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<P>
<H2 ALIGN="center"><A NAME="SECTION00000000000000000000">
Math examples</A>
</H2>
<BR>
<DIV ALIGN="CENTER">
<!-- 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 CELLPADDING="0" ALIGN="CENTER" 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 WIDTH="10" ALIGN="CENTER" NOWRAP><IMG
WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
SRC="img2.gif"
ALT="$\displaystyle =$"></TD>
<TD ALIGN="LEFT" NOWRAP><IMG
WIDTH="220" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
SRC="img3.gif"
ALT="$\displaystyle \frac{1} {2 \pi i}\int^{c+i\infty}_{c-i\infty}
\exp \left( u \ln u + \lambda u \right ) du$"> for <IMG
WIDTH="40" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
SRC="img4.gif"
ALT="$\displaystyle c \geq 0$"></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="img5.gif"
ALT="$\displaystyle \lambda$"></TD>
<TD WIDTH="10" ALIGN="CENTER" NOWRAP><IMG
WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
SRC="img2.gif"
ALT="$\displaystyle =$"></TD>
<TD ALIGN="LEFT" NOWRAP><IMG
WIDTH="185" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
SRC="img6.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="img7.gif"
ALT="$\displaystyle \gamma$"></TD>
<TD WIDTH="10" ALIGN="CENTER" NOWRAP><IMG
WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
SRC="img2.gif"
ALT="$\displaystyle =$"></TD>
<TD ALIGN="LEFT" NOWRAP><IMG
WIDTH="235" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
SRC="img8.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="img9.gif"
ALT="$\displaystyle \gamma'$"></TD>
<TD WIDTH="10" ALIGN="CENTER" NOWRAP><IMG
WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
SRC="img2.gif"
ALT="$\displaystyle =$"></TD>
<TD ALIGN="LEFT" NOWRAP><IMG
WIDTH="143" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
SRC="img10.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="img11.gif"
ALT="$\displaystyle \epsilon , \bar{\epsilon}$"></TD>
<TD WIDTH="10" ALIGN="CENTER" NOWRAP><IMG
WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
SRC="img2.gif"
ALT="$\displaystyle =$"></TD>
<TD ALIGN="LEFT" NOWRAP>actual/average energy loss</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(5)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
Since (<A HREF="sampleAMS.html#eqn:stress-sr">6</A>) or (<A HREF="sampleAMS.html#gdef">7d</A>) should 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
<!-- MATH
$a^2+b^2=c^2$
-->
<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
<!-- MATH
$a^n+b^n=c^n$
-->
<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>2$"> has long
been suspected (<A NAME="tex2html5"
HREF="http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html">Fermat, c.1637</A>). Only during the current decade has this been proved (<A NAME="tex2html6"
HREF="http://www.pbs.org:80/wgbh/nova/proof">Wiles, 1995</A>).
<P>
<BR>
<DIV ALIGN="CENTER"><A NAME="eqn:stress-sr"></A>
<!-- MATH
\begin{eqnarray}
V \mathbf{\pi}^{sr} & = & \left< \sum_i M_i \mathbf{V}_i \mathbf{V}_i
+ \sum_i \sum_{j>i} \mathbf{R}_{ij} \mathbf{F}_{ij}\right> \\\nonumber
& = & \left< \sum_i M_i \mathbf{V}_i \mathbf{V}_i
+ \sum_{i}\sum_{j>i}\sum_\alpha\sum_\beta \mathbf{r}_{i\alpha j\beta}\mathbf{f}_{i\alpha j\beta}
- \sum_i \sum_\alpha \mathbf{p}_{i\alpha} \mathbf{f}_{i\alpha} \right>
\end{eqnarray}
-->
<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="40" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img18.gif"
ALT="$\displaystyle V \mathbf{\pi}^{sr}$"></TD>
<TD WIDTH="10" ALIGN="CENTER" NOWRAP><IMG
WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
SRC="img2.gif"
ALT="$\displaystyle =$"></TD>
<TD ALIGN="LEFT" NOWRAP><IMG
WIDTH="233" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
SRC="img19.gif"
ALT="$\displaystyle \left< \sum_i M_i \mathbf{V}_i \mathbf{V}_i
+ \sum_i \sum_{j>i} \mathbf{R}_{ij} \mathbf{F}_{ij}\right>$"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(6)</TD></TR>
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"> </TD>
<TD WIDTH="10" ALIGN="CENTER" NOWRAP><IMG
WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
SRC="img2.gif"
ALT="$\displaystyle =$"></TD>
<TD ALIGN="LEFT" NOWRAP><IMG
WIDTH="418" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
SRC="img20.gif"
ALT="$\displaystyle \left< \sum_i M_i \mathbf{V}_i \mathbf{V}_i
+ \sum_{i}\sum_{j>i}\...
... j\beta}
- \sum_i \sum_\alpha \mathbf{p}_{i\alpha} \mathbf{f}_{i\alpha} \right>$"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
<DIV ALIGN="CENTER"><A NAME="bgdefs"></A><A NAME="bdef"></A><A NAME="b0def"></A><A NAME="budef"></A><A NAME="gdef"></A>
<!-- MATH
\begin{subequations}
\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"><IMG
WIDTH="559" HEIGHT="159" ALIGN="BOTTOM" BORDER="0"
SRC="img21.gif"
ALT="\begin{subequations}\begin{align}B_{ij}^\alpha & = \left(B_{ij}^\alpha\right)_0 ...
...\beta}{\d X_i} \frac{\d N_k^\alpha}{\d X_j} \right)\end{align}\end{subequations}"></TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL">
<BR><HR>
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<ADDRESS>
<I>Michel Goossens</I>
<BR><I>1999-03-30</I>
</ADDRESS>
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