summaryrefslogtreecommitdiff
path: root/info/examples/lwc/ch3/sampleAMS.tex
diff options
context:
space:
mode:
authorNorbert Preining <norbert@preining.info>2019-09-02 13:46:59 +0900
committerNorbert Preining <norbert@preining.info>2019-09-02 13:46:59 +0900
commite0c6872cf40896c7be36b11dcc744620f10adf1d (patch)
tree60335e10d2f4354b0674ec22d7b53f0f8abee672 /info/examples/lwc/ch3/sampleAMS.tex
Initial commit
Diffstat (limited to 'info/examples/lwc/ch3/sampleAMS.tex')
-rw-r--r--info/examples/lwc/ch3/sampleAMS.tex54
1 files changed, 54 insertions, 0 deletions
diff --git a/info/examples/lwc/ch3/sampleAMS.tex b/info/examples/lwc/ch3/sampleAMS.tex
new file mode 100644
index 0000000000..7a0f0ea462
--- /dev/null
+++ b/info/examples/lwc/ch3/sampleAMS.tex
@@ -0,0 +1,54 @@
+\documentclass[a4paper,twoside]{article}
+\usepackage{html}
+\usepackage{amsmath}
+
+\renewcommand{\d}{\partial}\providecommand{\bm}[1]{\mathbf{#1}}
+\providecommand{\Range}{\mathcal{R}}\providecommand{\Ker}{\mathcal{N}}
+\providecommand{\Quat}{\vec{\mathbf{Q}}}
+
+\newcommand{\StAndrews}{\htmlurl{http://www-groups.dcs.st-and.ac.uk/~history}}%
+\newcommand{\Pythagorians}{\htmladdnormallink
+ {Pythagorians}{\StAndrews/Mathematicians/Pythagoras.html}}
+\newcommand{\Fermat}{\htmladdnormallink
+ {Fermat, c.1637}{\StAndrews/HistTopics/Fermat's_last_theorem.html}}
+\newcommand{\Wiles}{\htmladdnormallink
+ {Wiles, 1995}{\htmlurl{http://www.pbs.org:80/wgbh/nova/proof}}}
+
+\begin{document}
+\htmlhead[center]{section}{Math examples}
+\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}
+
+Since~\eqref{eqn:stress-sr} or~\eqref{gdef} should hold for arbitrary $\delta\bm{c}$%
+-vectors, it is clear that $\Ker(A) = \Range(B)$ and that when $y=B(x)$ one has...\\
+...the \Pythagorians{} knew infinitely many solutions in integers to $a^2+b^2=c^2$.
+That no non-trivial integer solutions exist for $a^n+b^n=c^n$ with integers $n>2$ has long
+been suspected (\Fermat). Only during the current decade has this been proved (\Wiles).
+
+\begin{eqnarray}\label{eqn:stress-sr}
+ V \bm{\pi}^{sr} & = & \left< \sum_i M_i \bm{V}_i \bm{V}_i
+ + \sum_i \sum_{j>i} \bm{R}_{ij} \bm{F}_{ij}\right> \\ \nonumber
+ & = & \left< \sum_i M_i \bm{V}_i \bm{V}_i
+ + \sum_{i}\sum_{j>i}\sum_\alpha\sum_\beta \bm{r}_{i\alpha j\beta}\bm{f}_{i\alpha j\beta}
+ - \sum_i \sum_\alpha \bm{p}_{i\alpha} \bm{f}_{i\alpha} \right>
+\end{eqnarray}
+
+\begin{subequations}\label{bgdefs}
+\begin{align} B_{ij}^\alpha & =
+ \left(B_{ij}^\alpha\right)_0 + \left(B_{ij}^\alpha\right)_a \label{bdef} \\
+ \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) \label{b0def} \\
+ \left(B_{ij}^\alpha\right)_a & = H_{ij}^{\alpha \beta} a^\beta \label{budef} \\
+ 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) \label{gdef}
+\end{align}
+\end{subequations}
+\end{document}