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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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-<!--Navigation Panel-->
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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>