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author | Karl Berry <karl@freefriends.org> | 2006-01-09 00:44:40 +0000 |
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committer | Karl Berry <karl@freefriends.org> | 2006-01-09 00:44:40 +0000 |
commit | b4fc5f639874db951177ec539299d20908adb654 (patch) | |
tree | 52f08823ca58fffe3db6a9b075635038c567626c /Master/texmf-dist/doc/latex/apl/sample.tex | |
parent | dec3d98ebe442d7ea93efbaa8dd2e2be8149a467 (diff) |
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git-svn-id: svn://tug.org/texlive/trunk@80 c570f23f-e606-0410-a88d-b1316a301751
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diff --git a/Master/texmf-dist/doc/latex/apl/sample.tex b/Master/texmf-dist/doc/latex/apl/sample.tex new file mode 100644 index 00000000000..85481d7c943 --- /dev/null +++ b/Master/texmf-dist/doc/latex/apl/sample.tex @@ -0,0 +1,124 @@ + +%============================================================================ +% 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 |