%% This document created by Scientific Notebook (R) Version 3.0


\documentclass[12pt,thmsa]{article}
\usepackage{amssymb}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{makeidx}
\usepackage{sw20jart}

%TCIDATA{TCIstyle=article/art4.lat,jart,sw20jart}

%TCIDATA{<META NAME="ViewSettings" CONTENT="23">}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="Title" CONTENT="Applied Modern Algebra">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 16:56:09 1997}
%TCIDATA{CSTFile=MathHelp.cst}
%TCIDATA{PageSetup=72,72,36,36,0}
%TCIDATA{<LINK REL="begin" HREF="DM12-1.tex">}
%TCIDATA{<LINK REL="contents" HREF="DM0-0.tex">}
%TCIDATA{<LINK REL="end" HREF="DM12-6.tex">}
%TCIDATA{<LINK REL="next" HREF="DM12-2.tex">}
%TCIDATA{<LINK REL="parent" HREF="DM12.tex">}
%TCIDATA{<LINK REL="previous" HREF="DM11-9.tex">}
%TCIDATA{<LINK REL="top" HREF="DM0-0.tex">}

\input{tcilatex}
\begin{document}


\section{Applied Modern Algebra\label{Applied Modern Algebra}}

In this chapter we give examples of problems in applied algebra where you
can make use of \textsl{Scientific Notebook}. The examples include integer
equations, finite rings and fields, coding theory, and linear programming. 
\index{Integer solutions@Integer solutions}

\subsection{\ Solving Equations}

Many techniques in algebra are designed to solve equations, from integer
equations to polynomial equations to matrix equations. In this section, we
describe a few of the methods that can be applied to such problems.

\subsection{\ Integer Solutions\label{Solve, integer}}

The operation\textsf{\ Integer} on the \textsf{Solve} submenu finds\label%
{Integer solutions} 
\index{Solve@Solve!integer@integer}integer solutions to equations and
systems of equations.\medskip

\begin{quote}
$\blacktriangleright $ \textsf{Solve + Integer}
\end{quote}

\begin{quotation}
$3x+4y=10$, Solution is : $\left\{ x=4N_{1}-10,y=-3N_{1}+10\right\} $
\end{quotation}

Thus, one solution is given by $x=-10$ and $y=10$, and every solution is of
the form $x=4N_1-10$ and $y=-3N_1+10,$ where $N_1$ is any integer. As a
check, note that, if $x=4N_1-10$ and $y=-3N_1+10,$ then $%
3x+4y=3(4N_1-10)+4(-3N_1+10)=-30+40=10$.

In a similar manner, a system of equations can be solved for integer
solutions.\medskip

\begin{quote}
$\blacktriangleright $ \textsf{Solve + Integer}
\end{quote}

\begin{quotation}
$
\begin{array}{c}
3x+2y=5 \\ 
3x-z=1
\end{array}
$, Solution is : $\left\{ x=5-2N_{1},y=-5+3N_{1},z=-6N_{1}+14\right\} $
\end{quotation}

Indeed, if $x$, $y$, and $z$ are given by the stated equations, then, for
any integer $N_{1}$, we have $3x+2y=3(5-2N_{1})+2\left( -5+3N_{1}\right) =5$
and $3x-z=3\left( 5-2N_{1}\right) -\left( -6N_{1}+14\right) =1$.

\strut

\subsection{\ \label{Recursive definitions}Recursive Solutions\label%
{Solve, recursion}}

\textsf{Recursion} finds solutions to a 
\index{Solve@Solve!recursion@recursion}recursion or a system of recursions.
For example, to solve the recursion 
\index{Recursive definitions@Recursive definitions} $y(n+2)+3y(n+1)+2y(n)=0$%
, choose \textsf{Recursion} from the \textsf{Solve} submenu.\medskip

\begin{quote}
$\blacktriangleright $ \textsf{Solve} \textsf{+ Recursion}
\end{quote}

\begin{quotation}
$y\left( n+2\right) +3y\left( n+1\right) +2y\left( n\right) =0$,

Solution is : $\left\{ y\left( n\right) =\left( 2y\left( 0\right) +y\left(
1\right) \right) \left( -1\right) ^{n}+\left( -y\left( 0\right) -y\left(
1\right) \right) \left( -2\right) ^{n}\right\} $
\end{quotation}

You can also specify the initial conditions by listing a system of equations
in a column matrix.\medskip

\begin{quote}
$\blacktriangleright $ \textsf{Solve + Recursion}
\end{quote}

\begin{quotation}
$
\begin{array}{c}
y\left( n+2\right) +3y\left( n+1\right) +2y\left( n\right) =0 \\ 
y\left( 0\right) =-2 \\ 
y\left( 1\right) =1
\end{array}
$, Solution is : $\left\{ y\left( n\right) =-3\left( -1\right) ^{n}+\left(
-2\right) ^{n}\right\} $
\end{quotation}

This closed-form solution now makes it easy to find specific terms. For
example, if you define $y(n)=-3(-1)^n+(-2)^n$, then $y(n)$ can be directly
evaluated.\medskip

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$y\left( 20\right) =\allowbreak 1048573$
\end{quotation}

Systems of recursive equations also can be solved.\medskip

\begin{quote}
$\blacktriangleright $ \textsf{Solve + Recursion}
\end{quote}

\begin{quotation}
$
\begin{array}{r}
y(n+1)+z(n)=n+2^{n+1} \\ 
z(n+1)-y(n)=n+3-2^{n} \\ 
y(1)=1 \\ 
z(1)=2
\end{array}
$, Solution is : $\left\{ y\left( n\right) =-1+2^{n},z\left( n\right)
=n+1\right\} $
\end{quotation}

In particular, $z(10)=11$ and $y(10)=1023$.\medskip

\subsection{\ Related topics}

\hyperref{Integers Modulo m}{}{}{DM12-2.tex#Integers modulo m}

\hyperref{Other Systems Modulo m}{}{}{DM12-3.tex#Matrices modulo m}

\hyperref{Polynomials Modulo Polynomials}{}{}{DM12-4.tex#Polynomials modulo
polynomials}

\hyperref{Linear Programming}{}{}{DM12-5.tex#Linear programming}

\hyperref{Exercises and Solutions}{}{}{DM12-6.tex#Exercises}

\rule{0.67in}{0.01in}

\hyperref{\FRAME{itbpF}{0.5275in}{0.3399in}{0.0519in}{}{}{index.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";display "PICT";valid-file
"F";width 0.5275in;height 0.3399in;depth 0.0519in;original-width
36.875pt;original-height 22.5625pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'index.wmf';file-properties "XNPEU";}}}{}{}{%
DMIndex.tex#Math Index}\quad {\small Index entries: \hyperref{Applied modern
algebra}{}{}{DMIndex.tex#Applied modern algebra}, \hyperref{Solve}{}{}{%
DMIndex.tex#Solve}}

\end{document}
