TITLE: Is there any potential associated with magnetism?
QUESTION [10 upvotes]: Can anybody please tell me if magnetism is a conservative force or if there is a field associated with it? How to reason? One thing I know is that the work done by a magnetic force is equal to $0$.

REPLY [12 votes]: The force $q\vec v\times \vec B$ acting on a charged particle is clearly not conservative because it depends on the velocity. Conservative forces are those that integrate to a fixed work – energy difference that is independent of the path – between two points so they may only depend on the location. We can't associate a potential with a force that is velocity-dependent.
On the other hand, the magnetic force acting e.g. on a small piece of ferromagnet deserves a special discussion. If the magnet has magnetic moment $\vec m$, the energy of this magnet in an external magnetic field is simply
$$ U = -\vec m\cdot \vec B. $$
One may view this term in the energy as a potential energy that depends on the orientation of the magnet. This force is trying to orient $\vec m$ in the same direction as the external $\vec B$. Note that the field $\vec B$ above may depend on the position too, which is why a magnet that is already oriented to minimize the energy may be dragged to the region where the magnetic field gets stronger.
In general, whether or not $\vec B$ is the gradient of something depends on ${\rm curl}\, \vec B$ and this is governed by one of Maxwell's equations, namely
$$ \nabla \times \mathbf{B} = \mu_0\left(\mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t} \right) .$$
It's called Ampère's circuital law (with Maxwell's correction: the last term with the time derivative – also known as displacement current). You see that the curl doesn't vanish in the presence of currents and/or time-dependent electric fields. However, in the absence of currents and time-varying electric fields, the curl is equal to zero, and you may write $\vec B$ as the gradient of a potential in these regions.
However, this doesn't immediately translate to the conservative character of any magnetic force because the magnetic forces we have measured aren't proportional to $\vec B$ as vectors. We would have $\vec F = c\vec B$ which would be analogous to the electric force acting on charges but such a form of the force only applies to magnetic monopoles, something that may exist in Nature but only in the form of extremely heavy elementary particles that we haven't observed yet.
So because $\vec F$ isn't $c\vec B$ for any known force, the question of whether $\vec B$ is a gradient of something has no impact on the question of whether a known force related to magnetism is conservative or not. For this reason, it isn't even too useful to consider potentials $\Phi_B$ such that $\vec B = -\nabla \Phi_B$ even though in some cases and regions, we could find such a potential (when the curl vanishes).
Instead, it is useful to write $\vec B = {\rm curl}\,\vec A$ where $\vec A$ is called the vector potential. As long as there are no magnetic monopoles, $\vec B$ may always be written in this way because the only obstruction that could prevent us from rewriting $\vec B$ in this way would be a nonzero ${\rm div}\,\vec B$ but this divergence vanishes due to a simple Maxwell's equation.
However, the vector potential $\vec A$ can't be "immediately" interpreted as the potential energy of a unit charge etc. Instead, it appears in the Lagrangian and the Hamiltonian (with various signs) in the combination $\vec j\cdot \vec A$ where $\vec j$ is the electric current. In this sense, the vector potential is the potential energy per unit current (both of them are vectors).
This is a bit formal description. In order to get the actual energy, one would have to integrate over the whole paths of the currents. And it may be done. For example, if you have a small loop of electric current, it behaves like a magnet (electromagnet) and the contour integral $\oint \vec{d\ell}\cdot \vec A$ of $\vec A$ over this loop is nothing else than $\vec B\cdot \vec{dS}$ – by Stokes' law – where $\vec{dS}$ is the infinitesimal area of the loop with the normal direction added to turn it into a vector (a right-hand rule is implicit here). So this tells you that the potential energy of the loop of current i.e. a small electromagnet is nothing else than the $-\vec m\cdot \vec B$ potential energy I mentioned at the beginning.


TITLE: Find an estimator for the probability of a given random variable
QUESTION [3 upvotes]: Let $X$ be a random variable with a Binomial distribution $X \sim B(100,p)$. We want to find the probability $P\{X<4\}$ by estimating the parameter $p$ using the method of moments and maximum likelihood method.

I'm stuck. All I could do was conclude that using the method of moments, an estimator for $p$ would be $\bar X/100$. What am I supposed to do with this information and how can I use the ML method for the binomial distribution?

REPLY [1 votes]: If I understand correctly, the sample size is $1$.  The first sample moment is the mean of the observations in the sample, and since there is only one such observation, it is just the value of that observation, $X$.  The first population moment for $B(100,p)$ is $100p$.  In the method of moments one then equates the two and solves for $p$, thus:
$$
X=100p
$$
$$
p = \frac X {100}.
$$
That is the estimate of $p$ by the method of moments.  The likelihood function is
$$
L(p) = \binom {100} X p^X (1-p)^{100-X}.
$$
Finding the value of $p$ that maximizes this is the same as finding the value of $p$ that maximizes its logarithm because the logarithm function is increasing.  The logarithm is
$$
\ell(p) = \log\binom{100} X + \left( X\log p + (100-X)\log(1-p) \right).
$$
Then we have
$$
\ell\,'(p) = \frac X p - \frac{100-X}{1-p} = \frac{-100\left( p - \dfrac X {100} \right)}{p(1-p)} \quad \begin{cases} \ge 0 & \text{if }0<p\le 100/X, \\[8pt]  \le 0 & \text{if }100/X \le p < 1. \end{cases}
$$
Hence the maximizing value of $p$ is $\dfrac X {100}$.
Both methods yield the same estimate in this case.
If you use that estimate in place of $p$ in $\Pr(X<4)$ then you get
\begin{align}
& \Pr(X\in\{0,1,2,3\}) \\[10pt]
= {} & \binom {100} 0 \hat p^0(1-\hat p)^{100} + \binom{100} 1 \hat p^1(1-\hat p)^{99} + \binom{100} 2 p^2 (1-\hat p)^{98} + \binom{100} 3 \hat p^3(1-\hat p)^{97}
\end{align}
where $\hat p = X/100$ is the estimate.


TITLE: every number $n\in \mathbb{Z} $ can be represented as sum of different powers of $2$
QUESTION [1 upvotes]: Using generating function prove that every number $n\in \mathbb{Z} $ can be represented as sum of different power of $2$,
I mean, that for every $n\in \mathbb{Z}$ $$n=2^{k_1} +2^{k_2} +2^{k_3} +...  $$ where $k_i \neq k_j  $
Please give me some advices.

REPLY [4 votes]: Hint Prove that $\displaystyle  \prod_{\nu\geqslant 0}(1+x^{2^{\nu}})=\frac{1}{1-x}=1+x+x^2+x^3+\cdots.$ Note that this proves existence and uniqueness.


TITLE: In Rotational Dynamics do we have to consider the torque of pseudo force in Instantaneous Axis of Rotation frame?
QUESTION [1 upvotes]: In the attached image have I done everything right? If yes then why there is no torque of Pseudo Force?

REPLY [0 votes]: There will be a pseudo force acting on the COM of that body. How ever this will not provide any torque because, the instantaneous acceleration of the Point of contact will be $\frac{v^2}{R}$ acting upwards. So when the pseudo force is applied on the centre of mass, it will be pointed downwards passing through the IAOR. Hence no torque will be generated.


text:
\chapter{Upper Bound}

In this chapter we prove an upper bound on the number $N^K_{\mathrm{sd}}(\lambda)$ of self-dual cuspforms on $GL_N(\mathbb{A_Q})$.
The idea of the proof is to consider the functorial descent of each self-dual $\Pi$ to an appropriate classical group.
First we collect some information about functoriality in Section \ref{upper tools}. The proof is then in Section \ref{upper proof}.

Throughout this chapter, all groups and representations are adelic, unless stated otherwise.

\section{Tools}\label{upper tools}

Throughout this chapter, let $K=K_\infty\times \prod K_p$ be a compact subgroup of $GL_N(\mathbb A)$ with $K_\infty=O(N)$.
Mostly we will be working with the principal congruence subgroup $K=K(M)$,
for which $K_p$ is the group of matrices $A\in GL_N(\mathcal O_p)$ satisfying $A\equiv I\pmod {p^{m(p)}}$
(when $m(p)=0$ this just means that $K_p=GL_N(\mathcal O_p))$, where $M=\prod p^{m(p)}$.

Let us first establish global analogues of the local results from Chapter 4.

\begin{proposition}\label{upper K fixed}
a) For every $K$ there is a constant $C_K$ (depending only on $K$) such that for every automorphic representation $\Pi$ of $GL_N(\mathbb A)$ one has $\dim\Pi^K\leq C_K$.

b) Let $\mathbf G$ be a quasisplit classical group.
For each $M\in\mathbb N$ there exists an arithmetic compact subgroup $K^\prime\subset G(\mathbb{A})$ such that:

For each self-dual cuspidal automorphic representation $\Pi$ of $GL_N(\mathbb {A})$ which descends to a cuspidal automorphic representation $\pi$ of $G(\mathbb {A})$
and satisfies $\Pi^{K(M)}\neq 0$, one has $\pi^{K^\prime}\neq 0$.
\end{proposition}

\begin{proof}
a) Take $\Pi$ such that $\Pi^K\neq 0$ and write it as a restricted tensor product $\Pi=\otimes\Pi_v$.
Then $\Pi^K=\prod \Pi_v^{K_v}$, and so $\dim\Pi^K=\prod \dim\Pi_v^{K_v}$.
When $K_v$ is maximal and $\Pi_v$ is unramified (or when $v=\infty$), then $\dim\Pi_v^{K_v}=1$, and so this is just a finite product.

At each of the remaining finitely many places $v=p$, we have a constant $c_p=c_{p, K_p}$ (independent of $\Pi_p$) from Theorem \ref{Bernstein} such that $\dim\Pi_p^{K_p}\leq c_p$.
Let $C_K$ be the product of $c_p$ over these ramified places. Then $\dim\Pi^K\leq C_K$ and $C_K$ depends only on $K$.

b) Take $\Pi=\otimes\Pi_v$ such that $\Pi^K\neq 0$ and which descends to a cuspidal automorphic representation $\pi$ of $\mathbf G(\mathbb {A_Q})$.
When $K_p$ is maximal, $\Pi_p$ is unramified. Thus also $\pi_p$ is unramified and we can set $K^\prime_p$ to be a special maximal compact subgroup adopted to a standard minimal parabolic so that we have the corresponding Iwasawa decomposition (when $\mathbf G$ is split, we can take $K^\prime_p=\mathbf G(\mathcal O_p)$). Likewise, we can define $K^\prime_\infty$ to be a maximal compact subgroup of $\mathbf G(\mathbb R)$ (again chosen suitable to provide Iwasawa decomposition).

At the remaining places $v=p$, we use Corollary \ref{depth cor}, which gives us a
compact open subgroup $K_p^\prime\subset G$ with $\pi_p^{K^\prime_p}\neq 0$.
Then $K^\prime=K^\prime_\infty\times\prod K^\prime_p$ satisfies the conditions of the proposition.
\end{proof}

Let us next discuss to which classical groups $\mathbf G$ can $\Pi$ descend.
When $N=2n+1$ is odd, the only possibility is $\mathbf G=Sp_{2n}$. But when $N=2n$ is even, $\mathbf G$ can be split $SO_{2n}$ or $SO_{2n+1}$,
or one of the infinitely many non-split quasisplit groups $SO_{2n}^\ast$. However, for a fixed compact subgroup $K$, only finitely many of these can occur:

\begin{lemma}\label{upper sk}
Let $K=K(M)$. There is a finite set $S_K$ of quasisplit classical groups $\mathbf G$ such that every self-dual cuspidal automorphic representation $\Pi$ of $GL_N(\aq)$ with $\Pi^K\neq\{0\}$ descends to some $\mathbf G\in S_K$.
\end{lemma}

\begin{proof}
We just need to consider the case when $N=2n$. Let $\Pi$ be a self-dual cuspidal automorphic representation of $GL_N(\aq)$ with $\Pi^K\neq\{0\}$ and let $\chi$ be the central character of $\Pi$.
This is a character of the center $Z\simeq GL_1(\aq)$ of $GL_N(\aq)$.
Assume moreover that $\Pi$ descends to one of the non-split groups $SO_{2n}^\ast$. This group is then uniquely determined by the central character $\chi$.

We have $\chi(Z\cap K)=1$, which implies that the conductor of $\chi$ divides $M$.
By class field theory, there are only finitely many characters with conductor dividing $M$. Thus there are also only finitely many possibilities for the group $SO_{2n}^\ast$.
\end{proof}

\section{Statement and Proof of Theorem \ref{upper upper}}\label{upper proof}

We are now ready to prove the upper bound.

For $\lambda>0$ we have defined
$N^K(\lambda)=N_{GL(N)}^K(\lambda)=\sum_{\lambda(\Pi)\leq\lambda} \dim \Pi^K$, where the sum is over
all cuspidal automorphic representations $\Pi$ of $GL_N(\mathbb A)$ such that the
Laplacian eigenvalue $\lambda(\Pi)$ is at most $\lambda$.
Similarly define $N^K_{\mathrm{sd}}(\lambda)$ to be a similar sum $\sum_{\lambda(\Pi)\leq\lambda} \dim\Pi^K$, but this time ranging only over self-dual representations $\Pi$.

\begin{theorem}\label{upper upper}
Let $K=K(M)$.
There are positive constants $c$ and $\lambda_0$ (both depending on $K$) such that $N^K_{\mathrm{sd}}(\lambda)\leq c\lambda^{d/2}$ for all $\lambda>\lambda_0$.
Here $d=n^2+n$ for $N=2n+\varepsilon$ with $\varepsilon=0, 1$.
\end{theorem}

\begin{proof}
First of all, when $N=2$, we can just use the upper bound from (full) Weyl's Law for $GL_2$, since the dimension $d$ is 2 in both cases. Assume from now on that $N\neq 2$.

We need to estimate $$N^K_{\mathrm{sd}}(\lambda)=\sum_{\lambda(\Pi)\leq\lambda,\ \Pi\mathrm{\ sd}} \dim\Pi^K.$$

By Lemma \ref{upper sk}, there is a finite set $S=S_K$ such that each of our self-dual $\Pi$ descends to some $\mathbf G\in S$.
Hence we can write $N^K_{\mathrm{sd}}(\lambda)=\sum_{\mathbf G\in S}N^K_{\mathrm{sd}, \mathbf G}(\lambda)$, where $N^K_{\mathrm{sd}, \mathbf G}(\lambda)$ is the sum over all $\Pi$ which descend to $\mathbf G$.
It thus suffices to obtain an upper bound for each $N^K_{\mathrm{sd}, \mathbf G}(\lambda)$.

By Proposition \ref{upper K fixed}a), there is $c_1$ such that $\dim\Pi^K\leq c_1$ for each such $\Pi$. Hence
$$N^K_{\mathrm{sd}, \mathbf G}(\lambda)\leq c_1\cdot\#\{\Pi\ |\ \lambda(\Pi)\leq\lambda,\ \Pi^K\neq 0,\ \Pi\mathrm{\ descends\ to\ }\mathbf G\}.$$

Let $\Pi$ be such that $\lambda(\Pi)\leq\lambda,\ \Pi^K\neq 0$, and $\Pi$ descends to $\mathbf G$, and let $\pi$ be its descent, which is a cuspidal automorphic representation of $\mathbf G(\aq)$.
By Theorem \ref{main laplace}, the Laplacian eigenvalue of $\Pi$ is $\lambda(\Pi)=c_G\lambda(\pi)+d_G$. Hence there is $c_2$ such that for sufficiently large $\lambda$ we have $\lambda(\pi)\leq c_2\lambda$.
By Proposition \ref{upper K fixed}b), there is a compact subgroup $K^\prime$ of $\mathbf G(\aq)$ such that $\pi^{K^\prime}\neq 0$. (Note that neither $c_2$ nor $K^\prime$ depend on $\Pi$.)

Thus we see that
$$\#\{\Pi\ |\ \lambda(\Pi)\leq\lambda,\ \Pi^K\neq 0,\Pi\mathrm{\ descends\ to\ }\mathbf G\}\leq$$
$$\leq\#\{\pi\ |\ \pi\mathrm{\ repre\-sen\-tation\ of\ }\mathbf G(\aq),\ \lambda(\pi)\leq c_2\lambda,\ \pi^{K^\prime}\neq 0\}.$$
This is at most $N_\mathbf{G}^{K^\prime}(c_2\lambda)$.

Since $N\neq 2$, $\mathbf G$ is not $SO_2^\ast$. Hence $\mathbf G^1(\mathbb R)$ is not compact and we can use Donnelly's upper bound from Proposition \ref{adelic donnelly}. It gives us $N_\mathbf{G}^{K^\prime}(c_2\lambda)\leq c_3\lambda^{d(G)/2}$ for $\lambda$ large enough and some $c_3$ as we needed. Here
$d(G)$ is the dimension of the symmetric space for $\mathbf G$, given by the table preceding Proposition \ref{adelic donnelly}.
\end{proof}

config: arxiv


TITLE: "Universal" versus "nonuniversal" in the topological entanglement entropy
QUESTION [0 upvotes]: In their seminal paper, Kitaev and Preskill consider a 2D area law for entanglement of the form
$$
S(\rho) = \alpha |\partial A| - \gamma + \cdots.
$$
Here $\rho$ denotes the density matrix associated with the ground state of the system within a circular region $A$, $\alpha$ and $\gamma$ are constants, $|\partial A|$ is the circumference of the circle, and $\cdots$ represents corrections that vanish as the circular region is taken to be large.
Preskill and Kitaev describe $\alpha$ as "nonuniversal and ultraviolet divergent" and $\gamma$ is a "universal additive constant characterizing a global feature of the entanglement in the ground state". They then go on to detail different systems which have different values of $\gamma$.
I am confused in one what sense $\alpha$ is nonuniversal and $\gamma$ is. Both constants take different values for different systems. The constant $\gamma$ is more robust and has more physical meaning as it is a topological invariant, related to the total quantum dimension. However, I still find this distinction between "universal" and "nonuniversal" to be puzzling as both values are system-dependent. In what sense are these words being used in this context?

REPLY [2 votes]: The idea is that $\gamma$ is constant throughout a phase - i.e., it can only change at a phase transition, i.e. when the gap closes. Thus, it can be used as a signature of the phase - in this sense, it is universal.
On the other hand,  $\alpha$ will vary continuously throughout the phase and can take any possible value, and it will also not be robust against local perturbations. In particular, this means it cannot be used as a signature to identify a phase (since it can take any value in any phase). (Of course, this does not mean it is completely useless - on the contrary, it quantifies the amount of entanglement per unit volume in the state.)
Maybe the most clear way to see this is to check what happens to $S(\rho)$ when you rescale, i.e., renormalize, the system: Under rescaling by a factor $\lambda$, $\alpha\mapsto\lambda \alpha$, while $\gamma$ will not change.


TITLE: Dual of Banach-valued $L^p$
QUESTION [6 upvotes]: Let $X$ be an infinite-dimensional  Banach space and let $p\in(1,+\infty)$. We may define $L^p(\mathbb R;X)$. Is it always true that the topological dual of $L^p(\mathbb R;X)$ is $L^{p'}(\mathbb R;X^*)$? Maybe some reflexivity or separability is needed for the Radon-Nikodym argument to work.
Is there a simple realization of  the dual of $L^1(\mathbb R;X)$?

REPLY [9 votes]: Diestel-Uhl, Vector measures, Section IV.1., Theorem 1:
Let $(\Omega,\mu)$ be a $\sigma$-finite measure space, $1\leq p<\infty$ and $\frac{1}{p}+\frac{1}{p'}=1$.
The dual of $L^p(\mu;X)$ is $L^{p'}(\mu,X^\ast)$ if and only if $X^\ast$ has the Radon-Nikodym property.
This is especially the case if $X$ is reflexive.


TITLE: Diagonalisability of $AA^*$
QUESTION [1 upvotes]: How can one prove that for every matrix $A$, $AA^*$ [$^*$ denotes conjugate transpose] can be diagonalised by using a unitary transformation?
Thanks for the help.

REPLY [2 votes]: The spectral theorem states that every Hermitian matrix (and for that matter, every normal matrix) can be diagonalized using a unitary transformation.
It therefore suffices to show that $(AA^*)^* = AA^*$.

REPLY [2 votes]: The matrix $AA^*$ is Hermitian and therefore normal. Therefore, the spectral theorem says that it is diagonalizable.


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     \leftskip\@flushglue
	 \rightskip\@flushglue
	 \parindent\z@
	 \parfillskip\z@skip
     \let\QCTOptA\empty
     \let\QCTOptB\empty
     \let\QCBOptA\empty
     \let\QCBOptB\empty
	 \vbox\bgroup
        \ifOverFrame
           #5\QCTOptA\par
        \fi
        \GRAPHIC{#4}{#3}{#1}{#2}{\z@}
        \ifUnderFrame
           \break#5\QCBOptA
        \fi
	 \egroup
  \egroup
  \vspace\topsep
  \break
}

\def\FFRAME#1#2#3#4#5#6#7{

  \@ifundefined{floatstyle}
    {
     \begin{figure}[#1]
    }
    {
	 \ifx#1h
      \begin{figure}[H]
	 \else
      \begin{figure}[#1]
	 \fi
	}
  \let\QCTOptA\empty
  \let\QCTOptB\empty
  \let\QCBOptA\empty
  \let\QCBOptB\empty
  \ifOverFrame
    #4
    \ifx\QCTOptA\empty
    \else
      \ifx\QCTOptB\empty
        \caption{\QCTOptA}
      \else
        \caption[\QCTOptB]{\QCTOptA}
      \fi
    \fi
    \ifUnderFrame\else
      \label{#5}
    \fi
  \else
    \UnderFrametrue
  \fi
  \begin{center}\GRAPHIC{#7}{#6}{#2}{#3}{\z@}\end{center}
  \ifUnderFrame
    #4
    \ifx\QCBOptA\empty
      \caption{}
    \else
      \ifx\QCBOptB\empty
        \caption{\QCBOptA}
      \else
        \caption[\QCBOptB]{\QCBOptA}
      \fi
    \fi
    \label{#5}
  \fi
  \end{figure}
 }

\newcount\dispkind

\def\makeactives{
  \catcode`\"=\active
  \catcode`\;=\active
  \catcode`\:=\active
  \catcode`\'=\active
  \catcode`\~=\active
}
\bgroup
   \makeactives
   \gdef\activesoff{
      \def"{\string"}
      \def;{\string;}
      \def:{\string:}
      \def'{\string'}
      \def~{\string~}

    }
\egroup

\def\FRAME#1#2#3#4#5#6#7#8{
 \bgroup
 \ifnum\@msidraft=\@ne
   \wasdrafttrue
 \else
   \wasdraftfalse
 \fi
 \def\LaTeXparams{}
 \dispkind=\z@
 \def\LaTeXparams{}
 \doFRAMEparams{#1}
 \ifnum\dispkind=\z@\IFRAME{#2}{#3}{#4}{#7}{#8}{#5}\else
  \ifnum\dispkind=\@ne\DFRAME{#2}{#3}{#7}{#8}{#5}\else
   \ifnum\dispkind=\tw@
    \edef\@tempa{\noexpand\FFRAME{\LaTeXparams}}
    \@tempa{#2}{#3}{#5}{#6}{#7}{#8}
    \fi
   \fi
  \fi
  \ifwasdraft\@msidraft=1\else\@msidraft=0\fi{}
  \egroup
 }

\def\TEXUX#1{"texux"}

\def\BF#1{{\bf {#1}}}
\def\NEG#1{\leavevmode\hbox{\rlap{\thinspace/}{$#1$}}}

\def\limfunc#1{\mathop{\rm #1}}
\def\func#1{\mathop{\rm #1}\nolimits}

\def\unit#1{\mathord{\thinspace\rm #1}}

\long\def\QQQ#1#2{
     \long\expandafter\def\csname#1\endcsname{#2}}
\@ifundefined{QTP}{\def\QTP#1{}}{}
\@ifundefined{QEXCLUDE}{\def\QEXCLUDE#1{}}{}
\@ifundefined{Qlb}{\def\Qlb#1{#1}}{}
\@ifundefined{Qlt}{\def\Qlt#1{#1}}{}
\def\QWE{}
\long\def\QQA#1#2{}
\def\QTR#1#2{{\csname#1\endcsname {#2}}}
\long\def\TeXButton#1#2{#2}
\long\def\QSubDoc#1#2{#2}
\def\EXPAND#1[#2]#3{}
\def\NOEXPAND#1[#2]#3{}
\def\PROTECTED{}
\def\LaTeXparent#1{}
\def\ChildStyles#1{}
\def\ChildDefaults#1{}
\def\QTagDef#1#2#3{}

\@ifundefined{correctchoice}{\def\correctchoice{\relax}}{}
\@ifundefined{HTML}{\def\HTML#1{\relax}}{}
\@ifundefined{TCIIcon}{\def\TCIIcon#1#2#3#4{\relax}}{}
\if@compatibility
  \typeout{Not defining UNICODE  U or CustomNote commands for LaTeX 2.09.}
\else
  \providecommand{\UNICODE}[2][]{\protect\rule{.1in}{.1in}}
  \providecommand{\U}[1]{\protect\rule{.1in}{.1in}}
  \providecommand{\CustomNote}[3][]{\marginpar{#3}}
\fi

\@ifundefined{lambdabar}{
      \def\lambdabar{\errmessage{You have used the lambdabar symbol.
                      This is available for typesetting only in RevTeX styles.}}
   }{}

\@ifundefined{StyleEditBeginDoc}{\def\StyleEditBeginDoc{\relax}}{}

\def\QQfnmark#1{\footnotemark}
\def\QQfntext#1#2{\addtocounter{footnote}{#1}\footnotetext{#2}}

\@ifundefined{TCIMAKEINDEX}{}{\makeindex}

\@ifundefined{abstract}{
 \def\abstract{
  \if@twocolumn
   \section*{Abstract (Not appropriate in this style!)}
   \else \small
   \begin{center}{\bf Abstract\vspace{-.5em}\vspace{\z@}}\end{center}
   \quotation
   \fi
  }
 }{
 }
\@ifundefined{endabstract}{\def\endabstract
  {\if@twocolumn\else\endquotation\fi}}{}
\@ifundefined{maketitle}{\def\maketitle#1{}}{}
\@ifundefined{affiliation}{\def\affiliation#1{}}{}
\@ifundefined{proof}{\def\proof{\noindent{\bfseries Proof. }}}{}
\@ifundefined{endproof}{\def\endproof{\mbox{\ \rule{.1in}{.1in}}}}{}
\@ifundefined{newfield}{\def\newfield#1#2{}}{}
\@ifundefined{chapter}{\def\chapter#1{\par(Chapter head:)#1\par }
 \newcount\c@chapter}{}
\@ifundefined{part}{\def\part#1{\par(Part head:)#1\par }}{}
\@ifundefined{section}{\def\section#1{\par(Section head:)#1\par }}{}
\@ifundefined{subsection}{\def\subsection#1
 {\par(Subsection head:)#1\par }}{}
\@ifundefined{subsubsection}{\def\subsubsection#1
 {\par(Subsubsection head:)#1\par }}{}
\@ifundefined{paragraph}{\def\paragraph#1
 {\par(Subsubsubsection head:)#1\par }}{}
\@ifundefined{subparagraph}{\def\subparagraph#1
 {\par(Subsubsubsubsection head:)#1\par }}{}

\@ifundefined{therefore}{\def\therefore{}}{}
\@ifundefined{backepsilon}{\def\backepsilon{}}{}
\@ifundefined{yen}{\def\yen{\hbox{\rm\rlap=Y}}}{}
\@ifundefined{registered}{
   \def\registered{\relax\ifmmode{}\r@gistered
                    \else$\m@th\r@gistered$\fi}
 \def\r@gistered{^{\ooalign
  {\hfil\raise.07ex\hbox{$\scriptstyle\rm\text{R}$}\hfil\crcr
  \mathhexbox20D}}}}{}
\@ifundefined{Eth}{\def\Eth{}}{}
\@ifundefined{eth}{\def\eth{}}{}
\@ifundefined{Thorn}{\def\Thorn{}}{}
\@ifundefined{thorn}{\def\thorn{}}{}

\def\TEXTsymbol#1{\mbox{$#1$}}
\@ifundefined{degree}{\def\degree{{}^{\circ}}}{}

\newdimen\theight
\@ifundefined{Column}{\def\Column{
 \vadjust{\setbox\z@=\hbox{\scriptsize\quad\quad tcol}
  \theight=\ht\z@\advance\theight by \dp\z@\advance\theight by \lineskip
  \kern -\theight \vbox to \theight{
   \rightline{\rlap{\box\z@}}
   \vss
   }
  }
 }}{}

\@ifundefined{qed}{\def\qed{
 \ifhmode\unskip\nobreak\fi\ifmmode\ifinner\else\hskip5\p@\fi\fi
 \hbox{\hskip5\p@\vrule width4\p@ height6\p@ depth1.5\p@\hskip\p@}
 }}{}

\@ifundefined{cents}{\def\cents{\hbox{\rm\rlap c/}}}{}
\@ifundefined{tciLaplace}{\def\tciLaplace{\ensuremath{\mathcal{L}}}}{}
\@ifundefined{tciFourier}{\def\tciFourier{\ensuremath{\mathcal{F}}}}{}
\@ifundefined{textcurrency}{\def\textcurrency{\hbox{\rm\rlap xo}}}{}
\@ifundefined{texteuro}{\def\texteuro{\hbox{\rm\rlap C=}}}{}
\@ifundefined{euro}{\def\euro{\hbox{\rm\rlap C=}}}{}
\@ifundefined{textfranc}{\def\textfranc{\hbox{\rm\rlap-F}}}{}
\@ifundefined{textlira}{\def\textlira{\hbox{\rm\rlap L=}}}{}
\@ifundefined{textpeseta}{\def\textpeseta{\hbox{\rm P\negthinspace s}}}{}

\@ifundefined{miss}{\def\miss{\hbox{\vrule height2\p@ width 2\p@ depth\z@}}}{}

\@ifundefined{vvert}{\def\vvert{\Vert}}{}

\@ifundefined{tcol}{\def\tcol#1{{\baselineskip=6\p@ \vcenter{#1}} \Column}}{}

\@ifundefined{dB}{\def\dB{\hbox{{}}}}{}
\@ifundefined{mB}{\def\mB#1{\hbox{$#1$}}}{}
\@ifundefined{nB}{\def\nB#1{\hbox{#1}}}{}

\@ifundefined{note}{\def\note{$^{\dag}}}{}

\def\newfmtname{LaTeX2e}

\ifx\fmtname\newfmtname
  \DeclareOldFontCommand{\rm}{\normalfont\rmfamily}{\mathrm}
  \DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf}
  \DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt}
  \DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf}
  \DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit}
  \DeclareOldFontCommand{\sl}{\normalfont\slshape}{\@nomath\sl}
  \DeclareOldFontCommand{\sc}{\normalfont\scshape}{\@nomath\sc}
\fi

\def\alpha{{\Greekmath 010B}}
\def\beta{{\Greekmath 010C}}
\def\gamma{{\Greekmath 010D}}
\def\delta{{\Greekmath 010E}}
\def\epsilon{{\Greekmath 010F}}
\def\zeta{{\Greekmath 0110}}
\def\eta{{\Greekmath 0111}}
\def\theta{{\Greekmath 0112}}
\def\iota{{\Greekmath 0113}}
\def\kappa{{\Greekmath 0114}}
\def\lambda{{\Greekmath 0115}}
\def\mu{{\Greekmath 0116}}
\def\nu{{\Greekmath 0117}}
\def\xi{{\Greekmath 0118}}
\def\pi{{\Greekmath 0119}}
\def\rho{{\Greekmath 011A}}
\def\sigma{{\Greekmath 011B}}
\def\tau{{\Greekmath 011C}}
\def\upsilon{{\Greekmath 011D}}
\def\phi{{\Greekmath 011E}}
\def\chi{{\Greekmath 011F}}
\def\psi{{\Greekmath 0120}}
\def\omega{{\Greekmath 0121}}
\def\varepsilon{{\Greekmath 0122}}
\def\vartheta{{\Greekmath 0123}}
\def\varpi{{\Greekmath 0124}}
\def\varrho{{\Greekmath 0125}}
\def\varsigma{{\Greekmath 0126}}
\def\varphi{{\Greekmath 0127}}

\def\nabla{{\Greekmath 0272}}
\def\FindBoldGroup{
   {\setbox0=\hbox{$\mathbf{x\global\edef\theboldgroup{\the\mathgroup}}$}}
}

\def\Greekmath#1#2#3#4{
    \if@compatibility
        \ifnum\mathgroup=\symbold
           \mathchoice{\mbox{\boldmath$\displaystyle\mathchar"#1#2#3#4$}}
                      {\mbox{\boldmath$\textstyle\mathchar"#1#2#3#4$}}
                      {\mbox{\boldmath$\scriptstyle\mathchar"#1#2#3#4$}}
                      {\mbox{\boldmath$\scriptscriptstyle\mathchar"#1#2#3#4$}}
        \else
           \mathchar"#1#2#3#4
        \fi
    \else
        \FindBoldGroup
        \ifnum\mathgroup=\theboldgroup
           \mathchoice{\mbox{\boldmath$\displaystyle\mathchar"#1#2#3#4$}}
                      {\mbox{\boldmath$\textstyle\mathchar"#1#2#3#4$}}
                      {\mbox{\boldmath$\scriptstyle\mathchar"#1#2#3#4$}}
                      {\mbox{\boldmath$\scriptscriptstyle\mathchar"#1#2#3#4$}}
        \else
           \mathchar"#1#2#3#4
        \fi
	  \fi}

\newif\ifGreekBold  \GreekBoldfalse
\let\SAVEPBF=\pbf
\def\pbf{\GreekBoldtrue\SAVEPBF}

\@ifundefined{theorem}{\newtheorem{theorem}{Theorem}}{}
\@ifundefined{lemma}{\newtheorem{lemma}[theorem]{Lemma}}{}
\@ifundefined{corollary}{\newtheorem{corollary}[theorem]{Corollary}}{}
\@ifundefined{conjecture}{\newtheorem{conjecture}[theorem]{Conjecture}}{}
\@ifundefined{proposition}{\newtheorem{proposition}[theorem]{Proposition}}{}
\@ifundefined{axiom}{\newtheorem{axiom}{Axiom}}{}
\@ifundefined{remark}{\newtheorem{remark}{Remark}}{}
\@ifundefined{example}{\newtheorem{example}{Example}}{}
\@ifundefined{exercise}{\newtheorem{exercise}{Exercise}}{}
\@ifundefined{definition}{\newtheorem{definition}{Definition}}{}

\@ifundefined{mathletters}{

  \newcounter{equationnumber}
  \def\mathletters{
     \addtocounter{equation}{1}
     \edef\@currentlabel{\theequation}
     \setcounter{equationnumber}{\c@equation}
     \setcounter{equation}{0}
     \edef\theequation{\@currentlabel\noexpand\alph{equation}}
  }
  \def\endmathletters{
     \setcounter{equation}{\value{equationnumber}}
  }
}{}

\@ifundefined{BibTeX}{
    \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em
                 T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}}{}
\@ifundefined{AmS}
    {\def\AmS{{\protect\usefont{OMS}{cmsy}{m}{n}
                A\kern-.1667em\lower.5ex\hbox{M}\kern-.125emS}}}{}
\@ifundefined{AmSTeX}{\def\AmSTeX{\protect\AmS-\protect\TeX\@}}{}

\def\@@eqncr{\let\@tempa\relax
    \ifcase\@eqcnt \def\@tempa{& & &}\or \def\@tempa{& &}
      \else \def\@tempa{&}\fi
     \@tempa
     \if@eqnsw
        \iftag@
           \@taggnum
        \else
           \@eqnnum\stepcounter{equation}
        \fi
     \fi
     \global\tag@false
     \global\@eqnswtrue
     \global\@eqcnt\z@\cr}

\def\TCItag{\@ifnextchar*{\@TCItagstar}{\@TCItag}}
\def\@TCItag#1{
    \global\tag@true
    \global\def\@taggnum{(#1)}}
\def\@TCItagstar*#1{
    \global\tag@true
    \global\def\@taggnum{#1}}

\def\QATOP#1#2{{#1 \atop #2}}
\def\QTATOP#1#2{{\textstyle {#1 \atop #2}}}
\def\QDATOP#1#2{{\displaystyle {#1 \atop #2}}}
\def\QABOVE#1#2#3{{#2 \above#1 #3}}
\def\QTABOVE#1#2#3{{\textstyle {#2 \above#1 #3}}}
\def\QDABOVE#1#2#3{{\displaystyle {#2 \above#1 #3}}}
\def\QOVERD#1#2#3#4{{#3 \overwithdelims#1#2 #4}}
\def\QTOVERD#1#2#3#4{{\textstyle {#3 \overwithdelims#1#2 #4}}}
\def\QDOVERD#1#2#3#4{{\displaystyle {#3 \overwithdelims#1#2 #4}}}
\def\QATOPD#1#2#3#4{{#3 \atopwithdelims#1#2 #4}}
\def\QTATOPD#1#2#3#4{{\textstyle {#3 \atopwithdelims#1#2 #4}}}
\def\QDATOPD#1#2#3#4{{\displaystyle {#3 \atopwithdelims#1#2 #4}}}
\def\QABOVED#1#2#3#4#5{{#4 \abovewithdelims#1#2#3 #5}}
\def\QTABOVED#1#2#3#4#5{{\textstyle
   {#4 \abovewithdelims#1#2#3 #5}}}
\def\QDABOVED#1#2#3#4#5{{\displaystyle
   {#4 \abovewithdelims#1#2#3 #5}}}

\def\tint{\mathop{\textstyle \int}}
\def\tiint{\mathop{\textstyle \iint }}
\def\tiiint{\mathop{\textstyle \iiint }}
\def\tiiiint{\mathop{\textstyle \iiiint }}
\def\tidotsint{\mathop{\textstyle \idotsint }}
\def\toint{\mathop{\textstyle \oint}}
\def\tsum{\mathop{\textstyle \sum }}
\def\tprod{\mathop{\textstyle \prod }}
\def\tbigcap{\mathop{\textstyle \bigcap }}
\def\tbigwedge{\mathop{\textstyle \bigwedge }}
\def\tbigoplus{\mathop{\textstyle \bigoplus }}
\def\tbigodot{\mathop{\textstyle \bigodot }}
\def\tbigsqcup{\mathop{\textstyle \bigsqcup }}
\def\tcoprod{\mathop{\textstyle \coprod }}
\def\tbigcup{\mathop{\textstyle \bigcup }}
\def\tbigvee{\mathop{\textstyle \bigvee }}
\def\tbigotimes{\mathop{\textstyle \bigotimes }}
\def\tbiguplus{\mathop{\textstyle \biguplus }}

\def\dint{\mathop{\displaystyle \int}}
\def\diint{\mathop{\displaystyle \iint}}
\def\diiint{\mathop{\displaystyle \iiint}}
\def\diiiint{\mathop{\displaystyle \iiiint }}
\def\didotsint{\mathop{\displaystyle \idotsint }}
\def\doint{\mathop{\displaystyle \oint}}
\def\dsum{\mathop{\displaystyle \sum }}
\def\dprod{\mathop{\displaystyle \prod }}
\def\dbigcap{\mathop{\displaystyle \bigcap }}
\def\dbigwedge{\mathop{\displaystyle \bigwedge }}
\def\dbigoplus{\mathop{\displaystyle \bigoplus }}
\def\dbigodot{\mathop{\displaystyle \bigodot }}
\def\dbigsqcup{\mathop{\displaystyle \bigsqcup }}
\def\dcoprod{\mathop{\displaystyle \coprod }}
\def\dbigcup{\mathop{\displaystyle \bigcup }}
\def\dbigvee{\mathop{\displaystyle \bigvee }}
\def\dbigotimes{\mathop{\displaystyle \bigotimes }}
\def\dbiguplus{\mathop{\displaystyle \biguplus }}

\if@compatibility\else

  \RequirePackage{amsmath}
\fi

\def\ExitTCILatex{\makeatother\endinput}

\bgroup
\ifx\ds@amstex\relax
   \message{amstex already loaded}\aftergroup\ExitTCILatex
\else
   \@ifpackageloaded{amsmath}
      {\if@compatibility\message{amsmath already loaded}\fi\aftergroup\ExitTCILatex}
      {}
   \@ifpackageloaded{amstex}
      {\if@compatibility\message{amstex already loaded}\fi\aftergroup\ExitTCILatex}
      {}
   \@ifpackageloaded{amsgen}
      {\if@compatibility\message{amsgen already loaded}\fi\aftergroup\ExitTCILatex}
      {}
\fi
\egroup

\typeout{TCILATEX defining AMS-like constructs in LaTeX 2.09 COMPATIBILITY MODE}

\let\DOTSI\relax
\def\RIfM@{\relax\ifmmode}
\def\FN@{\futurelet\next}
\newcount\intno@
\def\iint{\DOTSI\intno@\tw@\FN@\ints@}
\def\iiint{\DOTSI\intno@\thr@@\FN@\ints@}
\def\iiiint{\DOTSI\intno@4 \FN@\ints@}
\def\idotsint{\DOTSI\intno@\z@\FN@\ints@}
\def\ints@{\findlimits@\ints@@}
\newif\iflimtoken@
\newif\iflimits@
\def\findlimits@{\limtoken@true\ifx\next\limits\limits@true
 \else\ifx\next\nolimits\limits@false\else
 \limtoken@false\ifx\ilimits@\nolimits\limits@false\else
 \ifinner\limits@false\else\limits@true\fi\fi\fi\fi}
\def\multint@{\int\ifnum\intno@=\z@\intdots@
 \else\intkern@\fi
 \ifnum\intno@>\tw@\int\intkern@\fi
 \ifnum\intno@>\thr@@\int\intkern@\fi
 \int}
\def\multintlimits@{\intop\ifnum\intno@=\z@\intdots@\else\intkern@\fi
 \ifnum\intno@>\tw@\intop\intkern@\fi
 \ifnum\intno@>\thr@@\intop\intkern@\fi\intop}
\def\intic@{
    \mathchoice{\hskip.5em}{\hskip.4em}{\hskip.4em}{\hskip.4em}}
\def\negintic@{\mathchoice
 {\hskip-.5em}{\hskip-.4em}{\hskip-.4em}{\hskip-.4em}}
\def\ints@@{\iflimtoken@
 \def\ints@@@{\iflimits@\negintic@
   \mathop{\intic@\multintlimits@}\limits
  \else\multint@\nolimits\fi
  \eat@}
 \else
 \def\ints@@@{\iflimits@\negintic@
  \mathop{\intic@\multintlimits@}\limits\else
  \multint@\nolimits\fi}\fi\ints@@@}
\def\intkern@{\mathchoice{\!\!\!}{\!\!}{\!\!}{\!\!}}
\def\plaincdots@{\mathinner{\cdotp\cdotp\cdotp}}
\def\intdots@{\mathchoice{\plaincdots@}
 {{\cdotp}\mkern1.5mu{\cdotp}\mkern1.5mu{\cdotp}}
 {{\cdotp}\mkern1mu{\cdotp}\mkern1mu{\cdotp}}
 {{\cdotp}\mkern1mu{\cdotp}\mkern1mu{\cdotp}}}

\def\RIfM@{\relax\protect\ifmmode}
\def\text{\RIfM@\expandafter\text@\else\expandafter\mbox\fi}
\let\nfss@text\text
\def\text@#1{\mathchoice
   {\textdef@\displaystyle\f@size{#1}}
   {\textdef@\textstyle\tf@size{\firstchoice@false #1}}
   {\textdef@\textstyle\sf@size{\firstchoice@false #1}}
   {\textdef@\textstyle \ssf@size{\firstchoice@false #1}}
   \glb@settings}

\def\textdef@#1#2#3{\hbox{{
                    \everymath{#1}
                    \let\f@size#2\selectfont
                    #3}}}
\newif\iffirstchoice@
\firstchoice@true

\def\Let@{\relax\iffalse{\fi\let\\=\cr\iffalse}\fi}
\def\vspace@{\def\vspace##1{\crcr\noalign{\vskip##1\relax}}}
\def\multilimits@{\bgroup\vspace@\Let@
 \baselineskip\fontdimen10 \scriptfont\tw@
 \advance\baselineskip\fontdimen12 \scriptfont\tw@
 \lineskip\thr@@\fontdimen8 \scriptfont\thr@@
 \lineskiplimit\lineskip
 \vbox\bgroup\ialign\bgroup\hfil$\m@th\scriptstyle{##}$\hfil\crcr}
\def\Sb{_\multilimits@}
\def\endSb{\crcr\egroup\egroup\egroup}
\def\Sp{^\multilimits@}
\let\endSp\endSb

\newdimen\ex@
\ex@.2326ex
\def\rightarrowfill@#1{$#1\m@th\mathord-\mkern-6mu\cleaders
 \hbox{$#1\mkern-2mu\mathord-\mkern-2mu$}\hfill
 \mkern-6mu\mathord\rightarrow$}
\def\leftarrowfill@#1{$#1\m@th\mathord\leftarrow\mkern-6mu\cleaders
 \hbox{$#1\mkern-2mu\mathord-\mkern-2mu$}\hfill\mkern-6mu\mathord-$}
\def\leftrightarrowfill@#1{$#1\m@th\mathord\leftarrow
\mkern-6mu\cleaders
 \hbox{$#1\mkern-2mu\mathord-\mkern-2mu$}\hfill
 \mkern-6mu\mathord\rightarrow$}
\def\overrightarrow{\mathpalette\overrightarrow@}
\def\overrightarrow@#1#2{\vbox{\ialign{##\crcr\rightarrowfill@#1\crcr
 \noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#1#2\hfil$\crcr}}}
\let\overarrow\overrightarrow
\def\overleftarrow{\mathpalette\overleftarrow@}
\def\overleftarrow@#1#2{\vbox{\ialign{##\crcr\leftarrowfill@#1\crcr
 \noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#1#2\hfil$\crcr}}}
\def\overleftrightarrow{\mathpalette\overleftrightarrow@}
\def\overleftrightarrow@#1#2{\vbox{\ialign{##\crcr
   \leftrightarrowfill@#1\crcr
 \noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#1#2\hfil$\crcr}}}
\def\underrightarrow{\mathpalette\underrightarrow@}
\def\underrightarrow@#1#2{\vtop{\ialign{##\crcr$\m@th\hfil#1#2\hfil
  $\crcr\noalign{\nointerlineskip}\rightarrowfill@#1\crcr}}}
\let\underarrow\underrightarrow
\def\underleftarrow{\mathpalette\underleftarrow@}
\def\underleftarrow@#1#2{\vtop{\ialign{##\crcr$\m@th\hfil#1#2\hfil
  $\crcr\noalign{\nointerlineskip}\leftarrowfill@#1\crcr}}}
\def\underleftrightarrow{\mathpalette\underleftrightarrow@}
\def\underleftrightarrow@#1#2{\vtop{\ialign{##\crcr$\m@th
  \hfil#1#2\hfil$\crcr
 \noalign{\nointerlineskip}\leftrightarrowfill@#1\crcr}}}

\def\qopnamewl@#1{\mathop{\operator@font#1}\nlimits@}
\let\nlimits@\displaylimits
\def\setboxz@h{\setbox\z@\hbox}

\def\varlim@#1#2{\mathop{\vtop{\ialign{##\crcr
 \hfil$#1\m@th\operator@font lim$\hfil\crcr
 \noalign{\nointerlineskip}#2#1\crcr
 \noalign{\nointerlineskip\kern-\ex@}\crcr}}}}

 \def\rightarrowfill@#1{\m@th\setboxz@h{$#1-$}\ht\z@\z@
  $#1\copy\z@\mkern-6mu\cleaders
  \hbox{$#1\mkern-2mu\box\z@\mkern-2mu$}\hfill
  \mkern-6mu\mathord\rightarrow$}
\def\leftarrowfill@#1{\m@th\setboxz@h{$#1-$}\ht\z@\z@
  $#1\mathord\leftarrow\mkern-6mu\cleaders
  \hbox{$#1\mkern-2mu\copy\z@\mkern-2mu$}\hfill
  \mkern-6mu\box\z@$}

\def\projlim{\qopnamewl@{proj\,lim}}
\def\injlim{\qopnamewl@{inj\,lim}}
\def\varinjlim{\mathpalette\varlim@\rightarrowfill@}
\def\varprojlim{\mathpalette\varlim@\leftarrowfill@}
\def\varliminf{\mathpalette\varliminf@{}}
\def\varliminf@#1{\mathop{\underline{\vrule\@depth.2\ex@\@width\z@
   \hbox{$#1\m@th\operator@font lim$}}}}
\def\varlimsup{\mathpalette\varlimsup@{}}
\def\varlimsup@#1{\mathop{\overline
  {\hbox{$#1\m@th\operator@font lim$}}}}

\def\stackunder#1#2{\mathrel{\mathop{#2}\limits_{#1}}}

\begingroup \catcode `|=0 \catcode `[= 1
\catcode`]=2 \catcode `\{=12 \catcode `\}=12
\catcode`\\=12
|gdef|@alignverbatim#1\end{align}[#1|end[align]]
|gdef|@salignverbatim#1\end{align*}[#1|end[align*]]

|gdef|@alignatverbatim#1\end{alignat}[#1|end[alignat]]
|gdef|@salignatverbatim#1\end{alignat*}[#1|end[alignat*]]

|gdef|@xalignatverbatim#1\end{xalignat}[#1|end[xalignat]]
|gdef|@sxalignatverbatim#1\end{xalignat*}[#1|end[xalignat*]]

|gdef|@gatherverbatim#1\end{gather}[#1|end[gather]]
|gdef|@sgatherverbatim#1\end{gather*}[#1|end[gather*]]

|gdef|@gatherverbatim#1\end{gather}[#1|end[gather]]
|gdef|@sgatherverbatim#1\end{gather*}[#1|end[gather*]]

|gdef|@multilineverbatim#1\end{multiline}[#1|end[multiline]]
|gdef|@smultilineverbatim#1\end{multiline*}[#1|end[multiline*]]

|gdef|@arraxverbatim#1\end{arrax}[#1|end[arrax]]
|gdef|@sarraxverbatim#1\end{arrax*}[#1|end[arrax*]]

|gdef|@tabulaxverbatim#1\end{tabulax}[#1|end[tabulax]]
|gdef|@stabulaxverbatim#1\end{tabulax*}[#1|end[tabulax*]]

|endgroup

\def\align{\@verbatim \frenchspacing\@vobeyspaces \@alignverbatim
You are using the "align" environment in a style in which it is not defined.}
\let\endalign=\endtrivlist

\@namedef{align*}{\@verbatim\@salignverbatim
You are using the "align*" environment in a style in which it is not defined.}
\expandafter\let\csname endalign*\endcsname =\endtrivlist

\def\alignat{\@verbatim \frenchspacing\@vobeyspaces \@alignatverbatim
You are using the "alignat" environment in a style in which it is not defined.}
\let\endalignat=\endtrivlist

\@namedef{alignat*}{\@verbatim\@salignatverbatim
You are using the "alignat*" environment in a style in which it is not defined.}
\expandafter\let\csname endalignat*\endcsname =\endtrivlist

\def\xalignat{\@verbatim \frenchspacing\@vobeyspaces \@xalignatverbatim
You are using the "xalignat" environment in a style in which it is not defined.}
\let\endxalignat=\endtrivlist

\@namedef{xalignat*}{\@verbatim\@sxalignatverbatim
You are using the "xalignat*" environment in a style in which it is not defined.}
\expandafter\let\csname endxalignat*\endcsname =\endtrivlist

\def\gather{\@verbatim \frenchspacing\@vobeyspaces \@gatherverbatim
You are using the "gather" environment in a style in which it is not defined.}
\let\endgather=\endtrivlist

\@namedef{gather*}{\@verbatim\@sgatherverbatim
You are using the "gather*" environment in a style in which it is not defined.}
\expandafter\let\csname endgather*\endcsname =\endtrivlist

\def\multiline{\@verbatim \frenchspacing\@vobeyspaces \@multilineverbatim
You are using the "multiline" environment in a style in which it is not defined.}
\let\endmultiline=\endtrivlist

\@namedef{multiline*}{\@verbatim\@smultilineverbatim
You are using the "multiline*" environment in a style in which it is not defined.}
\expandafter\let\csname endmultiline*\endcsname =\endtrivlist

\def\arrax{\@verbatim \frenchspacing\@vobeyspaces \@arraxverbatim
You are using a type of "array" construct that is only allowed in AmS-LaTeX.}
\let\endarrax=\endtrivlist

\def\tabulax{\@verbatim \frenchspacing\@vobeyspaces \@tabulaxverbatim
You are using a type of "tabular" construct that is only allowed in AmS-LaTeX.}
\let\endtabulax=\endtrivlist

\@namedef{arrax*}{\@verbatim\@sarraxverbatim
You are using a type of "array*" construct that is only allowed in AmS-LaTeX.}
\expandafter\let\csname endarrax*\endcsname =\endtrivlist

\@namedef{tabulax*}{\@verbatim\@stabulaxverbatim
You are using a type of "tabular*" construct that is only allowed in AmS-LaTeX.}
\expandafter\let\csname endtabulax*\endcsname =\endtrivlist

 \def\endequation{
     \ifmmode\ifinner
      \iftag@
        \addtocounter{equation}{-1}
        $\hfil
           \displaywidth\linewidth\@taggnum\egroup \endtrivlist
        \global\tag@false
        \global\@ignoretrue
      \else
        $\hfil
           \displaywidth\linewidth\@eqnnum\egroup \endtrivlist
        \global\tag@false
        \global\@ignoretrue
      \fi
     \else
      \iftag@
        \addtocounter{equation}{-1}
        \eqno \hbox{\@taggnum}
        \global\tag@false
        $$\global\@ignoretrue
      \else
        \eqno \hbox{\@eqnnum}
        $$\global\@ignoretrue
      \fi
     \fi\fi
 }

 \newif\iftag@ \tag@false

 \def\TCItag{\@ifnextchar*{\@TCItagstar}{\@TCItag}}
 \def\@TCItag#1{
     \global\tag@true
     \global\def\@taggnum{(#1)}}
 \def\@TCItagstar*#1{
     \global\tag@true
     \global\def\@taggnum{#1}}

  \@ifundefined{tag}{
     \def\tag{\@ifnextchar*{\@tagstar}{\@tag}}
     \def\@tag#1{
         \global\tag@true
         \global\def\@taggnum{(#1)}}
     \def\@tagstar*#1{
         \global\tag@true
         \global\def\@taggnum{#1}}
  }{}

\def\tfrac#1#2{{\textstyle {#1 \over #2}}}
\def\dfrac#1#2{{\displaystyle {#1 \over #2}}}
\def\binom#1#2{{#1 \choose #2}}
\def\tbinom#1#2{{\textstyle {#1 \choose #2}}}
\def\dbinom#1#2{{\displaystyle {#1 \choose #2}}}

\makeatother
\endinput