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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>sampleMath</TITLE>
-<META NAME="description" CONTENT="sampleMath">
-<META NAME="keywords" CONTENT="sampleMath">
-<META NAME="resource-type" CONTENT="document">
-<META NAME="distribution" CONTENT="global">
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-
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-
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-</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"
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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>
-<BR>
-<DIV ALIGN="CENTER"><A NAME="xiEmax"></A>
-
-<!-- MATH
- \begin{eqnarray}
-\phi(\lambda) & = & \frac{1} {2 \pi i}\int^{c+i\infty}_{c-i\infty}
- \exp \left( u \ln u + \lambda u \right ) du \hspace{1cm}\mbox{for } c \geq 0 \\
- \lambda & = & \frac{\epsilon -\bar{\epsilon} }{\xi}
- - \gamma' - \beta^2 - \ln \frac{\xi} {E_{\rm max}}\\
- \gamma & = & 0.577215\dots \mathrm{\hspace{5mm}(Euler's\ constant)} \\
- \gamma' & = & 0.422784\dots = 1 - \gamma \\
- \epsilon , \bar{\epsilon} & = & \mbox{actual/average energy loss}
-\end{eqnarray}
- -->
-
-<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
-<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
- WIDTH="35" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
- SRC="img1.gif"
- ALT="$\displaystyle \phi(\lambda)$"></TD>
-<TD ALIGN="CENTER" NOWRAP><IMG
- WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
- SRC="img2.gif"
- ALT="$\textstyle =$"></TD>
-<TD ALIGN="LEFT" NOWRAP><IMG
- WIDTH="325" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
- SRC="img3.gif"
- ALT="$\displaystyle \frac{1} {2 \pi i}\int^{c+i\infty}_{c-i\infty}
-\exp \left( u \ln u + \lambda u \right ) du \hspace{1cm}\mbox{for } c \geq 0$"></TD>
-<TD WIDTH=10 ALIGN="RIGHT">
-(1)</TD></TR>
-<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
- WIDTH="13" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
- SRC="img4.gif"
- ALT="$\displaystyle \lambda$"></TD>
-<TD ALIGN="CENTER" NOWRAP><IMG
- WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
- SRC="img2.gif"
- ALT="$\textstyle =$"></TD>
-<TD ALIGN="LEFT" NOWRAP><IMG
- WIDTH="185" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
- SRC="img5.gif"
- ALT="$\displaystyle \frac{\epsilon -\bar{\epsilon} }{\xi}
-- \gamma' - \beta^2 - \ln \frac{\xi} {E_{\rm max}}$"></TD>
-<TD WIDTH=10 ALIGN="RIGHT">
-(2)</TD></TR>
-<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
- WIDTH="13" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
- SRC="img6.gif"
- ALT="$\displaystyle \gamma$"></TD>
-<TD ALIGN="CENTER" NOWRAP><IMG
- WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
- SRC="img2.gif"
- ALT="$\textstyle =$"></TD>
-<TD ALIGN="LEFT" NOWRAP><IMG
- WIDTH="235" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img7.gif"
- ALT="$\displaystyle 0.577215\dots \mathrm{\hspace{5mm}(Euler's\ constant)}$"></TD>
-<TD WIDTH=10 ALIGN="RIGHT">
-(3)</TD></TR>
-<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
- WIDTH="18" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img8.gif"
- ALT="$\displaystyle \gamma'$"></TD>
-<TD ALIGN="CENTER" NOWRAP><IMG
- WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
- SRC="img2.gif"
- ALT="$\textstyle =$"></TD>
-<TD ALIGN="LEFT" NOWRAP><IMG
- WIDTH="143" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
- SRC="img9.gif"
- ALT="$\displaystyle 0.422784\dots = 1 - \gamma$"></TD>
-<TD WIDTH=10 ALIGN="RIGHT">
-(4)</TD></TR>
-<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
- WIDTH="24" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
- SRC="img10.gif"
- ALT="$\displaystyle \epsilon , \bar{\epsilon}$"></TD>
-<TD ALIGN="CENTER" NOWRAP><IMG
- WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
- SRC="img2.gif"
- ALT="$\textstyle =$"></TD>
-<TD ALIGN="LEFT" NOWRAP><IMG
- WIDTH="187" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
- SRC="img11.gif"
- ALT="$\displaystyle \mbox{actual/average energy loss}$"></TD>
-<TD WIDTH=10 ALIGN="RIGHT">
-(5)</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-
-<P>
-Since&nbsp;<A HREF="sampleMath.html#xiEmax">2</A> and <A HREF="sampleMath.html#eqn:stress-sr">6</A> hold for arbitrary
-<!-- MATH
- $\delta\mathbf{c}$
- -->
-<IMG
- WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
- SRC="img12.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="img13.gif"
- ALT="$\mathcal{N}(A) = \mathcal{R}(B)$"> and that when <IMG
- WIDTH="68" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
- SRC="img14.gif"
- ALT="$y=B(x)$"> one has...
-<BR>
-...the <A NAME="tex2html4"
- HREF="http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html">Pythagorians</A> knew infinitely many solutions in integers to <IMG
- WIDTH="88" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
- SRC="img15.gif"
- ALT="$a^2+b^2=c^2$">.
-That no non-trivial integer solutions exist for <IMG
- WIDTH="93" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
- SRC="img16.gif"
- ALT="$a^n+b^n=c^n$"> with integers <IMG
- WIDTH="43" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
- SRC="img17.gif"
- ALT="$n&gt;2$"> has long
-been suspected (<A NAME="tex2html5"
- 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="tex2html6"
- HREF="http://www.pbs.org:80/wgbh/nova/proof">Wiles, 1995</A>).
-
-<P>
-<BR>
-<DIV ALIGN="CENTER"><A NAME="eqn:stress-sr"></A>
-
-<!-- MATH
- \begin{eqnarray}
-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}
- -->
-
-<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
-<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
- WIDTH="40" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
- SRC="img18.gif"
- ALT="$\displaystyle V \mathbf{\pi}^{sr}$"></TD>
-<TD ALIGN="CENTER" NOWRAP><IMG
- WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
- SRC="img2.gif"
- ALT="$\textstyle =$"></TD>
-<TD ALIGN="LEFT" NOWRAP><IMG
- WIDTH="233" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
- SRC="img19.gif"
- ALT="$\displaystyle \left&lt; \sum_i M_i \mathbf{V}_i \mathbf{V}_i
-+ \sum_i \sum_{j&gt;i} \mathbf{R}_{ij} \mathbf{F}_{ij}\right&gt;$"></TD>
-<TD WIDTH=10 ALIGN="RIGHT">
-(6)</TD></TR>
-<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT">&nbsp;</TD>
-<TD ALIGN="CENTER" NOWRAP><IMG
- WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
- SRC="img2.gif"
- ALT="$\textstyle =$"></TD>
-<TD ALIGN="LEFT" NOWRAP><IMG
- WIDTH="418" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
- SRC="img20.gif"
- ALT="$\displaystyle \left&lt; \sum_i M_i \mathbf{V}_i \mathbf{V}_i
-+ \sum_{i}\sum_{j&gt;i}\...
-... j\beta}
-- \sum_i \sum_\alpha \mathbf{p}_{i\alpha} \mathbf{f}_{i\alpha} \right&gt;$"></TD>
-<TD WIDTH=10 ALIGN="RIGHT">
-&nbsp;</TD></TR>
-</TABLE></DIV>
-<BR CLEAR="ALL"><P></P>
-<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>