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diff --git a/Master/texmf-dist/source/fonts/apl/sample.tex b/Master/texmf-dist/source/fonts/apl/sample.tex deleted file mode 100644 index 85481d7c943..00000000000 --- a/Master/texmf-dist/source/fonts/apl/sample.tex +++ /dev/null @@ -1,124 +0,0 @@ - -%============================================================================ -% 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 |