\documentclass[a4paper]{article}
% Most of the macros below were created by myself in the process of writing my 2002 book "The theory of composites", that took 14 years to write 
%(Although I did not work on it continuously). They saved a lot of time with my one/two finger typing. 
%Some additional ones were created by my friends Patrick Bardsley, Nelson Beebe, and Sasha Movchan.
% Use at your own risk, keeping in mind that the macros here may sometimes conflict with macros used in other templates. In that case
% one simply redefines the macro to avoid the conflict.
% It is necessary to include the packages that you think you might use:
\usepackage[top=2cm, bottom=2cm, left=2cm, right=2cm]{geometry} 
%\usepackage[french]{babel}
%\usepackage[latin1]{inputenc}
\usepackage[dvipsnames,svgnames]{xcolor}
\usepackage{tikz,pgf}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{mathtools}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathrsfs}
\usepackage{float}
\usepackage{graphicx}
\usepackage{subcaption}
\usepackage{color}
\usepackage{pstricks-add}
\usepackage{hyperref}
%\usepackage{refcheck}
%\usepackage[active]{srcltx}
\renewcommand{\baselinestretch}{1}
\oddsidemargin -0.1truein \textwidth 7.0truein \evensidemargin
-0.1truein \topmargin -1truein \textheight 9.0truein
% The two commands below are needed if you want equation numbering (1.1),(1.2)....(2.1),(2.2) etc. If you want the numbering to go (1),(2), (3), just comment out them
\setcounter{page}{1}
\def\theequation{\thesection.\arabic{equation}}
	% Math environments
	\newtheorem{definition}{Definition}[section]
%	\newtheorem{lemma}{Lemma}[section]
	\newtheorem{theorem}{Theorem}[section]
	\newtheorem{corollary}{Corollary}[section]
	\newtheorem{remark}{Remark}[section]

%De-verbose the LaTeX math environment
          \def\half{{\scriptstyle{1\over 2}}}
	\newcommand\defn[1] {\ref{def:#1}}
	\newcommand\thm[1] {\ref{thm:#1}}
	\newcommand\sect[1] {\ref{sec:#1}}
	\newcommand\fig[1] {\ref{fig:#1}}
	\newcommand\labdefn[1] {\label{def:#1}}
	\newcommand\labthm[1] {\label{thm:#1}}
	\newcommand\labsect[1] {\label{sec:#1}}
	\newcommand\labfig[1] {\label{fig:#1}}
	\newcommand\etal{{\it et al.}}
% Use \eq{} for referencing equations
	\newcommand\eq[1] {(\ref{#1})}
	\newcommand{\bfm}[1]{\mbox{\boldmath ${#1}$}}
% Shorthand for widetilde and widehat
          \newcommand{\wt}[1]{\widetilde{#1}}
          \newcommand{\wh}[1]{\widehat{#1}}
% Below \nonum is used in the equationarray environment (\beqa,\eeqa{}) when one want to start a new line, but without the equation number. \beqn and \eeqn begin and end an equation array, but without the final number.
	\newcommand{\nonum}{\nonumber \\}
	\newcommand{\beqa}{\begin{eqnarray}}
	\newcommand{\eeqa}[1]{\label{#1}\end{eqnarray}}
	\newcommand{\beq}{\begin{equation}}
	\newcommand{\eeq}[1]{\label{#1}\end{equation}}
         \newcommand{\beqn}{\begin{eqnarray*}}
         \newcommand{\eeqn}{\end{eqnarray*}}
% Start and end matrices with \pbm and \epm
	\newcommand{\bpm}{\begin{pmatrix}}
	\newcommand{\epm}{\end{pmatrix}}
%Differential operators in PDE:
	\newcommand{\Grad}{\nabla}
	\newcommand{\Curl}{\nabla \times}
	\newcommand{\Div}{\nabla \cdot}
         \def\divergence{\mathop{\operator@font div}\nolimits}
%Underlines, double underlines, overlines, underbraces
         \newcommand{\dund}[1]{\underline{\underline{#1}}}
          \newcommand{\und}[1]{\smash{\underline{#1}}}
	\newcommand{\ov}[1]{\overline{#1}}
          \newcommand{\ubr}[1]{\underbrace{#1}}
%\rprp is a matrix for a 90 degree rotation
	\newcommand{\rprp}{\bfm{R}_{\perp}}
%Real part, Imaginary part, trace, angular brackets , implies
	\newcommand{\Real}{\mathop{\rm Re}\nolimits}
	\newcommand{\Imag}{\mathop{\rm Im}\nolimits}
	\newcommand{\Tr}{\mathop{\rm Tr}\nolimits}
	\newcommand{\lang}{\langle}
	\newcommand{\rang}{\rangle}
          \def\imp{{\, \Rightarrow\,}}
% \Md in partial derivatives, also to denote the boundary of a region
	\newcommand{\Md}{\partial}
	\newcommand{\MP}{\bigoplus}
	\newcommand{\MT}{\bigotimes}
	\newcommand{\MU}{\bigcup}
	\newcommand{\MI}{\bigcap}
	\newcommand{\MV}{\bigvee}
	\newcommand{\MW}{\bigwedge}
% Special math symbols, for complex numbers, natural numbers, reals, integers.
	\def\C{{\mathbb C}}
         \def\N{{\mathbb N}}
         \def\R{{\mathbb R}}
         \def\Z{{\mathbb Z}}
	% Abbreviate definitions of greek symbols
	\newcommand{\Ga}{\alpha}
	\newcommand{\Gb}{\beta}
	\newcommand{\Gd}{\delta}
	\newcommand{\Ge}{\epsilon}
	\newcommand{\Gve}{\varepsilon}
 	\newcommand{\Gf}{\phi}
	\newcommand{\Gvf}{\varphi}
	\newcommand{\Gg}{\gamma}
	\newcommand{\Gc}{\chi}
	\newcommand{\Gi}{\iota}
	\newcommand{\Gk}{\kappa}
	\newcommand{\Gvk}{\varkappa}
	\newcommand{\Gl}{\lambda}
	\newcommand{\Gn}{\eta}
	\newcommand{\Gm}{\mu}
	\newcommand{\Gv}{\nu}
	\newcommand{\Gp}{\pi}
	\newcommand{\Gt}{\theta}
	\newcommand{\Gvt}{\vartheta}
	\newcommand{\Gr}{\rho}
	\newcommand{\Gvr}{\varrho}
	\newcommand{\Gs}{\sigma}
	\newcommand{\Gvs}{\varsigma}
	\newcommand{\Gj}{\tau}
	\newcommand{\Gu}{\upsilon}
	\newcommand{\Go}{\omega}
	\newcommand{\Gx}{\xi}
	\newcommand{\Gy}{\psi}
	\newcommand{\Gz}{\zeta}
	\newcommand{\GD}{\Delta}
	\newcommand{\GF}{\Phi}
	\newcommand{\GG}{\Gamma}
	\newcommand{\GL}{\Lambda}
	\newcommand{\GP}{\Pi}
	\newcommand{\GT}{\Theta}
	\newcommand{\GS}{\Sigma}
	\newcommand{\GU}{\Upsilon}
	\newcommand{\GO}{\Omega}
	\newcommand{\GX}{\Xi}
	\newcommand{\GY}{\Psi}
        % Abbreviate definitions of bold greek symbols: note that defs.tex is
	% needed first
	\newcommand{\BGa}{\bfm\alpha}
	\newcommand{\BGb}{\bfm\beta}
	\newcommand{\BGd}{\bfm\delta}
	\newcommand{\BGe}{\bfm\epsilon}
	\newcommand{\BGve}{\bfm\varepsilon}
	\newcommand{\BGf}{\bfm\phi}
	\newcommand{\BGvf}{\bfm\varphi}
	\newcommand{\BGg}{\bfm\gamma}
	\newcommand{\BGc}{\bfm\chi}
	\newcommand{\BGi}{\bfm\iota}
	\newcommand{\BGk}{\bfm\kappa}
	\newcommand{\BGl}{\bfm\lambda}
	\newcommand{\BGn}{\bfm\eta}
	\newcommand{\BGm}{\bfm\mu}
	\newcommand{\BGv}{\bfm\nu}
	\newcommand{\BGp}{\bfm\pi}
	\newcommand{\BGt}{\bfm\theta}
	\newcommand{\BGvt}{\bfm\vartheta}
	\newcommand{\BGr}{\bfm\rho}
	\newcommand{\BGvr}{\bfm\varrho}
	\newcommand{\BGs}{\bfm\sigma}
	\newcommand{\BGvs}{\bfm\varsigma}
	\newcommand{\BGj}{\bfm\tau}
	\newcommand{\BGu}{\bfm\upsilon}
	\newcommand{\BGo}{\bfm\omega}
	\newcommand{\BGx}{\bfm\xi}
	\newcommand{\BGy}{\bfm\psi}
	\newcommand{\BGz}{\bfm\zeta}
	\newcommand{\BGD}{\bfm\Delta}
	\newcommand{\BGF}{\bfm\Phi}
	\newcommand{\BGG}{\bfm\Gamma}
	\newcommand{\BGL}{\bfm\Lambda}
	\newcommand{\BGP}{\bfm\Pi}
	\newcommand{\BGT}{\bfm\Theta}
	\newcommand{\BGS}{\bfm\Sigma}
	\newcommand{\BGU}{\bfm\Upsilon}
	\newcommand{\BGO}{\bfm\Omega}
	\newcommand{\BGX}{\bfm\Xi}
	\newcommand{\BGY}{\bfm\Psi}
		% Abbreviate symbols for caligraphic letters
	\newcommand{\Ca}{{\cal a}}
	\newcommand{\Cb}{{\cal b}}
	\newcommand{\Cc}{{\cal c}}
	\newcommand{\Cd}{{\cal d}}
	\newcommand{\Ce}{{\cal e}}
	\newcommand{\Cf}{{\cal f}}
	\newcommand{\Cg}{{\cal g}}
	\newcommand{\Ch}{{\cal h}}
	\newcommand{\Ci}{{\cal i}}
	\newcommand{\Cj}{{\cal j}}
	\newcommand{\Ck}{{\cal k}}
	\newcommand{\Cl}{{\cal l}}
	\newcommand{\Cm}{{\cal m}}
	\newcommand{\Cn}{{\cal n}}
	\newcommand{\Co}{{\cal o}}
	\newcommand{\Cp}{{\cal p}}
	\newcommand{\Cq}{{\cal q}}
	\newcommand{\Cr}{{\cal r}}
	\newcommand{\Cs}{{\cal s}}
	\newcommand{\Ct}{{\cal t}}
	\newcommand{\Cu}{{\cal u}}
	\newcommand{\Cv}{{\cal v}}
	\newcommand{\Cx}{{\cal x}}
	\newcommand{\Cy}{{\cal y}}
	\newcommand{\Cz}{{\cal z}}
	\newcommand{\CA}{{\cal A}}
	\newcommand{\CB}{{\cal B}}
	\newcommand{\CC}{{\cal C}}
	\newcommand{\CD}{{\cal D}}
	\newcommand{\CE}{{\cal E}}
	\newcommand{\CF}{{\cal F}}
	\newcommand{\CG}{{\cal G}}
	\newcommand{\CH}{{\cal H}}
	\newcommand{\CI}{{\cal I}}
	\newcommand{\CJ}{{\cal J}}
	\newcommand{\CK}{{\cal K}}
	\newcommand{\CL}{{\cal L}}
	\newcommand{\CM}{{\cal M}}
	\newcommand{\CN}{{\cal N}}
	\newcommand{\CO}{{\cal O}}
	\newcommand{\CP}{{\cal P}}
	\newcommand{\CQ}{{\cal Q}}
	\newcommand{\CR}{{\cal R}}
	\newcommand{\CS}{{\cal S}}
	\newcommand{\CT}{{\cal T}}
	\newcommand{\CU}{{\cal U}}
	\newcommand{\CV}{{\cal V}}
	\newcommand{\CW}{{\cal W}}
	\newcommand{\CX}{{\cal X}}
	\newcommand{\CY}{{\cal Y}}
	\newcommand{\CZ}{{\cal Z}}

	% Abbreviate symbols for bold caligraphic letters
	\newcommand{\BCa}{{\bfm{\cal a}}}
	\newcommand{\BCb}{{\bfm{\cal b}}}
	\newcommand{\BCc}{{\bfm{\cal c}}}
	\newcommand{\BCd}{{\bfm{\cal d}}}
	\newcommand{\BCe}{{\bfm{\cal e}}}
	\newcommand{\BCf}{{\bfm{\cal f}}}
	\newcommand{\BCg}{{\bfm{\cal g}}}
	\newcommand{\BCh}{{\bfm{\cal h}}}
	\newcommand{\BCi}{{\bfm{\cal i}}}
	\newcommand{\BCj}{{\bfm{\cal j}}}
	\newcommand{\BCk}{{\bfm{\cal k}}}
	\newcommand{\BCl}{{\bfm{\cal l}}}
	\newcommand{\BCm}{{\bfm{\cal m}}}
	\newcommand{\BCn}{{\bfm{\cal n}}}
	\newcommand{\BCo}{{\bfm{\cal o}}}
	\newcommand{\BCp}{{\bfm{\cal p}}}
	\newcommand{\BCq}{{\bfm{\cal q}}}
	\newcommand{\BCr}{{\bfm{\cal r}}}
	\newcommand{\BCs}{{\bfm{\cal s}}}
	\newcommand{\BCt}{{\bfm{\cal t}}}
	\newcommand{\BCu}{{\bfm{\cal u}}}
	\newcommand{\BCv}{{\bfm{\cal v}}}
	\newcommand{\BCx}{{\bfm{\cal x}}}
	\newcommand{\BCy}{{\bfm{\cal y}}}
	\newcommand{\BCz}{{\bfm{\cal z}}}
	\newcommand{\BCA}{{\bfm{\cal A}}}
	\newcommand{\BCB}{{\bfm{\cal B}}}
	\newcommand{\BCC}{{\bfm{\cal C}}}
	\newcommand{\BCD}{{\bfm{\cal D}}}
	\newcommand{\BCE}{{\bfm{\cal E}}}
	\newcommand{\BCF}{{\bfm{\cal F}}}
	\newcommand{\BCG}{{\bfm{\cal G}}}
	\newcommand{\BCH}{{\bfm{\cal H}}}
	\newcommand{\BCI}{{\bfm{\cal I}}}
	\newcommand{\BCJ}{{\bfm{\cal J}}}
	\newcommand{\BCK}{{\bfm{\cal K}}}
	\newcommand{\BCL}{{\bfm{\cal L}}}
	\newcommand{\BCM}{{\bfm{\cal M}}}
	\newcommand{\BCN}{{\bfm{\cal N}}}
	\newcommand{\BCO}{{\bfm{\cal O}}}
	\newcommand{\BCP}{{\bfm{\cal P}}}
	\newcommand{\BCQ}{{\bfm{\cal Q}}}
	\newcommand{\BCR}{{\bfm{\cal R}}}
	\newcommand{\BCS}{{\bfm{\cal S}}}
	\newcommand{\BCT}{{\bfm{\cal T}}}
	\newcommand{\BCU}{{\bfm{\cal U}}}
	\newcommand{\BCV}{{\bfm{\cal V}}}
	\newcommand{\BCW}{{\bfm{\cal W}}}
	\newcommand{\BCX}{{\bfm{\cal X}}}
	\newcommand{\BCY}{{\bfm{\cal Y}}}
	\newcommand{\BCZ}{{\bfm{\cal Z}}}
		% Abbreviate symbols for bold letters
	\def\Ba{{\bf a}}
	\def\Bb{{\bf b}}
	\def\Bc{{\bf c}}
	\def\Bd{{\bf d}}
	\def\Be{{\bf e}}
	\def\Bf{{\bf f}}
	\def\Bg{{\bf g}}
	\def\Bh{{\bf h}}
	\def\Bi{{\bf i}}
	\def\Bj{{\bf j}}
	\def\Bk{{\bf k}}
	\def\Bl{{\bf l}}
	\def\Bm{{\bf m}}
	\def\Bn{{\bf n}}
	\def\Bo{{\bf o}}
	\def\Bp{{\bf p}}
	\def\Bq{{\bf q}}
	\def\Br{{\bf r}}
	\def\Bs{{\bf s}}
	\def\Bt{{\bf t}}
	\def\Bu{{\bf u}}
	\def\Bv{{\bf v}}
	\def\Bw{{\bf w}}
	\def\Bx{{\bf x}}
	\def\By{{\bf y}}
	\def\Bz{{\bf z}}
	\def\BA{{\bf A}}
	\def\BB{{\bf B}}
	\def\BC{{\bf C}}
	\def\BD{{\bf D}}
	\def\BE{{\bf E}}
	\def\BF{{\bf F}}
	\def\BG{{\bf G}}
	\def\BH{{\bf H}}
	\def\BI{{\bf I}}
	\def\BJ{{\bf J}}
	\def\BK{{\bf K}}
	\def\BL{{\bf L}}
	\def\BM{{\bf M}}
	\def\BN{{\bf N}}
	\def\BO{{\bf O}}
	\def\BP{{\bf P}}
	\def\BQ{{\bf Q}}
	\def\BR{{\bf R}}
	\def\BS{{\bf S}}
	\def\BT{{\bf T}}
	\def\BU{{\bf U}}
	\def\BV{{\bf V}}
	\def\BW{{\bf W}}
	\def\BX{{\bf X}}
	\def\BY{{\bf Y}}
	\def\BZ{{\bf Z}}
	\def\B0{{\bf 0}}
% Abbreviations for symbols ending with (x), with x being bold. 
% These macros, and the following ones, may be ignored as its almost as simple to write the longer versions, and there is less to remember.
\newcommand{\Xa}{ a(\bfm{x})}
\newcommand{\Xb}{ b(\bfm{x})}
\newcommand{\Xc}{ c(\bfm{x})}
\newcommand{\Xd}{ d(\bfm{x})}
\newcommand{\Xe}{ e(\bfm{x})}
\newcommand{\Xf}{ f(\bfm{x})}
\newcommand{\Xg}{ g(\bfm{x})}
\newcommand{\Xh}{ h(\bfm{x})}
\newcommand{\Xii}{ i(\bfm{x})}
\newcommand{\Xj}{ j(\bfm{x})}
\newcommand{\Xk}{ k(\bfm{x})}
\newcommand{\Xl}{ l(\bfm{x})}
\newcommand{\Xm}{ m(\bfm{x})}
\newcommand{\Xn}{ n(\bfm{x})}
\newcommand{\Xo}{ o(\bfm{x})}
\newcommand{\Xp}{ p(\bfm{x})}
\newcommand{\Xq}{ q(\bfm{x})}
\newcommand{\Xr}{ r(\bfm{x})}
\newcommand{\Xs}{ s(\bfm{x})}
\newcommand{\Xt}{ t(\bfm{x})}
\newcommand{\Xu}{ u(\bfm{x})}
\newcommand{\Xv}{ v(\bfm{x})}
\newcommand{\Xw}{ w(\bfm{x})}
\newcommand{\Xx}{ x(\bfm{x})}
\newcommand{\Xy}{ y(\bfm{x})}
\newcommand{\Xz}{ z(\bfm{x})}
\newcommand{\XA}{ A(\bfm{x})}
\newcommand{\XB}{ B(\bfm{x})}
\newcommand{\XC}{ C(\bfm{x})}
\newcommand{\XD}{ D(\bfm{x})}
\newcommand{\XE}{ E(\bfm{x})}
\newcommand{\XF}{ F(\bfm{x})}
\newcommand{\XG}{ G(\bfm{x})}
\newcommand{\XH}{ H(\bfm{x})}
\newcommand{\XI}{ I(\bfm{x})}
\newcommand{\XJ}{ J(\bfm{x})}
\newcommand{\XK}{ K(\bfm{x})}
\newcommand{\XL}{ L(\bfm{x})}
\newcommand{\XM}{ M(\bfm{x})}
\newcommand{\XN}{ N(\bfm{x})}
\newcommand{\XO}{ O(\bfm{x})}
\newcommand{\XP}{ P(\bfm{x})}
\newcommand{\XQ}{ Q(\bfm{x})}
\newcommand{\XR}{ R(\bfm{x})}
\newcommand{\XS}{ S(\bfm{x})}
\newcommand{\XT}{ T(\bfm{x})}
\newcommand{\XU}{ U(\bfm{x})}
\newcommand{\XV}{ V(\bfm{x})}
\newcommand{\XW}{ W(\bfm{x})}
\newcommand{\XX}{ X(\bfm{x})}
\newcommand{\XY}{ Y(\bfm{x})}
\newcommand{\XZ}{ Z(\bfm{x})}
% Abbreviate definitions of greek symbols followed by (x) with a bold x.
\newcommand{\XGa}{\alpha(\bfm{x})}
\newcommand{\XGb}{\beta(\bfm{x})}
\newcommand{\XGd}{\delta(\bfm{x})}
\newcommand{\XGe}{\epsilon(\bfm{x})}
\newcommand{\XGve}{\varepsilon(\bfm{x})}
\newcommand{\XGf}{\phi(\bfm{x})}
\newcommand{\XGvf}{\varphi(\bfm{x})}
\newcommand{\XGg}{\gamma(\bfm{x})}
\newcommand{\XGc}{\chi(\bfm{x})}
\newcommand{\XGi}{\iota(\bfm{x})}
\newcommand{\XGk}{\kappa(\bfm{x})}
\newcommand{\XGl}{\lambda(\bfm{x})}
\newcommand{\XGn}{\eta(\bfm{x})}
\newcommand{\XGm}{\mu(\bfm{x})}
\newcommand{\XGv}{\nu(\bfm{x})}
\newcommand{\XGp}{\pi(\bfm{x})}
\newcommand{\XGt}{\theta(\bfm{x})}
\newcommand{\XGvt}{\vartheta(\bfm{x})}
\newcommand{\XGr}{\rho(\bfm{x})}
\newcommand{\XGvr}{\varrho(\bfm{x})}
\newcommand{\XGs}{\sigma(\bfm{x})}
\newcommand{\XGvs}{\varsigma(\bfm{x})}
\newcommand{\XGj}{\tau(\bfm{x})}
\newcommand{\XGu}{\upsilon(\bfm{x})}
\newcommand{\XGo}{\omega(\bfm{x})}
\newcommand{\XGx}{\xi(\bfm{x})}
\newcommand{\XGy}{\psi(\bfm{x})}
\newcommand{\XGz}{\zeta(\bfm{x})}
\newcommand{\XGD}{\Delta(\bfm{x})}
\newcommand{\XGF}{\Phi(\bfm{x})}
\newcommand{\XGG}{\Gamma(\bfm{x})}
\newcommand{\XGL}{\Lambda(\bfm{x})}
\newcommand{\XGP}{\Pi(\bfm{x})}
\newcommand{\XGQ}{\Theta(\bfm{x})}
\newcommand{\XGS}{\Sigma(\bfm{x})}
\newcommand{\XGU}{\Upsilon(\bfm{x})}
\newcommand{\XGO}{\Omega(\bfm{x})}
\newcommand{\XGX}{\Xi(\bfm{x})}
\newcommand{\XGY}{\Psi(\bfm{x})}
% Abbreviate definitions of bold greek symbols followed by (x) with a bold x: note that defs.tex is
% needed first
\newcommand{\XBGa}{\bfm{\alpha(x)}}
\newcommand{\XBGb}{\bfm{\beta(x)}}
\newcommand{\XBGd}{\bfm{\delta(x)}}
\newcommand{\XBGe}{\bfm{\epsilon(x)}}
\newcommand{\XBGve}{\bfm{\varepsilon(x)}}
\newcommand{\XBGf}{\bfm{\phi(x)}}
\newcommand{\XBGvf}{\bfm{\varphi(x)}}
\newcommand{\XBGg}{\bfm{\gamma(x)}}
\newcommand{\XBGc}{\bfm{\chi(x)}}
\newcommand{\XBGi}{\bfm{\iota(x)}}
\newcommand{\XBGk}{\bfm{\kappa(x)}}
\newcommand{\XBGl}{\bfm{\lambda(x)}}
\newcommand{\XBGn}{\bfm{\eta(x)}}
\newcommand{\XBGm}{\bfm{\mu(x)}}
\newcommand{\XBGv}{\bfm{\nu(x)}}
\newcommand{\XBGp}{\bfm{\pi(x)}}
\newcommand{\XBGt}{\bfm{\theta(x)}}
\newcommand{\XBGvt}{\bfm{\vartheta(x)}}
\newcommand{\XBGr}{\bfm{\rho(x)}}
\newcommand{\XBGvr}{\bfm{\varrho(x)}}
\newcommand{\XBGs}{\bfm{\sigma(x)}}
\newcommand{\XBGvs}{\bfm{\varsigma(x)}}
\newcommand{\XBGj}{\bfm{\tau(x)}}
\newcommand{\XBGu}{\bfm{\upsilon(x)}}
\newcommand{\XBGo}{\bfm{\omega(x)}}
\newcommand{\XBGx}{\bfm{\xi(x)}}
\newcommand{\XBGy}{\bfm{\psi(x)}}
\newcommand{\XBGz}{\bfm{\zeta(x)}}
\newcommand{\XBGD}{\bfm{\Delta(x)}}
\newcommand{\XBGF}{\bfm{\Phi(x)}}
\newcommand{\XBGG}{\bfm{\Gamma(x)}}
\newcommand{\XBGL}{\bfm{\Lambda(x)}}
\newcommand{\XBGP}{\bfm{\Pi(x)}}
\newcommand{\XBGT}{\bfm{\Theta(x)}}
\newcommand{\XBGS}{\bfm{\Sigma(x)}}
\newcommand{\XBGU}{\bfm{\Upsilon(x)}}
\newcommand{\XBGO}{\bfm{\Omega(x)}}
\newcommand{\XBGX}{\bfm{\Xi(x)}}
\newcommand{\XBGY}{\bfm{\Psi(x)}}
%Abbreviate symbols for bold-face letters followed by (x) with x being bold: note you need defs.tex first
\newcommand{\XBa}{{\bfm a}(\bfm{x})}
\newcommand{\XBb}{{\bfm b}(\bfm{x})}
\newcommand{\XBc}{{\bfm c}(\bfm{x})}
\newcommand{\XBd}{{\bfm d}(\bfm{x})}
\newcommand{\XBe}{{\bfm e}(\bfm{x})}
\newcommand{\XBf}{{\bfm f}(\bfm{x})}
\newcommand{\XBg}{{\bfm g}(\bfm{x})}
\newcommand{\XBh}{{\bfm h}(\bfm{x})}
\newcommand{\XBi}{{\bfm i}(\bfm{x})}
\newcommand{\XBj}{{\bfm j}(\bfm{x})}
\newcommand{\XBk}{{\bfm k}(\bfm{x})}
\newcommand{\XBl}{{\bfm l}(\bfm{x})}
\newcommand{\XBm}{{\bfm m}(\bfm{x})}
\newcommand{\XBn}{{\bfm n}(\bfm{x})}
\newcommand{\XBo}{{\bfm o}(\bfm{x})}
\newcommand{\XBp}{{\bfm p}(\bfm{x})}
\newcommand{\XBq}{{\bfm q}(\bfm{x})}
\newcommand{\XBr}{{\bfm r}(\bfm{x})}
\newcommand{\XBs}{{\bfm s}(\bfm{x})}
\newcommand{\XBt}{{\bfm t}(\bfm{x})}
\newcommand{\XBu}{{\bfm u}(\bfm{x})}
\newcommand{\XBv}{{\bfm v}(\bfm{x})}
\newcommand{\XBw}{{\bfm w}(\bfm{x})}
\newcommand{\XBx}{{\bfm x}(\bfm{x})}
\newcommand{\XBy}{{\bfm y}(\bfm{x})}
\newcommand{\XBz}{{\bfm z}(\bfm{x})}
\newcommand{\XBA}{{\bfm A}(\bfm{x})}
\newcommand{\XBB}{{\bfm B}(\bfm{x})}
\newcommand{\XBC}{{\bfm C}(\bfm{x})}
\newcommand{\XBD}{{\bfm D}(\bfm{x})}
\newcommand{\XBE}{{\bfm E}(\bfm{x})}
\newcommand{\XBF}{{\bfm F}(\bfm{x})}
\newcommand{\XBG}{{\bfm G}(\bfm{x})}
\newcommand{\XBH}{{\bfm H}(\bfm{x})}
\newcommand{\XBI}{{\bfm I}(\bfm{x})}
\newcommand{\XBJ}{{\bfm J}(\bfm{x})}
\newcommand{\XBK}{{\bfm K}(\bfm{x})}
\newcommand{\XBL}{{\bfm L}(\bfm{x})}
\newcommand{\XBM}{{\bfm M}(\bfm{x})}
\newcommand{\XBN}{{\bfm N}(\bfm{x})}
\newcommand{\XBO}{{\bfm O}(\bfm{x})}
\newcommand{\XBP}{{\bfm P}(\bfm{x})}
\newcommand{\XBQ}{{\bfm Q}(\bfm{x})}
\newcommand{\XBR}{{\bfm R}(\bfm{x})}
\newcommand{\XBS}{{\bfm S}(\bfm{x})}
\newcommand{\XBT}{{\bfm T}(\bfm{x})}
\newcommand{\XBU}{{\bfm U}(\bfm{x})}
\newcommand{\XBV}{{\bfm V}(\bfm{x})}
\newcommand{\XBW}{{\bfm W}(\bfm{x})}
\newcommand{\XBX}{{\bfm X}(\bfm{x})}
\newcommand{\XBY}{{\bfm Y}(\bfm{x})}
\newcommand{\XBZ}{{\bfm Z}(\bfm{x})}
% Abbreviate symbols for caligraphic-face letters ending with (x) with x being bold:
\newcommand{\XCa}{{\cal a}(\bfm{x})}
\newcommand{\XCb}{{\cal b}(\bfm{x})}
\newcommand{\XCc}{{\cal c}(\bfm{x})}
\newcommand{\XCd}{{\cal d}(\bfm{x})}
\newcommand{\XCe}{{\cal e}(\bfm{x})}
\newcommand{\XCf}{{\cal f}(\bfm{x})}
\newcommand{\XCg}{{\cal g}(\bfm{x})}
\newcommand{\XCh}{{\cal h}(\bfm{x})}
\newcommand{\XCi}{{\cal i}(\bfm{x})}
\newcommand{\XCj}{{\cal j}(\bfm{x})}
\newcommand{\XCk}{{\cal k}(\bfm{x})}
\newcommand{\XCl}{{\cal l}(\bfm{x})}
\newcommand{\XCm}{{\cal m}(\bfm{x})}
\newcommand{\XCn}{{\cal n}(\bfm{x})}
\newcommand{\XCo}{{\cal o}(\bfm{x})}
\newcommand{\XCp}{{\cal p}(\bfm{x})}
\newcommand{\XCq}{{\cal q}(\bfm{x})}
\newcommand{\XCr}{{\cal r}(\bfm{x})}
\newcommand{\XCs}{{\cal s}(\bfm{x})}
\newcommand{\XCt}{{\cal t}(\bfm{x})}
\newcommand{\XCu}{{\cal u}(\bfm{x})}
\newcommand{\XCv}{{\cal v}(\bfm{x})}
\newcommand{\XCw}{{\cal w}(\bfm{x})}
\newcommand{\XCx}{{\cal x}(\bfm{x})}
\newcommand{\XCy}{{\cal y}(\bfm{x})}
\newcommand{\XCz}{{\cal z}(\bfm{x})}
\newcommand{\XCA}{{\cal A}(\bfm{x})}
\newcommand{\XCB}{{\cal B}(\bfm{x})}
\newcommand{\XCC}{{\cal C}(\bfm{x})}
\newcommand{\XCD}{{\cal D}(\bfm{x})}
\newcommand{\XCE}{{\cal E}(\bfm{x})}
\newcommand{\XCF}{{\cal F}(\bfm{x})}
\newcommand{\XCG}{{\cal G}(\bfm{x})}
\newcommand{\XCH}{{\cal H}(\bfm{x})}
\newcommand{\XCI}{{\cal I}(\bfm{x})}
\newcommand{\XCJ}{{\cal J}(\bfm{x})}
\newcommand{\XCK}{{\cal K}(\bfm{x})}
\newcommand{\XCL}{{\cal L}(\bfm{x})}
\newcommand{\XCM}{{\cal M}(\bfm{x})}
\newcommand{\XCN}{{\cal N}(\bfm{x})}
\newcommand{\XCO}{{\cal O}(\bfm{x})}
\newcommand{\XCP}{{\cal P}(\bfm{x})}
\newcommand{\XCQ}{{\cal Q}(\bfm{x})}
\newcommand{\XCR}{{\cal R}(\bfm{x})}
\newcommand{\XCS}{{\cal S}(\bfm{x})}
\newcommand{\XCT}{{\cal T}(\bfm{x})}
\newcommand{\XCU}{{\cal U}(\bfm{x})}
\newcommand{\XCV}{{\cal V}(\bfm{x})}
\newcommand{\XCW}{{\cal W}(\bfm{x})}
\newcommand{\XCX}{{\cal X}(\bfm{x})}
\newcommand{\XCY}{{\cal Y}(\bfm{x})}
\newcommand{\XCZ}{{\cal Z}(\bfm{x})}
% Use \ar{}, \ab{} and \ag{} to create sections of text in red, blue, or green, to mark changes for coauthors or referees.
	\usepackage{fancyhdr}
		\usepackage{color}
                 \newcommand{\ar}[1]{\textcolor{red}{#1}}
                 \newcommand{\ab}[1]{\textcolor{blue}{#1}}
          \newcommand{\ag}[1]{\textcolor{green}{#1}}
	\newtheorem {Thm} {Theorem} [section]
	\newtheorem {Cor} [Thm] {Corollary}
	\newtheorem {Lem} [Thm] {Lemma}
	\newtheorem {Pro} [Thm] {Proposition}
	\newtheorem {Adef} [Thm] {Definition}
	\newenvironment {Def} {\begin{Adef}\rm}{\end{Adef}}
	\newtheorem {Arem} [Thm] {Remark}
	\newenvironment {Rem} {\begin{Arem}\rm}{\end{Arem}}
	\newtheorem {Aexa} [Thm] {Example}
	\newenvironment {Exa} {\begin{Aexa}\rm}{\end{Aexa}}
	\newtheorem {Anot} [Thm] {Notation}
	\newenvironment {Not} {\begin{Anot}\rm}{\end{Anot}}
	\newtheorem{question}{Question}
	% Math terms that should be in Roman:
	\def\ii{{\rm i}}
	\def\dd{{\rm d}}
	\newcommand{\diag}{\mbox{diag}}
         \newcommand{\rank}{\mbox{rank}}
	\newcommand{\sign}{\mbox{sign}}
	\newcommand{\lnn}{\mbox{ln}}
	\newcommand{\Ker}{{\rm Ker}}
	\newcommand{\dist}{\mbox{dist}}
\def\implies{{\, \Rightarrow\,}}
\newcommand{\detail}[1] {}
\makeatletter
\makeatother
	%%%%%%%%%%%%%%%%%%%%%%
	% Keywords, AMS, PACS
	%%%%%%%%%%%%%%%%%%%%%%
	\newenvironment{keywords}
	{\noindent{\bf Key words.}\small}{\par\vspace{1ex}}
	\newenvironment{AMS}
	{\noindent{\bf AMS subject classifications 2000.}\small}{\par}
	\newenvironment{PACS}{\noindent{\bf PACS number(s):}\small}{\par}
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%%%%%%%TITLE %%%TITLE %%%TITLE %%%TITLE %%%TITLE %%%TITLE %%%%%%%%
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	
	\title{\bf A latex template with helpful macros.}
	
	\author{Graeme W. Milton \\ \\Department of Mathematics,
            University of Utah, e-mail: milton@math.utah.edu}
	
	\begin{document}
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%%%%%%%BEGIN DOCUMENT %%%%BEGIN DOCUMENT %%%%BEGIN DOCUMENT%%%%%%%
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	
	\maketitle
	
	\tableofcontents
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%%%%%%%%ABSTRACT%%%%%%%%%%%
	%%%%%%%%%%%%%%%%%%%%%%%%%%%
	
	\begin{abstract}
The abstract goes here.
	\end{abstract}
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%%%%%%%%%KEYWORDS%%%%%%%%%%
	%%%%%%%%%%%%%%%%%%%%%%%%%%%
	
	\begin{keywords}
	    Latex Macros, Latex Template
	\end{keywords}
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%
	%%%   AMS or PACS   %%%%%%
	%%%%%%%%%%%%%%%%%%%%%%%%%%
	
	\begin{AMS}
	12345, 12346, 12347
	\end{AMS}
	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\date{}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\setcounter{equation}{0}
\labsect{intro}
%% Here the setcounter resets the equation numbering so it goes *.1, *.2, etc. where * is the section number
%% It is not necessary to label all sections with \labsect just those that you refer to.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The main purpose of this is to provide a handy shortcut to creating latex math symbols, using the macros provided in the source code. Besides well
known books on tex and latex,
such as those of Donald Knuth \cite{Knuth:1986:T}
and Leslie Lamport \cite{Lamport:1985:LDP} a  good source of information is the website \url{http://www.math.utah.edu/faq/texmf/} of Nelson Beebe. For putting
latex equations into powerpoint presentations see
the webpage \url{http://www.sci.utah.edu/~macleod/talks/equations-ppt.html}
of Rob MacLeod.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Examples of figures}
\setcounter{equation}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Figure \fig{first} shows the covers of the two books, "The Theory of Composites" \cite{Milton:2002:TOC}
and "Extending the Theory of Composites to Other Areas of Science" \cite{Milton:2016:ETC}. 
% Note that the \protect is not needed as the citations are outside the figure environment
\begin{figure}[!ht]
\centering
\includegraphics[width=0.9\textwidth]{mybooks.pdf}
% Below we have citations within a figure caption environment. The command \protect helps protect the citation in this environment. 
\caption{An example of a figure. (a) shows my first book \protect{\cite{Milton:2002:TOC}} published in 2002 that was reviewed by Gr\'egoire Allaire in Math Reviews MR1899805 (2003d:74077).
The second book (b)  \protect{\cite{Milton:2016:ETC}} published in 2016, has 4 chapters coauthored with Maxence Cassier, Ornella Mattei, Moti Milgrom, and Aaron Welters, and has been reviewed by Pradeep Sharma
in Applied Mechanics Reviews \protect{\cite{Sharma:2017:BRE}}  and by Yury Grabovsky in SIAM book reviews \protect{\cite{Grabovsky:2018:BRE}}. Note the
use of protect\{\} in figure captions when citing articles}
\labfig{first}
\end{figure}
Figure \fig{second} shows some covers associated with the reversal of the Hall coefficients. It is an example of the use of subfigures. 
\begin{figure}[ht!]
	\centering
	\begin{subfigure}{.45\linewidth}
		\includegraphics[width=\textwidth]{PRLCover.pdf}
		\caption{Physical Review Letters Cover}
	\end{subfigure}
	\hskip2em
	\begin{subfigure}{.45\linewidth}
		\includegraphics[width=\textwidth]{PhysTodayCover.pdf}
		\caption{Physics Today Cover}
	\end{subfigure}
	\begin{subfigure}{.45\linewidth}
		\includegraphics[width=\textwidth]{NatureCov.pdf}
		\caption{Nature Reviews Cover}
	\end{subfigure}
	\caption{An example of the use of subfigures. These cover figures
          show a simplified version \protect{\cite{Kadic:2015:HES}}
          of the metamaterial that we proved would reverse the sign of the Hall coefficient \protect{\cite{Briane:2009:HTD}}, experimentally confirmed by the group of
          Martin Wegener \protect{\cite{Kern:2016:EES}}. 
        }
	\labfig{second}
      \end{figure}

      In the coming Section \sect{exampeqn} we give some examples of equations.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      \subsection{Examples of some equations}
      \labsect{exampeqn}
% We do not use \setcounter{equation}{0} here as it is a subsection.
%% Here the setcounter resets the equation numbering so it goes *.1, *.2, etc. where * is the section number
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Here is an equation that shows the usefulness of underbraces, the pmatrix
environment, and the shorthand for bold roman characters and bold greek characters.
\beq \ubr{\bpm \Bj_1(\Bx) \\ \Bj_2(\Bx) \epm}_{\BJ(\Bx)}=
\ubr{\bpm \BGa(\Bx) & \BGb(\Bx) \\ \BGg(\Bx) & \BGd(\Bx) \epm}_{\BL(\Bx)}
\ubr{\bpm \Be_1(\Bx) \\ \Be_2(\Bx) \epm}_{\BE(\Bx)}.
\eeq{2.1a}
% Note that 2.1a is just a label and does not reflect the final numbering. Generally, following the advice of Michael E. Fisher ,
% one should try to number all equations. This makes it easier for other people who may want to refer to your equations
% and it avoids them having to say state "the third equation after equation (5.6)".
The equation \eq{2.1a} shows a constitutive law in a composite material where $\BL(\Bx)$ denotes the local
tensor field. Now we use the equation array environment, to write the differential constraints on the fields
\beqa \Div\Bj_1=0,\quad\Div\Bj_2=0, \nonum
\Be_1=-\Grad V_1,\quad \Be_2=-\Grad V_2 
\eeqa{2.1b}
In a body $\GO$ the equations \eq{2.1a} are satisified in the interior. At the boundary of $\GO$ one may
specify the boundary values of the potential:
\beq V_1(\Bx)=V_1^0(\Bx),\quad V_2(\Bx)=V_2^0(\Bx) \text{ for all } \Bx\in\Md\GO.
\eeq{2.2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
      \section{Another Section}
      \labsect{anothChap}
      \setcounter{equation}{0}
%%% Here thanks to "\setcounter{equation}{0}" the equatiions begin with ".1". 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      Here we give an elasticity equation with no number:
      $$ \BGs(\Bx)=\BCC(\Bx) \BGe(\Bx), \quad \text{where}
      \BGe=\tfrac{1}{2}[\Grad\Bu+(\Grad\Bu)^T]. $$
      and one with a number, reflecting the new Section \sect{anothChap}:
      \beq \Div\BGs=0.\eeq{stress}
% We do not use \setcounter{equation}{0} here as it is a subsection.
%% Here the setcounter resets the equation numbering so it goes *.1, *.2, etc. where * is the section number
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $\Md\GO$ represents the boundary of the region $\GO$. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgements}
%Here \section* is used as the Acknowledgements section number is not needed
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The author is greatful to Nelson Beebe, a fountain of knowledge about
Tex and Latex, for his help and friendly advice and, in particular, for his
generous assistance in producing the books shown in figure \fig{first}. Additionally
the author is grateful to the National Science Foundation for supporting his research.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Bibliography}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For references bibtex is recommended as it saves having to reformat
references according to the journal style. Instead one uses a .sty file
appropriate to the journal to which you are submitting your paper.
A lot of bibtex entries associated with the fields of my research are
in the files tcbook.bib and newref.bib that can be found on my website.

Here we use just a plain bibliography stlye file.
Substitute for plain.sty, whatever style file you require. The bibtex files
tcbook.bib and newref.bib contain the bibliography references: tcbook.bib
are those referenced in my book ``The theory of composites''
\cite{Milton:2002:TOC}, and  newref.bib are those referenced since that
book was finished. Both files are available on my website. 
\bibliographystyle{plain}
\bibliography{tcbook,newref}
\end{document}


