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+<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
+
+<!--Converted with LaTeX2HTML 99.1 release (March 30, 1999)
+original version by: Nikos Drakos, CBLU, University of Leeds
+* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
+* with significant contributions from:
+ Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
+<HTML>
+<HEAD>
+<TITLE>sampleMathThumb</TITLE>
+<META NAME="description" CONTENT="sampleMathThumb">
+<META NAME="keywords" CONTENT="sampleMathThumb">
+<META NAME="resource-type" CONTENT="document">
+<META NAME="distribution" CONTENT="global">
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+<META NAME="Generator" CONTENT="LaTeX2HTML v99.1 release">
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+
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+
+<LINK REL="next" HREF="node1.html">
+<LINK REL="next" HREF="node1.html">
+</HEAD>
+
+<BODY >
+<!--Navigation Panel-->
+<A NAME="tex2html9"
+ HREF="node1.html">
+<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
+ SRC="/afs/cern.ch/project/tex/new/latex2html/icons.gif/next_motif.gif"></A>
+<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
+ SRC="/afs/cern.ch/project/tex/new/latex2html/icons.gif/up_motif_gr.gif">
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+ SRC="/afs/cern.ch/project/tex/new/latex2html/icons.gif/previous_motif_gr.gif">
+<BR>
+<B> Next:</B> <A NAME="tex2html10"
+ HREF="node1.html">About this document ...</A>
+<BR>
+<BR>
+<!--End of Navigation Panel-->
+<P>
+<H2 ALIGN="center"><A NAME="SECTION00000000000000000000">
+Math examples</A>
+</H2>
+<DIV ALIGN="RIGHT">
+<IMG
+ WIDTH="474" HEIGHT="160" ALIGN="BOTTOM" BORDER="0"
+ SRC="img1.gif"
+ ALT="\begin{eqnarray}
+\phi(\lambda) &amp; = &amp; \frac{1} {2 \pi i}\int^{c+i\infty}_{c-i\in...
+...\epsilon , \bar{\epsilon} &amp; = &amp; \mbox{actual/average energy loss}
+\end{eqnarray}">
+<BR>
+
+</DIV>
+
+<P>
+Since&nbsp;(<A HREF="sampleMathThumb.html#eqn:stress-sr">6</A>) or&nbsp;(<A HREF="sampleMathThumb.html#gdef">7d</A>) should hold for arbitrary
+<!-- MATH
+ $\delta\mathbf{c}$
+ -->
+<IMG
+ WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
+ SRC="img2.gif"
+ ALT="$ \delta\mathbf{c}$">-vectors, it is clear that
+<!-- MATH
+ $\mathcal{N}(A) = \mathcal{R}(B)$
+ -->
+<IMG
+ WIDTH="104" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
+ SRC="img3.gif"
+ ALT="$ \mathcal{N}(A) = \mathcal{R}(B)$"> and that when <IMG
+ WIDTH="68" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
+ SRC="img4.gif"
+ ALT="$ y=B(x)$"> one has...
+<BR>
+...the <A NAME="tex2html3"
+ HREF="http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html">Pythagorians</A> knew infinitely many solutions in integers to
+<!-- MATH
+ $a^2+b^2=c^2$
+ -->
+<IMG
+ WIDTH="88" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
+ SRC="img5.gif"
+ ALT="$ a^2+b^2=c^2$">.
+That no non-trivial integer solutions exist for
+<!-- MATH
+ $a^n+b^n=c^n$
+ -->
+<IMG
+ WIDTH="93" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
+ SRC="img6.gif"
+ ALT="$ a^n+b^n=c^n$"> with integers <IMG
+ WIDTH="43" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
+ SRC="img7.gif"
+ ALT="$ n&gt;2$"> has long
+been suspected (<A NAME="tex2html4"
+ HREF="http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html">Fermat, c.1637</A>). Only during the current decade has this been proved (<A NAME="tex2html5"
+ HREF="http://www.pbs.org:80/wgbh/nova/proof">Wiles, 1995</A>).
+
+<P>
+<DIV ALIGN="RIGHT">
+</DIV>
+<BR>
+<DIV ALIGN="RIGHT">
+<P ALIGN="RIGHT"><A NAME="eqn:stress-sr"></A>
+<!-- MATH
+ \begin{eqnarray}
+\htmlimage{thumbnail=.4} V \mathbf{\pi}^{sr} & = & \left< \sum_i M_i \mathbf{V}_i \mathbf{V}_i
++ \sum_i \sum_{j>i} \mathbf{R}_{ij} \mathbf{F}_{ij}\right> \\\nonumber
+ & = & \left< \sum_i M_i \mathbf{V}_i \mathbf{V}_i
+ + \sum_{i}\sum_{j>i}\sum_\alpha\sum_\beta \mathbf{r}_{i\alpha j\beta}\mathbf{f}_{i\alpha j\beta}
+ - \sum_i \sum_\alpha \mathbf{p}_{i\alpha} \mathbf{f}_{i\alpha} \right>
+\end{eqnarray}
+ -->
+<A NAME="eqn:stress-sr"></A><A NAME="eqnarray52"
+ HREF="img8.gif"><IMG
+ WIDTH="140" HEIGHT="29" ALIGN="LEFT"
+ SRC="Timg8.gif"
+ ALT="\begin{eqnarray}V \mathbf{\pi}^{sr} &amp; = &amp; \left&lt; \sum_i M_i \mathbf{V}_i \mathbf...
+...m_i \sum_\alpha \mathbf{p}_{i\alpha} \mathbf{f}_{i\alpha} \right&gt;
+\end{eqnarray}"></A>
+<BR CLEAR="ALL">
+<P>
+<BR CLEAR="ALL"></DIV><P></P>
+<DIV ALIGN="RIGHT">
+</DIV>
+
+<P>
+<DIV ALIGN="RIGHT">
+</DIV><A NAME="bgdefs"></A><A NAME="bdef"></A><A NAME="b0def"></A><A NAME="budef"></A><A NAME="gdef"></A>
+<!-- MATH
+ \begin{subequations}
+\htmlimage{thumbnail=.3}\begin{align} B_{ij}^\alpha & =
+\left(B_{ij}^\alpha\right)_0 + \left(B_{ij}^\alpha\right)_a\\
+ \left(B_{ij}^\alpha\right)_0 & = \frac{1}{2}\left(\frac{\d N_i^\alpha}{\d X_j}
+ + \frac{\d N_j^\alpha} {\d X_i} \right)\\
+ \left(B_{ij}^\alpha\right)_a & = H_{ij}^{\alpha \beta} a^\beta\\
+ H_{ij}^{\alpha \beta} & =
+ \frac{1}{2}\left( \frac{\d N_k^\alpha}{\d X_i} \frac{\d N_k^\beta}{\d X_j}
+ + \frac{\d N_k^\beta}{\d X_i} \frac{\d N_k^\alpha}{\d X_j} \right)\end{align}
+\end{subequations}
+ -->
+<TABLE CELLPADDING="0" WIDTH="100%" ALIGN="CENTER">
+<TR VALIGN="MIDDLE">
+<TD NOWRAP ALIGN="CENTER"><A NAME="tex2html_wrap_indisplay268"
+ HREF="img9.gif"><IMG
+ WIDTH="1" HEIGHT="6" ALIGN="MIDDLE" BORDER="0"
+ SRC="Timg9.gif"
+ ALT="$\displaystyle \begin{align}B_{ij}^\alpha &amp; = \left(B_{ij}^\alpha\right)_0 + \le...
+...} + \frac{\d N_k^\beta}{\d X_i} \frac{\d N_k^\alpha}{\d X_j} \right)\end{align}$"></A></TD></TR>
+</TABLE>
+<DIV ALIGN="RIGHT">
+</DIV>
+<BR><HR>
+<!--Table of Child-Links-->
+<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
+
+<UL>
+<LI><A NAME="tex2html11"
+ HREF="node1.html">About this document ...</A>
+</UL>
+<!--End of Table of Child-Links-->
+<BR><HR>
+<ADDRESS>
+<I>Michel Goossens</I>
+<BR><I>1999-03-30</I>
+</ADDRESS>
+</BODY>
+</HTML>