text:
\begin{document}

\title{Lie symmetries and singularity analysis for generalized shallow-water
equations}
\author{Andronikos Paliathanasis\thanks{
Email: anpaliat@phys.uoa.gr} \\

{\ \textit{Institute of Systems Science, Durban University of Technology }}\\
{\ \textit{PO Box 1334, Durban 4000, Republic of South Africa}}}
\maketitle

\begin{abstract}
We perform a complete study by using the theory of invariant point
transformations and the singularity analysis for the generalized
Camassa-Holm equation and the generalized Benjamin-Bono-Mahoney equation.
From the Lie theory we find that the two equations are invariant under the
same three-dimensional Lie algebra which is the same Lie algebra admitted by
the Camassa-Holm equation. We determine the one-dimensional optimal system
for the admitted Lie symmetries and we perform a complete classification of
the similarity solutions for the two equations of our study. The reduced
equations are studied by using the point symmetries or the singularity
analysis. Finally, the singularity analysis is directly applied on the
partial differential equations from where we infer that the generalized
equations of our study pass the singularity test and are integrable.

Keywords: Lie symmetries; invariants; shallow water; Camassa-Holm;
Benjamin-Bono-Mahoney
\end{abstract}

\section{Introduction}

The Lie symmetry analysis plays a significant role in the study of nonlinear
differential equations. The existence of a Lie symmetry for a given
differential equation is equivalent with the existence of one-parameter
point transformation which leaves the differential equation invariant. The
later property can be used to reduce the number of independent variables on
the case of partial differential equations (PDE), or reduce the order of an
ordinary differential equation (ODE) \cite{bluman}, that is achieved thought
the Lie invariants. In addition, Lie symmetries can been used for the
determination of conservation laws. One the most well-know applications of
the latter are the two theorems of E. Noether \cite{bluman}. However, there
are also alternative methods to determine the conservation laws by using the
Lie point symmetries without imposing a Lagrange function, some of these
alternative approaches are described in \cite{bluman,ref2,ref3,ref4,ref5}
and references therein.

The are many applications of the Lie symmetries on the analysis of
differential equations, for the determination of exact solutions, to
determine conservation laws, study the integrability of dynamical systems or
classify algebraic equivalent systems \cite
{rep1,rep2,rep3,rep4,rep5,rep6,rep6a,rep6b}. Integrability is a very
important property of dynamical systems, hence it worth to investigate if a
given dynamical system is integrable \cite
{rep7,rep8,rep9,rep10,rep11,rep12,rep13,rep14}.

An alternative approach for the study of the integrability of nonlinear
differential equations is the singularity analysis. In contrary with the
symmetry analysis, singularity analysis is based on the existence of a pole
for the differential equation. The first major result of the singularity
analysis is the determination of the third integrable case of Euler's
equations for a spinning by Kowalevskaya \cite{Kowalevski88}. Since then,
the have been many contributions of the singularity analysis, mainly by the
French school led by Painlev\'{e} \cite{pp1,pp2,pp3} and many others \cite
{b1,b2,b3,b4,b5}. Nowadays, the application of singularity analysis is
summarized in the ARS algorithm \cite{ars1,ars2,ars3} which has made the
singularity analysis a routine tool for the practising applied
mathematicians.

Singularity analysis and symmetry analysis have been applied in a wide range
of differential equations arising from all areas of applied mathematics, for
instance see \cite{c01,c02,c04,c06,c07,c08,c10,c11,c12,c13,c14,c15,c16} and
references therein.\ The two methods are supplementary, on the study of
integrability of differential equations. Usually, the symmetry method is
applied to reduce the given differential equation into an algebraic
equation, or into another well-known integrable differential equation. On
the other hand, for differential equation which posses the Painlev\'{e}
property, i.e. it pass the singularity test, its solution is written in
terms of Laurent expansions, a recent comparison of the two methods is
presented in \cite{leachan}.

In this work, we study the integrability of generalized Camassa-Holm (CH)
and Benjamin-Bono-Mahoney (BBM) equations \cite{ch1,ch2} by using the Lie
point symmetries and the singularity analysis. These two equations describe
shallow-water phenomena.

The Camassa-Holm equation is a well-known integrable equation. It was
originally discovered by Fuchssteiner et al. in \cite{focas}, however become
popular a decade later by the study of Camassa and Holm where they proved
the existence of peaked solutions, also known as peakons. On the other hand,
BBM equation also known as regularized long-wave equation discovered in \cite
{bb01} and it is an extension of the KdV equation. The two equations are
related, in the sense they have a common operator and partial common
Hamiltonian structure. The plan of the paper is as follows.

In Section \ref{sec2} we present the basic elements on the mathematical
tools of our consideration, that is, the Lie point symmetries and the
singularity analysis. Our main analysis is included in Sections \ref{sec3}
and \ref{sec4} where we study the existence of similarity solutions for the
generalized CH and BBM equations, as also we prove the integrability of
these two equations by using the singularity analysis. Finally, we discuss
our results and draw our conclusions in Section \ref{sec5}.

\section{Preliminaries}

\label{sec2}

In this section we briefly discuss the application of Lie's theory on
differential equations as also the main steps of the singularity analysis.

\subsection{Lie symmetries}

Consider the vector field
\begin{equation}
X=\xi ^{i}\left( x^{k},u\right) \partial _{i}+\eta \left( x^{k},u\right)
\partial _{u},  \label{go.10}
\end{equation}
to be the generator of the local infinitesimal one-parameter point
transformation, \
\begin{align}
\bar{x}^{k}& =x^{k}+\varepsilon \xi ^{i}\left( x^{k},u\right) , \\
\bar{\eta}& =\eta +\varepsilon \eta \left( x^{k},u\right) .
\end{align}

Then $X$ is called a Lie symmetry for the differential equation, $\mathbf{H}
\left( y^{i},u,u_{i},u_{ij},...,u_{i_{1}i_{2}...i_{n}}\right) $, if there
exists a function $\lambda $ such that the following condition to hold
\begin{equation}
X^{\left[ n\right] }H=\lambda H  \label{go.11}
\end{equation}
where $X^{\left[ n\right] }$ is called the second prolongation/extension in
the jet-space and is defined as
\begin{equation}
X^{\left[ n\right] }=X+\left( D_{i}\eta -u_{,k}D_{i}\xi ^{k}\right) \partial
_{u_{i}}+\left( D_{i}\eta _{j}^{\left[ i\right] }-u_{jk}D_{i}\xi ^{k}\right)
\partial _{u_{ij}}+...+\left( D_{i}\eta _{i_{1}i_{2}...i_{n-1}}^{\left[ i
\right] }-u_{i_{1}i_{2}...k}D_{i_{n}}\xi ^{k}\right) \partial
_{u_{i_{1}i_{2}...i_{n}}}.  \label{go.13}
\end{equation}

The novelty of Lie symmetries is that they can be used to determine
similarity transformations, i.e. differential transformations where the
number of independent variables is reduced \cite{bluman}. The similarity
transformation is calculated with the use of the associated Lagrange's
system,
\begin{equation}
\frac{dx^{i}}{\xi ^{i}}=\frac{du}{u}=\frac{du_{i}}{u_{\left[ i\right] }}=...=
\frac{du_{ij..i_{n}}}{u_{\left[ ij...i_{n}\right] }}.
\end{equation}

The similarity transformation in the case of PDEs is used to reduce the
number of indepedent variables. The solutions derived by the application of
Lie invariants are called similarity solutions.

\subsection{Singularity analysis}

The modern treatment of the singularity analysis is summarized in the ARS
algorithm, established by Ablowitz, Ramani and Segur in \cite{ars1,ars2,ars3}
. There are three basic steps which are summarized as follows: (a) determine
the leading-order term which describes the behaviour of the solution near
the singularity, (b) find the position of the resonances \ which shows the
existence and the position of the integration constants and (c) write a
Laurent expansion with leading-order term determined in step (a) and perform
the consistency test. More details on the ARS algorithm as also on the
conditions which should hold at every step we refer the reader in n the
review of Ramani et al. \cite{buntis}, where illustrated applications are
presented.

It is important to mention that when a differential equation passes the
conditions and requirement of the ARS algorithm we can infer that the given
differential equation is algebraically integrable.

\section{Generalized Camassa-Holm equation}

\label{sec3}

We work with the generalized CH equation defined in \cite{ch1,ch2}
\begin{equation}
u_{t}-u_{xxt}+\frac{\left( k+2\right) \left( k+1\right) }{2}
u^{k}u_{x}=\left( \frac{k}{2}u^{k-1}u_{x}^{2}+u^{k}u_{xx}\right) _{x}~,~
\label{ch.01}
\end{equation}
where $k\geq 1$ is a positive integer number, while when $k=1$ CH equation
is recovered. The Lie symmetry analysis for the CH equation presented before
in \cite{c04}. It was found that the CH is invariant under a three
dimensional Lie algebra.

For the generalized CH equation (\ref{ch.01}) the application of Lie's
theory provides us that the admitted Lie point symmetries are three, and
more specifically they are
\begin{equation*}
X_{1}=\partial _{t}~,~X_{2}=\partial _{x}~\text{and }X_{3}=t\partial
_{t}-ku\partial _{u}.
\end{equation*}
The commutators and the adjoint representation of the Lie point symmetries
are presented in Tables \ref{tab1} and \ref{tab2} respectively.

The results presented in Tables \ref{tab1} and \ref{tab2} can be used to
classify the admitted Lie algebra as also to determine the one-dimensional
optimal systems \cite{olverb}. A necessary analysis to perform a complete
classification of the similarity solutions.

As far as the admitted Lie algebra is concerned, from Table \ref{tab1} it is
find to be the $\left\{ 2A_{1}\otimes _{s}A_{1}\right\} $ in the
Morozov-Mubarakzyanov Classification Scheme \cite
{Morozov58a,Mubarakzyanov63a,Mubarakzyanov63b,Mubarakzyanov63c}.

In order to find the one-dimensional optimal systems we consider the generic
symmetry vector
\begin{equation}
\mathbf{X}=a_{1}X_{1}+a_{2}X_{2}+a_{3}X_{3},  \label{ad.02}
\end{equation}
from where we find the equivalent symmetry by considering the adjoint
representation. We remark that the adjoint action admits two invariant
functions the $\phi _{1}\left( a_{i}\right) =a_{3}$ and $\phi _{2}\left(
a_{i}\right) =a_{3}$ which are necessary to simplify the calculations on the
derivation of the one-dimensional systems. More specifically there are four
possible cases, $\left\{ \phi _{1}\phi _{2}\neq 0\right\} $,$~\left\{ \phi
_{1}\neq 0~,~\phi _{2}=0\right\} ~,~\left\{ \phi _{1}=0~,~\phi _{2}\neq
0\right\} $ and $\left\{ \phi _{1}=0~,~\phi _{2}=0\right\} $.

Consequently, with the use of the invariant functions $\phi _{1}$ and $\phi
_{2}$ and Table \ref{tab2} we find that the possible one-dimensional optimal
systems are
\begin{equation}
X_{1},~X_{2}~,~X_{3}~,~cX_{1}+X_{2}~\text{and }X_{2}+\alpha C_{3}.
\label{ad.03}
\end{equation}
We proceed with the application of the latter one-dimensional system in
order to reduce the PDE (\ref{ch.01}) into an ODE.

\begin{table}[tbp] \centering

\caption{Commutators of the admitted Lie point symmetries by the
differential equation (\ref{ch.01})}
\begin{tabular}{cccc}
\hline\hline
$\left[ ,\right] $ & $\mathbf{X}_{1}$ & $\mathbf{X}_{2}$ & $\mathbf{X}_{3}$
\\ \hline
$\mathbf{X}_{1}$ & $0$ & $0$ & $kX_{1}$ \\
$\mathbf{X}_{2}$ & $0$ & $0$ & $0$ \\
$\mathbf{X}_{3}$ & $-kX_{1}$ & $0$ & $0$ \\ \hline\hline
\end{tabular}
\label{tab1}

\end{table}

\begin{table}[tbp] \centering

\caption{Adjoint representation for the Lie point symmetries of the
differential equation (\ref{ch.01})}
\begin{tabular}{cccc}
\hline\hline
$Ad\left( \exp \left( \varepsilon X_{i}\right) \right) X_{j}$ & $\mathbf{X}
_{1}$ & $\mathbf{X}_{2}$ & $\mathbf{X}_{3}$ \\ \hline
$\mathbf{X}_{1}$ & $X_{1}$ & $X_{2}$ & $X_{3}-k\varepsilon X_{1}$ \\
$\mathbf{X}_{2}$ & $X_{1}$ & $X_{2}$ & $X_{1}$ \\
$\mathbf{X}_{3}$ & $e^{\varepsilon k}X_{1}$ & $X_{2}$ & $X_{3}$ \\
\hline\hline
\end{tabular}
\label{tab2}

\end{table}

\subsection{Analytic solutions}

In this section we proceed with the application of the\ Lie point symmetries
to the nonlinear generalized CH equation. In order to solve the reduced
equation we apply the Lie point symmetries and when it is no possible to
proceed the reduction process, we consider the singularity analysis by
applying the ARS algorithm.

\subsubsection{Reduction with $X_{1}:$ Static solution}

The application of the Lie point symmetry vector $X_{1}$ indicates that the
solution $u$ is static, i.e. $u=U\left( x\right) $ where now function $
U\left( x\right) $ satisfies the third-order ODE
\begin{equation}
\frac{\left( k+2\right) \left( k+1\right) }{2}U^{k}U_{x}-\left( \frac{k}{2}
U^{k-1}U_{x}^{2}+U^{k}U_{xx}\right) _{x}=0~.  \label{ch.04}
\end{equation}
The latter equation can be easily integrated, and be written in the
equivalent form
\begin{equation}
U_{xx}+\frac{k}{2U}U_{x}^{2}-\frac{U_{0}}{U^{k}}+\frac{\left( k+2\right) }{2}
U=0,  \label{ch.05}
\end{equation}
where $U_{0}$ is a constant of integration. Equation (\ref{ch.05}) admits
the following conservation law
\begin{equation}
\frac{1}{2}U^{k}\left( U_{x}\right) ^{2}-U_{0}U-\frac{U^{K+2}}{2}=U_{1},
\label{ch.06}
\end{equation}
in which $U_{1}$ is a second constant of integration. Equation (\ref{ch.06})
can be integrated by quadratures.

Conservation law (\ref{ch.06}) is nothing else than the Hamiltonian function
of the second-order ODE (\ref{ch.05}). Before we proceed with another
reduction let us now apply the singularity analysis to determine the
analytic solution of equation (\ref{ch.04}).

\paragraph{Singularity analysis}

We substitute $U\left( x\right) =U_{0}\chi ^{p},~\chi =x-x_{0},$ in (\ref
{ch.04}) and we find the polynomial expression
\begin{equation}
\left( k+1\right) \left( k+2\right) \chi ^{p\left( k+1\right) -1}+\left(
p\left( k+1\right) -2\right) \left( p\left( k+2\right) -2\right) \chi
^{p\left( k+1\right) -3}=0,  \label{ch.07}
\end{equation}
Hence we can infer that the only possible leading-order behaviour to the
terms with $x^{p\left( k+1\right) -3}$, where from the requirement
\begin{equation}
\left( p\left( k+1\right) -2\right) \left( p\left( k+2\right) -2\right) =0,
\label{ch.08}
\end{equation}
provides
\begin{equation}
p_{1}=\frac{2}{k+2}~\text{or }p_{2}=\frac{2}{k+1},  \label{ch.09}
\end{equation}
while constant $U_{0}$ is undetermined.

Consider now the leading order term $p_{1}$. In order to find the resonances
we replace $U\left( x\right) =U_{0}\chi ^{p_{1}}+\mu \chi ^{p_{1}+s}$ in (
\ref{ch.09}) and we linearize around the $\mu =0$. From the linear terms of $
\mu $, the coefficient of the leading order terms $\chi ^{-\frac{2}{2+k}+s}$
are $s\left( s+1\right) \left( s\left( k+2\right) -2\right) $, where the
requirement the latter expression to be zero provides the three resonances
\begin{equation}
s_{1}=-1~,~s_{2}=0\text{ and }s_{3}=\frac{2}{k+2}.
\end{equation}
We remark that because $k$ is always a positive integer number, $p_{1}$ and $
s_{3}$ are always rational numbers. Resonance $s_{1}$ indicates that the
singularity is movable, the position of the singularity is one of the three
integration constants. Resonances $s_{2}$ shows that the coefficient
constant of the leading-order term should be arbitrary since it is also one
of the integration constants of the problem. The third constant it is given
at the position of the resonance $s_{3}$ and depends on the value $k$.
Moreover, because all the resonances are positive the solution will be given
by a Right Painlev\'{e} Series. In order to complete the ARS algorithm we
should perform the consistency test. For that we select a special value of $
k $.

We select $k=2$, and we consider the right Laurent expansion
\begin{equation}
U\left( x\right) =U_{0}\chi ^{\frac{1}{2}}+U_{1}\chi +U_{2}\chi ^{\frac{3}{2}
}+\dsum\limits_{i=3}^{\infty }U_{i}\chi ^{\frac{1}{2}+\frac{i}{2}},
\label{ch.10}
\end{equation}
we find that $U_{1}$ is the third integration constant while the first
coefficient constants are
\begin{equation}
U_{2}\left( U_{0},U_{1}\right) =-\frac{7}{8}\frac{\left( U_{1}\right) ^{2}}{
U_{0}},~U_{3}=\frac{5}{4}\frac{\left( U_{1}\right) ^{3}}{\left( U_{0}\right)
^{2}}~,~U_{4}=-\frac{273}{128}\frac{\left( U_{1}\right) ^{4}}{\left(
U_{0}\right) ^{3}}+\frac{U_{0}}{3}~,~...;
\end{equation}
Hence, the consistency test is satisfied and expression (\ref{ch.10}) is one
solution of the third-order ODE (\ref{ch.04}).

We work similar and for the second leading-order behaviour $p_{2}=\frac{2}{
k+1}$, the resonances are derived to be
\begin{equation}
s_{1}=-1~,~s_{2}=0~,~s_{3}=-\frac{2}{k+1},
\end{equation}
from where we infer that the solution is given by a Mixed Painlev\'{e}
Series. However, we perform the consistency test for various values of the
integer number $k$, and we conclude that for this leading-order behaviour
the differential equation does not pass the singularity test.

In order to understand better why only the leading-order behaviour $p_{1}$
passes the singularity test, let us perform the same analysis for the
second-order ODE (\ref{ch.05}). By replacing in $U\left( x\right) =U_{0}\chi
^{p}$ in (\ref{ch.05}) we find that the unique leading-order behaviour is
that with $p=\frac{2}{k+2}$ with arbitrary $U_{0}.$ The two resonances now
are calculated to be $s_{1}=-1$ and $s_{2}=0$, from where we can infer that
the solution is given by a Right Painlev\'{e} series and the two integration
constants is the $U_{0}$ and the position of the singularity. In that case,
since we know the two integration constants it is not necessary to perform
the consistency test.

\subsubsection{Reduction with $X_{2}$: Stationary solution}

Reduction with the vector field $X_{2}$ provides the stationary solution $
u\left( t,x\right) =U\left( t\right) $, where $U_{t}=0$, that is $u\left(
t,x\right) =u_{0}$. This is the trivial solution.

\subsubsection{Reduction with $X_{3}$: Scaling solution I}

From the symmetry vector $X_{3}$ we derive the Lie invariants
\begin{equation}
u=U\left( x\right) t^{-\frac{1}{k}}~,~x  \label{ch.12}
\end{equation}
hence, by replacing in (\ref{ch.01}) we end up with the following
third-order ODE
\begin{equation}
2kU^{k}U_{xxx}-2\left( 1-2k^{2}U^{k-1}U_{x}\right) U_{xx}+k\left( \left(
1-k\right) kU^{k-2}\left( U_{x}\right) ^{2}-\left( k+2\right) \left(
k+1\right) U^{k}\right) U_{x}+2U=0,  \label{ch.13}
\end{equation}
The latter equation admits only one Lie point symmetry, the vector field $
X_{2}.$ The latter vector field can be used to reduce equation (\ref{ch.13})
into a second-order nonautonomous ODE, with no symmetries. Hence, we proceed
with the application of the singularity analysis for equation (\ref{ch.13}).

We replace $U\left( x\right) =U_{0}\chi ^{p}$ in (\ref{ch.13}) where we find
the following expression
\begin{equation}
-2U_{0}\chi ^{p}+2U_{0}p\left( p-1\right) \chi ^{p-2}+U_{0}^{k+1}k\left(
k+2\right) \left( k+1\right) \chi ^{p\left( k+1\right)
-1}-U_{0}^{k+1}pk\left( p\left( k+2\right) -2\right) \left( p\left(
k+1\right) -2\right) \chi ^{\left( p\left( k+1\right) -3\right) }=0.
\end{equation}
From the latter term we find that the only possible leading terms with $k$
positive integer number are $p-2=\left( p\left( k+1\right) -3\right) ~$from
where we find that $p=\frac{1}{k}$ while $U_{0}$ is given by the following
expression
\begin{equation}
U_{0}^{-k}=2\left( 2-k\right) ,
\end{equation}
from where we infer that there is a leading-order behaviour only for $k\neq
2 $.

The resonances are calculated to be
\begin{equation}
s_{1}=-1~,~s_{2}=\frac{k-1}{k}~,~s_{2}=\frac{k-2}{2k},
\end{equation}
from where we can infer that for $k>2$, the solution is given by a Right
Painlev\'{e} Series. We perform the consistency test by choosing $k=3$.
Hence the Laurent expansion is written as
\begin{equation}
U\left( \chi \right) =U_{0}x^{\frac{1}{3}}+U_{1}x^{\frac{1}{2}
}+\dsum\limits_{i=2}^{\infty }U_{i}\chi ^{\frac{1+2i}{3}},
\end{equation}
where $U_{1}$ and $U_{4}$ are two integration constants of the solution
while the rest coefficient constants are $U_{i}=U_{i}\left(
U_{1},U_{4}\right) $.

\subsubsection{Reduction with $cX_{1}+X_{2}$: Travel-wave solution}

The travel-wave similarity solution is determined by the application of the
Lie invariants of the symmetry vector $cX_{1}+X_{2}$ where $c^{-1}$ is the
travel-wave speed. The invariant functions for that vector field are
determined to be
\begin{equation}
u\left( t,x\right) =U\left( \xi \right) ~,~\xi =x-c^{-1}t,
\end{equation}
where $U\left( \xi \right) $ satisfies the following third-order ODE
\begin{equation}
2\left( 1-cU^{k}\right) U_{\xi \xi \xi }-4ckU^{k-1}U_{\xi }U_{\xi \xi
}-\left( c\left( k-1\right) U^{k-1}U_{\xi \xi }+2-c\left( k+2\right) \left(
k+1\right) U^{k}\right) U_{\xi }=0.  \label{ch.14a}
\end{equation}

Equation (\ref{ch.14a}) is autonomous and admit only one symmetry vector the
$\partial _{\xi }$. It can easily integrated as follows
\begin{equation}
2\left( 2-cU^{k}\right) U_{\xi \xi }-c\left( 2U^{k-2}-U^{k-1}\right) \left(
U_{\xi }\right) ^{2}+c\left( k+2\right) U^{k-3}-2U+U_{0}=0,  \label{ch.14}
\end{equation}
which can be solved by quadratures.

Let us now apply the singularity analysis to write the analytic solution of
equation in (\ref{ch.14}) by using Laurent expansions. We apply the ARS
algorithm and we find the leading order term $U\left( \xi \right)
=U_{0}\left( \xi -\xi _{0}\right) ^{p}$ with $p=\frac{2}{k+2}$ and $U_{0}$
arbitrary. The resonances are calculated to be $s_{1}=0$ and $s_{2}=0$,
which means that the solution is given by a Right Painlev\'{e} Series with
integration constants the position of the singularity $\xi _{0}$ and the
coefficient constant of the leading order term $U_{0}$. The step of the
Painlev\'{e} Series \ depends on the value of $k$, for instance for $k=2$, \
$p=\frac{1}{2}$ and the step is $\frac{1}{2}$, while for $k=3,~p=\frac{2}{5}$
and the step is $\frac{1}{5}$.

\subsubsection{Reduction with $X_{2}+\protect\alpha X_{3}$: Scaling solution
II}

We complete our analysis by determine the similarity solution given by the
symmetry vector $X_{2}+\alpha X_{3}$. The that specific symmetry the Lie
invariants are calculated
\begin{equation}
u\left( t,x\right) =U\left( \xi \right) t^{-\frac{1}{k}}~,~\xi =x+\frac{1}{
\alpha k}\ln t.
\end{equation}

Therefore, by selecting $\xi $ to be the new independent variable and $
U\left( \xi \right) $ the new dependent variable we end up with the
third-order ODE
\begin{equation}
2\left( 1+\alpha k\right) U_{\xi \xi \xi }-2\alpha \left(
1-2k^{2}U^{k-1}U_{\xi }\right) U_{\xi \xi }+\left( a\left( k-1\right)
U^{k-2}\left( U_{\xi }\right) ^{2}-2-\alpha kU^{k}k\left( k+2\right) \left(
k+1\right) \right) U_{\xi }+2\alpha U=0.  \label{ch.16}
\end{equation}

The latter equation is autonomous and admit only one point symmetry, the
vector field $\partial _{\xi }$, which can be used to reduce by one the
order of the ODE. The resulting second-order ODE has no symmetries. Hence,
the singularity analysis is applied to study the integrability of (\ref
{ch.16}).

In order to perform the singularity analysis we do the change of variable $
V=U^{-1}$. Hence by replacing $V\left( \xi \right) =V_{0}\left( \xi -\xi
_{0}\right) ^{p}$ in (\ref{ch.16}) we find the leading-order terms
\begin{equation}
p_{1}=-1\text{ and }p_{2}=-2~~\text{for }k>1,
\end{equation}
while $V_{0}$ is arbitrary.

The resonances are calculated to be

\begin{eqnarray}
p_{1} &:&s_{1}=-1~,~s_{2}=0\text{ and }s_{3}=1;  \label{ch.17} \\
p_{2} &:&s_{1}=-1~,~s_{2}=0~\text{and}~s_{3}=-2.  \label{ch.18}
\end{eqnarray}
We apply the consistency test where we find that only only the leading-order
term $p_{1}$ provides a solution, which is given by the following Right
Painlev\'{e} Series
\begin{equation*}
V\left( \xi \right) =V_{0}\left( \xi -\xi _{0}\right)
^{-1}+\dsum\limits_{i=1}^{\infty }V_{i}\left( \xi -\xi _{0}\right) ^{-1+i}.
\end{equation*}

\subsection{Singularity analysis}

\label{sinch}

Until now we applied the singularity analysis to study the integrability of
the ODEs which follow by the similarity reduction for the generalized CH
equation. However, it is possible to apply the singularity analysis directly
in the PDE. We follow the steps presented in \cite{b5}.

Before we proceed with the application of the ARS algorithm we make the
change of transformation $u\left( t,x\right) =v\left( t,x\right) ^{-1}$ in (
\ref{ch.01}). For the new variable we search for a singular behaviour of the
form $v\left( t,x\right) =v_{0}\left( t,x\right) \phi \left( t,x\right) ^{p}$
, where $v_{0}\left( t,x\right) $ is the coefficient function and $\phi
\left( t,x\right) ^{p}$ is the leading-order term which describe the
singularity.

The first step of the ARS algorithm provides two values of $p,$ $p_{1}=-1$
and~$p_{2}=-2,~$\ where $v_{0}\left( t,x\right) $ is arbitrary. A necessary
and sufficient condition in order these two leading-order terms to exists is
$\phi _{,t}\phi _{,x}\neq 0$. Otherwise other leading-order terms follow,
however these possible cases studied before. The resonances for these two
leading-order terms are those given in (\ref{ch.17}) and (\ref{ch.19}).

Consequently, the following two Painlev\'{e} Series should be studied for
the consistency test
\begin{equation}
v\left( t,x\right) =v_{0}\left( t,x\right) \phi \left( t,x\right)
^{-1}+\dsum\limits_{i=1}^{\infty }v_{i}\left( t,x\right) \phi \left(
t,x\right) ^{-1+i},  \label{ch.19}
\end{equation}
\begin{equation}
v\left( t,x\right) =v_{0}\left( t,x\right) \phi \left( t,x\right)
^{-2}+\dsum\limits_{i=1}^{\infty }v_{i}\left( t,x\right) \phi \left(
t,x\right) ^{-2+i}.  \label{ch.20}
\end{equation}

By replacing (\ref{ch.19}) we find that the second integration constant is $
v_{1}\left( t,x\right) $. On the other hand, the series (\ref{ch.20}) does
not pass the consistency test. We conclude that that the generalized CH
equation passes the singularity test and it is an integrable equation.

We proceed our analysis with the BBM equation.

\section{Generalized Benjamin-Bono-Mahoney equation}

\label{sec4}

The generalized BBM equation is
\begin{equation}
u_{t}-u_{xxt}+\beta u^{k}u_{x}=0,  \label{bbm.01}
\end{equation}
where $k$ is a positive integer number. Equation (\ref{bbm.01}) can been
seen as the lhs of (\ref{ch.01}) when $\beta =\frac{\left( k+2\right) \left(
k+1\right) }{2}$ and reduce to the BBM equation when $k=1$. For the case of $
k=1$ the Lie symmetry analysis for the BBM equation presented recently in
\cite{bb02a,bb02}.

We apply the Lie theory in order to determine the point transformations
which leave equation (\ref{bbm.01}) invariant. We found that the equation (
\ref{bbm.01}) admits three point symmetries which are the vector fields $
X_{1},~X_{2},~X_{3}$ presented in\ Section \ref{sec3}. Hence, the admitted
Lie algebra is the $2A_{1}\otimes _{s}A_{1}$ and there are five
one-dimensional optimal systems as presented in (\ref{ad.03}). We proceed
with the application of the Lie point symmetries for the determination of
similarity solutions.

\subsection{Analytic solutions}

For equation (\ref{bbm.01}) the application of the Lie symmetries $X_{1}$
and $X_{2}$ provide the trivial solution $u\left( t,x\right) =u_{0}$ for
both cases.

\subsubsection{Reduction with $X_{3}$: Scaling solution I}

The application of the Lie invariants which given by the symmetry vector $
X_{3}$ gives~$u\left( t,x\right) =U\left( x\right) t^{\frac{1}{k}}$ where $
U\left( x\right) $ satisfies the second-order ODE
\begin{equation}
U_{xx}+\beta kU^{k}U_{x}-U=0.  \label{bbm.02}
\end{equation}
The latter equation is autonomous and admit the point symmetry $\partial
_{x} $ which can be used to reduce equation (\ref{bbm.02}) into the
following first-order ODE
\begin{equation}
y_{z}=\beta kz^{k}y^{2}-zy^{3},
\end{equation}
where $y\left( z\right) =\left( U_{x}\right) ^{-1}$ and $z=U\left( x\right) $
.

However, equation (\ref{bbm.02}) can be easily solved analytical by using
the singularity analysis. Indeed from the ARS algorithm we find the
leading-order behaviour
\begin{equation}
U\left( \chi \right) =\left( \frac{k+1}{\beta k^{2}}\right) ^{\frac{1}{k}
}x^{-\frac{1}{k}},
\end{equation}
with resonances
\begin{equation}
s_{1}=-1\text{ and~}s_{2}=\frac{1+k}{k}.
\end{equation}

In order to perform the consistency test we have to select specific value
for the parameter $k$. Indeed for $k=2$ we write the Laurent expansion
\begin{equation}
U\left( \chi \right) =\left( \frac{3}{4\beta }\right) ^{\frac{1}{2}}x^{-
\frac{1}{2}}+\dsum\limits_{i=1}^{\infty }U_{i}x^{-\frac{1+i}{2}},
\end{equation}
and by replacing in (\ref{bbm.02}) we find that
\begin{equation}
U_{1}=0~,~U_{2}=0~,~U_{4}=-\sqrt{\frac{1}{3\beta }}~,~U_{5}=0,~U_{6}=-\frac{
\sqrt{3\beta }}{4}\left( U_{3}\right) ^{2}~,~....
\end{equation}
where $U_{3}$ is the second integration constant. We conclude that that the
equation (\ref{bbm.02}) passes the Painlev\'{e} test.

\subsubsection{Reduction with $cX_{1}+X_{2}$: Travel-wave solution}

The travel-wave solution of the generalized BBM equation is $u=U\left( \xi
\right) $, where $\xi =x-c^{-1}t$ and $U\left( \xi \right) $ satisfies the
differential equation
\begin{equation}
U_{\xi \xi \xi }+\left( c\beta U^{k}-1\right) U_{\xi }=0.
\end{equation}
The latter equation can be integrated easily
\begin{equation}
U_{\xi \xi }+\left( \frac{c\beta }{k+1}U^{k+1}-U\right) +U_{0}=0,
\label{bhm.01}
\end{equation}
that is
\begin{equation}
\frac{1}{2}\left( U_{\xi }\right) ^{2}+\left( \frac{c\beta }{\left(
k+1\right) \left( k+2\right) }U^{k+2}-\frac{U^{2}}{2}\right) +U_{0}U-U_{1}=0,
\end{equation}
where $U_{0},~U_{1}$ are two integration constants. The latter differential
equation can be solved easily by quadratures.

As far as the singularity analysis is concerned for equation (\ref{bhm.01}),
the ARS algorithm provides the leading-order behaviour
\begin{equation}
U\left( \xi \right) =U_{0}\left( \xi -\xi _{0}\right) ^{-\frac{2}{k}
}~,~~U_{0}^{k}=-2\frac{\left( k+1\right) \left( k+2\right) }{\beta ck^{2}},
\end{equation}
with resonances
\begin{equation*}
s_{1}=-1\text{ and }s_{2}=\frac{2\left( k+2\right) }{k}.
\end{equation*}
The consistency test has been applied for various values of the positive
integer $k$, and we can infer that equation (\ref{bhm.01}) is integrable
according to the singularity analysis.

\subsubsection{Reduction with $X_{2}+\protect\alpha X_{3}$: Scaling solution
II}

From the Lie symmetry $X_{2}+\alpha X_{3}$ we find the similarity reduction $
u\left( t,x\right) =U\left( \xi \right) t^{-\frac{1}{k}}~,~\xi =x+\frac{1}{
\alpha k}\ln t$ where $U\left( \xi \right) $ is a solution of the following
differential equation
\begin{equation}
U_{\xi \xi \xi }-\alpha U_{\xi \xi }-\left( \alpha \beta U^{k}+1\right)
U_{\xi }+\alpha U=0.  \label{bbm.03}
\end{equation}
Equation\ (\ref{bbm.03}) can be reduced to the following second-order ODE by
use of the point symmetry vector $\partial _{\xi }$,
\begin{equation}
z^{2}y_{zz}+z\left( y_{z}\right) ^{2}-\alpha zy_{z}-\left( \alpha \beta
z^{k}+1\right) y+\alpha z=0,  \label{bbm.04}
\end{equation}
where $z=U\left( x\right) $ and $y\left( z\right) =U_{x}$.

We apply the ARS algorithm for equation (\ref{bbm.03}) and we find that it
passes the singularity test for the leading order behaviour
\begin{equation}
U\left( \xi \right) =U_{0}\left( \xi -\xi _{0}\right) ^{-\frac{2}{k}
}~,~U_{0}^{k}=2\frac{\left( k+2\right) \left( k+1\right) }{\alpha \beta k^{2}
},
\end{equation}
with resonances
\begin{equation}
s_{1}=-1~,~s_{2}=\frac{2\left( k+1\right) }{k}~,~s_{3}=\frac{2\left(
k+2\right) }{k}.  \label{bbm.05}
\end{equation}

\subsection{Singularity analysis}

We complete our analysis by applying the singularity test in the generalized
BBM equation in a similar way as we did in Section \ref{sinch} for the
generalized CH equation. Indeed we find the leading order term
\begin{equation}
u\left( t,x\right) =v_{0}\left( t,x\right) \phi \left( t,x\right) ^{-\frac{2
}{k}}~,~\left( v_{0}\left( t,x\right) \right) ^{k}=2\frac{\left( k+2\right)
\left( k+1\right) }{\alpha \beta k^{2}}\phi _{t}\phi _{x},  \label{bbm.06}
\end{equation}
and resonances those given in (\ref{bbm.05}). We performed the consistency
test and we infer that the generalized BBM equation passes the singularity
test for any value of the positive integer parameter $k$.

\section{Conclusion}

\label{sec5}

In this work we studied the existence of similarity solutions of the
generalized CH and generalized\ BBM equation. The approached that we used is
that of the Lie point symmetries. We determined the admitted invariant point
transformations for the differential equations of our consideration and we
determined the one-dimensional optimal systems by using the adjoint
representation of the admitted Lie algebra. The two differential equations
of our consideration are invariant under the same Lie symmetry vectors which
form the same Lie algebra with the CH and the BBM equations.

For each of the equations we perform five different similarity reductions
where the PDEs are reduced to third-order ODEs. The integrability of the
resulting equations is studied by using symmetries and/or the singularity
analysis. In the case of the generalized CH equation most of the reduced
ODEs can not be solved by using Lie symmetries, hence the application of the
singularity analysis was necessary to determine the analytic solutions of
the reduced equations.

Finally, we study the integrability of the PDEs of our consideration by
applying the singularity analysis directly on the PDEs and not on the
reduced equations. From the latter analysis we found that the generalized CH
and BBM equations pass the singularity analysis and their solutions are
given by Right Painlev\'{e} Series.

This work contribute to the subject of the integrability of generalized
equations describe shallow-water phenomena. The physical implication of the
new analytic solutions will be presented in a future communication.


TITLE: How much energy does proton - carbon 12 fusion produce?
QUESTION [3 upvotes]: Page 25 of this document from the California Institute of Technology says that proton-carbon 12 fusion releases 7.54 MeV, while Wikipedia says it releases 1.95 MeV. Which one is correct?

REPLY [3 votes]: OK, you are referring to the fusion reaction
$^{12}$C + $^{1}$H $\rightarrow$ $^{13}$N + $\gamma$.
Let's neglect the $\gamma$ and insert the atomic masses
$12\cdot u + 1.00782503223\cdot u = 13.005738609\cdot u + x$,
where $x$ is an upper boundary for the released energy (since we neglected the $\gamma$). The data for the atomic masses is from the NIST website and from wikipedia.
Solving that equation yields
$x=0.00208642323\cdot u$
which corresponds to ($1\cdot u$ corresponds approximately to $931.5$ MeV)
$x\approx 1.94 $ MeV.
Therefore, the wikipedia value seems to be right.


TITLE: Littlewood's Inequality
QUESTION [3 upvotes]: Working on an exercise from Shorack's Probability for Statisticians, Ex 3.4.3 (Littlewood's inequality).  Can't seem to find any material for this.  Help is appreciated.

Prove:Exercise 4.3 (Littlewood's inequality) Let $m_r \equiv \mathbb{E}|X|^r$ denote the $r$th absolute moment.  Then for $r \ge s \ge t \ge 0$, we have $m^{r-s}_t m^{s-t}_r \ge m^{r-t}_s$.

Thank you very much!

REPLY [2 votes]: It's a consequence of Hölder's inequality: we can write $s=\alpha t+(1-\alpha)r$, where $\alpha\in [0,1]$ (you can deal with the case $\alpha\in \{0,1\}$ by hand). We have
$$m_s^{r-t}=(E|X|^s)^{r-t}=(E[|X|^{t\alpha}|X|^{r(1-\alpha)}])^{r-t}.$$
We apply Hölder's inequality to $f=|X|^t$, $g=|X|^r$ and $p=\frac 1{\alpha}>1$, $q=\frac 1{1-\alpha}$. We get
$$m_s^{r-t}\leq (E|X|^t)^{(r-t)\alpha}(E|X|^r)^{(r-t)(1-\alpha)}=m_t^{(r-t)\alpha}m_r^{(r-t)(1-\alpha)}.$$
But $\alpha$ can be computed to see the exponents are what we expected.


TITLE: Describe a neighborhood of a given interior point
QUESTION [1 upvotes]: Before I ask the question, I would like to define some terms.
A point $z_0$ is said to be an interior point of a subset $S$ of the complex plane if there exists some neighborhood of $z_0$ that lies entirely within $S$. If every point $z$ of a set $S$ is an interior point, then $S$ is said to be an open set. For example, the inequality $\Re(z) > 1$ defines a right half-plane, which is an open set. All complex numbers $z = x + iy$ for which $x > 1$ are in this set.
But does not it depend on the value of $y$? Suppose $z_0=1.1+2i$. How can we describe a neighborhood in this case? Clearly the length between $z_0$ and some points should be less than some value, but what is this value? Please help me.

REPLY [2 votes]: One kind of neighborhood is an open disk around $z_0$.  In your example, $z_0$ is $0.1$ from the line $Re(z)=1$, so any disk of radius less than $0.1$ will work.  It doesn't depend on the imaginary part ($y$) because the division line doesn't depend on $y$.
If the set were the points for which $2Re(z) \gt Im(z)$, it is again an open set.  Your $z_0=1.1+2i$ is in this set too.  But the distance from the dividing line depends on $y$ in this case because the line is sloping.


TITLE: Electron Muon scattering in the CM frame, question about the energy
QUESTION [1 upvotes]: I calculated the spin-averaged amplitude for the $e^{-}\mu^{-} \rightarrow e^{-}\mu^{-}$ scattering in the CM frame in the high energy regime($m_{e}$,$m_{\mu} \rightarrow 0$) following the hints provided in Griffiths, and found the correct expression:
$$\left<|M|^2\right> = 2{g_e}^2 \left(\frac{1+cos^4(\theta/2)}{sin^4(\theta/2)}\right)$$
Plugging this in the expression for the differential cross-section of two bodys in the CM frame($1+2\rightarrow 1+2$), ie
$$\frac{d \sigma}{d\Omega} = \left(\frac{\hbar c}{8 \pi}\right)^2 \frac{S\left<|M|^2\right>}{(E_1+E_1)^2} \frac{|\pmb{p_i}|}{|\pmb{p_f}|}$$
I found the expression Griffths gives as the answer:
$$\frac{d \sigma}{d\Omega} = \left(\frac{\hbar c}{8 \pi}\right)^2 \frac{{g_e}^4}{2E^2} \left(\frac{1+cos^4(\theta/2)}{sin^4(\theta/2)}\right)$$
where $E$ is the energy of the electron.
My question is: in order to obtain this last relation I used that $E_1=E_2$, or in other notation, $E_{e}=E_{\mu}$. This gives me the correct expression but I'm not sure why I can do that. I found a lecture about the
$e^{-}e^{+} \rightarrow \mu^{-}\mu^{+}$ scattering who says:

because of working in the center of mass frame at the massless limit:
$$E_{e^+}=E_{e^-}=E_{\mu^-}=E_{\mu^+}$$

which is similar to what I have in the scattering I calculated.
Does anyone understand why we can set $E_{\mu}=E_{e}$??

REPLY [1 votes]: Reading Rob's answer I think I've come to an understanding.
The relativistic dispersion relation states that:
$$E^2 = m^2c^4 + p^2c^2$$
So, outside the high energy regime, we have
$$E^2_{e} = m_{e}^2c^4 + p_{e}^2c^2$$
$$E^2_{\mu} = m_{\mu}^2c^4 + p_{\mu}^2c^2$$
As Rob have said, in CM the total momentum vanishes so:
$$\pmb{p_{\mu}} = - \pmb{p_{e}} \Rightarrow p_{\mu} = p_{e} = p_i$$
and from the dispersion relation, we get
$$E^2_{e} = m_{e}^2c^4 + p_i^2c^2$$
$$E^2_{\mu} = m_{\mu}^2c^4 + p_i^2c^2$$
But, in the high energy regime, we have
$$m_e, m_{\mu} \rightarrow 0$$
So
$$E^2_{e} =   p_i^2c^2$$
$$E^2_{\mu} =  p_i^2c^2$$
and because of this, we can see that
$$E_{e} =  E_{\mu}$$


TITLE: Uniqueness of classical solution (Incompressible Euler Equations)
QUESTION [1 upvotes]: I am struggling at understanding some steps of the uniqueness proof of classical solutions of the incompressible Euler equations
$$
\partial_t v(x,t)+\text{div }(v(x,t)\otimes v(x,t))+\nabla p(x,t)=0,\qquad \text{div }v(x,t)=0.
$$
Here, $v(x,t)\otimes v(x,t)$ is the matrix with the entries $v_iv_j$ and the divergence of it is taken row-wise.

Assume that v and u are two classical solutions with pressure fields $p$ and $q$, respectively and with the same initial data $v_0$. Then, $v\equiv u$.

Here is the proof I have found in this document on page 3 (which is page 13 of the pdf document):
(Here, $\nabla_{sym}v=\frac{1}{2}(\nabla v+\nabla^T v)$ is the symmetric gradient and $d^-(v)$ is the negative part of its smallest eigenvalue).

I have mainly three questions.
(1) Where does the second equality come from?
Using the equations, we get
$$
2\int_{\mathbb{R}^d}(v-u)\cdot\partial_t(v-u)\, dx=-2\int_{\mathbb{R}^d}\underbrace{(v-u)\cdot(\text{div }(v(x,t)\otimes v(x,t))-\text{div }(u(x,t)\otimes u(x,t)))}_{=:A}\, dx -2\int_{\mathbb{R}^d}(v-u)\cdot\nabla(p-q)\, dx
$$
But I do not see that
$$
A=(v-u)\cdot(v\cdot\nabla v-u\cdot\nabla u).
$$
(2) Why does the second integral in the second row disappear, i.e. why do we have
$$
2\int_{\mathbb{R}^d}(v-u)\cdot\nabla(p-q)\, dx=0?
$$
(3) Why is
$$
v\cdot\nabla v-u\cdot\nabla u=\nabla_{sym}v(v-u)
$$
and what is the idea/ theory about this symmetric gradient and this $d^-(v)$ (i.e. why do we need the symmetric gradient and $d^-(v)$, where does this come from and what does this norm $\lVert d^{-}(v)\rVert_{L_x^{\infty}}$ mean)?
The Gronwall thing is then again clear to me.
Maybe you can help me a bit, would be great!

REPLY [0 votes]: (1) Let's compute the divergence of $v\otimes v$. Using the product rule, its  $i$th component becomes
$$\sum_j \frac{\partial}{\partial x_j}(v_iv_j)
= \sum_j v_j \frac{\partial v_i}{\partial x_j} + \sum_j v_i \frac{\partial v_j}{\partial x_j}
$$
The second term on the right is $v_i \operatorname{div}v = 0$ since $v$ is divergence-free. The first term is $v\cdot \nabla v_i$. This is the $i$th component, so the whole $\operatorname{div}(v\otimes v)$ can be written as $v\cdot \nabla v$.
(2) Integrate by parts, moving $\nabla $ to the other term, where it becomes divergence, and the divergence of $v-u$ is zero:
$$\int_{\mathbb{R}^d}(v-u)\cdot\nabla(p-q)\, dx
= - \int_{\mathbb{R}^d} \operatorname{div}(v-u) \ (p-q)\, dx
=0$$
This is used so often that it's worth remembering: integration of a divergence-free field against a gradient field gives zero (under suitable boundary conditions, so that no boundary terms emerge).
(3) By the fact mentioned in (2), we have $\int u \nabla ((v-u)^2) = 0$. Expand the gradient by the chain rule:  $\int (v-u) u (\nabla v - \nabla u) = 0$.  This fact implies
$$
\int (v-u)\cdot (v\cdot \nabla v - u \cdot \nabla u)
= \int (v-u)\cdot (v\cdot \nabla v - u \cdot \nabla v)
$$
(the difference of two terms is zero). The latter can be rewritten as
$\int (v-u)\cdot \nabla v (v-u)$, which is the same as $\int (v-u)\cdot \nabla_{sym} v (v-u)$ because the antisymmetric part of the gradient contributes zero. (Every matrix can be written as symmetric+antisymmetric, and $w\cdot Aw=0$ for every vector $w$ and every antisymmetric matrix $A$).
Finally, $(v-u)\cdot \nabla_{sym}v (v-u)$ is a quadratic form with matrix $\nabla_{sym}v$. These can be bounded by the largest and smallest eigenvalues:
$\lambda_{\min}(A)|w|^2\le w\cdot Aw\le \lambda_{\max}(A)|w|^2$ for any symmetric matrix $A$. (Think of diagonalizing the form.) In our case we use the lowest eigenvalue, since there is a minus sign in front of the integral:
$$ - (v-u)\cdot \nabla_{sym}v (v-u) \le -\lambda_{\min}(\nabla_{sym}v ) |v-u|^2 $$
Lastly, we pull the eigenvalue out of the integral, using the maximal value of $-\lambda_{\min}(\nabla_{sym}v )$; so only the negative values of $\lambda_{\min}(\nabla_{sym}v )$ are of concern, and the estimate uses the maximum of those.


TITLE: Identity theorem for holomorphic functions
QUESTION [0 upvotes]: I have to check whether there exists a holomorphic function $$f:U_{1+\epsilon}(0)\rightarrow \mathbb{C}$$ for all $n\in \mathbb{N}_{\ge 2}$ such that
$$
| f^{(n)}(0)|\ge (n!)^2.
$$
I'm sure that I have to use the identity theorem, but I don't know how to start.
Any help would be great!
Thyrð

REPLY [2 votes]: I assume that $U_{1+\varepsilon}(0)$ is the open disk of radius $1+\varepsilon$ centered at $0$. According to the Cauchy integral formula,
$$ f^{(n)}(0)=\frac{n!}{2\pi i}\int_{|z|=1}\frac{f(z)}{z^{n+1}}\;dz $$
which implies that
$$ |f^{(n)}(0)|\leq n!\sup_{|z|=1}|f(z)| $$
This expression will be less than $(n!)^2$ for all sufficiently large $n$ (depending on $f$), so no such function exists.


TITLE: If $\int _c^d f(x)\text{dx}=0$ for all $a\le c<d\le b$ then $f=0a.e.$
QUESTION [1 upvotes]: Let $f:[a,b]\to \Bbb R$ be a measurable function.Then show that

If $\int _c^d f(x)\text{dx}=0$ for all $a\le c<d\le b$ then $f=0a.e.$
If $\int _a ^c f(x) \text{dx}$
for all $a\le c\le b$ then $f=0a.e.$

My try:
Let $A^{+}=\{x:f(x)\ge 0\}$ and $A^{-}=\{x:f(x)<0\}$
Let $\phi $ be a simple function such that $\phi\le f$ then  $\int _c^d f(x)\text{dx}=0\implies \int _c ^d \phi=0\implies \sum a_im(E_i)=0$ where $E_i=\{x:\phi(x)=a_i\}$
But I don't know how to show that $f=0$ a.e. from here.Please give some hints

REPLY [1 votes]: First observe that $\int_E f(x)dx=0$ for all Borel measureble sets $E\subseteq[a,b]$. To see this, let
$$\mathcal{S}=\{E\subseteq[a,b]: E\text{ Borel measurable },\int_E f(x)dx=0\}.$$
It is easy to show that $\mathcal{S}$ is a $\sigma-$algebra on $[a,b]$ which contains all sub-intervals of $[a,b]$. Since the Borel $\sigma-$algebra is the smallest $\sigma-$algebra containing intervals, the claim follows.
With this observation, note that
$$\{f(x)\neq 0\}=\left(\bigcup_{n=1}^\infty E_n\right)\cup\left(\bigcup_{n=1}^\infty F_n\right)$$
where
$$E_n=\{f(x)>\frac{1}{n}\},\quad F_n=\{f(x)<-\frac{1}{n}\}$$
Assume $f$ is not $0$ a.e., then without loss of generality at least one $E_n$ (or $F_n$) has possitive measure, whis means
$$\int_{E_n}f(x)dx\geq \frac{1}{n}m(E_n)>0$$
where $m$ denotes the Lebesgue measure. This contradicts the claim we proved at first. Hence the result follows.


text:
\section{Preliminaries and Problem Statement}

In this section, we consider a distributed optimization problem on a network of $M$ sub-systems (nodes). The sub-systems communicate according to a fixed undirected graph $G =(\mathcal{V},\mathcal{E})$. The vertex set $\mathcal{V} = \{1,2,\cdots,M\}$ represents the sub-systems and the edge set $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$ specifies pairs of sub-systems that can communicate. If $(i,j)\in \mathcal{E}$, we say that sub-systems $i$ and $j$ are neighbors, and we denote by $\mathcal{N}_i = \{j| (i,j)\in \mathcal{E}\}$ the set of the neighbors of sub-system $i$. Note that $\mathcal{N}_i$ includes $i$. The cardinality of $\mathcal{N}_i$ is denoted by $|\mathcal{N}_i|$. We use a vector $[v]_i$ to denote the local variable of subsystem $i$ and $[v]_i$ can be of different dimensions for different $i$.
The collection of these local variables is denoted as $v=[v^{T}_1,\cdots,v^{T}_M]^{T}$.
Furthermore, the concatenation of the local variable $[v]_i$ of sub-system $i$ and the variables of its neighbors $[v]_j, j\in\mathcal N_i$ is denoted by $z_{i}$.
With appropriate selection matrices $E_i$ and $F_{ji}$, the variables have the following relationship: $z_{i} = E_iv$ and $[v]_i = F_{ji}z_{j}$, $j\in\mathcal{N}_i $, which implies the relation between the local variable $[v]_i$ and the global variable $v$, i.e. $[v]_i = F_{ji}E_jv$, $j\in\mathcal{N}_i $. We consider the following distributed optimization problem:

\begin{problem}[Distributed Optimization]
\begin{align}
\min_{z,v}  & \quad \sum^{M}_{i=1} f_{i}(z_i)\\
s.t. &\quad  z_i\in \mathbb{C}_{i}, \quad z_i = E_iv, \quad i=1,2,\cdots ,M \,,
\end{align}
\label{pr:dist_opt}
\end{problem}
\noindent where $f_i$ is the local cost function for node $i$ which is assumed to be strongly convex  with a convexity modulus $\rho_{f_i}>0$, and to have a Lipschitz continuous gradient with a Lipschitz constant $L(\nabla f_i)>0$.   The constraint $\mathbb{C}_i$ is assumed to be a convex set which represents a convex local constraint on $z_{i}$, i.e. the concatenation of the variables of sub-system $i$ and the variables of its neighbors.

The above problem formulation is fairly general and can represent a large class of problems in practice. In particular  it includes the  following quadratic programming problem, which we study as a particular instance in our applications.
\begin{problem}[Distributed Quadratic Problem]
\begin{align}
\min_{z,v}  & \quad \sum^{M}_{i=1} z^{T}_iH_iz_i + h^{T}_iz_i \label{pr:quadratic}\\
s.t. &\quad  C_iz_i\leq c_i, \quad z_i = E_iv, \quad i=1,2,\cdots ,M\,, \nonumber
\end{align}
\label{pr:dist_quad_opt}
\end{problem}
\noindent where $H_i \succ 0$.  In particular, we will assume that the smallest eigenvalue of $H_i$ satisfies $\lambda_{min}(H_i) := \lambda_{min}^{(i)} > 0$.

\subsection{Local Differential Privacy}

We present definitions and properties for differential privacy.

Let $\Pp, \Pp'$ be two databases with private parameters in some space $\mathcal X$ containing information relevant in executing an algorithm.
Let $\text{adj}:\mathcal X\times\mathcal X\mapsto [0,\infty)$ denote a metric defined on $\mathcal X$ that encodes the adjacency or distance of two elements.
A mechanism or algorithm $\mathcal A$ is a mapping from $\mathcal X$ to some set denoting its output space.

\begin{definition}[Differential Privacy]. A randomized algorithm $\mathcal{A}$ is $\epsilon$-differential private if for all $\mathcal{S}\subseteq \mbox{range}(\mathcal{A})$ and for all database $\Pp, \Pp'$ satisfying $\text{adj}(\Pp,\Pp')\leq 1$, it holds that
\begin{equation}
\mbox{Pr}[\mathcal{A}(\Pp) \in \mathcal{S}] \leq e^{\epsilon}\cdot\mbox{Pr}[\mathcal{A}(\Pp') \in \mathcal{S}] \,,
\end{equation}
where the probability space is over the mechanism $\mathcal{A}$.
\label{def: Differential privacy}
\end{definition}

\noindent This definition of differential privacy is suitable for cases where one uniform level of privacy needs to across all elements in the databases.
We now consider a distributed algorithm $\mathcal A(\Pp_1,\ldots, \Pp_M)$ in a network with $M$ agents for solving an optimization problem in a collaborative way, where $\Pp_i$ denotes the private parameters of agent $i$. The outputs of the mechanism are the message exchanged between nodes in the network over the time horizon of iterations.
This mechanism induces $M$ local mechanisms $\mathcal A_1(\Pp_1,\ldots, \Pp_M),\ldots, \mathcal A_M(\Pp_1,\ldots, \Pp_M)$, each executed by one agent. The output of one local mechanism $\mathcal A_i$ is the message sent out by node $i$, i.e. $\text{range}(\mathcal A_i)\subseteq\text{range}(\mathcal A)$. It is important to realize that although one local mechanism, say $\mathcal A_i$, does not necessarily have direct access to the input/database $\Pp_j, j\neq i$ of other nodes, the output of $\mathcal A_i$ could still be affected by $\Pp_j, j\neq i$ because of the interactions among different nodes. For this reason, we explicitly write $\Pp_1,\ldots, \Pp_M$ as input to all local mechanisms.

We now let each agent~$i$ specify its own level of privacy~$\epsilon_i$.
To formalize this specification, we require a definition:
\begin{definition}[Local Differential Privacy]
Consider a (global) mechanism $\mathcal A$ for a network with $M$ nodes, and  $M$ local mechanisms $\mathcal A_i, i=1,\ldots, M$ induced by $\mathcal A$.  We say that the mechanism $\mathcal A$ is \textit{$\epsilon_i$-differentially locally private for node $i$}, if for any $\mathcal S_i\in\text{range}(\mathcal A_i)$ it satisfies that
\begin{align}
\frac{\pp{\mathcal A_i(\Pp_1,\ldots, \Pp_i,\ldots, \Pp_M)\in\mathcal S_i}}{ \pp{\mathcal A_i(\Pp_1,\ldots, \Pp_i',\ldots, \Pp_M)\in\mathcal S_i}}\leq e^{\epsilon_i} \,,
\end{align}
where $\text{adj}(\Pp_i,\Pp_i')\leq 1$. Moreover, we say that the mechanism $\mathcal A$ is \textit{$(\epsilon_1,\ldots, \epsilon_M)$-differentially private}, if  $\mathcal A$ is $\epsilon_i$-differentially locally private for all nodes, where $i=1,\ldots, M$.
\label{def:local_privacy}
\end{definition}

Figure~\ref{fig:concept} presents the concept of local differential privacy pictorially, showing the various considerations that can be taken when designing for local/customized privacy.
Firstly, one may desire to include a central node 0 that communicates with all subsystems or implement a fully distributed problem between the subsystems that does not rely on any central node.
The former will lead to better convergence properties as information spreads more easily throughout the network, the latter will benefit privacy by making it harder to collect information from across the network.
Regardless, the method allows for the privacy to be purely local, strengthening the notion developed in~\cite{han_differentially_2017}, which implements differential privacy in distributed optimization through a trusted central node, assuming a star-shaped communication structure with no agent-to-agent communication.
Secondly, subsystems may have varying levels of privacy. In  Figure~\ref{fig:concept}, subsystem 1 and 3 have a local privacy specification, while subsystems 0 and 2 do not. The systems with local differential privacy have outgoing messages perturbed by noise as indicated by the dashed arrows.
As such, the method is flexible to various forms of distributed optimization or control problems with heterogeneous privacy and control properties across its nodes/agents, extending the work in \cite{huang_differentially_2015}, which considers a similar fully distributed algorithm but specifies a uniform privacy level for all agents.
\begin{figure}[!hbt]
\centering
\includegraphics[width = 0.45\textwidth]{figures/local_privacy_v3.pdf}
\caption{A system with 4 subsystems. Arrows denote message directions in distributed optimization. Here, node 0 denotes a central node that communicates with all other nodes, which can be left out (shaded part). The other 3 nodes represent subsystems. Nodes that have a privacy specification are patterned with dots and send out messaged perturbed by noise, as indicated by dashed arrows.}
\label{fig:concept}
\end{figure}

config: arxiv


text:
\begin{document}

\title[images of  Galois representations attached  to Hilbert modular
forms]{Explicit determination of  images of  Galois representations
attached  to Hilbert modular forms}

\author{Luis Dieulefait and Mladen Dimitrov}
\date{\today}

\begin{abstract} In \cite{dim-hmv} the second author proved that the
image of the Galois representation mod $\lambda$  attached to a
Hilbert modular newform  is ``large" for
all but finitely many primes $\lambda$, if the form is not a theta
series.  In this brief note, we give an explicit
bound for this exceptional finite set of primes and determine the images in three different
examples. Our examples are of Hilbert newforms on real quadratic fields, of
parallel or non-parallel weight and of different levels.
\end{abstract}

\maketitle

\vspace{1cm}
\section{Introduction}

Serre  \cite{serre2} and Ribet \cite{ribet3,ribet6}   proved a large
image result for the  compatible family of
Galois representations attached to a classical  modular newform $f$, provided
that $f$ is not a theta series. The result is the following: let
$E$ be the number field generated by the eigenvalues of $f$ and for
every prime $\lambda$  in $E$, let $\rho_\lambda$ be the continuous
$\lambda$-adic Galois representation  attached to $f$ constructed by Deligne;
then for all but finitely many primes  $\lambda$ the image of $\rho_\lambda$
 is large, namely it contains  $\SL_2(\mathbb{Z}_\ell)$, with $\ell$ the rational
prime below $\lambda$.
  In fact, the exact value of the image for all but finitely many primes
$\lambda$ can be given
if the inner-twists  of $f$ (if any) are taken into account.
 The finite set of primes where the above result fails is usually called the
   ``exceptional set" of the family $\{ \rho_\lambda \}$.
    Explicit versions of  the result of Ribet are known and the exceptional sets
  have been explicitly computed
  for several examples of classical modular forms \cite{Di,DV}.
  We will now recall a result of \cite{dim-hmv} that is the natural generalization
  of Ribet's result to the case of Hilbert modular forms.

\medskip
Let $F$ be a totally real number field of degree $d$.
Let $k=(k_1,..,k_d)$ be an arithmetic weight ({\it i.e.} $k_i\geq 2$ are of the same parity)
and put $k_0=\max\{k_i|1\leq i\leq d\}$. Let $N$ be an integral ideal of $F$
and $\psi$   a Hecke character of $F$  of conductor dividing $N$ and  infinity
type $2-k_0$.  We consider a Hilbert modular newform $f$ over $F$ of weight $k$,
level $N$ and central character $\psi$. By a theorem of
Shimura, the Fourier coefficients $c(f,\pi)$ of $f$ ($\pi$ is a prime of $F$)
generate a number field $E$.
The  absolute Galois  group of a field $L$ is denoted  $\G_L$.

\medskip
By the work of Ohta, Carayol, Blasius-Rogawski and  Taylor
\cite{Ta}, for every prime  $\lambda$ of $E$ one  can associate
to $f$ an   absolutely irreducible,   totally odd $\lambda$-adic representation
$\rho_{\lambda}:\G_F\rightarrow \GL_2(E_\lambda)$,   unramified outside $N l$.
By a theorem of Carayol \cite{carayol}  the restriction of $\rho_{\lambda}$
to the decomposition group at a prime  $\pi$ of $F$ not dividing $\ell$ is determined
by the local Langlands correspondence for $\GL_2$. In particular, for
all  primes  $\pi$ of $F$ not dividing $N l$ we have :
$$\tr(\rho_\lambda(\Frob_\pi))=c(f,\pi) \qquad
\det(\rho_\lambda(\Frob_\pi))=\psi(\pi)\Nm(\pi),$$
where  $\Frob_\pi$ denotes a geometric Frobenius at $\pi$.

\medskip
Denote by $\F_\lambda$ the residue field of $E_\lambda$.
By taking a Galois stable lattice, we define
$\overline{\rho}_{\lambda}=\rho_{\lambda}
\mod{\lambda}:\G_F \rightarrow \GL_2(\F_\lambda)$,
whose  semi-simplification is independent of the particular choice of
a lattice.

By \cite[Prop.3.8]{dim-hmv} if $f$ is not a theta series, then
 for all primes $\lambda$ outside a finite set of ``exceptional primes"
 the image of  $\overline{\rho}_{\lambda}$ contains $\SL_2(\F_{\ell})$.

\medskip
In this article, we  determine explicitly   the images and  the finite exceptional sets
for three examples of  Hilbert modular newforms on real quadratic number fields,
of different weights and levels. Our results can be summarized in the
following table :

\bigskip
\begin{center}\begin{tabularx}{\linewidth}{|l|c|c|c|c|c|}
\cline{1-6}
  $f$ constructed by  &   $F$  & $E$  &   k   &
N   &  exceptional $\lambda$ divide  \\
\cline{1-6}
 Consani-Scholten  & $\Q(\sqrt{5})$  &   $\Q(\sqrt{5})$  &
$(2,4)$   & $2\cdot 3\cdot 5 $   & $2\cdot 3\cdot 5$  (Thm.\ref{cs})\\
\cline{1-6}
  Demb{\'e}l{\'e}  &   $\Q(\sqrt{5})$   &   $\Q(\sqrt{6})$  &
$(2,2)$   & $5\cdot(8+3\sqrt{5}) $
& $2\cdot 3\cdot 5\cdot 19$  (Thm.\ref{dembele}) \\
\cline{1-6}
  Okada  &   $\Q(\sqrt{257})$   &   $\Q(\sqrt{13})$  &
$(2,2)$   & $1$
& $2\cdot 3\cdot 257 $  (Thm.\ref{okada})\\
\cline{1-6}
\end{tabularx}
\end{center}

\bigskip
Section \ref{modularity} contains a potential application to the
modularity of a quintic threefold.

\section{Large image results for Hilbert modular forms.}

In this section we recall some of the  results of
\cite[\S3]{dim-hmv}.

\medskip
Let $v$ be a prime of $F$ above $\ell$ of residual degree $h$.
Put $\overline{\rho}=\overline{\rho}_{\lambda}|_{D_{v}}$.
The semisimplification $\overline{\rho}^{\mathrm{ss}}$ of
$\overline{\rho}$ is tamely ramified and the image by
$\overline{\rho}^{\mathrm{ss}}$ of the tame inertia $I_{v}^t$
is cyclic. Let $a$ be a generator of
$T=\overline{\rho}^{\mathrm{ss}}(I_{v}^t)$. Let $n$ be the
image of a Frobenuis element by $\overline{\rho}^{\mathrm{ss}}$.
Then $n$ belongs to the normalizer $N$ of
$T$ in  $\GL_2(\F_\lambda)$ and  $nan^{-1}=a^{\ell^h}$.
Since $N/T$ is of order $2$, we have

$\bullet$ either $n\in T$ and $a=a^{\ell^h}$. In this case
there exist integers $(p_i)_{0\leq i\leq 2h-1}$ such that
$\overline{\rho}^{\mathrm{ss}}\!:\! I_{v}^t\!\rightarrow
\GL_2(\F_\lambda)$ factors through the natural map
$I_{v}^t\rightarrow \F_{\ell^h}^\times$ followed by
$$F_{\ell^h}^\times \rightarrow \F_{\ell^h}^\times\times
\F_{\ell^h}^\times\enspace,\enspace
x\mapsto (x^{p_0+p_1\ell+...+p_{h-1}\ell^{h-1}},x^{p_h+p_{h+1}\ell+...+p_{2h-1}\ell^{h-1}}),$$

$\bullet$ either $n\notin T$ and $a=a^{\ell^{2h}}$. In this case
there exist integers $(p_i)_{0\leq i\leq 2h-1}$ such that
$\overline{\rho}^{\mathrm{ss}}\!:\! I_{v}^t\!\rightarrow
\GL_2(\F_\lambda)$ factors through the natural map
$I_{v}^t\rightarrow \F_{\ell^{2h}}^\times$ followed by
$$F_{\ell^{2h}}^\times \rightarrow \F_{\ell^{2h}}^\times
\times\F_{\ell^{2h}}^\times \enspace,\enspace
x\mapsto (x^{p_0+p_1\ell+...+p_{2h-1}\ell^{2h-1}},x^{\ell^h(p_0+p_1\ell+...+p_{2h-1}\ell^{2h-1})}).$$
\begin{prop}{\rm\cite[Cor.2.13]{dim-hmv}} \label{weights}
Assume  that  $\ell>k_0$ is unramified in $F$ and  does not divide $N$.
Then $\overline{\rho}_{\lambda}$ is crystalline at $\ell$, the
  multisets $\bigcup_{v|\ell} \{p_i| 0 \leq i\leq 2h-1\}$ and
$\{\frac{k_0-k_i}{2}, \frac{k_0+k_i-2}{2}| 1\leq i\leq d\}$ are equal
and $p_{i+h}=p_i$, for all $0\leq i\leq h-1$.
\end{prop}

Using this proposition, one can show :

\begin{prop}\label{image}
Assume  that  $\ell\!>\!k_0$ is unramified in $F$ and  does not divide $N$.

{\rm(i)} {\rm\cite[Prop.3.1(i)]{dim-hmv}}
For all but finitely many primes $\lambda$,
 $\overline{\rho}_{\lambda}$ is absolutely  irreducible.

\smallskip
{\rm(ii)} {\rm\cite[\S3.2]{dim-hmv}} If $d(\ell\!-\!1) >  5\sum_{i=1}^d(k_i\!-\!1)$
then the image of $\overline{\rho}_{\lambda}$ in $\PGL_2(\F_\lambda)$ is
not isomorphic  to one of the groups $A_4$, $S_4$ or $A_5$.

\smallskip
{\rm(iii)} {\rm\cite[Lemma 3.4]{dim-hmv}}
 Assume that  $\ell\neq 2k_i-1$ for all $1\leq i \leq d$.
Assume that  the image   of $\overline{\rho}_{\lambda}$ in
$\PGL_2(\F_\lambda)$ is dihedral, that is  $\overline{\rho}_{\lambda}\cong
\overline{\rho}_{\lambda}\otimes \varepsilon_{K/F}$ where
$\varepsilon_{K/F}$ denotes the character of some quadratic extension
$K/F$. Then $K/F$ is unramified outside $N$.
\end{prop}

\begin{cor} {\rm\cite[Prop.3.8]{dim-hmv}}
 If  $f$ is not a theta series then for
all but finitely many primes $\lambda$ the image of
 $\overline{\rho}_{\lambda}$ contains (a conjugate of)
$\SL_2(\F_{\ell})$.
\end{cor}

The proof uses Dickson's classification theorem of the subgroups of
$\GL_2(\F_\lambda)$. According to this theorem, the image in
 $\PGL_2(\F_\lambda)$ of an irreducible subgroup of
 $\GL_2(\F_\lambda)$ that does not contain (a conjugate of)
$\SL_2(\F_{\ell})$ is  isomorphic either to a dihedral group, either to
one of the groups $A_4$, $S_4$ or $A_5$.

\section{On an example of Consani-Scholten.}\label{ex1}

In \cite{CoSc} Consani and Scholten construct a Hilbert modular newform $f$
on $F=\Q(\sqrt{5})$ of weight $(2,4)$, conductor $30$ and  trivial
central character.
The  Fourier coefficients $c(f,\pi)$ of $f$ belong to $E=\Q(\sqrt{5})$
and  are computed for $7\leq \Nm(\pi)\leq 97$. By Prop.\ref{weights},
for all $\ell\geq 7$,  $\overline{\rho}_{\lambda}$  is crystalline and
the  semi-simplifications of its restrictions to the inertia groups at
$\ell$ satisfy :

$\bullet$ for  $\ell=vv'$  split in $F$  :
$$\overline{\rho}_{\lambda|}{}_{I_v}^{\mathrm{s.s.}}\simeq
 1 \oplus \omega^3 \text{ and }
\overline{\rho}_{\lambda|I_{v'}}^{\mathrm{s.s.}}\simeq
 \omega \oplus \omega^2 \text{, or }
\overline{\rho}_{\lambda|}{}_{I_v}^{\mathrm{s.s.}}\simeq
 \omega_2^3 \oplus \omega_2^{3\ell} \text{ and }
\overline{\rho}_{\lambda|}{}_{I_{v'}}^{\mathrm{s.s.}}\simeq
 \omega_2^{1+2\ell} \oplus \omega_2^{2+\ell} .$$

$\bullet$ for $\ell$ inert  in $F$  :
$$\overline{\rho}_{\lambda|}{}_{I_{\ell}}^{\mathrm{ s.s.}} \simeq
 \omega_2 \oplus \omega_2^{2+3\ell} \text{ or }
 \omega_2^{1+3\ell} \oplus \omega_2^2 \text{ or }
\omega_4^{a+3\ell+(3-a)\ell^2} \oplus \omega_4^{3-a+a\ell^2+3\ell^3},
\text{with } a=1 \text{ or } 2.$$

Here $\omega$ is the cyclotomic character modulo $\ell$ and $\omega_2$ (resp. $\omega_4$)
denotes a fundamental character of level $2$ (resp. $4$).

\medskip
The aim of this section is to  establish  the following

\begin{theo} \label{cs} Let  $\ell=\Nm(\lambda)\geq 7$ be a prime.
Then the image of
 $\overline{\rho}_{\lambda}$ is isomorphic to
$\begin{cases}
 \{\gamma\in\GL_2(\F_{\ell}),  \det(\gamma)\in \F_{\ell}^{\times 3}\}\text{, if }
\ell\equiv \pm 1\pmod{5}, \\
 \{\gamma\in \GL_2(\F_{\ell^2}),  \det(\gamma)\in \F_{\ell}^{\times 3}\}\text{, if }
\ell\equiv \pm 2\pmod{5}.\end{cases}$
\end{theo}

\noindent {\bf Proof :} The theorem would follow from
 \cite[Prop.3.9]{dim-hmv} once
we establish that the image of $\overline{\rho}_{\lambda}$ is
not reducible, nor of  dihedral type, nor of $A_4$, $S_4$ or  $A_5$ type,
nor isomorphic to
$\{\gamma\in \F_{\ell^2}^\times\GL_2(\F_{\ell})\text{ , } \det(\gamma)\in \F_{\ell}^{\times 3}\}$
(in the last case we say that $\overline{\rho}_{\lambda}$ has  inner twists).

$\relbar$ Denote by $\epsilon=\frac{1+\sqrt{5}}{2}$ the fundamental unit of $F$.
Then $\epsilon^{120}\equiv 1\pmod{30}$.  By
\cite[Prop.3.1(ii)]{dim-hmv} we obtain that $\overline{\rho}_{\lambda}$ is
absolutely irreducible unless $\epsilon^{240}-1\in \lambda$ that is
(recall that $\ell \geq 7$)
\begin{equation}\label{list1}
 \ell\in\{7,11,23,31,61,241,599,1553,2161,20641\}
\end{equation}
For odd representations irreducibility is equivalent to absolute irreducibility.
 Since we  are only interested in showing the irreducibility for the finite set
 of primes in (\ref{list1}) it is enough to prove that for each $\lambda$ in $E$
 dividing such a prime $\ell$  there exists a prime $\pi$ in $F$ relatively prime to
$30\ell$ such that the characteristic polynomial of $\bar{\rho}_{\lambda}
(\Frob_{\pi})$ is irreducible over  $\mathbb{F}_\lambda$.
 We have used the characteristic polynomials for the primes $\pi$
 in $F$ dividing one of the following  primes $p$:
\begin{equation}\label{list2}
 11, 19, 29, 31, 41, 59
\end{equation}
 Observe that all these primes split in $F$. For each of these primes $\pi$ the
 characteristic polynomial  $x^2- c(f,\pi) x + p^3$ has been
 computed in \cite{CoSc}. For every prime  $\ell \geq 7$
  in (\ref{list1}) and for every  $\lambda$ in $E$ dividing $\ell$
   we checked that some of these characteristic polynomials are irreducible
  modulo $\lambda$, and this proves that for every prime $\ell \geq 7$ the
  residual representations are absolutely irreducible.

\smallskip
$\relbar$ Assume that the image of $\overline{\rho}_{\lambda}$ is of dihedral type
and $\ell\geq 7$.  By the local behavior of $f$ at $2$ and $3$ (\cf \cite{carayol})
and by  Prop.\ref{image}(iii)  there exists a   quadratic  extension $K/F$ of discriminant  dividing
$5$  such that  for every prime $\pi$ of $F$ inert in $K/F$ and
relatively prime to $30\ell$, $\tr(\overline{\rho}_\lambda (\Frob_\pi))=0$.
Equivalently, for every such prime $\pi$ we
 have  $c(f,\pi)\in \lambda$.

Thus, the algorithm for bounding the set of dihedral primes runs as follows: for
each quadratic extension $K$ of $F$ unramified outside $5$ find a couple of primes
$\pi$ in $F$ verifying  $\pi$ inert in $K/F$ and $c(f,\pi) \neq 0$.
The only possibly
dihedral primes are those $\lambda$ dividing these $c(f,\pi)$. Using again the
eigenvalues $c(f,\pi)$ for the primes $p$ listed in (\ref{list2}) we have checked with this
method that there is no dihedral prime $\ell \geq 7$.

Let us describe in more detail the computations performed : the following list contains all quadratic extensions of
$F = \Q(\sqrt{5})$ unramified outside $\sqrt{5}$ (here we use the fact that the multiplicative group of units of $F$ is generated by
$-1$ and $\epsilon=\frac{1+\sqrt{5}}{2}$) :
$$ K_1 = F (\sqrt[4]{5}); \; K_2 = F\Bigl( \sqrt{  \epsilon \sqrt{5}} \Bigr); \;
K_3 = F\Bigl(\sqrt{-\sqrt{5}}\Bigr); \; K_4 = F\Bigl(\sqrt{- \epsilon \sqrt{5}}\Bigr)$$
\begin{rque} It is easy to see that $K_4$ is the cyclotomic
field of $5$-th roots of unity.
\end{rque}
The primes $\pi$ in $F$ dividing $p = 29 \text{ or } 41$
are inert in $K_1 / F$ and in $K_3 / F$.
The primes $\pi$ in $F$ dividing $p = 11 \text{ or } 29$ (resp.
$p = 29 \text{ or } 59$) are inert in $K_2 / F$ (resp. in $K_4 / F$).
All these primes $\pi$ verify $c(f, \pi) \neq 0$.
In each case, we search for all primes $\lambda$ dividing
$N_p := \N_{E/\Q} c(f, \pi)$ for both primes $p$ in the above pairs.
By computing the prime factors of each of the  $N_p$ :
$$ N_{11} : \{ 2,11,19 \} \enspace\enspace N_{29} : \{ 2,5,29 \} \enspace\enspace
 N_{41} : \{ 2,41,379 \} \enspace\enspace  N_{59} : \{ 2,5,59,71 \} $$
we conclude that there are no dihedral primes $\ell\geq 7$.

$\relbar$ By  Prop.\ref{image}(ii) the  image of $\overline{\rho}_{\lambda}$ cannot be
of $A_4$, $S_4$ or $A_5$ type  for $\ell\geq 11$ and a more careful
study  shows that the argument remains  valid for $\ell=7$.

$\relbar$ Finally, assume that $\overline{\rho}_{\lambda}$ has inner
twists, that is the  image of $\overline{\rho}_{\lambda}$ is isomorphic
to $\{\gamma\in \F_{\ell^2}^\times\GL_2(\F_{\ell})\text{ , } \det(\gamma)\in \F_{\ell}^{\times 3}\}$.
In this case $\ell$ should be  inert in $E$ and  the squares of the traces of the
elements of the image of $\overline{\rho}_{\lambda}$ should belong to $\F_{\ell}$
(\cf \cite[\S3.4]{dim-hmv}). In particular $c(f,\pi)^2\in \F_{\ell}$
for all the primes $\pi$ in $F$  not dividing $30\ell$. Taking $\pi=(5+\epsilon)$ we
find a contradiction. \hfill $\square$

\section{Towards the modularity of a quintic threefold.}\label{modularity}

Here we describe a potential application.  Consani and  Scholten
study  the middle degree {\'e}tale cohomology of a quintic threefold
$\widetilde{X}$
(a proper and smooth $\Z[\frac{1}{30}]$-scheme with Hodge numbers
 $h^{3,0}=h^{2,1}=1$, $h^{2,0}=h^{1,0}=0$ and $h^{1,1}=141$). They
show that the $\G_{\Q}$-representation $\rH^3(\widetilde{X}_{\overline{\Q}},\Q_{\ell})$
is induced from a two dimensional representation $\sigma_\lambda$ of $\G_F$,
and they conjecture that $\sigma_\lambda$ is isomorphic to the $\rho_{\lambda}$
from section \ref{ex1}.

\begin{prop} \label{modularite} Assume $\ell\geq 7$ and  $\overline{\rho}_{\lambda}=
\sigma_\lambda  \mod{\lambda}$. Then  $\sigma_\lambda$ is modular, and
in particular the $L$-function associated to
$\rH^3(\widetilde{X}_{\overline{\Q}},\Q_{\ell})$ has  an analytic continuation
to the whole complex plane and a functional equation.
\end{prop}

\noindent {\bf Proof : } By Thm.\ref{cs} and  \cite[Prop.3.13]{dim-hmv} the ``large''
image condition $\mathrm{{\bf (LI}_{Ind \overline{\rho}_{\lambda}}{\bf )}}$
of  \cite[\S0.1]{dim-hmv}  is fulfilled for all  $\ell\geq 7$. Then
the proposition follows from  \cite[Thm.0.2]{dim-ihara}.
\hfill $\square$

\begin{rque}
{\rm (i)} In order to prove that $\overline{\rho}_{\lambda}= \sigma_\lambda$ mod $\lambda$ it is
enough, by the Cebotarev Density Theorem, to check that for ``sufficiently many''
primes $\pi$ of $F$ we have $\tr(\sigma_\lambda(\Frob_\pi)) \equiv c(f,\pi) \pmod{\lambda}$.
Unfortunately, a relatively small number of these values have been computed.

{\rm (ii)} If  $\ell$ is inert in $E$, the image of
$\Ind_F^{\Q} \overline{\rho}_\lambda: \G_F \rightarrow
(\GL_2\times \GL_2)(\F_{\ell^2})$ is isomorphic to
$$\{(\gamma,\gamma^\sigma)\enspace|\enspace\gamma\in \GL_2(\F_{\ell^2}),\det(\gamma)\in \F_{\ell}^{\times 3}\}$$
where $\sigma$ denotes the non-trivial automorphism of $\F_{\ell^2}$. This is
a typical example of a representation with non-maximal image satisfying
$\mathrm{{\bf (LI}_{Ind \overline{\rho}_{\lambda}}{\bf )}}$.

{\rm (iii)} The results of Taylor\cite{Ta3}, if extended to the
non-parallel weights $k$, would give the meromorphic   continuation
to the whole complex plane and the functional equation.
\end{rque}

\section{On examples of Demb{\'e}l{\'e}.}

In \cite[p.60]{Dem} Demb{\'e}l{\'e}  constructs a  Hilbert modular newform $f=f_4$
on $F=\Q(\sqrt{5})$ of parallel weight $(2,2)$, level $N=\sqrt{5}^2(8+3\sqrt{5})$, and
Fourier coefficients in  $E=\Q(\sqrt{6})$.

By Prop.\ref{weights} for all $\ell$ not dividing $5\cdot 19$,   $\overline{\rho}_{\lambda}$
is crystalline at $\ell$ and
$$\overline{\rho}_{\lambda|}{}_{I_{\ell}}^{\mathrm{s.s.}}\simeq
 1 \oplus \omega \text{  or }  \omega_2 \oplus \omega_2^{l} \text{  or }
\omega_4^{1+l} \oplus \omega_4^{\ell^2+\ell^3}.$$

\begin{theo} \label{dembele} Assume  $\ell\geq 7$ and $\ell\neq
 19$. Then the image of   $\overline{\rho}_{\lambda}$
is isomorphic to $\begin{cases}  \GL_2(\F_{\ell}) \text{, if } \ell\equiv \pm 1,\pm 5\pmod{24}, \\
 \{\gamma\in \GL_2(\F_{\ell^2}),\det(\gamma)\in\F_{\ell}^{\times }\}\text{, otherwise}.\end{cases}$
\end{theo}

\noindent {\bf Proof : } As in Thm.\ref{cs},  we have to discard  the reducible,
dihedral, $A_4$, $S_4$, $A_5$ and inner twist cases.

$\relbar$ Assume that  $\overline{\rho}_{\lambda}$ is reducible :
$\overline{\rho}_{\lambda}^{\mathrm{s.s.}}=
\varphi\oplus\omega\varphi^{-1}$, where $\varphi$ is a mod $\lambda$ crystalline character of
$\G_F$. Using the compatibility between local and global Langlands correspondence
for $\GL_2$ (\cf \cite{carayol}) and the fact that $f$ has trivial central character,
we deduce that the prime-to-$\ell$ part of the conductor  of  $\varphi$ divides   $\sqrt{5}$.
It is easy to check that  $\epsilon^{4}\equiv 1\pmod{\sqrt{5}}$ and that
$\lambda$ does not divide $\epsilon^{4}-1$. By the proof of  \cite[Prop.3.1]{dim-hmv}
 it follows  that $\varphi$ is unramified at $\ell$,
and therefore corresponds (via global class field theory) to a character of the
ray class group $\Cl_{F,\sqrt{5}}^+$. Now we use that  $\Cl_{F,\sqrt{5}}^+$ is of
order $2$, because the narrow class group  $\Cl_{F}^+$ is trivial and  $\epsilon^2$
generates an index $2$ subgroup  of $(\Z[\epsilon]/(\sqrt{5}))^\times\cong
(\F_5[X]/X^2)^\times$.  Hence $\varphi$ is a trivial or quadratic character.
By evaluating
$\overline{\rho}_{\lambda}^{\mathrm{s.s.}}=\varphi\oplus\omega\varphi^{-1}$ at
$\Frob_7$ we find a contradiction (the conclusion does not apply to the prime
$\lambda=7$, however in the next few lines we give a second proof of the residual
irreducibility that also covers this prime).

Once we know that $\varphi$ is unramified at $\ell$ and its conductor divides $\sqrt{5}$,
an alternative method
consists (without determining the order of $\varphi$) of taking primes that are trivial
modulo $\sqrt{5}$, more precisely :
 find primes $\pi$ in $F$ having a generator $\alpha$ (in $F$ all ideals are principal)
 such that  $\alpha \equiv 1 \pmod{\sqrt{5}}$.
 It follows from class field theory that for all such primes $\pi$, the element $\Frob_\pi$ is totally
 split in the ray class field of $F$ of conductor $\sqrt{5}$, hence
 $\varphi$ is trivial on  $\Frob_\pi$.   This property is fulfilled by the two primes of $F$
dividing $p=11$ and $31$. In fact  $ 11 = (-4+\sqrt{5}) (-4-\sqrt{5})$ and $31 = (6+ \sqrt{5}) (6-
 \sqrt{5})$.   Therefore, equating traces in
$\overline{\rho}_{\lambda}^{\mathrm{s.s.}}= \varphi\oplus\omega\varphi^{-1}$
at these $\Frob_\pi$ we obtain :
$$ c(f,-4+\sqrt{5} ) = -2 \sqrt{6} \equiv 1 + 11 \pmod{\lambda}
\text{, } \enspace c(f,6+ \sqrt{5}) = 2 \equiv 1 + 31 \pmod{\lambda}$$
From this, we conclude again that the residual representation is irreducible for
 $\ell > 5$, $\ell \neq 19$.

$\relbar$  Assume that   $\overline{\rho}_{\lambda}$ is of dihedral type. As in the
previous example, by the local behavior of  $f$ at $ 8 + 3 \sqrt{5}$
and by  Prop.\ref{image}(iii)  we have to consider only  quadratic extensions of $F$ unramified
outside $\sqrt{5}$, that is  the same extensions $K_1,...,K_4$ that we considered in
section \ref{ex1}.

For each $1\leq i\leq 4$, we  take again two rational primes $p$ among $11$, $29$, $41$, $59$
decomposing as $\pi \pi'$ in $F$ with both $\pi$ and $\pi'$ inert in $K_i /F$.
As in the previous example, let us list the prime factors of the norms $N_p = \N_{E/\Q} c(f, \pi) $
for each of these $p$  and a suitably chosen $\pi \mid p$ :
$$ N_{11} : \{ 2,3 \} \qquad N_{29} : \{ 3,5 \}  \qquad
 N_{41} : \{ 2,3 \} \qquad  N_{59} : \{ 3,5 \} $$

Since no prime $\ell \geq 7$ divides any of these norms, we  conclude that
no prime $\ell\geq 7$, $\ell \neq 19$, is dihedral.

$\relbar$ By Prop.\ref{image}(ii) the image of $\overline{\rho}_{\lambda}$ cannot be
of $A_4$, $S_4$ or $A_5$ type  for any $\ell\geq 7$.

$\relbar$  As in the proof of Thm.\ref{cs} we see
that $\overline{\rho}_{\lambda}$ cannot have inner twists. \hfill $\square$

\begin{rque}
We applied this method to bound explicitly  the set of exceptional primes
for several other examples of Hilbert modular newforms from Demb{\'e}l{\'e}'s tables
and it always worked well.
\end{rque}

\section{On an example of Okada.}
Our last example, due to K. Okada \cite{Ok},  concerns a  Hilbert modular newform $f$ on
$F=\Q(\sqrt{257})$ of weight $(2,2)$, level $1$, and Fourier coefficients in  $E=\Q(\sqrt{13})$.
Here the class number of $F$ is $3$.

\begin{theo} \label{okada} Assume  $\ell\geq 5$ and $\ell\neq
  257$. Then the image of  $\overline{\rho}_{\lambda}$
is isomorphic to $\begin{cases}  \GL_2(\F_{\ell}) \text{, if } \ell\equiv \pm 1,\pm 3,\pm 4 \pmod{13}, \\
 \{\gamma\in \GL_2(\F_{\ell^2}),\det(\gamma)\in\F_{\ell}^{\times }\}\text{, otherwise}.\end{cases}$
\end{theo}

\noindent {\bf Proof : } Once again we have to discard  the reducible,  dihedral,
$A_4$, $S_4$, $A_5$  and inner twist cases.

$\relbar$ Assume that  $\overline{\rho}_{\lambda}$ is reducible :
$\overline{\rho}_{\lambda}^{\mathrm{s.s.}}=
\varphi\oplus\omega\varphi^{-1}$, where $\varphi$ is a mod $\lambda$ crystalline character of
$\G_F$. As $f$ is of level $1$, the same method as in  the previous section
shows that $\varphi$ is everywhere unramified (and therefore of  order dividing
the class number of $F$ - fact that we won't use).

Assume that $\ell\neq 61$. Let $\pi$ be one of the two prime ideals  of $F$ dividing $61$.
As  $\pi$ is  principal,
by class field theory we deduce that $\varphi$ is trivial on  $\Frob_\pi$.
By evaluating $\overline{\rho}_{\lambda}^{\mathrm{s.s.}}= \varphi\oplus\omega\varphi^{-1}$
at $\Frob_\pi$ and taking traces, we obtain :
$$  -1\pm\sqrt{13} \equiv 1 + 61 = 62 \pmod{\lambda}.$$
 When  $\ell=61$, by applying the same argument with
a principal prime ideal  $\pi$ of $F$ dividing $67$, we get
 $$  -3\pm 3\sqrt{13} \equiv 1 + 67 = 68 \pmod{\lambda}.$$
Both these  congruences are impossible. Therefore
 $\overline{\rho}_{\lambda}$ is irreducible when $\ell\geq 5$ and $\ell\neq 257$.

\medskip
$\relbar$  As $f$ is of level $1$, and $F$ does not have unramified quadratic
extensions, it follows from Prop.\ref{image}(iii)   that the image of   $\overline{\rho}_{\lambda}$ cannot be of  dihedral  type for $\ell\geq 5$,  $\ell\neq 257$.

$\relbar$ The image of $\overline{\rho}_{\lambda}$ cannot be
of $A_4$, $S_4$ or $A_5$ type for  $\ell\geq 7$,  $\ell\neq 257$
(\cf  Prop.\ref{image}(ii)).
When $\ell=5$, we observe that the order of the image of
$\overline{\rho}_{5}(\Frob_{17A})$ in $\PGL_2(\F_{25})$ is at least $6$.

$\relbar$ As in the proof of Thm.\ref{cs} we see
that $\overline{\rho}_{\lambda}$ cannot have inner twists. \hfill $\square$


TITLE: Every closed convex cone in $ \mathbb{R}^2 $ is polyhedral
QUESTION [1 upvotes]: I recently came across the following problem:

Let $ K \subseteq \mathbb{R}^2 $ be a closed convex cone (meaning K is closed under non negative linear combinations) and I am asked to show it is polyhedral meaning it is the intersection of finitely many half spaces, or alternatively that it is finitely generated $ K = cone(A) $ where $ A $ is a finite set and cone(A) denotes all possible non-negative linear combinations of elements of A, or alternatively that K has a finite number of extreme vectors meaning vectors $x\in K$ such that if $ x=y+z $ where $ y,z\in K $ are non-negative multiples of x.

I have given three possible definitions of polyhedral cone but for some reason I cannot seem to see why a convex cone which is closed in $\mathbb{R}^2$ satisfies either one of the three possible definitions. I certainly need and appreciate all the help I can get on this.

REPLY [2 votes]: Let $K$ be a closed cone in the plane. If $K$ is not contained in a closed halfspace, then $K$ coincides with the plane. Suppose then that it is contained in a closed halfspace, whose boundary is a line $L$. Let now $L'$ be a line parallel to $L$ contained in the interior of that halfspace. The intersection $K\cap L'$ is a convex closed subset of $L'$, so it is a closed interval. Now consider cases: the interval may be the whole of $L'$, a closed halfline, or a bounded closed interval.


TITLE: Reversible Ito Diffusions
QUESTION [3 upvotes]: I have given a diffusion equation
$$ dX_t = -\nabla V(X_t) \, dt + \sigma dB_t.$$
I found here(1) a characterization when $X_t$ is reversible, aslong as $\sigma=1$. Is this also true for $\sigma \neq 1$?
It is said here(2) that the general result for arbitrary $\sigma$ can be found here(3). Unfortunately I do not have access to this document, and I also assume that the notation will probably be unreadable for me and not as plain and clean as here(1).
Therefore, I would like to know if it is easy to get from here(1) to the general case where $\sigma \neq 1$.
Also I would like to know if the stationary measure $\mu$ is unique if $X_t$ is reversible.

REPLY [0 votes]: I would recommend one of the classic books
[1] Stochastic Differential Equations.
An Introduction with Applications
By B. Øksendal
http://www.springer.com/us/book/9783540047582
[2] Stochastic Partial Differential Equations.
A Modeling, White Noise Functional Approach
by Holden,   Øksendal, Ubøe and Zhang
http://link.springer.com/book/10.1007/978-1-4684-9215-6
[3] Numerical Solution of Stochastic Differential Equations
 by Kloeden, and Platen
http://www.springer.com/us/book/9783540540625


TITLE: Relationship between Training Neural Networks and Calculus of Variations
QUESTION [4 upvotes]: I was wondering since Calculus of Variations is about determining extremas and Training Neural Networks is about finding a set of weights such that the total error is minimized, is it possible to draw an analogy between the two and treat them as the same discipline.
For the sake of simplicity lets train using linear regression on a huge data set.
We are trying to determine a polynomial that minimizes the total error.
In this scenario, the input of the functional is coefficients(weights) that uniquely determine a function. After running the functional we are given an error rate.
To determine if its the lowest error rate we change the coefficients up a bit and rerun the functional. The new coefficients are determined using gradient descent.
In this way, we progress towards the lowest error rate.
I recognize that there are methods like Euler-Lagrange equation but none of them relate to deep-learning according to my knowledge.

REPLY [1 votes]: Neural nets and calculus of variation have a lot more in common than most people think.  The big difference is the space in which you try to find an answer.  In your typical neural network, you estimate your answer using piecewise linear functions. But at least one trick in the calculus of variations carry over to neural nets, that of handling constraints. Both use Lagrange multipliers to solve that problem.  Any problem that can be cast of a calculus of variation can probably use a neural net to get a numerical approximation to the answer.  As well, if you have at hand the specific form of the Euler equation of the problem, it may well guide you to a specific basis set for forming the approximation and what other computational tricks such as stochastic gradiant descent is appropriate for the problem.