TeX Implementation

This is probably one of the most important environments to implement! This is because it's required to show syntax to the reader who wishes to learn about \TeX\ and \LaTeX.


\begin{verbatim}
 begin{verbatim}
  some \LaTeX\ here will not get parsed!
 end{verbatim}
\end{verbatim}

which looks like this when rendered:

\begin{verbatim}
some \LaTeX\ here will not get parsed!
\end{verbatim}

\begin{example}
This method based on the Chinese Remainder Theorem, is used to factor a large number N.


Pick a random integer for a starting number  $ = y_0 = z_0 $ Then recursively apply the function $f(x) = x^2 + a \bmod N$  twice to $z$, and once to $y$, where $a$ is the random remainder to start with.

$$y_1 = f(y_0)$$
$$z_1 = f(f(y_0))$$

We use this process to generate remainders until we hit a loop and one of them is divisible by $p$, where $N=pq$. which has a good chance of being achieved by $\sqrt{p}$ factors. 


We test this at each iteration by taking the greatest common divisor of the difference between the values and the number you are trying to factor.

$$\gcd(| y_k-x_k| , N)$$

If the $\gcd$ 1, then find $x_k $ and $ y_k$ and recursively apply the algorithm. If you do not get a one, then the number is a factor of $N$.

An algorithm written in Scheme emulates this procedure:

\begin{verbatim}
(define (pollards n y z fn)
        (let* ((n-y (fn y))
                  (n-z (fn (fn z)))
                  (g (gcd (abs (- n-y n-z)) n)))
            (if (= g 1) (pollards n n-y n-z fn) g)))
                    
\end{verbatim}
\end{example}

Here is an example of a program written in an assembly language called MIPS:

\begin{example}
This emulates {\tt *x = *y } where {\tt x} and {\tt y} are pointers
\begin{verbatim}
        .data
x:      .word 4
y:      .word 5

        .text
main:
    la      $s0, x
    la      $s1, y
    lw      $t1, 0($s1) # read y into temp var
    sw      $t1, 0($s0) # store into x
    lw      $t1, 0($s0) # read from x into temp var
    move    $a0, $t1
    li $v0, 1
    syscall			
    li      $v0, 10		
    syscall			
\end{verbatim}
\end{example}

verbatim

Thanks to MathJax, we can render \LaTeX\ equations in the browser. Here is an example


Using ``\$'' symbols, you can create a math environement. Also using the {\tt align} environment also works (thanks to MathJAX).


You can produce math {\it within} sentences. 


We can see that differentiation of a rational function will produce a rational Z-transform. If you recall, the set of signals that produce rational Z-transforms are obtained from, and only from, linear combinations of $\alpha^n u(n)$, $\alpha^n u(-n)$, $n^k \alpha^n u(n)$, and $n^k \alpha^n u(-n)$, where $k \in \N$, and shifted versions thereof.


\begin{verbatim}
We can see that differentiation of a rational function will produce a rational Z-transform.
If you recall, the set of signals that produce rational Z-transforms are obtained from, and only from, linear combinations of
$\alpha^n u(n)$, $\alpha^n u(-n)$, $n^k \alpha^n u(n)$, and $n^k \alpha^n u(-n)$, where $k \in \N$, and shifted versions thereof.
\end{verbatim}

Using two ``\$'' symbols makes a centered math environment.




\begin{verbatim}
$$\int_0^\infty H(\omega)e^{i\omega x} dx$$
\end{verbatim}

which produces

$$\int_0^\infty H(\omega)e^{i\omega x} dx$$

more info on \TeX support here (read carefully!) \href{http://docs.mathjax.org/en/latest/tex.html}{http://docs.mathjax.org/en/latest/tex.html}




\begin{align*}
\hat{G}(z) &= \sum \limits_{n\in\Z} g(n) z^{-n} \\
\frac{d}{dz}\hat{G}(z) &= -\sum \limits_{n\in\Z} ng(n) z^{-n-1} \\
\frac{d}{dz}\hat{G}(z) &= -z^{-1}\sum \limits_{n\in\Z} ng(n) z^{-n} \\
\end{align*}


math

You can create ``nice'' boxes by using the {\tt nicebox} environment, which produces something like this:

\begin{nicebox}
$$ \frac{\delta}{\delta u} \int_{birth}^{death} f(life) du = \mbox{your life} $$
\end{nicebox}

nicebox

Using the {\tt quotation} environment, you can do quotation blocks!


\begin{verbatim}
\begin{quotation}
a very long quote here... \cite[10]{sicp}
\end{quotation}
\end{verbatim}

Here is an example of what it looks like (a great quote from Abelson and Sussman):


\begin{quotation}
Our traffic with the subject matter of this book involves us with three foci of phenomena: the human mind, collections of computer programs, and the computer. Every computer program is a model, hatched in the mind, of a real or mental process. These processes, arising from human experience and thought, are huge in number, intricate in detail, and at any time only partially understood. They are modeled to our permanent satisfaction rarely by our computer programs. Thus even though our programs are carefully handcrafted discrete collections of symbols, mosaics of interlocking functions, they continually evolve: we change them as our perception of the model deepens, enlarges, generalizes until the model ultimately attains a metastable place within still another model with which we struggle. The source of the exhilaration associated with computer programming is the continual unfolding within the mind and on the computer of mechanisms expressed as programs and the explosion of perception they generate. If art interprets our dreams, the computer executes them in the guise of programs! \cite[10]{sicp}
\end{quotation}



9.4 Environments