\usepackage{amsmath}
\[
f(x)=
\begin{cases}
\frac{1}{2(x-1)}, & x>1 \\
\frac{\Gamma(x)}{2^{x-1}}, & 0<x<1
\end{cases}
\]


The result looks like this:

The solution is to use the dcases environment provided by the mathtools package instead. The prefix `d’ means `display’. It will set the cases in displayed math style (exactly the same as \frac versus \dfrac). So if we write the following code

\usepackage{amsmath,mathtools}
\[
f(x)=
\begin{dcases}
\frac{1}{2(x-1)}, & x>1 \\
\frac{\Gamma(x)}{2^{x-1}}, & 0<x<1
\end{dcases}
\]


then we will see the result which looks like this:

Many fantastic PSTricks examples.

Let the characteristic polynomial of a matrix be

Then the coefficients can be computed by

To prove the formula, note that the coefficient is the summation of all possible products of j eigenvalues, i.e.,

where denotes the j-combination of numbers from 1 to n, and the trace of is the sum of the th power of the eigenvalues, i.e.,

In addition, we have

The above indicates that the first part of cancels the second part of , whereas the second part of cancels the first part of The rest of proof becomes obvious now.

In a nutshell, the way to use publish is to first type in the texts (as your usually LaTeX editing), including Matlab codes, in a single .m file. Let’s say the file name is example.m. Then, in Matlab, you issue the command

publish('example.m', struct('format','latex','outputDir','ltx-src'));

It means that you want Matlab to process example.m and output a LaTeX file example.tex (that you can compile to get pdf) in the sub-directory ltx-src. This is it. Instead of writing example.tex, you write a file example.m.

So, how should I write example.m? It is best to give an example. See the following:

%%
% <latex>
% The eigenvalues of a circulant matrix can be
% obtained by performing FFT on the first column
% of the matrix. First, let us construct a
% $5\times5$ circulant matrix \verb|C| whose first
% column \verb|c| is generated with random input:
% </latex>


c = rand(5,1);
% sad that Matlab does not provide a circulant()
% command...
C = toeplitz(c, c([1 end:-1:2]))

%%
% <latex>
% The eigenvalues of \verb|C| are nothing but
% </latex>

lambda = fft(c)

%%
% <latex>
% Check it out! The output is the same as using
% the \verb|eig| command:
% </latex>

eig(C)

%%
% Fun, isn't it?


It is nothing but a script file that Matlab can execute, right? The trick part is that all the texts and LaTeX markups are buried in comment blocks. How the Matlab command publish makes a LaTeX output is that whenever it meets a whole block of comments starting with ‘%%‘, it strips the comment signs and decorates the whole block using the pair \begin{par} and \end{par}. On the other hand, whenever it meets a block of codes that does not start with ‘%%‘, Matlab knows that they are executable commands. Matlab uses \begin{verbatim} and \end{verbatim} to typeset these command texts, and automatically add the Matlab outputs of the commands, which are also decorated by the \begin{verbatim} and \end{verbatim} pair, in the LaTeX file. Something I am not satisfied is that Matlab does not recognize LaTeX commands such as \verb||. I have to put <latex></latex> so that Matlab can do a verbatim copy of \verb||, instead of expand the text \verb|| in some weird way, in the output LaTeX file.

It is time to try the above example yourself. Have fun.

/* File: t1.c */
#include

void foo(double *a)
{
a[0] = 1;
}

void main(void)
{
double a[2] = {0};
foo(a);
printf("first element of array a = %g\n", a[0]);
}


% File: t2.m
function t2()

a = zeros(2,1);
foo(a);
fprintf(1, 'first element of array a = %g\n', a(1));

function foo(a)

a(1) = 1;


% File: t3.m
function t3()

a = zeros(2,1);
foo_mex(a);
fprintf(1, 'first element of array a = %g\n', a(1));


/* File: foo_mex.c */
#include "mex.h"

void mexFunction(int nlhs, mxArray *plhs[], int nrhs, const mxArray *prhs[])
{
double *a = mxGetPr(prhs[0]);
a[0] = 1;
}


function y = f(x)
y1 = x.* (x<0);
y2 = x.^2 .* (x>=0) .* (x<2);
y3 = 4 .* (x>=2);
y = y1 + y2 + y3;


PS: The answer to the above question of changing to sans serif fonts is a simply line of code:

\renewcommand{\rmdefault}{cmss}

Indeed, a large part of fonts talked about in this post are no longer TeX fonts. TeX only recognizes fonts from .tfm files—bounding boxes and kerning informations. METAFONT can produce glyph files (.gf), but fonts we commonly used today are Adobe fonts (.afm), TrueType fonts (.ttf), etc. Corresponding tools are designed to convert these fonts into TeX font metrics, such as afm2tfm, ttf2tfm, recognized by the latex program, as well as into glyph files (ttf2pk, etc) used by the dvi drivers (dvips, Yap, xdvi, etc). The relationships are illustrated in the following figure. (I slightly cheat in the figure in that the Adobe font metric .afm will only be converted to TeX font metric .tfm, but no information is deduced for glyph file.)

[To be finished..]

The font definition file ‹enc›‹family›.fd simply declares the existence of some font, and provides a pointer to the metric file that the latex program should consult when typesetting the document. For example, in the txfont package, which provides characters in the Adobe Times font for both texts and formulas, there is one file t1txr.fd defining the txr font family using T1 encoding. The second and third commands read:

\DeclareFontFamily{T1}{txr}{}
\DeclareFontShape{T1}{txr}{m}{n}{t1xr}{}

It essentially says that a txr font with T1 encoding is defined. Whenever a T1 txr medium normal font is requested, LaTeX should use the t1xr.tfm metric file for all font sizes ( means “for all sizes”).

The corresponding encoding definition file ‹enc›enc.def defines the encoding. Let us see an example first to have an idea about what “encoding” means. In t1enc.def, there are four lines saying:

\DeclareFontEncoding{T1}{}{}
\DeclareTextAccent{\'}{T1}{1}
\DeclareTextSymbol{\ae}{T1}{230}
\DeclareTextComposite{\"}{T1}{a}{228}

The first line declares the existence of T1 encoding. The following lines mean that when this encoding is in effect the \'{x} command should superimpose character 1 (accent acute) over the character x, that \ae should produce character 230 (the æ ligature), and that \"a should produce character 228 (the umlaut ä). It would be much clearer to compare the above three commands with the corresponding ones in OT1 encoding (ot1enc.def):

\DeclareTextAccent{\'}{OT1}{19}
\DeclareTextSymbol{\ae}{OT1}{26}
(no composite command \"a defined)

One can see that, under different encodings, the same character has different indices in the character table. The same command can be issued to reference the same character, while it’s also possible that some characters exist in one encoding but not in the other. The encoding defined in LY1enc.def used by Y&Y TeX systems allows access to all the characters in a standard Adobe roman font.

Finally, the TeX font metric file ‹family›[‹size›].tfm is a binary file that defines width, depth, and height of each character, as well as the kerning pairs and ligature substitutions to be used with the font. LaTeX consults the tfm file to decide where to put characters when typesetting. The tfm file name has an optional [‹size›] field. For example, the cmr family has metric files cmr5.tfm, cmr6.tfm, cmr7.tfm, cmr8.tfm, cmr8.tfm, cmr10.tfm, cmr12.tfm, and cmr17.tfm. Each file fine tunes the size of each character and is used when different font sizes are correspondingly declared.

To summarize, when typesetting the document, latex requires three important files to get the correct encoding and character size information. Latex computes the locations of each character and outputs the dvi file.