$$ % create the definition symbol \def\bydef{\stackrel{\Delta}{=}} %\def\circconv{\otimes} \def\circconv{\circledast} \newcommand{\qed}{\mbox{ } \Box} \newcommand{\infint}{\int_{-\infty}^{\infty}} % z transform \newcommand{\ztp}{ ~~ \mathop{\mathcal{Z}}\limits_{\longleftrightarrow} ~~ } \newcommand{\iztp}{ ~~ \mathop{\mathcal{Z}^{-1}}\limits_{\longleftrightarrow} ~~ } % fourier transform pair \newcommand{\ftp}{ ~~ \mathop{\mathcal{F}}\limits_{\longleftrightarrow} ~~ } \newcommand{\iftp}{ ~~ \mathop{\mathcal{F}^{-1}}\limits_{\longleftrightarrow} ~~ } % laplace transform \newcommand{\ltp}{ ~~ \mathop{\mathcal{L}}\limits_{\longleftrightarrow} ~~ } \newcommand{\iltp}{ ~~ \mathop{\mathcal{L}^{-1}}\limits_{\longleftrightarrow} ~~ } \newcommand{\ftrans}[1]{ \mathcal{F} \left\{#1\right\} } \newcommand{\iftrans}[1]{ \mathcal{F}^{-1} \left\{#1\right\} } \newcommand{\ztrans}[1]{ \mathcal{Z} \left\{#1\right\} } \newcommand{\iztrans}[1]{ \mathcal{Z}^{-1} \left\{#1\right\} } \newcommand{\ltrans}[1]{ \mathcal{L} \left\{#1\right\} } \newcommand{\iltrans}[1]{ \mathcal{L}^{-1} \left\{#1\right\} } % coordinate vector relative to a basis (linear algebra) \newcommand{\cvrb}[2]{\left[ \vec{#1} \right]_{#2} } % change of coordinate matrix (linear algebra) \newcommand{\cocm}[2]{ \mathop{P}\limits_{#2 \leftarrow #1} } % Transformed vector set \newcommand{\tset}[3]{\{#1\lr{\vec{#2}_1}, #1\lr{\vec{#2}_2}, \dots, #1\lr{\vec{#2}_{#3}}\}} % sum transformed vector set \newcommand{\tsetcsum}[4]{{#1}_1#2(\vec{#3}_1) + {#1}_2#2(\vec{#3}_2) + \cdots + {#1}_{#4}#2(\vec{#3}_{#4})} \newcommand{\tsetcsumall}[4]{#2\lr{{#1}_1\vec{#3}_1 + {#1}_2\vec{#3}_2 + \cdots + {#1}_{#4}\vec{#3}_{#4}}} \newcommand{\cvecsum}[3]{{#1}_1\vec{#2}_1 + {#1}_2\vec{#2}_2 + \cdots + {#1}_{#3}\vec{#2}_{#3}} % function def \newcommand{\fndef}[3]{#1:#2 \to #3} % vector set \newcommand{\vset}[2]{\{\vec{#1}_1, \vec{#1}_2, \dots, \vec{#1}_{#2}\}} % absolute value \newcommand{\abs}[1]{\left| #1 \right|} % vector norm \newcommand{\norm}[1]{\left|\left| #1 \right|\right|} % trans \newcommand{\trans}{\mapsto} % evaluate integral \newcommand{\evalint}[3]{\left. #1 \right|_{#2}^{#3}} % slist \newcommand{\slist}[2]{{#1}_{1},{#1}_{2},\dots,{#1}_{#2}} % vectors \newcommand{\vc}[1]{\textbf{#1}} % real \newcommand{\Real}[1]{{\Re \mit{e}\left\{{#1}\right\}}} % imaginary \newcommand{\Imag}[1]{{\Im \mit{m}\left\{{#1}\right\}}} \newcommand{\mcal}[1]{\mathcal{#1}} \newcommand{\bb}[1]{\mathbb{#1}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\I}{\mathbb{I}} \newcommand{\Th}[1]{\mathop\mathrm{Th(#1)}} \newcommand{\intersect}{\cap} \newcommand{\union}{\cup} \newcommand{\intersectop}{\bigcap} \newcommand{\unionop}{\bigcup} \newcommand{\setdiff}{\backslash} \newcommand{\iso}{\cong} \newcommand{\aut}[1]{\mathop{\mathrm{Aut(#1)}}} \newcommand{\inn}[1]{\mathop{\mathrm{Inn(#1)}}} \newcommand{\Ann}[1]{\mathop{\mathrm{Ann(#1)}}} \newcommand{\dom}[1]{\mathop{\mathrm{dom} #1}} \newcommand{\cod}[1]{\mathop{\mathrm{cod} #1}} \newcommand{\id}{\mathrm{id}} \newcommand{\st}{\ |\ } \newcommand{\mbf}[1]{\mathbf{#1}} \newcommand{\enclose}[1]{\left\langle #1\right\rangle} \newcommand{\lr}[1]{\left( #1\right)} \newcommand{\lrsq}[1]{\left[ #1\right]} \newcommand{\op}{\mathrm{op}} \newcommand{\dotarr}{\dot{\rightarrow}} %Category Names: \newcommand{\Grp}{\mathbf{Grp}} \newcommand{\Ab}{\mathbf{Ab}} \newcommand{\Set}{\mathbf{Set}} \newcommand{\Matr}{\mathbf{Matr}} \newcommand{\IntDom}{\mathbf{IntDom}} \newcommand{\Field}{\mathbf{Field}} \newcommand{\Vect}{\mathbf{Vect}} \newcommand{\thm}[1]{\begin{theorem} #1 \end{theorem}} \newcommand{\clm}[1]{\begin{claim} #1 \end{claim}} \newcommand{\cor}[1]{\begin{corollary} #1 \end{corollary}} \newcommand{\ex}[1]{\begin{example} #1 \end{example}} \newcommand{\prf}[1]{\begin{proof} #1 \end{proof}} \newcommand{\prbm}[1]{\begin{problem} #1 \end{problem}} \newcommand{\soln}[1]{\begin{solution} #1 \end{solution}} \newcommand{\rmk}[1]{\begin{remark} #1 \end{remark}} \newcommand{\defn}[1]{\begin{definition} #1 \end{definition}} \newcommand{\ifff}{\LeftRightArrow} <!-- For the set of reals and integers --> \newcommand{\rr}{\R} \newcommand{\reals}{\R} \newcommand{\ii}{\Z} \newcommand{\cc}{\C} \newcommand{\nn}{\N} \newcommand{\nats}{\N} <!-- For terms being indexed. Puts them in standard font face and creates an index entry. arg: The term being defined. \newcommand{\pointer}[1]{#1\index{#1}} --> <!-- For bold terms to be index, but defined elsewhere Puts them in bold face and creates an index entry. arg: The term being defined. --> \newcommand{\strong}[1]{\textbf{#1}} <!-- For set names. Puts them in italics. In math mode, yields decent spacing. arg: The name of the set. --> \newcommand{\set}[1]{\textit{#1}} $$