%  This is a Latex template file for articles submitted to Annales 
%  Mathématiques Blaise Pascal written in English.  It uses the 
%  cedram-ambp-new.cls class file, which is available at 
%          http://ambp.cedram.org/?lang=en
%  Authors  must use this template for the body of their article, and the
%  sample bibtex file, biblio.bib, for their references. To
%  process references properly, use the commands: latex file, bibtex
%  file, latex file. (You may have to do latex, bibtex, latex, latex.)
%
%  Comment lines, which start with a % sign, are inserted in this
%  file to help authors use various commands.  Please do not alter the
%  cedram-ambp-new.cls file or create your own theorem, lemma, corollary,
%  definition, remark, remarks, example, notation and proof environments.  
%  In particular, if you use a standard set of
%  environments, such as ``\newtheorem{thm}{theorem}[section]'' etc. in
%  your latex files, you MUST NOT use them inside this template.

% Begin of the file
%\documentclass[francais]{cedram-ambp-new}
\documentclass[english]{cedram-ambp-new}
% Authors must use the cedram-ambp.cls class file as above.
% The english option allows to write in english with english keywords and
% typography.
% The french option allows to write in french with french keywords and
% typography.
% The Babel package is automatically downloaded. All its commands
% can therefore be used.

%% inputenc allows to write accented characters directly from the
% keyboard and not in the oldfashion tex way.
\usepackage[utf8]{inputenc}

%Authors can add other packages at the only condition that they do not
%change the style.
% It is not necessary to add amsmath or mathtools since they are automatically
% downloaded.

%  Authors can add their own commands. For example, \newcommand{\BbC}{\mathbb{C}}
%  déefine \BbC that writes
%  ``C''  in mathemtical mode with ``blackboard'' font.
\newcommand{\BbP}{\mathbb{P}}
\newcommand{\BbC}{\mathbb{C}}
\newcommand{\BbZ}{\mathbb{Z}}
\newcommand{\scl}{(-\sqrt{-1}\hbar)}
% The following command defines \abs to print
% an absolute value or the cardinality of a set.
\DeclarePairedDelimiter\abs{\lvert}{\rvert}

%  The title of the article
\title%
%% The optionnal argument is the short version for headings.
[An Example of an Article]
%% The mandatory argument is for the title page, summaries, headings
%% if optionnal void.
{An Example of an Article for this Journal}

%% The French title of the paper.
\alttitle{Un exemple d'article pour les Annales Math\'ematiques Blaise Pascal}

% The list of authors.  Please comment out (i.e. place a % in front
% of) lines that have blank entries. 
\author
%% The optional argument is for shortened version appearing in the headings
[T. \lastname{Parker}]
% You must distinguish between first, middle and last names.
{\firstname{Timothy} \lastname{Parker}}

\address{Department of Mathematics\\
and Statistics\\
Boston University\\
1110 Cummington St.\\
Boston, MA 02215\\
USA}
% The next line must be commented out if there is no support for the first author.
\thanks{This work is founded by CNRS}
\email{tparkerl@buniv.edu}
%
% Comment out the commands concerning additionnal authors if there is only one author.
\author[S. H. \lastname{Murdock}]{\firstname{Steve} \middlename{H.} \lastname{Murdock}}
\address{Department of Mathematics\\
and Statistics\\
Boston University\\
1110 Cummington St.\\
Boston, MA 02215\\
USA}
% The next line must be commented out if there is no support for the second author.
\thanks{This work is founded by Mathdoc}
\email{smur@math.buniv.edu}


% Comment out the commands concerning the third author if there is none.
\author[C. \lastname{Zed}]{\firstname{Charles} \lastname{Zed}}
\address{Université Clermont Auvergne, CNRS\\ Laboratoire de Mathématiques Blaise Pascal\\ F-63000 CLERMONT-FERRAND\\ FRANCE}% Comment out the next line if there is no support for the third author.
%\thanks{}
\email{Charles.Zed@uca.fr}

%% Keywords
\keywords{Example, Annals, \LaTeX\ class}
  
%% French keywords
\altkeywords{Exemple, Annales, classe \LaTeX}

%% Mathématical classification   (2000)
\subjclass{00X99}


\begin{document}
% The abstract.
\begin{abstract}
    This is a sample abstract for our example article prepared for the
  journal \og Annales Mathématiques Blaise Pascal\fg. This sample
  article provides information about how to use the cedram-ambp-new.cls
  \LaTeX\ class file.
\end{abstract}

% French abstract
\begin{altabstract}
Ceci est un exemple de résumé de l'exemple d'article pour la revue  \og Annales Mathématiques Blaise Pascal \fg. Cet exemple d'article passe en revue des exemples d'utilisation des environnements utiles pour pouvoir écrire un vrai article.
\end{altabstract}

% Beware! maketitle *after* abstract
\maketitle

% The beginning of the text.
\section{Introduction}



% Note the reference to cited works below.  Givental's paper in biblio.bib
% is labeled G, and Liu-Lian-Yau's paper is labeled LLY.

The work of Givental \cite{G} and Liu-Lian-Yau \cite{LLY}
 on mirror symmetry 
relates Gromov-Witten
invariants of a quintic hypersurface in \(\BbP^3\) to period
integrals of K\"ahler structures on the 
mirror manifold.  Givental's method uses detailed
calculations of equivariant GW invariants to produce flat sections of
the Dubrovin connection on the tangent bundle to the even cohomology
of the hypersurface,
which are then related to solutions of the Picard-Fuchs ODE for the
 periods on the mirror.

We would like to thank Charles Kay 
for several helpful conversations.

Throughout the paper, we use the following notation:

% An example of the notation environment.

\begin{notation} Let
\(\mathbb{Z}\) denote the set of integers, and let \(\mathcal{A}\) denote
the set of all even integers which cannot be written as the sum of two
primes.  Finally, let \(\mathcal{B} = \mathcal{A} \cap [0,200].\)
\end{notation}


% An example of the definition environment.

\begin{definition}
A Hopf algebra \(H\) is \emph{co-commutative} if \(\Delta=\tau\circ\Delta\)
where \(\Delta\) is the co-multiplication and \(\tau\) is the switch map
defined by \(\tau(h_1\otimes h_2)=h_2\otimes h_1\).
\end{definition}

\section{The next section}

Let \(\ast\) be  an associative, commutative product 
on a complex vector space \(\mathcal{H}\).  
The associated  Dubrovin
connection on \(T\mathcal{H}\) is
%
% Here is a labeled equation for future reference. The equation is
% called ``dub'', but will appear in the text as equation (2.1).
%
\begin{equation}\label{dub}\nabla_Y X=dX(Y)+\sqrt{-1}Y\ast X
\end{equation}
with connection one-form \(\omega (Y)(X)=\sqrt{-1}Y\ast X\). If \(\{
T_0,\ldots,T_m\}\) is a basis of \(\mathcal{H}\) with 
\begin{equation}\label{multtable}
T_i\ast T_j=\Gamma_{ij}^k
T_k,
\end{equation}
 then \(\omega_j^i =\sqrt{-1} \Gamma_{\ell j}^i\ dt^\ell\) (where a typical
\(\alpha\in \mathcal{H}\) is \(\alpha =t_iT^i\)).  It is fundamental that
\(\nabla\) is flat because the product is associative and commutative.
Note that \(\nabla\) stays flat if we replace 
 \(\sqrt{-1}\) in \eqref{dub} by \(\hbar\in\BbC\).
% Note the format ``\eqref{dub}'' in the previous line to refer back 
% to the first labeled equation.

% An example of a proposition.

\begin{proposition} Let \(\ast\) be a commutative, associative product
on a complex vector space \(\mathcal{H}\).  Define matrices \(\Gamma_i,
\ i =
1,\ldots,\dim \ \mathcal{H}\) by \(T_i\ast T_j = (\Gamma_i)^k_j T_k\)
for a basis \(\{T_k\}\) of \(\mathcal{H}\).  Then
a basis of the flat sections 
of the associated flat Dubrovin connection on \(T\mathcal{H}\)  is
\[\left\{ \exp [- \sqrt{-1}t^\ell \Gamma_\ell ]\right\}_{\ell =0}^m.
\]
\end{proposition}

% Note that equations beginning and ending with \)\) (or beginning
% with \[ and ending with \]) are not labeled in the text.

% An example of a lemma.

\begin{lemma}
Except in characteristic one, \(0 \ne 1\).
\end{lemma}



% An example of a theorem. 

\begin{theorem}
  Even in characteristic one, \(1+1=2.\)
\end{theorem}

% An example of the environment for a proof.

\begin{proof}
The proof of this theorem is left for the reader.
\end{proof}


% An example of a corollary. 

\begin{corollary}
In all characteristics, we have
\begin{equation}\label{remarkable} 2+2=4. \end{equation}
\end{corollary}



The second paper in this series will be devoted to applications of
(\ref{remarkable}). 

% An example of the example environment.

\begin{example}
Let \(A\) consist of a set of two elements and \(B\) a set of two
elements.  If \(A\cap B = \emptyset\), then \(\abs{A\cup B} = 4.\)
\end{example}
The following remark can be made:

% An example of the remark environment.

\begin{remark}
The reader can easily generalize this example to the case of three sets.
\end{remark}
The following additional remarks can be made:

% An example of the remarks environment.
\begin{remarks}
  \begin{enumerate}
  \item In a sequel to this article,  we will analyse the case of 4
    sets. 
\item Note the dependance on the number of elements. 
  \end{enumerate}
\end{remarks}

The following conjecture has not been used.
\begin{conj}
There is infinitely many prime numbers \(p\) such that \(p+2\) is prime.
\end{conj}

% The next command determines the bibliography style. Please do not
% change this.

\bibliographystyle{plain}

%  This inserts the bib file, biblio.bib
\bibliography{biblio}

% This command signals the end of the file.
\end{document}











