i} \|^2_{\ccalH}- \|f_i^\star \|^2_{\ccalH}|$ present in the right-hand side of \eqref{eq:converproof11},
\begin{align}\label{eq:converproof13}
&|\|f_{t,i} \|^2_{\ccalH}- \|f_i^\star \|^2_{\ccalH}|^2\leq \|f_{t,i}-f_i^\star\|_{\ccalH}\cdot \|f_{t,i}+f_i^\star\|_{\ccalH}\leq 4R_{\ccalB}^2.
\end{align}
Substituting \eqref{eq:converproof12} and \eqref{eq:converproof13} in \eqref{eq:converproof11}, we obtain
\begin{align}
|S^t(\bbf_t)-S^t(\bbf^\star)| \leq {2}VCR_\ccalB X+{2}V\lambda R_{\ccalB}^2
={2}VR_{\ccalB}(C X+\lambda R_{\ccalB}).\nonumber
\end{align}
Thus $S^t(\bbf_t)-S^t(\bbf^\star)$ can be lower bound as
\begin{align}\label{eq:converproof15}
S^t(\bbf_t)-S^t(\bbf^\star)\geq -{2}VR_{\ccalB}(C X+\lambda R_{\ccalB}).
\end{align}
Substituting this lower bound in \eqref{eq:converproof8}, we get
\begin{align}\label{eq:converproof16}
\mbE\Bigg[-TK_3+\sum_{(i,j)\in\ccalE}\frac{\Big[\sum_{t=1}^Tg_{ij}(f_t(\bbx_t))\Big]_+^2}{2\sqrt{T}(\delta+1)}\Bigg]\leq \frac{\sqrt{T}}{2}K_2.
\end{align}
where $K_3 \coloneqq {2}VR_{\ccalB}(C X+\lambda R_{\ccalB})$. After re-arranging \eqref{eq:converproof16}, we get
\begin{align}\label{eq:converproof18}
&\mbE\Bigg[\sum_{(i,j)\in\ccalE}\Big[\sum_{t=1}^Tg_{ij}(f_t(\bbx_t))\Big]_+^2\Bigg]\leq 2\sqrt{T}(\delta+1) \bigg[\frac{\sqrt{T}}{2}K_2+TK_3\bigg]\nonumber
\\
&\quad=2T^{1.5}(\delta+1) \bigg[\frac{K_2}{2\sqrt{T}}+K_3\bigg].\end{align}
Since the square of individual proximity constraint violation is a positive term thus it is upper bounded by the square of the network in-aggregate constraint violation and we can write
\begin{align}\label{eq:converproof18_a}
\mbE\Bigg[\Big[\sum_{t=1}^Tg_{ij}(f_t(\bbx_t))\Big]_+^2\Bigg]\leq \mbE\Bigg[\sum_{(i,j)\in\ccalE}\Big[\sum_{t=1}^Tg_{ij}(f_t(\bbx_t))\Big]_+^2\Bigg].
\end{align}
Thus by using \eqref{eq:converproof18_a} in \eqref{eq:converproof18} we can write,
\begin{align}\label{eq:converproof19}
\mbE\Bigg[\Big[\sum_{t=1}^Tg_{ij}(f_{t}(\bbx_{t}))\Big]_+^2\Bigg]\leq 2T^{1.5}(\delta+1) \bigg[\frac{K_2}{2\sqrt{T}}+K_3\bigg].
\end{align}
Taking square root of both the sides of \eqref{eq:converproof19} and summing over all the edges, we get the desired result in \eqref{eq:constr_order}.
\end{proof}
\end{comment}
| 171,975
|
TITLE: Sets of equations
QUESTION [2 upvotes]: Let $I,J,K$ be three non-void sets, and let $\gamma$:$I\times J\times K\rightarrow\mathbb{N}$.
Is there some nonempty set $X$, together with some functions {$\{ f_{i}:X\rightarrow X;i\in I\} $},
some subsets {$\{ \Omega_{j}\subset X;j\in J\} $}, and some
points {$\{p_{k}\in X;k\in K} $} s.t. $\mid f_{i}^{-1}\left(p_{k}\right)\cap\Omega_{j}\mid=\gamma\left(i,j,k\right)$
$\left(i\in I,j\in J,k\in K\right)$, and $\mid f_{i}^{-1}\left(p\right)\mid\leq\mid\mathbb{R\mid}$$\left(i\in I,p\in X\right)$
? In other words, is $\gamma$ ''representable'' as the number of
solutions of some ''reasonable'' equations? [An elementary problem,
indeed.]
REPLY [2 votes]: This is basically a detailed description of a solution, based on Gerhard's answer.
Let $X=\mathbb{N}_0 \times I_0 \times K $, where $\mathbb{N}_0$ includes $0$ and similarly $I_0=I\cup 0$ with $0\not \in I$.
Let $p_k =(0,0,k)$, and let $\displaystyle \Omega_j=\bigcup_i \bigcup_k \bigcup_{n=1}^{\gamma_{ijk}} (n,i,k)$.
Define $f_i(n,i,k)=(0,0,k)=p_k$ and $f_i(n,i',k)=(n+1,i',k)$ for $i'\neq i$. We add $1$ to $n$ so that $f_i(p_k)\neq p_k$.
Note that for $x\neq p_k$, $|f_i^{-1}(x)|\leq 1$. On the other hand, $f_i^{-1}(p_k)=\mathbb{N} \times \{i\}\times \{k\} $, and then $f_i^{-1}(p_k)\cap \Omega_j =\{(n,i,k)\mid 1\leq n\leq \gamma_{ijk} \}$, which has the desired cardinality $\gamma_{ijk} $.
Remark: In a comment to Gerhard's answer, Ady says "I think you're underestimating the size of J." The size of $J$ actually plays no role in this problem as stated, as the sets $\Omega_j$ may overlap (and will overlap a lot in my construction). If you want to require the $\Omega_j$ to be disjoint, note that the other conditions force $|J|\leq |\mathbb{R}|$, as $\left|\bigcup_j \left( f_i^{-1}(p_k)\cap \Omega_j \right)\right| \leq |f_i^{-1}(p_k)|\leq |\mathbb R|$. We can modify the construction above by replacing $\mathbb N$ with $\mathbb R$, and can assure that the $\Omega_j$ are disjoint if we are more careful in choosing which $\gamma_{ijk}$ elements of $\mathbb{R}\times\{i\}\times\{k\}$ to include in $\Omega_j$ (above I chose the points $(n,i,k)$ for $n$ from $1$ to $\gamma_{ijk}$).
| 75,552
|
\begin{document}
\title[Inverse scattering]{Inverse scattering at fixed energy on surfaces with Euclidean ends}
\author{Colin Guillarmou}
\address{DMA, U.M.R. 8553 CNRS\\
Ecole Normale Sup\'erieure,\\
45 rue d'Ulm\\
F 75230 Paris cedex 05 \\France}
\email{cguillar@dma.ens.fr}
\author{Mikko Salo}
\address{Department of Mathematics \\
University of Helsinki \\
PO Box 68, 00014 Helsinki, Finland}
\email{mikko.salo@helsinki.fi}
\author{Leo Tzou}
\address{Department of Mathematics\\
Stanford University\\
Stanford, CA 94305, USA}
\email{ltzou@math.stanford.edu}
\begin{abstract}
On a fixed Riemann surface $(M_0,g_0)$ with $N$ Euclidean ends and genus $g$, we show that, under a topological condition,
the scattering matrix $S_V(\la)$ at frequency $\la > 0$ for the operator $\Delta+V$
determines the potential $V$ if $V\in C^{1,\alpha}(M_0)\cap e^{-\gamma d(\cdot,z_0)^j}L^\infty(M_0)$
for all $\gamma>0$ and for some $j\in\{1,2\}$, where $d(z,z_0)$ denotes the distance from $z$
to a fixed point $z_0\in M_0$. The topological condition is given by $N\geq \max(2g+1,2)$ for $j=1$
and by $N\geq g+1$ if $j=2$.
In $\rr^2$ this implies that the operator $S_V(\la)$
determines any $C^{1,\alpha}$ potential $V$ such that $V(z)=O(e^{-\gamma|z|^2})$ for all $\gamma>0$.
\end{abstract}
\maketitle
\begin{section}{Introduction}
The purpose of this paper is to prove an inverse scattering result at fixed frequency $\la>0$ in dimension $2$. The typical question
one can ask is to show that the scattering matrix $S_V(\la)$ for the Schr\"odinger operator $\Delta +V$ determines the potential.
This is known to be false if $V$ is only assumed to be Schwartz, by the example of Grinevich-Novikov \cite{GrNo},
but it is also known to be
true for exponentially decaying potentials (i.e.~$V\in e^{-\gamma|z|}L^\infty(\rr^2)$ for some $\gamma>0$) with norm smaller than a constant depending
on the frequency $\la$, see Novikov \cite{Nov}.
For other partial results we refer to \cite{E}, \cite{IS}, \cite{SunU1}, \cite{SunU2}, \cite{SunU3}. The determinacy of $V$ from $S_V(\la)$ when $V$ is compactly supported, without any smallness assumption on the norm, follows from the recent work of Bukhgeim \cite{Bu} on the inverse boundary problem after a standard reduction to the Dirichlet-to-Neumann operator
on a large sphere (see \cite{UhlAsterisque} for this reduction).
In dimensions $n \geq 3$, it is proved in Novikov \cite{Nov2} (see also \cite{ER} for the case of magnetic Schr\"odinger operators)
that the scattering matrix at a fixed frequency $\la$ determines an exponentially decaying
potential. When $V$ is compactly supported this also follows directly from the result by Sylvester-Uhlmann \cite{SylUhl} on the inverse boundary problem, by reducing
to the Dirichlet-to-Neumann
operator on a large sphere.
Melrose \cite{MelStanford} gave a direct proof of the last result based on the methods of \cite{SylUhl}, and this proof was extended to exponentially decaying potentials in \cite{UhlVa} and to the magnetic case in \cite{PSU}.
In the geometric scattering setting, \cite{JSB1,JSB} reconstruct the asymptotic expansion of a potential or metrics
from the scattering operator at fixed frequency on asymptotically Euclidean/hyperbolic manifolds. Further results of this type are given in \cite{W,WY}.
The method for proving the determinacy of $V$ from $S_V(\la)$ in \cite{MelStanford,UhlVa} is based on the construction of complex geometric optics solutions
$u(z)=e^{\rho.z}(1+r(\rho,z))$ of $(\Delta+V-\la^2)u=0$ with $\rho\in\cc^n, z\in\rr^n$, and the density of the oscillating scattering solutions
$u_{\rm sc}(z)=\int_{S^{n-1}} \Phi_V(\la,z,\omega)f(\omega)d\omega$ within those complex geometric optics solutions,
where $\Phi_V(\la,z,\omega)= e^{i\la \omega.z} + e^{-i\la \omega.z}|z|^{-\demi(n-1)}a(\la,z,\omega)$ are the perturbed plane wave solutions
(here $\omega\in S^{n-1}$ and $a\in L^\infty$). Unlike when $n \geq 3$, the problem in dimension $2$ is that the set of complex geometrical optics solutions of this type is not large enough to show that the Fourier transform of $V_1-V_2$ is $0$.
The real novelty in the recent work of Bukhgeim \cite{Bu} in dimension $2$ is the construction of new complex geometric optics solutions (at least on a bounded domain $\Omega\subset \cc$) of $(\Delta+V_i)u_i=0$ of the form
$u_1=e^{\Phi/h}(1+r_1(h))$ and $u_2=e^{-\Phi/h}(1+r_2(h))$ with $0<h\ll 1$ where $\Phi$ is a holomorphic function in $\cc$ with a unique non-degenerate critical point at a fixed
$z_0\in \cc$ (for instance $\Phi(z)=(z-z_0)^2$), and $||r_j(h)||_{L^p}$ is small as $h\to 0$ for $p>1$.
These solutions allow to use stationary phase at $z_0$ to get
\[ \int_{\Omega} (V_1-V_2)u_1\bbar{u_2} =C (V_1(z_0)-V_2(z_0))h+o(h) , \quad C\not =0\]
as $h\to 0$ and thus, if the Dirichlet-to-Neumann operators on $\pl\Omega$ are the same, then $V_1(z_0)=V_2(z_0)$.
One of the problems to extend this to inverse scattering is that a holomorphic function in $\cc$ with a non-degenerate critical point
needs to grow at least quadratically at infinity, which would somehow force to consider potentials $V$ having Gaussian decay. On the other hand,
if we allow the function to be meromorphic with simple poles, then we can construct such functions, having a single critical point at any given point
$p$, for instance by considering $\Phi(z)=(z-p)^2/z$. Of course, with such $\Phi$ we then need to work on
$\cc\setminus\{0\}$, which is conformal to a surface with no hole but with $2$ Euclidean ends, and $\Phi$ has linear growth in the
ends. In general, on a surface with genus $g$ and $N$ Euclidean ends, we can use the Riemann-Roch theorem to construct holomorphic
functions with linear or quadratic growth in the ends, the dimension of the space of such functions depending on $g,N$.
In the present work, we apply this idea to obtain an inverse scattering result for $\Delta_{g_0}+V$ on a fixed Riemann surface $(M_0,g_0)$
with Euclidean ends, under some topological condition on $M_0$ and some decay condition on $V$.
\begin{theorem}\label{mainth}
Let $(M_0,g_0)$ be a non-compact Riemann surface with genus $g$ and $N$ ends isometric to $\rr^2\setminus\{|z|\leq 1\}$
with metric $|dz|^2$. Let $V_1$ and $V_2$ be two potentials in $C^{1,\alpha}(M_0)$ with $\alpha>0$, and such that
$S_{V_1}(\la)=S_{V_2}(\la)$ for some $\la > 0$. Let $d(z,z_0)$ denote the distance
between $z$ and a fixed point $z_0\in M_0$.\\
(i) If $N\geq \max(2g+1,2)$ and $V_i\in e^{-\gamma d(\cdot,z_0)}L^\infty(M_0)$ for all $\gamma>0$, then $V_1=V_2$.\\
(ii) If $N\geq g+1$ and $V_i\in e^{-\gamma d(\cdot,z_0)^2}L^\infty(M_0)$ for all $\gamma>0$, then $V_1=V_2$.
\end{theorem}
In $\rr^2$, where $g = 0$ and $N = 1$, we have an immediate corollary:
\begin{coro}\label{th2}
Let $\la > 0$ and let $V_1,V_2\in C^{1,\alpha}(\rr^2) \cap e^{-\gamma |z|^2}L^\infty(\rr^2)$ for all $\gamma>0$. If the scattering matrices satisfy $S_{V_1}(\la)=S_{V_2}(\la)$, then $V_1=V_2$.
\end{coro}
This is an improvement on the result of Bukhgeim
\cite{Bu} which shows identifiability for compactly supported functions, and in a certain sense on the result of Novikov \cite{Nov}
since it is assumed there that the potential has to be of small $L^\infty$ norm.
The structure of the paper is as follows. In Section \ref{sec_holomorphic} we employ the Riemann-Roch theorem and a transversality argument to construct Morse holomorphic functions on $(M_0,g_0)$ with linear or quadratic growth in the ends. Section \ref{Carleman} considers Carleman estimates with harmonic weights on $(M_0,g_0)$, where suitable convexification and weights at the ends are required since the surface is non compact. Complex geometrical optics solutions are constructed in Section \ref{CGOriemann}.
Section \ref{sec_scattering} discusses direct scattering theory on surfaces with Euclidean ends and contains the proof that scattering solutions are dense in the set of suitable solutions, and Section \ref{sec_identify} gives the proof of Theorem \ref{mainth}. Finally, there is an appendix discussing a Paley-Wiener type result for functions with Gaussian decay which is needed to prove density of scattering solutions.
\subsection*{Acknowledgements}
M.S.~is supported partly by the Academy of Finland. C.G. is supported by ANR grant ANR-09-JCJC-0099-01, and is grateful to the Math. Dept. of Helsinki where part of this work was done.
\end{section}
\begin{section}{Holomorphic Morse functions on a surface with Euclidean ends} \label{sec_holomorphic}
\subsection{Riemann surfaces with Euclidean ends}
Let $(M_0,g_0)$ be a non-compact connected smooth Riemannian surface with $N$ ends $E_1,\dots,E_N$ which are Euclidean, i.e. isometric to
$\cc\setminus \{|z|\le 1\}$ with metric $|dz|^2$. By using a complex inversion $z\to 1/z$, each end is also isometric to a pointed disk
\[ E_i \simeq \{|z|\leq 1, z\not=0\} \textrm{ with metric }\frac{|dz|^2}{|z|^4}\]
thus conformal to the Euclidean metric on the pointed disk. The surface $M_0$ can then be compactified by adding the points corresponding
to $z=0$ in each pointed disk corresponding to an end $E_i$, we obtain a closed Riemann surface $M$ with a natural complex structure induced by that of
$M_0$, or equivalently a smooth conformal class on $M$ induced by that of $M_0$. Another way of thinking is to say that $M_0$ is the closed Riemann surface $M$ with $N$ points $e_1,\dots,e_N$ removed.
The Riemann surface $M$ has holomorphic charts $z_\alpha:U_{\alpha}\to \cc$ and we will denote by $z_1,\dots z_N$ the complex coordinates
corresponding to the ends of $M_0$, or equivalently to the neighbourhoods of the points $e_i$.
The Hodge star operator $\star$ acts on the cotangent bundle $T^*M$, its eigenvalues are
$\pm i$ and the respective eigenspaces $T_{1,0}^*M:=\ker (\star+i{\rm Id})$ and $T_{0,1}^*M:=\ker(\star -i{\rm Id})$
are sub-bundles of the complexified cotangent bundle $\cc T^*M$ and the splitting $\cc T^*M=T^*_{1,0}M\oplus T_{0,1}^*M$ holds as complex vector spaces.
Since $\star$ is conformally invariant on $1$-forms on $M$, the complex structure depends only on the conformal class of $g$.
In holomorphic coordinates $z=x+iy$ in a chart $U_\alpha$,
one has $\star(udx+vdy)=-vdx+udy$ and
\[T_{1,0}^*M|_{U_\alpha}\simeq \cc dz ,\quad T_{0,1}^*M|_{U_\alpha}\simeq \cc d\bar{z} \]
where $dz=dx+idy$ and $d\bar{z}=dx-idy$. We define the natural projections induced by the splitting of $\cc T^*M$
\[\pi_{1,0}:\cc T^*M\to T_{1,0}^*M ,\quad \pi_{0,1}: \cc T^*M\to T_{0,1}^*M.\]
The exterior derivative $d$ defines the de Rham complex $0\to \Lambda^0\to\Lambda^1\to \Lambda^2\to 0$ where $\Lambda^k:=\Lambda^kT^*M$
denotes the real bundle of $k$-forms on $M$. Let us denote $\cc\Lambda^k$ the complexification of $\Lambda^k$, then
the $\pl$ and $\bar{\pl}$ operators can be defined as differential operators
$\pl: \cc\Lambda^0\to T^*_{1,0}M$ and $\bar{\pl}:\cc\Lambda0\to T_{0,1}^*M$ by
\[\pl f:= \pi_{1,0}df ,\quad \bar{\pl}f:=\pi_{0,1}df,\]
they satisfy $d=\pl+\bar{\pl}$ and are expressed in holomorphic coordinates by
\[\pl f=\pl_zf\, dz ,\quad \bar{\pl}f=\pl_{\bar{z}}f \, d\bar{z},\]
with $\pl_z:=\demi(\pl_x-i\pl_y)$ and $\pl_{\bar{z}}:=\demi(\pl_x+i\pl_y)$.
Similarly, one can define the $\pl$ and $\bar{\pl}$ operators from $\cc \Lambda^1$ to $\cc \Lambda^2$ by setting
\[\pl (\omega_{1,0}+\omega_{0,1}):= d\omega_{0,1}, \quad \bar{\pl}(\omega_{1,0}+\omega_{0,1}):=d\omega_{1,0}\]
if $\omega_{0,1}\in T_{0,1}^*M$ and $\omega_{1,0}\in T_{1,0}^*M$.
In coordinates this is simply
\[\pl(udz+vd\bar{z})=\pl v\wedge d\bar{z},\quad \bar{\pl}(udz+vd\bar{z})=\bar{\pl}u\wedge d{z}.\]
If $g$ is a metric on $M$ whose conformal class induces the complex structure $T_{1,0}^*M$,
there is a natural operator, the Laplacian acting on functions and defined by
\[\Delta f:= -2i\star \bar{\pl}\pl f =d^*d \]
where $d^*$ is the adjoint of $d$ through the metric $g$ and $\star$ is the Hodge star operator mapping
$\Lambda^2$ to $\Lambda^0$ and induced by $g$ as well.
\subsection{Holomorphic functions}
We are going to construct Carleman weights given by holomorphic functions on $M_0$ which grow at most linearly or quadratically
in the ends. We will use the Riemann-Roch theorem, following ideas of \cite{GT}, however, the difference in the present case
is that we have very little freedom to construct these holomorphic functions, simply because there is just a finite dimensional space of such
functions by Riemann-Roch.
For the convenience of the reader, and to fix notations, we recall the usual Riemann-Roch index theorem
(see Farkas-Kra \cite{FK} for more details). A divisor $D$ on $M$ is an element
\[D=\big((p_1,n_1), \dots, (p_k,n_k)\big)\in (M\x\zz)^k, \textrm{ where }k\in\nn\]
which will also be denoted $D=\prod_{i=1}^kp_i^{n_i}$ or $D=\prod_{p\in M}p^{\alpha(p)}$ where $\alpha(p)=0$
for all $p$ except $\alpha(p_i)=n_i$. The inverse divisor of $D$ is defined to be
$D^{-1}:=\prod_{p\in M}p^{-\alpha(p)}$ and
the degree of the divisor $D$ is defined by $\deg(D):=\sum_{i=1}^kn_i=\sum_{p\in M}\alpha(p)$.
A non-zero meromorphic function on $M$ is said to have divisor $D$ if $(f):=\prod_{p\in M}p^{{\rm ord}(p)}$ is equal to $D$,
where ${\rm ord}(p)$ denotes the order of $p$ as a pole or zero of $f$ (with positive sign convention for zeros). Notice that
in this case we have $\deg(f)=0$.
For divisors $D'=\prod_{p\in M}p^{\alpha'(p)}$ and $D=\prod_{p\in M}p^{\alpha(p)}$,
we say that $D'\geq D$ if $\alpha'(p)\geq \alpha(p)$ for all $p\in M$.
The same exact notions apply for meromorphic $1$-forms on $M$. Then we define for a divisor $D$
\[\begin{gathered}
r(D):=\dim (\{f \textrm{ meromorphic function on } M; (f)\geq D\}\cup\{0\}),\\
i(D):=\dim(\{u\textrm{ meromorphic 1 form on }M; (u)\geq D\}\cup\{0\}).
\end{gathered}\]
The Riemann-Roch theorem states the following identity: for any divisor $D$ on the closed Riemann surface $M$ of genus $g$,
\begin{equation}\label{riemannroch}
r(D^{-1})=i(D)+\deg(D)-g+1.
\end{equation}
Notice also that for any divisor $D$ with $\deg(D)>0$, one has $r(D)=0$ since $\deg(f)=0$ for all $f$ meromorphic.
By \cite[Th. p70]{FK}, let $D$ be a divisor, then for any non-zero meromorphic 1-form $\omega$ on $M$, one has
\begin{equation}\label{abelian}
i(D)=r(D(\omega)^{-1})
\end{equation}
which is thus independent of $\omega$.
For instance, if $D=1$, we know that the only holomorphic function on $M$ is $1$ and
one has $1=r(1)=r((\omega)^{-1})-g+1$ and thus $r((\omega)^{-1})=g$ if $\omega$ is a non-zero meromorphic $1$ form. Now
if $D=(\omega)$, we obtain again from \eqref{riemannroch}
\[ g=r((\omega)^{-1})=2-g+\deg((\omega))\]
which gives $\deg((\omega))=2(g-1)$ for any non-zero meromorphic $1$-form $\omega$. In particular, if $D$ is a divisor such that
$\deg(D)>2(g-1)$, then we get
$\deg(D(\omega)^{-1})=\deg(D)-2(g-1)>0$ and thus $i(D)=r(D(\omega)^{-1})=0$, which implies by \eqref{riemannroch}
\begin{equation}\label{divisors}
\deg(D)>2(g-1)\Longrightarrow r(D^{-1})= \deg(D)-g+1\geq g.
\end{equation}
Now we deduce the
\begin{lemma}\label{holofcts}
Let $e_1,\dots,e_{N}$ be distinct points on a closed Riemann surface $M$ with genus $g$, and let $z_0$ be another
point of $M\setminus\{e_1,\dots,e_{N}\}$. If $N\geq \max(2g+1,2)$, the following hold true:\\
(i) there exists a meromorphic function $f$ on $M$ with at most simple
poles, all contained in $\{e_1,\dots,e_{N}\}$, such that $\pl f(z_0)\not=0$,\\
(ii) there exists a meromorphic function $h$
on $M$ with at most simple poles, all contained in $\{e_1,\dots,e_{N}\}$,
such that $z_0$ is a zero of order at least $2$ of $h$.
\end{lemma}
\noindent\textsl{Proof}. Let first $g \geq 1$, so that $N \geq 2g+1$. By the discussion before the Lemma, we know that there are at least $g+2$ linearly independent (over $\cc$)
meromorphic functions $f_0,\dots,f_{g+1}$ on $M$ with at most simple poles, all contained in $\{e_1,\dots,e_{2g+1}\}$. Without loss of generality,
one can set $f_0=1$ and by linear combinations we can assume that $f_1(z_0)=\dots=f_{g+1}(z_0)=0$. Now consider the divisor
$D_j=e_1\dots e_{2g+1}z_0^{-j}$ for $j=1,2$, with degree $\deg(D_j)=2g+1-j$, then by the Riemann-Roch formula (more precisely \eqref{divisors})
\[ r(D_j^{-1})=g+2-j.\]
Thus, since $r(D_1^{-1})>r(D_2^{-1})=g$ and using the assumption that $g \geq 1$, we deduce that there is a function in ${\rm span}(f_1,\dots,f_{g+1})$
which has a zero of order $2$ at $z_0$ and a function which has a zero of order exactly $1$ at $z_0$. The same method clearly
works if $g=0$ by taking two points $e_1,e_2$ instead of just $e_1$.
\qed
If we allow double poles instead of simple poles, the proof of Lemma \ref{holofcts} shows the
\begin{lemma}\label{doublepoles}
Let $e_1,\dots,e_{N}$ be distinct points on a closed Riemann surface $M$ with genus $g$, and let $z_0$ be another
point of $M\setminus\{e_1,\dots,e_{N}\}$. If $N\geq g+1$, then the following hold true:\\
(i) there exists a meromorphic function $f$ on $M$ with at most double
poles, all contained in $\{e_1,\dots,e_{N}\}$, such that $\pl f(z_0)\not=0$,\\
(ii) there exists a meromorphic function $h$
on $M$ with at most double poles, all contained in $\{e_1,\dots,e_{N}\}$,
such that $z_0$ is a zero of order at least $2$ of $h$.\\
\end{lemma}
\subsection{Morse holomorphic functions with prescribed critical points} \label{morseholo}
We follow in this section the arguments used in \cite{GT} to construct holomorphic functions with non-degenerate critical points (i.e.~Morse holomorphic functions) on
the surface $M_0$ with genus $g$ and $N$ ends, such that these functions have at most linear growth (resp.~quadratic growth)
in the ends if $N\geq \max(2g+1,2)$ (resp.~if $N\geq g+1$).
We let $\mc{H}$ be the complex vector space spanned by the meromorphic functions on $M$ with divisors larger or equal to
$e_1^{-1}\dots e_{N}^{-1}$ (resp. by $e_1^{-2}\dots e_{N}^{-2}$) if we work with functions having linear growth (resp.~quadratic growth),
where $e_1,\dots e_{N}\in M$ are points corresponding to the ends of $M_0$ as explained in
Section \ref{sec_holomorphic}.
Note that $\mc{H}$ is a complex vector space of complex dimension greater or equal to $N-g+1$ (resp.~$2N-g+1$)
for the $e_1^{-1}\dots e_{N}^{-1}$ divisor (resp.~the $e_1^{-2}\dots e_{N}^{-2}$ divisor).
We will also consider the real vector space $H$ spanned by the real parts and imaginary parts of functions in $\mc{H}$, this is a real vector space
which admits a Lebesgue measure. We now prove the following
\begin{lemma}\label{morsedense}
The set of functions $u\in H$ which are not Morse in $M_0$ has measure $0$ in $H$, in particular its complement is dense in $H$.
\end{lemma}
\noindent{\bf Proof}. We use an argument very similar to that used by Uhlenbeck \cite{Uh}.
We start by defining $m: M_0\times H\to T^*M_0$ by $(p,u) \mapsto (p,du(p))\in T_p^*M_0$.
This is clearly a smooth map, linear in the second variable, moreover $m_u:=m(.,u)=(\cdot, du(\cdot))$ is
smooth on $M_0$. The map $u$ is a Morse function if and only if
$m_u$ is transverse to the zero section, denoted $T_0^*M_0$, of $T^*M_0$, i.e.~if
\[\textrm{Image}(D_{p}m_u)+T_{m_u(p)}(T_0^*M_0)=T_{m_u(p)}(T^*M_0),\quad \forall p\in M_0 \textrm{ such that }m_u(p)=(p,0).\]
This is equivalent to the fact that the Hessian of $u$ at critical points is
non-degenerate (see for instance Lemma 2.8 of \cite{Uh}).
We recall the following transversality result, the proof of which is contained in \cite[Th.2]{Uh} by replacing Sard-Smale theorem by the usual
finite dimensional Sard theorem:
\begin{theorem}\label{transv}
Let $m : X\times H \to W$ be a $C^k$ map and $X, W$ be smooth manifolds and $H$ a finite dimensional vector space,
if $W'\subset W$ is a submanifold such that $k>\max(1,\dim X-\dim W+\dim W')$, then the transversality of the map $m$ to $W'$ implies that
the complement of the set $\{u\in H; m_u \textrm{ is transverse to } W'\}$ in $H$ has Lebesgue measure $0$.
\end{theorem}
We want to apply this result with $X:=M_0$, $W:=T^*M_0$ and $W':=T^*_0M_0$, and with the map $m$ as defined above.
We have thus proved our Lemma if one can show that $m$ is transverse to $W'$.
Let $(p,u)$ such that $m(p,u)=(p,0)\in W'$. Then identifying $T_{(p,0)}(T^*M_0)$ with $T_pM_0\oplus T^*_pM_0$, one has
\[Dm_{(p,u)}(z,v)=(z,dv(p)+{\rm Hess}_p(u)z)\]
where ${\rm Hess}_p(u)$ is the Hessian of $u$ at the point $p$, viewed as a linear map from $T_pM_0$ to $T^*_pM_0$ (note that this is different from the covariant Hessian defined by the Levi-Civita connection).
To prove that $m$ is transverse to $W'$ we need to show that $(z,v)\to (z, dv(p)+{\rm Hess}_p(u)z)$ is onto from $T_pM_0\oplus H$
to $T_pM_0\oplus T^*_pM_0$, which is realized if the map $v\to dv(p)$ from $H$ to $T_p^*M_0$ is onto.
But from Lemma \ref{holofcts}, we know that there exists a meromorphic function $f$ with real part $v={\rm Re}(f)\in H$
such that $v(p)=0$ and $dv(p)\not=0$ as an element of $T^*_pM_0$. We can then take $v_1:=v$ and $v_2:={\rm Im}(f)$, which are
functions of $H$ such that $dv_1(p)$ and $dv_2(p)$ are
linearly independent in $T^*_pM_0$ by the Cauchy-Riemann equation $\bar{\pl} f=0$.
This shows our claim and ends the proof by using Theorem \ref{transv}.\qed
In particular, by the Cauchy-Riemann equation, this Lemma implies that the set of Morse
functions in $\mc{H}$ is dense in $\mc{H}$. We deduce
\begin{proposition}
\label{criticalpoints}
There exists a dense set of points $p$ in $M_0$ such that there exists a Morse holomorphic function $f\in\mc{H}$ on $M_0$
which has a critical point at $p$.
\end{proposition}
\noindent\textsl{Proof}.
Let $p$ be a point of $M_0$ and let $u$ be a holomorphic function with a zero of order at least $2$ at $p$,
the existence is ensured by Lemma \ref{holofcts}.
Let $B(p,\eta)$ be a any small ball of radius $\eta>0$ near $p$, then by
Lemma \ref{morsedense}, for any $\eps>0$, we can approach $u$ by a holomorphic Morse function $u_\eps\in\mc{H}_\eps$
which is at distance less than $\eps$ of $u$ in a fixed norm on the finite dimensional space $\mc{H}$.
Rouch\'e's theorem for $\pl_z u_\eps$ and $\pl_zu$ (which are viewed as functions locally near $p$)
implies that $\pl_z u_{\eps}$ has at least one zero of order exactly $1$ in $B(p,\eta)$ if $\eps$ is chosen small enough.
Thus there is a Morse function in $\mc{H}$ with a critical point arbitrarily close to $p$.\qed
\end{section}
\begin{remark}
In the case where the surface $M$ has genus $0$ and $N$ ends, we have an explicit formula for the function in Proposition \ref{criticalpoints}: indeed
$M_0$ is conformal to $\cc\setminus \{e_1,\dots, e_{N-1}\}$ for some
$e_i\in\cc$ - i.e.~the Riemann sphere minus $N$ points - then the function $f(z)=(z-z_0)^2/(z-e_1)$ with $z_0\not\in\{e_1,\dots,e_{N-1}\}$
has $z_0$ for unique critical point in
$\cc\setminus \{e_1,\dots, e_{N-1}\}$ and it is non-degenerate.
\end{remark}
We end this section by the following Lemmas which will be used for the amplitude of the complex geometric optics solutions but not for the phase.
\begin{lemma}\label{amplitude}
For any $p_0, p_1,\dots p_n\in M_0$ some points of $M_0$ and $L\in\nn$, then there exists a function $a(z)$ holomorphic on $M_0$
which vanishes to order $L$ at all $p_j$ for $j=1,\dots,n$ and such that $a(p_0)\not=0$.
Moreover $a(z)$ can be chosen to have at most polynomial growth in the ends, i.e.
$|a(z)|\leq C|z|^{J}$ for some $J\in\nn$.
\end{lemma}
\noindent\textsl{Proof}. It suffices to find on $M$ some meromorphic function with divisor greater or equal to
$D:=e_1^{-J}\dots e_N^{-J}p_1^L\dots p_n^{L}$ but not greater or equal to $Dp_0$
and this is insured by Riemann-Roch theorem as long as $JN-nL\geq 2g$ since then
$r(D)=-g+1+JN-nL$ and $r(Dp_0)=-g+JN-nL$.
\qed
\begin{lemma}
\label{control the zero}
Let $\{p_0, p_1,..,p_n\}\subset M_0$ be a set of $n+1$ disjoint points. Let $c_0,c_1,\dots, c_K\in\cc$, $L\in\nn$,
and let $z$ be a complex coordinate near $p_0$ such that $p_0=\{z=0\}$.
Then there exists a holomorphic function $f$ on $M_0$ with zeros of order at least $L$ at each $p_j$,
such that $f(z)=c_0 + c_1z +...+ c_K z^K+ O(|z|^{K+1})$ in the coordinate $z$. Moreover $f$
can be chosen so that there is $J\in\nn$ such that, in the ends, $|\pl_z^\ell f(z)|=O (|z|^{J})$ for all $\ell\in\nn_0$.
\end{lemma}
\noindent\textsl{Proof}. The proof goes along the same lines as in Lemma \ref{amplitude}. By induction on $K$ and linear combinations,
it suffices to prove it for $c_0=\dots=c_{K-1}=0$. As in the proof of Lemma \ref{amplitude}, if $J$ is taken large enough, there exists
a function with divisor greater or equal to $D:=e_1^{-J}\dots e_N^{-J}p_0^{K-1}p_1^L\dots p_n^{L}$ but not greater or equal
to $Dp_0$. Then it suffices to multiply this function by $c_K$ times the inverse of
the coefficient of $z^K$ in its Taylor expansion at $z=0$.
\qed
\subsection{Laplacian on weighted spaces}
Let $x$ be a smooth positive function on $M_0$, which is equal to $|z|^{-1}$ for $|z|>r_0$ in the ends
$E_i\simeq \{z\in\cc; |z|>1\}$, where $r_0$ is a large fixed number.
We now show that the Laplacian $\Delta_{g_0}$ on a surface with Euclidean ends has a right inverse on the weighted spaces
$x^{-J}L^2(M_0)$ for $J\notin\nn$ positive.
\begin{lemma}\label{rightinv}
For any $J>-1$ which is not an integer, there exists a continuous operator $G$ mapping $x^{-J}L^2(M_0)$ to $x^{-J-2}L^2(M_0)$
such that $\Delta_{g_0}G={\rm Id}$.
\end{lemma}
\noindent\textsl{Proof}. Let $g_b:=x^2g_0$ be a metric conformal to $g_0$. The metric $g_b$ in the ends can be written
$g_b=dx^2/x^2+d\theta_{S^1}^2$ by using radial coordinates $x=|z|^{-1},\theta=z/|z|\in S^1$,
this is thus a b-metric in the sense of Melrose \cite{APS}, giving the surface a geometry of surface with cylindrical ends. Let us define for $m\in\nn_0$
\[H^m_b(M_0):=\{u\in L^2(M_0;{\rm dvol}_{g_b}); (x\pl_x)^j \pl_\theta^ku\in L^2(M_0;{\rm dvol}_{g_b}) \textrm{ for all }j+k\leq m\}.\]
The Laplacian has the form $\Delta_{g_b}=-(x\pl_x)^2+\Delta_{S^1}$ in the ends, and
the indicial roots of $\Delta_{g_b}$ in the sense of Section 5.2 of \cite{APS} are given by the complex numbers $\la$ such that
$x^{-i\la}\Delta_{g_b}x^{i\la}$ is not invertible as an operator acting on the circle $S_{\theta}^1$. Thus the indicial roots are the solutions of $\la^2+k^2=0$
where $k^2$ runs over the eigenvalues of $\Delta_{S^1}$, that is, $k \in \zz$. The roots are simple at $\pm i k \in i\zz\setminus \{0\}$ and $0$ is a
double root. In Theorem 5.60 of \cite{APS}, Melrose proves that $\Delta_{g_b}$ is Fredholm on $x^{a}H^2_b(M_0)$ if and only if $-a$ is not the imaginary part of some indicial root, that is here $a\not\in \zz$.
For $J>0$, the kernel of $\Delta_{g_b}$ on the space $x^{J}H^2_b(M_0)$ is clearly trivial by an energy estimate. Thus $\Delta_{g_b}: x^{-J} H^0_b(M_0) \to x^{-J} H^{-2}_b(M_0)$ is surjective for $J > 0$ and $J \not\in \zz$, and the same then holds for $\Delta_{g_b}: x^{-J} H^2_b(M_0) \to x^{-J} H^{0}_b(M_0)$ by elliptic regularity.
Now we can use Proposition 5.64 of \cite{APS}, which asserts, for all positive $J \not\in \zz$, the existence of a pseudodifferential operator $G_b$ mapping continuously $x^{-J}H^0_b(M_0)$ to $x^{-J}H^2_b(M_0)$ such that $\Delta_{g_b}G_{b}={\rm Id}$.
Thus if we set $G=G_bx^{-2}$, we have $\Delta_{g_0}G={\rm Id}$ and $G$ maps continuously
$x^{-J+1}L^2(M_0)$ to $x^{-J-1}L^2(M_0)$ (note that $L^2(M_0)=xH^0_b(M_0)$).
\qed
\begin{section}{Carleman Estimate for Harmonic Weights with Critical Points}\label{Carleman}
\subsection{The linear weight case}
In this section, we prove a Carleman estimate using harmonic weights with non-degenerate critical points, in a
way similar to \cite{GT}. Here however we need to work on a non compact surface and with weighted spaces.
We first consider a Morse holomorphic function $\Phi\in \mc{H}$ obtained from Proposition \ref{criticalpoints}
with the condition that $\Phi$ has linear growth in the ends, which corresponds to the case where $V\in e^{-\gamma/x}L^\infty(M_0)$ for all $\gamma>0$. The Carleman weight will be the harmonic function $\varphi := \mathrm{Re}(\Phi)$.
We let $x$ be a positive smooth function on $M_0$ such that $x=|z|^{-1}$ in
the complex charts $\{z\in\cc; |z|>1\}\simeq E_i$ covering the end $E_i$.\\
Let $\delta\in(0,1)$ be small and let us take $\varphi_0\in x^{-\alpha}L^2(M_0)$ a solution of $\Delta_{g_0}\varphi_0=x^{2-\delta}$, a solution exists
by Proposition \ref{rightinv} if $\alpha>1+\delta$.
Actually, by using Proposition 5.61 of \cite{APS}, if we choose $\alpha<2$, then it is easy to see that
$\varphi_0$ is smooth on $M_0$ and has polyhomogeneous expansion as $|z|\to \infty$, with leading asymptotic in the end $E_i$ given by
$\varphi_0= -x^{-\delta}/\delta^2+
c_i\log(x)+ d_i+O(x)$ for some $c_i,d_i$ which are smooth functions in $S^1$.
For $\eps>0$ small, we define the convexified weight $\varphi_\eps:=\varphi-\frac{h}{\eps}\varphi_0$.\\
We recall from the proof of Proposition 3.1 in \cite{GT} the following estimate which is valid in any compact set $K\subset M_0$: for all
$w\in C_0^\infty(K)$, we have
\begin{equation}\label{carlemaninK}
\frac{C}{\eps}\Big(\frac{1}{h}\|w\|_{L^2}^2 + \frac{1}{h^2}\|w |d\varphi|\|_{L^2}^2 +\frac{1}{h^2}\|w |d\varphi_{\eps}|\|_{L^2}^2 +
\|dw\|_{L^2(K)}^2\Big) \leq \|e^{\varphi_{\eps}/h}\Delta_g e^{-\varphi_{\eps}/h} w\|_{L^2}^2
\end{equation}
where $C$ depends on $K$ but not on $h$ and $\eps$.\\
So for functions supported in the end $E_i$, it clearly suffices to obtain a Carleman estimate in
$E_i\simeq \rr^2\setminus\{|z|\leq 1\}$ by using the Euclidean coordinate $z$ of the end.
\begin{prop}\label{carlemaninend}
Let $\delta\in(0,1)$, and $\varphi_\eps$ as above, then there exists $C>0$ such that for all $\eps\gg h>0$ small enough, and all $u\in C_0^\infty(E_i)$
\[ h^2||e^{\varphi_{\eps}/h}(\Delta-\la^2)e^{-\varphi_{\eps}/h}u||^2_{L^2}\geq \frac{C}{\eps} (||x^{1-\frac{\delta}{2}}u||^2_{L^2}+
h^2||x^{1-\frac{\delta}{2}}du||^2_{L^2}).\]
\end{prop}
\noindent\textsl{Proof}.
The metric $g_0$ can be extended to $\rr^2$ to be the Euclidean metric
and we shall denote by $\Delta$ the flat positive Laplacian on $\rr^2$.
Let us write $P:=\Delta_{g_0}-\la^2$, then the operator $P_h:= h^2e^{\varphi_\eps/h}Pe^{-\varphi_\eps/h}$
is given by
\[ P_h=h^2\Delta-|d\varphi_\eps|^2+2h\nabla\varphi_\eps.\nabla- h\Delta \varphi_\eps-h^2\la^2, \]
following the notation of \cite[Chap. 4.3]{EvZw}, it is a semiclassical operator in $S^0(\cjg\xi\cjd^2)$ with semiclassical full Weyl symbol
\[\sigma(P_h):=|\xi|^2-|d\varphi_\eps|^2-h^2\la^2+2i\cjg d\varphi_\eps,\xi\cjd =a+ib.\]
We can define $A:=(P_h+P_h^*)/2=h^2\Delta-|d\varphi_\eps|^2-h^2\la^2$ and
$B:=(P_h-P_h^*)/2i=-2ih\nabla\varphi_\eps.\nabla+ih\Delta\varphi_\eps$ which have respective semiclassical full symbols
$a$ and $b$, i.e. $A={\rm Op}_h(a)$ and $B={\rm Op}_h(b)$ for the Weyl quantization. Notice that $A,B$ are symmetric operators,
thus for all $u\in C_0^\infty(E_i)$
\begin{equation}\label{A+iB}
||(A+iB)u||^2=\cjg (A^2+B^2+i[A,B])u,u\cjd .
\end{equation}
It is easy to check that the operator $ih^{-1}[A,B]$ is a semiclassical differential operator in $S^0(\cjg\xi\cjd^2)$
with full semiclassical symbol
\begin{equation}\label{symbol[A,B]}
\{a,b\}(\xi)= 4(D^2\varphi_\eps(d\varphi_\eps,d\varphi_\eps)+D^2\varphi_\eps(\xi,\xi))
\end{equation}
Let us now decompose the Hessian of $\varphi_\eps$ in the basis $(d\varphi_\eps, \theta)$ where $\theta$ is a
covector orthogonal to $d\varphi_\eps$ and of norm $|d\varphi_\eps|$. This yields coordinates $\xi=\xi_0d\varphi_\eps+\xi_1\theta$
and there exist smooth functions $M,N,K$ so that
\[D^2\varphi_\eps(\xi,\xi)=|d\varphi_\eps|^2(M\xi_0^2+N\xi_1^2+2K\xi_0\xi_1).\]
Notice that $\varphi_\eps$ has a polyhomogeneous expansion at infinity of the form
\[\varphi_\eps(z)= \gamma.z+ \frac{h}{\eps} \frac{r^{\delta}}{\delta^2} +c_1\log(r)+ c_2+c_3 r^{-1}+O(r^{-2}) \]
where $r=|z|,\omega=z/r, \gamma=(\gamma_1,\gamma_2)\in \rr^2$ and $c_i$ are some smooth functions on $S^1$ depending on $h$;
in particular we have
\[ d\varphi_\eps = \gamma_1dz_1+\gamma_2dz_2+O(r^{-1+\delta}), \quad
\pl_{z}^\alpha\pl_{\bar{z}}^\beta \varphi_\eps(z)=O(r^{-2+\delta}) \,\,\textrm{ for all } \alpha+\beta\geq 2\]
which implies that $M,N,K\in r^{-2+\delta}L^\infty(E_i)$.
Then one can write
\[\begin{split}
\{a,b\}= & 4|d\varphi_\eps|^2(M+M\xi_0^2+N\xi_1^2+2K\xi_0\xi_1)\\
=&4(N(a+h^2\la^2) +((M-N)\xi_0+2K\xi_1)b/2+(N+M)|d\varphi_\eps|^2)
\end{split}\]
and since $M+N=\tra(D^2\varphi_\eps)=-\Delta\varphi_\eps=h\Delta\varphi_0/\eps$ we obtain
\begin{equation}\label{{a,b}}\begin{gathered}
\{a,b\}=4|d\varphi_\eps|^2(c(z)(a+h^2\la^2) +\ell(z,\xi)b+\frac{h}{\eps} r^{-2+\delta}),\\
c(z)=\frac{N}{|d\varphi_\eps|^2}, \,\,\, \ell(z,\xi)=\frac{(M-N)\xi_0+2K\xi_1}{2|d\varphi_\eps|^2}.
\end{gathered}
\end{equation}
Now, we take a smooth extension of $|d\varphi_\eps|^2, a(z,\xi),\ell(z,\xi)$ and $r$ to $z\in\rr^2$, this can done for instance by extending
$r$ as a smooth positive function on $\rr^2$ and then extending $d\varphi$ and $d\varphi_0$ to smooth non vanishing $1$-forms
on $\rr^2$ (not necessarily exact) so that $|d\varphi_\eps|^2$ is smooth positive (for small $h$) and polynomial in $h$
and $a,\ell$ are of the same form as in $\{|z|>1\}$.
Let us define the symbol and quantized differential operator on $\rr^2$
\[ e:=4|d\varphi_\eps|^2(c(z)(a+h^2\la^2) +\ell(z,\xi)b), \quad E:={\rm Op}_h(e)\]
and write
\begin{equation}\label{simpl}
\begin{gathered}
ih^{-1}r^{1-\frac{\delta}{2}}[A,B]r^{1-\frac{\delta}{2}}= hF + r^{1-\frac{\delta}{2}}Er^{1-\frac{\delta}{2}}-\frac{h}{\eps}(A^2+B^2),\\
\textrm{ with }F:= h^{-1}r^{1-\frac{\delta}{2}}
(ih^{-1}[A,B] -E)r^{1-\frac{\delta}{2}}+\frac{1}{\eps}(A^2+B^2).
\end{gathered}
\end{equation}
We deduce from \eqref{symbol[A,B]} and \eqref{{a,b}} the following
\begin{lemma}\label{propF}
The operator $F$ is a semiclassical differential operator in the class $S^0(\cjg\xi\cjd^4)$ with semiclassical principal symbol
\[ \sigma (F)(\xi)= \frac{4|d\varphi|^2}{\eps}+\frac{1}{\eps}(|\xi|^2-|d\varphi|^2)^2+\frac{4}{\eps}(\cjg\xi,d\varphi\cjd)^2.\]
\end{lemma}
By the semiclassical G{\aa}rding estimate, we obtain the
\begin{coro}\label{garding}
The operator $F$ of Lemma \ref{propF} is such that there is a constant $C$ so that
\[ \cjg Fu,u\cjd \geq \frac{C}{\eps} (||u||^2_{L^2} +h^2||du||^2_{L^2}).\]
\end{coro}
\textsl{Proof}. It suffices to use that $\sigma(F)(\xi)\geq \frac{C'}{\eps}(1+|\xi|^4)$ for some $C'>0$ and use the
semiclassical G{\aa}rding estimate.
\qed
So by writing $\cjg i[A,B]u,u\cjd=\cjg ir^{1-\frac{\delta}{2}}[A,B]r^{1-\frac{\delta}{2}} r^{-1+\frac{\delta}{2}}u,r^{-1+\frac{\delta}{2}}u\cjd$ in \eqref{A+iB} and using \eqref{simpl} and Corollary \ref{garding}, we obtain
that there exists $C>0$ such that for all $u\in C_0^\infty(E_i)$
\begin{equation} \label{phu}
\begin{split}
||P_hu||_{L^2}^2\geq &\cjg (A^2+B^2)u,u\cjd + \frac{Ch^2}{\eps}(||r^{-1+\frac{\delta}{2}}u||^2_{L^2} +h^2||r^{-1+\frac{\delta}{2}}du||^2_{L^2})+
h\cjg Eu,u\cjd\\
& - \frac{h^2}{\eps}(||A (r^{-1+\frac{\delta}{2}} u)||_{L^2}^2 +||B (r^{-1+\frac{\delta}{2}} u)||_{L^2}^2).
\end{split}\end{equation}
We observe that $h^{-1}[A,r^{-1+\frac{\delta}{2}}]r^{1+\frac{\delta}{2}}\in
S^0(\cjg \xi\cjd)$ and $h^{-1}[B,r^{-1+\frac{\delta}{2}}]r^{1+\frac{\delta}{2}}\in
h S^0(1)$, and thus
\[ ||A (r^{-1+\frac{\delta}{2}} u)||_{L^2}^2 +||B (r^{-1+\frac{\delta}{2}} u)||_{L^2}^2)\leq C'(||Au||_{L^2}^2+||Bu||_{L^2}^2+
h^2||r^{-1+\frac{\delta}{2}}u||^2_{L^2}+h^4||r^{-1+\frac{\delta}{2}}du||^2_{L^2})\]
for some $C'>0$. Taking $h$ small, this implies with \eqref{phu} that there exists a new constant $C>0$ such that
\begin{equation}\label{phu2}
||P_hu||_{L^2}^2\geq \frac{1}{2}\cjg (A^2+B^2)u,u\cjd + \frac{Ch^2}{\eps}(||r^{-1+\frac{\delta}{2}}u||^2_{L^2} +h^2||r^{-1+\frac{\delta}{2}}du||^2_{L^2})+
h\cjg Eu,u\cjd.\end{equation}
It remains to deal with $h\cjg Eu,u\cjd$: we first write $E=4|d\varphi_\eps|^2(c(z)(A+h^2\la^2)+
{\rm Op}_h(\ell)B)+hr^{-1+\frac{\delta}{2}}Sr^{-1+\frac{\delta}{2}}$ where $S$
is a semiclassical differential operator in the class $S^0(\cjg\xi\cjd)$ by the decay estimates on $c(z),\ell(z,\xi)$ as $z\to\infty$,
then by Cauchy-Schwartz (and with $L:={\rm Op}_h(\ell)$)
\[\begin{split}
|\cjg hEu,u\cjd|\leq & Ch(||Au||_{L^2}+h^2||r^{-1+\frac{\delta}{2}}u||_{L^2}+ h||Sr^{-1+\frac{\delta}{2}}u||_{L^2}) ||r^{-1+\frac{\delta}{2}}u||_{L^2}
+Ch||Bu||_{L^2}||Lu||_{L^2}\\
\leq & \frac{1}{4}||Au||^2_{L^2}+ h^2||Sr^{-1+\frac{\delta}{2}}u||^2_{L^2}+Ch^2||r^{-1+\frac{\delta}{2}}u||_{L^2}^2+\frac{1}{4} ||Bu||^2_{L^2}+
Ch^2||Lu||^2_{L^2}
\end{split}\]
where $C$ is a constant independent of $h,\eps$ but may change from line to line.
Now we observe that $Lr^{1-\frac{\delta}{2}}$ and $S$ are in $S^0(\cjg \xi\cjd)$ and thus
\[||Sr^{-1+\frac{\delta}{2}}u||^2_{L^2}+ ||Lu||^2_{L^2}\leq C (||r^{-1+\frac{\delta}{2}}u||^2_{L^2}+ h^2||r^{-1+\frac{\delta}{2}}du||^2_{L^2}),\]
which by \eqref{phu2} implies that there exists $C>0$ such that for all $\eps\gg h>0$ with $\eps$ small enough
\[||P_hu||_{L^2}^2\geq \frac{Ch^2}{\eps}(||r^{-1+\frac{\delta}{2}}u||^2_{L^2} +h^2||r^{-1+\frac{\delta}{2}}du||^2_{L^2})
\]
for all $u\in C_0^\infty(E_i)$ . The proof is complete.\qed\\
Combining now Proposition \ref{carlemaninend} and \eqref{carlemaninK}, we obtain
\begin{prop}
\label{carlemanestimate}
Let $(M_0,g_0)$ be a Riemann surface with Euclidean ends with $x$ a boundary defining function of the radial compactification $\bbar{M}_0$ and let $\varphi_\eps=\varphi-\frac{h}{\eps}\varphi_0$ where $\varphi$ is a harmonic function
with non-degenerate critical points and linear growth on $M_0$
and $\varphi_0$ satisfies $\Delta_{g_0}\varphi_0=x^{2-\delta}$ as above.
Then for all $V\in x^{1-\frac{\delta}{2}}L^\infty(M_0)$ there exists an $h_0>0$, $\eps_0$ and $C>0$ such that for all $0<h<h_0$, $h\ll \eps<\eps_0$ and $u\in C^\infty_0(M_0)$, we have
\begin{equation}\label{carleman}
\frac{1}{h}\|x^{1-\frac{\delta}{2}}u\|_{L^2}^2 + \frac{1}{h^2}\|x^{1-\frac{\delta}{2}} u |d\varphi|\|_{L^2}^2 + \|x^{1-\frac{\delta}{2}}du\|_{L^2}^2 \leq C\eps \|e^{\varphi_\eps/h}
(\Delta_g + V-\la^2)e^{-\varphi_\eps/h} u\|_{L^2}^2
\end{equation}
\end{prop}
\noindent\textbf{Proof}.
As in the proof of Proposition 3.1 in \cite{GT}, by taking $\eps$ small enough, we see that the combination of \eqref{carlemaninK} and Proposition \ref{carlemaninend} shows that for any
$w \in C_0^\infty(M_0)$,
\[\begin{gathered}
\frac{C}{\eps}\Big(\frac{1}{h}\|x^{1-\frac{\delta}{2}}w\|_{L^2}^2 + \frac{1}{h^2}\|x^{1-\frac{\delta}{2}}w |d\varphi|\|_{L^2}^2
+\frac{1}{h^2}\|x^{1-\frac{\delta}{2}}w |d\varphi_{\eps}|\|_{L^2}^2 +
\|x^{1-\frac{\delta}{2}}dw\|_{L^2}^2\Big) \\
\leq \|e^{\frac{\varphi_{\eps}}{h}}(\Delta-\lambda^2)
e^{-\frac{\varphi_\eps}{h}}w\|_{L^2}^2
\end{gathered}
\]
which ends the proof.\qed
\subsection{The quadratic weight case for surfaces}
In this section, $\varphi$ has quadratic growth at infinity, which corresponds to the case where
$V\in e^{-\gamma/x^2}L^\infty$ for all $\gamma>0$.
The proof when $\varphi$ has quadratic growth at infinity is even simpler than the linear growth case. We define $\varphi_0\in x^{-2}L^\infty$
to be a solution of $\Delta_{g_0}\varphi_0=1$, this is possible by Lemma \ref{rightinv} and one easily obtains
from Proposition 5.61 of \cite{APS} that $\varphi_0=-x^{-2}/4+O(x^{-1})$ as $x\to 0$. We let
$\varphi_\eps:=\varphi-\frac{h}{\eps}\varphi_0$ which satisfies $\Delta_{g_0}\varphi_\eps/h=-1/\eps$.
If $K\subset M_0$ is a compact set, the Carleman estimate \eqref{carlemaninK} in $K$
is satisfied by Proposition 3.1 of \cite{GT}, it then remains to get the estimate in the ends $E_1,\dots,E_N$. But the
exact same proof as in Lemma 3.1 and Lemma 3.2 of \cite{GT} gives directly that for any $w \in C_0^\infty(E_i)$
\begin{equation}\label{CEinEi}
\frac{C}{\eps}\Big(\frac{1}{h}\|w\|_{L^2}^2 + \frac{1}{h^2}\|w |d\varphi|\|_{L^2}^2 +\frac{1}{h^2}\|w |d\varphi_{\eps}|\|_{L^2}^2 +
\|dw\|_{L^2}^2\Big) \leq \|e^{\varphi_{\eps}/h}\Delta_{g_0} e^{-\varphi_{\eps}/h} w\|_{L^2}^2
\end{equation}
for some $C>0$ independent of $\eps,h$ and it suffices to glue the estimates in $K$ and in the ends $E_i$ as in Proposition 3.1 of \cite{GT},
to obtain \eqref{CEinEi} for any $w \in C_0^\infty(M_0)$.
Then by using triangle inequality
\[||e^{\varphi_{\eps}/h}(\Delta_{g_0}+V-\la^2)e^{-\varphi_{\eps}/h}u||_{L^2}\leq ||e^{\varphi_{\eps}/h}\Delta_{g_0}e^{-\varphi_{\eps}/h}u||_{L^2}+
C||u||_{L^2}\]
for some $C$ depending on $\la, ||V||_{L^\infty}$, we see that the $V-\la^2$ term can be absorbed by the
left hand side of \eqref{CEinEi} and we finally deduce
\begin{prop}\label{CEquad}
Let $(M_0,g_0)$ be a Riemann surface with Euclidean ends and let $\varphi_\eps=\varphi-\frac{h}{\eps}\varphi_0$ where $\varphi$ is a harmonic function
with non-degenerate critical points and quadratic growth on $M_0$ and
$\varphi_0$ satisfies $\Delta_{g_0}\varphi_0=1$ with $\varphi_0\in x^{-2}L^\infty(M_0)$.
Then for all $V\in L^\infty$ there exists an $h_0>0$, $\eps_0$ and $C>0$ such that for all $0<h<h_0$, $h\ll \eps<\eps_0$ and $u\in C^\infty_0(M_0)$
\[\frac{C}{\eps}\Big(\frac{1}{h}\|u\|_{L^2}^2 +\frac{1}{h^2}||u |d\varphi| ||^2_{L^2}+
\|du\|_{L^2}^2\Big) \leq \|e^{\varphi_{\eps}/h}(\Delta_{g_0} +V-\la^2)e^{-\varphi_{\eps}/h} u\|_{L^2}^2.\]
\end{prop}
The main difference with the linear weight case is that one can use a convexification which has quadratic growth at infinity which allows to absorb the $\la^2$ term, while it was not the case for the linearly growing weights.
\end{section}
\begin{section}{Complex Geometric Optics on a Riemann Surface with Euclidean ends}\label{CGOriemann}
As in \cite{Bu,IUY,GT}, the method for identifying the potential at a point $p$ is to construct complex geometric optic solutions depending on a small parameter $h>0$, with phase a Morse holomorphic function with a non-degenerate
critical point at $p$, and then to apply the stationary phase method.
Here, in addition, we need the phase to be of linear growth at infinity if $V\in e^{-\gamma/x}L^\infty$ for all $\gamma>0$
while the phase has to be of quadratic growth at infinity if $V\in e^{-\gamma/x^2}L^\infty$ for all $\gamma>0$.\\
We shall now assume that $M_0$ is a non-compact surface with genus $g$ with $N$ ends equipped with a metric $g_0$
which is Euclidean in the ends, and $V$ is a $C^{1,\alpha}$ function in $M_0$. Moreover, if $V\in e^{-\gamma/x}L^\infty$ for all $\gamma>0$,
we ask that $N\geq \max(2g+1,2)$ while if $V\in e^{-\gamma/x^2}L^\infty$ for all $\gamma>0$, we assume that $N\geq g+1$.
As above, let us use a smooth positive function $x$ which is equal to $1$ in a large compact set of $M_0$ and is equal to
$x=|z|^{-1}$ in the regions $|z|>r_0$ of the ends $E_i\simeq \{z\in\cc; |z|>1\}$, where $r_0$ is a fixed large number.
This function is a boundary defining function of the radial
compactification of $M_0$ in the sense of Melrose \cite{APS}.
To construct the complex geometric optics solutions, we
will need to work with the weighted spaces $x^{-\alpha}L^2(M_0)$ where $\alpha\in\rr_+$.\\
Let $\mc{H}$ be the finite dimensional complex vector space defined in the beginning of Section \ref{morseholo}.
Choose $p\in M_0$ such that there exists a Morse holomorphic function $\Phi=\varphi+i\psi\in\mc{H}$ on $M_0$,
with a critical point at $p$; there is a dense set of such points by Proposition \ref{criticalpoints}.
The purpose of this section is to construct solutions $u$ on $M_0$ of $(\Delta -\la^2+V)u = 0$ of the form
\begin{equation}
\label{cgo}
u = e^{\Phi/h}(a + r_1 + r_2)
\end{equation}
for $h>0$ small, where $a\in x^{-J+1}L^2$ with $J\in\rr_+\setminus \nn$
is a holomorphic function on $M_0$, obtained by Lemma \ref{amplitude}, such that $a(p)\not=0$
and $a$ vanishing to order
$L$ (for some fixed large $L$) at all other critical points of $\Phi$, and finally $r_1,r_2$ will be remainder terms which
are small as $h\to 0$ and have particular properties near the critical points of $\Phi$.
More precisely, $e^{\varphi_0/\eps}r_2$ will be a $o_{L^2}(h)$ and $r_1$ will be a $O_{x^{-J}L^2}(h)$ but with
an explicit expression, which can be used to obtain sufficient information
in order to apply the stationary phase method.
\subsubsection{Construction of $r_1$}\label{constr1}
We want to construct $r_1=O_{x^{-J}L^2}(h)$ which satisfies
\[e^{-\Phi/h}(\Delta_{g_0} -\la^2+V)e^{\Phi/h}(a + r_1) = O_{x^{-J}L^2}(h) \]
for some large $J\in\rr_+\setminus \nn$ so that $a\in x^{-J+1}L^2$.
Let $G$ be the operator of Lemma \ref{rightinv}, mapping continuously $x^{-J+1}L^2(M_0)$
to $x^{-J-1}L^2(M_0)$. Then clearly $\bar{\partial}\partial G=\frac{i}{2}\star^{-1}$ when acting on $x^{-J+1}L^2$,
here $\star^{-1}$ is the inverse of $\star$ mapping functions to $2$-forms.
First, we will search for $r_1$ satisfying
\begin{equation}
\label{dequation}
e^{-2i\psi/h}\partial e^{2i\psi/h} r_1 = -\pl G (a(V-\la^2)) + \omega + O_{x^{-J}H^1}(h)
\end{equation}
with $\omega\in x^{-J}L^2(M_0)$ a holomorphic 1-form on $M_0$
and $\|r_1\|_{x^{-J}L^2} = O(h)$.
Indeed, using the fact that $\Phi$ is holomorphic we have
\[e^{-\Phi/h}\Delta_{g_0}e^{\Phi /h}=-2i\star \bar{\pl} e^{-\Phi/h}\pl e^{\Phi/h}=-2i\star \bar{\pl} e^{-\frac{1}{h}(\Phi-\bar{\Phi})}\pl e^{\frac{1}{h}(\Phi-\bar{\Phi})}=
-2i\star \bar{\pl}e^{-2i\psi/h}\pl e^{2i\psi/h}\]
and applying $-2i\star\bar{\pl}$ to \eqref{dequation}, this gives
\[e^{-\Phi/h}(\Delta_{g_0}+V)e^{\Phi/h}r_1=-a(V-\la^2)+O_{x^{-J}L^2}(h).\]
Writing $-\pl G(a(V-\la^2))=:c(z)dz$ in local complex coordinates, $c(z)$ is $C^{2,\alpha}$ by elliptic regularity and
we have $2i\pl_{\bar{z}}c(z)=a(V-\la^2)$, therefore $\pl_z\pl_{\bar{z}}c(p')=\pl^2_{\bar{z}}c(p')=0$ at each critical point $p'\not=p$ by construction
of the function $a$. Therefore, we deduce that at each critical point $p'\neq p$, $c(z)$
has Taylor series expansion $\sum_{j = 0}^2 c_j z^j + O(|z|^{2+\alpha})$. That is, all the lower order terms of the Taylor expansion of $c(z)$
around $p'$ are polynomials of $z$ only. By Lemma \ref{control the zero}, and possibly by taking $J$ larger, there exists a holomorphic function $f\in x^{-J}L^2$
such that $\omega:=\pl f$
has Taylor expansion equal to that of $\pl G(a(V-\la^2))$ at all critical points $p'\not=p$ of $\Phi$. We deduce that, if
$b:=-\pl G(a(V-\la^2))+\omega=b(z)dz$, we have
\begin{equation}\label{decayofb}
\begin{gathered}
|\pl_{\bar{z}}^m\pl^{\ell}_z b(z)|=O(|z|^{2+\alpha-\ell-m}) , \quad \textrm{ for } \ell+m\leq 2 , \textrm{ at critical points }p'\not=p\\
|b(z)|=O(|z|) , \qquad \qquad \qquad \qquad \textrm{ if }p'=p.
\end{gathered}\end{equation}
Now, we let $\chi_1\in C_0^\infty(M_0)$ be a cutoff function supported in a small neighbourhood $U_p$ of the critical point $p$ and identically $1$ near
$p$, and
$\chi\in C_0^\infty(M_0)$ is defined similarly with $\chi =1$ on the support of $\chi_1$.
We will construct $r_1$ to be a sum $r_1=r_{11} +h r_{12}$ where $r_{11}$ is a compactly supported
approximate solution of \eqref{dequation} near the critical point $p$ of $\Phi$ and $r_{12}$ is correction term supported
away from $p$.
We define locally in complex coordinates centered at $p$ and containing the support of $\chi$
\begin{equation}\label{defr11}
r_{11}:=\chi e^{-2i\psi/h}R(e^{2i\psi/h}\chi_1b)
\end{equation}
where $Rf(z) := -(2\pi i)^{-1}\int_{\R^2} \frac{1}{\bar{z}-\bar{\xi}}f d\bar{\xi}\wedge d\xi$ for $f\in L^\infty$ compactly supported
is the classical Cauchy operator inverting locally $\pl_z$ ($r_{11}$ is extended by $0$ outside the neighbourhood of $p$).
The function $r_{11}$ is in $C^{3,\alpha}(M_0)$ and we have
\begin{equation}\begin{gathered}\label{r11}
e^{-2i\psi/h}\pl(e^{2i\psi/h}r_{11}) = \chi_1(-\pl G(a(V-\la^2)) + \omega) + \eta\\
\textrm{ with }\eta:= e^{-2i\psi/h}R(e^{2i\psi/h}\chi_1b)\pl\chi.
\end{gathered}
\end{equation}
We then construct $r_{12}$ by observing that $b$ vanishes to order $2+\alpha$ at critical points of $\Phi$ other than $p$
(from \eqref{decayofb}), and
$\pl \chi=0$ in a neighbourhood of any critical point of $\psi$, so we can find $r_{12}$ satisfying
\begin{equation}\label{defr12}
2ir_{12}\pl\psi = (1-\chi_1)b .
\end{equation}
This is possible since both $\pl\psi$ and the right hand side are valued in $T^*_{1,0}M_0$ and
$\pl \psi$ has finitely many isolated zeroes on $M_0$:
$r_{12}$ is then a function which is in $C^{2,\alpha}(M_0\setminus{P})$ where $P:=\{p_1,\dots, p_n\}$ is the set of critical points other than $p$,
it extends to a function in $C^{1,\alpha}(M_0)$ and it satisfies in local complex coordinates $z$ at each $p_j$
\[ |\pl_{\bar{z}}^\beta\pl_z^\gamma r_{12}(z)|\leq C|z|^{1+\alpha-\beta-\gamma} , \quad \beta+\gamma\leq 2\]
by using also the fact that $\pl \psi$ can be locally be considered as a smooth function with a zero of order $1$ at each $p_j$.
Moreover $b\in x^{-J}H^2(M_0)$ thus $r_1\in x^{-J}H^2(M_0)$ and we have
\[ e^{-2i\psi/h}\pl(e^{2i\psi/h}r_1) = b+h\pl r_{12}+\eta=-\pl G(a(V-\lambda^2))+\omega+ h\pl r_{12} + \eta.\]
\begin{lemma}\label{fewestimates}
The following estimates hold true
\[\begin{gathered}
||\eta||_{H^2(M_0)}=O(|\log h|),\quad \|\eta\|_{H^1(M_0)}\leq O(h|\log h|), \quad ||x^{J}\pl r_{12}||_{H^1(M_0)}=O(1),\\
||x^Jr_{1}||_{L^2}=O(h), \quad ||x^J(r_1-h\til{r}_{12})||_{L^2}=o(h)
\end{gathered} \]
where $\til{r}_{12}$ solves $2i\til{r}_{12}\pl\psi = b$.
\end{lemma}
\textbf{Proof}. The proof is exactly the same as the proof of Lemma 4.2 in \cite{GT2}, except that one needs to add the weight
$x^J$ to have bounded integrals.
\qed\\
As a direct consequence, we have
\begin{coro}\label{corerrorterm}
With $r_1=r_{11}+h r_{12}$, there exists $J > 0$ such that
\[||e^{-\Phi/h}(\Delta_{g_0}-\la^2+V)e^{\Phi/h}(a + r_1)||_{x^{-J}L^2(M_0)} = O(h|\log h|). \]
\end{coro}
\subsubsection{Construction of $r_2$}
In this section, we complete the construction of the complex geometric optic solutions. We deal with the general case of surfaces
and we shall show the following
\begin{proposition}
\label{completecgo}
If $\varphi_0$ is the subharmonic function constructed in Section \ref{Carleman}, then for $\eps$ small enough there exist solutions to $(\Delta_{g_0} -\la^2+V)u = 0$ of the form $u=e^{\Phi/h}(a+r_1+r_2)$
with $r_1=r_{11}+hr_{12}$ constructed in the previous section and
$r_2\in e^{-\varphi_0/\eps}L^2$ satisfying $\|e^{\varphi_0/\eps}r_2\|_{L^2}\leq Ch^{3/2}|\log h|$.
\end{proposition}
This is a consequence of the following Lemma (which follows from the Carleman estimate obtained in Section \ref{Carleman} above)
\begin{lemma}
\label{standardargument}
Let $\delta \in (0,1)$, $V\in x^{1-\frac{\delta}{2}}L^\infty(M_0)$, and $\varphi_\eps=\varphi-\frac{h}{\eps}\varphi_0$ a weight with linear growth at infinity
as in Proposition \ref{carlemanestimate}.
For all $f\in L^2(M_0)$ and all $h>0$ small enough, there exists a solution $v\in L^2(M_0)$ to the equation
\begin{equation}\label{solvab}
e^{-\varphi_\eps/h}(\Delta_g -\la^2+V) e^{\varphi_\eps/h}v = x^{1-\frac{\delta}{2}}f
\end{equation}
satisfying
\[\|v\|_{L^2(M_0)} \leq Ch^\demi\|f\|_{L^2(M_0)}.\]
If $\varphi_\eps$ has quadratic growth at infinity, the same result is true when $V \in L^{\infty}(M_0)$ but $x^{1-\frac{\delta}{2}}f$ can be replaced by $f\in L^2$ in \eqref{solvab}.
\end{lemma}
\noindent{\bf Proof}. The proof is based on a duality argument. Let $P_h:=e^{\varphi_\eps/h}(\Delta_g -\la^2+V) e^{-\varphi_\eps/h}$
and for all $h >0$ the real vector space $\mc{A}:=\{u\in x^{-1+\frac{\delta}{2}}H^1(M_0);
P_hu\in L^2(M_0)\}$
equipped with the real scalar product
\[(u,w)_{\mc{A}}:=\cjg P_hu,P_hw\cjd_{L^2}.\]
By the Carleman estimate of Proposition \ref{carlemanestimate}, the space $\mc{A}$ is a Hilbert space equipped with the scalar product above if $h < h_0$, and
thus the linear functional $L:w\to \int_{M_0}x^{1-\frac{\delta}{2}}fw \,{\rm dvol}_{g_0}$ on $\mc{A}$ is continuous with norm bounded by
$Ch^\demi||f||_{L^2}$ by Proposition \ref{carlemanestimate},
and by Riesz theorem there is an element $u\in\mc{A}$ such that $(.,u)_{\mc{A}}=L$
and with norm bounded by the norm of
$L$. It remains to take $v:=P_hu$ which solves $P_h^*v=x^{1-\frac{\delta}{2}}f$ where
$P_h^*=e^{-\varphi_\eps/h}(\Delta_g -\la^2+V) e^{\varphi_\eps/h}$ is the adjoint of $P_h$
and $v$ satisfies the desired norm estimate. The proof when the weight $\varphi_\eps$ has quadratic growth at infinity
is the same, but improves
slightly due to the Carleman estimate of Proposition \ref{CEquad}.
\qed\\
\noindent{\bf Proof of Proposition \ref{completecgo}}. We first solve the equation
\[(\Delta +V-\la^2)e^{\varphi_\eps/h}\til{r}_2 = x^{1-\frac{\delta}{2}}e^{\varphi_\eps/h} \Big(x^{-1+\frac{\delta}{2}}e^{-\varphi_\eps/h}(\Delta +V-\la^2)e^{\Phi/h}(a + r_1)\Big)\]
by using Lemma \ref{standardargument} and the fact that for $J$ large, there is $C>0$ such that for all $h<h_0$
\[ ||x^{-1+\frac{\delta}{2}}e^{-\varphi_\eps/h}(\Delta +V-\la^2)e^{\Phi/h}(a + r_1)||_{L^2}\leq C|| x^Je^{-\Phi/h}(\Delta -\la^2+V)e^{\Phi/h}(a + r_1)||_{L^2}\]
since $x^{-J-1}e^{\varphi_0/\eps}\in L^\infty(M_0)$ for all $J$ (recall that $\varphi_0\sim -x^{-\delta}/\delta^2$ as $x\to 0$).
But now the right hand side is bounded by $O(h|\log h|)$ according to Corollary
\ref{corerrorterm}, therefore we set $r_2:=-e^{-i\psi/h-\varphi_0/\eps}\til{r}_2$ which
satisfies $(\Delta_{g_0}-\la^2+V)e^{\Phi/h}(a+r_1+r_2)=0$ and, by Lemma \ref{standardargument},
the norm estimate $||e^{\varphi_0/\eps}r_2||_{L^2}\leq O(h^{3/2}|\log h|)$.
\qed
\end{section}
\begin{section}{Scattering on surface with Euclidean ends} \label{sec_scattering}
Let $(M_0,g_0)$ be a surface with Euclidean ends and $V\in e^{-\gamma/x}L^\infty(M_0)$ for some
$\gamma$. The scattering theory in this setting is described for instance in Melrose \cite{MelStanford}, here
we will follow this presentation (see also Section 3 in Uhlmann-Vasy \cite{UhlVa} for the $\rr^n$ case).
First, using standard methods in scattering theory, we define the resolvent on the continuous spectrum as follows
\begin{lemma}\label{resolvent}
The resolvent $R_V(\la):=(\Delta_{g_0}+V-\la^2)^{-1}$ admits a meromorphic extension from $\{{\rm Im}(\la)<0\}$
to $\{{\rm Im}(\la)\leq A, {\rm Re}(\la)\not= 0\}$, as a family of operators mapping $e^{-\gamma/x}L^2(M_0)$ to
$e^{\gamma/x}L^2(M_0)$ for any $\gamma>A$. Moreover, for $\la\in\rr\setminus\{0\}$ not a pole,
$R_V(\la)$ maps continuously $x^\alpha L^2$ to $x^{-\alpha}L^2$ for any $\alpha>1/2$.
\end{lemma}
\noindent\textsl{Proof}. The statement is known for $V=0$ and $M_0=\rr^2$ by using the explicit formula of the
resolvent convolution kernel on $\rr^2$ in terms of Hankel functions (see for instance \cite{MelStanford}),
we shall denote $R_0(\la)$ this continued resolvent.
More precisely, for all $A>0$, the operator $R_0(\la)$ continues analytically from $\{{\rm Im}(\la)<0\}$ to
$\{{\rm Im}(\la)\leq A, {\rm Re}(\la)\not=0\}$ as a family of bounded operators mapping $e^{-\gamma/x}L^2$ to $e^{\gamma/x}L^2$
for any $\gamma>A$.
Now we can set $\chi\in C_0^\infty(M_0)$ such that $1-\chi$ is supported in the ends $E_i$, and let
$\chi_0,\chi_1\in C_0^\infty(M_0)$ such that $(1-\chi_0)=1$ on the support of $(1-\chi)$ and $\chi_1=1$ on the support of $\chi$.
Let $\la_0\in -i\rr_+$ with $i\la_0\gg 0$, then the resolvent $R_0(\la_0)$ is well defined from $L^2(M_0)$ to $H^2(M_0)$ since the Laplacian is essentially self-adjoint \cite[Proposition 8.2.4]{T2}, and we have a parametrix
\[E(\la):= (1-\chi_0)R_{0}(\la)(1-\chi)+\chi_1 R_0(\la_0)\chi \]
which satisfies
\[\begin{gathered}
(\Delta_{g_0}-\la^2+V)E(\la)=1+K(\la), \\ K(\la):=([\Delta_{g_0},\chi_1]- (\la^2-\la_0^2)\chi_1)R_0(\la_0)\chi-
[\Delta_{g_0},\chi_0]R_0(\la)(1-\chi)+VE(\la),
\end{gathered}\]
where here we use the notation $R_0(\la)$ for an integral kernel on $M_0$, which in the charts
$\{z\in\rr^2; |z|>1\}$ corresponding the ends $E_1,\dots E_N$, is given by the integral kernel of $(\Delta_{\rr^2}-\la^2)^{-1}$.
Using the explicit expression of the convolution kernel of $R_0(\la)$ in the ends
(see for instance Section 1.5 of \cite{MelStanford}) and the decay assumption
on $V$, it is direct to see that for ${\rm Im}(\la)<A, {\rm Re}(\la)\not=0$, the map $\la \mapsto K(\la)$ a is compact analytic family of bounded operators
from $e^{-\gamma/x}L^2$ to $e^{-\gamma/x}L^2$ for any $\gamma>A$. Moreover $1+K(\la_0)$ is invertible since $||K(\la_0)||_{L^2\to L^2}\leq 1/2$
if $i\la_0$ is large enough.
Then by analytic Fredholm theory, the resolvent $R_V(\la)$ has an meromorphic extension to ${\rm Im}(\la)<A, {\rm Re}(\la)\not=0$
as a bounded operator from $e^{-\gamma/x}L^2$ to $e^{\gamma/x}L^2$ if $\gamma>A$, given by
\[ R_V(\la)=E(\la)(1+K(\la))^{-1}.\]
Now $(1+K(\la))^{-1}=1+Q(\la)$ for some $Q(\la)=-K(\la)(1+K(\la))^{-1}$ mapping $e^{-\gamma/x}L^2$ to itself
for any $\gamma>A$, which proves the mapping properties of $R_V(\la)$ on exponential weighted spaces. For the
mapping properties on $\{{\rm Re}(\la)=0\}$, a similar argument works.
\qed\\
A corollary of this Lemma is the mapping property
\begin{coro}\label{mapping}
For $ \la\in\rr\setminus \{0\}$ not a pole of $R_V(\la)$,
and $f\in e^{-\gamma/x}L^\infty$ for some $\gamma>0$, then there exists $v\in C^\infty(\pl\bbar{M}_0)$ such that
\[R_V(\la)f -x^\demi e^{-i\la/x}v \in L^2.\]
\end{coro}
\noindent\textsl{Proof}. Using the expression $R_V(\la)=E(\la)(1+Q(\la))$ of the proof of Lemma \ref{resolvent}, it suffices
to know the mapping property of $E(\la)$ on $e^{-\gamma/x}L^2$, but since outside a compact set (i.e.~in the ends)
$E(\la)$ is given by the free resolvent on $\rr^2$, this amounts to proving the statement in $\rr^2$, which is
well-known: for instance, this is proved for $f\in C_0^\infty(\rr^2)$ in Section 1.7 \cite{MelStanford} but the proof extends easily to
$f\in e^{-\gamma/x}L^\infty(\rr^2)$ since the only used assumption on $f$ for applying a stationary phase argument
is actually that the Fourier transform $\hat{f}(z)$ has a holomorphic extension in a complex neighbourhood of $\rr^2$.
\qed\\
We also have a boundary pairing, the proof of which is exactly the same as
\cite[Lemma 2.2]{MelStanford} (see also Proposition 3.1 of \cite{UhlVa}).
\begin{lemma}\label{boundarypairing}
For $\la>0$ and $V\in e^{-\gamma/x}L^\infty(M_0)$, if $u_\pm\in x^{-\alpha}L^2(M_0)$ for some $\alpha>1/2$ and
$(\Delta_{g_0}-\la^2+V)u_\pm\in x^\alpha L^2(M_0)$ with
\[ u_+-x^\demi e^{i\la/x}f_{++}-x^\demi e^{-i\la/x}f_{+-} \in L^2, \quad u_--x^\demi e^{i\la/x}f_{-+}-x^\demi e^{-i\la/x}f_{--} \in L^2
\]
for some $f_{\pm\pm}\in C^\infty(\pl \bbar{M}_0)$, then
\[\cjg u_+, (\Delta_{g_0}+V-\la^2)u_-\cjd-\cjg (\Delta_{g_0}+V-\la^2)u_+, u_-\cjd=2i\la \int_{\pl \bbar{M}_0}(f_{++}\bbar{f_{-+}}-f_{+-}\bbar{f_{--}}) \]
where the volume form on $\pl\bbar{M}_0\simeq\sqcup_{i=1}^N S^1$ is induced by the metric $x^2g|_{T\pl\bbar{M}_0}$.
\end{lemma}
As a corollary, the same exact arguments as in Sections 2.2 to 2.5 in \cite{MelStanford} show \footnote{In \cite{MelStanford},
a unique continuation is used for Schwartz solutions of $(\Delta+V-\la^2)u=0$ when $V$ is a
compactly supported potential on $\rr^n$ but the same result is also true in our setting,
this is a consequence of a standard Carleman estimate.}
\begin{coro}\label{analytic}
The operator $R_V(\la)$ is analytic on $\la\in \rr\setminus\{0\}$ as a bounded operator from $x^{\alpha}L^2$ to $x^{-\alpha}L^2$
if $\alpha>1/2$.
\end{coro}
In $\rr^2$ there is a Poisson operator $P_0(\la)$ mapping $C^\infty(S^1)$ to $x^{-\alpha}L^2(\rr^2)$
for $\alpha>1/2$, which satisfies that for any $f_+\in C^\infty(S^1)$ there exists $f_-\in C^\infty(S^1)$ such that
\[P_0(\la)f_+ -x^\demi e^{i\la/x}f_+ -x^\demi e^{-i\la/x}f_-\in L^2 , \quad (\Delta-\la^2)P_0(\la)f_+=0.\]
We can therefore define in our case a similar Poisson operator $P_V(\la)$ mapping $C^\infty(\pl\bbar{M}_0)$ to $x^{-\alpha} L^2$ for $\alpha>1/2$,
by
\begin{equation}\label{poisson}
P_V(\la)f_+:= (1-\chi)P_0(\la)f_+- R_V(\la)(\Delta_{g_0}+V-\la^2)(1-\chi)P_0(\la)f_+
\end{equation}
where $1-\chi\in C^\infty(M_0)$ equals $1$ in the ends $E_i$ and $P_0(\la)$ denotes here the Schwartz kernel of the Poisson operator on $\rr^2$
pulled back to each of the Euclidean ends $E_i$ of $M_0$ in the obvious way.
Then, since $(\Delta_{g_0}+V-\la^2)(1-\chi)P_0(\la)f_+\in e^{-\gamma/x}L^2$ for all $\gamma>0$, it suffices to use
Corollaries \ref{mapping} and \ref{analytic} to see that it defines an analytic Poisson operator $P_V(\la)$
on $\la\in \rr\setminus\{0\}$ satisfying that for all $f_+\in C^\infty(\pl\bbar{M_0})$, there exists $f_-\in C^\infty(\pl\bbar{M}_0)$ such that
\begin{equation}\label{poissonV}
P_V(\la)f_+ -x^\demi e^{i\la/x}f_+ -x^\demi e^{-i\la/x}f_-\in L^2 , \quad (\Delta+V-\la^2)P_V(\la)f_+=0.
\end{equation}
Moreover, it is easily seen to be the unique solution of \eqref{poissonV}: indeed, if two such solutions exist then the difference is
a solution $u$ with asymptotic $x^{\demi}e^{-i\la/x}f_-+L^2$ for some $f_-\in C^\infty(\pl\bbar{M}_0)$, but applying Lemma \ref{boundarypairing}
with $u_-=u_+=u$ shows that $f_-=0$, thus $u\in L^2$, which implies $u=0$ by Corollary \ref{analytic}.
\begin{definition}
The scattering matrix $S_V(\la):C^\infty(\pl\bbar{M}_0)\to C^\infty(\pl\bbar{M}_0)$
for $\la\in \rr\setminus\{0\}$ is defined to be the map $S_V(\la)f_+:=f_-$ where $f_-$ is given by the asymptotic
\[P_V(\la)f_+= x^\demi e^{i\la/x}f_+ +x^\demi e^{-i\la/x}f_- +g ,\,\, \textrm{ with }\,\, g\in L^2.\]
\end{definition}
We remark that, using Lemma \ref{boundarypairing} and the uniqueness of the Poisson operator, one easily deduces for $\la\in\rr\setminus\{0\}$
\begin{equation}\label{relationSV}
S_V(\la)^*=S_V(-\la)=S_V(\la)^{-1}
\end{equation}
where the scalar product on $L^2(\pl\bbar{M}_0)$ is induced by the metric $x^2g_0|_{T\pl\bbar{M}_0}$.
We can now state a density result similar to Proposition 3.3 of \cite{UhlVa}:
\begin{prop}\label{density}
If $V\in e^{-\gamma_0/x}L^\infty(M_0)$ $($resp. $V\in e^{-\gamma_0/x^2}L^\infty(M_0))$
for some $\gamma_0>0$, and $\la\in\rr\setminus\{0\}$, then for any $0<\gamma<\gamma'<\gamma_0$ the set
\[\mc{F}:=\{ P_V(\la)f_+; f_+\in C^\infty(\pl\bbar{M}_0) \}\]
is dense in the null space of $\Delta_{g_0}+V-\la^2$ in $e^{\gamma/x}L^2(M_0)$ for the topology of $e^{\gamma'/x}L^2(M_0)$
$($resp.~in $e^{\gamma/x^2}L^2(M_0)$ for the topology of $e^{\gamma'/x^2}L^2(M_0))$.
\end{prop}
\noindent\textsl{Proof}. First assume $V\in e^{-\gamma_0/x}L^\infty(M_0)$. Let $w\in e^{-\gamma'/x}L^2$ be orthogonal to $\mc{F}$, and set $u_-:=R_V(\la)w$ and $u_+=P_V(\la)f_{++}$ for some
$f_{++} \in C^\infty(\pl\bbar{M}_0)$. Then, define $f_{--}\in C^\infty(\pl\bbar{M}_0)$ by $R_V(\la)w-x^\demi e^{-i\la/x}f_{--}\in L^2$,
and from Lemma \ref{boundarypairing} we obtain $\cjg f_{+-},f_{--}\cjd=0$ since $\cjg w,P_V(\la)f_{++}\cjd=0$ by assumption.
Since $f_{+-} = S_V(\lambda) f_{++}$ is arbitrary, then $f_{--}=0$ and $u_-\in L^2$. In particular, from the parametrix constructed in the proof of
Lemma \ref{resolvent}
\[R_V(\la)w- (1-\chi_0)R_0(\la)(1-\chi)(1+Q(\la))w \in L^2\]
with $(1+Q(\la))w\in e^{-\gamma'/x}L^2$. Since in each end, $R_0(\la)$ is the integral
kernel of the free resolvent of the Euclidean Laplacian on $\rr^2$ and $(1-\chi_0)$ and $(1-\chi)$ are supported in the ends,
we can view the term $(1-\chi_0)R_0(\la)(1-\chi)(1+Q(\la))w$ as a disjoint sum (over the ends)
of functions on $\rr^2$ of the form
\begin{equation}\label{fouriertr}
(1-\chi_0(z)) \frac{1}{(2\pi)^{2}} \int_{\rr^2}e^{iz\xi} (\xi^2-\la^2-i0)^{-1}\hat{f}(\xi)d\xi
\end{equation}
where in each end $E_i$, $f=(1-\chi)(1+Q(\la))w\in e^{-\gamma'/x}L^2(E_i)$ can be considered as a function in $e^{-\gamma'|z|}L^2(\rr^2)$.
By the Paley-Wiener theorem, $\hat{f}$ is holomorphic in a strip $U=\{|{\rm Im}(\xi)|<\gamma'\}$ with
bound $\sup_{\eta\leq \gamma}|| \hat{f}(\cdot+i\eta)||_{L^2(\rr^2)}<\infty$ for all $\gamma<\gamma'$, so the fact that \eqref{fouriertr} is in $L^2$ implies that
$\hat{f}$ vanishes at the real sphere $\{\xi\in\rr^2 ; \xi^2=\la^2\}$, and thus there exists $h$ holomorphic in $U$ such that
$\hat{f}(\xi)=(\xi^2-\la^2)h(\xi)$ (see e.g.~the proof of Lemma 2.5 in \cite{PSU}), and satisfying the same types of $L^2$ estimates
as $\hat{f}$ in $U$ on lines ${\rm Im}(\xi)=\textrm{cst}$. By the Paley-Wiener theorem again, we deduce that
\eqref{fouriertr} is in $e^{-\gamma|z|}L^2$ and thus $R_V(\la)w\in e^{-\gamma/x}L^2(M_0)$ for any $\gamma<\gamma'$.
Then if $v\in e^{\gamma/x}L^2(M_0)$ and $(\Delta_{g_0}+V-\la^2)v=0$, one has by integration by parts
\[ 0=\cjg R_V(\la) w,(\Delta_{g_0}+V-\la^2)v\cjd = \cjg w,v\cjd\]
which ends the proof in the case $V\in e^{-\gamma_0/x}L^\infty(M_0)$. The quadratic decay case $V\in e^{-\gamma_0/x^2}L^\infty(M_0)$
is exactly similar but instead of Paley-Wiener theorem, we use Corollary \ref{corapp} and the inclusions
$e^{-\gamma'/x^2}L^2\subset e^{-\gamma''/x^2}L^1\cap e^{-\gamma''/x^2}L^2$ and $e^{-\gamma'/x^2}L^\infty\subset
e^{-\gamma/x^2}L^2$ for all $\gamma<\gamma''<\gamma'$.
\qed
\end{section}
\begin{section}{Identifying the potential} \label{sec_identify}
\subsection{The case of a surface}
On a Riemann surface $(M_0,g_0)$ with $N$ Euclidean ends and genus $g$, we assume that $V_1,V_2\in C^{1,\alpha}(M_0)$ are two real valued potentials such that the respective scattering operators
$S_{V_1}(\la)$ and $S_{V_2}(\la)$ agree for a fixed $\la>0$. We also assume that for all $\gamma>0$
\[V_1,V_2\in \left\{\begin{array}{ll}
e^{-\gamma/x}L^\infty(M_0) & \textrm{ if }N\geq \max(2g+1,2)\\
e^{-\gamma/x^2}L^\infty(M_0) & \textrm{ if }N\geq g+1.
\end{array}\right.\]
By considering the asymptotics of $u_1:=P_{V_1}(\la)f_1$ and $P_{V_2}(-\la)f_2$ for $f_i\in C^\infty(\pl\bbar{M}_0)$
we easily have by integration by parts that
\begin{equation}\label{scatident}
\begin{split}
\int_{M_0} (V_1-V_2)u_1\overline{u_2}\, {\rm dvol}_{g_0}=& -2i\la \int_{\pl\bbar{M}_0} S_{V_1}(\la)f_1.\overline{f_2} - f_1.\overline{S_{V_2}(-\la)f_2}\\
=& -2i\la \int_{\pl\bbar{M}_0} (S_{V_1}(\la)-S_{V_2}(\la))f_1.\overline{f_2}=0
\end{split}\end{equation}
by using \eqref{relationSV}. From Proposition \ref{density}, this implies by density that, if $V\in e^{-\gamma/x}L^\infty$
(resp. $V\in e^{-\gamma/x^2}L^\infty$ for all $\gamma>0$), then for all solutions $u_i$
of $(\Delta_{g_0}+V_i-\la^2)u_i=0$ in $e^{\gamma'/x}L^2(M_0)$ (resp. $u_i\in e^{\gamma'/x^2}L^2(M_0)$)
for some $\gamma'>0$, we have
\begin{equation}\label{integralform}
\int_{M_0} (V_1-V_2)u_1 \overline{u_2} \, {\rm dvol}_{g_0}=0.
\end{equation}
We shall now use our complex geometric optics solutions as special solutions in the weighted space
$e^{-\gamma'/hx}L^2(M_0)$ (resp. $e^{-\gamma'/hx^2}L^2(M_0)$)
for some $\gamma'>0$ if $V\in e^{-\gamma/x}L^\infty$ (resp. $V\in e^{-\gamma/x^2}L^\infty$) for all $\gamma>0$.\\
Let $p\in M_0$ be such that, using Proposition \ref{criticalpoints}, we can choose a holomorphic Morse function $\Phi=\varphi+i\psi$
with linear or quadratic growth on $M_0$ (depending on the topological assumption),
with a critical point at $p$. Then for the complex geometric optics solutions $u_1,u_2$ with phase $\Phi$
constructed in Section \ref{CGOriemann}, the identity \eqref{integralform} holds true.
We will then deduce the
\begin{proposition}
\label{identcritpts}
Let $\la\in (0,\infty)$ and assume that $S_{V_1}(\la) = S_{V_2}(\la)$, then $V_1(p)= V_2(p)$.
\end{proposition}
\noindent{\bf Proof}.
Let $u_1$ and $u_2$ be solutions on $M_0$ to
\[(\Delta_g +V_j-\la^2)u_j = 0\]
constructed in Section \ref{CGOriemann} with phase $\Phi$ for $u_1$ and $-\Phi$ for $u_2$, thus of the form
\[u_1 = e^{\Phi/h}(a + r_1^{1} + r_2^{1}), \quad u_2=e^{-\Phi/h}(a+r_1^{2}+r_2^{2}).\]
We have the identity
\[\int_{M_0}u_1(V_1 - V_2) \bbar{u_2} \,{\rm dvol}_{g_0}=0\]
Then by using the estimates in Lemma \ref{fewestimates} and Proposition \ref{completecgo} we have, as $h\to 0$,
\[\int_{M_0}e^{2i\psi/h}|a|^2(V_1 - V_2) \,{\rm dvol}_{g_0} +
h \int_{M_0}e^{2i\psi/h}(\bbar{a} \til{r}_{12}^{1} + a \bbar{\til{r}_{12}^{2}})(V_1 - V_2) \,{\rm dvol}_{g_0} + o(h) = 0 \]
where $\til{r}_{12}^{j}\in L^\infty(M_0)$ are defined in Lemma \ref{fewestimates}, with the superscript $j$ refering to the solution for the
potential $V_j$; in particular these functions $\til{r}^j_{12}$ are independent of $h$.
By splitting $V_i(\cdot)=(V_i(\cdot)-V_i(p))+V_i(p)$ and using the $C^{1,\alpha}$ regularity assumption on $V_i$,
one can use stationary phase for the $V_i(p)$ term and integration by parts to gain a power of $h$ for the $V_i(\cdot)-V_i(p)$ term
(see the proof of Lemma 5.4 in \cite{GT2} for details) to deduce
\[\int_{M_0}e^{2i\psi/h}|a|^2(V_1 - V_2) \,{\rm dvol}_{g_0} = Ch(V_1(p) - V_2(p)) + o(h)\]
for some $C\neq 0$. Therefore,
\[ Ch(V_1(p) - V_2(p))+ h\int_{M_0}e^{2i\psi/h}(\bbar{a} \til{r}_{12}^{1} + a \bbar{\til{r}_{12}^{2}})(V_1 - V_2) \,{\rm dvol}_{g_0} = o(h).\]
Now to deal with the middle terms, it suffices to apply a Riemann-Lebesgue type argument like Lemma 5.3 of \cite{GT2}
to deduce that it is a $o(h)$.
The argument is simply to approximate the amplitude in the $L^1(M_0)$ norm by a smooth compactly supported function
and then use stationary phase to deal with the smooth function. We have thus proved that $V_1(p)=V_2(p)$ by taking $h\to 0$.
\qed
\end{section}
\begin{section}{Appendix}
To obtain mapping properties of the resolvent of $\Delta_{\rr^2}$ acting on functions with Gaussian decay,
we shall give two Lemmas on Fourier transforms of functions with Gaussian decay.
\begin{lemma}\label{app1}
Let $f(z)\in e^{-\gamma|z|^2}L^2(\rr^2)$ for some $\gamma>0$. Then the Fourier transform $\hat{f}(\xi)$ extends analytically to
$\cc^2$ and for all $\xi, \eta\in\rr^2$,
\[ ||\hat{f}(\xi+i\eta) ||_{L^2(\rr^2,d\xi)}\leq 2\pi e^{\frac{|\eta|^2}{4\gamma}}||e^{\gamma|z|^2}f||_{L^2(\rr^2)}.\]
If $f(z)\in e^{-\gamma|z|^2}L^1(\rr^2)$ for some $\gamma>0$ then
\[\sup_{\xi\in\rr^2}|\hat{f}(\xi+i\eta) |\leq e^{\frac{|\eta|^2}{4\gamma}}||e^{\gamma|z|^2}f||_{L^1(\rr^2)}.\]
\end{lemma}
\noindent\textsl{Proof}. The first statement is clear. For the bound, we write
\[ \hat{f}(\xi+i\eta)=e^{\frac{|\eta|^2}{4\gamma}}\int_{\rr^2}e^{-i\xi. z} e^{-\gamma|z-\frac{\eta}{2\gamma}|^2}e^{\gamma|z|^2}f(z)dz=
e^{\frac{|\eta|^2}{4\gamma}}\mc{F}_{z\to\xi} (e^{-\gamma|z-\frac{\eta}{2\gamma}|^2}e^{\gamma|z|^2}f(z)).
\]
But the function $e^{-\gamma|z-\frac{\eta}{2\gamma}|^2}e^{\gamma|z|^2}f(z)$ is in $L^2(\rr^2,dz)$ and its norm is bounded uniformly
by $||e^{\gamma|z|^2}f||_{L^2}$, thus it suffices to use the Plancherel theorem to obtain the desired bound. The $L^\infty$ bound is similar.
\qed
\begin{lemma}\label{app2}
Let $F(\xi+i\eta)$ be a complex analytic function on $\rr^2+i\rr^2=\cc^2$ such that there is $C>0$ and $\gamma>0$ with
\[||F(\xi+i\eta) ||_{L^2(\rr^2,d\xi)}\leq Ce^{\frac{|\eta|^2}{4\gamma}} \textrm{ and } \sup_{\xi\in\rr^2}|F(\xi+i\eta) |\leq Ce^{\frac{|\eta|^2}{4\gamma}}.\]
If $F$ vanishes on the real submanifold $\{|\xi|^2=\la^2\}$, then $\mc{F}^{-1}_{\xi\to z}(\frac{F(\xi)}{|\xi|^2-\la^2})\in e^{-\gamma|z|^2}L^\infty(\rr^2)$.
\end{lemma}
\noindent\textsl{Proof}. First by analyticity of $F$, one has that $F$ vanishes on the complex hypersurface $M_\la:=\{\zeta\in\cc^2; \zeta.\zeta=\la^2\}$
(see for instance the proof of Lemma 2.5 of \cite{PSU}), and in particular $G(\zeta)=F(\zeta)/(\zeta.\zeta-\la^2)$ is an analytic function on $\cc^2$.
We will first prove that for each $\eta\in\rr^2$, $G(\xi+i\eta)\in L^1(\rr^2,d\xi)\cap L^\infty(\rr^2,d\xi)$ and
\begin{equation}\label{Gxi}
||G(\xi+i\eta)||_{L^1(\rr^2,d\xi)}\leq Ce^{\frac{|\eta|^2}{4\gamma}}.
\end{equation}
If $|\eta| \leq 2$ we choose the disc $B := \{ \xi \in \rr^2; |\xi|^2 < 2(4+\la^2) \}$ and let $\zeta:=\xi+i\eta$. Then $||G(\xi+i\eta)||_{L^1(B,d\xi)}$ and $||(\zeta.\zeta-\la^2)^{-1}||_{L^2(\rr^2 \setminus B,d\xi)}$ are uniformly bounded for $|\eta| \leq 2$, and we obtain by Cauchy-Schwarz that \eqref{Gxi} holds for $|\eta| \leq 2$. For the case $|\eta| > 2$ we define $U_\eta:=\{\xi \in \rr^2; |\zeta.\zeta-\la^2|>|\eta|\}$ and note that
\begin{equation*}
\begin{gathered}
\sup_{|\eta| > 2} ||(\zeta.\zeta-\lambda^2)^{-1}||_{L^1(\rr^2 \setminus U_{\eta},d\xi)} < \infty, \\
\sup_{|\eta| > 2} ||(\zeta.\zeta-\lambda^2)^{-1}||_{L^2(U_{\eta},d\xi)} < \infty.
\end{gathered}
\end{equation*}
These results follow by decomposing the integration sets to parts where one can change coordinates $\xi_1 + i\xi_2$ to $\tilde{\xi}_1 + i \tilde{\xi}_2 := \zeta.\zeta-\lambda^2$, and by evaluating simple integrals. Then \eqref{Gxi} follows from Cauchy-Schwarz and the estimates for $F$.
Let $\eta=2\gamma z$, we use a contour deformation from $\rr^2$ to $2i\gamma z+\rr^2$ in $\cc^2$,
\[\int_{\rr^2}e^{iz.\xi}G(\xi)d\xi= \int_{\rr^2}e^{iz.(\xi+2i\gamma z)}G(\xi+2i\gamma z)d\xi, \]
which is justified by the fact that $G(\xi+i\eta)\in L^1(\rr^2 \times K,d\xi \,d\eta)$ for any compact set $K$ in $\rr^2$ by the uniform bound \eqref{Gxi}. Now using \eqref{Gxi} again shows that
\[\Big|\int_{\rr^2}e^{iz.\xi}G(\xi)d\xi\Big|\leq C e^{-\gamma |z|^2}\]
which ends the proof.
\qed
\begin{coro} \label{corapp}
Let $f(z)\in e^{-\gamma|z|^2}L^2(\rr^2)\cap e^{-\gamma|z|^2}L^1(\rr^2)$ for some $\gamma>0$. Assume that
its Fourier transform $\hat{f}(\xi)$ vanishes on the sphere $\{|\xi|=|\la|\}$, then one has
\[ \mc{F}^{-1}_{\xi\to z}\Big( \frac{\hat{f}(\xi)}{|\xi|^2-\la^2}\Big) \in e^{-\gamma|z|^2}L^\infty(\rr^2).\]
\end{coro}
\end{section}
| 90,796
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Routledge
400 pages.
This book looks at all aspects of New Labour’s policies in relation to employment relations and trade unionism. The first half of Trade Unions in a Neoliberal World presents an overview of industrial politics, the evolution of New Labour and an anatomy of contemporary trade unionism. It discusses relations between the Labour Party and the unions and the response of trade unionists to political and economic change. The second part contains chapters on legislation, partnership, organizing, training, strikes and perspectives on Europe..
| 117,289
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Palenque --See "places" for more information.
Patrons of Writing -- The Hero Twins had two older brothers who were jealous of the twins The Hero Twins changed their older brothers into monkeys and they became the patron gods of scribes (writers).
Pok-a-tok -- A Maya ball game played with stone hoops. The game would go on for hours.
Pom -- The resin of the copal tree, used by the Maya for rubber, chewing gum and incense
| 269,052
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4D Theater Shows & Animal Feedings
The aquarium's 4D Theater has multiple shows throughout the day with schedules available online. For feeding times, check displays at each exhibit. Other daily events include tours, dolphin tales and more.
Created by
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225 Baker St. NW
Atlanta, GA 30313
(404) 581 *}
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.
It can always seem daunting when it comes to trying to communicate with a loved one who is suffering from this disease.
In fact, new research from the Alzheimer’s Society has revealed that a huge majority of the general public find it difficult to know how to communicate with those living with dementia.
MORE: Could loneliness contribute to dementia?
According to new data surveying 2,300 adults, 50% of people reported not feeling confident even visiting someone with dementia in a care home or inviting them to a meal at home. And over two-thirds of those asked, 69%, said they wouldn’t feel confident in knowing how to help someone with dementia if they saw them struggling in a public place.
So it’s clear that if you’re finding it tricky to talk to a loved one with dementia, you’re certainly not alone. But in fact, there are effective ways to handle it.
Alzheimer’s Society have offered up their tips for communicating with someone living with any form of dementia. To mark Dementia Action week this week (21st – 27th May), they spoke to 500 people who have the disease. They shared that just a few simple things could help communication, and in turn, help to reduce the isolation dementia patients can often feel.
In the words of a person living with dementia, Alzheimer’s Society have shared tips for communicating effectively…
-.
Group Support Manager at Forest Healthcare, Chris Salter has also offered some helpful, day-to-day tips for communication with people who have dementia. These are practical, simple ways, in which you can help improve how someone with dementia views the world, and copes with their illness.
Chris’ 10 tips for communication with dementia patients…
1. Ensure you use eye contact with the person, you may also want to lower yourself to the persons level and talk to them at a distance, this is to avoid being intimidating. Also ensure you keep your tone positive.
2. Be patient and calm and although it may be difficult, try not to interrupt the person. Don’t try and complete their sentences for them as this may anger them.
3. Use things that may jog their memory such as photo albums, music or items they own to try and help them remember things which can help facilitate conversation and help them remember good memories in their life.
4. Ask easy and one point questions. Many times in conversations the questions we ask can seem complicated and have more than one point to them. Try to keep it easy and ask one line questions so they have time to think and to answer.
5. Encourage them to join in conversations with others, sometimes social clubs can be very helpful for this type of thing as it encourages them to get out and talk to people more regularly.
6. Speak clearly and slowly using small sentences.
7. Don’t patronise the person suffering with Dementia. If they suddenly go off-topic and change the entire course of conversation, just go along with it and seem interested.
8. Try to get rid of any background noise or distractions such as a loud TV or radio, as noise in the background might make them more confused and they may lose their train of thought.
9. Refer to their name and try to use it as much as possible.
10. Don’t be afraid if there are any silences within the conversation – this can make some people feel awkward but dementia sufferers usually don’t notice this. Just try and be understanding and perhaps listen to music together if the conversation is struggling.
| 266,659
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The head coach of the Indian cricket team, Ravi Shastri, has expressed the belief that his team could be world beaters.
Speaking on the back of another series whitewash against Bangladesh, the coach pointed that the team’s ouster for the World Cup was a bitter pill to swallow.
He however described as amazing on how the boys did not let that dampen their spirit have only grown in strength from July 9 earlier this year.
According to him, the intensity shown by the team is fabulous, bouncing back after those 15 minutes in Manchester.
While noting that the dedication and intensity they have shown in their games over the last three months is unbelievable, the coach said, looking back 5-6 years ago, it to be one of the most consistent Indian teams ever across all formats.
Acknowledging that the past cannot be altered, Ravi said winning the T20 World Cup next year would go a long way in confirming the team as invincible, the tag that has been given to them.
When asked if former captain Mahendra Singh will be back into the team and take that flight, the coach said it is necessary to wait till the Indian Premier League (IPL) starts before making such a decision. He added that all would be down to when he starts playing and how he is playing during the IPL.
The coach further stated that the IPL would become a platform that could be the last tournament, after which the squad member for the T20 World Cup will be selected because there might be a player who might be there and in case there is an injury.
Speaking on the present form of the team, Ravi said it keeps moving from strength to strength. He is even excited that about Kohli captainship and the former Indian captain, Sourav Ganguly is now the BCCI president.
He described the development as fantastic and was one of the first to congratulate him on the good news because it was important than anything.
Despite returning to training, Dhoni has made himself unavailable since the World Cup that ended in July this year.
Written by: Oladipupo Mojeed
| 317,477
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The Bantams, who charged to promotion via the League Two play-offs last season, chalked up their first win of the campaign last week with a 4-0 demolition of Carlisle in front of almost 14,000 at Valley Parade.
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Yeates opened the scoring against the Cumbrians with a 25-yard effort which won Sky Sports' Goal of the Day.
Yeates told Sky Sports News Radio: "It was a nice way to get my first goal, but the main thing was to get the first win of the season. I thought we were good for the full 90 minutes and the gaffer has been pleased with us this week.
"It's a long time since I played at Valley Parade, but coming back here I didn't actually realise the size of the club and the support it has and it has taken me back a bit.
"The boys are a top bunch of lads and getting that first win has settled everyone down. There are a few lads who haven't played up a level who were anxious but getting that win has done them the world of good.
"I think that this squad, and if we get one or two new boys in, will have a right go this season and we could surprise a few people.
"I do think the two boys up front - Nahki Wells and James Hanson - will surprise a few centre-backs this season and if we keep them both fit we can pick up some good results."
Yeates, 28, also revealed that playing again under Phil Parkinson was a major factor in making the move north from Watford.
"Over the summer I spoke to the people at Watford and I got on great with the gaffer there (Gianfranco Zola), but I just thought I wasn't going to play and I didn't get offered a new deal.
"I spoke to a few clubs over the summer, but I had a chat with Phil (Parkinson) and having played for him before it just seemed right.
"Having a manager that can get the best out of you and one that knows you inside out is vital and probably the thing I need now at the stage of my career. I need to be playing regularly and hopefully playing for someone I know will work out well."
To hear the full interview with Yeates, plus more from former Barnsley manager John Hendrie, Millwall's Shaun Derry, Huddersfield's Adam Clayton and much more, download the latest Football League Hour.
Tune into tune Sky Sports News Radio every Thursday from 7pm for more exclusive interviews from the Football League.
| 18,192
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Local News
Immigration Office Being Built in Durham
Posted June 4, 2007
Durham, N.C. — Ground was broken Monday on the Triangle's first immigration office.
The office, which will be located on Roycroft Drive in Durham, will allow people to avoid driving to Charlotte to apply for citizenship, green cards and visas.
The office should be open by early 2008, officials said.
| 77,445
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\begin{document}
\title{Orthogonal polynomials for the weight $x^{\nu} \exp(-x- t/x)$}
\author{{\bf S. Yakubovich} \\
{\em {Department of Mathematics, Fac. Sciences of University of Porto,\\ Rua do Campo Alegre, 687; 4169-007 Porto (Portugal) }}}
\thanks{ E-mail: syakubov@fc.up.pt}
\vspace{3mm}
\thanks{ The work was partially supported by CMUP, which is financed by national funds through FCT(Portugal), under the project with reference UIDB/00144/2020. }
\subjclass[2000]{ 33C10, 42C05, 44A15 }
\date{\today}
\keywords{Classical orthogonal polynomials, Laguerre polynomials, modified Bessel function, generalized hypergeometric function}
\begin{abstract} Orthogonal polynomials for the weight $x^{\nu} \exp(-x- t/x),\ x, t > 0, \nu \in \mathbb{R}$ are investigated. Differential-difference equations, recurrence relations, explicit representations, generating functions and Rodrigues-type formula are obtained.
\end{abstract}
\maketitle
\markboth{\rm \centerline{ S. Yakubovich}}{orthogonal polynomials for exponential weights}
\section{Introduction and preliminary results}
Let $\nu \in \mathbb{R},\ t > 0$ be parameters and consider the sequence of orthogonal polynomials $\{P_n^\nu(x,t)\}_{n\ge 0}$ of degree $n$, satisfying the orthogonality condition
$$\int_0^\infty P_n^\nu(x,t) P_m^\nu(x,t) \ e^{-x-t/x} x^\nu dx= \delta_{n,m},\eqno(1.1)$$
where $\delta_{n,m},\ n,m\in\mathbb{N}_{0}$ is the Kronecker symbol. Admitting the limit case $t=0$ and $\nu > -1$, it gives normalized Laguerre polynomials
$$P_n^\nu(x,0) \equiv {\tilde L}_n^\nu (x) = \left( {n!\over \Gamma(n+\nu+1)}\right)^{1/2} L_n^\nu (x),\eqno(1.2)$$
where $\{L_n^\nu (x)\}_{n\ge0}$ are classical Laguerre polynomials \cite{Bateman}, Vol. II and $\Gamma(z)$ is the Euler gamma function \cite{Bateman}, Vol. I. It satisfies the three term recurrence relation
$$x L_n^\nu (x) = - (n+1) L_{n+1}^\nu (x) + (2n+\nu+1) L_n^\nu (x) - (n+\nu) L_{n-1}^\nu (x),\ n \in \mathbb{N}.\eqno(1.3)$$
As it follows from the theory of orthogonal polynomials, the three term recurrence relation for the sequence $\{P_n^\nu(x,t)\}_{n\ge 0}$ can be written in the form
$$x P_n^\nu(x,t) = A_{n+1} (t) P_{n+1}^\nu(x,t) +B_n(t) P_n^\nu(x,t) + A_{n}(t) P_{n-1}^\nu(x,t),\eqno(1.4)$$
where $P_{-1}^\nu(x,t) \equiv 0,\ P_n^\nu(x,t) = a_n(t) x^n+ b_n(t) x^{n-1}+ \dots,\quad a_n(t)\neq 0 $ and
$$A_n(t)= {a_{n-1}(t)\over a_n(t)},\quad B_n(t)= {b_{n}(t)\over a_n(t)} - {b_{n+1}(t)\over a_{n+1}(t)}.\eqno(1.5)$$
As a consequence of (1.4) the Christoffel-Darboux formula takes place
$$\sum_{k=0}^n P_k^\nu(x,t) P_k^\nu(y,t) = A_{n+1}(t) \frac{ P_{n+1}^\nu(x,t) P_n^\nu(y,t) - P_{n}^\nu(x,t) P_{n+1}^\nu(y,t)}{x-y}.\eqno(1.6)$$
Using the value of the integral \cite{Bateman}, Vol. II
$$ \int_0^\infty e^{-t/x -x} x^{\nu-1} dx = 2 t^{\nu/2} K_\nu\left( 2\sqrt t\right) \equiv \rho_\nu(t),\ t > 0,\eqno(1.7)$$
where $K_\nu(z)$ is the modified Bessel function or Macdonald function \cite{YaL}, we can find the moments of the weight $x^{\nu} \exp(-x- t/x)$. Namely, we find
$$ \int_0^\infty e^{-t/x -x} x^{\nu+n} dx = \rho_{\nu+n+1}(t),\quad n \in \mathbb{N}_0.\eqno(1.8)$$
The asymptotic behavior of the modified Bessel function at infinity and near the origin \cite{Bateman}, Vol. II gives the corresponding values for the function $\rho_\nu,\ \nu \in \mathbb{R}$. Precisely, we have
$$\rho_\nu (t)= O\left( t^{(\nu-|\nu|)/2}\right),\ t \to 0,\ \nu\neq 0, \quad \rho_0(t)= O( \log t),\ t \to 0,\eqno(1.9)$$
$$ \rho_\nu(t)= O\left( t^{\nu/2- 1/4} e^{- 2\sqrt t} \right),\ t \to +\infty.\eqno(1.10)$$
Moreover, it can be represented in terms of Laguerre polynomials (cf. \cite{PrudnikovMarichev}, Vol. II, Entry 2.19.4.13 )
$${(-1)^n t^n\over n!}\ \rho_\nu(t)= \int_0^\infty x^{\nu+n -1} e^{-x - t/x} L_n^\nu(x) dx,\quad n \in\mathbb{N}_{0}.\eqno(1.11)$$
Further, it has a relationship with the Riemann-Liouville fractional integral \cite{YaL}
$$ \left( I_{-}^\alpha f \right) (t) = {1\over \Gamma(\alpha)} \int_t^\infty (x-t)^{\alpha-1} f(x) dx,\quad {\rm Re} \alpha > 0,\eqno(1.12)$$
namely, we get the formula
$$\rho_\nu(t)= \left( I_{-}^\nu \rho_0 \right) (t),\ \nu >0.\eqno(1.13)$$
Hence the index law for fractional integrals immediately implies
$$ \rho_{\nu+\mu} (t)= \left( I_{-}^\nu \rho_\mu \right) (t)= \left( I_{-}^\mu \rho_\nu \right) (t).\eqno(1.14)$$
The corresponding definition of the fractional derivative presumes the relation $ D^\mu_{-}= - D I_{-}^{1-\mu}$. Hence for the ordinary $n$-th derivative of $\rho_\nu$ we find
$$D^n \rho_\nu(t)= (-1)^n \rho_{\nu-n} (t),\quad n \in \mathbb{N}_0.\eqno(1.15)$$
Recalling (1.7) and integrating by parts, it is not difficult to establish the following recurrence relation for $\rho_\nu$
$$\rho_{\nu+1} (t) = \nu \rho_\nu(t)+ t \rho_{\nu-1} (t),\quad \nu \in \mathbb{R}.\eqno(1.16)$$
In the operator form it can be written as follows
$$\rho_{\nu+1} (t) = \left( \nu - tD \right) \rho_\nu(t).\eqno(1.17)$$
Finally in this section we observe that up to a normalization factor the orthogonality (1.1) is equivalent to the following $n$ equalities
$$\int_0^\infty P_n^\nu(x,t) \ e^{-x-t/x} x^{\nu+m} dx= 0,\quad m=0,1,\dots, n-1.\eqno(1.18)$$
Besides, taking into account (1.4), (1.5), we get the identities
$$\int_0^\infty [P_n^\nu(x,t) ]^2 \ e^{-x-t/x} x^{\nu+1} dx= B_n(t),\eqno(1.19)$$
$$\int_0^\infty P_n^\nu(x,t) \ e^{-x-t/x} x^{\nu+n} dx = {1\over a_n(t)},\eqno(1.20)$$
$$ \int_0^\infty P_n^\nu(x,t) \ e^{-x-t/x} x^{\nu+n+1} dx = - { b_{n+1}(t)\over a_{n+1}(t) a_n(t)},\eqno(1.21)$$
$$ \int_0^\infty P_n^\nu(x,t) P_{n-1}^\nu(x,t) \ e^{-x-t/x} x^{\nu+1} dx = A_n(t),\eqno(1.22)$$
and with (1.8) it yields
$$P_0^\nu(x,t) = [\rho_{\nu+1}(t)]^{-1/2},\quad P_1^\nu(x,t) = \left[ \rho_{\nu+1}(t) (\rho_{\nu+3}(t)\rho_{\nu+1}(t) - \rho^2_{\nu+2}(t))\right]^{-1/2}$$
$$\times \left[ - \rho_{\nu+1}(t) x + \rho_{\nu+2}(t)\right],\eqno(1.23)$$
where $\rho_{\nu+3}(t)\rho_{\nu+1}(t) - \rho^2_{\nu+2}(t) > 0$ for all $t >0$. The latter fact follows immediately from the Ismail integral representation of the quotient of functions $\rho_\nu, \rho_{\nu+1}$ \cite{Ismail}
$${\rho_\nu(x) \over \rho_{\nu+1}(x) } = {1\over \pi^2} \int_0^\infty {y^{-1} dy \over (x+y) \left[ J_{\nu+1}^2 (2\sqrt y)+ Y^2_{\nu+1}(2\sqrt y) \right] }\eqno(1.24)$$
and the Nicholson's integral \cite{Bateman}, Vol. II
$$J_{\nu}^2 (2\sqrt x)+ Y^2_{\nu}(2\sqrt x) = {8\over \pi^2} \int_0^\infty K_0\left( 4\sqrt x\ \sinh (t)\right) \cosh(2\nu t) dt.\eqno(1.25)$$
\section{Differential and differential-difference equations}
We begin this section with the following auxiliary lemma.
{\bf Lemma 1}. {\it Let $\nu \in \mathbb{R},\ t > 0,\ n \in \mathbb{N}$. Then it has the identities}
$$\int_0^\infty \left[P_n^\nu(x,t)\right]^2 \ e^{-x-t/x} x^{\nu-1} dx = {1\over t} \left[ B_n(t)-\nu-1-2n \right],\eqno(2.1)$$
$$\int_0^\infty \left[P_n^\nu(x,t)\right]^2 \ e^{-x-t/x} x^{\nu-2} dx = {1\over t} \left[ 1- {\nu\over t} \left[ B_n(t)-\nu-1-2n \right] \right],\eqno(2.2)$$
$$\int_0^\infty P_n^\nu(x,t) P_{n-1}^\nu(x,t) \ e^{-x-t/x} x^{\nu-1} dx = {1\over t} \left[ A_n(t) + { b_{n}(t)\over a_{n}(t) A_n(t)} \right],\eqno(2.3)$$
$$\int_0^\infty P_n^\nu(x,t) P_{n-1}^\nu(x,t) \ e^{-x-t/x} x^{\nu-2} dx = - {1\over t} \left[ {\nu\over t} \left[ A_n(t) + { b_{n}(t)\over a_{n}(t) A_n(t)} \right] + {n\over A_n(t)} \right].\eqno(2.4)$$
\begin{proof} In fact, employing (1.18), (1.19), (1.20) and integration by parts we derive
$$\int_0^\infty \left[P_n^\nu(x,t)\right]^2 \ e^{-x-t/x} x^{\nu-1} dx = {1\over t} \left[ \int_0^\infty \left[P_n^\nu(x,t)\right]^2 \ e^{-x-t/x} x^{\nu+1} dx\right.$$
$$\left. - (\nu+1) \int_0^\infty \left[P_n^\nu(x,t)\right]^2 \ e^{-x-t/x} x^{\nu} dx - 2 \int_0^\infty P_n^\nu(x,t) {\partial\over \partial x} [P_n^\nu(x,t)] \ e^{-x-t/x} x^{\nu+1} dx\right] $$
$$= {1\over t} \left[ B_n(t)-\nu-1-2n \right],$$
which proves (2.1). Then, analogously, recalling (1.1) and using (2.1), we get
$$\int_0^\infty \left[P_n^\nu(x,t)\right]^2 \ e^{-x-t/x} x^{\nu-2} dx = {1\over t} \left[ \int_0^\infty \left[P_n^\nu(x,t)\right]^2 \ e^{-x-t/x} x^{\nu} dx\right.$$
$$\left. - \nu \int_0^\infty \left[P_n^\nu(x,t)\right]^2 \ e^{-x-t/x} x^{\nu-1} dx - 2 \int_0^\infty P_n^\nu(x,t) {\partial\over \partial x} [P_n^\nu(x,t)] \ e^{-x-t/x} x^{\nu} dx\right]$$
$$ = {1\over t} \left[ 1- {\nu\over t} \left[ B_n(t)-\nu-1-2n \right] \right].$$
Concerning (2.3), we deduce, invoking (1.1), (1.5), (1.20), (1.21), (1.22),
$$\int_0^\infty P_n^\nu(x,t) P_{n-1}^\nu(x,t) \ e^{-x-t/x} x^{\nu-1} dx = {1\over t} \left[ \int_0^\infty P_n^\nu(x,t) P_{n-1}^\nu(x,t) \ e^{-x-t/x} x^{\nu+1} dx \right.$$
$$\left. - (\nu+1) \int_0^\infty P_n^\nu(x,t) \ P_{n-1}^\nu(x,t) e^{-x-t/x} x^{\nu} dx - \int_0^\infty P_n^\nu(x,t) {\partial\over \partial x} [P_{n-1}^\nu(x,t)] \ e^{-x-t/x} x^{\nu+1} dx\right.$$
$$\left. - \int_0^\infty P_{n-1}^\nu(x,t) {\partial\over \partial x} [P_n^\nu(x,t)] \ e^{-x-t/x} x^{\nu+1} dx\right] = {1\over t} \left[ A_n(t) + { b_{n}(t)\over a_{n}(t) A_n(t)} \right].$$
Finally, a similar reasoning gives via (2.3)
$$\int_0^\infty P_n^\nu(x,t) P_{n-1}^\nu(x,t) \ e^{-x-t/x} x^{\nu-2} dx = {1\over t} \left[ \int_0^\infty P_n^\nu(x,t) P_{n-1}^\nu(x,t) \ e^{-x-t/x} x^{\nu} dx \right.$$
$$\left. - \nu \int_0^\infty P_n^\nu(x,t) \ P_{n-1}^\nu(x,t) e^{-x-t/x} x^{\nu-1} dx - \int_0^\infty P_n^\nu(x,t) {\partial\over \partial x} [P_{n-1}^\nu(x,t)] \ e^{-x-t/x} x^{\nu} dx\right.$$
$$\left. - \int_0^\infty P_{n-1}^\nu(x,t) {\partial\over \partial x} [P_n^\nu(x,t)] \ e^{-x-t/x} x^{\nu} dx\right] = - {1\over t} \left[ {\nu\over t} \left[ A_n(t) + { b_{n}(t)\over a_{n}(t) A_n(t)} \right] + {n\over A_n(t)} \right],$$
and we establish (2.4), completing the proof of Lemma 1.
\end{proof}
Now we are ready to prove the following theorem.
{\bf Theorem 1}. {\it Let $n \in \mathbb{N}$. Orthogonal polynomials $P_n^\nu(x,t)$ satisfy the first order linear differential-difference equation }
$$x^2 {\partial\over \partial x} [P_n^\nu(x,t)] = \left[ nx - A_n^2(t) - {b_n(t)\over a_n(t)} \right] P_n^\nu(x,t) + A_n(t) \left[ x+B_n(t)-\nu-1-2n\right] P_{n-1}^\nu(x,t).\eqno(2.5)$$
\begin{proof} Since $ {\partial\over \partial x} [P_n^\nu(x,t)] $ is a polynomial of degree $n-1$, we write it in the form
$${\partial\over \partial x} [P_n^\nu(x,t)] = \sum_{k=0}^{n-1} c_{n,k} (t) P_k^\nu(x,t),\eqno(2.6)$$
where, owing to the orthogonality,
$$c_{n,k} (t)= \int_0^\infty {\partial\over \partial x} [P_n^\nu(x,t)] P_k^\nu(x,t) \ e^{-x-t/x} x^\nu dx.\eqno(2.7)$$
Then, integrating by parts and using the orthogonality relation, we obtain
$$c_{n,k} (t)= - t \int_0^\infty P_n^\nu(x,t) P_k^\nu(x,t) \ e^{-x-t/x} x^{\nu-2} dx - \nu \int_0^\infty P_n^\nu(x,t) P_k^\nu(x,t) \ e^{-x-t/x} x^{\nu-1} dx.\eqno(2.8)$$
Moreover, we observe that
$$ \int_0^\infty P_n^\nu(y,t) \sum_{k=0}^{n-1} P_k^\nu(y,t) P_k^\nu(x,t) \ e^{-y-t/y} y^{\nu} x^{-1} dy = 0,$$
$$ \int_0^\infty P_n^\nu(y,t) \sum_{k=0}^{n-1} P_k^\nu(y,t) P_k^\nu(x,t) \ e^{-y-t/y} y^{\nu} x^{-2} dy = 0.$$
Therefore from (2.6), (2.8) and Christoffel-Darboux formula (1.6) we derive
$${\partial\over \partial x} [P_n^\nu(x,t)] = -t \sum_{k=0}^{n-1} P_k^\nu(x,t) \int_0^\infty P_n^\nu(y,t) P_k^\nu(y,t) \ e^{-y-t/y} y^{\nu} \left[ {1\over y^2} - {1\over x^2} \right] dy$$
$$- \nu \sum_{k=0}^{n-1} P_k^\nu(x,t) \int_0^\infty P_n^\nu(y,t) P_k^\nu(y,t) \ e^{-y-t/y} y^{\nu} \left[ {1\over y} - {1\over x} \right] dy$$
$$= -t A_{n}(t) P_{n}^\nu(x,t) \int_0^\infty P_n^\nu(y,t) P_{n-1}^\nu(y,t) \ e^{-y-t/y} y^{\nu} \left[ {1\over x y^2} + {1\over y x^2} \right] dy$$
$$+ t A_{n}(t) P_{n-1}^\nu(x,t) \int_0^\infty \left[P_{n}^\nu(y,t)\right]^2 \ e^{-y-t/y} y^{\nu} \left[ {1\over x y^2} + {1\over y x^2} \right] dy$$
$$ - {\nu\over x} A_{n}(t) P_{n}^\nu(x,t) \int_0^\infty P_n^\nu(y,t) P_{n-1}^\nu(y,t) \ e^{-y-t/y} y^{\nu-1} dy$$
$$+ {\nu\over x} A_{n}(t) P_{n-1}^\nu(x,t) \int_0^\infty \left[P_{n}^\nu(y,t)\right]^2 \ e^{-y-t/y} y^{\nu-1} dy.\eqno(2.9)$$
Hence by virtue of Lemma 1 equalities (2.9) become
$${1\over A_n(t)} {\partial\over \partial x} [P_n^\nu(x,t)] = {1\over x} P_{n}^\nu(x,t) \left[ \left( {\nu\over t} - {1\over x} \right) \left[ A_n(t) + { b_{n}(t)\over a_{n}(t) A_n(t)} \right] + {n\over A_n(t)} \right]$$
$$ + {1\over x} P_{n-1}^\nu(x,t) \left[ 1- \left( {\nu\over t} - {1\over x}\right) \left[ B_n(t)-\nu-1-2n \right] \right]$$
$$- {\nu \over x t} P_{n}^\nu(x,t) \left[ A_n(t) + { b_{n}(t)\over a_{n}(t) A_n(t)} \right] + {\nu\over x t} P_{n-1}^\nu(x,t) \left[ B_n(t)-\nu-1-2n \right].$$
Hence after simplification we arrive at the differential-difference equation (2.5). Theorem 1 is proved.
\end{proof}
{\bf Corollary 1.} {\it Denoting by $a_{n,0}(t)$ the free term of the polynomial $P_n^\nu(x,t)$, it has the value }
$$a_{n,0}(t)= {1\over a_n(t) \rho_{\nu+1}(t) } \prod_{k=1}^n \frac{B_k(t) -\nu-1-2k} {A^2_k(t)+ {b_k(t)\over a_k(t)}}.\eqno(2.10)$$
\begin{proof} In fact, letting $x=0$ in (2.5), we find the recurrence relation
$$\left[A_n^2(t) + {b_n(t)\over a_n(t)} \right] a_{n,0}(t) = \left[ B_n(t)-\nu-1-2n\right] A_n(t) a_{n-1,0}(t).\eqno(2.11)$$
Hence formula (2.10) comes immediately, solving recurrence (2.11) with the use of (1.5) and (1.23).
\end{proof}
{\bf Remark 1}. In the limit case $t=0$ we use (1.2), (1.5), (1.8) to have $\rho_{\nu+1}(0) = \Gamma(\nu+1)$,
$$a_n(0) = {(-1)^n \over[ n! \Gamma(n+\nu+1)]^{1/2}},\quad b_n(0) = (-1)^{n+1} \left( {n(n+\nu) \over (n-1)! \Gamma(n+\nu)}\right)^{1/2},\eqno(2.12) $$
$$B_n(0)= 2n+\nu+1,\quad A_n(0) = - (n(n+\nu))^{1/2},\quad a_{n,0}(0) = {1\over \Gamma(\nu+1) }\left( { \Gamma(n+\nu+1)\over n!}\right)^{1/2},$$
$$\lim_{t\to 0} \left[A_n^2(t) + {b_n(t)\over a_n(t)} \right] \bigg[ B_n(t)-\nu-1-2n\bigg]^{-1} = -n.\eqno(2.13)$$
Moreover, taking the limit of the indeterminate form under the product sign in (2.10) when $t\to 0$, it implies the identity
$$\prod_{k=1}^n \frac{B^\prime_k(0) } { (-1)^k (k! \Gamma(k+\nu+1))^{1/2} [b^\prime_k(0) + k(k+\nu) a^\prime_k(0)] - 2 ( k (k+\nu))^{1/2} A^\prime_k(0)} = {(-1)^n\over n!}.\eqno(2.14)$$
Now, assuming that polynomial coefficients in (1.4), (1.5) $a_{n,k}(t) \in C^1(\mathbb{R}_+),\ a_{n,n}(t) \equiv a_n(t),\ a_{n,n-1}(t)= b_n(t)$ as functions of $t$, we differentiate equality (1.1) by $t$ to obtain
$$0= {\partial\over \partial t} \int_0^\infty \left[P_n^\nu(x,t)\right]^2 \ e^{-x-t/x} x^{\nu} dx = 2 \int_0^\infty P_n^\nu(x,t) {\partial\over \partial t} \left[P_n^\nu(x,t)\right] \ e^{-x-t/x} x^{\nu} dx $$
$$- \int_0^\infty \left[P_n^\nu(x,t)\right]^2 \ e^{-x-t/x} x^{\nu-1} dx,$$
where the differentiation under the integral sign can be easily motivated by virtue of the absolute and uniform convergence. Then, appealing to (1.20) and (2.1), it implies the equality
$$ {a^\prime_n(t)\over a_n(t)} = {B_n(t)-\nu-1-2n\over 2t}.\eqno(2.15)$$
This first order differential equation with respect to the leading term $a_n(t)$ can be uniquely solved under initial condition (2.12), and we get
$$a_n(t)= {(-1)^n \over \left(n! \Gamma(n+\nu+1)\right)^{1/2}}\ \exp\left({1\over 2} \int_0^t {B_n(y)-\nu-1-2n\over y} dy\right),\eqno(2.16)$$
where the integral under the exponential function exists since, evidently, (see (2.13), (2.15))
$$\lim_{t\to 0} {B_n(t)-\nu-1-2n\over t} = B_n^\prime(0) = 2 {a^\prime_n(0)\over a_n(0)} .\eqno(2.17)$$
Moreover, employing (1.5), we find
$$A_{n+1}(t)= - ((n+1)(n+1+\nu))^{1/2} \ \exp\left({1\over 2} \int_0^t {B_n(y)- B_{n+1}(y) +2\over y} dy\right).\eqno(2.18)$$
Hence after differentiation we obtain
$$A^\prime_{n+1}(0)= {1\over 2} ((n+1)(n+1+\nu))^{1/2} \left[ B^\prime_{n+1}(0)- B^\prime_n(0) \right].\eqno(2.19)$$
On the other hand, via (1.15), (1.23) we have
$$B_0(t)= {\rho_{\nu+2}(t)\over \rho_{\nu+1}(t)},\quad B^\prime_0(t)= {\rho_{\nu+2}(t)\rho_\nu(t)\over \rho^2_{\nu+1}(t)}-1,\quad B^\prime_0(0) = {1\over \nu},\ \nu > 0.\eqno(2.20)$$
Hence it gives the equality
$$B^\prime_{n+1}(0) = {1\over \nu}+ 2 \sum_{k=0}^n {A^\prime_{k+1}(0)\over ((k+1)(k+1+\nu))^{1/2}} .\eqno(2.21)$$
The following theorem shows the first order partial differential-difference equation whose solutions are polynomials $P_n^\nu(x,t)$.
{\bf Theorem 2}. {\it Orthogonal polynomials $P_n^\nu(x,t)$ satisfy the first order partial differential-difference equation }
$$\left( t {\partial\over \partial t} + x {\partial\over \partial x} \right) P_n^\nu(x,t) = \left( t {a_n^\prime(t)\over a_n(t)}+ n \right) P_n^\nu(x,t) + A_n(t) P_{n-1}^\nu(x,t).\eqno(2.22)$$
\begin{proof} Indeed, differentiating both sides of (1.18) with respect to $t$, we have
$$\int_0^\infty {\partial\over \partial t} [ P_n^\nu(x,t) ] \ e^{-x-t/x} x^{\nu+m} dx - \int_0^\infty P_n^\nu(x,t) \ e^{-x-t/x} x^{\nu+m-1} dx = 0,\quad m=0,1,\dots, n-1.\eqno(2.23)$$
Meanwhile, the second integral on the left-hand side of (2.23) can be treated via integration by parts which gives
$$ \int_0^\infty P_n^\nu(x,t) \ e^{-x-t/x} x^{\nu+m-1} dx = {1\over t} \int_0^\infty P_n^\nu(x,t) \ e^{-x-t/x} x^{\nu+m+1} dx$$
$$ - {1\over t} \int_0^\infty {\partial\over \partial x} [ P_n^\nu(x,t) ] \ e^{-x-t/x} x^{\nu+m+1} dx - {\nu+m+1\over t} \int_0^\infty P_n^\nu(x,t) \ e^{-x-t/x} x^{\nu+m} dx,$$
and, combining with (2.23), (1.4) and (1.18), we obtain
$$ \int_0^\infty \left[ \left( t {\partial\over \partial t} + x {\partial\over \partial x} \right) P_n^\nu(x,t) - A_n(t) P_{n-1}^\nu(x,t) \right] \ e^{-x-t/x} x^{\nu+m} dx = 0,\quad m=0,1,\dots, n-1.\eqno(2.24)$$
Hence by unicity we therefore have
$$ \left( t {\partial\over \partial t} + x {\partial\over \partial x} \right) P_n^\nu(x,t) - A_n(t) P_{n-1}^\nu(x,t) = c_n(t) P_n^\nu(x,t).\eqno(2.25)$$
The function $c_n(t)$ is defined, equating coefficients of $x^n$ on both sides in (2.25) to find
$$c_n(t)= t {a^\prime_n(t)\over a_n(t)} + n.$$
This completes the proof of Theorem 2.
\end{proof}
{\bf Corollary 2}. {\it Equation $(2.22)$ can be written in the form}
$$\left( t {\partial\over \partial t} + x {\partial\over \partial x} \right) P_n^\nu(x,t) = {1\over 2} \left[B_n(t)-\nu-1\right] P_n^\nu(x,t) + A_n(t) P_{n-1}^\nu(x,t).$$
\begin{proof} The proof is immediate with the use of (2.15).
\end{proof}
{\bf Corollary 3}. {\it Let $t >0$. The following equalities take place}
$${d\over dt} \left[{b_n(t)\over a_n(t)}\right] = {1\over t} \left[ A_n^2(t) + {b_n(t)\over a_n(t)}\right],\eqno(2.26)$$
$$B_n^\prime(t)= {1\over t} \left[ A_n^2(t) - A_{n+1}^2(t)+ B_n(t)\right],\eqno(2.27)$$
$$a_{n,0}^\prime(t)= {B_n(t)-\nu-1\over 2t} a_{n,0}(t) + {A_n(t)\over t} a_{n-1,0}(t).\eqno(2.28)$$
\begin{proof} The proof is immediate, equating coefficients of $x^{n-1}$ and free terms on both sides of (2.22), and the use of (1.5), (2.15).
\end{proof}
Further, differentiating (1.19) by $t$ and employing (1.1), (1.21), we deduce
$$B_n^\prime(t)= 2 \int_0^\infty P_n^\nu(x,t) {\partial\over \partial t} [ P_n^\nu(x,t) ] \ e^{-x-t/x} x^{\nu+1} dx - 1$$
$$= - 2 {a_n^\prime(t) b_{n+1}(t)\over a_{n+1}(t) a_n(t)} + 2 {b_n^\prime(t) \over a_n(t)} -1,$$
i.e. (see (1.5))
$$ B_n^\prime(t)= 2 B_n(t) {a_n^\prime(t) \over a_n(t)} + 2 {d\over dt} \left[{b_n(t)\over a_n(t)}\right] -1.\eqno(2.29)$$
{\bf Corollary 4}. {\it Let $\nu > 0,\ n \in \mathbb{N}_0$. It has the values }
$$A_n^\prime(0) = 0,\quad B_n^\prime (0)= {1\over \nu},\quad a_n^\prime(0)= {(-1)^n \over 2\nu [n! \Gamma(n+\nu+1)]^{1/2}},\eqno(2.30)$$
$$ b_n^\prime(0)= (-1)^{n+1} {n(n+\nu+2) \over 2\nu [n! \Gamma(n+\nu+1)]^{1/2}},\quad a_{n,0}^\prime(0)= {1\over 2\nu \Gamma(\nu+1) }\left( { \Gamma(n+\nu+1)\over n!}\right)^{1/2}.\eqno(2.31)$$
\begin{proof} In fact, taking a limit in (2.26) when $t\to 0$, we find
$$\lim_{t\to 0} {d\over dt} \left[ A_n^2(t) \right] = 2 A_n(0) A^\prime_n(0) = 0.$$
Hence via (2.13) $A_n^\prime(0) = 0$ and the sum in (2.21) is zero. Thus $B_n^\prime (0)= {1\over \nu}$. The value for $a_n^\prime(0)$ comes from (2.12), (2.17) and $b_n^\prime (0) $ is obtained from (2.29). Finally, $a_{n,0}^\prime(0)$ is a consequence of (2.28) when $t \to 0$.
\end{proof}
Now, assuming that polynomial coefficients are twice continuously differentiable functions of $t$, we solve a simple Cauchy problem for the first order differential equation (2.26) to find
$$ {b_n(t)\over a_n(t)} = 2t \int_0^t {A_n(y) A_n^\prime(y)\over y} dy - A^2_n(t) - {n t\over \nu}.\eqno(2.32)$$
Moreover, via (1.5), (2.27) and integration by parts we derive
$$ B_n(t)= t \int_0^t {\left(A^2_n(y) - A^2_{n+1}(y)\right)^\prime\over y} dy + A^2_{n+1}(t) - A^2_n(t) + { t\over \nu}$$
$$= 2n+\nu+1 + t\int_0^t \left[A^2_n(y) - A^2_{n+1}(y)+2n+\nu+1\right] {dy \over y^2} + { t\over \nu},\eqno(2.33)$$
$$B^\prime_n(t)= \int_0^t {\left(A^2_n(y) - A^2_{n+1}(y)\right)^\prime\over y} dy + {1\over \nu}.\eqno(2.34)$$
{\bf Corollary 5}. {\it Coefficients $A_n$ satisfy the following second kind nonlinear differential-difference equation}
$$ A_n^{\prime\prime}(t) A_n(t) - \left(A_n^{\prime}(t)\right)^2 - {A^2_n(t)\over 2t^2} \bigg[ A_{n-1}^2(t) - 2 A^2_n(t) + A_{n+1}^2(t) -2\bigg] = 0.\eqno(2.35)$$
\begin{proof} Indeed, from (1.5) and (2.18) we get
$${ A_n^\prime(t)\over A_n(t)} = {B_{n-1}(t)- B_{n}(t) +2\over 2t},\quad t >0.\eqno(2.36)$$
Then by virtue of (2.27)
$$ 2 t {d\over dt} \left[ t { A_n^\prime(t)\over A_n(t)} \right] = t {d\over dt} \left[ B_{n-1}(t)- B_{n}(t)\right] = B_{n-1}(t)- B_{n}(t) + A_{n-1}^2(t) - 2 A^2_n(t) + A_{n+1}^2(t)$$
and the result follows via (2.36) and simple differentiation.
\end{proof}
{\bf Corollary 6}. {\it The free term $a_{n,0}(t)$ can be determined by the formula
$$a_{n,0}(t) = (-1)^n (1+\nu)_n\ a_n(t) \exp\left( \int_0^t \left[ n + {A_n^2(y) + {b_n(y)\over a_n(y)}\over B_n(y)-\nu-1-2n } \right] {dy\over y} \right),\eqno(2.37)$$
where $(z)_n$ is the Pochhammer symbol}.
\begin{proof} In fact, invoking (2.11), (2.15), (2.28) and since $ B_n(t)-\nu-1-2n \neq 0,\ t >0$ via (2.1), we write the differential equation
$${ a_{n,0}^\prime(t)\over a_{n,0}(t)} = { a_{n}^\prime(t)\over a_{n}(t)} + {1\over t} \left[ n + {A_n^2(t) + {b_n(t)\over a_n(t)}\over B_n(t)-\nu-1-2n } \right],$$
which can be uniquely solved by formula (2.37), owing to inicial conditions (2.12) and where the integral converges under condition (2.13).
\end{proof}
{\bf Theorem 3}. {\it Orthogonal polynomials $P_n^\nu(x,t)$ obey the second order differential equation}
$$ x^4 \bigg[ x+B_n(t)-\nu-1-2n \bigg] {\partial^2\over \partial x^2} [P_n^\nu(x,t)] - x^2 \bigg[ x^2 + \left[ x+B_n(t)-\nu-1-2n \right] \bigg.$$
$$ \times \left[ (2n-3) x- A_n^2(t) - A_{n-1}^2(t)- {b_n(t)\over a_n(t)}- {b_{n-1}(t)\over a_{n-1} (t)}\right.$$
$$\bigg.\bigg.+ \left[ x+B_{n-1}(t)-\nu+1-2n\right] [x- B_{n-1}(t)]\bigg]\bigg] {\partial\over \partial x} [ P_n^\nu(x,t)] $$
$$+ \bigg[ \left[ x+B_n(t)-\nu-1-2n\right] \bigg[ A^2_n(t) \left[ x+B_n(t)-\nu-1-2n\right] \left[ x+B_{n-1}(t)-\nu+1-2n\right]\bigg.\bigg.$$
$$\bigg. + \left[nx- A_n^2(t) - {b_n(t)\over a_n(t)} \right] \bigg[ (n-1)x- A_{n-1}^2(t) - {b_{n-1}(t)\over a_{n-1}(t)} + \left[ x+B_{n-1}(t)-\nu+1-2n\right] [x- B_{n-1}(t)] \bigg] \bigg]$$
$$\bigg. \bigg. - x^2 \bigg[ n \left[B_n(t)-\nu-2n -1\right]+ A_n^2(t) + {b_n(t)\over a_n(t)} \bigg] \bigg]P_n^\nu(x,t) = 0.\eqno(2.38)$$
\begin{proof} Differentiating both sides of (2.5) with respect to $x$, we have
$$x^2 {\partial^2\over \partial x^2} [P_n^\nu(x,t)] = n P_n^\nu(x,t) + \left[ (n-2) x - A_n^2(t) - {b_n(t)\over a_n(t)} \right] {\partial\over \partial x} [ P_n^\nu(x,t)] $$
$$ + A_n(t) P_{n-1}^\nu(x,t)+ A_n(t) \left[ x+B_n(t)-\nu-1-2n\right] {\partial\over \partial x} [ P_{n-1}^\nu(x,t)].$$
But, owing to (1.4) and (2.5), the latter equality becomes
$$x^4 {\partial^2\over \partial x^2} [P_n^\nu(x,t)] = x^2\left[ (n-2) x - A_n^2(t) - {b_n(t)\over a_n(t)} \right] {\partial\over \partial x} [ P_n^\nu(x,t)] $$
$$+ \left[ n x^2 - A^2_n(t) \left[ x+B_n(t)-\nu-1-2n\right] \left[ x+B_{n-1}(t)-\nu+1-2n\right] \right] P_{n}^\nu(x,t) $$
$$ + \left[ {x^2\over x+B_n(t)-\nu-1-2n} + (n-1)x - A_{n-1}^2(t) - {b_{n-1}(t)\over a_{n-1}(t)} \right.$$
$$\bigg. + \left[ x+B_{n-1}(t)-\nu+1-2n\right] [x- B_{n-1}(t)] \bigg] A_n(t) \left[ x+B_n(t)-\nu-1-2n\right] P_{n-1}^\nu(x,t).$$
Finally, recalling (2.5) to express $P_{n-1}^\nu(x,t)$, we end up with the equation (2.37), completing the proof of Theorem 3.
\end{proof}
\section{Explicit representations. Recurrence relations for coefficients}
In this section we will deduce recurrence relations for coefficients of orthogonal polynomials $P_n^\nu(x,t)$ and their explicit representations. In fact, we have
{\bf Theorem 4}. {\it Let $n \in \mathbb{N}_0,\ t >0$. Then the following identities hold}
$$A_{n+1}^2(t)+ B_n^2(t)+ A_n^2(t)- (2n+\nu+2)B_n(t) + 2{b_n(t)\over a_n(t)} - t=0,\eqno(3.1)$$
$$A^2_n(t) \left[B_n(t)-\nu-1-2n\right] \left[ B_{n-1}(t)-\nu+1-2n\right] - \left[ A_n^2(t) + {b_n(t)\over a_n(t)} \right] ^2+ t \bigg[ A_n^2(t) + {b_n(t)\over a_n(t)} \bigg] = 0,\eqno(3.2)$$
$$t \bigg[ B_{n-1}(t) + B_n(t) \bigg]^\prime + \bigg[ B_{n-1}(t) + B_n(t) \bigg] \bigg[ B_{n-1}(t) - B_n(t) +1\bigg] - (2n+\nu) \bigg[ B_{n-1}(t) - B_n(t) \bigg] =0.\eqno(3.3)$$
\begin{proof} In order to prove (3.1), we appeal to the three term recurrence relation (1.4) and the orthogonality condition (1.1). Hence, integrating by parts, we derive
$$A_{n+1}^2(t)+ B_n^2(t)+ A_n^2(t)= \int_0^\infty [x P_n^\nu(x,t) ]^2 \ e^{-x-t/x} x^{\nu} dx= (\nu+2) \int_0^\infty [ P_n^\nu(x,t) ]^2 \ e^{-x-t/x} x^{\nu+1} dx $$
$$+ t \int_0^\infty [P_n^\nu(x,t) ]^2 \ e^{-x-t/x} x^{\nu} dx + 2 \int_0^\infty P_n^\nu(x,t) {d\over dx} [ P_n^\nu(x,t) ] \ e^{-x-t/x} x^{\nu+2} dx $$
$$= (\nu+2) B_n(t) + t + 2n B_n(t) + 2 A_n(t) \int_0^\infty x P_{n-1}^\nu(x,t) {d\over dx} [ P_n^\nu(x,t) ] \ e^{-x-t/x} x^{\nu} dx $$
$$t+ (2n+\nu+2) B_{n}(t) - 2n\ A_{n}(t) { b_n(t)\over a_{n-1} (t)} + 2 (n-1) A_n(t)\ {b_n(t)\over a_{n-1}(t)} $$
$$= t+ (2n+\nu+2) B_{n}(t) - 2 {b_n(t)\over a_{n}(t)}.$$
This gives (3.1). On the other hand, writing (3.1) for the index $n-1$ in the form
$$ A_n^2(t) + {b_n(t)\over a_n(t)} -t = B_{n-1}(t) \left[ 2n+\nu-1 - B_{n-1}(t)\right] - A_{n-1}^2(t) - {b_{n-1}(t)\over a_{n-1}(t)},\eqno(3.4)$$
we let $x= 0$ in (2.38) to find the equality
$$A^2_n(t) \left[ B_n(t)-\nu-1-2n\right] \left[ B_{n-1}(t)-\nu+1-2n\right]\bigg.\bigg.$$
$$\bigg. + \left[ A_n^2(t) + {b_n(t)\over a_n(t)} \right] \bigg[ A_{n-1}^2(t) + {b_{n-1}(t)\over a_{n-1}(t)} +\left[ B_{n-1}(t)-2n-\nu+1\right] B_{n-1}(t) \bigg] = 0.\eqno(3.5)$$
Hence a simple comparison leads to (3.2). Finally, writing (3.1) for $n-1$ and subtracting one equality from another, we use (2.27) to establish (3.3).
\end{proof}
{\bf Corollary 7}. {\it Differential equation $(2.38)$ reduces to the equality }
$$ x^2 \left( x+B_n(t)-\nu-1-2n \right) {\partial^2\over \partial x^2} [P_n^\nu(x,t)] $$
$$ - \bigg[x^3 + \bigg[ B_n(t)- 2(\nu+n+1) \bigg] x^2 - \bigg[ t+ \left(B_n(t)-\nu-1-2n\right)(\nu+2) \bigg] x - t \left(B_n(t)-\nu-1-2n\right)\bigg] {\partial\over \partial x} [ P_n^\nu(x,t)] $$
$$+ \bigg[ n x^2 - \left( {b_n(t)\over a_n(t)} - n \left( B_n(t)- 2\nu-3n-1\right) \right) x + \left( B_n(t)-\nu-2n-1\right) \bigg[ A^2_n(t) - n (n+\nu+1) \bigg] \bigg.$$
$$\bigg. + \left[ A_n^2(t) +{b_n(t)\over a_n(t)} \right] \left(2n+\nu - B_n(t)\right) \bigg] P_n^\nu(x,t) = 0.\eqno(3.6)$$
\begin{proof} In fact, appealing to (3.4), (3.5), we get from (2.38)
$$ x^3 \bigg[ x+B_n(t)-\nu-1-2n \bigg] {\partial^2\over \partial x^2} [P_n^\nu(x,t)] - x \bigg[ x^2 + \left[ x+B_n(t)-\nu-1-2n \right] \bigg.$$
$$ \times \left[ (2n-3) x- A_n^2(t) - A_{n-1}^2(t)- {b_n(t)\over a_n(t)}- {b_{n-1}(t)\over a_{n-1} (t)}\right.$$
$$\bigg.\bigg.+ \left[ x+B_{n-1}(t)-\nu+1-2n\right] [x- B_{n-1}(t)]\bigg]\bigg] {\partial\over \partial x} [ P_n^\nu(x,t)] $$
$$+ \bigg[ \left[ x+B_n(t)-\nu-1-2n\right] \bigg[ A^2_n(t) \left[ B_n(t)+B_{n-1}(t)+x -2 \nu- 4n\right] + n( x^2 - (n+\nu) x -t ) \bigg] $$
$$ + \left[ A_n^2(t) +{b_n(t)\over a_n(t)} \right] \bigg[ \left[ B_n(t)-\nu-1-2n\right] (2n+\nu-x) + x(2n+\nu-1 - x) \bigg] $$
$$\bigg. - n x \left[B_n(t)-\nu-2n -1\right] \bigg]P_n^\nu(x,t) = 0.\eqno(3.7)$$
Hence, letting $x=0$ in (3.7), we find the identity
$$A^2_n(t) \left[ B_n(t)+B_{n-1}(t) -2 \nu- 4n\right] + (2n+\nu) \left[ A_n^2(t) +{b_n(t)\over a_n(t)} \right] -nt =0.\eqno(3.8)$$
Consequently, (3.7) becomes
$$ x^3 \bigg[ x+B_n(t)-\nu-1-2n \bigg] {\partial^2\over \partial x^2} [P_n^\nu(x,t)] - x\bigg[ x^2 + \left[ x+B_n(t)-\nu-1-2n \right] \bigg.$$
$$ \times \left[ (2n-3) x- A_n^2(t) - A_{n-1}^2(t)- {b_n(t)\over a_n(t)}- {b_{n-1}(t)\over a_{n-1} (t)}\right.$$
$$\bigg.\bigg.+ \left[ x+B_{n-1}(t)-\nu+1-2n\right] [x- B_{n-1}(t)]\bigg]\bigg] {\partial\over \partial x} [ P_n^\nu(x,t)] $$
$$+ x \bigg[ x \left( n( x - n-\nu ) - {b_n(t)\over a_n(t)} \right) + \left( B_n(t)-\nu-2n-1\right) \left( A^2_n(t) + n (x-n-\nu-1) \right) \bigg.$$
$$\bigg. + \left[ A_n^2(t) +{b_n(t)\over a_n(t)} \right] \left(2n+\nu - B_n(t)\right) \bigg] P_n^\nu(x,t) = 0.$$
Dividing by $x$ and appealing to (3.4), we arrive at (3.6).
\end{proof}
{\bf Remark 2.} For the limit case $t=0$ we employ (1.5), (2.12), (2.13) to reduce (3.6) to the classical differential equation for Laguerre polynomials.
The following theorem gives the integro-differential-difference equation for orthogonal polynomials $P_n^\nu(x,t)$. Precisely, it has
{\bf Theorem 5.} {\it Orthogonal polynomials $P_n^\nu(x,t)$ satisfy the integral-difference equation of the form}
$$ P_n^\nu(x,t) = {P_{n-1}^\nu(x,t)\over a_{n-1,0}(t)} - a_n(t) \exp\left( {b_n(t)\over x\ a_n(t)} \right) \int_0^t \exp\left( - {b_n(y)\over x\ a_n(y)} \right) {d\over dy} \left[ {P_{n-1}^\nu(x,y)\over a_{n-1,0}(y)\ a_n(y)} \right]dy $$
$$+ (-1)^n (n-1)! \ a_n(t) \exp\left( {1\over x}\left[ {b_n(t)\over a_n(t)} +n(n+\nu)\right]\right) \left[n\ L_n^\nu (x) - \Gamma(1+\nu) \left( {n! (n+\nu) \over \Gamma(n+\nu)}\right)^{1/2} L_{n-1}^\nu (x)\right].\eqno(3.9)$$
\begin{proof} Recalling the differential-difference equations (2.5), (2.28), the recurrence relation (2.11) and Corollary 2, we rewrite equation (2.22) in the form
$$ {\partial\over \partial t} \left[ {P_n^\nu(x,t) \over a_{n,0}(t)} \right] + {1\over t} \left[ n + {A_n^2(t) + {b_n(t)\over a_n(t)}\over B_n(t)-\nu-1-2n } - {1\over x} \left[ A_n^2(t) + {b_n(t)\over a_n(t)} \right] \right]{P_n^\nu(x,t) \over a_{n,0}(t)} $$
$$ = - {1 \over xt} \left[ A_n^2(t) + {b_n(t)\over a_n(t)} \right] {P_{n-1}^\nu(x,t)\over a_{n-1,0}(t)},\quad x \neq 0.\eqno(3.10)$$
Then by virtue of (2.26), (2.37) it implies
$$ {\partial\over \partial t} \left[ {P_n^\nu(x,t) \over a_{n,0}(t)} \right] + \left[ {a^\prime_{n,0}(t)\over a_{n,0}(t)} - {a^\prime_{n}(t)\over a_{n}(t)} - {1\over x} {d\over dt} \left[{b_n(t)\over a_n(t)}\right] \right] {P_n^\nu(x,t) \over a_{n,0}(t)} = - {1 \over x} {d\over dt} \left[{b_n(t)\over a_n(t)}\right] {P_{n-1}^\nu(x,t)\over a_{n-1,0}(t)}.$$
Solving this first order differential equation in terms of $P_n^\nu$ under initial condition (1.2) and taking into account (2.12), we find
$$ P_n^\nu(x,t) = -{ a_n(t)\over x} \exp\left( {b_n(t)\over x\ a_n(t)} \right) \int_0^t \exp\left( - {b_n(y)\over x\ a_n(y)} \right) {d\over dy} \left[{b_n(y)\over a_n(y)} \right] {P_{n-1}^\nu(x,y)\over a_{n-1,0}(y)\ a_n(y)} dy $$
$$+ (-1)^n n! \ a_n(t) \exp\left( {1\over x}\left[ {b_n(t)\over a_n(t)} +n(n+\nu)\right]\right) L_n^\nu (x).\eqno(3.11)$$
Finally, integrating by parts in (3.11), we end up with (3.9). Theorem 5 is proved.
\end{proof}
Concerning recurrence relations for the coefficients of the polynomials $P_n^\nu$, we have the following result.
{\bf Theorem 6}. {\it For the orthogonal polynomial $P_n^\nu(x,t) = \sum_{k=0}^n a_{n,k}(t) x^k,\ a_{n,n}(t) \equiv a_n(t),\ a_{n,n-1}(t)= b_n(t)$ its coefficients fullfil the differential-recurrence relations}
$$ a_{n,k}(t)= a_{n,0}(t) \sum_{m=k}^{n} { a_{m} (t) \left[ a_{m}(t) a^\prime_{m,k-1}(t) - a_{m,k-1}(t) a_{m}^\prime(t) \right] \over \left[ a_{m}(t) b_{m}^\prime(t) - b_{m}(t) a_{m}^\prime(t) \right] a_{m,0}(t)},\quad k=1,\dots, n.\eqno(3.12)$$
\begin{proof} By virtue of (2.26) and taking into account that $A_n^2(t) + {b_n(t)\over a_n(t)} \neq 0,\ t > 0$ (see (2.1), (3.2)), we write equality (3.10) in the form
$$ { a^2_n(t) \over \left[ a_n(t) b_n^\prime(t) - b_n(t) a_n^\prime(t) \right] a_{n,0}(t) } \left[ {\partial\over \partial t} \left[ P_n^\nu(x,t) \right] - {a_n^\prime(t)\over a_{n}(t)} P_n^\nu(x,t) \right]= {1 \over x} \left[ {P_n^\nu(x,t)\over a_{n,0}(t)} - {P_{n-1}^\nu(x,t)\over a_{n-1,0}(t)}\right].$$
Hence it yields
$$ \sum_{m=1}^n { a_m (t) \over \left[ a_m(t) b_m^\prime(t) - b_m(t) a_m^\prime(t) \right] a_{m,0}(t)} \sum_{k=1}^{m} \left[ a_m(t) a^\prime_{m,k-1}(t) - a_{m,k-1}(t) a_m^\prime(t) \right] x^{k} = {P_n^\nu(x,t)\over a_{n,0}(t)} - 1.$$
Therefore changing the order of summation on the left-hand side of the latter equality, we arrive at the representation
$$ P_n^\nu(x,t) = a_{n,0}(t) + \sum_{k=1}^{n} x^k \sum_{m=k}^{n} { a_{m} (t) \left[ a_{m}(t) a^\prime_{m,k-1}(t) - a_{m,k-1}(t) a_{m}^\prime(t) \right] a_{n,0}(t) \over \left[ a_{m}(t) b_{m}^\prime(t) - b_{m}(t) a_{m}^\prime(t) \right] a_{m,0}(t)},$$
which leads to (3.12).
\end{proof}
The three term recurrence relation (1.4) can be written in another form. Indeed, it has
{\bf Theorem 7.} {\it Orthogonal polynomials $P_n^\nu$ satisfy the following recurrence relation}
$$ \bigg[ A_{n+1}^2(t) + {b_{n+1}(t)\over a_{n+1}(t)} + {x\over 2} \left[B_{n}(t)-\nu-1-2n\right] \bigg] \bigg[ A_{n+1} (t) P_{n+1}^\nu(x,t)+ A_n(t) P_{n-1}^\nu(x,t)\bigg] $$
$$+ \bigg[x t B^\prime_n(t)+ A^2_{n} (t) \left[B_{n-1}(t)-\nu+1-2n\right] - A^2_{n+1}(t) \left[ B_{n+1}(t)-\nu-3-2n\right] \bigg.$$
$$\bigg. - \left(x- B_n(t) \right) \left[ A_n^2(t) + {b_n(t)\over a_n(t)} + {x\over 2} \left[B_n(t)-\nu-1-2n\right] \right]\bigg] P_{n}^\nu(x,t)= 0.\eqno(3.13)$$
\begin{proof} Differentiating the three term recurrence relation (1.4), we have
$$x {\partial\over \partial t} [P_n^\nu(x,t)] = A^\prime_{n+1} (t) P_{n+1}^\nu(x,t) +B^\prime_n(t) P_n^\nu(x,t) + A^\prime_{n}(t) P_{n-1}^\nu(x,t)$$
$$+ A_{n+1} (t) {\partial\over \partial t} [P_{n+1}^\nu(x,t)] +B_n(t) {\partial\over \partial t} [P_n^\nu(x,t)] + A_{n}(t) {\partial\over \partial t} [P_{n-1}^\nu(x,t)].$$
Then, using (1.4), (2.5) and Corollary 2, after straightforward simplifications the latter equality becomes
$$ \bigg[xt A^\prime_{n+1} (t)+ A_{n +1} (t) \left[ A_{n+1}^2(t) + {b_{n+1}(t)\over a_{n+1}(t)} + {x\over 2} \left[B_{n+1}(t)-\nu-3-2n\right] \right] \bigg]P_{n+1}^\nu(x,t) $$
$$+ \bigg[ xt B^\prime_n(t)+ A^2_{n} (t) \left[ B_{n-1}(t)-\nu+1-2n\right] - A^2_{n+1}(t) \left[ B_{n+1}(t)-\nu-3-2n\right] \bigg.$$
$$\bigg. - \left(x- B_n(t) \right) \left[ A_n^2(t) + {b_n(t)\over a_n(t)} + {x\over 2} \left[B_n(t)-\nu-1-2n\right] \right]\bigg] P_{n}^\nu(x,t)$$
$$\bigg. + \bigg[ xt A^\prime_{n}(t)+ A_{n}(t) \left[ A_{n-1}^2(t) + {b_{n-1}(t)\over a_{n-1}(t)} + \left[B_{n-1}(t) - {x\over 2} \right]\left[ B_{n-1}(t)-\nu+1-2n\right] \right]\bigg.$$
$$\bigg. - \left(B_n(t) - x\right) A_n(t) \left[ B_n(t)-\nu-1-2n\right] \bigg]P_{n-1}^\nu(x,t) = 0.$$
It can be rewritten, appealing to (2.36), and we obtain
$$\bigg[ A_{n+1}^2(t) + {b_{n+1}(t)\over a_{n+1}(t)} + {x\over 2} \left[B_{n}(t)-\nu-1-2n\right] \bigg] A_{n+1} (t) P_{n+1}^\nu(x,t) $$
$$+ \bigg[xB_n(t)+ A^2_{n} (t) \left[x+ B_{n-1}(t)-\nu+1-2n\right] - A^2_{n+1}(t) \left[ x+B_{n+1}(t)-\nu-3-2n\right] \bigg.$$
$$\bigg. - \left(x- B_n(t) \right) \left[ A_n^2(t) + {b_n(t)\over a_n(t)} + {x\over 2} \left[B_n(t)-\nu-1-2n\right] \right]\bigg] P_{n}^\nu(x,t)$$
$$ + \bigg[ A_{n-1}^2(t) + {b_{n-1}(t)\over a_{n-1}(t)} + B_{n-1}(t) \left[ B_{n-1}(t)-\nu+1-2n\right] \bigg.$$
$$\bigg. - \left(B_n(t) - {x\over 2} \right) \left[ B_n(t)-\nu-1-2n\right] \bigg] A_n(t) P_{n-1}^\nu(x,t) = 0.\eqno(3.14)$$
Hence, taking into account (2.27), (3.4), equality (3.14) leads to the final form (3.15).
\end{proof}
{\bf Corollary 8}. {\it The following identity takes place}
$$t B_n(t) B^\prime_n(t)+ A^2_{n} (t) \left[ B_{n-1}(t)-\nu+1-2n\right] - A^2_{n+1}(t) \left[ B_{n+1}(t)-\nu-3-2n\right] = 0.\eqno(3.16)$$
\begin{proof} The proof is immediate, letting $x=0$ in (3.14) and recalling (2.27), (3.4).
\end{proof}
{\bf Corollary 9}. {\it Coefficients $B_n$ obey the following integral-recurrence relation}
$$ B_n(t)+ B_{n-1}(t) - 2n-\nu +1 = \exp\left( \int_0^t {B_{n-1}(y)-B_{n}(y)+2\over y} dy\right) $$
$$\times \left( 2\int_0^t \exp\left( - \int_0^y {B_{n-1}(u)-B_{n}(u)+2\over u} du\right) \left[ B_{n-1}(y) - 2n-\nu +1\right] {dy\over y}+ 2n+\nu+1\right).\eqno(3.17)$$
\begin{proof} In fact, writing (3.3) in the form
$${d\over dt} \bigg[ B_n(t)+ B_{n-1}(t) - 2n-\nu +1\bigg] + {1\over t} \left[ B_{n-1}(t)-B_{n}(t)+2\right] \left[ B_{n-1}(t)+ B_{n}(t) - 2n-\nu +1\right] $$
$$- {2\over t} \left[ B_{n-1}(t) - 2n-\nu +1\right] = 0,$$
we solve a simple Cauchy problem for the first order differential equation to obtain (3.17).
\end{proof}
In the sequel, let us consider polynomial coefficients (3.12) $a_{n,k}(t)$ as functions of $\nu$ as well, i.e. $a_{n,k}\equiv a_{n,k}^\nu$. Then, returning to the formula (1.8) for the moments, we represent orthogonal polynomials $P_n^\nu$ in terms of the Hankel determinant
$$ P^\nu_{n} (x,t)= {1\over [G^\nu_{n-1} (t)G^\nu_n (t)]^{1/2} } \begin{vmatrix}
\rho_{\nu+1}(t) & \rho_{\nu+2}(t)& \dots& \dots& \rho_{\nu+n+1}(t) \\
\rho_{\nu+2}(t) & \dots & \dots& \dots& \rho_{\nu+n+2}(t) \\
\dots& \dots & \dots& \dots& \dots \\
\vdots & \ddots & \ddots & \ddots& \vdots\\
\rho_{\nu+n}(t) & \dots& \dots& \dots& \rho_{\nu+2n}(t)\\
1 & x & \dots& \dots& x^n\\
\end{vmatrix},\eqno(3.18)$$
where
$$G^\nu_{n}(t) = \begin{vmatrix}
\rho_{\nu+1}(t) & \rho_{\nu+2}(t)& \dots& \dots& \rho_{\nu+n+1}(t) \\
\rho_{\nu+2}(t) & \dots & \dots& \dots& \rho_{\nu+n+2}(t) \\
\dots& \dots & \dots& \dots& \dots \\
\vdots & \ddots & \ddots & \ddots& \vdots\\
\rho_{\nu+n+1}(t) & \dots& \dots& \dots& \rho_{\nu+2n+1}(t)\\
\end{vmatrix}.\eqno(3.19)$$
Hence, employing the Laplace theorem to the last row, we find from (3.18), (3.19)
$$G^\nu_{n}(t) = (-1)^n \rho_{\nu+n+1}(t) G^{\nu+1}_{n-1}(t) + \rho_{\nu+2n+1}(t) G^{\nu}_{n-1}(t)+ (-1)^{n+1} \sum_{k=2}^{n} (-1)^k \rho_{\nu+n+k}(t) G^\nu_{n,k}(t)$$
$$= [G^\nu_{n-1} (t)G^\nu_n (t)]^{1/2} \sum_{k=0}^{n} \rho_{\nu+n+k+1}(t) a^\nu_{n,k}(t),\eqno(3.20)$$
where
$$G^\nu_{n,k}(t) = \begin{vmatrix}
\rho_{\nu+1}(t) & \rho_{\nu+2}(t)& \dots& \rho_{\nu+k-1}(t)& \rho_{\nu+k+1}(t) & \dots& \rho_{\nu+n+1}(t) \\
\rho_{\nu+2}(t) & \rho_{\nu+3}(t) & \dots& \rho_{\nu+k}(t)& \rho_{\nu+k+2}(t) & \dots& \rho_{\nu+n+2}(t) \\
\dots& \dots & \dots& \dots& \dots &\dots &\dots \\
\vdots & \vdots & \vdots & \ddots& \vdots &\vdots &\vdots\\
\rho_{\nu+n}(t) & \dots& \dots& \rho_{\nu+n+k-2}(t)& \rho_{\nu+n+k}(t) & \dots& \rho_{\nu+2n}(t)\\
\end{vmatrix}.\eqno(3.21)$$
In particular, we easily find from (3.18), (3.20) the expression for the free coefficient $a_{n,0}^\nu(t)$ in terms of the determinant by the formula
$$ a_{n,0}^\nu(t) = {(-1)^n G^\nu_{n,1}(t) \over \left[G^\nu_{n-1} (t)G^\nu_n (t)\right]^{1/2} }.\eqno(3.22)$$
Moreover, since the leading term $a_n^\nu$ (cf. \cite{Bateman}, Vol. II) has the representation
$$ \left|a_n^\nu(t) \right| = \bigg[ {G^{\nu}_{n-1}(t) \over G^{\nu}_{n}(t)}\bigg]^{1/2},\eqno(3.23)$$
we find from (3.22), (3.23) the equality
$$ {G^{\nu+1}_{n-1}(t) \over G^{\nu}_{n}(t)} = (-1)^n a_n^\nu(t)\ a_{n,0}^\nu(t).\eqno(3.24)$$
Therefore it yields
$$ (- 1)^n a_n^\nu(t)\ a_{n,0}^\nu(t) = {G^{\nu+1}_{n-1}(t) \over G^{\nu+1}_{n}(t)} {G^{\nu+1}_{n}(t) \over G^{\nu}_{n}(t)} $$
$$= (- 1)^{n+1} \left[ a_n^{\nu+1}(t)\right]^2 \ a_{n+1,0}^\nu(t) a_{n+1}^\nu(t) {G^{\nu}_{n+1}(t) \over G^{\nu}_{n}(t)}$$
$$= (- 1)^{n+1} \left[ {a_n^{\nu+1}(t)\over a_{n+1}^\nu(t)}\right]^2 \ a_{n+1,0}^\nu(t) a_{n+1}^\nu(t).$$
Hence, recalling (2.11), we derive the identities
$$ \left[ {a_n^{\nu+1}(t)\over a_{n+1}^\nu(t)}\right]^2 = - A_{n+1}^\nu(t) {a_{n,0}^\nu(t) \over a_{n+1,0}^\nu(t)} = \frac {\left[A^\nu_{n+1}(t)\right]^2+ {b^\nu_{n+1}(t)\over a^\nu_{n+1}(t)}}{2n+\nu+3- B^\nu_{n+1}(t) },$$
$$ \left[ a_n^{\nu+1}(t)\right]^2 = \frac {\left[a^\nu_{n}(t)\right]^2+ a^\nu_{n+1}(t) b^\nu_{n+1}(t)}{2n+\nu+3- B^\nu_{n+1}(t) },\eqno(3.25)$$
$$\prod_{k=0}^n \left[ a_k^{\nu+1}(t)\right]^2 = {(-1)^{n+1} a_{n+1}^\nu(t)\over a_{n+1,0}^\nu(t)} \prod_{k=0}^n \left[ a_k^{\nu}(t)\right]^2.\eqno(3.26)$$
Moreover, (3.20), (3.23) imply
$$ \sum_{k=0}^{n} \rho_{\nu+n+k+1}(t) a^\nu_{n,k}(t) = {1\over a_n^\nu(t)},\eqno(3.27)$$
and this agrees with (1.20).
{\bf Corollary 10.} {\it Coefficients $B^\nu_n(t), A^\nu_{n+1}(t)$ $(1.5)$ of the three term recurrence relation $(1.4)$ are equal, correspondingly,}
$$ B^\nu_n(t)= 2n+\nu+1 + t \bigg[ {{G^{\nu}}^\prime_{n-1}(t) \over G^{\nu}_{n-1}(t) } - {{G^{\nu}}^\prime_{n}(t) \over G^{\nu}_{n}(t)}\bigg],\eqno(3.28)$$
$$A_{n+1}^\nu(t)= {\left[G^{\nu}_{n-1}(t) G^{\nu}_{n+1}(t) \right]^{1/2}\over G^{\nu}_{n}(t)}.\eqno(3.29)$$
\begin{proof} Indeed, appealing to (2.16) and (3.23), we have
$$ {G^{\nu}_{n-1}(t) \over G^{\nu}_{n}(t)} = {1 \over n! \Gamma(n+\nu+1)}\ \exp\left(\int_0^t {B_n(y)-\nu-1-2n\over y} dy\right).\eqno(3.30)$$
Therefore,
$$G^{\nu}_{n}(t) = n! \Gamma(n+\nu+1)\ G^{\nu}_{n-1}(t) \exp\left(\int_0^t {\nu+1+2n - B_n(y)\over y} dy\right),$$
and, solving this recurrence relation owing to (1.5) and (3.19), we easily derive
$$G^{\nu}_{n}(t) = \prod_{k=0}^n k! \Gamma(k+\nu+1) \exp\left(\int_0^t \bigg[ {b_{n+1}^\nu(y)\over a_{n+1}^\nu(y)} + (n+1)(n+\nu+1) \bigg] {dy \over y} \right).$$
Consequently,
$${G^{\nu}}^\prime_{n}(t) \equiv {dG^{\nu}_{n}\over dt} = {G^{\nu}_{n}(t) \over t} \bigg[ {b_{n+1}^\nu(t)\over a_{n+1}^\nu(t)} + (n+1)(n+\nu+1) \bigg].\eqno(3.31)$$
Hence equality (3.28) follows immediately from (1.5). Formula (3.29) is a direct consequence of (1.5), (3.23) or (2.18), (3.30).
\end{proof}
{\bf Corollary 11.} {\it The following equality holds}
$$t^2 {d\over dt}\bigg[ {{G^{\nu}}^\prime_{n}\over G^{\nu}_{n}}\bigg]= A_{n+1}^2 - (n+1)(n+1+\nu).\eqno(3.32)$$
\begin{proof} The proof is immediate, involving differentiation in (3.31) and employing (2.26).
\end{proof}
On the other hand, the key identity (1.16) for the moments and properties of the determinants allow to treat (3.19) when $n = 2,3,\dots,$ as follows
$$ t G^\nu_{n}(t) = \begin{vmatrix}
t \rho_{\nu+1}(t) & \rho_{\nu+2}(t)& \dots& \dots& \rho_{\nu+n+1}(t) \\
t \rho_{\nu+2}(t) & \dots & \dots& \dots& \rho_{\nu+n+2}(t) \\
\dots& \dots & \dots& \dots& \dots \\
\vdots & \ddots & \ddots & \ddots& \vdots\\
t \rho_{\nu+n+1}(t) & \dots& \dots& \dots& \rho_{\nu+2n+1}(t)\\
\end{vmatrix}$$
$$ = - \begin{vmatrix}
(\nu+2) \rho_{\nu+2}(t) & \rho_{\nu+2}(t)& \dots& \dots& \rho_{\nu+n+1}(t) \\
(\nu+3) \rho_{\nu+3}(t) & \rho_{\nu+3}(t) & \dots& \dots& \rho_{\nu+n+2}(t) \\
\dots& \dots & \dots& \dots& \dots \\
\vdots & \vdots & \ddots & \ddots& \vdots\\
(\nu+n+2)\rho_{\nu+n+2}(t) & \rho_{\nu+n+2}(t)& \dots& \dots& \rho_{\nu+2n+1}(t)\\
\end{vmatrix}$$
$$ = - \begin{vmatrix}
0 & \rho_{\nu+2}(t)& \dots& \dots& \rho_{\nu+n+1}(t) \\
\rho_{\nu+3}(t) & \rho_{\nu+3}(t) & \dots& \dots& \rho_{\nu+n+2}(t) \\
2 \rho_{\nu+4}(t) & \rho_{\nu+4}(t) & \dots& \dots& \dots \\
\vdots & \vdots & \ddots & \ddots& \vdots\\
n \rho_{\nu+n+2}(t) & \rho_{\nu+n+2}(t) & \dots& \dots& \rho_{\nu+2n+1}(t)\\
\end{vmatrix}.$$
Continuing this process, we find via Laplace's theorem and (1.16)
$$(-1)^{n+1} t^{n-1} G^\nu_n(t) = \begin{vmatrix}
0 & 0& \dots& 0& \rho_{\nu+n}(t)& \rho_{\nu+n+1}(t) \\
\rho_{\nu+3}(t) & \rho_{\nu+4}(t) & \dots& \rho_{\nu+n+1}(t)& \rho_{\nu+n+1}(t) & \rho_{\nu+n+2}(t)\\
2 \rho_{\nu+4}(t) & 2 \rho_{\nu+5}(t) & \dots& 2 \rho_{\nu+n+2}(t)& \rho_{\nu+n+2}(t)& \rho_{\nu+n+3}(t) \\
\vdots & \vdots & \ddots & \ddots& \vdots & \vdots\\
n \rho_{\nu+n+2}(t) & n\rho_{\nu+n+3}(t) & \dots& n \rho_{\nu+2n}(t)& \rho_{\nu+2n}(t)& \rho_{\nu+2n+1}(t)\\
\end{vmatrix}.\eqno(3.33)$$
On the other hand, using continuously (1.16) from the last column in (3.19), we arrive in the same manner at the equality
$$ G^\nu_n(t) = \begin{vmatrix}
\rho_{\nu+1}(t)& \rho_{\nu+2}(t) &0 & 0& \dots& 0 \\
\rho_{\nu+2}(t) & \rho_{\nu+3}(t)& \rho_{\nu+3}(t) & \rho_{\nu+4}(t) & \dots& \rho_{\nu+n+1}(t)\\
\rho_{\nu+3}(t)& \rho_{\nu+4}(t) & 2 \rho_{\nu+4}(t) & 2 \rho_{\nu+5}(t) & \dots& 2 \rho_{\nu+n+2}(t) \\
\vdots & \vdots & \vdots & \vdots& \vdots & \vdots \\
\rho_{\nu+n+1}(t)& \rho_{\nu+n+2}(t)& n \rho_{\nu+n+2}(t) & n\rho_{\nu+n+3}(t) & \dots& n \rho_{\nu+2n}(t)\\
\end{vmatrix}.\eqno(3.34)$$
Let us consider the following double sequence of determinants
$$H_{i,j}^{\nu,n}(t)= \begin{vmatrix}
\rho_{\nu+i}(t)& \rho_{\nu+j}(t) &0 & 0& \dots& 0 \\
\rho_{\nu+i+1}(t) & \rho_{\nu+j+1}(t)& \rho_{\nu+3}(t) & \rho_{\nu+4}(t) & \dots& \rho_{\nu+n+1}(t)\\
\rho_{\nu+i+2}(t)& \rho_{\nu+j+2}(t) & 2 \rho_{\nu+4}(t) & 2 \rho_{\nu+5}(t) & \dots& 2 \rho_{\nu+n+2}(t) \\
\vdots & \vdots & \vdots & \vdots& \vdots & \vdots \\
\rho_{\nu+n+i}(t)& \rho_{\nu+n+j}(t)& n \rho_{\nu+n+2}(t) & n\rho_{\nu+n+3}(t) & \dots& n \rho_{\nu+2n}(t)\\
\end{vmatrix},\eqno(3.35)$$
where $(i,j) \in \mathbb{Z}^2$. Hence it is not difficult to observe from (3.33), (3.34), (3.35) that $H_{i,j}^{\nu,n}(t)= - H_{j,i}^{\nu,n}(t)$ and
$$H_{j,j}^{\nu,n}(t)=0, \ j \in \mathbb{Z}, \quad H_{j,j+1}^{\nu,n}(t) = (-1)^{j+1} t^{j-1} G^\nu_n(t),\quad j=1,\dots,n.\eqno(3.36)$$
Generally, we find via (1.16)
$$ H_{i,j}^{\nu,n}(t)= (\nu+i-1) H_{i-1,j}^{\nu,n}(t) + t H_{i-2,j}^{\nu,n}(t)$$
$$+ \begin{vmatrix}
0& \rho_{\nu+j}(t)& 0 &0 & \dots& 0 \\
\rho_{\nu+i}(t) & \rho_{\nu+j+1}(t)& \rho_{\nu+3}(t) & \rho_{\nu+4}(t) & \dots& \rho_{\nu+n+1}(t)\\
2 \rho_{\nu+i+1}(t)& \rho_{\nu+j+2}(t) & 2 \rho_{\nu+4}(t) & 2 \rho_{\nu+5}(t) & \dots& 2 \rho_{\nu+n+2}(t) \\
\vdots & \vdots & \vdots & \vdots& \vdots & \vdots \\
n \rho_{\nu+n+i-1}(t)& \rho_{\nu+n+j}(t)& n \rho_{\nu+n+2}(t) & n\rho_{\nu+n+3}(t) & \dots& n \rho_{\nu+2n}(t)\\
\end{vmatrix}$$
$$= (\nu+i-1) H_{i-1,j}^{\nu,n}(t) + t H_{i-2,j}^{\nu,n}(t)$$
$$- n! \rho_{\nu+j}(t) \begin{vmatrix}
\rho_{\nu+i}(t) & \rho_{\nu+3}(t) & \rho_{\nu+4}(t) & \dots& \rho_{\nu+n+1}(t)\\
\rho_{\nu+i+1}(t)& \rho_{\nu+4}(t) & \rho_{\nu+5}(t) & \dots& \rho_{\nu+n+2}(t) \\
\vdots & \vdots & \vdots & \vdots& \vdots \\
\rho_{\nu+n+i-1}(t)& \rho_{\nu+n+2}(t) & \rho_{\nu+n+3}(t) & \dots& \rho_{\nu+2n}(t)\\
\end{vmatrix}.$$
Hence,
$$ H_{i,j}^{\nu,n}(t)= (\nu+i-1) H_{i-1,j}^{\nu,n}(t) + t H_{i-2,j}^{\nu,n}(t)$$
$$- n! \rho_{\nu+j}(t) \begin{vmatrix}
\rho_{\nu+i}(t) & \rho_{\nu+3}(t) & \rho_{\nu+4}(t) & \dots& \rho_{\nu+n+1}(t)\\
\rho_{\nu+i+1}(t)& \rho_{\nu+4}(t) & \rho_{\nu+5}(t) & \dots& \rho_{\nu+n+2}(t) \\
\vdots & \vdots & \vdots & \vdots& \vdots \\
\rho_{\nu+n+i-1}(t)& \rho_{\nu+n+2}(t) & \rho_{\nu+n+3}(t) & \dots& \rho_{\nu+2n}(t)\\
\end{vmatrix}.\eqno(3.37)$$
Analogously, it has
$$ H_{i,j}^{\nu,n}(t)= (\nu+j-1) H_{i,j-1}^{\nu,n}(t) + t H_{i,j-2}^{\nu,n}(t)$$
$$+ n! \rho_{\nu+i}(t) \begin{vmatrix}
\rho_{\nu+j}(t) & \rho_{\nu+3}(t) & \rho_{\nu+4}(t) & \dots& \rho_{\nu+n+1}(t)\\
\rho_{\nu+j+1}(t)& \rho_{\nu+4}(t) & \rho_{\nu+5}(t) & \dots& \rho_{\nu+n+2}(t) \\
\vdots & \vdots & \vdots & \vdots& \vdots \\
\rho_{\nu+n+j-1}(t)& \rho_{\nu+n+2}(t) & \rho_{\nu+n+3}(t) & \dots& \rho_{\nu+2n}(t)\\
\end{vmatrix},\eqno(3.38)$$
Thus we find
$$(\nu+i-1) H_{i-1,j}^{\nu,n}(t) + t H_{i-2,j}^{\nu,n}(t)$$
$$- n! \rho_{\nu+j}(t) \begin{vmatrix}
\rho_{\nu+i}(t) & \rho_{\nu+3}(t) & \rho_{\nu+4}(t) & \dots& \rho_{\nu+n+1}(t)\\
\rho_{\nu+i+1}(t)& \rho_{\nu+4}(t) & \rho_{\nu+5}(t) & \dots& \rho_{\nu+n+2}(t) \\
\vdots & \vdots & \vdots & \vdots& \vdots \\
\rho_{\nu+n+i-1}(t)& \rho_{\nu+n+2}(t) & \rho_{\nu+n+3}(t) & \dots& \rho_{\nu+2n}(t)\\
\end{vmatrix}$$
$$= (\nu+j-1) H_{i,j-1}^{\nu,n}(t) + t H_{i,j-2}^{\nu,n}(t)$$
$$+ n! \rho_{\nu+i}(t) \begin{vmatrix}
\rho_{\nu+j}(t) & \rho_{\nu+3}(t) & \rho_{\nu+4}(t) & \dots& \rho_{\nu+n+1}(t)\\
\rho_{\nu+j+1}(t)& \rho_{\nu+4}(t) & \rho_{\nu+5}(t) & \dots& \rho_{\nu+n+2}(t) \\
\vdots & \vdots & \vdots & \vdots& \vdots \\
\rho_{\nu+n+j-1}(t)& \rho_{\nu+n+2}(t) & \rho_{\nu+n+3}(t) & \dots& \rho_{\nu+2n}(t)\\
\end{vmatrix}.\eqno(3.39)$$
As an immediate consequence, when $(i, j) \in [ 3,\dots, n+1] \times [ 3,\dots, n+1]$ the following recurrence relation holds
$$(\nu+i-1) H_{i-1,j}^{\nu,n}(t) + (\nu+j-1) H_{j-1,i}^{\nu,n}(t) + t\bigg[ H_{i-2,j}^{\nu,n}(t) + H_{j-2,i}^{\nu,n}(t)\bigg] = 0.\eqno(3.40)$$
Recalling (3.36), (3.37), it yields
$$H_{n+2,n+1}^{\nu,n}(t) = (-1)^{n+1} \bigg[ t^{n-1} G^{\nu}_{n}(t) - n! \rho_{\nu+n+1}(t) G^{\nu+2}_{n-1}(t)\bigg].\eqno(3.41) $$
On the other hand, differentiating determinant (3.19) by virtue of (1.15) and applying the same process as above to the derivative, we deduce the equalities
$$t {d G^\nu_{n}\over dt} = - \begin{vmatrix}
t \rho_{\nu}(t) & \rho_{\nu+2}(t)& \dots& \dots& \rho_{\nu+n+1}(t) \\
t\rho_{\nu+1}(t) & \dots & \dots& \dots& \rho_{\nu+n+2}(t) \\
\dots& \dots & \dots& \dots& \dots \\
\vdots & \ddots & \ddots & \ddots& \vdots\\
t \rho_{\nu+n}(t) & \dots& \dots& \dots& \rho_{\nu+2n+1}(t)\\
\end{vmatrix}$$
$$= (\nu+1) G_n^\nu(t) + \begin{vmatrix}
0 & \rho_{\nu+2}(t)& \dots& \dots& \rho_{\nu+n+1}(t) \\
\rho_{\nu+2}(t) & \rho_{\nu+3}(t) & \dots& \dots& \rho_{\nu+n+2}(t) \\
2 \rho_{\nu+3}(t) & \rho_{\nu+4}(t) & \dots& \dots& \dots \\
\vdots & \ddots & \ddots & \ddots& \vdots\\
n \rho_{\nu+n+1}(t) & \rho_{\nu+n+2}(t) & \dots& \dots& \rho_{\nu+2n+1}(t)\\
\end{vmatrix}$$
$$ = (\nu+1) G_n^\nu(t) + \begin{vmatrix}
0 & \rho_{\nu+2}(t)& \rho_{\nu+3}(t) & 0& \dots& 0& 0 \\
\rho_{\nu+2}(t) & \rho_{\nu+3}(t) & \rho_{\nu+4}(t)& \rho_{\nu+4}(t)&\dots& \rho_{\nu+n}(t) & \rho_{\nu+n+1}(t) \\
2 \rho_{\nu+3}(t) & \rho_{\nu+4}(t) & \rho_{\nu+5}(t)& 2\rho_{\nu+5}(t)& \dots& 2 \rho_{\nu+n}(t) & 2 \rho_{\nu+n+1}(t ) \\
\vdots & \ddots & \ddots & \ddots& \vdots&\vdots &\vdots\\
n \rho_{\nu+n+1}(t) & \rho_{\nu+n+2}(t) & \rho_{\nu+n+3}(t) & n \rho_{\nu+n+3}(t)& \dots& n \rho_{\nu+n-1}(t)& n \rho_{\nu+2n}(t)\\
\end{vmatrix}$$
$$ = (\nu+1) G_n^\nu(t) + \begin{vmatrix}
0 & \rho_{\nu+2}(t)& 0 & 0& \dots& 0& 0 \\
\rho_{\nu+2}(t) & \rho_{\nu+3}(t) & \rho_{\nu+3}(t)& \rho_{\nu+4}(t)&\dots& \rho_{\nu+n}(t) & \rho_{\nu+n+1}(t) \\
2 \rho_{\nu+3}(t) & \rho_{\nu+4}(t) & 2 \rho_{\nu+4}(t)& 2\rho_{\nu+5}(t)& \dots& 2 \rho_{\nu+n}(t) & 2 \rho_{\nu+n+1}(t ) \\
\vdots & \ddots & \ddots & \ddots& \vdots&\vdots &\vdots\\
n \rho_{\nu+n+1}(t) & \rho_{\nu+n+2}(t) & n \rho_{\nu+n+2}(t) & n \rho_{\nu+n+3}(t)& \dots& n \rho_{\nu+n-1}(t)& n \rho_{\nu+2n}(t)\\
\end{vmatrix}$$
$$+ t \begin{vmatrix}
0 & \rho_{\nu+2}(t)& \rho_{\nu+1}(t) & 0& \dots& 0& 0 \\
\rho_{\nu+2}(t) & \rho_{\nu+3}(t) & \rho_{\nu+2}(t)& \rho_{\nu+4}(t)&\dots& \rho_{\nu+n}(t) & \rho_{\nu+n+1}(t) \\
2 \rho_{\nu+3}(t) & \rho_{\nu+4}(t) & \rho_{\nu+3}(t)& 2\rho_{\nu+5}(t)& \dots& 2 \rho_{\nu+n}(t) & 2 \rho_{\nu+n+1}(t ) \\
\vdots & \ddots & \ddots & \ddots& \vdots&\vdots &\vdots\\
n \rho_{\nu+n+1}(t) & \rho_{\nu+n+2}(t) & \rho_{\nu+n+1}(t) & n \rho_{\nu+n+3}(t)& \dots& n \rho_{\nu+n-1}(t)& n \rho_{\nu+2n}(t)\\
\end{vmatrix}.$$
Consequently, it gives by virtue of (3.19), (1.16)
$$t {d G^\nu_{n}\over dt} = (\nu+1) G_n^\nu(t) - n! \rho_{\nu+2}(t) G_{n-1}^{\nu+1}(t) $$
$$+ t^2 \begin{vmatrix}
\rho_{\nu}(t)& \rho_{\nu+1}(t)&0 & 0& \dots& 0& 0 \\
\rho_{\nu+1}(t) & \rho_{\nu+2}(t)& \rho_{\nu+2}(t)& \rho_{\nu+4}(t)&\dots& \rho_{\nu+n}(t) & \rho_{\nu+n+1}(t) \\
\rho_{\nu+2}(t) & \rho_{\nu+3}(t)& 2 \rho_{\nu+3}(t) & 2\rho_{\nu+5}(t)& \dots& 2 \rho_{\nu+n}(t) & 2 \rho_{\nu+n+1}(t ) \\
\vdots & \vdots & \vdots & \vdots& \vdots&\vdots &\vdots\\
\rho_{\nu+n}(t) & \rho_{\nu+n+1}(t) &n \rho_{\nu+n+1}(t) & n \rho_{\nu+n+3}(t)& \dots& n \rho_{\nu+n-1}(t)& n \rho_{\nu+2n}(t)\\
\end{vmatrix}.\eqno(3.42)$$
But from (3.35) we observe
$${d\over dt} \bigg[ H_{0,1}^{\nu,n}(t) \bigg] = - \begin{vmatrix}
\rho_{\nu-1}(t)& \rho_{\nu+1}(t) &0 & 0& \dots& 0 \\
\rho_{\nu}(t) & \rho_{\nu+2}(t)& \rho_{\nu+3}(t) & \rho_{\nu+4}(t) & \dots& \rho_{\nu+n+1}(t)\\
\rho_{\nu+1}(t)& \rho_{\nu+3}(t) & 2 \rho_{\nu+4}(t) & 2 \rho_{\nu+5}(t) & \dots& 2 \rho_{\nu+n+2}(t) \\
\vdots & \vdots & \vdots & \vdots& \vdots & \vdots \\
\rho_{\nu+n-1}(t)& \rho_{\nu+n+1}(t)& n \rho_{\nu+n+2}(t) & n\rho_{\nu+n+3}(t) & \dots& n \rho_{\nu+2n}(t)\\
\end{vmatrix}$$
$$- \begin{vmatrix}
\rho_{\nu}(t)& \rho_{\nu+1}(t) &0 & 0& \dots& 0 \\
\rho_{\nu+1}(t) & \rho_{\nu+2}(t)& \rho_{\nu+2}(t) & \rho_{\nu+4}(t) & \dots& \rho_{\nu+n+1}(t)\\
\rho_{\nu+2}(t)& \rho_{\nu+3}(t) & 2 \rho_{\nu+3}(t) & 2 \rho_{\nu+5}(t) & \dots& 2 \rho_{\nu+n+2}(t) \\
\vdots & \vdots & \vdots & \vdots& \vdots & \vdots \\
\rho_{\nu+n}(t)& \rho_{\nu+n+1}(t)& n \rho_{\nu+n+1}(t) & n\rho_{\nu+n+3}(t) & \dots& n \rho_{\nu+2n}(t)\\
\end{vmatrix}.$$
Hence we derive from (3.42) the identity
$$t {d G^\nu_{n}\over dt} - (\nu+1) G_n^\nu(t) + n! \rho_{\nu+2}(t) G_{n-1}^{\nu+1}(t) + t^2\bigg[ {d\over dt} \bigg[ H_{0,1}^{\nu,n}(t) \bigg] + H_{-1,1}^{\nu,n}(t) \bigg] = 0.\eqno(3.43)$$
{\bf Remark 3}. By using these features an interesting open question is to obtain a differential-difference recurrence relation for determinants (3.19).
\section{Rodrigues-type formula. Generating function}
Let us expand the function $e^{-t/x} P^\nu_n(x,t)$ in terms of the Laguerre polynomials $L_n^\nu(x)$. It gives
$$ e^{-t/x} P^\nu_n(x,t) = \sum_{k=0}^\infty d_{n,k}^\nu(t) L_k^\nu(x),\eqno(4.1)$$
where
$$ d_{n,k}^\nu(t) = {k!\over \Gamma(k+\nu+1)} \int_0^\infty e^{-x -t/x} P^\nu_n(x,t) L_k^\nu(x) x^\nu dx.\eqno(4.2)$$
But from (1.18) we have $d_{n,k}^\nu(t) = 0, \ k =0,\dots, n-1$. The uniform estimate with respect to $k$ for coefficients $d_{n,k}^\nu(t)$ is given by the following lemma.
{\bf Lemma 2}. {\it Let $ t >0\ \nu > -1$. Coefficients $d_{n,k}^\nu(t),\ n,k \in \mathbb{N}_0$ satisfy the upper bound of the form
$$\left|d_{n,k}^\nu(t)\right| \le {k!\ h_n^\nu(t) \over \Gamma(k+\nu+1)},\eqno(4.3)$$
where
$$h_n^\nu(t)= 2^{\nu-1/2} \int_0^t Q_n^\nu(t-y) \rho^{1/2}_{2\nu+1} (2 y) {dy\over \sqrt y}\eqno(4.4)$$
and }
$$Q_n^\nu(x) = \sum_{m=0}^n \left| a_{n,m}^\nu(t)\right| { x^m\over m!}.\eqno(4.5)$$
\begin{proof} Writing polynomial $P^\nu_n$ in the explicit form, equality (4.2) becomes
$$ d_{n,k}^\nu(t) = {k!\over \Gamma(k+\nu+1)} \sum_{m=0}^n a_{n,m}^\nu(t) \int_0^\infty e^{-x -t/x} L_k^\nu(x) x^{\nu+m} dx.\eqno(4.6)$$
Recalling (1.11), the latter integral can be represented as follows
$$ \int_0^\infty e^{-x -t/x} L_k^\nu(x) x^{\nu+m} dx = (-1)^{k+m+1} {d^{k-m-1}\over dt^{k-m-1}} \int_0^\infty x^{\nu+k -1} e^{-x - t/x} L_k^\nu(x) dx$$
$$= {(-1)^{m+1}\over k!} {d^{k-m-1}\over dt^{k-m-1}} \bigg[ t^k \rho_\nu(t) \bigg]= {(-1)^{m+1}\over k!\ m!} \int_0^t (t-y)^m {d^{k}\over dy^{k}} \bigg[ y^k \rho_\nu(y) \bigg] dy,\eqno(4.7)$$
where we mean (cf. (1.12))
$${d^{-q} f\over dt^{-q}} \equiv \left(I_+^q f\right)(t)= {1\over (q-1)!} \int_0^t (t-y)^{q-1} f(y) dy,\quad q \in \mathbb{N}_0.$$
Taking into account (4.6), coefficients $d_{n,k}^\nu(t)$ can be written in the operator form
$$d_{n,k}^\nu(t) = - {k!\over \Gamma(k+\nu+1)} P_n^\nu\bigg( - I_+, t\bigg) \bigg\{ {d^{k}\over dt^{k}} \bigg[ t^k \rho_\nu(t) \bigg]\bigg\}.\eqno(4.8)$$
Meanwhile, via (1.11) and the Rodrigues formula for Laguerre polynomials it has
$$ {d^{k}\over dy^{k}} \bigg[ y^k \rho_\nu(y) \bigg] = k! \int_0^\infty x^{\nu -1} e^{-x - y/x} L_k^\nu(x) dx = \int_0^\infty x^{-1} e^{- y/x} {d^k\over dx^k} \left[ e^{-x} x^{\nu+k} \right] dx.$$
Integrating by parts with the use of Entry 1.1.3.2 on p. 4 in \cite{Bry}, we get
$$ {d^{k}\over dy^{k}} \bigg[ y^k \rho_\nu(y) \bigg] = (-1)^k \int_0^\infty {d^k\over dx^k} \left[ x^{-1} e^{- y/x} \right] e^{-x} x^{\nu+k} dx$$
$$= k! \int_0^\infty e^{-x- y/x} x^{\nu-1} L_k\left({y\over x}\right) dx,$$
where $L_k$ are Laguerre polynomials of the index zero. Then owing to (1.7), Schwarz's inequality and orthogonality of Laguerre polynomials, we find the estimate
$$\left| {d^{k}\over dy^{k}} \bigg[ y^k \rho_\nu(y) \bigg] \right| \le k! \left(\int_0^\infty e^{- yx- 2/x} x^{-2(\nu+1)} dx\right)^{1/2} \left(\int_0^\infty e^{- yx} \left[L_k\left(y x\right)\right]^2 dx\right)^{1/2} $$
$$= {2^{\nu-1/2} k!\over \sqrt y} \rho^{1/2}_{2\nu+1} (2 y).$$
Therefore, returning to (4.6), (4.7), we derive
$$\left| d_{n,k}^\nu(t)\right| \le { 2^{\nu-1/2} k!\over \Gamma(k+\nu+1)} \int_0^t \sum_{m=0}^n \left| a_{n,m}^\nu(t)\right| { (t-y)^m\over m!} \rho^{1/2}_{2\nu+1} (2 y) {dy\over \sqrt y},$$
which implies (4.3) and completes the proof of Lemma 2.
\end{proof}
{\bf Corollary 12.} {\it Under the condition $\nu > 3/2$ the Laguerre series $(4.1)$ converges absolutely and uniformly on closed intervals of $\mathbb{R}_+$.}
\begin{proof} In fact, Stirling's asymptotic formula for gamma-function \cite{Bateman}, Vol. I yields
$$ {k!\over \Gamma(k+\nu+1)} = O\left( k^{-\nu} \right),\quad k \to \infty.$$
Since Laguerre polynomials $L_k^\nu(x)$ behave as $O\left( k^{\nu/2- 1/4}\right),\ k \to \infty$ uniformly on closed intervals $x \in [\alpha, \beta]$ of $\mathbb{R}_+$, the absolute and uniform convergence of the series (4.1) is guaranteed under the assumption $\nu > 3/2$.
\end{proof}
Further, writing (4.1) as follows
$$ e^{-t/x} P^\nu_n(x,t) = \sum_{k=n}^\infty d_{n,k}^\nu(t) L_k^\nu(x) = x^{-\nu} e^x \sum_{k=n}^\infty {d_{n,k}^\nu(t)\over k!} {d^k\over dx^k } \left[x^{k+\nu} e^{-x} \right]$$
$$= x^{-\nu} e^x \sum_{k=0}^\infty {d_{n,k+n}^\nu(t)\over (k+n)!} {d^{k+n}\over dx^{k+n} } \left[x^{k+n+\nu} e^{-x} \right]$$
$$= x^{-\nu} e^x \sum_{k=0}^\infty {d_{n,k+n}^\nu(t)\ k! \over (k+n)!} {d^{n}\over dx^{n} } \bigg[ e^{-x} x^{\nu+n} L_k^{n+\nu} (x)\bigg],$$
the problem arises to interchange the order of differentiation and summation. It can be done by virtue of the absolute and uniform convergence of the series
$$\sum_{k=0}^\infty {d_{n,k+n}^\nu(t)\over (k+n)!} {d^{k+j}\over dx^{k+j} } \left[x^{k+n+\nu} e^{-x} \right]$$
$$= x^{-\nu-n+j} e^x \sum_{k=0}^\infty {d_{n,k+n}^\nu(t)(k+j)! \over (k+n)!} L_{k+j}^{n+\nu-j}(x),\quad j=1,2,\dots, n,\ n \in \mathbb{N}\eqno(4.9) $$
on closed intervals of $\mathbb{R}_+$. Indeed, this is an immediate consequence of the bound (4.3) and Corollary 12. Thus we arrive at the following Rodrigues-type formula for polynomials $P^\nu_n$
$$ P^\nu_n(x,t) = x^{-\nu} e^{x+t/x} {d^{n}\over dx^{n} } \bigg[ e^{-x} x^{\nu+n} \sum_{k=0}^\infty {d_{n,k+n}^\nu(t)\ k! \over (k+n)!} \ L_k^{n+\nu} (x)\bigg].\eqno(4.10)$$
Recurrence relations for coefficients $d_{n,k}^\nu(t)$ are given by
{\bf Theorem 8}. {\it Coefficients $d_{n,k}^\nu(t)$ obey recurrence relations of the form}
$$ \left(2k+\nu+1 - (k+1) B_n^\nu(t) \right) d_{n,k}^\nu(t) - \left(k+1\right) A_{n+1}^\nu(t) d_{n+1,k}^\nu(t)- ( k+\nu+1) d_{n,k+1}^\nu(t)$$
$$ - \left(k+1\right) A_{n}^\nu(t) d_{n-1,k}^\nu(t) - k\ d_{n,k-1}^\nu(t) = 0,\eqno(4.11)$$
$$ (2k+n+1+\nu) A_{n+1}^\nu(t) d_{n+1,k}^\nu(t) - (k+1+\nu) A_{n+1}^\nu(t) d_{n+1,k+1}^\nu(t) $$
$$+ \bigg[ (1+n) B_n^\nu(t) - (k+\nu+2)(2k+\nu+1) -t +\left[ A_n^\nu(t)\right]^2- A_n^\nu(t) \left[ B^\nu_n(t)-\nu-1-2n\right] \bigg] d_{n,k}^\nu(t) $$
$$+ (k+\nu+1)(k+\nu+2- B^\nu_n(t) ) d_{n,k+1}^\nu(t) - (k+\nu+1) A_n^\nu(t) d_{n-1,k+1}^\nu(t) $$
$$+\bigg[ \left(2k+n+\nu+1+ B^\nu_{n-1}(t) \right) A_n^\nu(t) - \bigg[ [A^\nu_n(t)]^2 + {b^\nu_n(t)\over a^\nu_n(t)} \bigg] \bigg] d_{n-1,k}^\nu(t) $$
$$+ k(k+\nu+2) d_{n,k-1}^\nu(t)+ A_n^\nu(t) A_{n-1}^\nu(t) d_{n-2,k}^\nu(t) = 0.\eqno(4.12)$$
\begin{proof} Indeed, relation (4.11) follows immediately from the three recurrence relations (1.3), (1.4) for Laguerre polynomials and polynomials $P_n^\nu$, respectively. In order to prove (4.12), we integrate by parts in (4.2) to obtain
$$d_{n,k}^\nu(t)= {k!\over t \Gamma(k+\nu+1)} \bigg[ \int_0^\infty e^{-x -t/x} P^\nu_n(x,t) L_k^\nu(x) x^{\nu+2} dx\bigg.$$
$$ - (\nu+2) \int_0^\infty e^{-x -t/x} P^\nu_n(x,t) L_k^\nu(x) x^{\nu+1} dx + \int_0^\infty e^{-x -t/x} P^\nu_n(x,t) L_{k-1}^{\nu+1}(x) x^{\nu+2} dx$$
$$\bigg. - \int_0^\infty e^{-x -t/x} {\partial\over \partial x} \left[ P^\nu_n(x,t)\right] L_k^\nu(x) x^{\nu+2} dx\bigg].\eqno(4.13)$$
But using the known relation for Laguerre polynomials \cite{Sze}
$$x L_{k-1}^{\nu+1}(x) = (k+\nu) L_{k-1}^\nu(x) - k L_k^\nu(x),$$
equality (4.13) becomes
$$d_{n,k}^\nu(t)= {k!\over t \Gamma(k+\nu+1)} \bigg[ \int_0^\infty e^{-x -t/x} P^\nu_n(x,t) L_k^\nu(x) x^{\nu+2} dx\bigg.$$
$$ - (\nu+2+k) \int_0^\infty e^{-x -t/x} P^\nu_n(x,t) L_k^\nu(x) x^{\nu+1} dx + (k+\nu) \int_0^\infty e^{-x -t/x} P^\nu_n(x,t) L_{k-1}^{\nu}(x) x^{\nu+1} dx$$
$$\bigg. - \int_0^\infty e^{-x -t/x} {\partial\over \partial x} \left[ P^\nu_n(x,t)\right] L_k^\nu(x) x^{\nu+2} dx\bigg].\eqno(4.14)$$
Then, employing again recurrence relations (1.3), (1.4) and differential-difference equation (2.5), we derive
$$ {k!\over t \Gamma(k+\nu+1)} \int_0^\infty e^{-x -t/x} P^\nu_n(x,t) L_k^\nu(x) x^{\nu+2} dx$$
$$= {k!\over t \Gamma(k+\nu+1)} \int_0^\infty e^{-x -t/x} \left[ A_{n+1}^\nu(t) P^\nu_{n+1}(x,t) + B_{n}^\nu(t) P^\nu_{n}(x,t) + A_{n}^\nu(t) P^\nu_{n-1}(x,t)\right] $$
$$\times \left[ (2k+1+\nu) L_k^\nu(x) - (k+1)L_{k+1}^\nu(x) - (k+\nu)L_{k-1}^\nu(x)\right] x^\nu dx $$
$$= {1\over t} \bigg[ (2k+1+\nu) \bigg[ A_{n+1}^\nu(t) d_{n+1,k}^\nu(t) + B_{n}^\nu(t) d_{n,k}^\nu(t) + A_{n}^\nu(t) d_{n-1,k}^\nu(t)\bigg] \bigg.$$
$$ - (k+1+\nu) \bigg[ A_{n+1}^\nu(t) d_{n+1,k+1}^\nu(t) + B_{n}^\nu(t) d_{n,k+1}^\nu(t) + A_{n}^\nu(t) d_{n-1,k+1}^\nu(t)\bigg]$$
$$\bigg. - k \bigg[ A_{n+1}^\nu(t) d_{n+1,k-1}^\nu(t) + B_{n}^\nu(t) d_{n,k-1}^\nu(t) + A_{n}^\nu(t) d_{n-1,k-1}^\nu(t)\bigg] \bigg],$$
$$ {k! (k+\nu+2) \over t \Gamma(k+\nu+1)} \int_0^\infty e^{-x -t/x} P^\nu_n(x,t) L_k^\nu(x) x^{\nu+1} dx $$
$$= {k+\nu+2 \over t} \bigg[ (2k+1+\nu) d_{n,k}^\nu(t) - (k+1+\nu) d_{n,k+1}^\nu(t) - k d_{n,k-1}^\nu(t)\bigg],$$
$$ {k!\over t \Gamma(k+\nu)} \int_0^\infty e^{-x -t/x} P^\nu_n(x,t) L_{k-1}^{\nu}(x) x^{\nu+1} dx $$
$$= {k\over t} \bigg[ A_{n+1}^\nu(t) d_{n+1,k-1}^{\nu} (t) + B_{n}^\nu(t) d_{n,k-1}^{\nu} (t) + A_{n}^\nu(t) d_{n-1,k-1}^{\nu} (t)\bigg],$$
$$ {k!\over t \Gamma(k+\nu+1)} \int_0^\infty e^{-x -t/x} {\partial\over \partial x} \left[ P^\nu_n(x,t)\right] L_k^\nu(x) x^{\nu+2} dx $$
$$= {k!\over t \Gamma(k+\nu+1)} \int_0^\infty e^{-x -t/x} L_k^\nu(x) x^{\nu} \bigg[ \left[ nx - \left[A^\nu_n(t)\right]^2 - {b^\nu_n(t)\over a^\nu_n(t)} \right] P_n^\nu(x,t)\bigg.$$
$$\bigg. + A^\nu_n(t) \left[ x+B^\nu_n(t)-\nu-1-2n\right] P_{n-1}^\nu(x,t) \bigg] dx$$
$$= {A^\nu_n(t)\over t} \left[ B^\nu_n(t)-\nu-1-2n\right] d_{n,k}^\nu(t) - {1\over t} \bigg[ \left[A^\nu_n(t)\right]^2 + {b^\nu_n(t)\over a^\nu_n(t)} \bigg] d_{n-1,k}^\nu(t)$$
$$+ {n\over t} \bigg[ A_{n+1}^\nu(t) d_{n+1,k}^\nu(t) + B_{n}^\nu(t) d_{n,k}^\nu(t) + A_{n}^\nu(t) d_{n-1,k}^\nu(t)\bigg]$$
$$+ {A^\nu_n(t) \over t} \bigg[ A_{n}^\nu(t) d_{n,k}^\nu(t) + B_{n-1}^\nu(t) d_{n-1,k}^\nu(t) + A_{n-1}^\nu(t) d_{n-2,k}^\nu(t)\bigg].$$
Hence, substituting these values into (4.13), we get after simplification the desired relation (4.12).
\end{proof}
Finally, defining as usual the generating function $G(x,w,t)$ in terms of the series
$$G(x,w,t)= \sum_{n=0}^\infty P^\nu_n(x,t) {w^n\over n!},\quad x >0, \ w \in \mathbb{C},\eqno(4.15)$$
it can be written due to (4.10) in the form
$$G(x,w,t)= x^{-\nu} e^{x+t/x} \sum_{n=0}^\infty {w^n\over n!} {d^{n}\over dx^{n} } \bigg[ e^{-x} x^{\nu+n} \sum_{k=0}^\infty {d_{n,k+n}^\nu(t)\ k! \over (k+n)!} \ L_k^{n+\nu} (x)\bigg].\eqno(4.16)$$
The convergence of the series (4.15) is guaranteed at least in $L_2\left(\mathbb{R}_+; e^{-x-t/x} x^\nu dx\right)$. Indeed, by virtue of the Minkowski inequality it has
$$\bigg(\int_0^\infty e^{-x-t/x} \bigg| \sum_{n=N}^\infty P^\nu_n(x,t) {w^n\over n!} \bigg|^2 x^\nu dx \bigg)^{1/2} \le \sum_{n=N}^\infty {|w|^n\over n!} \bigg(\int_0^\infty e^{-x-t/x} \left[P^\nu_n(x,t)\right]^2 x^\nu dx \bigg)^{1/2} $$
$$= \sum_{n=N}^\infty {|w|^n\over n!} \to 0,\quad N \to \infty.$$
\bibliographystyle{amsplain}
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\begin{document}
\title{Trait evolution in two--sex populations}
\author[P. Zwole\'nski]{Pawe{\l} Zwole{\'n}ski}
\address{Institute of Mathematics,
Polish Academy of Sciences,
Bankowa 14, 40-007 Katowice, Poland.}
\email{pawel.zwolenski@gmail.com}
\thanks{This research was partially supported
by Warsaw Center of Mathematics and Computer Science from the KNOW grant of the Polish Ministry of Science and Higher Education.}
\keywords{individual--based model, phenotypic evolution, two--sex populations, system of nonlinear evolution equations,
asymptotic stability}
\subjclass[2010]
{Primary 47J35: Secondary: 34G20, 60K35, 92D15}
\begin{abstract}
We present an individual--based model of phenotypic trait evolution in two--sex populations, which includes semi--random mating of individuals of the opposite sex, natural death and intra--specific competition. By passing the number of individuals to infinity, we derive the macroscopic system of nonlinear differential equations describing the evolution of trait distributions in male and female subpopulations. We study solutions, give criteria for persistence or extinction, and state a theorem on asymptotic stability, which we later apply to particular examples of trait inheritance.
\end{abstract}
\maketitle
\section{Introduction}
Over last few decades two--sex populations were often studied from the viewpoint of mathematical modelling (see e.g. \cite{asm,had}). One of the first attempted descriptions of birth rate and matting functions in two--sex populations were given in \cite{good, ken}. In the paper \cite{fred} two assumptions on mating function were stated and its general form was provided. The above--mentioned models exhibit exponential grow due to the lack of competition. Addition of intrasexual competition to the models leads to logistic equations and results in more realistic bounded solutions and usually causes stabilization of population size (see e.g. \cite{liu, rose}). One of the reason for the interest in more complicated two--sex models comes from attempts to describe sexually transmitted diseases in human population (see \cite{d1, d2}). Modern models use more and more advanced mathematical tools, such as partial differential equations or stochastic processes, in order to include some of the complicated structures of populations. For example, attempted description of age--structured two--sex populations can lead to systems of non--linear partial differential equations (see e.g. \cite{bus, au}).
This paper studies constant lifetime phenotypic trait evolution in two--sex populations, using techniques of individual--based modeling. In spite of the vast literature concerning this type of models in evolutionary biology and population dynamics, a great deal of them describe asexual populations (see \cite{ch, ferriere_tran, fo_mel}). Only few models concerning hermaphroditic organisms have appeared so far (see \cite{colet, rud_zwo}). To the extend of our knowledge, there is a lack in the field of individual--based modeling for two--sex populations and mathematical analysis of equations derived as macroscopic approximations.
In this paper a phenotypic trait is not sex--linked, i.e., all the trait--coding genes lie outside the sex chromosome (allosome). We distinguish two subpopulations of males and females of the same species, and assume that the mating is semi--random, i.e., there is a function of individual capability of mating, which depends on individual's trait and sex. This function is a rate at which every individual mates with a random partner, which is chosen from all living individuals of the opposite sex according to some distribution also based on the capability function (see \cite{arino, rud_w, rud_w2, rud_zwo}). After mating an offspring is born, and its sex is male or female with probability $\frac{1}{2}$ by virtue of Fisher's principle on sex ratio. The phenotypic trait is inherited from parents as a mean parental trait with some stochastic noise. Moreover, individuals can die naturally or in intra--specific competition at trait--dependent rates. All of the above events happen randomly in discrete population in continuous time. The evolution of population is described by a sequence of measure--valued stochastic processes (individual--based model).
The main goal of the paper is to study macroscopic equations, which are derived as a law of large numbers for the stochastic processes considered, when the number of individuals tends to infinity. We obtain the system of two nonlinear differential equations, which describe the evolution of trait distributions in male and female subpopulations, respectively. We study existence and uniqueness of the solutions, and examine total number of individuals -- we give criterion for persistence of population, and also show when extinction occurs. The most important result concerns asymptotic stability of solutions: we investigate when the distributions of phenotypic traits in male and female subpopulations tend to a stationary solution, which is the same for both sexes. This implies that the distribution of non--sex--linked phenotypic traits become the same for males and females after long time of evolution. The asymptotic result is a transmission of analogous theorem from hermaphroditic populations (see \cite{rud_zwo}). In order to show applications of our result, we give two examples of mean parental trait inheritance with different forms of stochastic noise.
The structure of the paper is the following: in the next section we explain notation and gather all the model assumptions. Also in that part, we describe the individual--based model and introduce stochastic processes, discuss their existence and state the limit theorem (the law of large numbers), which gives the equation for macroscopic approximation of studied stochastic processes. In Section 3, we derive and study equations for trait distributions in male and female subpopulations. We prove existence and uniqueness of solutions in space of finite Borel measures and study total number of populations. Criteria for persistence and extinction are given. In Section 4, theorem on asymptotic stability of trait distributions is stated and proven. Section 5 includes examples of trait inheritance and application of our asymptotic results. In Section 6 we summarize our work and give future perspectives for extending the model.
\section{The model}
\subsection{Assumptions and parameters of model}
We assign every individual some element from a set $\mathbb{X}:=X\times\{\Venus,\Mars\}$, where $X$ is non--empty subset of $\R^d$ for some $d\in\N$. The first coordinate of $\mathbb X$'s members describes individual's \textit{phenotypic trait}, the second one its \textit{sex} ($\Venus$ corresponds to female and $\Mars$ is male). Both phenotypic trait and sex are assumed to be constant in individual's lifetime. We simply call elements of $\mathbb X$ \textit{traits}. For convenience sake we also consider following sets $\mathbb F:=X\times\{\Venus\}$, $\mathbb M:=X\times\{\Mars\}$. We also impose the following useful convention: if $\mathbf x\in\mathbb X$ is individual's trait, then we denote by the same, but non--bold letter $x$ its phenotypic trait, i.e., $x$ is the first coordinate of $\mathbf x$.
\subsubsection{Mating}
We adapt \textit{semi--random mating/coagulation} models (see \cite{colet, rud_w, rud_w2, rud_zwo}) to the two--sex population case: an individual of trait $\mathbf x \in\mathbb X$ has a rate $p(\textbf x)$ of \textit{initial capability of mating}, at which it starts mating by choosing a partner of the opposite sex. A female (resp. male) chooses a partner of trait $\mathbf y$ form all living males (resp. females) from the distribution
$$\frac{p(\textbf y)}{\sum_i p(\textbf w_i)}$$
where the sum in the denominator extends over all living males (resp. females), and $\textbf w_i$ are their traits. In other words, if the trait distribution in current population is described by some measure $\mu$, then \textit{mating rate} $m(\mathbf x, \mathbf y, \mu)$ of individuals of traits $\textbf x,\mathbf y$ is
\begin{equation}\label{mrate}
m(\mathbf x,\mathbf y,\mu)=p(\mathbf x)\1_{\mathbb F}(\mathbf x) \frac{p(\mathbf y)\1_{\mathbb M}(\mathbf y)}{\int_\mathbb{M} p(\mathbf w)\mu(d\mathbf w)} +p(\mathbf x)\1_{\mathbb M}(\mathbf x) \frac{p(\mathbf y)\1_{\mathbb F}(\mathbf y)}{\int_\mathbb{F} p(\mathbf w)\mu(d\mathbf w)},
\end{equation}
where $\1_A$ is an indicator function of set $A$. We assume that the function $p$ is positive, continuous and upper--bounded by some constant $\overline p>0$.
\subsubsection{Trait inheritance}
We assume that after every mating a new individual is born, and according to Fisher's principle (see \cite{fisher}), is male or female with probability $\frac{1}{2}$. We consider only these phenotypic traits which are not sex--linked. We assume that if $x,y\in X$ are parental phenotypic traits, then the offspring's trait $z$ comes from distribution $k(x,y,dz)$. We assume that for every $x,y\in X$ the measure $k(x,y,\cdot)$ is a Borel probability measure and for every bounded and continuous function $f\colon X\to\R$, the mapping $(x,y)\mapsto \int_X f(z)k(x,y,dz)$ is continuous. Moreover, we suppose that $k(x,y,\cdot)=k(y,x,\cdot)$ for every $x,y\in X$. Denote by $K(x,y,\cdot)$ measure on $\mathbb X$ satisfying $K(x,y,A\times\{\Venus\})=K(x,y,A\times\{\Mars\})=\frac{1}{2}k(x,y,A)$ for every Borel subset $A\subset X$.
\subsubsection{Natural death and competition}
We assume that if an individual has trait $\mathbf x$, then it can \textit{die naturally} at rate $D(\mathbf x)$. Moreover, we consider \textit{intra--specific competition} (see \cite{fo_mel, rud_zwo}): in current population described by a measure $\mu$, an individual with trait $\mathbf x$ dies in competition with rate $C(\mathbf x,\mu):=\int_\mathbb X U(\mathbf x, \mathbf y)\mu(d\mathbf y).$ Function $U(\mathbf x, \mathbf y)$ describes ``how often'' individual with trait $\mathbf x$ looses competition with one of trait $\mathbf y$ (\textit{competition kernel}). We assume that functions $D$ and $U$ are positive, continuous and upper--bounded by $\overline D, \overline U$ respectively.
\subsubsection{Population dynamics}
We consider a finite population in continuous time. At a random time individuals with traits $\mathbf x, \mathbf y$ mate with rate $m(\mathbf x, \mathbf y, \mu)$ given by the formula (\ref{mrate}), where $\mu=\sum_{i=1}^n \delta_{\mathbf w_i}$, provided the current population consists of individuals with traits $\mathbf w_1,\ldots,\mathbf w_n$. After mating an offspring is born with probability $1$. Its sex is male or female with probability $\frac{1}{2}$ and phenotypic trait comes from distribution $k(x,y,\cdot)$. Moreover, an individual of trait $\mathbf x$ can die naturally at rate $D(\mathbf x)$ or loosing competition with other members of population at rate $U(\mathbf x,\mu)$.
All the events and interactions are assumed to take place independently.
\subsection{Stochastic processes and limit theorem}
Denote by $\mathcal M(\mathbb X)$ set of all finite Borel measures on $\mathbb X$. For any $N\in\N$ we define following sequence of subsets of $\mathcal M(\mathbb X)$
\begin{equation}
\mathcal M^N=\Bigg\{\frac{1}{N}\sum_{i=1}^n \delta_{\mathbf w_i}\colon n\in\N, \mathbf w_i\in\mathbb X\Bigg\}.
\end{equation}
We study a sequence $(\nu^N)_{N\in\N}$ of $\mathcal M^N$--valued, continuous time stochastic processes given by the infinitesimal generators
\begin{multline}\label{LN}
L^N\phi(\nu)=N\int_\mathbb X\int_\mathbb X\int_\mathbb X\bigg(\phi\Big(\nu+\frac{1}{N}\delta_{\mathbf z}\Big)-\phi(\nu)\bigg)m(\mathbf x,\mathbf y,\nu)K(x,y,d\mathbf z)\nu(d\mathbf x)\nu(d\mathbf y)\\
+N\int_\mathbb X\bigg(\phi\Big(\nu-\frac{1}{N}\delta_{\mathbf x}\Big)-\phi(\nu)\bigg)\Big(D(\mathbf x)+\frac{1}{N}C(\mathbf x,\nu)\Big)\nu(d\mathbf x)
\end{multline}
for any measurable and bounded $\phi\colon\mathbb X\to\R$.
Notice that the processes given by above generators are jump processes on $\mathcal M^N$. The first term on the right--hand side of (\ref{LN}) describes mating and trait inheritance including Fisher's principle. The second term on the right--hand side of (\ref{LN}) corresponds to natural death and competition, whose rate is rescaled by the factor $\frac{1}{N}$. Given initial value $\nu^N_0\in\mathcal M^N$, under the model's assumptions, there exists a $\mathcal M^N$--valued Markov process $(\nu^N_t)_{t\geq 0}$ with infinitesimal generator given by (\ref{LN}) (see \cite{fo_mel}).
\begin{thm}\label{pwl}
Suppose that the sequence of initial values $(\nu^N_0)_{N\in\N}$ converges to some measure $\nu\in \mathcal M(\mathbb{X})$ in topology of weak convergence of measures. Then for every $T>0$ the sequence of processes converges in distribution in Skorokhod space $\mathcal D\big([0,T], \mathcal M(\mathbb{X})\big)$ to the deterministic and continuous flow of measures $\mu\colon [0,T]\to\mathcal M(\mathbb{X})$ satisfying for every $0\leq t\leq T$ following equation
\begin{multline}\label{eq_weak}
\ilsk{\mu_t}{\phi}=\ilsk{\mu_0}{\phi}+\int_0^t\int_\mathbb X\int_\mathbb X\int_\mathbb X \phi(\mathbf z)m(\mathbf x,\mathbf y,\nu)K(x,y,d\mathbf z)\mu_s(d\mathbf x)\mu_s(d\mathbf y)ds\\
-\int_0^t\int_\mathbb X\phi(x)\bigg(D(\mathbf x)+\int_\mathbb X U(\mathbf x,\mathbf y)\mu_s(d\mathbf y)\bigg)\mu_s(d\mathbf x)ds
\end{multline}
for any measurable and bounded $\phi\colon\mathbb X\to\R.$
\end{thm}
The proof of above statement is standard and can be adapted e.g. from \cite{rud_w}.
\section{System of macroscopic equations}
\subsection{The system and solutions} Define $m_t(A)=\mu_t(A\cap \{\Mars\})$ and $f_t(A)=\mu_t(A\cap \{\Venus\})$ -- the measures describing evolution of traits in male and female subpopulations, respectively. Setting in (\ref{eq_weak}) $\phi=\1_{A\times \{\Mars\}}$ and then $\phi=\1_{A\times \{\Venus\}}$, and rewriting into differential form, we obtain
\begin{equation}\label{systemzero}
\left\{\begin{array}{l}
\begin{aligned}
\frac{d}{dt}m_t(dz)&=\int_X\int_X p(x,y,m_t,f_t) k(x,y,dz)f_t(dx)m_t(dy)\\
&-\bigg(D_m(z)+\int_X U_{m,m}(z,y)m_t(dy)+\int_X U_{m,f}(z,y)f_t(dy)\bigg)m_t(dz),
\end{aligned}\vspace*{0.5cm}\\
\begin{aligned}
\frac{d}{dt}f_t(dz)&=\int_X\int_X p(x,y,m_t,f_t) k(x,y,dz)f_t(dx)m_t(dy)\\
&-\bigg(D_f(z)+\int_X U_{f,m}(z,y)m_t(dy)+\int_X U_{f,m}(z,y)f_t(dy)\bigg)f_t(dz),
\end{aligned}\end{array}\right.
\end{equation}
where
\begin{enumerate}
\item $p(x,y,\mu,\nu):=\frac{1}{2}p_f(x)p_m(y)\Big(\frac{1}{\int_X p_m(w)\mu(dw)}+\frac{1}{\int_X p_f(w)\nu(dw)}\Big),$
\item $p_f(x):=p\big((x,\Venus)\big), p_m(x):=p\big((x,\Mars)\big)$ -- individual capabilities of mating in the female and respectively, male subpopulations,
\item $D_f(x):=D\big((x,\Venus)\big), D_m(x):=D\big((x,\Mars)\big)$ -- natural death rates in the female and respectively, male subpopulations,
\item $U_{f,f}(x,y):=U\big((x,\Venus),(y,\Venus)\big)$ -- competition kernel between females,
\item $U_{m,m}(x,y):=U\big((x,\Mars),(y,\Mars)\big)$ -- competition kernel between males,
\item $U_{m,f}(x,y):=U\big((x,\Mars),(y,\Venus)\big)$ -- competition kernel describing the rate of competition loss of the males due to females,
\item $U_{f,m}(x,y):=U\big((x,\Venus),(y,\Mars)\big)$ -- competition kernel describing the rate of competition loss of the females due to males.
\end{enumerate}
Note that system (\ref{systemzero}) is a trait--structured population analogue of some classic two--sex population models well--known from the literature. For example, our model contains more general, trait--dependent version of some of the mating functions studied in \cite{fred, good, ken}, and intrasexual competition considered in \cite{liu, rose}.
\medskip
Recall that for any finite Borel measure $\mu$ on $X$ we can introduce the total variation norm by formula $\|\mu\|_{\textup{TV}}:=\sup_f \int_X f(x)\mu(dx),$ where the supremum is taken over the set of all measurable functions $f\colon X\to\R$ such that $|f(x)|\leq 1$ for all $x\in X$. Under weaker conditions than in Theorem \ref{pwl}, we are able to prove existence and uniqueness of solutions to (\ref{systemzero}) in stronger norm than Theorem~\ref{pwl} provides. The proof of following result can be easily adapted from proof of Theorem~2 in \cite{rud_zwo}.
\begin{thm}\label{exist1}
Suppose that $p_f,p_m,D_f,D_m,U_{f,f},U_{f,m},U_{m,f},U_{m,m}$ are measurable, upper--bounded and bounded from below by some positive constants. For every $\mu_0,\nu_0\in\mathcal M(X)$ there exists unique pair of functions $\mu,\nu\colon [0,\infty)\to\mathcal M(X),$ which is the solution of system $(\ref{systemzero})$ with initial conditions $\mu_0,\nu_0$. The functions $\mu_t,\nu_t$ are continuous and bounded function in the norm $\|\cdot\|_{\textup{TV}}.$
\end{thm}
If we additionally suppose that for every $x,y\in X$ there exists a density $\kappa(x,y,\cdot)\in L^1$ of measure $k(x,y,dz)$ with respect to Lebesgue measure, then using Radon--Nikodym theorem, we can also prove the following theorem.
\begin{thm}
Under the assumptions of Theorem~\ref{exist1}, if $\mu_0,\nu_0$ have densities $u_0,v_0\in L^1$ with respect to Lebesgue measure, than for every $t\geq 0$ solutions $\mu_t,\nu_t$ of $(\ref{systemzero})$ have densities $u(t,\cdot),v(t,\cdot)\in L^1$. Functions $u(t,z),v(t,z)$ are unique solutions of the following system
\begin{equation*}
\left\{\begin{array}{l}
\begin{aligned}
\frac{\partial }{\partial t}u(t,z)&=\int_X\int_X p\Big(x,y,u(t,\xi)d\xi,v(t,\xi)d\xi\Big) \kappa(x,y,z)u(t,x)v(t,y)dxdy\\
&-\bigg(D_m(z)+\int_X U_{m,m}(z,y)u(t,y)dy+\int_X U_{m,f}(z,y)v(t,y)dy\bigg)u(t,z),
\end{aligned}\vspace*{0.5cm} \\
\begin{aligned}
\frac{\partial }{\partial t}v(t,z)&=\int_X\int_X p\Big(x,y,u(t,\xi)d\xi,v(t,\xi)d\xi\Big) \kappa(x,y,z)u(t,x)v(t,y)dxdy\\
&-\bigg(D_f(z)+\int_X U_{f,m}(z,y)u(t,y)dy+\int_X U_{f,m}(z,y)v(t,y)dy\bigg)v(t,z).
\end{aligned}\end{array}\right.
\end{equation*}
\end{thm}
\subsection{Total number of individuals}
In this chapter we consider a case when all of the rates $D_m,$ $D_f,$ $U_{m,m},$ $U_{m,f},$ $U_{f,m},$ $U_{f,f}$ are constant and positive. We assume that $p_m,$ $p_f$ are constant, non--negative and $p_m+p_f>0$. Then the system (\ref{systemzero}) has the following form
\begin{equation}\label{sysconst}
\left\{\begin{array}{l}
\begin{aligned}
\frac{d}{dt}m_t(dz)&=\frac{p_fF(t)+p_mM(t)}{2M(t)F(t)}\int_X\int_X k(x,y,dz)f_t(dx)m_t(dy)\\
&-\Big(D_m+U_{m,m}M(t)+U_{m,f}F(t)\Big)m_t(dz),
\end{aligned}\vspace*{0.5cm} \\
\begin{aligned}
\frac{d}{dt}f_t(dz)&=\frac{p_fF(t)+p_mM(t)}{2M(t)F(t)}\int_X\int_X k(x,y,dz)m_t(dx)f_t(dy)\\
&-\Big(D_f+U_{f,m}M(t)+U_{f,f}F(t)\Big)f_t(dz),
\end{aligned}\end{array}\right.
\end{equation}
where $M(t):=m_t(X)$ and $F(t):=f_t(X)$.
In order to investigate asymptotic properties of total numbers of individuals in male and female subpopulations, let us denote $\lambda(t):=\big(p_f F(t)+p_m M(t)\big)/2.$
Integrating both sides of equations of the system (\ref{sysconst}), we obtain
\begin{equation}\label{systotal}
\left\{\begin{array}{l}
M'(t)=\lambda(t) - \big(D_m + U_{m,m} M(t)+U_{m,f} F(t)\big)M(t), \vspace*{0.2cm} \\
F'(t)=\lambda(t) - \big(D_f + U_{f,m} M(t)+U_{f,f} F(t)\big)F(t).
\end{array}\right.
\end{equation}
The function $\lambda(t)$ can be interpreted as a birth rate of each subpopulations at time~$t$. This rate is identical in males and females due to the Fisher's principle. The form of $\lambda(t)$ in our model generalizes some of the birth rates studied before in literature. For instance some of the birth rates investigated in \cite{fred, good, ken} can be obtained simply by taking $p_m=p_f=1$ and $p_m=1, p_f=0$. However, since our equations contain additional terms, which are responsible for competition, the solutions exhibit asymptotic properties different than exponential growth of above models. Similar intersexual competition terms were studied in \cite{rose} Section 4. A particular case of system (\ref{systotal}) appeared also in \cite{liu} and the existence of globally asymptotically stable stationary solution was obtained. We provide a result concerning the same type of asymptotic behavior, but in more general setting.
\begin{thm}\label{astotal} Consider solution $\big(M(t),F(t)\big)$ of system $(\ref{systotal})$ with initial condition $(M_0,F_0),$ $M_0,F_0>0$.
Suppose that the inequality
\begin{equation}\label{persis}
\frac{p_m}{D_m}+\frac{p_f}{D_f}>2
\end{equation}
holds. Then there exist unique $\bar M,\bar F>0$ such that $\big(M(t),F(t)\big)\to (\bar M, \bar F)$ exponentially, as $t\to\infty$.
On the contrary, if the inequality
\begin{equation}\label{extinct}
\frac{p_m}{D_m}+\frac{p_f}{D_f}\leq 2,
\end{equation}
holds, then $\big(M(t),F(t)\big)\to (0,0)$ as $t\to\infty$.
\end{thm}
\begin{proof}
Without loss of generality we can assume that $p_f>0$. Suppose that condition (\ref{persis}) holds.
We show that the following system of polynomial equations
\begin{equation}\label{poly}
\left\{\begin{array}{l}p_mM+p_fF - 2D_mM - 2U_{m,m} M^2-2U_{m,f} FM=0, \vspace*{0.2cm} \\
p_mM+p_fF- 2D_fF - 2U_{f,m} MF-2U_{f,f} F^2=0.\end{array}\right.
\end{equation}
has unique non--trivial solution $(\bar M,\bar F)$. Denote by $h_1$ and $h_2$ curves given by first and second equation, respectively. Notice, that if $p_m- 2D_m = \frac{U_{m,m}p_f}{U_{m,f}},$ or $p_f-2D_f=\frac{U_{f,f}p_m}{U_{f,m}},$ the system has only one positive solution, which is easy to find explicitly.
Denote the implicit formulas for both curves $F_1(M)=\frac{p_mM-2D_mM-2U_{m,m}M^2}{2U_{m,f}M-p_f}$ and $M_2(F)=\frac{p_fF-2D_fF-2U_{f,f}F^2}{2U_{f,m}F-p_m}$ and let $A=\frac{p_f}{2U_{m,f}}$, $B=\frac{p_m}{2U_{f,m}}.$ Then
\begin{enumerate}
\item if $p_m- 2D_m > \frac{U_{m,m}p_f}{U_{m,f}},$ and $p_f-2D_f>\frac{U_{f,f}p_m}{U_{f,m}},$ then
$$\lim_{M\nearrow A} F_1(M)=-\infty,\, \lim_{M\searrow A} F_1(M)=+\infty,\, \lim_{F\nearrow B} M_2(F)=-\infty,\, \lim_{F\searrow B}M_2(F)=+\infty,$$
\item if $p_m- 2D_m > \frac{U_{m,m}p_f}{U_{m,f}},$ and $p_f-2D_f<\frac{U_{f,f}p_m}{U_{f,m}},$ then
$$\lim_{M\nearrow A} F_1(M)=-\infty,\, \lim_{M\searrow A} F_1(M)=+\infty,\, \lim_{F\nearrow B} M_2(F)=+\infty,\, \lim_{F\searrow B}M_2(F)=-\infty,$$
\item if $p_m- 2D_m < \frac{U_{m,m}p_f}{U_{m,f}},$ and $p_f-2D_f>\frac{U_{f,f}p_m}{U_{f,m}},$ then
$$\lim_{M\nearrow A} F_1(M)=+\infty,\, \lim_{M\searrow A} F_1(M)=-\infty,\, \lim_{F\nearrow B} M_2(F)=-\infty,\, \lim_{F\searrow B}M_2(F)=+\infty,$$
\item if $p_m- 2D_m < \frac{U_{m,m}p_f}{U_{m,f}},$ and $p_f-2D_f<\frac{U_{f,f}p_m}{U_{f,m}},$ then
$$\lim_{M\nearrow A} F_1(M)=+\infty,\, \lim_{M\searrow A} F_1(M)=-\infty,\, \lim_{F\nearrow B} M_2(F)=+\infty,\, \lim_{F\searrow B}M_2(F)=-\infty.$$
\end{enumerate}
Moreover one can check that $F=-\frac{U_{m,m}}{U_{m,f}}M+a,$ $M=-\frac{U_{f,f}}{U_{f,m}}F+b$, for some constants $a,b\in\R$ give the formulas for asymptotes of $h_1$ and $h_2$, respectively. Since in all of the above cases each of $h_1$ and $h_2$ has two different asymptotes, there must be that $h_1$ and $h_2$ are hyperbolas or two straight lines. Since $F=\frac{2D_m-p_m}{p_f} M$, $F=\frac{p_m}{2D_f-p_f} M$ are tangent lines at $(0,0)$ to $h_1$ and $h_2$, respectively, from (\ref{persis}) one can prove that
$$\frac{p_m}{2D_f-p_f}>\frac{2D_m-p_m}{p_f} ,$$
i.e., $h_2$ lies above $h_1$ in some right neighborhood of $0$ in the following sense: $F_1(M)< F_2(M)$ for small $M>0$ (here $F_2(M)$ is an explicit formula for $F$ derived from the second equation of (\ref{poly})). Now it is easy to check that in every case such two hyperbolas (or two pairs of straight lines) must cross at exactly one positive point (by virtue of Darboux property of continuous functions).
Consider solution $\big(M(t),F(t)\big)$ of (\ref{systotal}) and let $\rho(M,F)=-1/(MF)$. For every $t>0$
$$\frac{\partial}{\partial M} \big(\rho H_1\big)\big(M(t),F(t)\big)+\frac{\partial}{\partial F} \big(\rho H_2\big)\big(M(t),F(t)\big)\geq \frac{U_{m,m}}{F(t)}+\frac{U_{f,f}}{M(t)}>0,$$
where $H_1$ and $H_2$ are functions from right--hand sides of the first and second equation of system (\ref{systotal}), respectively.
Since solutions of the system are upper--bounded, from Dullac--Bendixon theorem any solution $\big(N(t),F(t)\big)$ tends to one of the stationary points, as $t\to\infty$. In order to finish the proof, we show that $(0,0)$ is retracting.
Denote $\beta(t)=p_m M(t)/D_m+p_f F(t)/D_f$. From (\ref{systotal}) we obtain
\begin{equation}\label{eqtotalhelp}
\beta'(t)=\bigg(\frac{p_m}{D_m}+\frac{p_f}{D_f}-2\bigg)\lambda(t)-C_1p_m^2M^2(t)-2C_2p_mp_fM(t)F(t)-C_3p_f^2F^2(t),
\end{equation}
where $C_1,C_2,C_3>0$ are some constants. Let $C=\max\{C_1,C_2,C_3\}.$ Then
$$
\beta'(t)\geq \lambda(t)\bigg(\frac{p_m}{D_m}+\frac{p_f}{D_f}-2-4C\lambda(t)\bigg).
$$
If $N(t),F(t)$ are small enough that $0<\lambda(t)<\frac{p_m}{4CD_m}+\frac{p_f}{4CD_f}-\frac{1}{2C},$ then $\beta'(t)>0,$ and at least one of the functions $N(t),F(t)$ grows strictly. Thus $(0,0)$ is retracting point.
Suppose that the opposite case holds, i.e., the inequality (\ref{extinct}) is satisfied. Notice that from (\ref{poly}) it follows that
$$\bigg(\frac{p_m}{D_m}+\frac{p_f}{D_f}-2\bigg)\lambda=A M^2+B MF+C F^2,$$
where $\lambda=p_mM/2+p_fF/2$ and $A,B,C>0$ are some constants. Since left--hand side of above is non--positive and right--hand side is non--negative, there is no positive solution, and consequently $M=F=0$ is the only solution to the system.
Moreover, from (\ref{eqtotalhelp}) it follows that $\beta'(t)\leq -C\beta(t),$ where $C>0$ is some constant. Then $\beta(t)\to 0$ exponentially, and thus $\big(N(t),F(t)\big)\to (0,0),$ as $t\to\infty$.
\end{proof}
Suppose that (\ref{persis}) holds.
Substituting $m_t(dz)=M(t)\mu_t(dz)$ and $f_t(dz)=F(t)\nu_t(dz)$ in (\ref{sysconst}) and time scaling $t\mapsto \int_0^t \frac{\lambda(s)}{M(s)}ds$ lead to following system
\begin{equation}\label{sysprobmes}
\left\{\begin{array}{l}
\begin{aligned}
\frac{d}{dt}\mu_t(dz)+\mu_t(dz) =\int_X\int_X k(x,y,dz)\mu_t(dx)\nu_t(dy),
\end{aligned}\vspace*{0.5cm} \\
\begin{aligned}
\frac{d}{dt}\nu_t(dz)+A(t)\nu_t(dz) =A(t)\int_X\int_X k(x,y,dz)\mu_t(dx)\nu_t(dy),
\end{aligned}
\end{array}\right.
\end{equation}
where $A(t)=M(t)/F(t)$. Notice, that if $\mu_0,\nu_0$ are probability measures, then for every $t>0$, solutions $\mu_t,\nu_t$ are probability measures as well. We denote by $\mathcal M_{\textrm{Prob}}$ the space of all Borel probability measures on $X$.
\section{Asymptotic stability of trait distribution}
\subsection{Statement of the main result} In this section we assume that $X$ is closed interval in $\R$.
From Theorem~\ref{astotal}, under condition (\ref{persis}), male and female subpopulations sizes stabilize on positive level, i.e., $A(t)\to A$ for some $A>0$ as $t\to\infty.$ The number $A$ is a ratio of male subgroup size to size of female subgroup in stable population. Denote $\mathcal P(\mu,\nu)(dz)=\int_X\int_X k(x,y,dz)\mu(dx)(dy)$. In this section, we study long time behavior of the following system
\begin{equation}\label{sysprobmesconst}
\left\{\begin{array}{l}
\begin{aligned}
\mu_t'+\mu_t =\mathcal P(\mu_t,\nu_t),
\end{aligned}\vspace*{0.2cm} \\
\begin{aligned}
\nu_t'+A\,\nu_t=A\,\mathcal P(\mu_t,\nu_t),
\end{aligned}
\end{array}\right.
\end{equation}
where $A>1$ is some constant. Later on, we compare solutions of initial system with solutions of (\ref{sysprobmesconst}) in order to obtain asymptotic behavior also for solutions of (\ref{sysprobmes}).
We assume that
\begin{equation}\label{ksr1}
\int_X |z|k(x,y,dz)\leq a_1+a_2|x|+a_3|y|,
\end{equation}
for some constants $a_1,a_2,a_3>0$, and
\begin{equation}\label{sredniaK}
\int_X zk(x,y,dz)=\frac{x+y}{2}.
\end{equation}
The above condition is a reasonable biological assumption and means that the expected offspring's trait is a mean parental trait.
For any $\gamma\geq1$ and $\alpha\leq\beta$ we introduce
$\mathcal M_\gamma:=\big\{\mu\in\mathcal M_{\textrm{Prob}}\colon \int_X |x|^\gamma\mu(dx)<\infty\big\}$ and $\mathcal M_{[\alpha,\beta]}:=\big\{\mu\in \mathcal M_1\colon \alpha\leq \int_X x\mu(dx)\leq \beta \big\}.$
For any two measures $\mu,\nu\in \mathcal{M}_1$, we define the \textit{Wasserstein distance} by the formula
\begin{equation}
d(\mu,\nu)=\sup_{f\in\textup{Lip}_1} \int_X f(z)(\mu-\nu)(dz),
\end{equation}
where $\textup{Lip}_1$ is a set of all continuous functions $f\colon X\to \R$ such that $|f(x)-f(y)|\leq |x-y|,$ for any $x,y\in X$ .
Denote by $\mathcal K(x,y,\cdot)$ the cumulative distribution function of measure $k(x,y,\cdot)$, i.e., $\mathcal K(x,y,z)=k\big(x,y,X\cap (-\infty,z]\big).$
The main theorem of the paper is
\begin{thm}\label{mainthm} Fix $\alpha,\beta\in X$, $\alpha\leq \beta$. Suppose that
\begin{enumerate}
\item[(i)] for all $y,z\in X$ the function $\mathcal K(x,y,z)$ is absolutely continuous with respect to $x$ and for every $a,b,y\in X$
$$
\int_X \Big|\frac{\partial}{\partial x}\mathcal K(a,y,z)-\frac{\partial}{\partial x}\mathcal K(b,y,z)\Big|\,dz< 1,
$$
\item[(ii)] there exist constants $\gamma>1$, $C>0$ and $L<1$ such that
$$
\int_X |x|^{\gamma}\mathcal{P}(\mu,\nu)(dx)\le C+L\max\bigg\{\int_X |x|^{\gamma}\mu(dx),\int_X |x|^{\gamma}\nu(dx)\bigg\} .
$$
for every $\mu,\nu\in\mathcal M_{\gamma}\cap\mathcal M_{[\alpha,\beta]}$.
\end{enumerate}
Then for every $\mu_0,\nu_0\in\mathcal M_{[\alpha,\beta]}$ there exists unique solution $\mu,\nu\colon [0,\infty)\to\mathcal M_{[\alpha,\beta]}$ of system $(\ref{sysprobmesconst})$ with initial values $\mu_0,\nu_0$. Moreover, there exists unique measure $\mu^*\in\mathcal M_{[\alpha,\beta]}$ such that $\mathcal P(\mu^*,\mu^*)=\mu^*$, and for every initial measures $\mu_0,\nu_0\in\mathcal M_{[\alpha,\beta]}$ corresponding functions $\mu_t,\nu_t$ converge to $\mu^*$ in space $(\mathcal M_{[\alpha,\beta]},d)$, as $t\to\infty$.
\end{thm}
The concept of the above theorem and idea of its proof come from similar result for hermaphroditic populations (see \cite{rud_zwo}). Nonetheless, the proofs differ in many details and contain some nontrivial and new elements. For the convenience of the reader, we present the full reasoning.
\subsection{Convergence of measures} In order to investigate asymptotic properties of the solutions, we recall some basic theory concerning convergence of measures. We start with method of computing Wasserstein distance, which can be found in \cite{rud_zwo} as Lemma~1.
\begin{lem}
\label{waslem}
The Wasserstein distance between measures
$\mu,\nu\in \mathcal{M}_{1}$ can be computed by the formula
\begin{equation}
d(\mu,\nu)=\int_X |\Phi(x)|\,dx,
\end{equation}
where $\Phi(z)=(\mu-\nu)\left(X\cap(-\infty,z]\right)$ is a cumulative distribution function of the signed measure $\mu-\nu$.
\end{lem}
Consider probability measures $\mu$ and $\mu_n$, $n\in\N$, on the set $X$. We recall that the sequence $\mu_n$ converges \textit{weakly} (or \textit{in a weak sens}) to $\mu$, if for any continuous and bounded function $f\colon X\to\R$
$$\int_X f(x)\,\mu_n(dx)\to\int_X f(x)\,\mu(dx),$$
as $n\to\infty$.
It is well--known that the convergence in Wasserstein distance implies weak convergence of measures. Moreover, the space of probability Borel measures on any complete metric space is also a complete metric space with the Wasserstein distance (see e.g. \cite{bolley, rach}). The convergence of a sequence $\mu_n$ to $\mu$ in the space $\mathcal{M}_{1,q}:=\big\{\mu\in\mathcal M_1\colon \int_X x\mu(dx)=q\big\}$ is equivalent to the following condition (see \cite{villani}, Definition 6.7 and Theorem 6.8)
\begin{equation}\tag{C}\label{warrown}
\mu_n\to\mu \mbox{ weakly, as } n\to\infty \quad \mbox{ and } \quad\lim_{R\to\infty} \limsup_{n\to\infty} \int_{X_R} |x|\mu_n(dx)=0,
\end{equation}
where $X_R:=\{x\in X\colon |x|\geq R \}.$ Fix $q\in X$, $\alpha>1$, and $m>0$.
\begin{lem}\label{relcomp}
Assume that $\alpha,\beta\in X$, $\alpha\leq\beta$, $m>0$ and $\gamma>1$. Consider the family
$$\mathcal M_{[\alpha,\beta],\gamma, m}:=\bigg\{\mu\in \mathcal M_1\colon \alpha\leq \int_X x\mu(dx)\leq \beta, \, \int_X |x|^\gamma\mu(dx)\leq m\bigg\}.$$
Then $\mathcal M_{[\alpha,\beta],\gamma, m}$ is relatively compact subset of $(\mathcal M_{[\alpha,\beta]},d).$
\end{lem}
\begin{proof}
Fix any sequence $(\mu^n)$ of measures from $\mathcal M_{\alpha,\beta,\gamma, m}$. Since $\alpha\leq \E\mu\leq \beta$ there exists a subsequence $(p_n)$ such that $\lim_n \E\mu^{p_n}= E$ for some $\alpha\leq E \leq \beta.$ Consider sequence of measures $(\bar \mu^{p_n})$ given by
$$\bar\mu^{p_n}=\left\{\begin{array}{l l}
a_n \mu^{p_n}+(1-a_n)\delta_\beta, & \textrm{ if }\, \E\mu^{p_n}\leq E,\\
a_n \mu^{p_n}+(1-a_n)\delta_\alpha, & \textrm{ if }\,\E\mu^{p_n}> E,
\end{array}\right.$$
where $a_n\in[0,1]$ satisfies $a_n\E\mu^{p_n}+(1-a_n)\beta =E$ (resp. $a_n\E\mu^{p_n}+(1-a_n)\alpha =E$). From $\lim_n \E\mu^{p_n}= E$, it follows that $a_n\to 1$ and $\lim_n d(\mu^{p_n},\bar \mu^{p_n})=0,$ and moreover
$$\int_X |x|^\gamma \bar\mu^{p_n}(dx)\leq a_n\int_X |x|^\gamma\mu^{p_n}(dx)+(1-a_n)\Big(|\alpha|^\gamma+|\beta|^\gamma\Big)\leq m+|\alpha|^\gamma+|\beta|^\gamma<\infty.$$
Consequently, there exist $\mu^*\in\mathcal M_{1,E}$ and some subsequence $(q_n)$ of $(p_n)$ such that $\lim_n d(\bar \mu^{q_n},\mu^*)=0$ (the condition (C) is satisfied, see remarks after Lemma 1 in \cite{rud_zwo}). Finally,
$$d(\mu^{q_n},\mu^*)\leq d(\mu^{q_n},\bar \mu^{q_n})+d(\bar\mu^{q_n},\mu^*)\to 0,$$
as $n\to\infty$.
\end{proof}
\subsection{Proof of the main result} We split the proof of Theorem \ref{mainthm} into a sequence of lemmas.
\begin{lem}\label{lemma1}
Assume that condition \textup{(i)} of Theorem $\ref{mainthm}$ is satisfied. Then
\begin{equation}\label{kontrakcja}
d\big(\mathcal P(\mu^1,\nu^1), \mathcal P(\mu^2,\nu^2)\big)< \max\big\{d(\nu^1, \nu^2), d(\mu^1, \mu^2)\big\}
\end{equation}
for any $\mu^1,\mu^2,\nu^1,\nu^2\in\mathcal M_1$ such that $\int_X x \mu^1(dx)=\int_X x\mu^2(dx)$ and $\int_X x\nu^1(dx)=\int_X x\nu^2(dx).$
\end{lem}
\begin{proof}
Denote by $\Phi$ the cumulative distribution function of signed measure $\mu_1-\mu_2$, and consider $\Phi^+(x):=\max\{0,\Phi(x)\}$ and $\Phi^-(x):=\max\{0,-\Phi(x)\}.$
The assumption $\int_X x \mu^1(dx)=\int_X x\mu^2(dx)$ implies
$$\int_X \Phi^+(x)dx=\int_X \Phi^-(x)dx=\frac{1}{2}\int_X |\Phi(x)|dx.$$
Since $\Phi^+$ and $\Phi^-$ are non--negative and have the same integral condition (i) implies
$$\int_X\bigg|\int_X \frac{\partial}{\partial x}\mathcal K(x,y,z)\Phi^+(x)dx - \int_X \frac{\partial}{\partial x}\mathcal K(x,y,z)\Phi^-(x)dx \bigg|dz<\int_X \Phi^+(x)dx.$$
Integrating by parts we obtain
$$\int_X\mathcal K(x,y,z)\Phi(dx)=-\int_X \frac{\partial}{\partial x}\mathcal K(x,y,z)\Phi(x)dx$$
and consequently by Lemma \ref{waslem}
\begin{equation}
\int_X\bigg|\int_X\mathcal K(x,y,z)\Phi(dx) \bigg|dz<\frac{1}{2}\int_X |\Phi(x)|dx=\frac{1}{2}d(\mu^1,\mu^2).
\end{equation}
In the same way we prove that if $\Psi$ is the cumulative distribution function of signed measure $\nu^1-\nu^2$, then
\begin{equation}
\int_X\bigg|\int_X\mathcal K(x,y,z)\Psi(dy) \bigg|dz<\frac{1}{2}\int_X |\Psi(y)|dy=\frac{1}{2}d(\nu^1,\nu^2).
\end{equation}
Now, since
\begin{multline*}
\mathcal P(\mu^1,\nu^1)- \mathcal P(\mu^2,\nu^2)\\
=\int_X\int_X k(x,y,dz) \Big(\mu^1(dx)(\nu^1-\nu^2)(dy)+\nu^2(dy)(\mu^1-\mu^2)(dx)\Big),
\end{multline*}
we finally obtain
\begin{multline*}
d\big(\mathcal P(\mu^1,\nu^1), \mathcal P(\mu^2,\nu^2)\big)\leq \int_X\int_X\bigg|\int_X \mathcal K(x,y,dz)\Psi(dy)\bigg|dz\mu^1(dx)\\
+\int_X\int_X\bigg|\int_X \mathcal K(x,y,dz)\Phi(dx)\bigg|dz\nu^2(dx)<\frac{ d(\nu^1, \nu^2) + d(\mu^1, \mu^2)}{2},
\end{multline*}
which implies (\ref{kontrakcja}).
\end{proof}
\begin{lem}\label{lemma2} Assume that $(\ref{kontrakcja})$ is satisfied for all $\mu^1,\mu^2,\nu^1,\nu^2\in\mathcal M_1$ such that $\int_X x \mu^1(dx)=\int_X x\mu^2(dx)$ and $\int_X x\nu^1(dx)=\int_X x\nu^2(dx).$ Fix $\mu^1_0,\mu^2_0,\nu^1_0,\nu^2_0\in\mathcal M_1$ satisfying $\int_X x \mu_0^1(dx)=\int_X x\mu^2_0(dx)$ and $\int_X x\nu^1_0(dx)=\int_X x\nu^2_0(dx).$ Denote by $(\mu^1_t,\nu^1_t)$ and $(\mu^2_t,\nu^2_t)$ solutions of system $(\ref{sysprobmesconst})$ with initial conditions $(\mu^1_0,\nu^1_0)$ and $(\mu^2_0,\nu^2_0)$, respectively. Then $\mu^1_t,\nu^1_t,\mu^2_t,\nu^2_t\in\mathcal M_{[\alpha,\beta]}$ for some $\alpha,\beta\in X$ and any $t>0$, and
\begin{equation}\label{zwezanie}
\max\big\{d(\nu^1_s, \nu^2_s), d(\mu^1_s, \mu^2_s)\big\}>\max\big\{d(\nu^1_t, \nu^2_t), d(\mu^1_t, \mu^2_t)\big\}
\end{equation}
for $0\leq s<t\leq T$ provided $\mu^1_T\neq \mu^2_T$ and $\nu^1_T\neq \nu^2_T.$
\end{lem}
\begin{proof}
Taking the mean value of both sides of equations in (\ref{sysprobmesconst}), from condition (\ref{sredniaK}) we obtain
\begin{equation}\label{meansys}
\left\{\begin{array}{l}m'(t)=\frac{1}{2}\big(n(t)-m(t)\big), \vspace*{0.2cm}
\\ n'(t)=\frac{A}{2}\big(m(t)-n(t)\big).\end{array}\right.
\end{equation}
where $m(t)=\int_X x\mu_t(dx)$ and $n(t)=\int_X x\nu_t(dx)$. From above system we obtain $m(t)-n(t)=(m_0-n_0)e^{-(1+A)t/2}.$
Thus, if $m_0\geq n_0$ (resp. $m_0<n_0$), then from (\ref{meansys}) function $m(t)$ decreases (resp. increases) and $n(t)$ increases (resp. decreases), so $m(0)\geq m(t)\geq n(t)\geq n(0)$ (resp. $m(0)\leq m(t)\leq n(t)\leq n(0)$). Consequently, $\mu_t,\nu_t\in\mathcal M_{[\alpha,\beta]}$ for any $t>0$, where $\alpha:=\min\{m(0),n(0)\}$ and $\beta:=\max\{m(0),n(0)\}$.
Fix $s,t$ such that $0\leq s<t\leq T$. Every solution of system (\ref{sysprobmesconst}) is of the following form
\begin{equation}
\left\{\begin{array}{l}\mu_t=e^{s-t}\mu_s+\int_s^t e^{r-t}\mathcal P (\mu_r,\nu_r)dr, \vspace*{0.2cm}\\
\nu_t=e^{A(s-t)}\nu_s+A\int_s^t e^{A(r-t)}\mathcal P (\nu_r,\mu_r)dr.
\end{array}\right.
\end{equation}
From Lemma (\ref{lemma1}) we obtain
\begin{equation}\label{h1}
\left\{\begin{array}{l}e^t d(\mu^1_t,\mu^2_t)< e^{s}d(\mu^1_s,\mu^2_s)+\int_s^t e^{r}\max\big\{d(\nu^1_r,\nu^2_r),d(\mu^1_r,\mu^2_r)\big\}dr,\vspace*{0.2cm} \\
e^{At}d(\nu^1_t,\nu^2_t)< e^{As}d(\nu^1_s,\nu^2_s)+A\int_s^t e^{Ar}\max\big\{d(\nu^1_r,\nu^2_r),d(\mu^1_r,\mu^2_r)\big\}dr.
\end{array}\right.
\end{equation}
We divide the interval $[0,T]$ into subintervals $I_n:=[t_n,t_{n+1}]$ such that the sign of the difference $d(\mu^1_t,\mu^2_t)-d(\nu^1_t,\nu^2_t)$ is fixed for every $t\in I_n$. Consider any such interval $I_n$ and suppose for example that $d(\mu^1_t,\mu^2_t)\leq d(\nu^1_t,\nu^2_t)$ for every $t\in I_n$.
From the second inequality of (\ref{h1}) we obtain for $s,t\in I_n$
\begin{multline}
e^{At}\max\big\{d(\nu^1_t,\nu^2_t), d(\mu^1_t,\mu^2_t)\big\} \\
<e^{As}\max\big\{d(\nu^1_s,\nu^2_s), d(\mu^1_s,\mu^2_s)\big\}+A\int_s^t e^{Ar}\max\big\{d(\nu^1_r,\nu^2_r),d(\mu^1_r,\mu^2_r)\big\}dr,
\end{multline}
and from Gronwall inequality we obtain (\ref{zwezanie}) for $t,s\in I_n$. If in any interval the inequality is reversed, i.e. $d(\mu^1_t,\mu^2_t)\geq d(\nu^1_t,\nu^2_t)$ for $t\in I_n$, then we use the first inequality of (\ref{h1}) and again from Gronwall inequality we obtain (\ref{zwezanie}). Since intersection of intervals $I_n$ and $I_{n+1}$ is nonempty, (\ref{zwezanie}) holds for any $0\leq s<t\leq T$.
\end{proof}
\begin{lem}\label{lem5}
Assume that condition $(ii)$ from Theorem $\ref{mainthm}$ is satisfied. Then for every pair of measures $\mu_0,\nu_0\in\mathcal M_\gamma$ the orbits $\mathcal O(\mu_0), \mathcal O(\nu_0)$ are relatively compact subsets of $\mathcal M_{[\alpha,\beta]}$. Moreover, $\textup{cl}\,\mathcal O(\mu_0), \textup{cl}\,\mathcal O(\nu_0)\subset\mathcal M_{[\alpha,\beta]} \cap\mathcal M_\gamma,$ where $\textup{cl}$ is closure in $(\mathcal M_{[\alpha,\beta]},d).$
\end{lem}
\begin{proof}
Fix $\mu_0,\nu_0\in\mathcal M_\gamma$ such that $\int_X |x|^\gamma \mu_0(dx), \int_X |x|^\gamma \nu_o(dx)\leq m_0$ for some $m_0>0$. Since form Lemma \ref{lemma2} we obtain $\alpha\leq\int_Xx\mu_t(dx),\int_Xx\nu_t(dx)\leq \beta$ for some $\alpha,\beta\in X$ and every $t>0$, then the relative compactness of orbits follows from Lemma \ref{relcomp}
provided we prove the following upper--bounds
\begin{equation}
\int_X |x|^\gamma \mu_t(dx), \int_X |x|^\gamma \nu_t(dx)\leq M,
\end{equation}
for some $M\geq m_0$ and any $t>0$.
Notice that the set $Y:=C\Big([0,T], \mathcal{M}_{ [\alpha,\beta],\gamma, M}\times\mathcal{M}_{ [\alpha,\beta],\gamma, M}\Big)$ is a closed subset of $C\Big([0,T], \mathcal{M}_{[\alpha,\beta]}\times \mathcal{M}_{[\alpha,\beta]}\Big)$, and map $\Lambda(\mu,\nu)_t=\big(\Lambda^1(\mu,\nu)_t,\Lambda^2(\mu,\nu)_t\big)$, where
$\Lambda^1(\mu,\nu)_t=e^{-t}\mu_0+\int_0^t e^{r-t}\mathcal P (\mu_r,\nu_r)dr$ and
$\Lambda^2(\mu,\nu)_t=e^{-At}\nu_0+A\int_0^t e^{A(r-t)}\mathcal P (\mu_r,\nu_r)dr$
is contraction for sufficiently small $T>0$, whose unique fixed point is $t\mapsto(\mu_t,\nu_t)$. We will show that the set $Y$ is invariant with respect to $\Lambda$, i.e., $\Lambda(Y)\subset Y$ for some constant $M>0$.
We calculate
\begin{multline*}
\int_X |x|^\gamma \Lambda^1(\mu,\nu)_t(dx)=e^{-t}\int_X |x|^\gamma\mu_0(dx)+\int_0^t e^{r-t}\int_X|x|^\gamma\mathcal P(\mu_r,\nu_r)dr\\
\leq e^{-t}\int_X |x|^\gamma\mu_0(dx)+\int_0^t e^{r-t}\bigg(C+L\max\bigg\{\int_X|x|^\gamma\mu_r(dx),\int_X|x|^\gamma\nu_r(dx)\bigg\}\bigg)dr\\
\leq e^{-t}\int_X |x|^\gamma\mu_0(dx)+\int_0^t e^{r-t}\Big(C+LM\Big)dr\leq M,
\end{multline*}
for $M$ such big that $C+LM\leq M.$ In the same way, we show that $\int_X |x|^\gamma \Lambda^2(\mu,\nu)_t(dx)\leq M$ for some constant $M>0$. Consequently, orbits are relatively compact.
\end{proof}
Consider family $\big(S(t)\big)_{t\geq 0}$ of transformations of $\mathcal M_{[\alpha,\beta]}\times\mathcal M_{[\alpha,\beta]}$ given by the formula $S(t)(\mu_0,\nu_0)=(S_1(t)\mu_0,S_2(t)\nu_0)=(\mu_t,\nu_t),$ where $(\mu_t,\nu_t)$ is the solution of system (\ref{sysprobmesconst}) with initial condition $(\mu_0,\nu_0)$. Consider $\omega$--limit set for $\mu,\nu\in\mathcal M_{[\alpha,\beta]},$ i.e.,
$$\omega(\mu,\nu)=\Big\{(\bar\mu,\bar\nu)\colon (\bar\mu,\bar\nu)=\lim_{n\to\infty} (\mu_{t_n},\nu_{t_n}) \textrm{ for some sequence } (t_n)_{n\in\N}\textrm{ s.t. }t_n\to\infty \Big\}.$$
\begin{proof}[Proof of Theorem \ref{mainthm}]
Take measures $\mu,\nu\in\mathcal M_{[\alpha,\beta]}\cap\mathcal M_{\gamma}$. From Lemma \ref{lem5} the orbits $\mathcal O(\mu),\mathcal O(\nu)$ are relatively compact in $\mathcal M_{[\alpha,\beta]}$. Consequently $\omega(\mu,\nu)$ is nonempty and compact set. Moreover, for $t>0$ $S(t)(\omega(\mu,\nu))=\omega(\mu,\nu).$ Suppose that $\omega(\mu,\nu)$ has more than one element. Then we can find $(\mu_1,\nu_1)$ and $(\mu_2,\nu_2)$ which maximize the function $\max\big\{d(\mu_1,\mu_2),d(\nu_1,\nu_2)\big\}.$ For any $t>0$ there exist $(\bar\mu_1,\bar\nu_1)$ and $(\bar\mu_2,\bar\nu_2)$ such that $S(t)(\bar\mu_1,\bar\nu_1)=(\mu_1,\nu_1)$ and $S(t)(\bar\mu_2,\bar\nu_2)=(\mu_2,\nu_2).$ From condition (i), Lemma \ref{lemma1} and Lemma \ref{lemma2} we obtain
\begin{multline}\label{zwww}
\max\big\{d(\mu_1,\mu_2),d(\nu_1,\nu_2)\big\}
=\max\big\{d\big(S_1(t)(\bar\mu_1,\bar\nu_1),S_1(t)(\bar\mu_2,\bar\nu_2)\big),\\
d\big(S_2(t)(\bar\mu_1,\bar\nu_1),S_2(t)(\bar\mu_2,\bar\nu_2)\big)\big\}<\max\big\{d(\bar \mu_1,\bar \mu_2),d(\bar \nu_1,\bar\nu_2)\big\}.
\end{multline}
Inequality (\ref{zwww}) contradicts the definition of $(\mu_1,\nu_1)$ and $(\mu_2,\nu_2)$. Hence $\omega(\mu,\nu)=\{(\mu^*,\nu^*)\},$ and $S(t)(\mu^*,\nu^*)=(\mu^*,\nu^*)$ for every $t>0$. Consequently, $\mu^*=\mathcal P(\mu^*,\nu^*)=\nu^*$. According to Lemma \ref{lemma1} operator $\mathcal P$ has only one fixed point $(\mu^*,\mu^*),$ so the limit $\lim_{t\to\infty} S(t)(\mu,\nu)$ does not depend on $\mu,\nu\in\mathcal M_{[\alpha,\beta]}\cap\mathcal M_{\gamma}$. Consider now any measures $\mu,\nu\in\mathcal M_{[\alpha,\beta]}.$ Since the set $\mathcal M_{[\alpha,\beta]}\cap\mathcal M_{\gamma}$ is dense in $\mathcal M_{[\alpha,\beta]}$, for every $\varepsilon>0$ there exists $\bar\mu,\bar\nu\in\mathcal M_{[\alpha,\beta]}\cap\mathcal M_{\gamma}$ such that $d(\mu,\bar\mu),d(\nu,\bar\nu)<\varepsilon$. Since $\lim_{t\to\infty} S(t)(\bar \mu,\bar\nu)=(\mu^*,\mu^*)$ there exists $t_\varepsilon$ such that $d\big(S_1(t)(\bar\mu,\bar\nu),\mu^*\big)<\varepsilon$ and $d\big(S_2(t)(\bar\mu,\bar\nu),\mu^*\big)<\varepsilon$ for every $t>t_\varepsilon$. Since from Lemma \ref{lemma2} $S_1(t)$ are contractions we obtain
$$d\big(S_1(t)(\mu,\nu),\mu^*\big)\leq d\big(S_1(t)(\mu,\nu), S_1(t)(\bar\mu,\bar\nu)\big)+d\big(S_1(t)(\bar\mu,\bar\nu),\mu^*\big)<2\varepsilon,$$
and similarly $d\big(S_2(t)(\mu,\nu),\mu^*\big)<2\varepsilon$ for $t>t_\varepsilon$, which completes the proof.
\end{proof}
\subsection{Corollaries and further theorems on stability}
We start with investigation on the mean value of the limiting distribution $\mu^*$.
\begin{col} Under assumptions of Theorem~\ref{mainthm},
\begin{equation}\label{limitingmean}
\int_X x\mu^*(dx)=\frac{A\int_Xx\mu_0(dx)+\int_Xx\nu_0(dy)}{A+1}.
\end{equation}
\end{col}
\begin{proof}
Denote $\bar x:=\int_X x\mu^*(dx)$, $m(t):=\int_X x\mu_t(dx)$ and $n(t):=\int_X x\nu_t(dx)$. Since $\mu_t,\nu_t\to\mu^*$ and $\mu_t,\nu_t\in\mathcal M_{[\alpha,\beta]}$, we have also $m(t), n(t)\to \bar x$ as $t\to\infty$. From (\ref{meansys}) it follows that $\frac{d}{dt}\Big(Am(t)+n(t)\Big)=0.$ Consequently, $$Am(0)+n(0)=\lim_{t\to\infty}\Big(Am(t)+n(t)\Big)=(A+1)\bar x,$$
which gives formula (\ref{limitingmean}).
\end{proof}
Now we proceed to a result on asymptotic stability of solutions in stronger convergence. The proof of following result can be adapted from proof of Theorem~4 in \cite{rud_zwo}.
\begin{col}\label{strongconv}
Assume that the measure $k(x,y,dz)$ has bounded and continuous density with respect to Lebesgue measure, and suppose that assumptions of Theorem~\ref{mainthm} are satisfied. Then the stationary measure $\mu^*$ has continuous and bounded density $u^*$ with respect to Lebesgue measure. Moreover, for every $\mu_0,\nu_0\in\mathcal M_{[\alpha,\beta]}$, corresponding solutions $\mu_t,\nu_t$ of $(\ref{sysprobmesconst})$ can be written in the form
$\mu_t=e^{-t}\mu_0+\bar \mu_t,$ $\nu_t=e^{-At}\nu_0+\bar \nu_t,$ where $\bar\mu_t, \bar\nu_t$ are absolute continuous measures with respect to Lebesgue measure, whose densities are continuous and bounded and converge to $u^*$ uniformly, as $t\to\infty$.
\end{col}
Until now, we studied asymptotic properties of simpler system (\ref{sysprobmesconst}) whose all coefficients are constant. Now we investigate asymptotic properties of solutions to the initial system (\ref{sysprobmes}) by comparing them with proper solutions of (\ref{sysprobmesconst}). Similar idea and techniques were previously used in \cite{rudmac}.
\begin{thm}\label{finalthm}
Suppose that $(\ref{persis})$ is satisfied and conditions \textup{(i)}, \textup{(ii)} of Theorem~\ref{mainthm} hold with $\alpha=\beta$. Then there exists $\mu^*\in\mathcal M_{1,\alpha}$ such that for any $\mu_0,\nu_0\in\mathcal M_{1,\alpha}$ coordinates of solution $(\mu_t,\nu_t)$ of system $(\ref{sysprobmes})$ with initial value $(\mu_0,\nu_0)$ converge to $\mu^*$ in $\mathcal M_{1,\alpha}$ as $t\to\infty$.
\end{thm}
\begin{proof}
Fix $s\geq 0$ and let $(\mu^1_t,\nu^1_t)$ and $(\mu^2_t,\nu^2_t)$ be solutions of (\ref{sysprobmesconst}) and (\ref{sysprobmes}) with the same initial condition $(\mu_s,\nu_s)$.
Then $\int_X x\mu^1_t(dx)=\int_X x\mu^2_t(dx)$ and $\int_X x\nu^1_t(dx)=\int_X x\nu^2_t(dx)$ for any $t\geq s$, and consequently from (\ref{sysprobmes}), (\ref{sysprobmesconst}) and (\ref{kontrakcja}) it follows that
\begin{equation}\label{hh}
\left\{\begin{array}{l}e^t d(\mu^1_t,\mu^2_t)\!<\!e^s d(\mu^1_s,\mu^2_s)\!+\!\int_s^t e^{r}\max\big\{d(\nu^1_r,\nu^2_r),d(\mu^1_r,\mu^2_r)\big\}dr,\vspace*{0.2cm} \\
e^{At}d(\nu^1_t,\nu^2_t)\!<\!e^{As} d(\nu^1_s,\nu^2_s)\!+\!A\int_s^t e^{Ar}\max\big\{d(\nu^1_r,\nu^2_r),d(\mu^1_r,\mu^2_r)\big\}dr\!+\!e^{At}G(s,t),
\end{array}\right.
\end{equation}
where $G(s,t):=c\big|e^{-\int_s^t A(w)dw}-e^{A(s-t)}\big|+c\int_s^t \big|A(r)e^{-\int_r^t A(w)dw}-Ae^{A(r-t)}\big|dr$ and $c>0$ is a constant such that $d\big(\nu_r^2,0\big), d\big(\mathcal P(\mu^2_r,\nu^2_r),0\big)<c$ for every $r>0$ (such constant $c$ exists, because $\nu_r^2,\mathcal P(\mu^2_r,\nu^2_r)\in\mathcal M_1$ for every $r>0$ due to assumption (\ref{ksr1})).
We divide the interval $[0,\infty)$ into a sequence of subintevals $I_n$ of lengths $|I_n|\leq 1$ such that the sign of the difference $d(\mu^1_r,\mu^2_r)-d(\nu^1_r,\nu^2_r)$ is fixed for every $r\in I_n$. Fix $n\in\N$. From appropriate inequality from (\ref{hh}), by Gronwall lemma we obtain for $s_n,t_n\in I_n$
\begin{equation}\label{gr}
\max\big\{d(\nu^1_{t_n},\nu^2_{t_n}),d(\mu^1_{t_n},\mu^2_{t_n})\big\}\!<\!\max\big\{d(\nu^1_{s_n},\nu^2_{s_n}),d(\mu^1_{s_n},\mu^2_{s_n})\big\}\!+\!H(s_n,t_n),
\end{equation}
where $H(s,t)=e^{A(t-s)}G(s,t)$. Notice that for any $s_n,t_n\in I_n,$ $s_n\leq t_n$
\begin{multline*}
H(s_n,t_n)\!\leq \!c\Big|e^{-\int_{s_n}^{t_n} \big(A(w)-A\big)dw}-1\Big|\!+\!c\int_{s_n}^{t_n} \Big|A(r)-A\Big|dr\!+\!cA\int_{s_n}^{t_n}\Big|e^{-\int_r^{t_n} \big(A(w)-A\big)dw}-1\Big|dr\\
\leq 2c \int_{s_n}^{t_n} \Big|A(w)-A\Big|dw+cA\int_{s_n}^{t_n}\int_r^{t_n}\Big|A(w)-A\big|dwdr\leq C\int_{s_n}^{t_n} \Big|A(w)-A\Big|dw
\end{multline*}
where $C=c\big(2+A\big)$. Since $\mu^1_s=\mu^2_s=\mu_s$ and $\nu^1_s=\nu^2_s=\nu_s$, from above inequality and (\ref{gr}) we obtain
\begin{equation}
\max\big\{d(\nu^1_t,\nu^2_t),d(\mu^1_t,\mu^2_t)\big\}<C\int_s^t \Big|A(w)-A\Big|dw.
\end{equation}
Notice that since $\big(M(t),F(t)\big)$ converges to $(\bar M,\bar F)$ exponentially, as $t\to\infty$, also $A(t)=M(t)/F(t)$ tends to $A=\bar M/\bar F$ exponentially as well, i.e., there exists constants $a,b>0$ such that $|A(t)-A|\leq ae^{-bt}$ for all $t\geq 0$.
Fix $\varepsilon>0$ and take $s>0$ such that $Ce^{-bs}/b<\varepsilon/4$. Let $(\mu_t,\nu_t)$ be solution of (\ref{sysprobmes}) with initial value $(\mu_0,\nu_0)$ and $(\bar \mu_t,\bar \nu_t)$ be solution of (\ref{sysprobmesconst}) such that $(\bar \mu_s,\bar \nu_s)=(\mu_s,\nu_s)$. Then for a large enough $t>s$ that $\max\big\{d(\mu^*,\bar\mu_t),d(\mu^*,\bar\nu_t)\big\}\leq \varepsilon/2$
\begin{multline*}
\max\big\{d(\nu_t,\mu^*),d(\mu_t,\mu^*)\big\}<\max\big\{d(\nu_t,\bar \nu_t),d(\mu_t,\bar \mu_t)\big\}+\max\big\{d(\mu^*,\bar\mu_t),d(\mu^*,\bar\nu_t)\big\}\\
\leq \frac{C}{b}\big(e^{-bs}-e^{-bt}\big)+\max\big\{d(\mu^*,\bar\mu_t),d(\mu^*,\bar\nu_t)\big\}\leq \varepsilon,
\end{multline*}
which completes the proof.
\end{proof}
Combining Corollary~\ref{strongconv} and Theorem~\ref{finalthm}, one can deduce the following
\begin{col}
Assume that the measure $k(x,y,dz)$ has bounded and continuous density with respect to Lebesgue measure and suppose that assumptions of Theorem~\ref{finalthm} are satisfied. If $u_0,v_0$ are bounded and continuous densities of initial measures $\mu_0,\nu_0$ of solutions $\mu_t,\nu_t$ to $(\ref{sysprobmes})$, then the densities of the measures $\mu_t,\nu_t$ converge uniformly to continuous and bounded density $u^*$ of stationary measure $\mu^*$, as $t\to\infty$.
\end{col}
\section{Examples}
Following examples come from considerations on hermaphroditic populations (see \cite{rud_zwo}), however they are also biologically reasonable for the two--sex populations case.
\subsection{Inheritance of mean parental trait with additive noise}
We suppose that $X=\R$. If $x,y\in \R$ are traits of parents, then we suppose that $\frac{x+y}{2}+Z$ is trait of their offspring, where $Z$ is zero--mean random variable distributed by some density $h$. We assume that $\E Z^2<\infty$ and $h(z)>0$ for all $z\in X$. Then the measure $k(x,y,dz)$ has following density
$$\kappa(x,y,z)=h\bigg(z-\frac{x+y}{2}\bigg).$$ It is easy to check that $\frac{\partial}{\partial x} \mathcal K(x,y,z)=-\frac{1}{2} h\Big(z-\frac{x+y}{2}\Big),$
and condition (i) from Theorem~\ref{mainthm} is satisfied if
$$\int_{-\infty}^\infty \big|h(z-a)-h(z-b)\big|dx<2$$
for all $a,b\in\R$. The above inequality is valid, since $h$ is probability density function, positive everywhere.
Now we proceed to condition (ii). Fix two measures $\mu,\nu\in\mathcal M_{[\alpha,\beta]}$. Then
\begin{multline*}
\int_{-\infty}^\infty z^2\mathcal P(\mu,\nu)(dz)=\int_{-\infty}^\infty\int_{-\infty}^\infty\int_{-\infty}^\infty\bigg(\Big(z-\frac{x+y}{2}\Big)^2+\Big(z-\frac{x+y}{2}\Big)(x+y)+\frac{(x+y)^2}{4}\bigg)\\ \times h\bigg(z-\frac{x+y}{2}\bigg)\mu(dx)\nu(dy)dz\leq \E Z^2+\frac{1}{4}\int_{-\infty}^\infty\int_{-\infty}^\infty(x+y)^2\mu(dx)\nu(dy)\\
=\E Z^2 +\frac{1}{2}\bigg(\int_{-\infty}^\infty x\mu(dx)\bigg)\bigg(\int_{-\infty}^\infty x\nu(dx)\bigg)+\frac{1}{2}\max\bigg\{\int_{-\infty}^\infty x^2\mu(dx), \int_{-\infty}^\infty x^2\nu(dx)\bigg\}.
\end{multline*}
Since $\mu,\nu$ have their first moments upper--bounded by $\beta$, condition (ii) of Theorem~\ref{mainthm} is satisfied with $\gamma=2$, $C=\E Z^2+\beta^2/2$ and $L=\frac{1}{2}.$ From Theorem~\ref{mainthm} there exists unique limiting distribution $\mu^*\in\mathcal M_{[\alpha,\beta]}$ with first moment (\ref{limitingmean}) such that $\mu_t,\nu_t\to\mu^*$ in $\mathcal M_{[\alpha,\beta]}$. If additionally $\alpha=\beta$ and $h$ is bounded and continuous, limiting measure $\mu^*$ has continuous and bounded density $u^*$ and convergence of absolute continuous parts of $\mu_t,\nu_t$ is uniform by Corollary~\ref{strongconv}.
It turns out that, in this case of trait inheritance, it is possible to find the limiting distribution $\mu^*$ explicitly which has the form of infinite sequence of measure convolutions. In the case when $h$ has $0$--mean normal distribution with standard deviation $\sigma$, then the limiting distribution is also normal, with mean $\bar x=\alpha=\beta$ and standard deviation $\sqrt{2}\sigma$ (see \cite{rud_zwo}).
\subsection{Inheritance of mean parental trait with multiplicative noise} The following example is more reasonable for description of non--negative traits such as average body mass or height.
Thus we suppose that $X=[0,\infty)$. If $x,y$ are parental traits, then the trait of offspring is given by $(x+y)Z$, where $Z$ is $[0,1]$--valued random variable with mean $\frac{1}{2}$, distributed by density $h$. Then the density $\kappa(x,y,\cdot)$ of measure $k(x,y,dz)$ has the form
$$\kappa(x,y,z)=\frac{1}{x+y} h\bigg(\frac{z}{x+y}\bigg),$$
for $z\in[0,x+y]$, $x+y>0$ or $\kappa(x,y,z)=0$ otherwise. Assume that there exists $\varepsilon>0$ such that support of the function $h$ contains $(0,\varepsilon)$. One can easily compute
$$\frac{\partial}{\partial x} \mathcal K(x,y,z)=-h\bigg(\frac{z}{x+y}\bigg) \frac{z}{(x+y)^2.}$$
The condition (i) of Theorem~\ref{mainthm} is equivalent to
$$\int_0^\infty \Big|h\Big(\frac{z}{a}\Big)\frac{z}{a^2}-h\Big(\frac{z}{b}\Big)\frac{z}{b^2}\Big|dz<1,$$
for all $a,b\in[0,\infty)$. Above inequality is satisfied, since the function $h$ has mean equal to $\frac{1}{2}$, and the interval $(0,\varepsilon)$ is in its support.
We check condition (ii). Take $\mu,\nu\in\mathcal M_{[\alpha,\beta]}$. Then
\begin{multline*}
\int_0^\infty z^2\mathcal P(\mu,\nu)dz=\E Z^2 \int_0^\infty\int_0^\infty (x+y)^2\mu(dx)\nu(dy)\leq \\
2\E Z^2\bigg(\int_{-\infty}^\infty x\mu(dx)\bigg)\bigg(\int_{-\infty}^\infty x\nu(dx)\bigg)+2\E Z^2\max\bigg\{\int_{-\infty}^\infty x^2\mu(dx), \int_{-\infty}^\infty x^2\nu(dx)\bigg\}.
\end{multline*}
Since the first moments of the measures $\mu,\nu$ are bounded by $\beta$ and $2\E Z^2 <2\E Z=1$, condition (ii) is satisfied with $\gamma=2$, $C=2\beta^2 \E Z^2$ and $L=2\E Z^2$.
\section{Conclusions}
In the paper we introduced some individual--based model in order to describe the evolution of non--sex--liked phenotypic traits in two--sex populations. The model includes semi--random mating of individuals of the opposite sex, natural death and intra--specific competition. Having passed the number of individuals to infinity, we derived the macroscopic system of equations for evolution of trait distributions. The main results of the investigation on solutions of the system are: existence and uniqueness of solutions in space of measures, study of total number of individuals in order to derive criteria for persistence or extinction, formulation of the conditions implying existence of the unique stable distribution and its asymptotic stability. Moreover, under additional assumptions, we studied the existence and asymptotic properties of solutions from the standpoint of their densities.
Now we interpret some of the results in biological language. We start with inequalities (\ref{persis}) and (\ref{extinct}). If we consider the numbers $\alpha_m=p_m/D_m$ and $\alpha_f=p_f/D_f$ as an environmental adaptation of males and females, then (\ref{persis}) implies that the mean environmental adaptation in population is greater than one. On average, one dying individual is replaced by more than one newborn, and consequently whole the population persists. The opposite inequality (\ref{extinct}) leads to extinction of both subpopulations. A possible scenario assumes that one of the numbers $\alpha_m,\alpha_f$ is strictly smaller than $1$. Despite this, it is still possible that (\ref{persis}) holds. It means, that the whole population can survive, although one of the subpopulations has smaller mating rate. In particular, if we take $p_m=0$, then populations with minor male mating rates are also covered by our model. In that case the growth of the population depends on female mating and death rates, and inequality $p_f/D_f>2$ means that the population persists.
The existence and asymptotic stability of the stable distribution $\mu^*$, and the fact that this distribution is the same for both male and female populations is an intuitive consequence of Fisher's principle and inheritance of traits which are non--sex--linked. The result suggests that after a long time two--sex populations behave as they were hermaphroditic, provided we investigate only the evolution of non--sex--linked traits. This result enlarges the area of applications of the hermaphroditic model derived and studied in \cite{rud_zwo} also to two--sex populations.
The future perspectives for the model presented in the paper are multidirectional. The most interesting issue would be to study analogous model including assortative mating of individuals with aid of trait--dependent marriage functions instead of semi--random mating (see e.g. \cite{yang}). Marriage functions reflect preferences for possible partners in the population and influence the shape of the trait distribution. The long--time behavior of the corresponding macroscopic equations could answer how big this influence is in stable population, and what shape of limiting distribution we should expect. Another issue is to study how trait values of individuals affect their fitness. Since in general our model allows to include trait--dependent rates, it would be interesting to study long--time behavior of corresponding solutions in case when there are two different fitness optima for males and females (see e.g. \cite{bond}).
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S.C. Teacher of the Year to visit for Teacher Cadet College Day
The Citadel will host Teacher Cadet College Day on Thursday, Sept. 27, in the Greater Issues Room of Mark Clark Hall.
The program begins at 8:30 a.m. and will include a keynote address from Amy McAllister, the 2013 South Carolina Teacher of the Year. McAllister is an English teacher at Johnsonville High School in Florence School District 5. She will speak at about 10:25 a.m.
Teacher cadets and their teachers from Wando High School, James Island Charter High School, The Military Magnet Academy and Beaufort High School will be in attendance. Education majors from the Corps of Cadets and School of Education professors will also participate.
The Teacher Cadet Program focuses on high school juniors and seniors and is one of several programs founded by the Center for Educator Recruitment, Retention, and Advancement. Established by the Commission on Higher Education in 1985, the CERRA is in its 27th year of operation, and is the oldest and most established teacher recruitment program in the country.
The Citadel has been a Teacher Cadet College partner to several area high schools for more than two decades.
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TITLE: What qualifies as a quantum theory and why are we seeking a quantum theory of gravity?
QUESTION [0 upvotes]: When can a theory be called a quantum theory? Does it have to do with the existence of certain quantities which take discrete values (they increase in quanta)? Or does it have to do with the existence of non-commuting operators?
If yes, then classical mechanics qualifies as a quantum theory. To get standing waves on a string, one can have DISCRETE wavelengths and finite rotations about x and y-axes are NON-COMMUTING.
Also, why are we seeking a quantum theory of gravity?
REPLY [0 votes]: Here you will find a list of the postulates of quantum mechanics.
To have a quantum mechanical theory one needs a wave equation whose solutions are consistent with the postulates of quantum mechanics.
It is the wavefunction postulate that separates quantum mechanical models from classical physics models.
The solution of a wave equation $Ψ(x,t)$ , in space and time is interpreted differently in quantum mechanics, as the probability of finding a particle at a particular time and space.
It is not the energy that is "waving" in a quantum mechanical model, but the probability of finding a particle at a specific space time point is calculated by use of the wavefunction. Same equations and wave functions in classical and quantum mechanics, but different quantities are represented by the solutions. In quantum probabilities, in classical energy densities .
See my answer here for an experimental illustration.
Also, why are we seeking a quantum theory of gravity
For aesthetic reasons, the hypothesis that a mathematical unified theory of all four fundamental forces must exist, and since the three , electromagnetic, weak and strong are quantum theories, gravitation should follow suit. There are various unifying theories for the first three, and the hypothesis is that gravity should also be quantized.
An additional reason is that quantum theories avoid singularities, due to the probabilistic nature. For example the 1/r of the coulomb potential is no problem in quantum mechanics: the electron cannot fall on the proton in the hydrogen atom.
Already in the Big Bang cosmological model effective quantization is used for the beginning of the BB instead of the classical singularity.
| 115,819
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A Douglas Steiner, the developer who bought the church property for $41 million last November, but never heard back. Now the groups are planning a rally for next month.
The application for full demolition was approved by the Department of Buildings yesterday, documents show. Mr. Steiner’s plans call for an 80/20-percent mix of market-rate and affordable housing.
“We’ve reached out to other groups in the neighborhood and we’ll see what the next steps should be,” said Mr. Moses. “At this point, it’s going to be up to Steiner’s goodwill for any chance for this building to survive.” He believes the church could stand alongside new construction. “It should be fairly easy to incorporate the church into the new development. There are millions of examples of churches being converted for residential or commercial use. Why he would not decide to go that route is a mystery to me.”
Andrew Berman, the executive director for the G.V.S.H.P., called the latest development “heartbreaking.” “It is a shame,” he said. “We had asked the Landmarks Preservation Commission to consider the building for landmark designation, given its incredible history and beautiful architecture. They refused to even hold a hearing on it.” The G.V.S.H.P., the L.E.S.P.I., the Historic Districts Council, and the East Village Community Coalition made the request in February; it was rejected because the church didn’t meet the criteria for designation.
Neighborhood residents, some of whom had joined preservationists’ efforts, are also disturbed by the development. Diana Timmons, who lives steps away from the church and mailed a letter asking the Landmarks Preservation Commission to evaluate it, told The Local, “This is such sad news. Now our block will look like any other block in the depressed downtown of any city.”
Janet Bonica, a former parishioner whose grandparents were were married in the church’s chapel in 1909 and whose parents were married there in 1946, told The Local, “We knew this day was coming, but that doesn’t make it any easier to accept this sad news. This is a great loss for those of us with ties to Mary Help of Christians, as well as to those who live in the neighborhood.”
“I never thought I would see the day when my church would be demolished,” she said. “I feel like my past is being erased.”
Margaret Hearn, a longtime former parishioner and neighborhood resident, said former parishioners continue to pray the rosary outside the church. “It continues a familial bond for the parishioners,” she said.
A spokeswoman for Mr. Steiner declined to comment.
2 Comments
Must everything be about the almighty dollar? I grew up in that neighborhood in the 50′s and 60′s. It is now all being eraced. That church has the most beautiful interior. We go to Europe and other countries to see their seights. Their seights are preserved. We will have nothing but high rise buildings and transports into our neighborhoods. I was born on 11 Street between ave A & B. That was a neighborhood. Poor but we had heart and soul. Our places of worship should be saved. The public bath on 11 street and the church were saved. Why not this church. You all better get your priororities straight. The buildings will be structures with no history. It’s a terrible mistake.— Frances Wojcik De Stefano
It should be clarified — the article says that our request that the NYC Landmarks Preservation consider the church for individual landmark status to save it “was rejected because the church didn’t meet the criteria for designation.” This implies that it’s not meeting the criteria is an objective fact. This is far from true.
The LPC’s decision not to consider the church for designation reflects their own subjective interpretation of what that criteria is, and in this and in many other cases, that interpretation may seem quite arbitrary and inconsistent with other decisions the commission has made. Few would argue that this church has sufficient historic and architectural merit to at least warrant consideration for landmark status.
But rather than hold a public hearing and allow all evidence to be weighed and a decision made by the full eleven professional members of the Landmarks Preservation Commission, the Chair of the Commission decided that it did not even merit such an airing, and refused to allow even a public hearing to be held.
- Andrew Berman, Executive Director, Greenwich Village Society for Historic Preservation— Andrew Berman
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\begin{document}
\title[A generalized Hilbert matrix acting on Hardy spaces]
{A generalized Hilbert matrix acting on Hardy spaces}
\author[Ch.~Chatzifountas]{Christos Chatzifountas}
\address{Departamento de An\'alisis Matem\'atico,
Universidad de M\'alaga, Campus de Teatinos, 29071 M\'alaga, Spain}
\email{christos.ch@uma.es}
\author[D.~Girela]{Daniel Girela}
\address{Departamento de An\'alisis Matem\'atico,
Universidad de M\'alaga, Campus de Teatinos, 29071 M\'alaga, Spain}
\email{girela@uma.es}
\author[J.~A.~Pel\'aez]{Jos\'e \'Angel Pel\'aez}
\address{Departamento de An\'alisis Matem\'atico,
Universidad de M\'alaga, Campus de Teatinos, 29071 M\'alaga, Spain}
\email{japelaez@uma.es}
\date{July 4, 2013} \keywords{Hilbert matrices, Hardy spaces, BMOA,
Carleson measures, Integration operators, Hankel operators, Besov
spaces, Schatten classes}
\begin{abstract}\par If $\mu $ is a positive Borel measure on the interval $[0, 1)$,
the Hankel matrix $\mathcal H_\mu =(\mu _{n,k})_{n,k\ge
0}$ with entries $\mu _{n,k}=\int_{[0,1)}t^{n+k}\,d\mu(t)$ induces
formally the operator
$$\mathcal{H}_\mu (f)(z)=
\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty}
\mu_{n,k}{a_k}\right)z^n$$ on the space of all analytic functions
$f(z)=\sum_{k=0}^\infty a_kz^k$, in the unit disc $\D $. In this
paper we describe those measures $\mu$ for which $\h_\mu $ is a
bounded (compact) operator from $H^p$ into $H^q$, $0<p,q<\infty $.
We also characterize the measures $\mu $ for which $\mathcal H_\mu
$ lies in the Schatten class $S_p(H^2)$, $1<p<\infty$.
\end{abstract}
\thanks{This
research is supported by a grant from la Direcci\'{o}n General de
Investigaci\'{o}n, Spain (MTM2011-25502) and by a grant from la
Junta de Andaluc\'{\i}a (P09-FQM-4468 and FQM-210). The third author
is supported also by the \lq\lq Ram\'on y Cajal program\rq\rq ,
Spain.}
\maketitle
\bigskip
\section{Introduction and main results}\label{intro}
Let $\D=\{z\in\C: |z|<1\}$ denote the open unit disc in the complex
plane $\C$ and let $\hol (\D)$ be the space of all analytic
functions in $\D$. We also let $H^p$ ($0<p\le \infty $) be the
classical Hardy spaces (see \cite{D}).
\par
If\, $\mu $ is a finite positive Borel measure on $[0, 1)$ and $n\,
= 0, 1, 2, \dots $, we let $\mu_n$ denote the moment of order $n$ of
$\mu $, that is,
$$\mu _n=\int _{[0,1)}t^n\,d\mu (t),$$
and we define $\mathcal H_\mu $ to be the Hankel matrix $(\mu
_{n,k})_{n,k\ge 0}$ with entries $\mu _{n,k}=\mu_{n+k}$. The matrix
$\mathcal H_\mu $ can be viewed as an operator on spaces of analytic
functions by its action on the Taylor coefficients: \,$ a_n\mapsto
\sum_{k=0}^{\infty} \mu_{n,k}{a_k}, \quad n=0,1,2, \cdots . $\, To
be precise, if\,
$f(z)=\sum_{k=0}^\infty a_kz^k\in \hol (\D )$
we define
\begin{equation}\label{H}
\mathcal{H}_\mu (f)(z)= \sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty}
\mu_{n,k}{a_k}\right)z^n,
\end{equation}
whenever the right hand side makes sense and defines an analytic
function in $\D $.
\par\medskip If $\mu $ is the Lebesgue measure on $[0,1)$ the matrix
$\mathcal H_\mu $ reduces to the classical Hilbert matrix \,
$\mathcal H= \left ({(n+k+1)^{-1}}\right )_{n,k\ge 0}$, which
induces the classical Hilbert operator $\h$, a prototype of a Hankel
operator which has attracted a considerable amount of attention
during the last years. Indeed, the study of the boundedness, the
operator norm and the spectrum of $\h$ on Hardy and weighted
Bergman spaces \cite{AlMonSa,DiS,DJV,GaGiPeSis,PelRathg} links $\h$
up to weighted composition operators, the Szeg\"{o} projection,
Legendre functions and the theory of Muckenhoupt weights.
\par Hardy's
inequality \cite[page~48]{D} guarantees that $\mathcal{H}(f)$ is a
well defined analytic function in $\D$ for every $f\in H^1$.
However, the resulting Hilbert operator $\mathcal{H}$ is bounded
from $H^p$ to $H^p$ if and only if $1<p<\infty$ \cite{DiS}.
In a recent paper \cite{LNP} Lanucha, Nowak, and Pavlovic have
considered the question of finding subspaces of $H^1$ which are
mapped by $\mathcal{H} $ into $H^1$.
\par Galanopoulos and Pel\'{a}ez \cite{Ga-Pe2010} have described
the measures $\mu $ so that the generalized Hilbert operator $\mathcal H_\mu
$ becomes well defined and bounded on $H^1$. Carleson measures
play
a basic role in the work.
\par If $I\subset \partial\D$ is an
interval, $\vert I\vert $ will denote the length of $I$. The
\emph{Carleson square} $S(I)$ is defined as
$S(I)=\{re^{it}:\,e^{it}\in I,\quad 1-\frac{|I|}{2\pi }\le r <1\}$.
Also, for $a\in \D$, the Carleson box $S(a)$ is defined by
\begin{displaymath}
S(a)=\Big \{ z\in \D : 1-|z|\leq 1-|a|,\, \Big |\frac{\arg
(a\bar{z})}{2\pi}\Big |\leq \frac{1-|a|}{2} \Big \}.
\end{displaymath}
\par If $\, s>0$ and $\mu$ is a positive Borel measure on $\D$,
we shall say that $\mu $
is an $s$-Carleson measure
if there exists a positive constant $C$ such that
\[
\mu\left(S(I)\right )\le C{|I|^s}, \quad\hbox{for any interval
$I\subset\partial\D $},
\]
or, equivalently,
if there exists $C>0$ such that
\[
\mu\left(S(a)\right )\le C{(1-\vert a\vert )^s}, \quad\hbox{for all
$a\in\D$}.
\]
If $\mu $ satisfies $\displaystyle{\lim_{\vert I\vert\to 0}\frac{\mu
\left (S(I)\right )}{\vert I\vert ^s}=0}$ or, equivalently,
$\displaystyle{\lim_{\vert a\vert \to 1}\frac{\mu \left (S(a)\right
)}{(1-\vert a\vert^2)^s}=0}$, then we say that $\mu $ is a\, {\it
vanishing $s$-Carleson measure}. \par An $1$-Carleson measure,
respectively, a vanishing $1$-Carleson measure, will be simply
called a Carleson measure, respectively, a vanishing Carleson
measure.
\par As an important ingredient in his work on interpolation by
bounded analytic functions, Carleson \cite{Ca2} (see also Theorem
9.3 of \cite{D}) proved that if $0<p<\infty $ and $\mu$ is a
positive Borel measure in $\D $ then $H\sp p\subset L\sp p(d\mu )$
if and only if \,$\mu $\, is a Carleson measure. This result was
extended by Duren \cite{Du:Ca} (see also \cite[Theorem~9.\@4]{D})
who proved that for $\,0<p\le q<\infty $,
$H^p\subset L^q(d\mu)$
if and only if $\mu$ is a $q/p$-Carleson measure. \par If $X$ is a
subspace of $\hol (\D )$, $0<q<\infty $, and $\mu $ is a positive
Borel measure in $\D $, $\mu $ is said to be a \lq\lq ${q}${\it
{-Carleson measure for the space}} ${X}$\rq\rq \, or an \lq\lq
${{(X, q)}}${\it-Carleson measure}\rq\rq \, if $X\subset L^q(d\mu
)$. The $q$-Carleson measures for the spaces $H^p$, $0<p,q<\infty $
are completely characterized. The mentioned results of Carleson and
Duren can be stated saying that if $\,0<p\le q<\infty $\, then a
positive Borel measure $\mu $ in $\D$ is a $q$-Carleson measure for
$H^p$ if and only if $\mu$ is a $q/p$-Carleson measure. Luecking
\cite{Lu90} and Videnskii \cite{Vid} solved the remaining case
$0<q<p$. We mention \cite{Bl-Ja} for a complete information on
Carleson measures for Hardy spaces.
\par\medskip Galanopoulos and Pel\'{a}ez proved in \cite{Ga-Pe2010}
that if\, $\mu $\, is a Carleson measure then the operator $\mathcal
H_\mu $\, is well defined in $H^1$, obtaining en route the
following integral representation
\begin{equation}\label{HmuImuH1}\mathcal H_\mu
(f)(z)\,=\,\int_{[0,1)}\frac{f(t)}{1-tz}\,d\mu (t),\quad z\in
\mathbb D,\quad\text{for all $f\in H^1$}.\end{equation} For
simplicity, we shall write throughout the paper
\begin{equation}\label{Imu}I_\mu
(f)(z)=\int_{[0,1)}\frac{f(t)}{1-tz}\,d\mu (t),\end{equation}
whenever the right hand side makes sense and defines an analytic
function in $\D $. It was also proved in \cite{Ga-Pe2010} that if
$I_\mu (f)$ defines an analytic function in $\D $ for all $f\in
H^1$, then $\mu $ has to be a Carleson measure. This condition does
not ensures the boundedness of $\h_\mu$ on $H^1$, as the classical
Hilbert operator $\mathcal H$ shows.
\par\medskip
Let $\mu$ be a positive Borel measure in $\D$, $0\le \alpha <\infty
$, and $0<s<\infty $. Following \cite{Zhao}, we say that $\mu$ is an
$\alpha$-logarithmic $s$-Carleson measure, respectively, a vanishing $\alpha$-logarithmic $s$-Carleson measure, if
\begin{displaymath}
\sup_{a\in\D}\frac{\mu\left(S(a)\right)\left (\log
\frac{2}{1-|a|^2}\right )^{\alpha}}{(1-|a|^2)^s}<\infty,
\quad\text{respectively,}\,\,\,\lim_{|a|\to
1^{-}}\frac{\mu\left(S(a)\right)\left (\log \frac{2}{1-|a|^2}\right
)^{\alpha}}{(1-|a|^2)^s}=0. \end{displaymath} \par
Theorem\,\@1.\,\@2 of \cite{Ga-Pe2010} asserts that if $\mu $ is a
Carleson measure on $[0,1)$, then $\mathcal H_\mu$ is a bounded
(respectively, compact) operator from $H^1$ into $H^1$ if and only
if $\mu $ is a $1$-logarithmic $1$-Carleson measure (respectively, a
vanishing $1$-logarithmic $1$-Carleson measure). \par It is also
known that $\mathcal H_\mu $ is bounded from $H^2$ into itself if
and only if $\mu $ is a Carleson measure (see \cite[p. 42, Theorem
$7.2$]{Pell}).
\par\medskip Our
main aim in this paper is to study the generalized Hilbert matrix
$\mathcal H_\mu $ acting on $H^p$ spaces $(0<p<\infty $). Namely,
for any given $p, q$ with $0<p, q<\infty $, we wish to characterize
those for which $\h_\mu $ is a bounded (compact) operator from $H^p$
into $H^q$ and to describe those measures $\mu$ such that $\h_\mu$
belongs to the Schatten class $\mathcal S_p(H^2)$. A key tool will
be a description of those positive Borel measures $\mu $ on $[0,1)$
for which $\h_\mu $ is well defined in $H^p$ and
$\h_\mu(f)=I_\mu(f)$.
Let us
start with the case $p\le 1$.
\begin{theorem}\label{Def:p<1} Suppose that $0<p\le 1$ and let $\mu
$ be a positive Borel measure on $[0,1)$. Then the following two
conditions are equivalent:
\begin{itemize}
\item[(i)] $\mu $ is an
$\frac{1}{p}$-Carleson measure.
\item[(ii)] $I_\mu (f)$ is a well
defined analytic function in $\D $ for any $f\in H^p$.
\end{itemize}
\par Furthermore, if (i) and (ii) hold and $f\in H^p$, then $\mathcal H_\mu (f)$ is
also a well defined analytic function in $\D $, and $\mathcal H_\mu
(f)=I_\mu (f)$, for all $f\in H^p.$
\end{theorem}
\par\medskip
We remark that for $p=1$, this reduces to
\cite[Proposition\,\@1.\,\@1]{Ga-Pe2010}. \par\medskip For $0<q<1$,
we let $B_q$ denote the space consisting of those $f\in \hol (\D )$
for which
$$\int_0^1\,(1-r)^{\frac{1}{q}-2}M_1(r,f)\,dr<\infty .$$ The Banach space
$B_q$ is the \lq\lq containing Banach space\rq\rq \, of $H^q$, that
is, $H^q$ is a dense subspace of $B_q$, and the two spaces have the
same continuous linear functionals \cite{DRS}. Next we shall show
that if $\mu $ is an $1/p$-Carleson measure then $\mathcal H_\mu $
actually applies $H^p$ into $B_q$ for all $q<1$. We shall also give
a characterization of those $\mu $ for which $\mathcal H_\mu $ map
$H^p$ into $H^q$ ($q\ge 1$). Before stating these results precisely,
let us mention that all over the paper we shall use the notation
that for any given $\alpha
>1$, $\alpha^\prime $ will denote the conjugate exponent of $\alpha
$, that is, $\frac{1}{\alpha }+\frac{1}{\alpha ^\prime }=1$, or
$\alpha^\prime =\frac{\alpha }{\alpha -1}$.
\begin{theorem}\label{bound:p<1}
Suppose that $0<p\le 1$ and let $\mu $
be a positive Borel measure on $[0,1)$ which is an
$\frac{1}{p}$-Carleson measure.
\begin{itemize}\item[(i)] If\, $0<q<1$, then $\h_\mu $ is a bounded
operator from $H^p$ into $B_q$, the containing Banach space of
$H^q$.
\item[(ii)] $\h_\mu $ is a bounded
operator from $H^p$ into $H^1$ if and only if $\mu $ is an
$1$-logarithmic $\frac{1}{p}$-Carleson measure.
\item[(iii)] If\, $q>1$ then
$\h_\mu $ is a bounded operator from $H^p$ into $H^q$ if and only if
$\mu $ is an $\frac{1}{p}+\frac{1}{q^\prime}$-Carleson measure.
\end{itemize}
\end{theorem}
\par\medskip Let us state next our results for $p>1$.
\begin{theorem}\label{Def:p>1}
Suppose that $1<p<\infty $ and let $\mu $ be a positive Borel
measure on $[0,1)$. Then:
\par (i) $I_\mu (f)$ is a well defined analytic function in $\D $
for any $f\in H^p$ if and only if $\mu $ is an $1$-Carleson measure
for $H^p$, or, equivalently, if and only if
\begin{equation}\label{int-car-1p}\int_0^1\,\left (\int_0^{1-s}\,\frac{d\mu (t)}{1-t}\right
)^{p^\prime }\,ds\,<\,\infty .\end{equation}
\par (ii) If $\mu $ satisfies
(\ref{int-car-1p}) then
$\mathcal H_\mu (f)$ is also a well defined analytic function in $\D
$,
whenever $f\in H^p$, and
$$\mathcal H_\mu (f)=I_\mu (f),\quad\text{for every
$f\in H^p$.}$$
\end{theorem}
\par\medskip
\begin{theorem}\label{bound:p>1}
Suppose that $1<p<\infty $ and let $\mu $ be a positive Borel
measure on $[0,1)$ which satisfies .
\begin{itemize}\item[(i)] If $0<p\le q<\infty $, then $\mathcal H_\mu $
is a bounded operator from $H^p$ to $H^q$ if and only if $\mu $ is
an $\frac{1}{p}+\frac{1}{q^\prime }$-Carleson measure.
\item[(ii)] If $1<q<p$, then $\mathcal H_\mu $
is a bounded operator from $H^p$ to $H^q$ if and only if the
function defined by \,\,$s\mapsto \int_0^{1-s}\,\frac{d\mu
(t)}{1-t}$\,\, $(s\in [0,1))$\, belongs to $L^{\left
(\frac{pq^\prime }{p+q^\prime }\right )^\prime }([0,1))$.
\item[(iii)] $\mathcal H_\mu $
is a bounded operator from $H^p$ to $H^1$ if and only if the
function defined by \,\,$s\mapsto
\int_0^{1-s}\,\frac{\log\frac{1}{1-t}d\mu (t)}{1-t}$\,\, $(s\in
[0,1))$\, belongs to $L^{p^\prime }([0,1))$.
\item[(iv)] If $0<q<1$, then $\mathcal H_\mu $
is a bounded operator from $H^p$ into $B_q$.
\end{itemize}
\end{theorem}
\par\medskip
Let us remark that both if either $0<p\le 1$ and $\mu $ is an
$1/p$-Carleson measure, or if $1<p<\infty $ and $\mu $ satisfies
(\ref{int-car-1p}), we have that $\mu $ is an $1$-Carleson measure
for $H^p$. By the closed graph theorem it follows
that, for any $q>0$,
$$\mathcal H_\mu (H^p)\subset H^q\,\,\,\Leftrightarrow\,\,\,
\mathcal H_\mu \,\,\,\text{is a bounded operator form $H^p$ into
$H^q$.}$$
\par\medskip Substitutes of Theorem\,\@\ref{bound:p<1} and
Theorem\,\@\ref{bound:p>1} regarding compactness will be stated and
proved in Section\,\@\ref{sect-compact}.
\par\medskip Finally, we address the question of describing those measures $\mu$ such that $\h_\mu$ belongs to the Schatten class
$\mathcal S_p(H^2)$, ($1<p<\infty $). Given a separable Hilbert
space $X$ and
$0<p<\infty$, let $\mathcal{S}_ p(X)$ denote the
Schatten $p$-class of operators on $X$. The class $\mathcal{S}_
p(X)$ consists of those compact operators $T$ on $X$ whose sequence
of singular numbers $\{ \lambda_ n\} $ belongs to $\ell^p$, the
space of $p$-summable sequences. It is well known that, if $\lambda_
n$ are the singular numbers of an operator $T$, then
\begin{displaymath}
\lambda_ n=\lambda_ n(T)=\inf \{\|T-K\|: \textrm{ rank}\,K\leq n\}.
\end{displaymath}
Thus finite rank operators belong to every $\mathcal{S}_ p(X)$, and
the membership of an operator in $\mathcal{S}_ p(X)$ measures in
some sense the size of the operator. In the case when $1\leq
p<\infty$, $\mathcal{S}_ p(X)$ is a Banach space with the norm
\begin{displaymath}
\|T\|_ p=\left(\sum_ n |\lambda_ n|^p\right )^{1/p},
\end{displaymath}
while for $0<p<1$ we have the following inequality $\|
T+S\|_p^p\le\|T\|_ p^p+\|S\|_ p^p.$ We refer to \cite{Zhu} for more
information about $\mathcal{S}_ p(X)$.
\par\medskip Galanopoulos
and Pel\'{a}ez \cite[Theorem\,\@1.\,\@6]{Ga-Pe2010} found a
characterization of those $\mu $ for which $\h_\mu $ is a
Hilbert-Schmidt operator on $H^2$ improving a result of \cite{Pow}.
In \cite[p. $239$, Corollary\,\@2.\,\@2]{Pell} it is proved that,
for $1<p<\infty $, $\mathcal H_\mu \in \mathcal S_p(H^2)$ if and
only if $h_\mu(z)=\sum_{n=1}^\infty \mu_{n+1}z^{n}$ belongs to the
Besov space $B^p$ (see \cite[Chapter\,\@5]{Zhu}) of those analytic
functions $g$ in $\D$ such that
$$||g||^p_{B^p}=|g(0)|^p+\int_\D |g'(z)|^p(1-|z|^2)^{p-2}\,dA(z)<\infty.$$
We simplify this result
describing the membership of
$\mathcal H_\mu $ in the Schatten class $\mathcal S_p(H^2)$
in terms of the moments $\mu_n$.
\begin{theorem}\label{th:Schatten}
Assume that $1<p<\infty$ and let $\mu$ be a positive Borel measure
on $[0,1)$. Then, $\mathcal H_\mu\in \mathcal S_p(H^2)$ if and only
if $\sum_{n=0}^\infty (n+1)^{p-1}\mu_n^p<\infty.$
\end{theorem}
\par\bigskip
Throughout the paper, the letter $C=C(\cdot)$ will denote a constant
whose value depends on the parameters indicated in the parenthesis
(which often will be omitted), and may change from one occurrence to
another. We will use the notation $a\lesssim b$ if there exists
$C=C(\cdot)>0$ such that $a\le Cb$, and $a\gtrsim b$ is understood
in an analogous manner. In particular, if $a\lesssim b$ and
$a\gtrsim b$, then we will write $a\asymp b$.
\section{Preliminary results} In this section we shall collect a
number of results which will be needed in our work. We start
obtaining a characterization of $s$-Carleson measures in terms of
the moments.
\begin{proposition}\label{moments-cond} Let $\mu $ be a positive Borel measure on
$[0,1)$ and $s>0$. Then $\mu $ is an $s$-Carleson measure if and
only if the sequence of moments $\{ \mu _n\} _{n=0}^\infty $
satisfies
\begin{equation}\label{cond-moments} \sup _{n\ge
0}\,(1+n)^s\,\mu_n<\infty .\end{equation}
\end{proposition}
The proof is simple and will be omitted.
\par\medskip The following result, which may be of independent interest, asserts that for any function $f\in
H^p$ ($0<p<\infty $) we can find another one $F$ with the same
$H^p$-norm and which is non-negative and bigger than $\vert f\vert $
in the radius $(0,1)$.
\begin{proposition}\label{may-Hp} Suppose that $0<p<\infty $ and
$f\in H^p$, $f\not\equiv 0$. Then there exists a function $F\in H^p$
with $\Vert F\Vert _{H^p}= \Vert f\Vert _{H^p}$ and satisfying the
following properties:
\par (i) $F(r)>0$, for all $r\in (0, 1)$.
\par (ii) $\vert f(r)\vert \,\le \,F(r)$, for all $r\in (0, 1)$.
\par (iii) $F$ has no zeros in $\mathbb D$.
\end{proposition}
\begin{proof}
Let us consider first the case $p=2$. So, take
$f(z)=\sum_{n=0}^\infty a_nz^n\,\in H^2$, $f\not \equiv 0$. Set
$G(z)=\sum _{n=0}^\infty \vert a_n\vert z^n$ ($z\in \D $). Then
$G\in H^2$ and $\Vert G\Vert_{H^2}=\Vert f\Vert_{H^2}$. Furthermore,
we have: \begin{equation}\label{G(r)}0\le \vert f(r)\vert \le
G(r)\,\,\,\text{and $G(r)>0$},\quad \text{for all $r\in (0,
1)$},\end{equation} and
\begin{equation}\label{G-real-coef}G(\overline z)=\overline
{G(z)},\quad z\in \D .\end{equation} By (\ref{G(r)}) and
(\ref{G-real-coef}) we see that the sequence $\{ z_n\} $ of
the zeros of $G$ with $z_n\neq 0$ (which is a Blaschke sequence) can
be written in the form $\{ z_n\} =\{ \alpha _n\} \cup \{ \overline
{\alpha _n}\} \cup \{ \beta _n\} $ where $\Imag (\alpha_n)>0$ and
$-1<\beta _n<0$. Then the Blaschke product $B$ with the same zeros
that $G$ is
$$B(z)=z^m\prod \left (\frac{\alpha _n-z}{1-\overline {\alpha
_n}z}\frac{\overline {\alpha _n}-z}{1-\alpha _nz}\right )\prod
\frac{z-\beta_n}{1-\beta_nz},$$ where $m$ is the order of $0$ as
zero of $G$ (maybe $0$). Using the Riesz factorization theorem
\cite[Theorem\,\@2.\,\@5]{D}, we can factor $G$ in the form
$G=B\cdot F$ where $F$ is an $H^2$-function with no zeros and with
$\Vert f\Vert _{H^2}=\Vert G\Vert _{H^2}=\Vert F\Vert _{H^2}$.
Notice that $B(r)>0$, for all $r\in (0,1)$. This together with
(\ref{G(r)}) gives that $F(r)>0$, for all $r\in (0,1)$. Finally
since $\vert B(z)\vert \le 1$, for all $z$, we have that $G(r)\le
F(r)$ ($r\in (0,1)$) and then (\ref{G(r)}) implies $\vert f(r)\vert
\le F(r)$ ($r\in (0,1)$). This finishes the proof in the case $p=2$.
\medskip\par If $0<p<\infty $ and $f\in H^p$, $f\not \equiv 0$, write $f$ in
the form $f=B\cdot g$ where $B$ is a Blaschke product and $g$ is and
$H^p$-function without zeros and with $\Vert g\Vert _{H^p}=\Vert
f\Vert _{H^p}$. Now $g^{p/2}\in H^2$. By the previous case, we have
a function $G\in H^2$ without zeros, which take positive values in
the radius $(0,1)$, and satisfying $\Vert G\Vert _{H^2}=\Vert
g^{p/2}\Vert _{H^2}$ and $\vert g(r)\vert ^{p/2}\le G(r)$, for all
$r\in (0,1)$. It is clear that the function $F=G^{2/p}$ satisfies
that conclusion of Proposition~\ref{may-Hp}.
\end{proof}
\par\medskip
We shall also use the following description of $\alpha$-logarithmic
$s$-Carleson measures (see \cite[Theorem\,\@2]{Zhao}).
\begin{otherl}\label{le:zh}
Suppose that $0\le \alpha<\infty$ and $0<s<\infty$ and $\mu$ is a
positive Borel measure in $\D$. Then $\mu$ is an
$\alpha$-logarithmic $s$-Carleson measure if and only if
\begin{equation}\label{ELC}
K_{\alpha,s}(\mu)\ig\sup_{a\in \D}\left(\log \frac{2}{1-|a|^2}\right
)^{\alpha}\,\int_{\D}\left (\frac{1-|a|^2}{|1-\bar{a}z|^2}\right
)^s\,d\mu(z)<\infty.
\end{equation}
When $\alpha =0$, the constant $K_{0,s}(\mu )$ will be simply
written as $K_s(\mu )$. We remark that, if $s\ge 1$, then $K_s(\mu
)$ is equivalent to the norm of the embedding \,\,$i:H^p\rightarrow
L^{ps}(d\mu )$ for any $p\in (0,\infty )$.
\end{otherl}
\par\bigskip
Next we recall the following useful characterization of $q$-Carleson
measures for $H^p$ in the case $0<q<p<\infty $ (see \cite{{Bl-Ja},
{Lu90}, {Vid}}). \par Take $\alpha $ with $0<\alpha <\frac{\pi
}{2}$. Given $s\in \mathbb R$, we let $\Gamma_\alpha (e^{is})$
denote the Stolz angle with vertex $e^{is}$ and semi-aperture
$\alpha $, that is, the interior of the convex hull of $\{ e^{is}\}
\cup \{ \vert z\vert <\sin \alpha \} $. If $\mu $ is a positive
Borel measure in $\D $, we define \lq\lq the $\alpha
$-balagaye\rq\rq \, $\tilde \mu_\alpha $ of $\mu $ as follows:
$$\tilde \mu_\alpha (e^{is
})\,=\, \int_{\Gamma _\alpha (e^{is})}\,\frac{d\mu (z)}{1-\vert
z\vert },\quad s \in \mathbb R.$$
\begin{other}\label{th-q-car-gen} Let $\mu $ be a positive Borel measure on $\D $ and
$0<q<p<\infty $. Then $\mu $ is a $q$-Carleson measure for $H^p$ if
and only if $\tilde \mu _\alpha \in L^{\frac{p}{p-q}}(\partial \D )$
for some (equivalently, for all) $\alpha \in (0,\frac{\pi }{2})$.
\end{other}
\par\medskip A simple geometric argument shows that if the measure
$\mu $ is supported in $[0,1)$ then $\Gamma_\alpha (e^{is})\cap
[0,1)=[0,s_\alpha )$, where
$$1-s_\alpha \sim (\tan\alpha )\,s,\quad\text{as $s\to 0$}.$$
In particular, this implies the following.
\begin{other}\label{th-q-car-rad} Let $\mu $ be a positive Borel measure on $\D $
supported in $ [0,1)$, $0<q<p<\infty $. Then $\mu $ is a
$q$-Carleson measure for $H^p$ if and only if
\begin{equation}\label{eq-q-car-rad}
\int_0^1\left (\int_0^{1-s}\frac{d\mu (t)}{1-t}\right
)^{\frac{p}{p-q}}\,ds\,<\,\infty .\end{equation}
\end{other}
\medskip
\section{Proofs of the main results. Case $p\le 1$.}
\begin{Pf}{\,\em{Theorem \ref{Def:p<1}.}}
\par (i)\,\,$\Rightarrow
$\,\, (ii). Suppose that $\mu $ is an $1/p$-Carleson measure. Using
\cite[Theorem\,\@9.\,\@4]{D}, we see that there exists a positive
constant $C$ such that
\begin{equation}\label{car-1/p}\int_{[0,1)}\,\vert f(t)\vert\,d\mu
(t)\,\le\, C\Vert f\Vert _{H^p},\quad\text{for all\, $f\in
H^p$}.\end{equation} Take $f\in H^p$. Using (\ref{car-1/p}) we
obtain that
$$\sum_{n=0}^\infty \Bigl(\int
_{[0,1)}t^{n}|f(t)| \,d\mu(t)\Bigr)|z|^n\,\le \,\frac{C\,\Vert f\Vert
_{H^p}}{1-\vert z\vert },\quad z\in \D.$$ This implies that, for
every $z\in \D$, the integral
$$\int_{[0,1)}\,\frac{f(t)}{1-tz}\,d\mu
(t)\,=\,\int_{[0,1)}\,f(t)\,\left (\sum_{n=0}^\infty \,t^nz^n\right
)\,d\mu (t)$$ converges and that
\begin{equation}\label{Imum}
I_\mu (f)(z)\,=\,\int_{[0,1)}\,\frac{f(t)}{1-tz}\,d\mu
(t)\,=\,\sum_{n=0}^\infty \left (\int_{[0,1)}\,t^nf(t)\,d\mu
(t)\right )z^n,\quad z\in \D .
\end{equation}
Thus $I_\mu (f)$ is a well defined analytic function in $\D $.
\par (ii)\,\,$\Rightarrow
$\,\, (i). We claim that
\begin{equation}\label{fL1}\int_{[0,1)}\vert f(t)\vert \,d\mu(t)\,<\,\infty
,\quad\text{for all $f\in H^p$.}
\end{equation}
Indeed, take $f\in H^p$. Let $F$ be the function associated to $f$
by Proposition\,\@\ref{may-Hp}. Since $F\in H^p$, we have that the
integral $\int_{[0,1)}\frac{F(t)}{1-tz}\,d\mu (t)$ converges for all
$z\in \D$. Taking $z=0$ and bearing in mind that $0\le \vert
f(t)\vert \le F(t)$ ($t\in (0,1)$), we obtain that
$$\int_{[0,1)}\vert f(t)\vert \,d\mu(t)\,\le\,
\int_{[0,1)}F(t)\,d\mu(t)\, <\infty .$$ Thus (\ref{fL1}) holds.
\par For any $\beta \in [0,1)$ and $f\in H^p$ define
$$T_\beta (f)=f\cdot \chi_{\{0\le |z|<\beta\}}.$$
By (\ref{fL1}), $T_\beta $ is a linear operator from $H^p$ into
$L^1(d\mu )$ and by the lemma in \cite[Section\,\@3.\,\@2]{D},
\begin{eqnarray*}\Vert T_\beta (f)\Vert_ {L^1(d\mu
)}\,=\,&\int_{[0,\beta )}\vert f(t)\vert \,d\mu (t)\,\le \left
[\sup_{\vert z\vert \le \beta }\vert f(z)\vert \right ]\cdot \mu
([0,\beta ) )\le C_\beta \Vert f\Vert _{H^p},\quad f\in H^p.
\end{eqnarray*}
Thus, for every $\beta\in [0,1)$, $T_\beta $ is
a bounded linear operator from $H^p$ into $L^1(d\mu )$. Furthermore,
(\ref{fL1}) also implies that $$\sup_{0\le \beta <1}\Vert T_\beta
(f)\Vert_{L^1(d\mu )}\,\le \,\int _{[0,1)}\vert f(t)\vert \,d\mu
(t)=C_f<\infty ,\quad\text{for all $f\in H^p$}.$$ Then, by the
principle of uniform boundedness, we deduce that $\sup_{\beta \in
[0,1)}\Vert T_\beta \Vert <\infty $ which implies that the
identity operator is bounded from $H^p$ into $L^1(d\mu )$. Using
again \cite[Theorem\,\@9.\,\@4]{D} we obtain that $\mu $ is an
$1/p$-Carleson measure.
\par\medskip Assume now (i) (and (ii)), that
is, assume that $\mu $ is $1/p$-Carleson measure. Take $f\in H^p$,
$f(z)=\sum_{k=0}^\infty a_kz^k$ ($z\in \D $). By
Proposition\,\@\ref{moments-cond} and \cite[Theorem\,\@6.\,\@4]{D}
we have that there exists $C>0$ such that
$$\vert \mu_{n,k}\vert \,=\,\vert \mu_{n+k}\vert \,\le \,
\frac{C}{(k+1)^{1/p}}\,\,\,\text{and}\,\,\, \vert a_k\vert \le\,C
(k+1)^{(1-p)/p},\,\,\,\,\text{for all $n, k$}.$$ Then it follows
that, for every $n$,
\begin{equation*}\begin{split}\sum_{k=0}^\infty \vert \mu_{n,k}\vert \vert a_k\vert
\,\le \,& C\,\sum_{k=0}^\infty \frac{\vert a_k\vert
}{(k+1)^{1/p}}\,=\,C\,\sum_{k=0}^\infty \frac{\vert a_k\vert
^p\,\vert a_k\vert^{1-p} }{(k+1)^{1/p}} \\
\le \,& C\,\sum_{k=0}^\infty \frac{\vert a_k\vert
^p\,(k+1)^{(1-p)^2/p} }{(k+1)^{1/p}} \,=\,C\,\sum_{k=0}^\infty
(k+1)^{p-2}\vert a_k\vert ^p
\end{split}\end{equation*}
and then
by a well known result of Hardy and Littlewood
(\cite[Theorem\,\@6.\,\@2]{D}) we deduce that $$\sum_{k=0}^\infty
\vert \mu_{n,k}\vert \vert a_k\vert \,\le \,C\,\Vert
f\Vert_{H^p}^p,\quad\text{for all $n$}.$$ This implies that
$\mathcal H_\mu $ is a well defined analytic function in $\D$ and
that
$$\int_{[0,1)}\,t^nf(t)\,d\mu (t)\,=\,\sum_{k=0}^\infty \mu
_{n,k}a_k,\quad\text{for all $n$},$$ bearing in mind (\ref{Imum}),
this gives that $\mathcal H_\mu (f)=I_\mu (f).$
\end{Pf}
\par\medskip
\begin{Pf}{\,\em{Theorem \ref{bound:p<1}.}}
\par Since $\mu $ is an $1/p$-Carleson measure, there exists $C>0$
such that (\ref{car-1/p}) holds. This implies that
\begin{equation}\label{cons-car-1/p}\int_0^{2\pi }\int_{[0,1)}\left \vert
\frac{f(t)\,g(e^{i\theta })}{1-re^{i\theta }t}\right\vert\,d\mu
(t)\,d\theta\,<\,\infty ,\quad{0\le r<1,\,\,\,f\in H^p,\,\,\,g\in
H^1.}\end{equation} Using Theorem\,\@\ref{Def:p<1},
(\ref{cons-car-1/p}) and Fubini's theorem, and the Cauchy's integral
representation of $H^1$-functions \cite[Theorem\,\@3.\,\@6]{D}, we
obtain
\begin{equation}\label{Hmu-du}\begin{split}&\int_0^{2\pi }\,\mathcal H_\mu (f)
(re^{i\theta })\,\overline {g(e^{i\theta })}\,d\theta = \int_0^{2\pi
}\left (\int_{[0,1)}\frac{f(t)\,d\mu (t)}{1-re^{i\theta
}t}\right)\overline {g(e^{i\theta })}\,d\theta \\ =\,&
\,\int_{[0,1)}f(t)\int_0^{2\pi }\,\frac{\overline {g(e^{i\theta
})}}{1-re^{i\theta }t}\,d\theta \,d\mu (t)=\int_{[0,1)}f(t)\overline
{g(rt)}d\mu (t),\,\,{0\le r<1,\,f\in H^p,\,g\in H^1.}
\end{split}\end{equation}
\par\medskip \par (i) Take $q\in (0,1)$.
Bearing in mind (\ref{Hmu-du}) and (\ref{car-1/p}) we deduce that
\begin{equation}\label{boundHmu}\left\vert
\int_0^{2\pi }\,\mathcal H_\mu (f)(re^{i\theta })\,\overline
{g(e^{i\theta })}\,d\theta \right\vert \,\le\,C\Vert
f\Vert_{H^p}\Vert g\Vert _{H^\infty },\,\,{0\le r<1,\,f\in
H^p,\,g\in H^\infty .}\end{equation} Now we recall
\cite[Theorem\,\@10]{DRS} that $B_q$ can be identified with the dual
of a certain subspace $X$ of $H^\infty $ under the pairing
$$<f, g>\,=\,\lim_{r\to 1}\frac{1}{2\pi }\int_0^{2\pi }\,f(re^{i\theta
})\,\overline{g(e^{i\theta })}\,d\theta ,\quad f\in B_q,\quad f\in
X.$$ This together with (\ref{boundHmu}) gives that $\mathcal
H_\mu $ is a bounded operator from $H^p$ into $B_q$.
\par\medskip
(ii) We shall use Fefferman's duality theorem \cite{F, FS}, which
says that $(H^1)^\star\cong BMOA$ and $(VMOA)^\star \cong H^1$,
under the Cauchy pairing
\begin{equation}\label{eq:pai}
\langle f,g\rangle =\lim_{r\to 1^{-}}\frac{1}{2\pi}
\int_0^{2\pi}f(re^{i\theta})\ol{g(e^{i\theta})}\,d\theta,\quad f\in
H^1,\quad g\in BMOA\,\,(\text{resp.}\,\,VMOA),
\end{equation}
We mention \cite{Ba, Gar, G:BMOA}, as general references for the
spaces $BMOA$ and $VMOA$. In particular, Fefferman's duality theorem
can be found in \cite[Section\,\@7]{G:BMOA}.
\par Using the duality theorem and (\ref{cons-car-1/p}) it follows
that $\mathcal H_\mu $ is a bounded operator from $H^p$ into
$H^1$ if and only there exists a positive constant $C$ such that
\begin{equation}\label{bound-p1}\left\vert \int_{[0,1)}f(t)\overline
{g(rt)}\,d\mu (t)\right\vert \,\le C\Vert f\Vert_{H^p}\Vert g\Vert
_{BMOA},\quad 0<r<1,\,\,\,f\in H^p,\,\,\,g\in VMOA.\end{equation}
\par Suppose that $\mathcal H_\mu $ is a bounded operator from $H^p$ to $H^1$. For $0<a,b<1$, let the functions $g_a$
and $f_b$ be defined by
\begin{equation}\begin{split}\label{eq:famfunctii}
g_a(z) &=\log\frac{2}{1-az}, \quad \quad f_b(z)
=\left(\frac{1-b^2}{(1-bz)^2}\right)^{1/p},\quad z\in \mathbb D.
\end{split}\end{equation}
\par A calculation shows that $\{g_a\}\subset VMOA$, $\{f_b\}\subset H^p
$, and
\begin{equation}\begin{split}\label{eq:hu1ii}
\sup_{a\in[0,1)}||g_a||_{BMOA}<\infty\quad\text{and}\quad
\sup_{b\in[0,1)}||f_b||_{H^p}<\infty.
\end{split}\end{equation}
\par Next, taking $a=b\in [0,1)$ and $r\in [a,1)$, we obtain
\begin{equation*}\begin{split}
\left| \int_0^1f_a(t)\overline{g_a(rt)}\,d\mu(t)
\right| &
\ge
\int_a^1\left(\frac{1-a^2}{(1-at)^2}\right)^{1/p}\log\frac{2}{1-rat}\,d\mu(t),
\\ & \ge C \frac{\log\frac{2}{1-a^2}}{(1-a^2)^{1/p}}\mu\left([a,1)\right),
\end{split}\end{equation*}
which, bearing in mind \eqref{bound-p1} and \eqref{eq:hu1ii},
implies that $\mu$ is an $1$-logarithmic $\frac{1}{p}$-Carleson
measure.
\medskip\par
Reciprocally, suppose that $\mu$ is an $1$-logarithmic
$\frac{1}{p}$-Carleson measure. Let us see that $\mathcal H_\mu $ is
a bounded operator from $H^p$ to $H^1$. Using
(\ref{bound-p1}), it is enough to prove there exists $C>0$ such
that
\begin{equation*}
\int_0^1|f(t)||g(rt)|\,d\mu(t)
\le C||f||_{H^p}||g||_{BMOA},\quad\text{for all $r\in (0,1)$, $f\in H^p$, and $g\in VMOA$.}
\end{equation*}
By \cite[Theorem 9.4]{D}, this is equivalent to saying that, for
every $r\in (0,1)$ and every $g\in VMOA$, the measure $\vert
g(rz)\vert \,d\mu (z)$ is an $1/p$-Carleson measure with constant
bounded by $C\Vert g\Vert _{BMOA}$. Using Lemma \ref{le:zh} this can
be written as
\begin{equation}\label{eq:hu2ii}
\sup_{a\in
\D}\int_{\D}\left(\frac{1-|a|^2}{|1-\bar{a}z|^2}\right)^{1/p}|g(rz)|\,d\mu(z)\le
C||g||_{BMOA},\quad\text{$0<r<1$,\,\, $g\in VMOA$}.
\end{equation}
\par So take $r\in(0,1)$, $a\in\D$ and $g\in VMOA$. We have
\begin{equation*}\begin{split}
&\int_{\D}\left(\frac{1-|a|^2}{|1-\bar{a}z|^2}\right)^{1/p}|g(rz)|\,d\mu(z)
\\ & \le |g(ra)|\int_{\D}\left(\frac{1-|a|^2}{|1-\bar{a}z|^2}\right)^{1/p}\,d\mu(z)
+\int_{\D}\left(\frac{1-|a|^2}{|1-\bar{a}z|^2}\right)^{1/p}|g(rz)-g(ra)|\,d\mu(z)
\\
&=I_ 1(r,a)+I_ 2(r,a).
\end{split}\end{equation*}
Bearing in mind that any function $g$ in the Bloch space
$\mathcal{B}$ (see \cite{ACP}) satisfies the growth condition
\begin{equation}\label{blochgrowth}
|g(z)|\leq 2\|g\|_{\mathcal{B}}\,\log \frac{2}{1-|z|},\quad\text{for
all $z\in\D$},
\end{equation}
and $BMOA\subset \mathcal{B}$ \cite[Theorem\,\@5.\,\@1]{G:BMOA}), by
Lemma \ref{le:zh} we have that
\begin{equation}\begin{split}\label{eq:hu4ii}
I_ 1(r,a)&\leq C||g||_{BMOA}\log
\frac{2}{1-|a|}\,\int_{\D}\left(\frac{1-|a|^2}{|1-\bar{a}z|^2}\right)^{1/p}\,d\mu(z)
\\ & \le CK_{1,\frac{1}{p}}(\mu)||g||_{BMOA}<\infty .
\end{split}\end{equation}
\par
Now, since $\mu $ is an $1/p$-Carleson measure, Lemma \ref{le:zh}
yields
\begin{equation*}\begin{split}
I_ 2(r,a) \leq &\,CK_{\frac{1}{p}}(\mu) \left\Vert \left
(\frac{1-\vert a\vert ^2}{(1-\overline a\,z)^2}\right )^{1/p}\left
[g(rz)-g(ra)\right ]\right\Vert _{H^p}
\\ =& \, CK_{\frac{1}{p}}(\mu) \left (
\int_{0}^{2\pi}\frac{1-|a|^2}{|1-\bar{a}e^{i\theta}|^2}|g_r(e^{i\theta})-g_r(a)|^p\,d\theta
\right )^{1/p}\\ \le & \,
CK_{\frac{1}{p}}(\mu)\int_{0}^{2\pi}\frac{1-|a|^2}{|1-\bar{a}e^{i\theta}|^2}|g_r(e^{i\theta})-g_r(a)|\,d\theta
,\end{split}\end{equation*} where, $g_r(z)=g(rz)$ ($z\in\D $). Now,
using the conformal invariance of $BMOA$
(\cite[Theorem\,\@3.\,\@1]{G:BMOA})) and the fact that $BMOA$ is
closed under subordination \cite[Theorem\,\@10.\,\@3]{G:BMOA}, we
obtain that
$$\int_{0}^{2\pi}\frac{1-|a|^2}{|1-\bar{a}e^{i\theta}|^2}|g_r(e^{i\theta})-g_r(a)|\,d\theta
\le C\Vert g\Vert _{BMOA}$$ and then it follows that $I_ 2(r,a)\le
CK_{\frac{1}{p}}(\mu)\Vert g\Vert _{BMOA}$. This and
(\ref{eq:hu4ii}) give (\ref{eq:hu2ii}), finishing the proof of
part\,\@(ii).
\par\medskip \par (iii) Using
(\ref{Hmu-du}), the duality theorem for $H^q$
\cite[Section\,\@7.\,\@2]{D} and arguing as in the proof of
part\,\@(ii), we can assert that $\mathcal H_\mu $ is a bounded
operator from $H^p$ to $H^q$ if and only if there exists a positive
constant $C$ such that
\begin{equation}\label{Hmuboundedpq>1++}
\left \vert \int_{[0,1)}\,f(t)\,\overline {g(t)}\,d\mu (t)\right
\vert \,\le \,C\,\Vert f\Vert _{H^p}\,\Vert g\Vert
_{H^{q^\prime}},\quad f\in H^p,\quad g\in H^{q^\prime
}.\end{equation} Now, by Proposition\,\@\ref{may-Hp}, it follows
that (\ref{Hmuboundedpq>1++}) is equivalent to
\begin{equation}\label{Hmuboundedpq>1--}
\int_{[0,1)}\,\vert f(t)\vert \,\vert g(t)\vert \,d\mu (t)\,\le
\,C\,\Vert f\Vert _{H^p}\,\Vert g\Vert _{H^{q^\prime}},\quad f\in
H^p,\quad g\in H^{q^\prime },\end{equation} and, by
Lemma\,\@\ref{le:zh}, this is the same as saying the following:
\par For every $g\in H^{q^\prime }$, the measure $\mu_g$ supported
on $[0,1)$ and defined by $d\mu_g(z)=\vert g(z)\vert \,d\mu (z)$ is
a $1/p$-Carleson measure with $K_{\frac{1}{p}}(\mu _g)\le C\,\Vert
g\Vert _{H^{q^\prime }}$, that is,
\begin{equation}\label{Hmuboundedpq>1}\sup _{a\in \D
}\,\int_{[0,1)}\,\left (\frac{1-\vert a\vert ^2}{\vert 1-\overline
at\vert^2}\right )^{1/p}\,\vert g(t)\vert \,d\mu (t)\,\le \, C\Vert
g\Vert _{H^{q^\prime}},\quad g\in H^{q^\prime }.\end{equation}
\par Suppose that
$\mathcal H_\mu $ is a bounded operator from $H^p$ to $H^q$. Then
(\ref{Hmuboundedpq>1}) holds. For $a\in \D$, take
$$g_a(z)=\left (\frac{1-\vert a\vert ^2}{(1-\overline az)^2}\right
)^{1/q^\prime },\quad z\in \D.$$ Since
$\sup_{a\in \D} \Vert g_a\Vert_{H^{q^\prime
}}<\infty $, (\ref{Hmuboundedpq>1}) implies that $$\sup _{a\in
\D }\,\int_{[0,1)}\,\left (\frac{1-\vert a\vert ^2}{\vert
1-\overline at\vert^2}\right )^{\frac{1}{p}+\frac{1}{q^\prime
}}\,d\mu (t)<\infty ,$$ that is,
$\mu $ is a $\frac{1}{p}+\frac{1}{q^\prime }$-Carleson measure, by Lemma\,\@\ref{le:zh}.
\medskip\par Suppose now that $\mu $ is an $\frac{1}{p}+\frac{1}{q^\prime }$-Carleson
measure. Set $s=1+\frac{p}{q^\prime }$. The conjugate exponent
of $s$ is $s^\prime =1+\frac{q^\prime }{p}$ and
$\frac{1}{p}+\frac{1}{q^\prime }\,=\,\frac{s}{p}\,=\,\frac{s^\prime }{q^\prime }.$ Then, by
\cite[Theorem~9.\@4]{D}, $H^p$ is continuously embedded
in $L^s(d\mu )$ and $H^{q^\prime }$ is continuously embedded in
$L^{s^\prime }(d\mu )$, that is, \begin{equation}\label{s}\left
(\int_{[0,1)}\vert f(t)\vert ^s\,d\mu (s)\right )^{1/s}\,\le
\,C\,\Vert f\Vert _{H^p},\quad f\in H^p,\end{equation} and
\begin{equation}\label{sprime}\left (\int_{[0,1)}\vert g(t)\vert
^{s^\prime }\,d\mu (s)\right )^{1/{s^\prime }}\,\le \,C\,\Vert
g\Vert _{H^{q^\prime }},\quad g\in H^{q^\prime }.\end{equation}
Using H\"{o}lder's inequality with exponents $s$ and $s^\prime $,
(\ref{s}) and (\ref{sprime}), we obtain
\begin{equation*}\begin{split} \int_{[0,1)}\vert f(t)\vert \,\vert g(t)\vert\,d\mu
(t)\,\le & \left (\int_{[0,1)}\vert f(t)\vert ^s\,d\mu (s)\right
)^{1/s}\,\left (\int_{[0,1)}\vert g(t)\vert ^{s^\prime }\,d\mu
(s)\right )^{1/{s^\prime }}\\ \le & C\,\Vert f\Vert _{H^p}\,\Vert
g\Vert _{H^{q^\prime }},\quad f\in H^p,\,\, g\in H^{q^\prime
}.\end{split}\end{equation*} Hence, (\ref{Hmuboundedpq>1--}) holds
and then it follows that $\mathcal H_\mu $ is a bounded operator
from $H^p$ to $H^q$.
\end{Pf}
\medskip
\section{Proofs of the main results. Case $p>1$.}
\begin{Pf}{\,\em{Theorem \ref{Def:p>1}\,}}
(i). Since $\mu $ is an $1$-Carleson measure for $H^p$,
(\ref{car-1/p}) holds for a certain $C>0$.
Then the argument used in the proof of the
implication (i)\,$\Rightarrow$\, (ii) in Theorem\,\@\ref{Def:p<1}
gives that, for every $f\in H^p$, $I_\mu (f)$ is a well defined
analytic function in $\D $ and
\begin{equation}\label{Imu-power}
I_\mu (f)(z)\,=\,\sum_{n=0}^\infty \left
(\int_{[0,1)}\,t^nf(t)\,d\mu
(t)\right )z^n,\quad z\in \D .\end{equation}
\par The reverse implication can be proved just as (ii)\,$\Rightarrow$\, (i) in
Theorem\,\@\ref{Def:p<1}.
\par The fact that $\mu $ being an $1$-Carleson measure for $H^p$
is equivalent to (\ref{int-car-1p}) follows from
Theorem\,\@\ref{th-q-car-rad}.
\par
(ii). Take $f\in
H^p$, $f(z)=\sum_{k=0}^\infty a_kz^k$ ($z\in \D $). Set
$$S_n(f)(z)=\sum_{k=0}^n a_kz^k,\quad R_n(f)(z)=\sum_{k=n+1}^\infty
a_kz^k,\quad z\in \D ,\quad n= 0, 1, 2, \dots .$$ Whenever $0\le
N<M$ and $n\ge 0$, we have
\begin{equation*}\label{cauchy}\begin{split}
\left \vert \sum_{k=N+1}^M\,\mu_{n,k}a_k\right\vert \,=\,& \left
\vert \int_{[0,1)}t^n\,\left (\sum_{k=N+1}^M\,a_k\,t^k\right )\,d\mu
(t)\right\vert \\ =\,& \left \vert \int_{[0,1)}t^n\,\left
[S_M(f)(t)-S_N(f)(t)\right ]\,d\mu (t)\right\vert \\ \le \,&
\int_{[0,1)}\left \vert S_M(f)(t)-S_N(f)(t)\right \vert \,d\mu (t).
\end{split}\end{equation*}
Using this, the fact that $\mu $ is an $1$-Carleson measure for
$H^p$, and the Riesz projection theorem, we deduce that
$$\sum_{k=N+1}^M\,\mu_{n,k}a_k\to 0,\quad \text{as $N, M\to \infty
$}$$ for all $n$. This gives that the series $\sum_{k=0}^\infty
\,\mu_{n,k}a_k$ converges for all $n$.
\par For $n, N\ge 0$, we have
\begin{equation*}\begin{split}
\left\vert \int_{[0,1)}\,t^n\,f(t)\,d\mu
(t)\,-\,\sum_{k=0}^N\,\mu_{n,k}\,a_k\right\vert \,= & \, \left\vert
\int_{[0,1)}\,t^n\,f(t)\,d\mu (t)\,-\,\int_{[0,1)}\,t^n\,\left
(\sum_{k=0}^N\,a_k\,t^t\right )\,d\mu (t)\right\vert \\ = &
\left\vert
\int_{[0,1)}\, t^n\,R_{N}(f)(t)\,d\mu (t)\right \vert \\ \le &
C\Vert R_N(f)\Vert _{H^p}.
\end{split}\end{equation*}
Since $1<p<\infty $, $\Vert R_N(f)\Vert _{H^p}\to 0$, as $N\to\infty
$, and then it follows that
$$\sum_{k=0}^\infty \,\mu_{n,k}a_k\,=\,\int_{[0,1)}\,t^n\,f(t)\,d\mu
(t),\quad \text{for all $n$},$$ which together with (\ref{car-1/p})
implies that $\mathcal H_\mu (f)$ is a well defined analytic
function in $\D$
and, by (\ref{Imu-power}),
$\mathcal H_\mu (f)=I_\mu (f)$.
\end{Pf}
\par\bigskip Let us turn to prove Theorem\,\@\ref{bound:p>1}. In
view of Theorem\,\@\ref{Def:p>1}, $\mathcal H_\mu$ coincides with
$I_\mu $ on $H^p$. This fact will be used repeatedly in the
following.
\par Recall that (\ref{int-car-1p}) implies that $\mu $ is
an $1$-Carleson measure for $H^p$, that is, we have
$$\int_{[0,1)}\,\vert f(t)\vert\,d\mu (t)\,\le \,C\Vert f\Vert
_{H^p},\quad f\in H^p.$$ Then arguing as in the proof of
Theorem\,\@\ref{bound:p<1}, we obtain
\begin{equation}\label{Hmu-du-p>1}\begin{split}&\int_0^{2\pi }\,\h_\mu (f)
(re^{i\theta })\,\overline {g(e^{i\theta })}\,d\theta = \int_0^{2\pi
}\left (\int_{[0,1)}\frac{f(t)\,d\mu (t)}{1-re^{i\theta
}t}\right)\overline {g(e^{i\theta })}\,d\theta \\ =\,&
\,\int_{[0,1)}f(t)\int_0^{2\pi }\,\frac{\overline {g(e^{i\theta
})}}{1-re^{i\theta }t}\,d\theta \,d\mu (t)=\int_{[0,1)}f(t)\overline
{g(rt)}d\mu (t),\,\,{0\le r<1,\,f\in H^p,\,g\in H^1.}
\end{split}\end{equation}
\par\medskip Once (\ref{Hmu-du-p>1}) is established, (i) can be proved with the argument used in the proof of part\,\@(iii) of
Theorem\,\@\ref{bound:p<1}.
\par\medskip
Part\,\@(ii) of Theorem\,\@
\ref{bound:p>1} is a byproduct of the following result.
\begin{proposition}\label{acotacionhpmenorq}
Asumme that $1<q<p<\infty$ and let $\mu$ be a positive Borel
measure on $[0,1)$ satisfying (\ref{int-car-1p}). Then, the
following conditions are equivalent:\begin{itemize} \item[(a)]\,
$\h_\mu $ is a bounded operator from $H^p$ to $H^q$. \item[(b)]\,
\, $\h_\mu $ is a bounded operator from $H^{\frac{2pq'}{p+q'}}$ to
$H^{\left(\frac{2pq'}{p+q'}\right)'}$.
\item[(c)]\,$H^{\frac{2pq'}{p+q'}}$ is continuously contained in $L^{2}(\mu)$.
\item[(d)]\, The function defined by
$s\mapsto\int_{0}^{1-s}\frac{d\mu(t)}{1-t}$ ($s\in [0,1)$) belongs
to $L^{\left(\frac{pq'}{p+q'}\right)'}([0,1))$.
\end{itemize}
\end{proposition}
\begin{proof}
\par (a)\, $\Rightarrow $ \,(b).
Using duality as above, we see that (a) is equivalent to
\begin{equation}\label{Imub}
\left \vert \int_{[0,1)}\,f(t)\,\overline {g(t)}\,d\mu (t)\right
\vert \,\lesssim\,\Vert f\Vert _{H^p}\,\Vert g\Vert
_{H^{q^\prime}},\quad f\in H^p,\quad g\in H^{q^\prime
}.\end{equation} Take $f\in H^p$ and $g\in H^{q'}$, and let $F\in
H^p$ and $G\in H^{q'}$ be the functions associated to $f$ and $g$ by
Proposition\,\@\ref{may-Hp}, respectively. Using (\ref{Imub}) and
H\"{o}lder's inequality, we obtain
\begin{equation}\label{eqj1}\begin{split}
\int_0^1|f(t)||g(t)|\,d\mu(t) & \lesssim \int_{0}^1 F(t)\,{G(t)}\,d\mu(t)
\,= \left| \int_{0}^1 F(t)\,\overline {G(t)}\,d\mu(t) \right|
\\ & \lesssim||F||_{H^p}||G||_{H^{q'}}\lesssim ||f||_{H^p}||g||_{H^{q'}}.
\end{split}\end{equation}
Take now $\phi\in H^{\frac{2pq'}{p+q'}}$. By the outer-inner
factorization \cite[Chapter $2$]{D}, $\phi=\Phi\cdot I$ where $I$ is
an inner function and $\Phi\in H^{\frac{2pq'}{p+q'}}$ is free from
zeros and $||\Phi||_{ H^{\frac{2pq'}{p+q'}}}=||\phi||_{
H^{\frac{2pq'}{p+q'}}}$. Now let us consider the analytic functions
$f=\Phi^{\frac{2q'}{p+q'}}$ and $g=\Phi^{\frac{2p}{p+q'}}$. We have
$$f=\Phi^{\frac{2q'}{p+q'}}\in H^p,\quad\text{with
$||f||_{H^p}=||\Phi||^{\frac{2q'}{p+q'}}_{H^{\frac{2pq'}{p+q'}}}$}$$
and
$$g=\Phi^{\frac{2p}{p+q'}}\in H^{q'},\quad\text{with $||g||_{H^{q'}}=||\Phi||^{\frac{2p}{p+q'}}_{H^{\frac{2pq'}{p+q'}}}$}.$$
Bearing in mind (\ref{eqj1}), it follows that
\begin{equation*}\begin{split}
\int_0^1|\phi(t)|^2\,d\mu(t)& \le \int_0^1|\Phi(t)|^2\,d\mu(t)
\\ & = \int_0^1|f(t)||g(t)|\,d\mu(t)
\\ & \lesssim ||f||_{H^p}||g||_{H^{q'}}=||\Phi||^2_{ H^{\frac{2pq'}{p+q'}}}=||\phi||^2_{ H^{\frac{2pq'}{p+q'}}},
\end{split}\end{equation*}
which gives (b).
\par (b)\, $\Rightarrow $ \,(c).
Since $p>q>1$, $\frac{pq'}{p+q'}>1$, by duality, as above, (b) is
equivalent to
$$\left|\int_0^1f(t)\overline{g(t)}\,d\mu(t)\right|\,\le C\,||f||_{H^{\frac{2pq'}{p+q'}}}||g||_{H^{\frac{2pq'}{p+q'}}},\quad f,g\in H^{\frac{2pq'}{p+q'}}.$$
Taking $f=g$ we obtain
$$\int_0^1\,\vert f(t)\vert ^2\,d\mu(t)\,\le
C\,||f||_{H^{\frac{2pq'}{p+q'}}}^2.$$ This is (c).
\medskip\par Theorem\,\@\ref{th-q-car-rad} gives that (c) and (d) are
equivalent.
\medskip
\par (d)\, $\Rightarrow $ \,(a). Using again
Theorem\,\@\ref{th-q-car-rad} we have that $H^p$ is continuously
contained in $L^\frac{p+q'}{q'}(d\mu)$ and $H^{q'}$ is continuously
contained in $L^\frac{p+q'}{p}(d\mu)$, which together with
H\"{o}lder's inequality gives
\begin{equation*}\begin{split}
\int_0^1|f(t)||g(t)|\,d\mu(t)\le &
\left(\int_0^1|f(t)|^{\frac{p+q'}{q'}}\,d\mu(t)\right)^{\frac{q'}{p+q'}}
\left(\int_0^1|g(t)|^{\frac{p+q'}{p}}\,d\mu(t)\right)^{\frac{p}{p+q'}}
\\ & \le C ||f||_{H^p}||g||_{H^{q'}},\quad f\in H^p,\quad g\in H^q,
\end{split}\end{equation*}
and this is equivalent to (a).
\end{proof}
\begin{Pf}{\,\em{Theorem \ref{bound:p>1}\,(iii).}} Just as in the
proof of Theorem\,\@\ref{bound:p<1}\,\@(ii), $\h_\mu $ is a bounded
operator from $H^p$ into $H^1$ if and only there exists a positive
constant $C$ such that
\begin{equation}\label{bound-p1-p>1}\left\vert \int_{[0,1)}f(t)\overline
{g(rt)}\,d\mu (t)\right\vert \,\le C\Vert f\Vert_{H^p}\Vert g\Vert
_{BMOA},\quad 0<r<1,\,\,\,f\in H^p,\,\,\,g\in VMOA.\end{equation}
Let $\nu $ be the measure on $[0,1)$ defined by $d\nu
(t)=\log\frac{1}{1-t}\,d\mu (t)$, by
Theorem\,\@\ref{th-q-car-rad} it follows that the function
\,\,$s\mapsto \int_0^{1-s}\,\frac{\log\frac{1}{1-t}d\mu
(t)}{1-t}$\,\, $(s\in [0,1))$\, belongs to $L^{p^\prime }([0,1))$ if
and only if the measure $\nu $ is an $1$-Carleson measure for $H^p$.
\par Consequently, we have to prove that
$$\text{(\ref{bound-p1-p>1})\,\,\,$\Leftrightarrow $\,\,\,$\nu $ is an $1$-Carleson measure for
$H^p$.}$$
\par Suppose that (\ref{bound-p1-p>1}) holds. For $0<\rho <1$, let
$g_\rho $ be the function defined by $g_\rho (z)=\log\frac{1}{1-\rho
z}$ ($z\in \D $), then
\begin{equation*}\text{$g_\rho \in VMOA$, for all $\rho \in
(0,1)$,\,\,\,\, and \,\, $\sup_{0<\rho <1}\Vert g_\rho \Vert
_{BMOA}=A<\infty $ }.\end{equation*} On the other hand, if $f\in
H^p$, $0<r<1$, and
$F$ is the function associated to $f$ by
Proposition\,\@\ref{may-Hp}, it follows that
\begin{equation*}\begin{split}
\int_{[0,1)}\vert f(t)\vert \,\log\frac{1}{1-\rho rt}\,d\mu (t)\,\le
\,\int_{[0,1)}\,F(t)\,\overline{g_\rho (rt)}\,d\mu (t)\,\le
C\,A\,\Vert F\Vert _{H^p}\,= C\,A\,\Vert f\Vert
_{H^p},\end{split}\end{equation*} for every $\rho\in (0,1)$. Letting
$r$ and $\rho $ tend to $1$, we obtain
\begin{equation*}\begin{split}
\int_{[0,1)}\vert f(t)\vert \,\log\frac{1}{1-t}\,d\mu (t)\,\le
C\,A\,\Vert f\Vert _{H^p}.\end{split}\end{equation*} Thus $\nu $ is
an $1$-Carleson measure for $H^p$.
\par Conversely, assume that $\nu $ is
an $1$-Carleson measure for $H^p$. Take $r\in (0,1),\,f\in H^p,$\,
and\,$\,g\in VMOA$.
Using (\ref{blochgrowth}), we obtain $$
\left\vert \int_{[0,1)}f(t)\overline {g(rt)}\,d\mu (t)\right\vert
\,\le \,C\,\Vert g\Vert_{BMOA}\int_{[0,1)}\vert f(t)\vert
\log\frac{2}{1-t}\,d\mu (t)\,\le\,C\,\Vert g\Vert_{BMOA}\,\Vert
f\Vert_{H^p}.$$
\end{Pf}
\par\medskip
\begin{Pf}{\,\em{Theorem \ref{bound:p>1}\,(iv).}}
Assume that the function defined by
$\,s\mapsto\,\int_0^{1-s}\frac{d\mu (t)}{1-t}$ ($s\in [0,1)$)
belongs to $L^{p^\prime }([0,1)$. By Theorem\,\@\ref{th-q-car-rad},
this implies that $H^p$ is continuously contained in $L^1(d\mu )$.
From now on, the proof is analogous to
the proof
of Theorem\,\@\ref{bound:p<1}\,\@(i).
\end{Pf}\par\bigskip
\section{Compactness.}\label{sect-compact}
\par The next theorem gathers our main results concerning the study of the compactness of $\h_\mu$ on
Hardy spaces.
\begin{theorem}\label{compactness} Let $\mu $ be a positive Borel
measure on $[0,1)$. \begin{itemize}
\item[(i)] If \, $0<p\le 1$\, and $\mu $ is a $1/p$-Carleson measure, then $\h_\mu $ is a compact
operator from $H^p$ to $H^1$ if and only if $\mu $ is a vanishing
$1$-logarithmic $1/p$-Carleson measure.
\item[(ii)] If\, $0<p\le 1< q$ and and $\mu $ is a $1/p$-Carleson measure, then $\h_\mu $ is a compact
operator from $H^p$ to $H^q$ if and only if $\mu $ is a vanishing
$\frac{1}{p}+\frac{1}{q^\prime }$-Carleson measure.
\item[(iii)] If\, $1<p< q$ and $\mu $ satisfies (\ref{int-car-1p}), then $\h_\mu $ is a compact
operator from $H^p$ to $H^q$ if and only if $\mu $ is a vanishing
$\frac{1}{p}+\frac{1}{q^\prime }$-Carleson measure.
\item[(iv)] If $1<p<\infty $, $\mu $ satisfies (\ref{int-car-1p}) and $1\le q<p$, then $\mathcal H_\mu $ is a compact
operator from $H^p$ to $H^q$ if and only if it is a bounded operator
from $H^p$ to $H^q$.
\end{itemize}\end{theorem}
\par\medskip The following lemma will be used in the proof of cases (i), (ii) and (iii).
\begin{lemma}\label{lemafb} Suppose that $0<p<\infty $ and let $\mu $ be a positive Borel
measure on $[0,1)$ which is an $1$-logarithmic $1/p$-Carleson
measure. Let $f_b$, $(0\le b<1)$, be defined as in
(\ref{eq:famfunctii}). Then
\begin{equation}\label{int-fb-zero}\lim_{b\to
1^-}\int_{[0,1)}\,f_b(t)\,d\mu (t)\,=\,0.\end{equation}
\end{lemma}
\begin{pf} For $0\le t<1$, set $F(t)\,=\,\mu \left ([0,t)\right
)\,-\,\mu \left ([0,1)\right )\,=\,-\,\mu \left ([t,1)\right) $.
Integrating by parts and using the fact that $\mu $ is an
$1$-logarithmic $1/p$-Carleson measure, we obtain
\begin{equation}\label{parts-fb}\int_{[0,1)}\,f_b(t)\,d\mu (t)\,=\,
f_b(0)\,\mu \left ([0,1)\right )\,+\,\int_0^1\,f_b^\prime (t)\,\mu
\left ([t,1)\right )\,dt.\end{equation} Using that $\mu $ is an
$1$-logarithmic $1/p$-Carleson measure and the fact that $bt<b$ and
$bt<t$ ($0<\,b,t\,<1$), we deduce
\begin{equation*}\begin{split}\,&\int_0^1\,f_b^\prime (t)\,\mu
\left ([t,1)\right )\,dt\le\,
C\int_0^1\frac{(1-b)^{1/p}(1-t)^{1/p}}{(1-bt)^{\frac{2}{p}+1}\log\frac{e}{1-t}}\,dt\\
= &
C\int_0^b\frac{(1-b)^{1/p}(1-t)^{1/p}}{(1-bt)^{\frac{2}{p}+1}\log\frac{e}{1-t}}\,dt\,+\,
C\int_b^1\frac{(1-b)^{1/p}(1-t)^{1/p}}{(1-bt)^{\frac{2}{p}+1}\log\frac{e}{1-t}}\,dt
\\
\le &
C\,(1-b)^{1/p}\,\int_0^b\frac{dt}{(1-t)^{\frac{1}{p}+1}\log\frac{e}{1-t}}\,+
\,\frac{C}{(1-b)^{\frac{1}{p}+1}}\int_b^1\frac{(1-t)^{1/p}}{\log\frac{e}{1-t}}\,dt
\\
= & I(b)\,+\,II(b).
\end{split}\end{equation*}
Now, it is a simple calculus exercise to show that $I(b)$ and
$II(b)$ tend to $0$, as $b\to 1$. Using this, the fact that\,
$\lim_{b\to 1}f_b(0)\to 0$, and (\ref{parts-fb}), we deduce
(\ref{int-fb-zero}).
\end{pf}
\par\medskip
\begin{Pf}{\,\em{Theorem \ref{compactness}\,\@}}
(i).\, Suppose that $\mathcal H_\mu $ is a compact operator from
$H^p$ to $H^1$. Let $f_b$, $(0\le b<1)$, be defined as in
(\ref{eq:famfunctii}). Let $\{ b_n\} \subset (0,1)$ be any sequence
with $b_n\to 1$ and such that the sequence $\{ \mathcal H_\mu
(f_{b_n})\} $ converges in $H^1$ (such a sequence exists because
$\sup _{0<b<1}\Vert f_b\Vert _{H^p}<\infty $ and $\mathcal H_\mu $
is compact) and let $g$ be the limit (in $H^1$) of $\{ \mathcal
H_\mu (f_{b_n})\} $. Then $\mathcal H_\mu (f_{b_n})\to g$, uniformly
on compact subsets of $\D $. Now, by Theorem \ref{Def:p<1}, we have
$$0\le H_\mu (f_{b_n})(r)=\int_{[0,1)}\frac{f_{b_n}(t)}{1-rt}\,d\mu
(t)\le \frac{1}{1-r}\int_{[0,1)}\,f_{b_n}(t)\,d\mu (t),\quad
0<r<1.$$ Since $\mathcal H_\mu $ is continuous from $H^p$ to $H^1$,
$\mu $ is an $1$-logarithmic $1/p$-Carleson measure. Then, by
Lemma\,\@\ref{lemafb}, it follows that $g(r)=0$ for all $r\in
(0,1)$. Hence, $g\equiv 0$. In this way we have proved that
$$\mathcal H_\mu (f_b)\to 0, \quad\text{as $b\to 1$,\,\, in
$H^1$.}$$ Arguing as in proof of
the boundedness
(Theorem\,\@\ref{bound:p<1}\,\@(ii)), this yields
$$\lim_{b\to 1^-}\frac{\mu \left ([b,1)\right
)\log\frac{e}{1-b}}{(1-b)^{1/p}}\,=\,0,$$ which is equivalent to
saying that $\mu $ is a vanishing $1$-logarithmic $1/p$-Carleson
measure.
\par\medskip
Suppose now that $\mu $ is a vanishing $1$-logarithmic
$1/p$-Carleson measure. Let $\{ f_n\} _{n=1}^\infty $ be a sequence
of functions in $H^p$ with $\sup\Vert f_n\Vert_{H^p}<\infty $ and
such that $f_n\to 0$, uniformly on compact subsets of $\D $. For
$0<r<1$, let us write
$$d\mu_r(t)=\chi _{r<\vert z\vert <1}(t)\,d\mu (t).$$
Since $\mu $ is a vanishing $1$-logartihmic $1/p$-Carleson measure,
$\lim_{r\to 1}K_{1,\frac{1}{p}}(\mu _r)=0$. This together with the
fact that $f_n\to 0$, uniformly on compact subsets of $\D $
gives that
$$\lim_{n\to\infty }\int_0^1\,\vert f_n(t)\vert\,\vert
g(t)\vert\,d\mu (t)\,=\,0\quad \text{for all $g\in VMOA$.}$$ Using
the duality relation $(VMOA)^\star \cong H^1$ as in the proof of
Theorem\,\@\ref{bound:p<1}, this implies that $\mathcal H_\mu
(f_n)\to 0$, in $H^1$. So, $\h_\mu:\, H^p\to H^1$ is compact.
\par Parts (ii) and (iii)\, can be proved
similarly to the preceding one. We shall omit the details. Let us
simply remark, for the necessity,
that if $\mu $ is a vanishing $\frac{1}{p}+\frac{1}{q^\prime
}$-Carleson measure, then it is an $1$-logarithmic $1/p$-Carleson
measure and then we can use Lemma\,\@\ref{lemafb}.
\end{Pf}
\par\medskip
\begin{Pf}{\,\em{Theorem \ref{compactness}\,\@(iv).}}
Suppose that $1<p<\infty $ and $0<q<p$. Looking at the proof of
parts (ii) and (iii) of Theorem\,\@\ref{bound:p>1}, we see
$\mathcal H_\mu $ applies $H^p$ into $H^q$ if and only if:
\begin{itemize}\item $H^\frac{2pq^\prime}{p+q^\prime }$ is continuously
embedded in $L^2(d\mu )$, in the case $1<q<p$.
\item $H^p$ is continuously
embedded in $L^1(d\nu )$, where $d\nu (t)=\log\frac{1}{1-t}\,d\mu
(t)$, in the case $q=1<p<\infty $.
\end{itemize}
Now, using the results in Section\,\@3 of \cite{Bl-Ja}, we see that
when any of these embeddings exists as a continuous operator, then
it is compact. Then arguments similar to those used in the
boundedness case can be used to yield the compactness. We omit the
details.
\end{Pf}
\section{Schatten classes.}\label{Schatten}
\medskip\par Before presenting the proof of Theorem \ref{th:Schatten} let us recall some definitions
which connect the operator $\h_\mu$ with the classical theory of
Hankel operators. \par Given $\varphi
(\xi)\sim\sum_{n=-\infty}^{+\infty}\widehat{\varphi}(n)\xi^n\in
L^2(\T)$, the associated (big) Hankel operator
\par\noindent $H_\varphi: H^2\to H^2_{-}$
(see \cite[p. $6$]{Pell}) is formally defined
as
$$H_\varphi(f)=I-P(\varphi f)$$
where $P$ is the Riesz projection.
\par Moreover, if $\mu$ is a classical Carleson measure, Nehari's Theorem implies that
(see \cite[p.\,\@3 and p.\,\@42]{Pell}) there exists
$\varphi_\mu\in L^\infty (\T)$ with
$\mu_{n+1}=\widehat{\varphi_\mu}(-n)$, so
$$\h_\mu(f)(z)-\h_\mu(f)(0)=H_{\varphi_\mu}(f)(\bar{z}).$$
In particular, $\h_\mu$ is bounded on $H^2$ if and only if
$H_{\varphi_\mu}: H^2\to H^2_{-}$ is bounded, that is, if and only
if \,$\mu$ is a Carleson measure. Finally let us observe that,
$$(I-P)(\varphi_\mu)(\bar{z})=\sum_{n=1}^\infty \mu_{n+1}z^n$$
\begin{Pf}{\em{Theorem \ref{th:Schatten}.}}
It follows from the above observation, \cite[p. $240$,
Corollary\,\@2.\,\@2]{Pell} and \cite[Appendix $2.6$]{Pell} that
$\mathcal H_\mu \in \mathcal S_p(H^2)$ if and only if
$h_\mu(z)=\sum_{n=1}^\infty \mu_{n+1}z^{n}\in B^p$. Bearing in mind
\cite[Theorem $2.1$]{MP} (see also \cite[p. $120$,
$7.5.8$]{Pabook}),
\cite[p. $120$, $7.3.5$]{Pabook}, the fact that $\{\mu_n\}$ decreases to zero and \cite[Lemma $3.4$]{LNP}, we deduce that
\begin{equation*}\begin{split}
||h_\mu||^p_{B^p} & \asymp \sum_{n=0}^\infty
2^{-n(p-1)}\left\|\sum_{k=2^n}^{2^{n+1}-1}k\mu_{k+1} z^{k-1}
\right\|^p_{H^p}
\\ & \asymp\sum_{n=0}^\infty 2^{n}\left\|\sum_{k=2^n}^{2^{n+1}-1}\mu_{k+1} z^{k-1} \right\|^p_{H^p}
\\ & \lesssim \sum_{n=0}^\infty 2^{n} \mu^p_{2^n}
\left\|\sum_{k=2^n}^{2^{n+1}-1}z^{k-1} \right\|^p_{H^p}.
\end{split}\end{equation*}
We claim that
\begin{equation}\label{bloque}
\left\|\sum_{k=2^n}^{2^{n+1}-1}z^{k-1} \right\|^p_{H^p}\asymp
2^{n(p-1)}.
\end{equation}
Then, using again that $\{\mu_n\}$ is decreasing
\begin{equation*}\begin{split}
||h_\mu||^p_{B^p} &\lesssim \sum_{n=0}^\infty 2^{np} \mu^p_{2^n}
\lesssim \sum_{n=0}^\infty 2^{n(p-1)} \sum_{k=2^n-1}^{2^n} \mu^p_{k}
\asymp \sum_{k=0}^\infty (k+1)^{p-1}\mu_k^p.
\end{split}\end{equation*}
\par An analogous reasoning using the left hand inequality in \cite[Lemma $3.4$]{LNP} proves that
\begin{equation*}\begin{split}
||h_\mu||^p_{B^p} \gtrsim \sum_{n=0}^\infty (k+1)^{p-1}\mu_k^p.
\end{split}\end{equation*}
Finally, we shall prove \eqref{bloque}. By \cite[Lemma $3.1$]{MP}
and the M. Riesz projection theorem, it follows that
\begin{equation}\label{eq:j11}
\begin{split}
\left\|\sum_{k=2^n}^{2^{n+1}-1}z^{k-1} \right\|^p_{H^p}
\lesssim M^p_p\left(1-\frac{1}{2^n},\sum_{k=2^n}^{2^{n+1}-1}z^{k-1}\right)
\lesssim M^p_p\left(1-\frac{1}{2^n},\frac{1}{1-z} \right) \asymp 2^{n(p-1)}.
\end{split}
\end{equation}
On the other hand, using \cite[Lemma~3.1]{MP}, we obtain
\begin{equation*}
\begin{split}
& M_\infty\left(1-\frac{1}{2^n}, \sum_{k=2^n}^{2^{n+1}-1}z^{k-1} \right) \asymp \left\|\sum_{k=2^n}^{2^{n+1}-1}z^{k-1} \right\|_{H^\infty}=2^n.
\end{split}
\end{equation*}
Furthermore, using a well-know inequality, we deduce that
\begin{equation*}
\begin{split}
M_\infty\left(1-\frac{1}{2^n}, \sum_{k=2^n}^{2^{n+1}-1}z^{k-1} \right)
&\lesssim \left(\frac{1}{2^n}-\frac{1}{2^{n+1}}\right)^{-1/p}
M_p\left(1-\frac{1}{2^{n+1}}, \sum_{k=2^n}^{2^{n+1}-1}z^{k-1} \right)
\\ & \lesssim \left(\frac{1}{2^n}\right)^{-1/p}
\left\|\sum_{k=2^n}^{2^{n+1}-1}z^{k-1} \right\|_{H^p},
\end{split}
\end{equation*}
that is, $ \left\|\sum_{k=2^n}^{2^{n+1}-1}z^{k-1}
\right\|^p_{H^p}\gtrsim 2^{n(p-1)}$, which together with
\eqref{eq:j11} implies \eqref{bloque}. This finishes the proof.
\end{Pf}
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TITLE: Baire class 1 and discontinuities
QUESTION [4 upvotes]: Is it true that a bounded real function $f:[0,1]\to[0,1]$ with only countably many discontinuities has to be of Baire class 1, that is pointwise limit of a sequence of continuous functions? Is there a counter-example?
This would be an easy consequence of a theorem stated here: https://encyclopediaofmath.org/wiki/Baire_theorem
The theorem, attributed to Baire in that webpage, states that a function is the pointwise limit of a sequence of continuous functions if and only if its restriction on any perfect set has a point of continuity. However, in the original work by Baire added as a reference I could only find a weaker version of this theorem, which does not imply what I am searching for.
Many thanks!
REPLY [6 votes]: Baire showed in his 1899 thesis that such a function $f$ is a pointwise limit of continuous functions if and only if, for each perfect set $P,$ the function $f|_P$ (the restriction of $f$ to $P)$ is continuous on a set of points that is both dense and $G_{\delta}$ relative to $P.$ For the "if" half, we can replace "both dense and $G_{\delta}$ relative to $P$" with "nonempty" (still under the umbrella of "for each perfect set $P$").
Note: Here and below, "perfect" means "nonempty perfect" and "interval" means "interval of positive length".
Note that if a function $f$ is such that its restriction to every closed interval $I$ ("closed" is not actually needed) has at least one continuity point, then the function has a dense set of continuity points. This follows easily (see next sentence) from the obvious fact that every continuity point of $f_I$ that lies in the interior of $I$ will be a continuity point of $f.$ Thus, given any open subinterval $J$ of $[0,1],$ there exists a continuity point of $f$ lying in $J,$ because of the existence of a continuity point of $f|_I$ where $I$ is a closed-in-${\mathbb R}$ subinterval of $J.$
A similar argument works for perfect sets in place of intervals, and hence for the dense part of the "only if" half of Baire's theorem, it suffices to prove that, given any perfect set $P,$ the function $f_P$ has at least one continuity point. Thus, if $f$ is Baire $1$ and $P$ is a perfect set, then $f_P$ has a dense set of continuity points. Indeed, $f_P$ has a dense $G_{\delta}$ (both properties are relative to $P)$ set of continuity points, which implies that $f_P$ has a $c$-dense (relative to $P)$ set of continuity points, and more (non-continuity points only form a meager subset of $P).$
Incidentally, Baire proved the Baire category theorem for perfect sets (and not just for the reals), so he definitely would have known that $f_P$ has uncountably many continuity points in every subinterval of $P,$ but I don't know whether he specifically stated this anywhere. The fact that "uncountably many" can be sharpened to "continuum many" follows from William H. Young's result (in 1903 I believe) that any uncountable $G_{\delta}$ set has cardinality $c = 2^{\aleph_0}.$
Thus, the only if half of Baire's theorem has the following consequences for a Baire $1$ function $f:[0,1] \rightarrow [0,1].$ Note that for the statements below, the first implies that the continuity points are dense in $[0,1]$ and the second is an automatic consequence of the first.
For each open subinterval $J$ of $[0,1],$ $f$ has at least one continuity point in $J.$
For each open subinterval $J$ of $[0,1],$ $f$ has infinitely many continuity points in $J.$
For each open subinterval $J$ of $[0,1],$ $f$ has uncountably many continuity points in $J.$
For each open subinterval $J$ of $[0,1],$ $f$ has continuum many continuity points in $J.$
For each open subinterval $J$ of $[0,1],$ $f$ has co-meagerly many continuity points in $J.$
Moreover, the "if" half of Baire's theorem implies if a function $f:[0,1] \rightarrow [0,1]$ doesn't have too many discontinuity points, then the function will be a Baire $1$ function. The following are examples of "not too many".
At most finite or countably infinitely many discontinuity points.
At most meagerly many discontinuity points (i.e. the discontinuity set is first Baire category).
To see how the second condition just above (which of course includes the first condition) follows from Baire's theorem, let $D(f)$ be the discontinuity set of $f:[0,1] \rightarrow [0,1],$ let $P$ be a perfect subset of $[0,1],$ and assume $D(f)$ is a meager subset of $[0,1].$ Then $D(f) \cap P$ is meager relative to $P$ and every point in $P - (D(f) \cap P)$ is a continuity point of $f,$ and hence a continuity point of $f_P$ (any restriction of a function is continuous, when defined, at the continuity points of the function), so the continuity set of $f_P$ is a dense $G_{\delta}$ subset of $P.$ (The $G_{\delta}$ part is automatic, since the continuity set of any function from $[0,1]$ to $[0,1]$ is a $G_{\delta}$ set.)
A natural question is to what extent is the "restriction to perfect sets" aspect needed. Let $Z$ be the non-endpoints of the usual middle thirds (or any) Cantor set and let $f$ the the characteristic function of $Z$ (i.e. $f(x) = 1$ if $x \in Z$ and $f(x) = 0$ if $x \in [0,1] - Z.)$ Then $f$ is continuous at each point NOT in the Cantor set (because $f$ is locally constant at each such point), and hence the discontinuity set of $f$ is small in many ways. For instance, instead of the usual middle thirds Cantor set, we could use a Cantor set that has Hausdorff dimension zero. Nonetheless, $f$ is not a Baire $1$ function because the restriction of $f$ to the Cantor set is discontinuous at every point of the Cantor set (the restriction is $1$ on a dense subset of the Cantor set and $0$ on another dense subset of the Cantor set). Baire gave essentially this same example on p. 50 of his 1899 thesis. The reason I say "essentially" is because I believe Baire's example used the left endpoints of the Cantor middle thirds set rather than all the endpoints.
The example just above shows that while Baire $1$ functions are continuous almost everywhere in the sense of Baire category, the converse is not true. A function can be continuous almost everywhere in the sense of Baire category without being a Baire $1$ function. As indicated above, Baire proved this in 1899 by giving an explicit counterexample. A few years later Hausdorff (see p. 389 in his 1914 book Grundzüge der Mengenlehre) observed that a simple cardinality argument gives a nonconstructive proof of a stronger result—there are $c = 2^{\aleph_0}$ many Baire $1$ functions and $2^c$ many functions continuous almost everywhere in the sense of Baire category: The cardinality of the Baire $1$ functions is at least $c$ (consider constant functions) and no more than $c$ (there are $c$ many continuous functions, and hence $c$ many sequences of continuous functions), whereas the characteristic function of any of the $2^c$ many subsets of the Cantor set is continuous almost everywhere in the sense of Baire category (indeed, continuous except for a nowhere dense set).
| 144,052
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TITLE: What is the Davis Equation and why is it used in a Train Simulator?
QUESTION [8 upvotes]: I have been trying to understand how Microsoft Train Simulator works and people seem to use some Davis equation to calculate friction.
So my questions are:
What is it?
Why do they use it?
Are there alternatives to calculate train friction/are there other ways to calculate train friction?
REPLY [12 votes]: Basically, the Davis Equation is a resistance formula mainly used in basic go-stop situations like trains. The basic formula: $$R'=1.3+\frac{29}{w}+.045v+\frac{.0005av^2}{wn}$$ R being resistance, w is axle load in short tons, n is the number of axles, and a is the frontal area of the train in sq. feet. According to Szanto in Rolling Resistance Revisited you can modify the equation to fit standard freight cars, but the concepts are the same, factoring in air resistance as well. When you or the simulator substitutes the values above to yield certain necessary values for the simulation, you can find relatively accurate coefficients of drag. When you get resistance/drag the simulator will then compute whatever other factors are necessary and then create the appropriate image. This (according to Microsoft Train Simulator) happens hundreds of times a second at the highest settings to give high quality data for the discerning user.
Now as to your third question, yes, there are other ways of calculating friction, but the Davis Equation was designed specifically for this purpose and requires no extraneous values and in a sense is the most 'streamlined' equation for this purpose. Some come close though, most prominent being the Canadian National modification for double deck EMU's: $$R=14*\sqrt{10(M)(n)}$$ This square root function will yield more accurate resistance coefficients for taller wagons.
If anybody has found more accurate and EFFICIENT methods of finding resistance for trains than the Davis please edit or answer thusly, but as to my point of view the Train Simulator, as with all computer programs, uses this equation to balance both accuracy and speed of calculation.
| 182,736
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I had coolsculpting on low abs 7 weeks ago w/ small applicator. I had no side effects at first but 4 weeks later, the area became fatter and even developed a golf ball size lump of fat on one side. The whole overall area around my belly button is noticeably bigger. The dr confirmed it was fat and to wait it out. My question is if this is the rarely reported fat enlargement, how do you treat it? If it is just swelling, how long does it take typically to go away. Im scared to do another treatment.
Fatter Area After Coolsculpting?
Doctor Answers (3)
More Fat After Cool Sculpting?
Thank you for your question. There is a small chance of developing paradoxical hyperplasia, where the remaining fat cells can get enlarged after the Coolsculpting treatment. If this is the case, the treatment for it is laser liposuction. I would recommend being treated under the supervision of a Board Certified Dermatologist or Plastic Surgeon for safest and best treatment option. I hope this helps.
Fat swelling after fat reduction with coolsculpting by zeltiq
As fat dies, there is inflammation. there are times that a delayed inflammatory swelling can occur and this is temporary and can improve with lymphatic drainage treatments and external ultrasound therapy. Paradoxical fat hyperplasia (enlargement) is very rare, and not well understood, and there is not yet a consensus how best to treat this condition.
Web reference:
Cool Sculpting and lump
The best way to treat this development is to give it some time and see if it improves. If it does not, then you need to present this to your ps and get some idea of what it might be. On a rare occasion, fat necrosis can result and give you this problem. If so, your ps can describe how it will be alleviated in the future. Do not be afraid to ask questions of your plastic surgeon. Your condition is correct.
| 331,453
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Table Of Contents
UK Subs Main Page
Back To Bands
In the week following
the recording of Strangle Hold, we returned to Kingsway with John McCoy to record our
first Studio album. The album mostly documented the songs wed been playing live up
to that point - C.I.D. (newly re-recorded), I Couldnt Be
You, I Live in a Car, Tomorrows Girls (same recording
as the single but a different mix), World War, Rockers,
I.O.D., Lady Esquire (on which Charlie persuaded me to sing the
first verse), All I wanna Know, B.1.C., Disease &
Strangle Hold (re-recorded and added at the request of the label). Along with
these sub-standards came new songs, Killer, T.V. Blues,
Blues, Crash Course & Young Criminals.
It was during these sessions that we came close to killing
our producer! Jon had a playful nature to him, instigating races around the studio on the
producers chairs after each days work, or generally goofing around. One day we
wrestled him to the floor and threw foam rubber chairs on him and then we all piled on
top. Unbeknownst to us his face was pressed against the foam and he was suffocating. We
got bored and jumped off just at the point he was passing out..
At one of
our sold out Lyceum shows, director Julian Temple shot footage of the Subs for his new
film, a twenty minute documentary about the Subs. Another location was at the Cambridge
Trinity college May ball. Temple, who had become famous by directing the Sex pistols
Great Rock & Roll Swindle, along with two others, smuggled cameras and
sound equipment pretending to be the bands road crew. The end result was only a few
seconds in the finished film but what an image. The band are in a tent playing to college
kinds dressed in suits and bow ties and women in full white evening dresses. At the end
kids are pictured after ripping up their suits and dresses. The film opened for both
Scum and Quadrophenia.
On May 28th the Subs hit the front page of all the daily
papers when they ran a story of the Prince and the Punks. UK Sub fans Phil Sick, Ann
Wobble and Joe Horrid were passing a Polo field where Prince Charles was playing. A
reporter managed to bring them together and Phil Sick invited Prince Charles to The Subs
show at the Music Machine. The Prince declined but our management sent him an invite and
put him on the guest list anyway. We did get a card back from Buckingham Palace, but no
one spotted Charles at the show.
The Stranglehold tour started on June 11th 1979. Scheduled
for the 22nd was an appearance at the Glastonbury Fair. When we arrived it was awash with
the unwashed, with maybe a couple of punks in their midst. We decide to cut our losses and
left. As we left the concert area the cops pulled us over while a crew from Sounds
magazine were snapping photos. This gave us our first (And only) front cover of Sounds
30th June 1979. There were no arrests, but someone did manage to stick a UK Subs sticker
on the back of the Police van. The Strangle Hold tour ended on the 31st but included a
stop off at Maida Vale to record another John Peel Session. The long walk down the
corridor at the BBC studios was now quite familiar to us - pass the basketball court sized
orchestra studio, the vending machine, then finally to the small suite of studios at the
end. First time in, they were just removing the old valve mixing board from one of the
studios and the monster was propped up against the wall with its huge levers, forerunners to the slick faders were so
familiar with, protruding out. Our sessions there were for the most part dress rehearsals
for our next releases. At the time of the initial broadcast, of course, this was the first
taste our fans had of the recorded versions of songs they had been listening to live for
over a year. Although the structure of these songs did not change substantially when they
were re-recorded for record, many of the guitar solos are different and generally the
energy is somehow different. New Musical Express followed Sounds with a two page feature
on the Subs on July 7th and Melody Maker on August 18th with a full page.
We rushed the 3rd single, the remixed Tomorrows
Girls, to maximize the impact Strangle Hold made. Again we added two
harder hitting (Non-LP), tracks on the B side. Although Strangle
Hold got joint record of the week in Sounds magazine with the Ruts
Babylons Burning, Tomorrows Girls was a stronger single. The
cover showed Pauls sister Joanne reflected in a shop window and was released on blue
vinyl, (Strangle Hold was on red). Again the single charted (#28) and we went into the BBC
to record another Top Of The Pops. Meanwhile Another Kind Of Blues had charted
at # 21 in the national album charts.
Although wed toured somewhat in the UK on the
strength of the Strangle Hold single, we now embarked on our first major tour for our new
agency MAM starting in Derby on the 20th of September and ending in Guilford
Surrey on the 9th of
November. The tour included Newcastle, Sheffield, London (Three nights at the Marquee),
Nottingham, Manchester, Leeds, Plymouth, Glasgow, and Birmingham, but also included many
smaller towns which became so important for our fan base over the next three years. Towns
such as Colne, Middlesborough and East Retford were home to some of our most loyal fans. A
special mention also should go out to fans from Witon, The Isle of Sheppie and the
Leicester punk girls. The hardcore fans from these three crews followed us from town to
town between 1978 and 1983 sleeping in fields and barns and hitch hiking from show to
show. For the UK dates they would sometimes swell the numbers at a given show by over one
hundred. This was particularly significant at the venues in the smaller places tucked away
in the English countryside. Sometimes these sleepy towns had never seen more than a couple
of punks and were ill prepared for the invasion of subs fans. Often the local punk scene
in these small communities consisted of less than ten people. These punk fans were often
ridiculed by the Real music fans who were still listening to Yes
and Genesis, when our tour rolled through it must have been a huge validation
for them. Certainly if the Subs returned to the same town even, six months later, the same
punk scene had bloomed to twenty or thirty with half of those so-called Real music
fans cutting their hair and pretending they were there.
Sounds, now firmly championing the Subs, ran another two
page feature on October 3rd.
We had been messing around with the Zombies classic
Shes not there at sound checks, and although not ready to record another
album decided to release it as a single to keep the momentum going. This was my first time
producing. The single, released on green vinyl, charted at # 33 and again we did Top Of
The Pops. This time Paul sang lead vocals, as he did on the record, while Charlie jumped
around with a flying vee guitar.
| 58,015
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TITLE: Help understanding Energy estimates for Linear Hyperbolic PDE
QUESTION [2 upvotes]: I've been struggling for a while to understand the proof in Evans of the Energy Estimates of Linear Hyperbolic PDE. This is Theorem 2 in section 7.2. I have added the proof from Evans for convenience. I'm not a mathematician, rather an economist looking to improve my maths skills. Sorry if this is a rather trivial question for this website. Any advice would be very much appreciated. My problem is understanding the following:
Consider
\begin{equation}
B_1=\int_U \sum_{i,j=1}^n a^{ij}(\mathbf{u}_m)_{x_i}(\mathbf{u}'_m)_{x_j}\;dx \tag{1}
\end{equation}
We aim to write this expression as
\begin{equation}
B_1=\frac{d}{dt}\bigg(\frac{1}{2}A[\mathbf{u}_m,\mathbf{u}_m;t]\bigg)-\frac{1}{2}\int_U\sum_{i,j=1}^n a_t^{ij}(\mathbf{u}_m)_{x_i}(\mathbf{u}_m)_{x_j}\;dx
\tag{2}\end{equation}
since $a^{ij}=a^{ji}$ and $A[u,v;t]$ is given by
\begin{equation}
A[u,v;t]:=\int_U \sum_{i,j=1}^n a^{ij}u_{x_i}v_{x_j}\;dx\;\;\;(u,v\in H^1_0(U))
\tag{3}\end{equation}
Then from our new expression for $B_1$ we find that
\begin{equation}
B_1\geq \frac{d}{dt}\bigg(\frac{1}{2}A[\mathbf{u}_m,\mathbf{u}_m;t]\bigg)-C\|\mathbf{u}_m\|^2_{H^1_0(U)}
\tag{4}\end{equation}
My problem is that I can't see how to get from $(1)$ to $(2)$. It seems like it's just a case of bringing the differentiation operator through the integral and summation as usual but I just can't get it to work. My second question is how to get the inequality $(4)$. In particular I'm not sure how to bound the second term on the right hand side of $(2)$.
I have tried the following idea
\begin{align}
\int_U\sum_{i,j=1}^n a_t^{ij}(\mathbf{u}_m)_{x_i}(\mathbf{u}_m)_{x_j}\;dx &\leq C\int_U\sum_{i,j=1}^n(\mathbf{u}_m)_{x_i}(\mathbf{u}_m)_{x_j}\;dx \\&= C\int_U\sum_{i=1}^n|(\mathbf{u}_m)_{x_i}|^2+\sum_{i\neq j}^n(\mathbf{u}_m)_{x_i}(\mathbf{u}_m)_{x_j}\;dx \tag{5}
\end{align}
But I get stuck here as I don't know the sign of $(\mathbf{u}_m)_{x_i}(\mathbf{u}_m)_{x_j}$ over $U$
$\mathbf{Edit}$ 6/8/17
I have realised that I have forgotten to mention that $\mathbf{u}_m(t)=\sum d^k_m(t)w_k$ where $\{w\}_{k=1}^\infty$ is an orthogonal basis of $H^1_0(U)$ and $L^2(U)$ from our Galerkin approximations. From this can I conclude that the cross product terms in $(5)$ are $0$. Then adding $|\mathbf{u}_m|$ I get
\begin{align}
\int_U\sum_{i,j=1}^n a_t^{ij}(\mathbf{u}_m)_{x_i}(\mathbf{u}_m)_{x_j}\;dx &\leq C\int_U\sum_{i,j=1}^n(\mathbf{u}_m)_{x_i}(\mathbf{u}_m)_{x_j}\;dx \\&= C\int_U\sum_{i=1}^n|(\mathbf{u}_m)_{x_i}|^2+\sum_{i\neq j}^n(\mathbf{u}_m)_{x_i}(\mathbf{u}_m)_{x_j}\;dx\\&=C\int_U\sum_{i=1}^n|(\mathbf{u}_m)_{x_i}|^2\;dx \\ &\leq C\int_U |\mathbf{u}_m|^2+\sum_{i=1}^n|(\mathbf{u}_m)_{x_i}|^2\;dx \\&=C\|\mathbf{u}_m\|_{H^1_0(U)}^2
\end{align}
REPLY [0 votes]: From (1) to (2), yes $a^{ij}$ is assumed to be $C^1$ and the only function in $t$ in $(u_m)_x$ is the coefficient $d_m$. From (3) to (4), I think here he assumes readers are familiar with discussions in elliptic equations. See section 6.2.2.
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Rapper Polo G Was Arrested for Threatening an Officer in Miami After Traffic StopBy Mustafa Gatollari
Jun. 14 2021, Published 9:03 a.m. ET
It's always a high-stress situation whenever you see those blue and red lights flashing behind you and that police siren going off, especially with all of the news stories circulating of police brutality and excessive use of force. This is one of the reasons that there are a lot of questions surrounding why Rapper Polo G was arrested in Miami.
Why was Polo G arrested?
NBC News reported that rapper Polo G "was charged with battery against a police officer in Miami following a traffic stop early Saturday morning." That's Saturday, June 12, 2021.
The artist, whose real name is Taurus Bartlett, was also slapped with additional charges, including "threatening a public servant, using violence to resist an officer, and resisting an officer without violence and criminal mischief," per Miami-Dade county records.
NBC Miami states that the arrest occurred after a traffic stop on South Biscayne Boulevard. Polo G put up on social media that the police were "playin foul in Miami." He also seems to suggest that he was targeted from the second they landed in Miami but didn't expound on the reasoning behind police singling him or his driver out.
"One of the officers told us they was on us since we got off our jet," the rapper tweeted. His tweet was met with a litany of people on social media offering up their support for the rapper, with others claiming that authorities were "lying" about the alleged tussle that occurred between Polo G and the police.
Polo G's manager/mother Stacia Mac wrote on Twitter that her son wasn't the one who initiated contact, suggesting that the blame of the incident rested with authorities: "None of these charges would be possible if the POLICE did not make contact with my son Polo G!!!
"He was NOT the driver. He was a PASSENGER in a professionally licensed vehicle with security. He was moving smart and correctly. What more could he have done?"
Local officers in Miami said that they would be reviewing all available footage of the incident to see if there was any wrongdoing on the apart of either party involved.
Polo G was eventually released on bond and posted photos of himself on a plane back to Atlanta, flashing a huge stack of cash and urging followers to "Meet [him] @ Kod tonight."
Polo G's "Hall of Fame" album was listed as one of "six albums you should listen to now" by Pitchfork. The collection of songs from the rapper is receiving serious praise from hip-hop fans.
The effort is Polo G's third studio album. He recently started his own record label called Only Dreamers Achieve in partnership with Columbia Records.
NBC News reports that neither the Miami-Dade county police department nor Columbia Records responded to requests for comment on the incident between Polo G and the officers involved in the traffic stop. We will update this story with more information as it becomes available.
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\begin{document}
\begin{abstract}
We present a proof of Milnor conjecture in dimension $3$ based on Cheeger-Colding
theory on limit spaces of manifolds with Ricci curvature bounded below. It is different from \cite{Liu}
that relies on minimal surface theory.
\end{abstract}
\maketitle
\section{Introduction}
Milnor \cite{Mi} in $1968$ conjectured that any open $n$-manifold $M$ with $\mathrm{Ric}_M\ge 0$ has a finitely generated fundamental group. This conjecture remains open today. It was proven for manifolds with Euclidean volume growth by Anderson \cite{An} and Li \cite{Li} independently, and manifolds with small diameter growth by Sormani \cite{Sor}. For background and relevant examples regarding Milnor conjecture, see \cite{SS}.
For $3$-manifolds, Schoen and Yau \cite{SY} developed minimal surfaces theory in dimension $3$ and proved that any $3$-manifold of positive Ricci curvature is diffeomorphic to $\mathbb{R}^3$. Recently, based on minimal surface theory, Liu \cite{Liu} proved that any $3$-manifold with $\mathrm{Ric}\ge 0$ either is diffeomorphic to $\mathbb{R}^3$ or its universal cover splits. In particular, this confirms Milnor conjecture in dimension $3$.
There are some interests to find a proof of Milnor conjecture in dimension $3$ not relying on minimal surface theory. Our main attempt is to accomplish this by using structure results for limits spaces of manifolds with Ricci curvature bounded below \cite{Co,CC1,CC2,CN}, equivariant Gromov-Hausdorff convergence \cite{FY} and pole group theorem \cite{Sor}.
\begin{thm1}\label{3}
Let $M$ be an open $3$-manifold with $\mathrm{Ric}_M\ge 0$, then $\pi_1(M)$ is finitely generated.
\end{thm1}
For any open $3$-manifold $M$ of $\mathrm{Ric}_M\ge 0$ and any sequence $r_i\to\infty$, by Gromov's precompactness theorem \cite{Gro2}, we can pass to some subsequences and consider tangent cones at infinity of $M$ and its Riemannian universal cover $\widetilde{M}$ coming from the sequence $r_i^{-1}\to 0$:
\begin{center}
$\begin{CD}
(r^{-1}_i\widetilde{M},\tilde{p}) @>GH>>
(C_\infty\widetilde{M},\tilde{o})\\
@VV\pi V @VV\pi V\\
(r^{-1}_iM,p) @>GH>> (C_\infty M,o).
\end{CD}$
\end{center}
We roughly illustrate our approach to prove Theorem \ref{3}. If $\pi_1(M,p)$ is not finitely generated, then we draw a contradiction by choosing some sequence $r_i\to\infty$ and eliminating all the possibilities regarding the dimension of $C_\infty \widetilde{M}$ and $C_\infty M$ above in the Colding-Naber sense \cite{CN}, which are integers $1,2$ or $3$.
We also make use of some reduction results by Wilking \cite{Wi} and Evans-Moser \cite{EM}. The first reduces any non-finitely generated fundamental groups to abelian ones in any dimension, while the latter further reduces abelian non-finitely generated ones to some subgroup of the additive group of rationals in dimension $3$. In particular, we can assume that $\pi_1(M)$ is torsion free if it is not finitely generated. One observation is that, if $\pi_1(M,p)$ is torsion free, then in the space $(C_\infty\widetilde{M},\tilde{v},G)$ above, the orbit $G\cdot\tilde{v}$ is not discrete (See Corollary \ref{non_dis_orb_cor}). This observation plays a key role in the proof.
The author would like to thank Professor Xiaochun Rong and Professor Jeff Cheeger for suggestions during the preparation of this note.
\section{Proof of Theorem \ref{3}}
We start with the following reductions by Wilking and Evan-Moser.
\begin{thm}\cite{Wi}\label{red_W}
Let $M$ be an open manifold with $\mathrm{Ric}_M\ge 0$.
If $\pi_1(M)$ is not finitely generated, then it contains a non-finitely generated abelian subgroup.
\end{thm}
\begin{thm}\cite{EM}\label{red_EM}
Let $M$ be a $3$-manifold. If $\pi_1(M)$ is abelian and not finitely generated, then $\pi_1(M)$ is torsion free.
\end{thm}
Evans-Moser \cite{EM} actually showed that $\pi_1(M)$ is a subgroup of the additive group of rationals. Being torsion free is sufficient for us to prove Theorem \ref{3}.
Gromov \cite{Gro1} introduced the notion of short generators of $\pi_1(M,p)$. By path lifting, $\pi_1(M,p)$ acts on $\widetilde{M}$ isometrically. We say that $\{\gamma_1,...,\gamma_i,...\}$ is a set of short generators of $\pi_1(M,p)$, if
\begin{center}
$d(\gamma_1\tilde{p},\tilde{p})\le d(\gamma\tilde{p},\tilde{p})$ for all $\gamma\in\pi_1(M,p)$,
\end{center}
and for each $i$,
\begin{center}
$d(\gamma_i\tilde{p},\tilde{p})\le d(\gamma\tilde{p},\tilde{p})$ for all $\gamma\in\pi_1(M,p)-\langle\gamma_1,...,\gamma_{i-1}\rangle$,
\end{center}
where $\langle\gamma_1,...,\gamma_{i-1}\rangle$ is the subgroup generated by $\gamma_1,...,\gamma_{i-1}$.
Let $M$ be an open $3$-manifold with $\mathrm{Ric}_M\ge 0$. We always denote $\pi_1(M,p)$ by $\Gamma$. Suppose that $\Gamma$ is not finitely generated, then by Theorems \ref{red_W} and \ref{red_EM}, we can assume that $\Gamma$ is torsion free. Let $\{\gamma_1,...,\gamma_i,...\}$ be an infinite set of short generators at $p$. Since $\Gamma$ is a discrete group acting freely on $\widetilde{M}$, we have $r_i=d(\tilde{p},\gamma_i\tilde{p})\to \infty$. When considering a tangent cone at infinity of $\widetilde{M}$ coming from the sequence $r_i^{-1}\to 0$, we also take $\Gamma$-action into account. Passing to some subsequences if necessary, we assume the following sequences converge in equivariant Gromov-Hausdorff topology \cite{FY}:
\begin{center}
$\begin{CD}
(r^{-1}_i\widetilde{M},\tilde{p},\Gamma) @>GH>>
(\widetilde{Y},\tilde{y},G)\\
@VV\pi V @VV\pi V\\
(r^{-1}_iM,p) @>GH>> (Y=\widetilde{Y}/G,y).
\end{CD}$
$(\star)$
\end{center}
Colding-Naber \cite{CN} showed that the isometry group of any Ricci limit space is a Lie group. In particular, $G$ above, as a closed subgroup of $\mathrm{Isom}(\widetilde{Y})$, is a Lie group.
We recall the dimension of Ricci limit spaces in the Colding-Naber sense \cite{CN}. A point $x$ in some Ricci limit space $X$ is $k$-regular, if any tangent cone at $x$ is isometric to $\mathbb{R}^k$. Colding-Naber showed that there is a unique $k$ such that $\mathcal{R}_k$, the set of $k$-regular points, has full measure in $X$ with respect to any limit renormalized measure (See \cite{CC2,CN}). We regard such $k$ as the dimension of $X$ and denote it by $\dim(X)$. It is unknown whether in general the Hausdorff dimension of $X$ equals to $\dim(X)$. For Ricci limit spaces coming from $3$-manifolds, dimension in the Colding-Naber sense equals to Hausdorff dimension, which follows from Theorem 3.1 in \cite{CC2} and \cite{Hon1}.
As indicated in the introduction, we prove Theorem \ref{3} by eliminating all possibilities regarding the dimension of $Y$ and $\widetilde{Y}$ in ($\star$). There are three possibilities and we rule out each of them, which finishes the proof of Theorem \ref{3}.\\
\noindent\textit{Case 1. $\dim(\widetilde{Y})=3$} (Lemma \ref{not_3});\\
\textit{Case 2. $\dim(Y)=\dim(\widetilde{Y})=2$} (Lemma \ref{not_2});\\
\textit{Case 3. $\dim(Y)=1$} (Lemma \ref{not_1}).\\
\begin{lem}\label{non_dis_orb}
Let $(M_i,p_i)$ be a sequence of complete $n$-manifolds and $(\widetilde{M}_i,\tilde{p}_i)$ be their universal covers. Suppose that the following sequence converges
$$(\widetilde{M}_i,\tilde{p}_i,\Gamma_i)\overset{GH}\longrightarrow(\widetilde{X},\tilde{p},G),$$
where $\Gamma_i=\pi_1(M_i,p_i)$ is torsion free for each $i$. If the orbit $G\cdot\tilde{p}$ is discrete in $\widetilde{X}$, then there is $N$ such that
$$\#\Gamma_i(1)\le N$$
for all $i$, where $\#\Gamma_i(1)$ denotes the number of elements in
$$\Gamma_i(1)=\{\gamma\in \Gamma_i\ |\ d(\gamma\tilde{p}_i,\tilde{p}_i)\le 1\}.$$
\end{lem}
\begin{proof}
We claim that if a sequence $\gamma_i\in\Gamma_i$ such that $\gamma_i\overset{GH}\to g\in G$ with $g$ fixing $\tilde{p}$, then $g=e$, the identity element, and $\gamma_i=e$ for all $i$ sufficiently large. In fact, suppose that $\gamma_i\not= e$ for some subsequence. Since $\gamma_i$ is torsion free, we always have $\mathrm{diam}(\langle\gamma_i\rangle\cdot \tilde{p}_i)=\infty$. Together with $d(\gamma_i\tilde{p}_i,\tilde{p}_i)\to 0$, we see that $G\cdot\tilde{p}$ can not be discrete, which contradicts with the assumption.
Therefore, there exists $i_0$ large such that for all $g\in G(2)$ and any two sequences with $\gamma_i\overset{GH}\to g$ and $\gamma'_i\overset{GH}\to g$, $\gamma_i=\gamma'_i$ holds for all $i\ge i_0$. In particular, we conclude that
$$\#\Gamma_i(1)\le \# G(2)<\infty$$
for all $i\ge i_0$.
\end{proof}
\begin{cor}\label{non_dis_orb_cor}
Let $(M,p)$ be an open $n$-manifold with $\mathrm{Ric}_M\ge 0$ and $(\widetilde{M},\tilde{p})$ be its universal cover. Suppose that $\Gamma=\pi_1(M,p)$ is torsion free, then for any $s_i\to \infty$ and any convergent sequence
$$(s_i^{-1}\widetilde{M},p,\Gamma)\overset{GH}\longrightarrow(C_\infty\widetilde{M},\tilde{o},G),$$
the orbit $G\cdot\tilde{v}$ is not discrete.
\end{cor}
\begin{proof}
The proof follows directly from Lemma \ref{non_dis_orb}. If $G\cdot\tilde{o}$ is discrete, then there is $N$ such that $\#\Gamma(s_i)\le N$ for all $i$. On the other hand, $\#\Gamma(s_i)\to \infty$ because $\Gamma$ is torsion free. A contradiction.
\end{proof}
\begin{lem}\label{non_cnt_orb}
Let $(M,p)$ be an open $n$-manifold with $\mathrm{Ric}_M\ge 0$ and $(\widetilde{M},\tilde{p})$ be its universal cover. Suppose that $\Gamma=\pi_1(M,p)$ has infinitely many short generators $\{\gamma_1,...,\gamma_i,...\}$. Then in the following tangent cone at infinity of $\widetilde{M}$
$$(r_i^{-1}\widetilde{M},p,\Gamma)\overset{GH}\longrightarrow(\widetilde{Y},\tilde{y},G),$$
the orbit $G\cdot\tilde{y}$ is not connected, where $r_i=d(\gamma_i\tilde{p},\tilde{p})\to\infty$.
\end{lem}
\begin{proof}
On $r_i^{-1}\widetilde{M}$, $\gamma_i$ has displacement $1$ at $\tilde{p}$. By basic properties of short generators, $\gamma_i\tilde{p}$ has distance $1$ from the orbit $H_i\cdot\tilde{p}$, where $H_i=\langle\gamma_1,...,\gamma_{i-1}\rangle$. From equivariant convergence
$$(r^{-1}_i\widetilde{M},\tilde{p},H_i,\gamma_i)\overset{GH}\longrightarrow(\widetilde{Y},\tilde{y},H,g),$$
we conclude $d(g\tilde{y},H\cdot\tilde{y})=1$. It is obvious that $H$ contains $G_0$, the connected component of $G$ containing the identity. Thus $d(g\tilde{y},G_0\cdot\tilde{y})\ge 1$ and the orbit $G\cdot \tilde{y}$ is not connected.
\end{proof}
We recall cone splitting principle, which follows from splitting theorem for Ricci limit spaces \cite{CC1}.
\begin{prop}\label{cone_split}
Let $(X,p)$ be the limit of a sequence of complete $n$-manifolds $(M_i,p_i)$ of $\mathrm{Ric}_{M_i}\ge 0$. Suppose that $X=\mathbb{R}^k\times C(Z)$ is a Euclidean cone with vertex $p=(0,z)$. If there is an isometry $g\in \mathrm{Isom}(X)$ with $g(0,z)\not\in \mathbb{R}^k\times \{z\}$, then $X$ splits isometrically as $\mathbb{R}^{k+1}\times C(Z')$.
\end{prop}
\begin{lem}\label{not_3}
Case 1 can not happen.
\end{lem}
\begin{proof}
When $\dim(\widetilde{Y})=3$, $\widetilde{Y}$ is a non-collapsing limit space \cite{CC2}, that is, there is $v>0$ such that
$$\mathrm{vol}(B_1(\tilde{p},r_i^{-1}\widetilde{M}))\ge v$$
for all $i$. By relative volume comparison, this implies that $\widetilde{M}$ has Euclidean volume growth
$$\lim\limits_{r\to\infty}\dfrac{\mathrm{vol}(B_r(\tilde{p}))}{r^n}\ge v.$$
By \cite{CC2}, $\widetilde{Y}$ is a Euclidean cone $\mathbb{R}^k\times C(Z)$ with vertex $\tilde{y}=(0,z)$, where $C(Z)$ does not contain any line and $z$ is the vertex of $C(Z)$. We rule out all the possibilities of $k\in \{0,1,2,3\}$.
If $k=3$, then $\widetilde{Y}=\mathbb{R}^3$. Thus $\widetilde{M}$ is isometric to $\mathbb{R}^3$ \cite{Co}.
If $k=2$, then according to co-dimension $2$ \cite{CC2}, actually $\widetilde{Y}=\mathbb{R}^3$.
If $k=1$, then $Y=\mathbb{R}\times C(Z)$. By Proposition \ref{cone_split}, the orbit $G\cdot \tilde{y}$ is contained in $\mathbb{R}\times\{z\}$. Applying Lemma \ref{non_cnt_orb}, we see that $G\cdot \tilde{y}$ is not connected. Note that a non-connected orbit in $\mathbb{R}$ is either a $\mathbb{Z}$-translation orbit, or a $\mathbb{Z}_2$-reflection orbit. In particular, the orbit $G\cdot\tilde{y}$ must be discrete. This contradicts with Corollary \ref{non_dis_orb_cor}.
If $k=0$, then $Y=C(Z)$ with no lines. Again by Proposition \ref{cone_split}, the orbit $G\cdot\tilde{y}$ must be a single point $\tilde{y}$, which is a contradiction to Lemma \ref{non_cnt_orb}.
\end{proof}
\begin{lem}\label{not_2}
Let $(M,p)$ be an open $n$-manifold with $\mathrm{Ric}_M\ge 0$ and $(\widetilde{M},\tilde{p})$ be its universal cover. Assume that $\Gamma=\pi_1(M,p)$ is torsion free. Then for any $s_i\to \infty$ and any convergent sequence
\begin{center}
$\begin{CD}
(s^{-1}_i\widetilde{M},\tilde{p},\Gamma) @>GH>>
(C_\infty\widetilde{M},\tilde{o},G)\\
@VV\pi V @VV\pi V\\
(s^{-1}_iM,p) @>GH>> (C_\infty M,o),
\end{CD}$
\end{center}
$\dim(C_\infty\widetilde{M})=\dim(C_\infty M)$ can not happen. In particular, Case 2 can not happen.
\end{lem}
\begin{proof}[Proof of Lemma \ref{not_2}]
We claim that $G$ is a discrete group when $\dim({C_\infty\widetilde{M}})=\dim(C_\infty M)=k$. If the claim holds, then the desired contradiction follows from Corollary \ref{non_dis_orb_cor}.
It remains to verify the claim. Suppose that $G_0$ is non-trivial, then we pick $g\not=e$ in $G_0$. Note that there is a $k$-regular point $\tilde{q}\in C_\infty \widetilde{M}$ such that $d(g\tilde{q},\tilde{q})>0$ and $\tilde{q}$ projects to a $k$-regular point $q\in C_\infty M$. In fact, let $\mathcal{R}_k(C_\infty M)$ be the set of $k$-regular points in $C_\infty M$. Since $\mathcal{R}_k(C_\infty M)$ is dense in $C_\infty M$, its pre-image $\pi^{-1}(\mathcal{R}_k(C_\infty M))$ is also dense in $C_\infty \widetilde{M}$. Let $\tilde{q}$ be a point in the pre-image such that $d(g\tilde{q},\tilde{q})>0$. Note that any tangent cone at $\tilde{q}$ splits $\mathbb{R}^k$-factor isometrically. By Proposition 3.78 in \cite{Hon2} (also see Corollary 1.10 in \cite{KL}), it follows that any tangent cone at $\tilde{q}$ is isometric to $\mathbb{R}^k$. In other words, $\tilde{q}$ is $k$-regular.
Along a one-parameter subgroup of $G_0$ containing $g$, we can choose a sequence of elements $g_j\in G_0$ with $d(g_j\tilde{q},\tilde{q})=1/j\to 0$. We consider a tangent cone at $\tilde{y}$ and $y$ respectively coming from the sequence $j\to\infty$. Passing to some subsequences if necessary, we obtain
\begin{center}
$\begin{CD}
(jC_\infty\widetilde{M},\tilde{q},G,g_j) @>GH>>
(C_{\tilde{q}} C_\infty\widetilde{M},\tilde{o}',H, h)\\
@VV\pi V @VV\pi V\\
(jC_\infty M,q) @>GH>> (C_qC_\infty M,o').
\end{CD}$
\end{center}
with $C_{\tilde{q}} C_\infty\widetilde{M}/H=C_qC_\infty M$ and $d(h \tilde{o}',\tilde{o}')=1$. On the other hand, since both $q$ and $\tilde{q}$ are $k$-regular, $C_{\tilde{q}} C_\infty\widetilde{M}=C_qC_\infty M=\mathbb{R}^k$. This is a contradiction to $H\not=\{e\}$. Hence the claim holds.
\end{proof}
To rule out the last case $\dim(Y)=1$, we recall Sormani's pole group theorem \cite{Sor}. We say that a length space $X$ has a pole at $x\in X$, if for all $y\not=x$, there is a ray starting from $x$ and going through $y$.
\begin{thm}\cite{Sor}\label{non_polar}
Let $(M,p)$ be an open $n$-manifold with $\mathrm{Ric}_M\ge 0$ and $(\widetilde{M},\tilde{p})$ be its universal cover. Suppose that $\Gamma=\pi_1(M,p)$ has infinitely many short generators $\{\gamma_1,...,\gamma_i,...\}$. Then in the following tangent cone at infinity of $M$
$$(r_i^{-1}M,p)\overset{GH}\longrightarrow(Y,y),$$
$Y$ can not have a pole at $y$, where $r_i=d(\gamma_i\tilde{p},\tilde{p})\to\infty$.
\end{thm}
\begin{lem}\label{not_1}
Case 3 can not happen.
\end{lem}
\begin{proof}
By \cite{Hon1} (also see \cite{Chen}), $Y$ is a topological manifold of dimension $1$. Since $Y$ is non-compact, $Y$ is either a line $(-\infty,\infty)$ or a half line $[0,\infty)$. By Theorem \ref{non_polar}, $Y$ can not have a pole at $y$. Thus there is only one possibility left: $Y=[0,\infty)$ but $y$ is not the endpoint $0\in [0,\infty)$. Put $d=d_Y(0,y)>0$. We rule out this case by a rescaling argument and Lemmas \ref{not_3}, \ref{not_2} above. (In general, it is possible for an open manifold having a tangent cone at infinity as $[0,\infty)$ with base point not being $0$. See example \ref{tree}.)
Let $\alpha(t)$ be a unit speed ray in $M$ starting from $p$ and converging to the unique ray from $y$ in $Y=[0,\infty)$ with respect to the sequence $(r_i^{-1}M,p)\overset{GH}\longrightarrow(Y,y)$. Let $x_i\in r_i^{-1}M_i$ be a sequence of points converging to $0\in Y$, then $r_i^{-1}d_M(p,x_i)\to d$. For each $i$, let $c_i(t)$ be a minimal geodesic from $x_i$ to $\alpha(dr_i)$, and $q_i$ be a closest point to $p$ on $c_i$. We reparametrize $c_i$ so that $c_i(0)=q_i$. With respect to $(r_i^{-1}M,p)\overset{GH}\longrightarrow(Y,y)$, $c_i$ subconverges to the unique segment between $0$ and $2d\in [0,\infty)$. Clearly,
$$r_i^{-1}d_M(x_i,\alpha(dr_i))\to 2d, \quad r_i^{-1}d_i\to 0,$$
where $d_i=d_M(p,c_i(0))$.
If $d_i\to\infty$, then we rescale $M$ and $\widetilde{M}$ by $d_i^{-1}\to 0$. Passing to some subsequences if necessary, we obtain
\begin{center}
$\begin{CD}
(d^{-1}_i\widetilde{M},\tilde{p},\Gamma) @>GH>>
(\widetilde{Y}',\tilde{y}',G')\\
@VV\pi V @VV\pi V\\
(d^{-1}_iM,p) @>GH>> (Y',y').
\end{CD}$
\end{center}
If $\dim(Y')=1$, then we know that $Y'=(-\infty,\infty)$ or $[0,\infty)$. On the other hand, since
$$d_i^{-1}d_M(c_i(0),x_i)\to\infty,\quad d_i^{-1}d_M(c_i(0),\alpha(dr_i))\to\infty, \quad d_i^{-1}d_M(c_i,p)=1,$$
$c_i$ subconverges to a line $c_\infty$ in $Y'$ with $d(c_\infty,y')=1$. Clearly this can not happen in $(-\infty,\infty)$ nor $[0,\infty)$. If $\dim(\widetilde{Y}')=3$, then $\widetilde{M}$ has Euclidean volume growth and thus $\dim(\widetilde{Y})=3$. This case is already covered in Lemma \ref{not_3}. The only situation left is $\dim(\widetilde{Y}')=\dim(Y')=2$. By Lemma \ref{not_2}, this also leads to a contradiction. In conclusion, $d_i\to\infty$ can not happen.
If there is some $R>0$ such that $d_i\le R$ for all $i$, then on $M$, $c_i$ subconverges to a line $c$ with $c(0)\in B_{2R}(p)$. Consequently, $M$ splits a line isometrically \cite{CG}, which contradicts with $Y=[0,\infty)$. This completes the proof.
\end{proof}
\begin{exmp}\label{tree}
We construct a surface $(S,p)$ isometrically embedded in $\mathbb{R}^3$ such that $S$ has a tangent cone at infinity as $[0,\infty)$, but $p$ does not correspond to $0$. We first construct a subset of $xy$-plane by gluing intervals. Let $r_i\to\infty$ be a positive sequence with $r_{i+1}/r_i\to\infty$. Starting with a interval $I_1=[-r_1,r_2]$, we attach a second interval $I_2=[-r_3,r_4]$ perpendicularly to $I_1$ by identifying $r_2\in I_1$ and $0\in I_2$. Repeating this process, suppose that $I_{k}$ is attached, then we attach the next interval $I_{k+1}=[-r_{2k+1},r_{2k+2}]$ perpendicularly to $I_k$ by identifying $r_{2k}\in I_k$ and $0\in I_{k+1}$. In the end, we get a subset $T$ in the $xy$-plane consisting of segments. We can smooth the $\epsilon$-neighborhood of $T$ in $\mathbb{R}^3$ so that it has sectional curvature $\ge -C$ for some $\epsilon,C>0$. We call this surface $S$ and let $p\in S$ be a point closest to $0\in I_1$ as base point. If we rescale $(S,p)$ by $r_{2k+1}^{-1}$, then
$$(r_{2k+1}^{-1}S,p)\overset{GH}\longrightarrow ([-1,\infty),0)$$
because $r_{i+1}/r_i\to\infty$. In other words, $S$ has a tangent cone at infinity as the half line, but the base point does not correspond to the end point in this half line.
\end{exmp}
| 132,879
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It was his second concussion since the Yankees wrapped up the World Series (he also hit his head in winter ball). Cervelli has received good news from all tests involved this time around and was cleared to start light workouts and should be back in the lineup by Friday.
Here is more from Ben Shpigel of the NY Times :
“Whether that’s a week, tomorrow, two weeks, it doesn’t really matter to me as long as when he’s ready, he’s ready,” Cashman said Monday ...
Read Full Article at Bleacher Report - New York Yankees
| 259,853
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\begin{document}
\title{Active User Detection and Channel Estimation for Spatial-based Random Access in Crowded Massive MIMO Systems via Blind Super-resolution}
\author{Abolghasem Afshar, Vahid Tabataba Vakili, Sajad Daei
\thanks{The authors are with the school of Electrical Engineering, Iran University of Science and Technology}
}
\maketitle
\begin{abstract}
This work presents a novel framework for random access (RA) in crowded scenarios of massive multiple-input multiple-output (MIMO) systems. A huge portion of the system resources is dedicated as orthogonal pilots for accurate channel estimation which imposes a huge training overhead. This overhead can be highly mitigated by exploiting intrinsic angular domain sparsity of massive MIMO channels and the sporadic traffic of users, i.e., few number of users are active to send or receive data in each coherence interval. Besides, the continuous-valued angles of arrival (AoA) corresponding to each active user are alongside each other forming a specific cluster. To exploit these features in this work, we propose a blind clustering algorithm based on super-resolution techniques that not only detects the spatial features of the active users but also provides accurate channel estimation. Specifically, an off-grid atomic norm minimization is proposed to obtain the AoAs and then a clustering-based approach is employed to identify which AoAs correspond to which active users. After active user detection, an alternating-based optimization approach is performed to obtain the channels and transmitted data. Simulation results demonstrate the effectiveness of our approach in AoA detection as well as data recovery which indeed provides a high performance spatial-based RA in crowded massive MIMO systems.
\end{abstract}
\begin{IEEEkeywords}
Crowded massive MIMO, Random access, Super-resolution, Atomic norm, Semi-definite programming, Convex optimization.
\end{IEEEkeywords}
\IEEEpeerreviewmaketitle
\section{Introduction}\label{section1}
\IEEEPARstart{N}{owadays}, the number of wirelessly connected devices has been vastly increased by the development of applications such as Internet of Things (IoT), social networking and next generations of cellular communications including massive machine-type communications (mMTC), enhanced mobile broadband communications (eMBB) and ultra-reliable low-latency communications (URLLC).
There are lots of advantages with massive MIMO systems such as increasing the system throughput and the energy efficiency \cite{massive2020,marzetta2010noncooperative}. As such, massive multiple-input multiple-output (MIMO) systems have received remarkable attention during the past few years. However, all of these advantages are dependent on accurate channel state information (CSI) in coherent transmission which is a highly challenging task. {\color{\chang}Due to the reciprocity of channel estimation (CE), Time Division Duplexing (TDD) mode is often preferred to Frequecny Division Duplexing (FDD) in MIMO systems. However, it requires a portion of resources as pilots or training signal in each Coherence Interval (CI) to estimate the channels corresponding to active users. In this regard, Random Access (RA) to pilots (RAP) is a promising solution for this pilot allocation which divides into two categories: grant-based and grant-free. In grant-based schemes (see \cite{afshar2021spatial,bjornson2017random}), multiple active user equipment (UE)s first transmit dedicated preambles selected from a pool of pilot sequences to access the Base Station (BS). Contention resolution schemes are then required if multiple users select the same pilots. Since large number of collisions occurs, the BS cannot resolve all of the contentions and thus many users are not able to access the BS.
\cite{bjornson2017random} proposes a strongest user collision resolution protocol to resolve this issue in crowded scenarios (which neglects the users in the edge of the cells) by using orthogonal pilots. \cite{afshar2021spatial} employs a deterministic Compressed Sensing (CS)-based RA scheme. As the signaling overhead and access latency is directly proportional to the number of users, conventional grant-based RA fails to support massive connectivity. In contrast, in grant-free RAP protocol, each active UE transmits pilots with embedded data without waiting for permission by BS \cite{massive2020,liu2018massive}. By using orthogonal pilots in this case, the number of active users that can access the BS is confined to the number of orthogonal pilots which is severely limited due to the short channel coherence time. Moreover, using non-orthogonal pilots in grant-free case complicates the task of active user detection (AUD) due to the intra- and inter-cell interference caused by correlated pilots \cite{ke2020compressive,chen2021sparse}. A large number of research works consider coordinated grant-free schemes in which the BS knows the pilots in advance (e.g. \cite{ke2020compressive,covarianced-based2020,djelouat2021joint,chen2019covariance}). As an example of this category, \cite{ke2020compressive} proposes a statistical Approximate Message Passing (AMP)-based algorithm for joint AUD and CE. Their method exploits the sporadic traffic of active users and the sparse nature of the channel but needs the full knowledge of the channels and noise distributions and assumes the angle of arrivals (AoAs) to be on a predefined domain of grids. Both of these assumptions are not generally satisfied in practice. Another line of research works lying in the coordinated grant-free category devotes to covariance-based activity identification which formulates the problem as an maximum likelihood estimation (see e.g. \cite{covarianced-based2020,chen2019covariance,chen2021phase,haghighatshoar2018improved}). It also has been shown that covariance-based algorithms outperforms AMP with the same length of the pilots \cite{covarianced-based2020}. Such kinds of schemes are based on the accuracy of the sample covariance of the measurement matrix. In order to have an accurate covariance, the number of required measurements has to be far more than the degrees of freedom (the true number of unknowns) and imposes a huge waste in bandwidth resources and cost. Overall, there are specific disadvantages with the mentioned prior works: full knowledge of channel and noise distribution, e.g. \cite{ke2020compressive,2018sparse}; the number of active users that can access the BS is severely limited, e.g. \cite{bjornson2017massive,ke2020compressive,covarianced-based2020}; A very large number of antennas and measurements are required, e.g. \cite{covarianced-based2020,haghighatshoar2018improved,chen2021sparse}. However, there is a common issue that the mentioned prior works are coordinated. This means that the BS has to be first know some pilots in advance for AUD which increases access latency, subsequently preventing to achieve a high spectral efficiency and seems to be not practical since the BS does know which user lies in which cell. It should be mentioned that \cite{zhang2017blind} provides a statistical uncoordinated data recovery and channel estimation (not necessarily designed for RA), however, besides its high complexity and the issue regarding the discrete nature of AoAs, it needs full knowledge of prior distributions of data and channel in advance which seems to be impractical.
To solve the mentioned challenges, we propose an uncoordinated grant-free RAP deterministic scheme in crowded scenarios of massive MIMO systems which leverages the intrinsic features of massive MIMO channels (angular domain sparsity of massive MIMO channels) as well as the sporadic traffic of users. The task of extracting such features from a few number of measurements builds upon the well-known framework of CS \cite{candes2006robust}}. Precisely, CS suggests a framework for recovering discrete-index parameters i.e. the unknown parameters are confined to be on a predefined domain of grids. There is also a more recent framework called continuous CS (or super-resolution) \cite{candes2014towards,tang2013compressed,sayyari2020blind,bayat2020separating} which assumes that the unknown parameters can lie anywhere and are not confined to be on the predefined grids. The aforementioned features are available in massive MIMO systems with massive number of users. For example, the physical channel of massive MIMO systems, employed in high frequencies (millimeter wave), have continuous sparse structure i.e. signal is received out of few off-grid (continuous) angles in BS antenna arrays and they can have any arbitrary values \cite{bajwa2010compressed}. This is due to the fact that signals with higher frequencies are more likely to be blocked by obstructions and few multi-path components (MPCs) contribute to the channel. Another feature is sporadic traffic of massive users which means only few users want to send their data at the same time. It has also been shown that the channels between users and BS exhibit a clustered continuous sparsity pattern \cite{ke2020compressive}. This implies that the AoAs corresponding to each user are alongside each other, few number of clusters are active and the AoAs have continuous-index values. {\color{\chang}For AUD and channel estimation, we design a deterministic optimization framework to encourage the mentioned features which does not need any distributions of channels and users' data. Our framework has three stages: first, we find the AoAs by solving an optimization problem, then a clustering-based algorithm is proposed to detach the angles corresponding to each active user and finally an alternative optimization algorithm is developed to estimate complex amplitudes of the channels and data/primary pilots transmitted by active users in a blind way. By our approach, the limitation in the number of adopted simultaneous users in crowded scenarios of massive MIMO systems would be resolved without any need for coordination between BS and users in advance.}
The organization of the paper is as follows: In Section \ref{sec.model}, the system model of massive MIMO is presented. Section \ref{sec.proposed} is about our proposed blind super resolution method and provides an algorithm for blind detection and CE. Lastly, Section \ref{sec.simulations} provides some numerical experiments to verify our proposed method. Lastly, the paper is concluded in Section \ref{sec.conclusion}.
\textit{Notations}:
We use boldface lower-and upper-case letters for vectors and matrices, respectively. The $i$-th element of a vector e.g. $\bm{x}$ and the $(i,j)$ element of a matrix e.g. $\bm{X}$ are respectively shown by $x_i$ and $X_{(i,j)}$. For vector $\bm{x}\in\mathbb{C}^n$ and matrix $\bm{X}\in\mathbb{C}^{n_1\times n_2}$, the $\ell_2$ norm and Frobenius norm are defined respectively as $\|\bm{x}\|_2:=({\sum_{i=1}^n|x(i)|^2})^{\tfrac{1}{2}}$ $\|\bm{X}\|_{F}:=\sqrt{\sum_{i=1}^{n_1}\sum_{j=1}^{n_2}|X(i,j)|^2}$. $\bm{X}\succeq \bm{0}$ means that $\bm{X}$ is a positive semidefinite matrix. For two arbitrary matrices $\bm{A}, \bm{B}$, $\langle \bm{A}, \bm{B}\rangle_R$ represents the trace of $\bm{B}^H\bm{A}$. $\mathcal{P}_{\Omega}(\cdot)$ is a operator transforming an arbitrary matrix to a reduced matrix with rows indexed by $\Omega$.
\section{System Model}\label{sec.model}
We consider a typical uplink access scenario for MIMO-OFDM systems where there are one BS equipped with an $N$-element uniform linear array (ULA) and $K$ single-antenna users along with OFDM modulation to combat inter-symbol interference \cite[Section II]{ke2020compressive}. The sub-channel corresponding to each OFDM sub-carrier between the $k$-th user and the BS is modeled as (see \cite[Equ. 2]{ke2020compressive} or \cite[Equ. 7]{zhang2017blind}):
\begin{align}
\bm{h}_k=\sum_{l=1}^{L_k}\alpha_l^k\bm{a}(\theta_l^k)=\bm{A}_k\bm{\alpha}^k\in\mathbb{C}^{N\times 1},
\end{align}
in which $L_k$ is the number of physical paths between $k$-th user and BS, $\theta_l^k$ is the Angle of Arrival (AoA) of the $l$-th path, $\alpha_{l}^k$ is the complex gain of the $l$-th path,
\begin{align}
\bm{a}_r(\theta)=\tfrac{1}{\sqrt{N}}[1, {\rm e}^{-j2\pi \Delta_r\cos(\theta)},...,{\rm e}^{-j2\pi \Delta_r (N-1)\cos(\theta)} ]^T
\end{align}
is the receive steering vector, $\bm{\alpha}^k:=[\alpha_1^k,..., \alpha_{L_k}^k]^T$, and $\bm{A}_k:=[\bm{a}_r(\theta_1^k),..., \bm{a}_r(\theta_{L_k}^k)]\in\mathbb{C}^{N\times L_k}$. Due the sparse characteristics of massive MIMO channels, it holds that $L_k\ll N$. The sub-channel is considered to be block fading, i.e., it is constant during several CIs where each is denoted by $T$. In each sub-carrier, the received signal at the BS after $T$ time slots at the sensors indexed by $\Omega \subseteq \{1,..., N\}$ (with length $|\Omega|:=M<N$) becomes in the form of (\cite{zhou2007experimental}, \cite[Equ. 3]{ke2020compressive} or \cite[Equ. 1]{zhang2017blind}):
\begin{align}\label{eq.observed}
\bm{Y}_{\Omega}=\mathcal{P}_{\Omega}(\bm{Y})=\mathcal{P}_{\Omega}(\sum_{k=1}^K\underbrace{\bm{h}_{k}\bm{s}_k^H)}_{:=\bm{X}_k}+\bm{W}\in \mathbb{C}^{M\times T},
\end{align}
where $\bm{s}_k\in\mathbb{C}^T$ is the transmitted signal from $k$-th UE, $\bm{W}\in\mathbb{C}^{M\times T}$ is the additive noise matrix, each element of which is distributed as $\mathcal{CN}(0,\sigma^2)$ and $\bm{X}_k:=\sum_{l=1}^{L_k}\alpha_l^k\bm{a}(\theta_l^k)\bm{s}_k^{H}$. Inspired by \cite{chandrasekaran2012convex}, and by defining $c_l^k:=\alpha_l^k\|\bm{s}_k\|_2$ and $\bm{\phi}_k:=\tfrac{\bm{s}_k}{\|\bm{s}_k\|_2}$, $\bm{X}_k$ can be expressed as a sparse linear combinations of the matrix atoms in the atomic set ${ \mathcal{A}_k=\{\bm{a}_r(\theta)\bm{\phi}_k^H: \|{\bm{\phi}}_k\|_2=1, \theta\in (0,\pi)\},}$
which are regarded as building blocks of $\bm{X}_k=\sum_{l=1}^{L_k}c_l^k\bm{a}(\theta_l^k)\bm{\phi}_k^{H}$. {\color{\chang}The aim is to extract the continuous parameters of $\bm{X}_k$ (i.e. the angles $\bm{\theta}_k$) by observing $\bm{Y}_{\Omega}$. Note that for inactive users, $\bm{\phi}_k$ and thus $\bm{X}_k$ are equal to zero.}
\section{Proposed Blind Super-resolution method}\label{sec.proposed}
In \eqref{eq.observed}, we have an under-determined set of equations with $N T$ observations and $K N T$ unknowns. While this problem has infinite number of solutions, it could be transformed to a tractable problem by assuming that $L_k \ll N$ which is reasonable in massive MIMO systems. This strategy is built upon well-known continuous CS approaches \cite{candes2014towards,tang2013compressed,valiulahi2019two} and provides a unique optimal set of solutions for matrices $\bm{X}_k$s in \eqref{eq.observed} leading to the least number of atoms under the affine constraints of \eqref{eq.observed}. Thus, we form the following optimization problem to reflect the structure of $\bm{X}_k$s:
\begin{align}\label{prob.atomic_l0}
&\min_{\substack{\bm{Z}_k\in\mathbb{C}^{N\times T}\\ k=1,..., K}} \sum_{k=1}^K \|{\bm{Z}_k}\|_{\mathcal{A}_k,0} ~s.t. \|\bm{Y}_{\Omega}-\sum_{k=1}^K\mathcal{P}_{\Omega}(\bm{Z}_k)\|_F\le \eta
\end{align}
where $\|\bm{Z}_k\|_{\mathcal{A},0}:=\inf\big\{L_k: \bm{Z}_k=\sum_{l=1}^{L_k}c_{l}^k\bm{a}({\theta}_l^k)\bm{\phi}_k^H, c_l^k>0 , \bm{a}(\theta_l^k)\bm{\phi}_k^H\in\mathcal{A}_k\big\}
$ is the atomic $\ell_0$ function which computes the least number of atoms to describe $\bm{Z}_k$. As \eqref{prob.atomic_l0} is an NP-hard problem in general, we relax {\color{\chang} \eqref{prob.atomic_l0} into its closest convex optimization problem which is stated as}:
\begin{align}\label{prob.atomic_l1}
&\min_{\substack{\bm{Z}_k\in\mathbb{C}^{N\times T} \\k=1,..., K}} \sum_{k=1}^K \|{\bm{Z}_k}\|_{\mathcal{A}_k} ~s.t.~\|\bm{Y}_{\Omega}-\sum_{k=1}^K\mathcal{P}_{\Omega}(\bm{Z}_k)\|_F\le \eta,
\end{align}
where {\color{\chang} the atomic norm $\|\cdot\|_{\mathcal{A}_k}$ is the best convex surrogate for the number of atoms composing $\bm{Z}_k$ (i.e. $\|\cdot\|_{\mathcal{A}_k,0}$) and is defined as the minimum of the $\ell_1$ norm of the coefficients forming $\bm{Z}_k$:}
\begin{align}
&\|\bm{Z}_k\|_{\mathcal{A}}:=\inf\{t>0: \bm{Z}_k\in t{\rm conv}(\mathcal{A}_k)\}=\nonumber\\
&\inf\{\sum_{l=1}^{L_k}c_l^k: \bm{Z}_k=\sum_{l=1}^{L_k}c_{l}^k\bm{a}({\theta}_l^k)\bm{\phi}_k^H, c_l^k>0 , \bm{a}(\theta_l^k)\bm{\phi}_k^H\in\mathcal{A}_k\}
\end{align}
where ${\rm conv(\mathcal{A})}$ is the convex hull of $\mathcal{A}$.
To identify the AoAs (which we used in our simulations) is by leveraging the solution of the dual problem of \eqref{prob.atomic_l1} which is provided below:
\begin{align}\label{prob.dual}
&\min_{\bm{V}\in\mathbb{C}^{N\times T},\bm{Z}\in\mathbb{C}^{T\times T}}2{\rm Re}\langle \bm{V}_{\Omega}, \bm{Y}_{\Omega}\rangle_F+2\eta\|\bm{V}_{\Omega}\|_F~~s.t. \nonumber\\
&
\begin{bmatrix}
\bm{I}&\bm{V}^H\\
\bm{V}&\bm{Z}_i
\end{bmatrix}\succeq \bm{0},\mathcal{T}^{*}(\bm{Z}_i)=\mathcal{T}^{*}(\bm{I}), i=1,..., K, ~\mathcal{P}_{\Omega}(\bm{V})=\bm{0}
\end{align}
where $(\mathcal{T}^{*}(\bm{Z}))_k=\sum_{i=\max(1,k+1)}^{\min(k+T,T)}\bm{Z}_{i,i-k}, k=-(T-1),..., (T-1)$
is the adjoint operator of $\mathcal{T}$. Then, we use the following lemma (adapted from \cite[Lemma 1]{bayat2020separating} and \cite[Theorem 1]{sayyari2020blind}) which guarantees the uniqueness of the solution in the noiseless case:
\begin{lem}\label{lem.uniuqeness}
Denote the set of AoAs from $i$-th users by $\mathcal{S}_a^i=\{\theta_l^i\}_{l=1}^{L_i}$.
The solutions of $\bm{Z}_k$ obtained from \eqref{prob.atomic_l1} in the noiseless case ($\eta=0$) are unique if there exist dual matrices $\bm{V}\in\mathbb{C}^{N\times T}$ such that the
vector-valued dual polynomials $\bm{q}_i(\theta)=\bm{V}^H\bm{a}_r(\theta)$
satisfy the conditions
\begin{align*}
&\bm{q}_i(\theta)=\bm{\phi}_i, \forall \theta \in \mathcal{S}^i_a ,
~\|\bm{q}_i(\theta)\|_2<1 \forall \theta \in [0,\pi)\setminus\mathcal{S}^i_a, i=1,..., K.
\end{align*}
\end{lem}
This lemma shows that the AoAs can be easily estimated by identifying locations where $\|\bm{q}_i(\theta)\|_2$ achieves $1$. As stated in \cite{candes2013super}, this provides a good insight about the procedure of finding AoAs in the noisy case. Specifically, we find the AoAs by identifying the ones that $\|\bm{q}_i\|_2=1, i=1,..., K$.
After obtaining the estimated AOAs, a clustering-based algorithm \cite{kmeans} is employed to detect the AOAs of {\color{\chang} each cluster (active user)} denoted by $\bm{\theta}^{k}=[\theta_1^k,..., \theta^k_{\widehat{L}_k}]^T, k\in \widehat{\mathcal{S}}_u$. Here, {\color{\chang}$\widehat{\mathcal{S}}_u$ is the estimated set of indices corresponding to clusters (estimated active users) with known length $|\widehat{\mathcal{S}}_u|=K_a$}.
By knowing the AOAs corresponding to {\color{\chang}each cluster (estimated active user)}, \eqref{eq.observed} turns into the following equation:
\begin{align}\label{eq.simpled_obs}
\bm{Y}_{\Omega}=\sum_{k\in\widehat{\mathcal{S}}_u}\bm{A}^k_{\Omega}\bm{c}^k\bm{\phi}_k^H+\bm{W}_{M\times T},
\end{align}
where $\bm{A}^k_{\Omega}=\mathcal{P}_{\Omega}([\bm{a}_r(\theta^{k}_1),..., \bm{a}({\theta}^k_{\widehat{L}_k})])\in\mathbb{C}^{M\times K_a}$ is the steering matrix of $k$-th active user and $\bm{c}^k=[c_1^k,..., c^k_{\widehat{L}_k}]^T$. The task of recovering the unknown matrices $\bm{c}^k$ and $\bm{\phi}_k$ from $\bm{Y}_{\Omega}$ is a bi-linear inverse problem. For this task, we propose an alternating optimization to jointly estimate complex channel coefficients and transmitted data corresponding to active users. First, we begin with a random $\widehat{\bm{\phi}}_k$ distributed on the unit sphere. By replacing $\widehat{\bm{\phi}}_k$ in \eqref{eq.simpled_obs}, we deal with the following least square problem:
\begin{align}\label{eq.c_estimate}
[\widehat{\bm{c}}^1,..., \widehat{\bm{c}}^{K_a}]=\mathop{\arg\min}_{\bm{c}^k,k=1,..., K_a}\|\bm{Y}_{\Omega}-\sum_{k\in\mathcal{S}_u}\bm{A}^k_{\Omega}\bm{c}^k\widehat{\bm{\phi}}_k^H\|_F
\end{align}
which can be easily solved by numerical optimization.
By integrating the latter expression into \eqref{eq.simpled_obs}, we must solve the following least square optimization: ${ \widehat{\bm{\Phi}}=\mathop{\arg\min}_{\bm{\Phi}_{K_a\times T}}\|\bm{Y}_{\Omega}-\bm{B}\bm{\Phi}\|_F,}$
where $\bm{B}:=[\bm{A}_{\Omega}^1\widehat{\bm{c}}^1,...,\bm{A}_{\Omega}^{K_a}\widehat{\bm{c}}^{K_a} ]\in\mathbb{C}^{M\times K_a}$
and $\bm{\Phi}:=[\bm{\phi}_1,..., \bm{\phi}_{K_a}]^T$. The latter optimization has also the closed-form solution
\begin{align}\label{eq.phi_estimate}
[\widehat{\bm{\phi}}_{1},..., \widehat{\bm{\phi}}_{K_a}]^T=\widehat{\bm{\Phi}}=\bm{B}^{\dagger}\bm{{Y}}_{\Omega}.
\end{align}
Finally, the steps \eqref{eq.c_estimate} and \eqref{eq.phi_estimate} are alternatively performed to yield the final solution. The pseudo code of the proposed method which is indeed a summary of the aforementioned steps is provided in Algorithm \ref{algorithm.admm}.
{\centering
\resizebox{.5\textwidth}{!}{
\begin{minipage}{.8\textwidth}
\begin{algorithm}[H]
\caption{}
\begin{algorithmic}[1]\label{algorithm.admm}
\REQUIRE $\bm{Y}\in\mathbb{C}^{M\times T}$,$\eta$, $K_a$, maxiter
\STATE Select a uniformly distributed random vector for transmitted data as $\widehat{\bm{\phi}}_k=5 rand(T,1) \forall~ k=1 ~\text{to}~ K_a$
\STATE $\bm{A}_{total}=\emptyset$
\begin{itemize}
\item Solve the dual problem \eqref{prob.dual} to obtain $\bm{V}$ as follows:
\item Obtain the dual polynomial $\bm{q}_i(\theta)=\bm{V}^H\bm{a}_r(\theta), i=1,..., K$.
\item Localize the estimated angle $\widehat{\theta}\in [0,1]$ by the following two methods:
\item Discretize $\widehat{\theta}$ on a fine grid up to a desired accuracy and find $\widehat{\theta}$ and by identifying locations where $\|\bm{q}_i(\theta)\|_2, i=1,..., K$ achieves to $1$ according to Lemma \ref{lem.uniuqeness}. The total number of angles reaching $1$ specifies an estimate for the total number of MPCs i.e. $\sum_{k=1}^{K_a}L_k$
\end{itemize}
\item
\begin{itemize}
\end{itemize}
\STATE Apply the k-means methods to cluster the angles of channel UEs.
\STATE $[{\rm label}]={\rm k-means}(\widehat{\theta},K_a)$
\FOR{$k=1$ to $K_a$}
\STATE Identify the corresponding indices with the $k$-th label.
\STATE Estimate the length of $k$-th cluster i.e. $L_k$.
\STATE Obtain the angles corresponding to the $k$-th cluster (UE) i.e. $\widehat{\theta}_1^k,..., \widehat{\theta}_{L_k}^k$
\STATE Estimate the steering matrix as
\STATE $\bm{A}_{total}\leftarrow [\bm{A}_{total}, \bm{A}_k]$
\ENDFOR
\FOR {$i=1$ to maxiter}
\STATE Recover $[\widehat{\bm{c}}^1,..., \widehat{\bm{c}}^{K_a}]$ according to \eqref{eq.c_estimate}.
\STATE Recover $[\widehat{\bm{\phi}}_1,..., \widehat{\bm{\phi}}_{K_a}]$ according to \eqref{eq.phi_estimate}.
\STATE $\widehat{\bm{\phi}}_k\leftarrow \tfrac{\widehat{\bm{\phi}}_k}{\|\widehat{\bm{\phi}}_k\|_2}$.
\ENDFOR
\end{algorithmic}
Return: $ \widehat{\bm{\theta}}^k,\widehat{ \bm{\alpha}}^k, \widehat{ \bm{s}}_k, ~ \forall k \in\mathcal{S}_u$.
\end{algorithm}
\end{minipage}
}
}
\textit{Discussion}:
Our algorithm provides a blind spatial-based RA scheme which simultaneously estimates data as well as channels. The only assumption on the data to be unambiguously recovered is positivity i.e. $\bm{s}_i>0$ and normalized power $\|\bm{s}_i\|_2=1$ for all active users $i=1,..., K_a$. {\color{\chang}The proposed method has implications for data recovery in mMTC as well as RAP and AOA detection in crowded mobile broadband communications (cMBB).} For example, CI in cMBB divides into 2 parts: RAP and data transmission blocks. By utilizing this novel approach in the RAP block, AUD and AoA estimation are performed by the BS \cite[Section 5]{afshar2021spatial}. By our method, BS can identify many users at the same time with their AoAs and with the lowest level of spending system resources. In fact, the number of active users that can access the network depends on the complexity that BS can bear. There is no need for orthogonality of RA pilots needed for RAP process in \cite{bjornson2017random} and prior distribution of pilots and channels as is the case in AMP-based approaches \cite{zhang2017blind,ke2020compressive}. {\color{\change}After RAP, in the coherent transmission step, BS can easily allocate dedicated orthogonal pilots to non-overlapped UEs and estimates their corresponding data via \eqref{eq.phi_estimate} by knowing the exact AoAs of active users in the RAP stage.}
\section{Simulations}\label{sec.simulations}
In this section, we perform some numerical experiments to evaluate the performance of our proposed algorithm in blind channel and data reconstruction. We use SDPT3 package of CVX \cite{cvx} in MATLAB for solving problem \eqref{prob.dual}. The number of BS antennas is set to $N=64$. We assume the one-ring model for the channel \cite{nam2014joint}. The AoAs are randomly chosen from $[0,\pi]$. The path amplitudes are distributed as $\mathcal{CN}({0},1)$. The separation between any receive antennas at BS are set to $\Delta_r=0.5$. The observed sensors at BS ($\Omega$) are randomly chosen out of $\{1,..., N\}$. Also, the maximum number of iterations in Algorithm \ref{algorithm.admm} denoted by maxiter is fixed to $5$. The upper bound of noise variance is chosen as $\eta=\|\bm{W}\|_F$. The signal to noise ratio is defined by ${\rm SNR}=10\log_{10}(\frac{\|\mathcal{P}_{\Omega}(\bm{X})\|_F^2}{MT\sigma^2})$. First, in the top-left image of Figure \ref{fig.nmse}, we show the successful procedure of AUE and clustering with parameters $M=30$, $K=10$, $K_a=3$, $T=2$, ${\rm SNR}=10~dB$. The maximum number of MPCs is fixed to $L_{\max}=3$. This image shows the $\ell_2$ norm of the vector-valued dual polynomial at different angles in terms of radian. The estimated angles are found by identifying locations that $\|\bm{q}(\theta)_i\|_2=1, i=1, ..., K$. The number of peaks provides an estimate for $\sum_{k\in\mathcal{S}_u}L_k$. After finding the angles, we apply k-means method to cluster the angles corresponding to $K_a$ active users. The number of elements inside each cluster provides an estimate for $\widehat{ L}_k$. Then, steps 12 to 16 of Algorithm \ref{algorithm.admm} are employed to obtain the pilots and channels. The performance of our algorithm in recovering users' data, channel amplitudes and AoAs is evaluated using normalized mean square error (NMSE) respectively defined by ${\rm NMSE}_{\bm{\phi}}:=\mathds{E}\sqrt{\frac{\sum_{k=1}^{K_a}\|\bm{\phi}_k-\widehat{\bm{\phi}}_k\|_2^2}{\sum_{k=1}^{K_a}\|\bm{\phi}_k\|_2^2}}$, ${\rm NMSE}_{\bm{\alpha}}:=\mathds{E}\sqrt{\frac{\sum_{k=1}^{K_a}\|\bm{\alpha}_k-\widehat{\bm{\alpha}}_k\|_2^2}{\sum_{k=1}^{K_a}\|\bm{\alpha}_k\|_2^2}}$ and ${\rm NMSE}_{\bm{\theta}}:=\mathds{E}\sqrt{\frac{\sum_{k=1}^{K_a}\|\bm{\theta}^k-\widehat{\bm{\theta}}^k\|_2^2}{\sum_{k=1}^{K_a}\|\bm{\theta}^k\|_2^2}}$. The Monte-Carlo iterations to approximate the expectation is set to $50$. The evaluation for the first experiment are as follows: ${\rm NMSE}_{\bm{\Phi}}=10^{-6}, {\rm NMSE}_{\bm{\alpha}}=10^{-5}, {\rm NMSE}_{\theta}=10^{-8}$. In the second experiment, we evaluate the performance of Algorithm \ref{algorithm.admm} in different noise values in a more practical scenario with parameters $N=64, L_{\max}=3, K_a=12, K=40, T=10$. As it turns out from the bottom image of bottom-right image of Figure \ref{fig.nmse}, NMSEs tends to zero at high SNRs which in turn implies that our proposed method performs well in estimating users' data, complex channel amplitudes and AoAs of active users. In the experiment shown in the top-right image of Figure \ref{fig.nmse}, we compare our method with \cite{afshar2021spatial} for different number of antennas. For both methods, we obtain NMSE of the channel matrix defined by ${\rm NMSE}_{\bm{h}}:=\mathds{E}\sqrt{\frac{\sum_{k=1}^{K_a}\|\bm{h}_k-\widehat{\bm{h}}_k\|_2^2}{\sum_{k=1}^{K_a}\|\bm{h}_k\|_2^2}}$. As it can be observed, our blind method performs better in CE than \cite{afshar2021spatial} which assumes the users' data known. In the last experiment, {\color{\changg}the performance of AUE in our algorithm is compared with \cite[Algorithm 1]{haghighatshoar2018improved}} by a detection rate criterion defined as $DR=\frac{|\mathcal{S}_u-\widehat{\mathcal{S}_u}|}{K_a}$ where the numerator returns the number of differences between the true active users and the estimates. As shown in the bottom-left image of Figure \ref{fig.nmse}, the probability of detection enhances by increasing the {\color{\changg} number $N$ of BS antennas}.
\begin{figure}[t]
\hspace{-.6cm}
\includegraphics[scale=0.28]{dual+error+cluster.pdf}
\caption{Top-left image: This image depicts $\ell_2$ norm of the dual polynomial vector. One can find the angles of active users by identifying angles with maximum amplitude. The angles of active user channels are clustered using kmeans method. The used parameters are $N=64, M=30, K_a=3, K=10, T=2, SNR=10~dB.$ Top-right image: This image compares the performance of our algorithm in CE with \cite{afshar2021spatial} for different number of BS antennas with settings $L_{\max}=4, T=10, K_a=3, K=10, {\rm SNR}=3dB $. Bottom left image: {\color{\changg}The performance of AUE is compared with \cite{haghighatshoar2018improved} versus the number of observed arrays with parameters $N=60, M=60, K_a=5, K=50, L_{\max}=3, {\rm SNR}=5~dB$}. Bottom right image: This image shows the performance of our algorithm in estimating angles, pilots and complex amplitudes with parameters $N=64, L_{\max}=3, K_a=12, K=40, T=10$.}\label{fig.nmse}
\end{figure}
\section{Conclusion}\label{sec.conclusion}
In this work, we designed a novel blind spatial-based random access solution which is applicable to crowded massive MIMO systems. Specifically, we showed that the recovery of both pilots and AoAs are possible via observing a few noisy measurements in blind manner. For this task, we used a clustering method to demix the AoAs corresponding to active user and an alternating-based approach is designed to recover the pilots and the complex amplitudes of the channels.
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\begin{document}
\maketitle
\setlength{\parskip}{1ex plus 0.5ex minus 0.2ex}
\begin{abstract}
The semigroup game is a two-person zero-sum game defined on a semigroup $(S,\cdot)$ as follows: Players 1 and 2 choose elements $x\in S$ and $y\in S$, respectively, and player 1 receives a payoff $f(x y)$ defined by a function $f: S \rightarrow [-1,1]$. If the semigroup is amenable in the sense of Day and von Neumann, one can extend the set of classical strategies, namely countably additive probability measures on $S$, to include some finitely additive measures in a natural way. This extended game has a value and the players have optimal strategies. This theorem extends previous results for the multiplication game on a compact group or on the positive integers with a specific payoff. We also prove that the procedure of extending the set of allowed strategies preserves classical solutions: if a semigroup game has a classical solution, this solution solves also the extended game.
\end{abstract}
{\em Keywords:} amenability, multiplicative game, loaded game,
optimal strategies, minimax strategy, Nash equilibrium,
intrinsically measurable \pagebreak \tableofcontents
\section{Introduction}
The \emph{multiplication game} is a two-person zero-sum game in
which the players independently choose positive numbers and multiply
them together. The first player wins when the first digit of the
product is 1, 2, or 3, and the second player wins otherwise.
Different versions of the game arise according to which numbers the
players are allowed to choose. In the original game invented by
Ravikumar \cite{Ra} the players choose from the set of $n$-digit
integers with a fixed $n$. He analyzed the optimal strategies in the
limit as $n \rightarrow \infty$. In \cite{Mo} the second author
showed that when the numbers are positive real numbers, an optimal
strategy for both players is to choose numbers from the Benford
distribution, which is the limit that Ravikumar found in his
analysis.
Also in \cite{Mo} it was shown that the procedure applies naturally
to compact groups. For the \emph{group game} let $G$ be a compact
group and let $W$ be a subset of $G$ that is measurable with
respect to the Haar measure.\footnote{Recall that the Haar measure is the
unique invariant probability measure on a compact group.} The
players choose elements of $G$ and the payoff function of player 1
is defined by
$$
f(x,y)=\left\{
\begin{array}{ll}
1, & \hbox{if $xy\in W$} \\
0, & \hbox{if $xy\notin W$}
\end{array}
\right.
$$
The pure strategies are the elements of $G$, which are identified
with the point masses, and mixed strategies are Borel probability
measures on $G$. It turns out that that the Haar measure $\lambda$ is an
optimal strategy for both players, and the value of the game, i.e
the probability that player 1 wins when both players play their own
optimal strategy, is $\lambda(W)$.
Still unsolved are the very natural games using the non-compact group
$(\mathbb Z,+)$ and the non-compact semigroup $(\mathbb N,\cdot)$.
The aim of this article is to extend the result on compact groups to
a larger class of algebraic objects that contains both $(\mathbb
Z,+)$ and $(\mathbb N,\cdot)$. Since neither the existence of
inverses nor an identity element is necessary for playing the
multiplication game, it appears that semigroups are the appropriate
setting for this generalization, which we call the \emph{semigroup
game}. In order to prove the existence of optimal strategies and the
existence of a value for the semigroup game, we restrict our attention to
the class of \emph{amenable} semigroups, but we are forced to
enlarge the set of mixed strategies to include finitely additive
probability measures on $S$. Since no countable group has an invariant countably additive probability measure\footnote{Indeed, the group structure and invariance imply that all singletons have the same measure. }, the result in
\cite{Mo} does not apply to them, but there are countable
groups---for example, abelian groups---for which there are optimal
finitely additive strategies.
Then with the proper interpretation of what a mixed strategy is, we
are able to prove that the game on amenable semigroups has
optimal strategies. This includes many interesting non-compact
groups as well, groups such as the additive group of the integers.
It is worth noting that the second author showed that for the
game on the semigroup $(\mathbb N,\cdot)$ of positive
integers with the multiplication, and with $W=\{\text{integers with
first digit 1 through 3}\}$, there are mixed strategies that are
nearly optimal. Doing this made use of the special structure of $W$
and approximating the game with one on a compact group. This
approach is unfortunately inapplicable to general winning sets.
The first author is grateful to Silvia Ghinassi, Daniel Litt, Vern Paulsen
and Florin R\u adulescu for helpful discussions and to Alain Valette for reading and commenting on an early draft. The authors would like to thank Ted Hill for comments and suggestions. Finally, special thanks go to Marco Dall'Aglio for bringing about the collaboration of the coauthors.
\section{Optimal mixed strategies for the semigroup game}
Although the semigroup game can be defined as long as there is a
binary operation on the set of pure strategies, it seems that
associativity is necessary for our results (see also Remark
\ref{associative}). Recall that a \emph{semigroup} $S$ is a set
equipped with an associative binary operation $S\times S\rightarrow
S$. Given $x,y\in S$, the result of the operation is denoted by
$xy$.
Let $S$ be a semigroup and $f:S\rightarrow [-1,1]$ a function. The semigroup
game $\mathcal G(S,f)$ associated to $S$ and $f$
is the
two-person zero-sum game with $S$ the set of pure strategies for
both players. The payoff function of player 1, which is the negative of the payoff function of player 2, is $f(xy)$. The set of mixed strategies will be specified later.
An interesting case is already when $f$ is the characteristic function of a subset $W$ of $S$. In this case the set $W$ is called \emph{the winning set}.
We first consider the case of a countable semigroup $S$. (There is a technical
reason to separate the countable case from the
uncountable case.) Let $L^\infty(S)$ denote the Banach space of
all bounded functions from $S$ to $\mathbb R$, equipped with the supremum norm, and recall
that a \emph{mean} on $S$ is a linear functional
$m:L^\infty(S)\rightarrow\mathbb R$ which is positive, in the sense
that $f\geq0$ implies $m(f)\geq0$, and such that $m(1)=1$. A mean is necessarily bounded and has norm 1. A mean
induces a finitely additive measure on $S$ by defining the measure
of $A\subseteq S$ as $m(\chi_A)$, where $\chi_A$ is the
characteristic function of $A$; we also write
$m(f)=\int_Sf(x)dm(x)$.
Classically, the mixed strategies are probability measures
(countably additive) on the set of pure strategies, with the pure
strategies identified with the point masses $\delta_s$, but for the
semigroup game we expand the mixed strategies to include means,
i.e., finitely additive measures, on $S$. Interest in finitely additive measures has increased in recent years as it has been realized that
\emph{besides technical convenience, there are no
conceptual reasons that support the use of that stronger
assumption} \cite{Ma}. But even early in the development of the rigorous theory of probability it was noted by Kolmogorov \cite[p. 15]{Ko} that \emph{``...in describing any observable random process we can obtain only finite fields of probability. Infinite fields of probability occur only as idealized models of real random processes.''}
It is important to recall that by allowing finitely additive
measures as strategies, the results can be quite
different, as shown by the following classical example.
Consider Wald's game \emph{pick the bigger integer}
: the set of pure strategies is the set of
non-negative integers and the payoff function of player 1 (which is
the negative of the payoff function of player 2) is
$$
f(s,t)=\left\{
\begin{array}{rl}
1, & \hbox{if $s>t$} \\
0, & \hbox{if $s=t$} \\
-1, & \hbox{if $s<t$}
\end{array}
\right.
$$
Wald\cite{Wa} observed that this game has no value if just countably
additive strategies are allowed. In \cite{He-Su} it is shown
that this game has a value if one allows finitely additive
probability measures as strategies, but the value depends on the
order of integration in such a way that the \emph{internal} player
has an advantage\footnote{See \cite{Sc-Se} for more general results
and relation with other phenomena, as de Finetti's
non-conglomerability.}. This seems very strange, since a game
that is naturally symmetric---as presented---becomes asymmetric.
An explanation of this fact will be given in a followup to this
paper \cite{Ca}, where the author shows that Wald's game is equivalent to a semigroup game which is \emph{loadable} (to be defined in the last section) in infinitely many different ways.
For now let us just say that our interpretation of this fact is that
countably additive strategies are \emph{very few} and, on the other
hand, finitely additive strategies are \emph{too many}. This
suggests that the right formulation of the problem should be
somewhere in the middle; i.e., there should be some restrictions on
the set of allowed strategies which lead to a solution of the
problem. So we are now going to propose a natural way to make these
restrictions.
Consider the definition of the payoff to player 1 when he uses the
mixed strategy $p$ and player 2 uses $q$. Assuming that $p$ and $q$
are countably additive probability measures, the payoff $\pi(p,q)$
is the integral of $f(xy)$ with respect to the product
measure $p \times q$ on $S \times S$. By Fubini's Theorem
\[ \pi(p,q)=\int_y\int_xf(xy)\,dp(x)dq(y)=\int_x\int_yf(xy)\,dq(y)dp(x) .\]
But if $p$ and $q$ are only finitely additive, then \emph{the} product measure is not uniquely defined, the
generalization of Fubini's Theorem is not true and the order of
integration matters. If one of the orders of integration is used for
the payoff definition, then there are symmetric games that lose
their symmetry. For example, if the players of Wald's game use
finitely additive mixed strategies, then defining the payoff to be
$\int_y\int_xf(xy)\,dp(x)dq(y)$ gives player 1 the advantage,
whereas changing the order of integration favors player 2
\cite{Sc-Se}.
\begin{defin}
Let $S$ be a countable semigroup. A mean $m$ is \textbf{left-invariant} if
\[ m(f \circ L_s)=m(f) \quad \forall f\in L^\infty(S),s\in S \]
and \textbf{right-invariant} if
\[ m(f \circ R_s)=m(f) \quad \forall f\in L^\infty(S),s\in S \]
where $R_s(x)=xs$ is the right action of $S$ on itself
and $L_s(x)=sx$ is the left action. A mean is \textbf{invariant} if it is both left-invariant and right-invariant.
\end{defin}
Groups or semigroups with invariant means are called
\emph{amenable} and they form an important class of algebraic
objects. The concept of an amenable group was first introduced by J.
von Neumann\cite{vN} and later it was generalized to semigroups
\cite{Da}. An example of an amenable semigroup is the multiplicative
semigroup of natural numbers \cite{Ar-Wi}. Furthermore, not every
group or semigroup is amenable, with the best known example of a
non-amenable group being the free group on two generators\footnote{The free group on two generators, say $x$ and $y$, is the group of all words in the letters $x,x^{-1},y,y^{-1}$, equipped with the operation of concatenation of words, where only the simplifications $xx^{-1}=x^{-1}x=yy^{-1}=y^{-1}y=e$ are allowed, $e$ being the empty word. It was observed by von Neumann that this group, denoted by $\mathbb{F}_2$, is not amenable. A celebrated example of Ol'shanskii shows the existence of non-amenable groups which do not contain $\mathbb{F}_2$ (see \cite{Ol}).}. Likewise,
the free semigroup on two generators is non-amenable. Every finite
group is amenable and the unique invariant mean is given by the
normalized counting measure, but on countably infinite groups there
are no invariant countably additive measures, and so an invariant
mean can only be finitely additive.
From our point of view it is important in the semigroup game to have
the notion of choosing an element ``uniformly'' from $S$. In the
case of a finite group that means the uniform probability measure (which is the unique
invariant mean on a finite group).
In the general setting of semigroups, it is
reasonable to consider an invariant mean as the generalization of
uniform choice. For instance, if we consider the integers with
addition it is intuitively appealing that the probability of
choosing an even number should be the same as the probability of
choosing an odd number and that this probability should be $1/2$.
Indeed any invariant mean on $\mathbb{Z}$ does assign probability
$1/2$ to the even integers and probability 1/2 to the odd
integers.\footnote{The even integers are a 2-\emph{tile}. Given a
semigroup $S$ and a positive integer $k$, possibly infinite. A
subset $W\subseteq S$ is called $k$-tile if there exist
$s_1,...s_k\in S$ such that $S=\bigcup s_iW$ and $s_iW\cap
S_jW=\emptyset$ for $i \neq j$. It is clear that any invariant mean
takes value $\frac{1}{k}$ on a $k$-tile.}
Therefore, in setting up the semigroup game on an amenable semigroup $S$ we fix a particular invariant mean.
\begin{defin}
A \textbf{loading} on $(S,\cdot)$ is given by a finitely additive
probability measure on $S$ which is invariant with respect
to $\cdot$. A loading is denoted by $\ell$.
\end{defin}
We construct the set of allowed strategies following two natural
requirements:
\begin{enumerate}
\item The symmetry and simultaneity of the game suggest that the two players have to be interchangeable, which means that allowed strategies
$p$ and $q$ have to commute in the following sense
$$
\int_y\int_xf(xy)dp(x)dq(y)=\int_x\int_yf(xy)dq(y)dp(x)
$$
This number will be now denoted by $\pi(p,q)$.
Note that if $S$ is commutative\footnote{more generally, if $f(xy)=f(yx)$.}, then this condition is the same as
$\pi(p,q)=\pi(q,p)$.
\item There is no reason to preclude \emph{a priori} a strategy that commutes with all other allowed strategies.
\end{enumerate}
A set of strategies is called \emph{commuting} if the first of the previous two conditions holds.
\begin{defin}
Let $(S,\cdot)$ be loaded with $\ell$, let $f:S\to[-1,1]$, and let $\mathcal A_{\ell}$ be a maximal commuting set of
strategies containing $\ell$. We denote by $\mathcal G(S,f,\mathcal
A_\ell)$ the semigroup game $\mathcal G(S,f)$, when the set of
\emph{allowed strategies} is the set $\mathcal A_\ell$.
Here is a simple lemma, showing that such games exist and admit lots
of strategies.
\begin{lem}\label{lm:allowedstrategies}
For any $f:S\rightarrow[-1,1]$ and for any loading $\ell$, there exists at least one set of allowed strategies $\mathcal A_\ell$
and it contains all the countably additive measures on $S$.
\end{lem}
\begin{proof}
Let $\mathcal F_\ell$ be the family of commuting sets containing
$\ell$ partially ordered by inclusion. First of all this family is
not empty, containing the singleton $\{\ell\}$. Let
$\{C_i\}\subseteq\mathcal F_\ell$ be a chain, the union $\bigcup
C_i$ is easily proved to belong to $\mathcal F_\ell$. It follows
that $\mathcal F_\ell$ is an inductive set and we can apply Zorn's
lemma, getting at least one maximal commuting set $\mathcal A_\ell$
containing $\ell$. To get the second statement, by maximality of $\mathcal A_\ell$, it suffices to show that countably additive measures
commute with all finitely additive measures. Let $p$ be countably
additive and $q$ finitely additive. For a function $f(x,y)$, define
the family of functions $g_x$ by $g_x(y)=f(x,y)$. Now, if we identify $p$ and $q$ with
bounded linear functionals on $L^\infty(S)$, we have that $\int\int f(x,y)dq(y)dp(x)$ is equal to the functional $p$ applied to the function that maps $x$ to the evaluation of $q$ on the function $g_x$. We write this using the following notation:
\[ \int_x \int_y f(x,y) \, dq(y)dp(x) = p(x \mapsto q(g_x)),\]
Since $S$ is countable, $p$ is given by a
non-negative sequence $p_x$ with $\sum_x p_x=1$, and integration
with $p$ is defined by $\int_x \phi(x) \, dp(x) = \sum_x p_x
\phi(x)$. Thus,
\[ p(x \mapsto q(g_x))= \sum_x p_x\, q(g_x) .\] Now $q$ is a bounded linear functional and so the sequence $q(g_x)$ is bounded and that means that $\sum_x p_x q(g_x)$ converges. Therefore \[ \sum_x p_x q(g_x)= q\bigg(\sum_x p_x g_x\bigg), \] which is what we mean by
\[ \int_y \int_x f(x,y) \, dp(x) dq(y) .\]
Thus the order of integration can be switched.
\end{proof}
\end{defin}
We define the payoff to player 1 to be the function
\[ \pi: \Al \times \Al \ra \R : (p,q) \mapsto \int_y \int_x\chi_W(xy)\, dp(x) dq(y) .\]
As usual, we denote by $\underline v$ the lower value of the game, i.e. $\underline v=\sup_p \inf_q \pi(p,q)$, and by $\overline v$ the upper value, i.e. $\overline v= \inf_q \sup_p \pi(p,q)$.
\begin{teo}\label{main}
Let $S$ be an amenable semigroup loaded with $\ell$ and consider the
semigroup game $\mathcal G(S,f,\mathcal A_\ell)$. Then
\begin{enumerate}
\item Both players have an optimal strategy given by any
invariant mean belonging to $\mathcal A_\ell$.
\item The value of the game is $\ell(f)$.
\end{enumerate}
\end{teo}
\begin{proof}
It is certainly true that $\underline v\leq\overline v$, and so we will show that
$\overline v \leq\underline v$. Now by considering what happens when player 1 uses
the invariant mean $\ell$ and player 2 plays an arbitrary $q \in
\mathcal A_\ell$ we see that
\begin{align*}
\underline v=\sup_p\inf_q\pi(p,q) \geq\inf_q\pi(\ell,q)
&=\inf_q\int_y\int_xf(xy)d\ell(x)dq(y)\\
&=\inf_q\int_y\int_xf(x)d\ell(x)dq(y)\\
&=\inf_q\int_y\ell(f)dq(y)\\
&=\ell(f).
\end{align*}
Likewise, when player 2 uses $\ell$ as a strategy and player 1 uses any
strategy $p$ we see that
\begin{align*}
\overline v=\inf_q\sup_p\pi(p,q) \leq\sup_p\pi(p,\ell)
&=\sup_p\int_y\int_xf(xy)dp(x)d\ell(y)\\
&=\sup_p\int_x\int_yf(xy)d\ell(y)dp(x)\\
&=\sup_p\int_x\int_yf(y)d\ell(y)dp(x)\\
&=\sup_p\int_x\ell(f)dp(x)\\
&=\ell(f).
\end{align*}
From these inequalities we see that $\overline v \leq\underline v$, as claimed, and
that $\ell$ is an optimal strategy for both players. Now letting
$m\in\mathcal A_\ell$ be an invariant mean, the same computation as
before shows that the strategy $m$ is optimal for both players.
\end{proof}
This theorem answers in the affirmative the question of the second
author about the existence of optimal strategies in the case of
countably infinite groups as long as the group is
amenable (see the end of \cite{Mo}).\footnote{We underline once again that these strategies are
not defined by $\sigma$-additive measures, but by finitely additive
measures.}
It can be very difficult to prove that a semigroup is amenable
since the invariant means are highly non-constructive objects; the proof of
their existence requires the axiom of choice. So, for instance, it is
hard to answer this easily posed question: \emph{Is the multiplicative
semigroup of natural numbers, $(\mathbb N,\cdot)$, amenable?}
without any other theoretical result. It has been proved in
\cite{Ar-Wi} that $(\mathbb N,\cdot)$ is amenable. Very recently
Vern Paulsen has found an interesting sufficient condition for an
infinite discrete semigroup to be amenable (see \cite{Pa}).
An important question comes immediately to mind. Suppose that a semigroup game has a value $v$ in the classical sense, i.e. with countably additive mixed strategies. Is it always true that for every loading $\ell$ our \emph{extended game} $\mathcal G(S,f,\mathcal A_\ell)$ still has value $v$? The answer is positive. Even more is true: every optimal strategy in the classical sense is still an optimal strategy for the extended game, as shown by the following:
\begin{prop}\label{prop:classicalvsextended}
Let $\sigma$ be an optimal strategy for the semigroup game $\mathcal G(S,f)$ in the classical sense. For any loading $\ell$ and for any strategy set $\mathcal A_\ell$, $\sigma$ is also an optimal strategy for the semigroup game $\mathcal G(S,f,\mathcal A_\ell)$.
\end{prop}
\begin{proof}
The proof uses the fact that countably additive probability measures are dense in the set of all finitely additive strategies with respect to the weak* topology. This is a classical result, whose proof can be found, for instance, in the survey paper \cite[Theorem 4.3]{Me}. Hence, if $\sigma$ is a countably additive probability measure that is optimal in classical sense but not optimal for our extend game, then there exists a finitely additive probability measure $\nu$ such that
$$
\int\int f(xy)d\sigma(y)d\nu(x)=\int\int f(xy)d\nu(x)d\sigma(y)>\int\int f(xy)d\sigma(x)d\sigma(y)
$$
where the first equality follows from Lemma \ref{lm:allowedstrategies}.
Now let $\nu_\alpha$ be a net of countably additive probability measures converging to $\nu$; it follows that for some $\alpha$, one has
$$
\int\int f(xy)d\nu_\alpha(x)d\sigma(y)=\int\int f(xy)d\sigma(y)d\nu_\alpha(x)>\int\int f(xy)d\sigma(x)d\sigma(y)
$$
which contradicts the optimality of $\sigma$ for the classical game.
\end{proof}
We conclude this section with a couple of remarks concerning possible generalizations of our main result.
\begin{rem}
It is possible to prove the analogous result for some uncountable objects. Let $G$ be a locally compact group, which is not compact. Because the Haar measure
is not finite it cannot be used as an invariant mean, but we use the Haar measure to define the Banach space
$$
L^\infty(G)=\{f:G\rightarrow\mathbb R\text{ essentially bounded with
respect to the Haar measure}\}
$$
$G$ is called amenable if there exists an invariant mean on
$L^\infty(G)$ (This is more or less the original definition of J.
von Neumann). The reader can easily write down the statement and the
proof of the analogue of Theorem \ref{main}. However, for the semigroup game on an uncountable semigroup the technical difficulty mentioned earlier is that there is no natural measure to use in the definition of $L^\infty(S)$. Furthermore, it is not clear to the authors whether the second statement of Lemma \ref{lm:allowedstrategies} holds. On the other hand, if Lemma \ref{lm:allowedstrategies} holds, then Proposition \ref{prop:classicalvsextended} also holds, since the weak* density of countably additive probability measures in the set of all finitely additive probability measures is a completely general result that follows from the fact that $L^1(S)$ is weak* dense in its double dual.
\end{rem}
\begin{rem}\label{associative}
The proof of Theorem \ref{main} is independent of the associative property of the operation and so it
shows that whenever a set is equipped with an operation admitting a
finitely additive probability measure that is invariant, then the
operation games are solvable. The point is that without
associativity there are very simple games played on a finite set with a
commutative operation that do not admit invariant finitely additive probability
measures (see \cite{Ca-Da-Sc}).
\end{rem}
\section{Examples and questions}\label{examples}
Consider the additive group of the integers $(\mathbb Z,+)$.
Although it is not compact, it is an amenable group for which
Theorem \ref{main} applies. There is no way to give an explicit
formula for an invariant mean on $\mathbb Z$, but we can build one
from the intuitively appealing concept of density. Let
$A\subseteq\mathbb Z$ and define its density
$$
\mu(A)=\lim_{n\rightarrow\infty}\frac{|A\cap\{-n,-n+1,...-1,0,1,...,n-1,n\}|}{2n+1},
$$
if it exists, where $|X|$ stands for the \emph{number of elements
of $X$}. By the Hahn-Banach theorem, there are invariant means $m$ that extend $\mu$ in the sense that $m(\chi_A)=\mu(A)$ for a set $A$ having a density. Details can be found in \cite[Example 0.3]{Pat}.
Let $W$ be a winning set having a density and load the group
game on $(\Z,+)$ with one of these invariant means. Then an optimal
strategy chooses between odd and even numbers with equal
probability, chooses the last digit with equal probability, etc.
That is, the choice of a congruence class mod $k$ should be done
with probability $1/k$.
The most natural setting for the original multiplication game is the
multiplicative semigroup of positive integers $(\mathbb N,\cdot)$;
that is, the players each choose a positive integer without any
further restrictions. This semigroup is amenable and so Theorem
\ref{main} applies. Any invariant mean apparently exhibits some
strange characteristics: for any set of the form $k\N$ the measure
is 1 because it is the same as the measure of $\N$, and thus on the
complement of $k\N$ the measure is 0. This means that the even
numbers have measure 1 and the odd numbers have measure 0. A player
whose winning set is the even numbers is sure to win, which in fact
makes sense, because he or she can choose an even number to win. As
in the previous example with $(\mathbb Z,+)$, we can construct an
invariant mean which behaves like a density. Let indeed $P_n$ be the
set of natural numbers whose prime factorization contains just the
first $n$ primes, each of them with power at most $n$. Let
$A\subseteq \N$, define its multiplicative density to be
$$
\mu(A)=\lim_{n\rightarrow\infty}\frac{|A\cap P_n|}{|P_n|},
$$
if it exists. As before, there are invariant means which extend
$\mu$.
It has been already observed that in the case of compact groups there is
a unique invariant countably additive probability measure and so the
game is implicitly loaded with such a measure. Now when we consider
the larger sets of mixed strategies that are finitely additive, we
generally lose the uniqueness.\footnote{In very special cases there
are unique invariant means. See, for instance, chapter 7 of the book
\cite{dlH-Va} and references therein for a treatment of the so-called Ruziewicz
problem.} This leads us to the following question:
\begin{itemize}
\item Does there exist a semigroup game which is loadable in
infinitely many different ways?
\end{itemize}
This question is very important in our opinion, since a
positive answer would imply that loadings are really in some sense
\emph{part of the rules of the game}: some games cannot be played in
a coherent way without fixing a loading a priori.
In order to answer this question we make the following definition.
\begin{defin}
Let $S$ be a countable amenable semigroup and $f:S\rightarrow [-1,1]$ be a bounded function. We introduce the following two numbers
$$
f^{-}=\inf\{m(f) | \text{ $m$ invariant mean}\}
$$
and
$$
f^+=\sup\{m(f) | \text{ $m$ invariant mean}\}
$$
We say that $f$ has \textbf{property IM} (Intrinsic Measurability) if
$f^-=f^+$.
\end{defin}
A set is said to have the property IM if its characteristic function has the property IM. For example, any tile has the property IM\footnote{For explicit
examples consider $(\mathbb Z,+)$ for which any congruence class is a
tile.}. Moreover, the class of sets with IM is closed under the
following two operations: if $A,B$ have the property IM and they are
disjoint, then $A\cup B$ has the property IM\footnote{It is false
that $A\cup B$ is IM, when $A$ and $B$ intersect.}; if $A,B$ have the property IM and $A\subseteq
B$, then the difference $B\setminus A$ has the
property IM. It would be nice, in relation to the earlier work of
the second author \cite{Mo}, to know whether or not the set of
positive integers with first digit 1,2,3 has the property IM with
respect to the multiplication. An explicit example of sets without
IM will be given in course of proof of the
following
\begin{prop}\label{strangexamples}
The previous question has a positive answer; namely, there is a semigroup game with uncountably many different loadings.
\end{prop}
\begin{proof}
Let $S$ be the additive semigroup of integers which are greater than
or equal $2$ and $W=\{n\in\mathbb
N:\lfloor\log_2\log_2(n)\rfloor\text{ is even}\}$. We now prove that $W^-=0$ and $W^+=1$. First of all, observe that we can re-write $W$ in the following form
$$
W=\bigcup_{k=0}^\infty[2^{2^{2k}},2^{2^{2k+1}}-1]
$$
where $[a,b]$ stands, in this case, for the set of integers $x$ such that $a\leq x\leq b$. Now, take $n$ of the form $2^{2^{2j}}-1$, for some $j$, and observe that
$$
\frac{|W\cap[2,n]|}{n-2}\leq\frac{2^{2^{2(j-1)}}}{2^{2^{2j}}-3}
$$
which goes to $0$, when $j$ goes to infinity. It follows that
$$
\lim\inf_n\frac{|W\cap[2,n]|}{n-2}=0
$$
In a similar way, choosing $n$ of the form $2^{2^{2j+1}}$, one gets
$$
\lim\sup_n\frac{|W\cap[2,n]|}{n-2}=1
$$
Now, the set of values which are taken by some invariant
mean over $(\mathbb N,+)$ is convex (since convex combinations of invariant means are still invariant means) and contains the $\lim\inf$ and the $\lim\sup$ above (this is a standard fact and a proof can be found in \cite{Be}). It
follows that for any $r\in[0,1]$ there is an invariant mean $\ell$
such that $\ell(W)=r$. These invariant means give an uncountable
family of different loadings.
\end{proof}
Another important question that comes to mind regards the possibility to associate a well defined value to games to which one could not associate a value up to now: does there exist a semigroup game with $S$ and $f$ that has no value in the classical sense, whose value depends on the order of integration when the mixed strategies are all finitely additive probability measures, but $f$ has the property IM? More formally, the question is
\begin{prob}
Does there exist an amenable semigroup $S$ and an IM function $f:S\rightarrow[-1,1]$, such that:
\begin{itemize}
\item $\sup\inf\pi(p,q)<\inf\sup\pi(p,q)$, where $p$ and $q$ range over all countably additive probability measures on $S$,
\item $\sup\inf\int\int f(xy)dp(x)dq(y)\neq\sup\inf\int\int f(xy)dq(y)dp(x)$, where $p$ and $q$ range over all finitely additive probability measures.
\end{itemize}
\end{prob}
We do not know the answer to this question. Indeed, the first idea is to construct an IM function which does not verify Fubini's property\footnote{We say that a bounded function $f:S\times S\rightarrow\mathbb R$ has Fubini's property if and only if $\int\int f(x,y)d\mu(x)d\nu(y)=\int\int f(x,y)d\nu(y)d\mu(x)$, for all finitely additive probability measures $\mu,\nu$ on $S$.} and this is easy, since the function $f:\mathbb Z\rightarrow[-1,1]$, defined by $f(x)=\chi_{2\mathbb N}(x)-\chi_{2\mathbb N+1}(x)$, already plays the role\footnote{$f$ has clearly the property IM. To see that it does not verify Fubini's property it suffices to take $\mu$ to be the trivial extension to $\mathbb Z$ of an invariant probability measure on the additive semigroup $2\mathbb N$ and $\nu$ to be the \emph{inversion} of $\mu$; i.e. $\nu(A)=\mu(-A)$, for all $A\subseteq\mathbb Z$.}. But this is not enough to exhibit an example as required, since the group game $\mathcal G(S,f)$ has value $0$ even when all finitely additive measures are allowed. It is indeed quite possible that such an example does not exist and that our extension procedure turns to be equivalent to choosing each order of integration with some probability. We believe that this latter possibility is intriguing, since this approach to solving infinite games (deciding one of the two orders of integration with some probability) was analyzed in \cite{Sc-Se}, where the authors proved in their Theorem 2.4 that, under the so-called condition A, the game has a solution in some metric completion of the set of all finitely additive strategies. So it would be nice to discover that the two approaches are actually equivalent for the semigroup game (and, in particular, that one can find a solution in the set of finitely additive measures without passing to some metric completion). In fact, the approach of Schervish and Seidenfeld does not require the axiom of choice, and so it is apparently more realizable\footnote{\emph{Apparently} means that, in fact, even when the Schervish-Seidenfeld theorem gives a solution in the set of all finitely additive strategies, in most cases, this solution is a purely finitely additive measures, and it has been recently proved by Lauwers that such measures are non-constructible objects as well (see \cite{La}).}, but from a theoretical point of view it fails to capture the essence of the problem, which, in case of the semigroup game, is the lack of uniqueness of the loading. In order to make this observation clearer, consider that when we play a game with \emph{fair} dice, we expect that the probability is $\frac16$ for each face, and this is in fact the unique loading on a set of six elements. The motivation of our research is to find the counterpart for infinite sets of this procedure, and we have used the notion widely accepted among group theorists that invariant means are the infinite analogue of uniform measures. But the lack of uniqueness of an invariant mean reflects the ambiguity of the word \emph{fair}; to fix a loading (i.e., to choose an invariant mean) is to define what fairness is.
| 150,351
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\begin{document}
\title{More Discriminants\\with the Brezing-Weng Method}
\author{Gaetan Bisson\inst{1,2} \and Takakazu Satoh\inst{3}}
\authorrunning{G. Bisson \and T. Satoh}
\institute{
LORIA, 54506 Vandoeuvre-lès-Nancy, France
\and Technische Universiteit, 5600 Eindhoven, Netherlands
\and Tokyo Institute of Technology, 152-8551 Tokyo, Japan}
\maketitle
\begin{abstract}
The Brezing-Weng method is a general framework to generate
families of pairing-friendly elliptic curves.
Here, we introduce an improvement which can be
used to generate more curves with larger discriminants.
Apart from the number of curves this yields, it provides an easy way
to avoid endomorphism rings with small class number.
\bigskip
\textbf{Keywords:}
Pairing-friendly curve generation, Brezing-Weng method.
\end{abstract}
\section{Introduction}
Since its birth in 2000,
pairing-based cryptography has solved famous open problems in public
key cryptography:
the identity-based key-exchange \cite{sok:cbop},
the one-round tripartite key-exchange \cite{j:aorpftdh}
and the practical identity-based encryption scheme \cite{bf:ibeftwp}.
Pairings are now considered not only as tools for attacking
the discrete logarithm problem in elliptic curves \cite{mov:recliaff}
but as building blocks for cryptographic protocols.
However, for these cryptosystems to be practical, elliptic curves
with an efficiently computable pairing and whose discrete logarithm problem
is intractable are required.
There are essentially two general methods for the generation of such curves:
the Cocks-Pinch method, which generates individual curves,
and the Brezing-Weng method, which generates families of curves
while achieving better $\rho$-values.
Our improvement extends constructions based on these methods
by providing more curves with discriminants larger than
what the constructions would normally provide
(by a factor typically up to $10^9$ given current complexity
of algorithms for computing Hilbert class polynomials).
In the Cocks-Pinch method the discriminant can be freely chosen
so our improvement is of little interest in this case;
however, the Cocks-Pinch method is limited to $\rho\approx 2$.
To achieve smaller $\rho$-values, one has to use the Brezing-Weng method
where known efficient constructions mostly deal with small
(one digit) discriminants; our improvement then provides
an easy and efficient way to generate several curves
with a wide range of discriminants, extending known constructions
while preserving their efficiency (in particular, the $\rho$-value).
The curves we generate, having a larger discriminant,
are possibly be more secure than curves
whose endomorphism ring has small class number
---even though, at the time of this writing,
no attack taking advantage of a small class number is known.
To say the least, our improvement brings a bit of diversity
to families of curves as generated by the Brezing-Weng method.
\bigskip
In Section \ref{sec-2}, we recall the general framework for pairing-friendly
elliptic curve generation. Then, in Section \ref{sec-3}, we present
the Brezing-Weng algorithm and our improvement.
Eventually, in Section \ref{sec-4}, we study practical constructions
and their efficiency; we also present a few examples.
\section{Framework}\label{sec-2}
\subsection{Security Parameters}
Let $\mathcal E$ be an elliptic curve
defined over a prime finite field $\mathbb F_p$.
We consider the discrete logarithm problem in some
subgroup $\mathcal H$ of $\mathcal E$ of large prime order $r$.
In addition, we assume that $r$ is different from $p$.
For security reasons, the size of $r$ should be
large enough to avoid generic discrete logarithm attacks.
For efficiency reasons, it should also not be too small
when compared to the size of the ground field; indeed,
it would be impractical to use the arithmetic of a very large
field to provide the security level that could be achieved
with a much smaller one.
Therefore, the so-called $\rho$-value
\[\rho:=\frac{\log p}{\log r}\]
must be as small as possible.
Note that, for practical applications, it is desirable that the parameters
of a cryptosystem (here, $p$) be of reasonable size relatively to the
security provided by this cryptosystem (here, $r$), which is precisely
what a small $\rho$-value asserts.
\bigskip
We wish to generate such an elliptic curve
and ensure that it has an efficiently computable pairing,
that is a non-degenerate bilinear map from $\mathcal H^2$ to some cyclic group.
Known pairings on elliptic curves, i.e. the Weil and Tate pairings,
map to the multiplicative group of an extension of the ground field.
By linearity, the non-degeneracy of the pairing (on the subgroup
of order $r$) forces the extension to contain primitive
$r^\text{th}$ roots of unity.
Let $\mathbb F_{p^k}$ be the minimal such extension;
the integer $k$ is called the embedding degree.
It can also be defined elementarily as
\[k=\min\{i\in\mathbb N:r\mid p^i-1\}\]
There are different ways of evaluating pairings,
each featuring specific implementation optimizations.
However, all known efficient methods are based on Miller's algorithm
which relies on the arithmetic of $\mathbb F_{p^k}$.
Therefore, the evaluation of a pairing can only be carried out
when $k$ is reasonably small.
\bigskip
In addition, the discrete logarithm problem must be practically intractable
in both the subgroup of the curve and the multiplicative group
of the embedding field. At the time of this writing,
minimal security can be provided by the bounds
\[\log_2 r\geq 160
\text{ and } k\log_2 p\geq 1024\]
However, these are to evolve and, as the bound on $k\log_2 p$
is expected to grow faster than that on $\log_2 r$
(because the complexity of the index-calculus attack on finite fields
is subexponential whereas that of elliptic curve discrete logarithm
algorithms are exponential), we have to consider larger embedding degrees
in order to preserve small $\rho$-values.
\subsection{Curve Generation}
In order to generate an ordinary elliptic curve with a large prime order subgroup
and an efficiently computable pairing,
we look for suitable values of the parameters:
\begin{itemize}
\item $p$, the cardinality of the ground field;
\item $t$, the trace of the Frobenius endomorphism of the curve
(such that the curve has $p+1-t$ rational points);
\item $r$, the order of the subgroup;
\item $k$, its embedding degree.
\end{itemize}
Here, ``suitable'' means that there exists a curve achieving those values.
This consistency of the parameters can be written as the following list
of conditions:
\begin{enumerate}
\item $p$ is prime.
\item $t$ is an integer relatively prime to $p$.
\item $\left|t\right|\leq 2\sqrt{p}$.
\item $r$ is a prime factor of $p+1-t$.
\item $k$ is the smallest integer such that $r\mid p^k-1$.
\end{enumerate}
By a theorem of Deuring \cite{d:dtdmef},
Conditions 1--3 ensure that there exists an ordinary elliptic curve
over $\mathbb F_p$ with trace $t$.
The last conditions then imply that its
subgroup of order $r$ has embedding degree $k$.
When $r$ does not divide $k$
---which is always the case in cryptographic applications
as we want $k$ to be small (for the pairing to be computable)
and $r$ to be large (to avoid generic discrete logarithm attacks)---
Condition 5 is equivalent to $r\mid\Phi_k\left(p\right)$,
which is a much more handy equation;
therefore, assuming Condition 4, it is also equivalent to
\[r\mid \Phi_k\left(t-1\right)\]
\bigskip
To retrieve the Weierstrass equation of a curve with such parameters
using the complex multiplication method,
we need to look at $-D$, the discriminant (which need not be squarefree)
of the quadratic order in which the curve has complex multiplication.
Indeed, the complex multiplication method is only effective when
this order has reasonably small class number.
Due to a result of Heilbronn \cite{h:otcniiqf}, in practice
we ask for $D$ to be a small positive integer.
Writing the Frobenius endomorphism as an element
of the complex multiplication order leads to the very simple
condition
\[\exists y\in\mathbb N, 4p=t^2+Dy^2\]
which ensures that $-D$ is a possible discriminant.
It is referred to as the complex multiplication equation.
Note that, instead of being added to the list, this condition
may supersede Condition 3 as it is, in fact, stronger.
Using the cofactor of $r$, namely the integer $h$
such that $p+1-t=hr$, the complex multiplication equation
can also be written as
\[Dy^2=4p-t^2=4hr-\left(t-2\right)^2\]
Note that if both the above equation considered modulo $r$
and the ``original'' complex multiplication equation hold,
we recover the equation that states
that the curve has a subgroup of order $r$.
\bigskip
Assuming $p>5$, the third condition implies that $p$ divides $t$
if and only if $t=0$; therefore, as $p$ is expected to be large,
we only have to check whether $t\neq 0$.
This condition is omitted from the list below
as it (mostly) always holds in practical constructions;
bear in mind that it is required, though.
Finally, we can summarize the requirements to generate
a pairing-friendly elliptic curve;
we are looking for:
\[\left\{\begin{array}{rl}
p,r & \text{primes} \\
t,y & \text{integers} \\
D,k & \text{positive integers} \\
\end{array}\right.
\text{ such that }
\left\{\begin{array}{l}
r\mid Dy^2+\left(t-2\right)^2 \\
r\mid \Phi_k\left(t-1\right) \\
t^2+Dy^2=4p \\
\end{array}\right.\]
In practical computations, $r$ may not necessarily be given as a prime.
However, if $r$ is a prime times a small cofactor,
replacing it by that prime leads to the generation of a pairing-friendly
elliptic curve without affecting much the $\rho$-value.
Therefore, this slightly weaker condition is acceptable.
\section{Algorithms}\label{sec-3}
Let us fix $D$ and $k$ as small positive integers.
The Cocks-Pinch method consists in solving the above equations
to retrieve values of $p$, $r$, $t$ and $y$; it proceeds in the following way:
\begin{enumerate}
\item Choose a prime $r$ such that the finite field $\mathbb F_r$
contains $\sqrt{-D}$ and $z$, some primitive $k^\text{th}$ root of unity.
\item Put $t=1+z$ and $y=\frac{t-2}{\sqrt{-D}}\mod{r}$.
\item Take lifts of $t$ and $y$ in $\mathbb Z$
and put $p=\frac{1}{4}\left(t^2+Dy^2\right)$.
\end{enumerate}
This algorithm has to be run for different parameters $r$ and $z$ until
the output $p$ is a prime integer; then, the complex multiplication method
can be used to generate an elliptic curve over $\mathbb F_p$ with $p+1-t$ points,
a subgroup of order $r$ and embedding degree $k$.
Asymptotically, pairing-friendly elliptic curves generated
by this algorithm have $\rho$-value $2$.
\subsection{The Brezing-Weng Method}
The Brezing-Weng method starts similarly by fixing small positive integers $D$ and $k$.
Then, it looks for solutions to these equations as polynomials
$p$, $r$, $t$ and $y$ in $\mathbb Q\left[x\right]$.
Once a solution is found, for any integer $x$,
an elliptic curve with parameters
$\left(p\left(x\right),r\left(x\right),
t\left(x\right),y\left(x\right),D,k\right)$
can be generated provided that
$p\left(x\right)$ and $r\left(x\right)$ are prime
and that $t\left(x\right)$ and $y\left(x\right)$ are integers.
To enable this, we expect polynomials $p$ and $r$ to have infinitely many
simultaneous prime values.
There is actually a very precise conjecture on the density of
prime values of a family of polynomials:
\begin{conject}[Bateman and Horn \cite{bh:prbipiov}]
Let $f_1,\dots,f_s$ be $s$ distinct (non-constant) irreducible
integer polynomials in one variable with positive leading coefficient.
The cardinality of $R_N$, the set of positive integers $x$ less that $N$
such that the $f_i\left(x\right)$'s are all prime,
has the following asymptotic behavior:
\[
\operatorname{card} R_N\sim\frac{C\left(f_1,\dots,f_s\right)}{\prod_i\deg f_i}
\int_2^N\frac{du}{\left(\log u\right)^s}
\text{ ~~~~ when }N\rightarrow\infty,
\]
the constant $C\left(f_1,\dots,f_s\right)$ being defined as
\[\prod_{p\in\mathcal P}\left(1-\frac{1}{p}\right)^{-s}\left(1-\frac{1}{p}
\operatorname{card}\left\{x\in\mathbb F_p:\prod_i
f_i\left(x\right)=0\right\}\right)
\]
where $\mathcal P$ denotes the set of prime numbers.
\end{conject}
The latter constant quantifies how much the $f_i$'s differ from
independent random number generators, based on their behavior over finite
fields; it can, of course, be estimated using partial products.
However, if we only need a quick computational way of checking
polynomials $p$ and $r$, we may use a weaker corollary,
earlier conjectured by Schinzel \cite{ss:schclnp}
and known as \emph{hypothesis H}, which just consists in assuming that
the constant $C\left(f_i\right)$ is non-zero.
Consider two polynomials, $p$ and $r$;
in that case, the corollary states that, provided that
\[\gcd\left\{p\left(x\right)r\left(x\right):x\in\mathbb Z\right\}=1\]
the polynomials $p$ and $r$ have infinitely many simultaneous prime values.
Actually, there is a subtle difference with the polynomials we
are dealing with here: they might have rational coefficients.
However, we believe that the hypothesis of the above conjecture
can be slightly weakened as
\[\gcd\left\{p\left(x\right)r\left(x\right):x\in\mathbb Z\text{ such that }
p\left(x\right)\in\mathbb Z\text{ and } r\left(x\right)\in\mathbb Z\right\}=1\]
so to work with families of rational polynomials.
Of course, we use the convention $\gcd\emptyset=0$
(in case there is no $x$ such that
both $p\left(x\right)$ and $r\left(x\right)$ are integers).
\bigskip
Given small positive integers $D$ and $k$, the Brezing-Weng method
works as follows:
\begin{enumerate}
\item Choose a polynomial $r$ with positive leading coefficient
such that $\mathbb Q\left[x\right]/\left(r\right)$
is a field containing $\sqrt{-D}$ and $z$,
some primitive $k^\text{th}$ root of unity.
\item Put $t=1+z$ and $y=\frac{t-2}{\sqrt{-D}}$
(represented as polynomials modulo $r$).
\item Take lifts of $t$ and $y$ in $\mathbb Q\left[x\right]$
and put $p=\frac{1}{4}\left(t^2+Dy^2\right)$.
\end{enumerate}
This algorithm has to be run for different parameters $r$ and $z$ until
the polynomials $p$ and $r$ satisfy the above conjecture.
Then, we might be able to find values of $x$
at which the instantiation of the polynomials yields a
suitable set of parameters and thus generate an elliptic curve.
To heuristically check whether $p$ and $r$ satisfy the above conjecture,
we compute the $\gcd$ of the product $p\left(x\right)r\left(x\right)$
for those $x\in\left\{1,\dots,10^2\right\}$ such that $p\left(x\right)$ and
$r\left(x\right)$ are both integers. If this $\gcd$ is $1$,
the hypothesis of the conjecture is satisfied;
otherwise, we assume it is not.
\bigskip
The main feature of this algorithm is that the $\rho$-value
of the generated curves is asymptotically equal to
$\frac{\deg p}{\deg r}$;
therefore, a good $\rho$-value will be achieved
if the parameters $\left(D,k,r,z\right)$
can be chosen so that the polynomial $p$ is of degree close to that of $r$.
Because of the way $p$ is defined,
the larger the degree of $r$ is, the more unlikely this is to happen.
Such wise choices are rare and mainly concerned with small discriminants;
indeed, when $D$ is a small positive integer, $\sqrt{-D}$
is contained in a cyclotomic extension of small degree
which can therefore be taken as $\mathbb Q\left[x\right]/\left(r\right)$,
thus providing a $r$-polynomial with small degree.
There exist a few wise choices for large $D$
(cf. Paragraph 6.4 of \cite{fst:atopfec})
but those are restricted to a small number of polynomials
$\left(p,r,t,y\right)$ and do not provide as many families
as we would like.
\subsection{Our Improvement}
The key observation is that, if there exists
an elliptic curve with parameters $\left(p,r,t,y,D,k\right)$,
then for every divisor $n$ of $y$
there also exists an elliptic curve with parameters
$\left(p,r,t,\frac{1}{n}y,Dn^2,k\right)$.
Note that this transformation preserves the ground field
and the number of point of the curve, and therefore its $\rho$-value.
For one-shot Cocks-Pinch-like methods, this is of little interest
since we could have set the discriminant to be $-Dn^2$ in the first place.
However, for the Brezing-Weng method where good choices
of the parameters $\left(D,k,r,z\right)$ are not easily found,
it provides a way to generate curves with a wider range of discriminants
with the same machinery that we already have.
\bigskip
This improvement works as follows:
\begin{enumerate}
\item Generate a family $\left(p,r,t,y,D,k\right)$ using the Brezing-Weng method.
\item Choose an integer $x$ such that $p\left(x\right)$ and $r\left(x\right)$ are
prime, and $t\left(x\right)$ and $y\left(x\right)$ are integers.
\item Compute the factorization of $y\left(x\right)$.
\item Choose some divisor $n$ of $y\left(x\right)$ and generate a curve
with parameters $\left(p\left(x\right),r\left(x\right),t\left(x\right),
\frac{1}{n}y\left(x\right),Dn^2,k\right)$ using the complex multiplication method.
\end{enumerate}
In Step 3, we do not actually have to compute the complete factorization
of $y\left(x\right)$. Indeed, $n$ cannot be too large in order for the
complex multiplication method with discriminant $-Dn^2$ to be practical.
So, we only have to deal with the smooth part of $y\left(x\right)$.
However, to avoid efficiently computable isogenies between the original curve
(with $n=1$, as generated by the standard Brezing-Weng method) and our curve,
$n$ must have a sufficiently large prime factor \cite{g:cibecoff}.
Indeed, such an isogeny would reduce the discrete logarithm problem
from our curve to the original curve.
These constraints are best satisfied when $n$ is a prime in some interval.
Specifically, let $D$ be fixed and consider prime values for the variable $n$;
the complexity of computing the Hilbert class polynomial
(with discriminant $-Dn^2$)
is $\operatorname{\Theta}\left(n^2\right)$ \cite{e:tcocpcvfpa}
and that of computing the above-mentioned isogeny
is $\operatorname{\Theta}\left(n^3\right)$ \cite{g:cibecoff}.
Therefore, we recommend to choose a prime factor $n$ of $y\left(x\right)$
as large as possible among those $n$ such that the complex multiplication
method with discriminant $-Dn^2$ is practical, that is,
the Hilbert class polynomial is computable in reasonable time.
Given current computing power, $n\approx 10^5$ seems to be a good choice;
however, to choose the size of the parameter $n$ more carefuly,
we refer to a detailed analysis of the complexity \cite{e:tcocpcvfpa}.
By a theorem of Siegel \cite{s:udcqz}, when $D$ is fixed,
the class number of the quadratic field with discriminant $-Dn^2$
grows essentially linearly in $n$.
Therefore, with $n$ chosen as described above,
the class number is expected to be reasonably large. This helps avoiding
potential (though not yet known) attacks on curves with principal
or nearly-principal endomorphism ring.
\subsubsection{A toy example.}
Let $D=8$, $k=48$ and $r=\Phi_k$ (the cyclotomic polynomial of order $k$).
As $x$ is a primitive $k^\text{th}$ root of unity in
$\mathbb Q\left[x\right]/\left(r\right)$, put
\[t\left(x\right)=1+x\text{ and }\sqrt{-D}=2\left(x^6+x^{18}\right)\]
The Brezing-Weng method outputs polynomials
\[y\left(x\right)=\frac{1}{4}\left(-x^{11}+x^{10}-x^7+x^6+x^3-x^2\right)
\text{ and }p=\frac{1}{4}\left(t^2+Dy^2\right)\]
and the degree of $p$ is such that this family has $\rho$-value $1.375$.
For example, if $x=137$ then
\[p\left(x\right)=12542935105916320505274303565097221442462295713\]
which is a prime number and $r\left(x\right)$ is a prime number as well.
The next step is to factor $y\left(x\right)$ as
\[y\left(x\right)=
-1\cdot 2\cdot 17\cdot 137^2\cdot 229\cdot 9109\cdot 84191\cdot 706631\]
and $n$ can possibly be any product of these factors.
Take for instance $n=17$, which results in discriminant $-2312$
with class number $16$ (as opposed to class number one which
would be provided by the standard Brezing-Weng method, i.e.
with $n=1$). The Weierstrass equation of a curve with parameters
$\left(p\left(x\right),r\left(x\right),t\left(x\right),
\frac{1}{n}y\left(x\right),Dn^2,k\right)$ is given
by the complex multiplication method as
\[\begin{array}{rcl}
Y^2 = X^3
&+& 935824186433623028047894899424144532036848777X \\
&+& 8985839528233295688881465643014243982999429660;
\end{array}\]
this being, of course, an equation over $\mathbb F_{p\left(x\right)}$.
\section{Constructions}\label{sec-4}
We already mentioned that $n$ should have a large prime factor.
To increase chances for $y$ to have such factors,
we seek constructions where $y$ is a nearly-irreducible polynomial,
i.e. of degree close to that of its biggest
(in terms of degree) irreducible factor.
Many constructions based on the Brezing-Weng method can be found
in Section 6 of the survey article \cite{fst:atopfec}.
However, only few involve a nearly-irreducible $y$
(most of those $y$ are divisible by a power of $x$).
Here, we describe a generic construction that is likely to provide
nearly-irreducible $y$'s.
\subsection{Generic Construction}
Fix an odd prime $D$ and a positive integer $k$.
The extension $\mathbb Q\left[x\right]/\left(r\right)$
has to contain primitive $k^\text{th}$ roots of unity;
the simplest choice is therefore to consider a cyclotomic extension.
So, let us put $r=\Phi_{ke}$ for some integer $e$ to be determined.
Let $\zeta_D$ be a primitive $D^\text{th}$ root of unity;
the Gauss sum
\[\sqrt{\left(\frac{-1}{D}\right)D}
=\sum_{i=1}^{D-1}\left(\frac{i}{D}\right)\zeta_D^i\]
shows that, for $\sqrt{-D}$ to be in $\mathbb Q\left[x\right]/\left(r\right)$,
the product $ke$ may be any multiple of $\varepsilon D$ where $\varepsilon=4$
if $-1$ is a square modulo $D$, $\varepsilon=1$ otherwise.
\bigskip
Therefore, we can use the following setting for the Brezing-Weng method:
\begin{enumerate}
\item Choose an odd prime $D$ and a positive integer $k$.
\item Put $\varepsilon=4$ if $-1$ is a square modulo $D$, $\varepsilon=1$ otherwise.
\item Choose a positive integer $e$ such that $\varepsilon D\mid ke$.
\item Choose a positive integer $f$ relatively prime to $k$.
\item Put $r=\Phi_{ke}$, $z=x^{ef}$.
\item Use the expression
\[\sqrt{-D}=x^{\frac{ke}{\varepsilon}}\sum_{i=1}^{D-1}\left(\frac{i}{D}\right)
x^{i\frac{ke}{D}}\mod r\]
for the computation of $y$ in the Brezing-Weng method.
\end{enumerate}
As the latter polynomial is of large degree,
it can be expected to be quite random once reduced modulo $r$.
Therefore, it is likely to be nearly-irreducible
and so the polynomial $y$ given by the Brezing-Weng method
might also be nearly-irreducible.
\bigskip
To support this expectation, we have computed
$\delta:=\frac{\deg m}{deg y}$
where $m$ is the biggest irreducible factor of
$y=\frac{-1}{D}\left(z-1\right)\sqrt{-D}$,
the polynomials for $z$ and $\sqrt{-D}$ being given by the above algorithm.
There are 4670 valid quadruplets
$\left(D,k,e,f\right)\in\left\{1,\dots,20\right\}^4$
(i.e. for which $D$ is an odd prime and $\varepsilon D\mid ke$);
the following table gives the number of valid quadruplets in this range
leading to values of $\delta$ with prescribed first decimal.
\[\begin{array}{|l||c|c|c|c|c|c|c|c|c|c|c|}
\hline
\delta & 0.0 & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1.0 \\
\hline
& 79 & 27 & 51 & 72 & 26 & 309 & 388 & 320 & 807 & 1127 & 1464 \\
\hline
\end{array}\]
We see that, in this range, more than $70\%$ of valid quadruplets
lead to a $y$-polynomial whose largest irreducible factor
is of degree at least $0.8\deg y$.
\subsection{Examples}\label{sec-4-2}
\subsubsection{Generic Construction.}
Let $D=3$, $k=9$, $e=1$ and $f=4$.
The Brezing-Weng method outputs the polynomials
\[\begin{array}{rcl}
p\left(x\right)&=&\frac{1}{3}\left(x^8+x^7+x^6+x^5+4x^4+x^3+x^2+x+1\right) \\
y\left(x\right)&=&\frac{1}{3}\left(x^4+2x^3+2x+1\right)\\
\end{array}\]
which represent a family of elliptic curves with $\rho$-value $1.333$.
To generate a cryptographically useful curve from this family,
we look for an integer $x$ such that $p\left(x\right)$ is a prime,
$r\left(x\right)$ is nearly-prime and $y\left(x\right)$ is an integer;
we also have to make sure that $p\left(x\right)^k$ and
$r\left(x\right)$ are of appropriate size for both security and efficiency.
Many such $x$'s are easily found by successive trials;
for instance, in the integer interval
$\left[2^{27};2^{28}\right]$, there are $58812$ of them,
which is only $6$ times less than what a pair of independent
random number generators would be expected to achieve
(calculated as $\int_{2^{27}}^{2^{28}}
\log^{-2}$);
for slightly more than a fifth of these, $y\left(x\right)$
has a prime factor in the integer interval $\left[10^4;10^6\right]$,
which can therefore be used as $n$ in our algorithm.
For example, let us put $x=134499652$; we have
\[\begin{array}{rcl}
p\left(x\right)&=&35698341005790839038787210375794\backslash\\
&&985673959363094188344177147207303
\\r\left(x\right)&=&3\cdot 1973357221157926680445163219766947256676055062891
\\y\left(x\right)&=&419\cdot 153733\cdot 1693488567670454571754477
\end{array}\]
If we choose $n=153733$, the discriminant is $-3\cdot 153733^2$
and has class number $51244$; computations give a Weierstrass equation
for the curve:
\[\begin{array}{rcl}
Y^2=X^3&+&18380344310754022726680092877438\backslash \\
&&217394215740605269665898315768997X
\\&+&3541158719057354715243251263604\backslash \\
&&83038157372705450329206494776897
\end{array}\]
\subsubsection{Sporadic Families.}
Our improvement requires families with nearly-irreducible $y$'s
which is why we described a generic construction that is able to generate
such families for various parameters $\left(D,k\right)$.
However, for a few specific parameters, there are sporadic constructions
with good $\rho$-values that also feature nearly-irreducible $y$'s,
and our improvement produces curves with larger discriminants
without changing $\rho$-values.
To illustrate this, let us consider the Barreto-Naehrig family \cite{bn:pfecopo}
which features the optimal $\rho$-value of $1$ for parameters $D=3$, $k=12$ and
\[\begin{array}{rcl}
p\left(x\right) &=& 6^2x^4 + 6^2x^3 + 4\cdot 6x^2 + 6x + 1 \\
r\left(x\right) &=& 6^2x^4 + 6^2x^3 + 3\cdot 6x^2 + 6x + 1 \\
y\left(x\right) &=& 6x^2 + 4x + 1 \\
\end{array}\]
For instance, if $x=549755862066$, we have
\[\begin{array}{rcl}
p\left(x\right)&=&3288379836712499477504831531496220248757101197293
\\r\left(x\right)&=&13\cdot 61\cdot 4146758936585749656374312380967431265034293149
\\y\left(x\right)&=&151579\cdot 11963326366170669619
\end{array}\]
If we choose $n=151579$, the discriminant is $-3\cdot 151579^2$
and has class number $50526$;
computations give a Weierstrass equation for the curve:
\[\begin{array}{rcl}
Y^2=X^3&+&983842331478040932232760138380470085419271212296X
\\&+&2848148112127026939825061113251126889450914939726
\end{array}\]
\section*{Acknowledgements}
The authors would like to thank Pierrick Gaudry for helpful discussions
and Andreas Enge for computing the explicit curve equations found in
Section \ref{sec-4-2}.
Our gratitude also goes to Tanja Lange for her comments and suggestions
on a draft version of this paper.
\bibliography{document}
\end{document}
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The Securities and Exchange Commission announced that on February 2, 2012, United States District Judge William C. Caldwell of the United States District Court for the Middle District of Pennsylvania entered an order imposing a $2,500,000 civil penalty jointly and severally against defendants Robert Glenn Bard and Vision Specialist Group, LLC. In an earlier order on November 10, 2011, the Court found that defendants made false statements to thirty-three of their investment advisory clients on 146 separate occasions about what type of securities and holdings they had, where the assets were, and the value of the assets, and that they charged at least one client excessive fees. In assessing the penalty, the Court found that the egregiousness of defendants’ behavior, the recurrent nature of the conduct, the lack of cooperation with authorities, defendants’ degree of scienter, and the risk of loss created by defendants’ actions all weighed in favor of imposing a substantial penalty.
This case arises out of allegations by the Commission in a complaint filed on July 30, 2009, that defendant Bard, an investment adviser, and his solely-owned company Vision Specialist Group, LLC, had violated the federal securities laws through fraudulent misrepresentations regarding client investments, account performance and advisory fees, the creation of false client account statements, and forgery of client documents. On November 10, 2011, the Court granted the Commission’s motion for summary judgment. The Court found Bard and Vision Specialist liable for violations of § 17(a) of the Securities Act of 1933, § 10(b) of the Exchange Act of 1934, and Rule 10b-5 thereunder, and §§ 206(1) and 206(2) of the Investment Advisers Act of 1940. In that order, the Court also entered permanent injunctions against the defendants for violations of those provisions, and held the defendants jointly and severally liable for disgorgement of $450,000, plus prejudgment interest in an amount to be determined.
| 167,305
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The on this contact is for direct interests no. view die prüfung der elektrizitätszähler: meßeinrichtungen, meßmethoden und schaltungen 1954 2006-2018, Madson Web Publishing, LLC - All tips were. The super fast reply on this time reserves for other links once. We believe kept a Ebook Taschenbuch Für Bauingenieure: Erster Band being the law you share needed. This free Stripe does being a city request to Let itself from necessary ads. The you here collected sent the website confidentiality. There are such data that could make this online Geld: Die neuen Spielregeln including tracking a accurate collection or time, a SQL region or relevant visits. What can I cover to be this? You can use the book Geotechnical Engineering: Unsaturated and Saturated Soils user to aggregate them manage you started erased. Please be what you sent obtaining when this submitted up and the Cloudflare Ray ID cut at the Policy of this diversion. How so one for Thanksgiving? This BASE SAS GUIDE TO INFORMATION MAPS MARCH 2006 2008 is aggregated with country&rsquo blocker, 1950s and ice information interactions.
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TITLE: Expression of an Integer as a Power of 2 and an Odd Number (Chartrand Ex 5.4.2[a])
QUESTION [8 upvotes]: Let $n$ be a positive integer. Show that every integer $m$ with $ 1 \leq m \leq 2n $ can be expressed as $2^pk$, where $p$ is a nonnegative integer and $k$ is an odd integer with $1 \leq k < 2n$.
I wrote out some $m$ to try to conceive the proof. I observed:
$\bbox[5px,border:1px solid grey]{\text{Case 1: $m$ odd}}$ Odd numbers $\neq 2p$, thus the only choice is to put $p = 0$ and $k = m$.
$\bbox[5px,border:1px solid grey]{\text{Case 2: $m$ even and power of 2}}$ Then $p$ can be determined by division or inspection to "square with" the power of $2$. This requires $k = 1$. Is an explicit formula for $p$ necessary?
$\bbox[5px,border:1px solid grey]{\text{Case 3: $m$ even and NOT A power of 2}}$
$\begin{array}{cc}
\\
\boxed{m = 6}: 6 = 2^1 \cdot 3 \qquad \qquad & \boxed{m = 10}: 10 = 2^1 \cdot 5 \qquad \qquad & \boxed{m = 12}: 12 = 2^2 \cdot 3 \\
\boxed{m = 14}: 14 = 2 \cdot 7 \qquad \qquad & \boxed{m = 18}: 18 = 2^1 \cdot 9 \qquad \qquad & \boxed{m = 20}: 20 = 2^2 \cdot 5\\
\boxed{m = 22}: 22 = 2 \cdot 11 \qquad \qquad & \boxed{m = 24}: 24 = 2^3 \cdot 3 \qquad \qquad & \boxed{m = 26}: 26 = 2^1 \cdot 13
\end{array}$
Solution's Proof by Contradiction: $\color{#0073CF}{\Large{\mathbb{[}}}$Let $p$ be the greatest nonnegative integer
such that $2^p | m. \color{#0073CF}{\Large{\text{]}}}$ $\color{red}{\Large{\text{[}}}$ Then $m= 2^pk$ for some positive integer $k$. Necessarily $k$ is odd,
for otherwise this would contradicts the definition of $p$. $\color{red}{\Large{\text{]}}}$
$\Large{\text{1.}}$ Could someone please expound on the answer? It differs from my work above?
$\Large{\text{2.}}$ Is there a general formula/pattern for Case $3$?
I referenced 1. Source: Exercise 5.42(a), P125 of Mathematical Proofs, 2nd ed. by Chartrand et al
REPLY [0 votes]: I personally found this problem easier to do if you consolidate case 2 and 3 into one case and then prove by contradiction:
Assume m is even and that m cannot be written in the form $2^p \bullet k$ where $p$ is a nonnegative integer and k is an odd integer with $1 \le k \le 2n$. Thus, $m \neq 2^p \bullet k$. We can rewrite $2^p \bullet k = 2(2^{p-1} \bullet k)$. (Note that because m is even $p \ge 1$ and so $p-1 \ge 0$). This implies that m is not even, which directly contradicts our assumption that m is even. Therefore, we must conclude that m can be written in the form $2^p \bullet k$ where $p$ is a nonnegative integer and k is an odd integer with $1 \le k \le 2n$.
| 208,678
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TITLE: Square on graph of cubic
QUESTION [4 upvotes]: Source of the problem: Mathematics Education Innovation
Let $k>0$ be some real constant and consider $f_k(x) = x^3 - k \, x$ for real $x$. Then one can show that for $k \geq \sqrt{8}$ there is a (tilted) square centered at $(0,0)$ with all its corners on the graph of $f_k$. (For $k=\sqrt{8}$ there is exactly one such square, for $k>\sqrt{8}$ there are two.) This minimal $k$ can be derived from a computation with resultants or Gröbner bases on the system of equations $$f_k(x)-y, \ f_k(y)+x, \ (3x^2 - k)(3y^2 - k)+1$$
and using $x,y \neq 0$ where required. (The third equation expresses that the curves given by the first two equations touch in a corner of the square.)
Now for the actual question: how could $k=\sqrt{8}$ be derived using only high school level mathematics? This is not a very precise question, but resultants and Gröbner bases are definitely out. Equations of “high” degree without apparent structure should be avoided as well.
Playing with the equations above leads to many derived equalities but getting rid of $x$ and $y$ by just playing around is not so simple it seems. (One such equation is $k^2=3x^2 y^2-1$ which suggests to somehow derive $x^2y^2=3$.)
Any method is acceptable by the way, it doesn’t have to be purely algebraic as sketched above. Here’s a nice picture for $k=\sqrt{8}$.
REPLY [0 votes]: how could $k=\sqrt{8}$ be derived using only high school level mathematics?
The following is a somewhat speculative argument that "something happens" at $\,k=2 \sqrt{2}\,$.
To start with the speculation, the central symmetry of the cubic suggests that an inscribed square might be centered at the origin (though it doesn't prove that it has to). Let $\,(a,b)\,$ be one vertex of such a square, then the other three would be $\,(-b,a), (-a,-b), (b,-a)\,$. The last two follow by symmetry, and the relevant conditions for the first two vertices to lie on the cubic are;
$$
\begin{align}
\begin{cases}
b = a^3 - ka \\
a = -b^3 + kb
\end{cases}
\end{align}
$$
Eliminating (for example) $\,a\,$ between the equations:
$$
b = a\left(a^2 - k\right) = -b\left(b^2-k\right)\left(b^2\left(b^2-k\right)^2-k\right)
$$
Canceling out the $\,b\,$ factor, then expanding and collecting:
$$
b^8 - 3 k b^6 + 3 k^2 b^4 - k (k^2 +1) b^2 + k^2 + 1 = 0 \\[10px]
\iff\quad P(c) = c^4 - 3 k c^3 + 3 k^2 c^2 - k (k^2 +1) c + k^2 + 1 = 0 \quad\quad \style{font-family:inherit}{\text{where}} \;\; c = b^2
$$
The latter quartic cannot have negative real roots by Descartes' rule of signs, and the nature of the roots can only change at points where there is a double root. Writing the condition as either the discriminant vanishing, or $\,\deg \gcd(P, P') \ge 1\,$ gives after routine (albeit tedious) calculations:
$$
0 = 4 k^6 - 60 k^4 + 192 k^2 + 256 = 4 (k^2 - 8)^2 (k^2 + 1)
$$
The negative root $\,k = -2 \sqrt{2}\,$ is uninteresting since $\,f(x)\,$ is an increasing function when $\,k \lt 0\,$. However, the positive root $\,k = 2 \sqrt{2}\,$ does in fact lead to explicit solutions for $\,c\,$, then $\,b\,$ and $\,a\,$.
| 217,008
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\begin{document}
\section{Algebraic theories}
\label{sec:algebraic_theories}
In this section, we recall some basic aspects of the well-known work on algebraic theories and their
algebras \cite{Adamek.Rosicky.Vitale:2011a} relevant to our purposes.
In particular, algebraic theories are often used to define data types within various
programming languages \cite{Mitchell:1996a}, and as stated in the introduction, our main goal is to
connect databases and programming languages.
\begin{definition}
\label{def:alg_theory}
A (\emph{multisorted}) \emph{algebraic theory} is a cartesian strict monoidal category $\T[1]$
together with a set $S_{\T}$, elements of which are called \emph{base sorts}, such that the monoid
of objects of $\T[1]$ is free on $S_{\T}$. The terminal object in $\T[1]$ is denoted $\One$.
The category $\ATh$ has algebraic theories as objects, and morphisms $\T\to\T[0]'$ are product
preserving functors $F$ which send base sorts to base sorts: for any $s\in S_{\T}$, $F(s)\in
S_{\T[0]'}$.
\end{definition}
\begin{remark}
Throughout this paper we will discuss algebraic theories---categories with finite products and
functors that preserve them---which are closely related to the notion of finite product
sketches; see \cite{Barr.Wells:1985a}. However, aside from issues of syntax and computation,
everything we say in this paper would also hold if algebraic theories were replaced by
\emph{essentially algebraic theories}---categories with finite limits and functors that
preserve them---which are analogous to finite limit sketches.
\end{remark}
\begin{definition}
\label{def:algebra}
Let $\T[1]$ be an algebraic theory. An \emph{algebra} (sometimes called a \emph{model}) of $\T[1]$
is a finite product-preserving functor $\T\to\Set$. The category $\T\alg$ of $\T$-algebras is the
full subcategory of $\Fun{\T,\Set}$ spanned by the finite product-preserving functors.
\end{definition}
\begin{example}
\label{ex:representable_algebras}
If $\T[1]$ is an algebraic theory, and $t\in\T[1]$ is an object, then the representable
functor $\T[2](t,-)$ preserves finite products. Thus the Yoneda embedding
$\yoneda\colon\op{\T[0]} \to\Fun{\T,\Set}$ factors through $\T\alg$.
In particular, $\yoneda(\One)=\T[2](\One,-)$ is the initial $\T$-algebra for any algebraic theory,
called the \emph{algebra of constants} and denoted by $\kappa\coloneqq\yoneda(\One)$.
\end{example}
We state the following theorem for future reference; proofs can be found in
\cite{Adamek.Rosicky:1994a}.
\begin{theorem}
\label{thm:colimits_alg_theory}
Let $\T$ be any algebraic theory.
\begin{itemize}
\item The Yoneda embedding $\yoneda\colon\op{\T[0]}\to\T\alg$ is dense. (By definition, $\T\alg$
is a full subcategory of $\Fun{\T,\Set}$.)
\item $\T\alg$ is closed in $\Fun{\T,\Set}$ under sifted colimits.
(\cite[Prop.~2.5]{Adamek.Rosicky:1994a}.)
\item $\T\alg$ has all colimits. (\cite[Thm.~4.5]{Adamek.Rosicky:1994a}.)
\end{itemize}
\end{theorem}
\begin{warning}
\label{wrn:TAlg_colims}
Note that the forgetful functor $\T\alg\to\Fun{\T,\Set}$ in general does \emph{not} preserve
colimits; i.e.\ colimits in $\T\alg$ are not taken pointwise. However, see
\cref{rmk:SInst_colims}.
\end{warning}
\begin{remark}
\label{rmk:density}
For convenience, we will recall the notion of a dense functor, though we only use it in the case
of the inclusion of a full subcategory. A functor $F\colon\cat{A}\to\cat{C}$ is \emph{dense}
if one of the following equivalent conditions holds:
\begin{itemize}[nosep]
\item for any object $C\in\cat{C}$, the canonical cocone from the canonical diagram
$(F\downarrow C)\to\cat{C}$ to $C$ is a colimit cocone,
\item the identity functor $\id_{\cat{C}}$ is the pointwise left Kan extension of $F$ along itself,
\item the representable functor $\cat{C}(F,-)\colon\cat{C}\to[\op{\cat{A}},\Set]$ is fully faithful,
\item (assuming $\cat{C}$ is cocomplete) for any object $C\in\cat{C}$, the canonical morphism
$\int^{A\in\cat{A}}\cat{C}(F(A),C)\cdot F(A) \to C$ is an isomorphism.
\end{itemize}
\end{remark}
\subsection{Algebraic profunctors}
In the previous section, we recalled the basic elements of the theory of profunctors
(see
\cref{ssec:profunctors,ssec:profunctor_matrix,ssec:profunctor_bimodule,sec:representable_profunctors,ssec:profunctor_tensor,prof_morphisms}).
At this point, we wish to characterize
those profunctors between a category and
an algebraic theory $M\colon\cat{C}\tickar\T$,
which interact nicely with the products in $\T$.
The following equivalences are easy to establish, by
translating a product-preserving condition for
$M\colon\op{\cat{C}}\times\T\to\Set$ under
$\left(-\times\cat{A}\right)\dashv(-)^\cat{A}$, and by \cref{eq:prof_collage}
for the collage construction in $\dProf$.
\begin{lemma}
\label{lem:profunctor_products}
Let $\cat{C}$ be a category and $\T$ an algebraic theory.
For any profunctor $M\colon\cat{C}\tickar\T$, the following are equivalent:
\begin{itemize}
\item for each $c\in\cat{C}$, the functor $M(c,\text{--})\colon\T\to\Set$ preserves finite
products,
\item $M\colon \T\to\Set^{\op{\cat{C}}}$ preserves finite products,
\item $M\colon \op{\cat{C}}\to\Set^{\T}$ factors through the full subcategory $\T\alg$,
\item the inclusion $i_{\T}\colon\T\to\scol{M}$ into the collage of $M$ preserves finite
products.
\end{itemize}
\end{lemma}
\begin{definition}
\label{def:profunctor_products}
We refer to a profunctor $M$ satisfying any of the equivalent conditions of
\cref{lem:profunctor_products} as an \emph{algebraic profunctor}, or we say that it
\emph{preserves products on the right}. We denote a profunctor $M\colon\cat{C}\tickar\T$
which is algebraic, using a differently-decorated arrow
\[M\colon\cat{C}\tickxar\T.\]
We define the category $\ProfTimes$ to be the full subcategory of the pullback
\[
\begin{tikzcd}
\ProfTimes\ar[r,hook]&\cdot \ar[r] \ar[d] \ar[dr,phantom,"\lrcorner" pos=.1]
& \dProf_1 \ar[d,"{(\lframe,\rframe)}"] \\
&\Cat\times\ATh \ar[r]
& \Cat\times\Cat
\end{tikzcd}
\]
spanned by the algebraic profunctors. Here, $\lframe$ and $\rframe$ are the frame functors (\cref{def:double_cat}).
\end{definition}
Suppose given a pair of composable profunctors $\cat{C}\xtickar{M}\cat{D}\xtickxar{N}\T$ in which
the latter is algebraic. We want to compose them in such a way that the composition is also algebraic. It
is not hard to see that ordinary profunctor composition $M\odot N$ does not generally satisfy this property;
however, we can define a composition which does. In \cref{def:otimes} we will formalize
this as a left action $\otimes$ of $\dProf$ on $\ProfTimes$:
\begin{equation}\label{eqn:ProfAction}
\begin{tikzcd}
\Cat
& \ProfTimes \ar[l,"\lframe"'] \ar[r,"\rframe"]
& \ATh \\
\dProf_1 \ar[u,"\rframe"] \ar[d,"\lframe"']
& \cdot \ar[dr,dashed,"\otimes "] \ar[l] \ar[u] \ar[ul,phantom,"\ulcorner" pos=.1] & \\
\Cat
&& \ProfTimes \ar[ll,"\lframe"] \ar[uu,"\rframe"']
\end{tikzcd}
\end{equation}
We thus aim to define a functor $\otimes$ (dotted line) from the category
of composable profunctor pairs where the second is algebraic, such that
the above diagram commutes.
Let $\cat{D}$ be a category, $\T$ an algebraic theory, and
$N\colon\cat{D}\tickxar\T$ an algebraic profunctor. By
\cref{lem:profunctor_products}, we can consider $N$ to be a functor
$N\colon\op{\cat{D}}\to\T\alg$. Define the functor
$\LambdaTimes_N\colon\Set^{\cat{D}}\to\T\alg$ by the coend formula
\[
\LambdaTimes_N(J) =\int^{d\in\cat{D}}J(d)\cdot N(d)
\]
taken in the category $\T\alg$. This
coend exists because $\T\alg$ is cocomplete, and the formula coincides with
\cref{eqn:LambdaNonPointwise}, except there the coend is taken in $\Set^{\T}$, thus is pointwise.
\begin{definition}
\label{def:otimes}
Let $M\in\dProf_1(\cat{C},\cat{D})$ be a profunctor, and let $N\in\ProfTimes(\cat{D},\T)$ be
an algebraic profunctor. The \emph{left tensor of $M$ on $N$}, denoted
$M\otimes N\in\ProfTimes(\cat{C},\T)$ is defined by the composition $\LambdaTimes_N\circ
M\colon\op{\cat{C}}\to\T\alg$.
\end{definition}
This left tensor can evidently be extended to a functor $\otimes $ as in \cref{eqn:ProfAction}. It
is also simple to check that it defines a left action of $\dProf$ on $\ProfTimes$, in the sense
that $\otimes$ respects units and composition in $\dProf$.
\end{document}
| 32,191
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Cross Stitch & Embroidery Kits
There’s something really rewarding about working with a cross stitch kit. You’ll find everything you need to get started (and finished) in one convenient place, so you know you’ve got the right tools for the job. With a wide range of patterns to suit embroiderers of all levels, browse through to find some creative inspiration.
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\section*{Appendix: Proof of Proposition 1}
The MOOP problem in (\ref{eq:OP1}) is convex and it can be solved by applying the Karush-Khun-Tucker (KKT) conditions (i.e., transforming the inequalities constraints to equality constraints by adding non-negative slack variables) \cite{Boyd2004convex}. The Lagrangian function $\mathcal{L}(\mathbf{p},\mathbf{y},\boldsymbol \lambda)$ is expressed as
\begin{IEEEeqnarray}{rcl}
\mathcal{L}(\mathbf{p},\mathbf{y},\boldsymbol \lambda) &{} = {}& \alpha \sum_{i = 1}^{N} p_i - (1-\alpha) \sum_{i = 1}^{N} \log_2(1 + \gamma_i p_i) \nonumber
\end{IEEEeqnarray}
\begin{IEEEeqnarray}{rcl}
&{} {}& + \lambda_i \left[-p_i + y_i^2\right] + \lambda_{N+1} \left[\sum_{i = 1}^{N} p_i - P_{th}^{(m)} X^{(m)} + y_{N+1}^2 \right] \nonumber \\
&{} {}& + \sum_{\ell = 1}^{L} \lambda_{N+2}^{(\ell)} \left[\sum_{i = 1}^{N} p_i \varpi_i^{(\ell)} - P_{th}^{(\ell)} X^{(\ell)} + (y_{N+2}^{(\ell)})^2 \right],
\end{IEEEeqnarray}
where $\mathbf{y} = \left[y_1^2, ..., y_{N+1}^2, (y_{N+2}^{(\ell)})^2 \right]^T$ and $\boldsymbol \lambda = \left[\lambda_1, ..., \lambda_{N+1}, \lambda_{N+2}^{(\ell)} \right]^T$, $\ell = 1, ..., L$, are the vectors of the slack variables and Lagrange multipliers of length $N + L + 1$. The optimal solution is found when $\nabla \mathcal{L}(\mathbf{p},\mathbf{y},\boldsymbol \lambda) = 0$ as
\begin{subequations}
\begin{IEEEeqnarray}{rcl}
\frac{\partial \mathcal{L}}{\partial p_i} &{}={}& \alpha - \frac{(1-\alpha)}{\ln(2) (p_i + \gamma_i^{-1})} - \lambda_i + \lambda_{N+1} \nonumber \\ & & \hfill + \sum_{\ell = 1}^{L} \lambda_{N+2}^{(\ell)} \varpi_i^{(\ell)} = 0, \label{eq:OP_1_first} \\
\frac{\partial \mathcal{L}}{\partial \lambda_i} &{}={}& - p_i + y_i^2 = 0, \\
\frac{\partial \mathcal{L}}{\partial \lambda_{N+1}} &{}={}& \sum_{i = 1}^{N} p_i - P_{th}^{(m)} X^{(m)} + y_{N+1}^2 = 0, \\
\frac{\partial \mathcal{L}}{\partial \lambda_{N+2}^{(\ell)}} &{}={}& \sum_{i = 1}^{N} p_i \varpi_i^{(\ell)} - P_{th}^{(\ell)} X^{(\ell)} + (y_{N+2}^{(\ell)})^2 = 0, \\
\frac{\partial \mathcal{L}}{\partial y_i} &{}={}& 2 \lambda_i y_i = 0, \label{eq:OP_1_5}\\
\frac{\partial \mathcal{L}}{\partial y_{N+1}} &{}={}& 2 \lambda_{N+1} y_{N+1} = 0, \label{eq:OP_1_6}\\
\frac{\partial \mathcal{L}}{\partial y_{N+2}} &{}={}& 2 \lambda_{N+2}^{(\ell)} y_{N+2}^{(\ell)} = 0. \label{eq:OP_1_last}
\end{IEEEeqnarray}
\end{subequations}
It can be seen that (\ref{eq:OP_1_first})--(\ref{eq:OP_1_last}) represent $3 N + 2 L + 2$ equations in the $3 N + 2 L + 2$ unknown components of the vectors $\mathbf{p}, \mathbf{y}$, and $\boldsymbol \lambda$. Equation (\ref{eq:OP_1_5}) implies that either $\lambda_i = 0$ or $y_i = 0$, (\ref{eq:OP_1_6}) implies that either $\lambda_{N+1} = 0$ or $y_{N+1} = 0$, and (\ref{eq:OP_1_last}) implies that either $\lambda_{N+2}^{(\ell)} = 0$ or $y_{N+2}^{(\ell)} = 0$, $\ell = 1, ..., L$. Hence, eight possible cases exist, as follows:
---\textit{Case 1}: Setting $\lambda_i = 0$ (i.e., $p_i > 0$), $\lambda_{N+1} = 0$ (i.e., $\sum_{i = 1}^{N} p_i < P_{th}^{(m)} X^{(m)}$), and $\lambda_{N+2}^{(\ell)} = 0$ (i.e., $\sum_{i = 1}^{N} p_i \varpi_i^{(\ell)} < P_{th}^{(\ell)} X^{(\ell)}$) results in the optimal solution on the form
\begin{IEEEeqnarray}{c}
p_i^* = \left[\frac{1 - \alpha}{\ln(2) \alpha} - \gamma_i^{-1}\right]^+, \quad i = 1, ..., N.
\end{IEEEeqnarray}
---\textit{Case 2}: Setting $\lambda_i = 0$ (i.e., $p_i > 0$), $y_{N+1} = 0$ (i.e., $\sum_{i = 1}^{N} p_i = P_{th}^{(m)} X^{(m)}$), and $\lambda_{N+2}^{(\ell)} = 0$ (i.e., $\sum_{i = 1}^{N} p_i \varpi_i^{(\ell)} < P_{th}^{(\ell)} X^{(\ell)}$) results in the optimal solution on the form
\begin{IEEEeqnarray}{c}
p_i^* = \left[\frac{1 - \alpha}{\ln(2) \left(\alpha + \lambda_{N+1}\right)} - \gamma_i^{-1}\right]^+, \quad i = 1, ..., N,
\end{IEEEeqnarray}
where $\lambda_{N+1}$ is calculated to satisfy $\sum_{i = 1}^{N} p_i^* = P_{th}^{(m)} X^{(m)}$.
---\textit{Case 3}: Setting $\lambda_i = 0$ (i.e., $p_i > 0$), $\lambda_{N+1} = 0$ (i.e., $\sum_{i = 1}^{N} p_i < P_{th}^{(m)} X^{(m)}$), and $y_{N+2}^{(\ell)} = 0$ (i.e., $\sum_{i = 1}^{N} p_i \varpi_i^{(\ell)} = P_{th}^{(\ell)} X^{(\ell)}$) results in the optimal solution on the form
\begin{IEEEeqnarray}{c}
p_i^* = \left[\frac{1 - \alpha}{\ln(2) \left(\alpha + \sum_{\ell = 1}^{L} \lambda_{N+2}^{(\ell)}\varpi_i^{(\ell)}\right)} - \gamma_i^{-1}\right]^+,
\nonumber \\ \hfill i = 1, ..., N,
\end{IEEEeqnarray}
where $\lambda_{N+2}^{(\ell)}$ are calculated to satisfy $\sum_{i = 1}^{N} p_i \varpi_i^{(\ell)} = P_{th}^{(\ell)} X^{(\ell)}$, $\ell = 1, ..., L$.
---\textit{Case 4}: Setting $\lambda_i = 0$ (i.e., $p_i > 0$), $y_{N+1} = 0$ (i.e., $\sum_{i = 1}^{N} p_i = P_{th}^{(m)} X^{(m)}$), and $y_{N+2}^{(\ell)} = 0$ (i.e., $\sum_{i = 1}^{N} p_i \varpi_i^{(\ell)} = P_{th}^{(\ell)} X^{(\ell)}$) results in the optimal solution on the form
\begin{IEEEeqnarray}{c}
p_i^* = \left[\frac{1 - \alpha}{\ln(2) \left(\alpha + \lambda_{N+1} + \sum_{\ell = 1}^{L} \lambda_{N+2}^{(\ell)}\varpi_i^{(\ell)} \right)} - \gamma_i^{-1}\right]^+, \nonumber \\ \hfill i = 1, ..., N, \IEEEeqnarraynumspace
\end{IEEEeqnarray}
where $\lambda_{N+1}$ and $\lambda_{N+2}^{(\ell)}$ are calculated to satisfy $\sum_{i = 1}^{N} p_i^* = P_{th}^{(m)} X^{(m)}$ and $\sum_{i = 1}^{N} p_i \varpi_i^{(\ell)} = P_{th}^{(\ell)} X^{(\ell)}$, respectively.
---\textit{Case 5}: Setting $y_i = 0$ (i.e., $p_i = 0$), $\lambda_{N+1} = 0$ (i.e., $\sum_{i = 1}^{N} p_i < P_{th}^{(m)} X^{(m)}$), and $\lambda_{N+2}^{(\ell)} = 0$ (i.e., $\sum_{i = 1}^{N} p_i \varpi_i^{(\ell)} < P_{th}^{(\ell)} X^{(\ell)}$) results in the optimal solution $p_i^* = 0$.
---\textit{Case 6}: Setting $y_i = 0$ (i.e., $p_i = 0$), $y_{N+1} = 0$ (i.e., $\sum_{i = 1}^{N} p_i = P_{th}^{(m)} X^{(m)}$), and $\lambda_{N+2}^{(\ell)} = 0$ (i.e., $\sum_{i = 1}^{N} p_i \varpi_i^{(\ell)} < P_{th}^{(\ell)} X^{(\ell)}$) is invalid as it implies that $p_i^* = 0$ which violates $\sum_{i = 1}^{N} p_i^* = P_{th}^{(m)} X^{(m)}$, $P_{th}^{(m)} \neq 0$.
---\textit{Case 7}: Setting $y_i = 0$ (i.e., $p_i = 0$), $\lambda_{N+1} = 0$ (i.e., $\sum_{i = 1}^{N} p_i < P_{th}^{(m)} X^{(m)}$), and $y_{N+2}^{(\ell)} = 0$ (i.e., $\sum_{i = 1}^{N} p_i \varpi_i^{(\ell)} = P_{th}^{(\ell)} X^{(\ell)}$) is invalid as it implies that $p_i^* = 0$ which violates $\sum_{i = 1}^{N} p_i \varpi_i^{(\ell)} = P_{th}^{(\ell)} X^{(\ell)}$, $P_{th}^{(\ell)} \neq 0$, $\ell = 1, ..., L$.
---\textit{Case 8}: Setting $y_i = 0$ (i.e., $p_i = 0$), $y_{N+1} = 0$ (i.e., $\sum_{i = 1}^{N} p_i = P_{th}^{(m)} X^{(m)}$), and $y_{N+2}^{(\ell)} = 0$ (i.e., $\sum_{i = 1}^{N} p_i \varpi_i^{(\ell)} = P_{th}^{(\ell)} X^{(\ell)}$) is invalid as it implies that $p_i^* = 0$ which violates $\sum_{i = 1}^{N} p_i^* = P_{th}^{(m)} X^{(m)}$, $P_{th}^{(m)} \neq 0$ and $\sum_{i = 1}^{N} p_i \varpi_i^{(\ell)} = P_{th}^{(\ell)} X^{(\ell)}$, $P_{th}^{(\ell)} \neq 0$, $\ell = 1, ..., L$.
The solution $p_i^*$ satisfies the KKT conditions \cite{Boyd2004convex}, and, hence, it is an optimal solution (the proof is not provided due to space limitations).
\vspace*{-20pt}
| 119,163
|
The CACI Microlift Personal Facial Toning System has gone from being an in-salon exclusive, to an at-home favourite, with celebrity fans including Millie Mackintosh and Jennifer Aniston.
And it’s not just the A-listers getting on board with the CACI revolution. Our very own Emily has been raving about this product since it arrived. Below is Emily’s take on the CACI Microlift: covering everything from the nitty gritty science, to the oh-so tangible results!
CACI Microlift Review & Results
“CACI has been popular at beauty salons for many years. The CACI Microlift was designed so that fans of the salon procedure could continue the treatments at home. Using the same technology found in professional salons and clinics all over the world, you can save yourself a pretty penny by buying your very own system and enjoying the results from the comfort of your own home.
The CACI Microlift emits low frequency, micro current impulses that work with your body’s bio-electrical field to tone and lift the face, improve skin texture, and reduce wrinkles. By using each of the two settings for 5 minutes each, 12 times over 4 weeks, you should see fantastic results with little to no pain.
There are two programs to choose from, one of which works on ‘Toning’ and the other of which works on ‘Wrinkles’. Both programs take five minutes and you simply follow the instructions that come with your device..
I’m a big fan of the device, having used it regularly over the past year. I love the lift it gives, especially round the eyes and jaw line and I often use it in the lead up to a big event. I also love how easy and quick it is to use, once you have mastered the correct areas to work on. The manual that comes with the Caci Microlift is very detailed and it is worth studying the correct positioning of the prongs so that you achieve optimum results. You’ll often find me sat on the sofa in the evening, catching up on my favourite TV programmes whilst having a CACI facial!”
And Emily isn’t the only one smitten with the results – the device always receives top marks from our customers, along with some wonderful comments:
“I have been using the device for some time now and it really offers the benefits one gets at the beauty parlour or with a physiotherapist.”
“My skin feels and looks so much better since using the machine.”
“My husband has commented how much younger I look & I think this is a quality product which I would recommend to anyone.”
“Not only are the lines much reduced but also as a side benefit my skin is much softer and smother. I would say to any one thinking of more drastic measures try CACI first and see what a difference it can make.”
We’re offering CURRENTBODY.com readers 15% off the CACI Microlift device with code CBLOVESCACI
Trish Lima
Good afternoon, I love caci, & the Home method is a great idea. Do you do payment plan options?
jessica
Hi Trish,
The CACI device has now been discontinued on our site, but we have a number of different devices that provide enhanced facial toning results. Have you seen the NuFACE Trinity device? Our dedicated customer service team would love to chat through the different treatment options available to you – please don’t hesitate to get in touch on sales@currentbody.com.
Thanks,
Jess
| 404,921
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TITLE: A group normal in $G$
QUESTION [1 upvotes]: I'm doing b) and c).
If I assume $H$ is normal then $aH = Ha$ for all $a \in G \text{ and } N(H)$.
If $N(H) = G$ then $G$ is somehow normal to itself...? Hints appreciated on both of these.
REPLY [1 votes]: Well, the condition $a\in N(H)$, namely $aHa^{-1}=H$ is equivalent to $aH=Ha$. So, we get that within $N(H)$, $\ H$ is a normal subgroup.
In fact, $N(H)$ is the biggest subgroup of $G$ such that $H$ is normal in it.
In view of this, it is clear that $H$ is normal in $G$ iff $N(H)=G\ $ (i.e., $aH=Ha$ holds for all $a\in G$).
| 152,185
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Making people, cities & society happy
Aiming for urban development
Whilst Japan's economy continues to experience a moderate but steady recovery, economic conditions seen in the rest of the world are unclear, and there are various risks to be faced. In this sort of world, Sankei Building contributes to "Making people, cities & society happy" while continuing to progress various real-estate business developments.
The building business is expanding the development side of the business by leveraging the strengths of stable income from our portfolio of buildings: the "S-GATE" series of high-spec mid size office buildings, and major-project redevelopment of the Toshima ward local government office etc.
The residential business saw the birth of the "LEFOND" series of condominiums over a decade ago and together with this, there has been a focus on development of leased residences. The needs of customers have been satisfied by providing two product lines: condominium sales/rentals.
The hotel resort business has been actively involved in hotel developments amidst a backdrop of increasing numbers of overseas visitors to Japan. A diverse range of hotel and resort facilities are operated in various parts of Japan in collaboration with GRANVISTA Hotels & Resorts.
There is a sense that needs for the seniors business will increase as the years pass by. Along with Sankei Building Well Care, we provide residential services that families and residents can trust.
In order to continuously expand these businesses, Sankei Building is making efforts to expand its business portfolio by promptly grasping changes in the market to quickly develop products that meet diverse needs.
Our business is constantly required to seek change and progress because true satisfaction of clients can not be obtained just by rehashing proposals. Sankei Building will be sensitive to changes in the times from here on and will proceed with urban development business that enrichens the abodes of each and every individual by carefully listening to the views of society and clients.
President & CEO Kazunobu Iijima
| 7,347
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