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%============================================================================
% S A M P L E . T E X
%============================================================================
%===================================================================
% Sample problems; solutions give examples on using APL style in TeX
% Taken from the course ``Mathematics on the Computer'', Fall 87
%===================================================================
\magnification = \magstep1
\advance\vsize by 3truecm
\input mssymb % for some math symbols only! This is the new
% symbol font for some standard and non-standard
% mathematical symbols. It is only used here for
% blackboard bold letters. If you dont have it,
% just define \def\Bbb{} etc.
\input aplstyle
\choosett{apl}
\font\sans = amss10
\font\sltt = amsltt10
\def\header{{\sans Sample problems 9.\ 10.\ 1987}}
% some of them come from Sims' ``Abstract Algebra, A Computational Approach''
\def\APL{{\sltt APL}}
\nopagenumbers
\tolerance = 300
\noindent
\header
\vskip 2cm
\item{1.} Let $N>1$ be an integer. Show that each of the following
matrices represents a binary operation on
$S(N)$ (we set locally \BX@IO_0@.) Which of them are
associative, which commutative?
\medskip
\itemitem{a)} @(@\IO@N)@\SO@.@\CE\IO@N@
\itemitem{b)} \AB@(@\IO@N)@\SO@.-@\IO@N@
\itemitem{c)} @N@\AB@(@\IO@N)@\SO@.+@\IO@N@
\itemitem{d)} @N@\AB@(@\IO@N)@\SO@.#@\IO@N@
\medskip
\item{} Here @x@\CE@y@ is $\max(x,y)$, @x@\AB@y@ is
$y\bmod x$ and \AB@x@ is the absolute value of $x$.
\bigskip
\item{2.} Write an \APL\ function @GPOWER@ that computes for a group
@G@ (global variable) the $n$-th power of a given element $x$.
(If $S(M)$ is a representation vector of @G@, then
@GPOWER@ is a map $S(M)\times \Bbb Z\to S(M)$. Simply
use iteration.)
\bigskip
\item{3.} (Continuing problem 2.) A faster algorithm is obtained by
decomposing $x^n$ into its 2--base form
$x^n = x^{i_0}\times x^{2i_1}\times
x^{4i_2}\times ... \times x^{{2^k}i_k}$, where $i_j\in\{0,1\}$. Show
that the complexity of this algorithm is $O(\log_2(n))$.
(Show that the number of necessary multiplications does
not exceed $2\log_2(n)$). How would you write the corresponding
function in \APL? (Note that the binary representation of $n$
can be obtained by applying iteratively the procedure $n\bmod 2$.)
\bigskip
\item{4.} Write an \APL\ function @GTSGP@ that computes for a given group @G@
(global variable) the subgroup generated by a given subset $A$. The
function @GTSGP@ has one argument (the vector @A@) and returns
a subset of the set $S(N)$ (as a vector). (Extend the set @A@
by the group operation until @A@ becomes closed with respect
to the operation.)
\bigskip
\item{5.} Write an \APL\ function @INV@ that returns for a group @G@
the vector of inverse elements as a vector $S(N)\to S(N)$ so
that the index of the inverse of $x_i$ is @(INV G)[I]@.
\bigskip
\item{6.} Let $(G,\theta)$ be a group and let $A$ be a subset of $G$. Program
the following algorithm in \APL\ to find the subgroup @H@
generated by @A@. Compare the perfomance of this algorithm
with the algorithm in Problem 4.
\medskip
\itemitem{a)} put $H$ and $Y$ equal to $\{e\}$.
\itemitem{b)} let $Y$ be $YA\smallsetminus H$.
\itemitem{c)} if $Y=\emptyset$, stop.
\itemitem{d)} put $H$ equal to $H\cup Y$ and
go to (b).
\medskip
\item{} ($e$ is the neutral element and $YA\smallsetminus H$
is the set--theoretical difference of $YA$ and $H$.
The product $YA$ is the set $\{y\theta a: y\in Y, a\in A\}$.)
\bigskip
\item{7.} Write an \APL\ function @PROD@ that returns for given groups
$(G_1,\theta_1)$ ja $(G_2,\theta_2)$ the {\sl direct product}
$(G_1\times G_2,\theta_1\times\theta_2)$ as a group table.
(The binary operation in the product is $(x,y)\theta_1\times\theta_2
(z,w) = (x\theta_1 z,y\theta_2 w)$).
\bigskip
\vfill\eject
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