%% Example file for UEF thesis - just to illustrate the layout
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\chapter{SOME EXAMPLES}
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\section{Mathematical equations}
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As I have shown in my first publication \myp{1} \emph{etc}. Unfortunately, there were some errors so I tried to correct them in my later publications \myp{2}--\myp{4}.

Other references can (and normally should) be  used in the review. Here are some articles \cite{poincare1,poincare2,poincare3,poincare4} and some more \cite{taylorwiles,goldschmidt}. Here are some books \cite{arnold,ovsienko,tarkhanov}. Symbolic computation system can be referenced like this \cite{DGPS}. Then proceedings or other collective works \cite{dudnikov,spencer} and thesis \cite{quillen}. Still some more \cite{nelder65,goodman68,gale91,nishihara87} and finally some sited www-pages \cite{uef,physmat,tenk}.
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\section{Some stuff}
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Let us consider a time-harmonic electromagnetic field of frequency $\omega$.
It is characterized by three-dimensional electric and magnetic fields of
the form
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\begin{align}
{E}({r},t)&=\Re\lbrace {E}({r}) \exp{(-\textrm{i}
\omega t)} \rbrace\ , \\
{H}({r},t)&=\Re\lbrace {H}({r}) \exp{(-\textrm{i}
\omega t)} \rbrace\ ,
\end{align}
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where ${r}=(x,y,z)$ is the  three-dimensional position vector and
$\Re$ means the real part. In a continuous medium, these time-harmonic
fields satisfy Maxwell's equations
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\begin{align}
\nabla \times {E}({r}) &= \textrm{i} \omega {B}({r})\ ,\\
\nabla \times {H}({r}) &= {J}({r}) -\textrm{i}
\omega {D}({r})\ , \\
\nabla \cdot {D}({r}) &= \rho({r})\ ,\\
\nabla \cdot {B}({r}) &= 0\ ,
\end{align}
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where ${D}({r})$, ${B}({r})$, ${J}({r})$,
and $\rho({r})$ are the electric displacement, the magnetic
induction, the electric current density and the electric charge density,
respectively. In linear isotropic media we have constitutive relations
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\begin{align}
{D}({r}) &= \epsilon({r}) {E}({r})\ ,\\
{B}({r}) &= \mu({r}) {H}({r})\ ,\\
{J}({r}) &= \sigma({r}) {E}({r})\ ,
\end{align}
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where $\epsilon({r})$, $\mu({r})$, and $\sigma({r})$
are known as the permittivity, the
magnetic permeability, and the conductivity, respectively.
The permittivity $\epsilon$ is written as
$\epsilon({r})=\epsilon_r({r})\epsilon_0$,
where $\epsilon_0$ is the permittivity in vacuum
and $\epsilon_r$ is the relative permittivity. The
refractive index of the medium is defined as
$n({r})=\sqrt{\epsilon_r({r})}$.
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\subsection{Splitting the equation}
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Here is one basic example:
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\begin{align}
\begin{split} % use split-environment to get just one equation number in align-environment
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U(x,y,z) &= \frac{n\exp(\textrm{i}  kn\Delta z)}{\textrm{i}  \lambda \Delta z}
\exp \left[\frac{\textrm{i} \pi n}{\lambda \Delta z} (x^2 +y^2)\right] \\
& \quad \times\int\!\!\!\int_{-\infty}^{\infty}
U(x^{\prime},y^{\prime},z_0) \exp \left[ \frac{\textrm{i} \pi n}{\lambda\Delta z}
({x^{\prime}}^2+{y^{\prime}}^2)\right]
d x^{\prime} d y^{\prime}\ .
\end{split}
\end{align}
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The following example is even more complicated:
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\begin{align}
\label{W_propa}
&W(x_1,y_1,z_1,x_2,y_2,z_2)  \nonumber \\  % \nonumber removes number from the line in align-environment
& \quad = \int\!\!\!\int\!\!\!\int\!\!\!\int_{-\infty}^{\infty}
A(\alpha_1,\beta_1,z_0,\alpha_2,\beta_2,z_0)
\exp \lbrack -\textrm{i}  2\pi
(x_1\alpha_1-x_2\alpha_2+y_1\beta_1-y_2\beta_2) \rbrack \\
& \quad\quad
\times \exp \{ -\textrm{i}  2\pi \lbrack w^{\ast}(\alpha_1,\beta_1)\Delta z_1-w(\alpha_2,\beta_2)
\Delta z_2\rbrack\} d \alpha_1 d \alpha_2 d \beta_1 d \beta_2\ . \nonumber
\end{align}
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Next example shows how to define one variable in two ways:
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\begin{equation}
\label{313}
w = \begin{cases}
\lbrack{(n/\lambda)^2 -(\alpha^2+\beta^2)}\rbrack^{1/2}\ ,&  \text{when}
\,\, \alpha^2+\beta^2 \leq (n/\lambda)^2\ , \\
\textrm{i} \lbrack{(\alpha^2+\beta^2)-(n/\lambda)^2}\rbrack^{1/2}  &
\text{otherwise.}
\end{cases}
\end{equation}
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How about defining equation group?
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\begin{equation}
\begin{cases}
x=r\cos\phi \\[-1mm]
%% NOTE \\[-1mm] have to be added, otherwise the result looks ugly.
%% Even the Latex-system is not perfect in all cases ... :)
y=r\sin\phi
\end{cases} ,
\phantom{12345}
\begin{cases}
x'=r\tan\phi\\[-1mm]
y'=r\cot\phi%
\end{cases}\ .
\end{equation}
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And finally a matrix vector product:
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\begin{equation}
\begin{pmatrix} D_1t_1&-a_{12}t_2&\dots&-a_{1n}t_n\\
-a_{21}t_1&D_2t_2&\dots&-a_{2n}t_n\\
\vdots&\vdots&\ddots&\vdots\\
-a_{n1}t_1& -a_{n2}t_2&\dots&D_nt_n
\end{pmatrix}
\begin{pmatrix}
c_1\\
c_2\\
c_3\\
\vdots\\
c_{n-2}\\
c_{n-1}\\
c_n
\end{pmatrix}\ .
\end{equation}
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\section{Theorems, etc.}
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\begin{definition}
Let $\mathbb{Z}$ be the integers.
\end{definition}
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\begin{theorem}
Here is a nice formula:
\[
  1+2+\dots+n=\frac{n(n+1)}{2}\ .
\]
\label{lause}
\end{theorem}
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\begin{proof}
By induction. The details are left to the reader.
\end{proof}
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\begin{lemma}
If $n\in \mathbb{Z}$, then $2n$ is even.
\end{lemma}
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\begin{proof}
This follows easily by Theorem \ref{lause}.
\end{proof}
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\blindmathpaper % This command just adds some generic mathematical text in this example
                % Remove this in the real case
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\section{Figures}
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Recommended way to include figures is to make them first in  \textsf{.eps format} or
\textsf{.jpeg format}. One picture in Figure \ref{figeps} and many pictures in Figure \ref{many}.
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\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{monitahokkaat.eps}
\caption{A picture.\label{figeps}}
\end{figure}
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\begin{figure}
\centering
\begin{tabular}{cc}
\includegraphics[width=50mm, height=50mm]{monitahokkaat.eps}&
\includegraphics[width=50mm, height=50mm]{monitahokkaat.eps}\\
\includegraphics[width=30mm, height=30mm]{monitahokkaat.eps}&
\includegraphics[width=30mm, height=30mm]{monitahokkaat.eps}
\end{tabular}
\caption{\blindtext\label{many}} % Command \blindtext generates some text
\end{figure}
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\clearpage
\section{Tables  and something strange}
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Well, in principle it is easy, but\ldots. Hopefully the tables shown below
illustrates a little bit what is going on, and how the tables should look like.
First, the basic table:
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\begin{table}[!ht]
\label{rigobs}
\caption{A short table caption.}
\centering
\begin{tabular}{ccc}
\hline
$sr_{\textrm d}$ & $sr_{\textrm o}$ & $t$\\
\hline
1.0000 & 1.0000 & 1.0000 \\
0.2500 & 0.2499 & ~0.9998 \\
\hline
\end{tabular}
\end{table}
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If you have to combine multiple columns, you can do that as shown in Table \ref{conf}.
The meaning of $\rightarrow $ in Table \ref{conf} is obvious, isn't it?
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\begin{table}[!hb]
\label{conf}
\caption{Comparison of diffraction efficiencies for three connections,
$1\rightarrow 3$, $1\rightarrow 5$ and $1\rightarrow 7$, calculated by the MWCW-theory.}\centering
\begin{tabular}{|r|cc|cc|cc|}
~&\multicolumn{2}{c|}{$1\rightarrow 3$}&\multicolumn{2}{c|}{$1\rightarrow 5$}&
\multicolumn{2}{c|}{$1\rightarrow 7$} \\
$i$ &$E_{3i}^m$ & $E_{3i}^r$ &$E_{3i}^m$ & $E_{3i}^r$ &$E_{3i}^m$
& $E_{3i}^r$  \\[1pt]
\hline
$0$    & 0.3235 & 0.3237 & 0.1907 & 0.1911 & 0.1413 & 0.1404 \\
$^+1$ & 0.3217 & 0.3217 & 0.1962 & 0.1960 & 0.1412 & 0.1403 \\
$^+2$ & 0.0065 & 0.0066 & 0.1931 & 0.1930 & 0.1408 & 0.1408 \\
$^+3$ & 0.0025 & 0.0024 & 0.0021 & 0.0020 & 0.1397 & 0.1401 \\
$^+4$ & 0.0007 & 0.0008 & 0.0066 & 0.0066 & 0.0007 & 0.0006 \\
$^+5$ & 0.0033 & 0.0034 & 0.0046 & 0.0047 & 0.0038 & 0.0038 \\
$^+6$ & 0.0001 & 0.0001 & 0.0039 & 0.0040 & 0.0016 & 0.0013 \\
$^+7$ & 0.0000 & 0.0000 & 0.0003 & 0.0002 & 0.0002 & ~0.0002
\end{tabular}
\end{table}
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\subsection{Additional section for more text}
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\blindtext % Command \blindtext generates some text
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\chapter{ANOTHER CHAPTER WITH A SO LONG TITLE THAT IT OCCUPIES AT LEAST TWO LINES}
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Of course such long titles do not look good so they should be avoided.
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\section{Another section}
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\blindtext
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\subsection{First case}
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\blindtext
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\section{One more section}
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\blindtext
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\subsection{Second case}
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Just some blabla here. Just some blabla here. Just some blabla here. Just some blabla here. Just some blabla here. Just some blabla here.Just some blabla here. Just some blabla here. Just some blabla here. Just some blabla here. Just some blabla here. Just some blabla here. Just some blabla here. Just some blabla here. Just some blabla here.Just some blabla here. Just some blabla here. Just some blabla here. Just some blabla here. Just some blabla here. Just some blabla here. Just some blabla here. Just some blabla here. Just some blabla here.Just some blabla here. Just some blabla here. Just some blabla here. Just some blabla here. Just some blabla here. Just some blabla here. Just some blabla here. Just some blabla here. Just some blabla here.Just some blabla here. Just some blabla here. Just some blabla here.
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\subsection{Third case}
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\blindtext
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\chapter{YET ANOTHER CHAPTER}
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\section{Another section}
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\Blindtext % Command \Blindtext generates some text
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\subsection{Small subsection}
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\blindtext
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\subsection{Another subsection with a very long title just to test TOC formatting}
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\blindtext
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