\newcommand{\symbolsfootnote}{The page where the
glossary entry is defined is given in angle brackets
(\hypertarget{ch:symbols}{\eg},
\examplebracketref{ch:symbols}).}\begin{theglossary}
\item[\symheader{*}\hfill] 

  \item [{[\texttt {xx}]}] see reference number \texttt {xx} in \hyperref [ch:bibliography]{the bibliography}
		\nopage{100}

  \indexspace
\item[\symheader{A}\hfill] 

  \item [$=$] \index {mathematics!equality ($=$)|indexglo}is equal to
		\nopage{100}
  \item [$\triangleq $] \index {mathematics!definition ($\triangleq $)|indexglo}defined as
		\nopage{100}
  \item [$\approx $] is approximately equal to\nopage{100}
  \item [$<$ ($>$)] strictly less (greater) than\nopage{100}
  \item [$\leq $ ($\geq $)] less (greater) than or equal to\nopage{100}
  \item [$x + y$] sum of $x$ and $y$\nopage{100}
  \item [$x \times y$] product of $x$ and $y$ (also denoted $xy$)
		\nopage{100}
  \item [$-x$] additive inverse of $x$\nopage{100}
  \item [$x - y$] difference of $x$ and $y$ (\ie {}, $x - y \triangleq x + -y$)
		\nopage{100}
  \item [$\sgn (x)$] sign function of $x$\nopage{100}
  \item [$\prod $] product of elements of a set\nopage{100}
  \item [$\sum $] sum of elements of a set\nopage{100}

  \indexspace
\item[\symheader{B}\hfill] 

  \item [$\N $] the set of the \index {mathematics!numbers!natural numbers ($\N $)|indexglo}natural numbers (\ie {}, $\{1,2,3,\dots \}$)
		\nopage{100}
  \item [$\W $] the set of the \index {mathematics!numbers!whole numbers ($\W $)|indexglo}whole numbers (\ie {}, $\{0,1,2,3,\dots \}$)
		\nopage{100}
  \item [$\Z $] the set of the \index {mathematics!numbers!integers ($\Z $)|indexglo}integers (\ie {}, $\{\dots ,-3,-2,-1,0,1,2,3,\dots \}$)
		\nopage{100}
  \item [$\Q $] the set of the \index {mathematics!numbers!rationals ($\Q $)|indexglo}rationals (\ie {}, ratios of integers)
		\nopage{100}
  \item [$\R $] the set of the \index {mathematics!numbers!real numbers ($\R $)|indexglo}real numbers
		\nopage{100}
  \item [$\R _{>0}$] the set of the \index {mathematics!numbers!real positive numbers ($\R _{>0}$)|indexglo}strictly positive real numbers
		\nopage{100}
  \item [$\R _{\geq 0}$] the set of the \index {mathematics!numbers!real non-negative numbers ($\R _{\geq 0}$)|indexglo}non-negative real numbers
		\nopage{100}
  \item [$\R _{<0}$] the set of the \index {mathematics!numbers!real negative numbers ($\R _{<0}$)|indexglo}strictly negative real numbers
		\nopage{100}
  \item [$\R _{\leq 0}$] the set of the \index {mathematics!numbers!real non-positive numbers ($\R _{\leq 0}$)|indexglo}non-positive real numbers
		\nopage{100}
  \item [$\R _{\neq 0}$] the set of the \index {mathematics!numbers!real non-zero numbers ($\R _{\neq 0}$)|indexglo}non-zero real numbers
		\nopage{100}
  \item [$\extR $] the set of the \index {mathematics!numbers!extended real numbers ($\extR $)|indexglo}extended real numbers (\ie {}, $\R \cup \{-\infty ,+\infty \}$)
		\nopage{100}
  \item [$\R ^n$] the \index {mathematics!numbers!Euclidean $n$-space ($\R ^n$)|indexglo}\aimention {Euclid}Euclidean $n$-space
		\nopage{100}
  \item [$\R ^{n \times m}$] space of $n$-by-$m$ real matrices
		\nopage{100}
  \item [$\R ^{n \times n}$] the unitary associative real algebra
		\nopage{100}
  \item [$e$] \aimention {Leonhard Euler}Euler's number (\ie {}, constant $e \approx 2.71828182845904523536$)
		\nopage{100}
  \item [$\log _b(x)$] logarithm of positive real number $x$ in base $b$ (\ie {}, $b^{\log _b(x)} = x$)
		\nopage{100}
  \item [$\log (x)$] common logarithm of positive real number $x$ (\ie {}, $10^{\log (x)} = x$)
		\nopage{100}
  \item [$\ln (x)$] natural logarithm of positive real number $x$ (\ie {}, $e^{\ln (x)} = x$)
		\nopage{100}
  \item [$\exp (x)$] exponential function (\ie {}, $\exp (x) \triangleq e^x$)
		\nopage{100}
  \item [$\lceil x \rceil $] the ceiling of real number $x$ (\ie {}, the least integer not less than $x$)
		\nopage{100}
  \item [$\lfloor x \rfloor $] the floor of real number $x$ (\ie {}, the greatest integer not greater than $x$)
		\nopage{100}

  \indexspace
\item[\symheader{C}\hfill] 

  \item [$\set {X}$] a set $\set {X}$\nopage{100}
  \item [$\{a,b,c\}$] a set of objects $a$, $b$, and $c$\nopage{100}
  \item [$\dots $] continue the established pattern \adinfinitum {} (\eg {}, the infinite set $\{1,2,3,\dots \}$)
		\nopage{100}
  \item [$\{ u : p \}$] set of all elements of $u$ such that $p$
		\nopage{100}
  \item [$\{ u : p, q, r \}$] set of all elements of $u$ such that $p$, $q$, and $r$
		\nopage{100}
  \item [$\emptyset $] \index {mathematics!sets!empty set ($\emptyset $)|indexglo}the empty set (\ie {}, $\{\}$)
		\nopage{100}
  \item [$\in $] is an element of (\ie {}, \index {mathematics!sets!set element ($\in $)|indexglo}set inclusion)
		\nopage{100}
  \item [$\notin $] is not an element of (\ie {}, \index {mathematics!sets!exclusion ($\notin $)|indexglo}set exclusion)
		\nopage{100}
  \item [$\subset $ ($\supset $)] is a \index {mathematics!sets!superset ($\supseteq $ or $\supset $)|indexglo}proper/strict subset (superset) of
		\nopage{100}
  \item [$\subseteq $ ($\supseteq $)] is a \index {mathematics!sets!subset ($\subseteq $ or $\subset $)|indexglo}subset (superset) of
		\nopage{100}
  \item [$\set {X} = \set {Y}$] set $\set {X}$ is equal to set $\set {Y}$ (\ie {}, $\set {X} \subseteq \set {Y}$ and $\set {Y} \subseteq \set {X}$)
		\nopage{100}
  \item [$\set {X} \neq \set {Y}$] set $\set {X}$ is not equal to set $\set {Y}$
		\nopage{100}
  \item [$\Pow (\set {U})$] \index {mathematics!sets!power set ($\Pow $)|indexglo}power set of set $\set {U}$ (\ie {}, the set of all subsets of $\set {U}$)
		\nopage{100}
  \item [$\pipe \set {X}\pipe $] cardinality of set $\set {X}$
		\nopage{100}
  \item [$\bigcap $] intersection of many sets (compare to $\sum $)
		\nopage{100}
  \item [$\bigcup $] union of many sets (compare to $\sum $)
		\nopage{100}
  \item [$\set {X} \cap \set {Y}$] \index {mathematics!sets!set intersection ($\cap $)|indexglo}set intersection (or meet) of sets $\set {X}$ and $\set {Y}$
		\nopage{100}
  \item [$\set {X} \cup \set {Y}$] \index {mathematics!sets!set union ($\cup $)|indexglo}set union (or join) of sets $\set {X}$ and $\set {Y}$
		\nopage{100}
  \item [$\set {X} \setdiff \set {Y}$] \index {mathematics!sets!set difference ($\setdiff $)|indexglo}difference of sets $\set {X}$ and $\set {Y}$
		\nopage{100}
  \item [$\set {X}^c$] \index {mathematics!sets!set complement (${}^c$)|indexglo}complement of set $\set {X}^c$ (\eg {}, $\set {U} \setdiff \set {X}$ where $\set {X} \subseteq \set {U}$)
		\nopage{100}
  \item [$(a,b)$] \index {mathematics!ordered pair|indexglo}ordered pair of objects $a$ and $b$ (\ie {}, $(a,b) \triangleq \{\{a\},\{a,b\}\}$)
		\nopage{100}
  \item [$(x_1,x_2,\dots ,x_n)$] $n$-tuple (\ie {}, \index {mathematics!n-tuple@$n$-tuple|indexglo}tuple of length $n \in \N $ with coordinates $x_1$, $x_2$,\dots ,$x_n$ in their respective order)
		\nopage{100}
  \item [$\set {X} \times \set {Y}$] \index {mathematics!sets!Cartesian product ($\times $)|(indexglo}(binary) \aimention {Ren\'{e} Descartes}Cartesian product of sets $\set {X}$ and $\set {Y}$ (\ie {}, $\set {X} \times \set {Y} \triangleq \{(x,y):x \in \set {X}, y \in \set {Y}\}$)
		\nopage{100}
  \item [$\set {X}_1 \times \cdots \times \set {X}_n$] \aimention {Ren\'{e} Descartes}Cartesian product of $n$ sets $\set {X}_1$, \dots , $\set {X}_n$ (\ie {}, $\set {X}_1 \times \cdots \times \set {X}_n \triangleq \{(x_1,\dots ,x_n):x_1 \in \set {X}_1, \dots , x_n \in \set {X}_n\}$)
		\nopage{100}
  \item [$\set {X}^n$] \aimention {Ren\'{e} Descartes}Cartesian product of set $\set {X}$ with itself $n$ times (\eg {}, $\set {X}^3 \triangleq \set {X} \times \set {X} \times \set {X}$)\index {mathematics!sets!Cartesian product ($\times $)|)indexglo}
		\nopage{100}
  \item [${[a,b]}$] \index {mathematics!numbers!real number intervals|(indexglo}interval $[a,b] \triangleq \{ x \in \set {X} : a \leq x \leq b \}$
		\nopage{100}
  \item [${(a,b]}$] interval $(a,b] \triangleq \{ x \in \set {X} : a < x \leq b \}$
		\nopage{100}
  \item [${[a,b)}$] interval $[a,b) \triangleq \{ x \in \set {X} : a \leq x < b \}$
		\nopage{100}
  \item [${(a,b)}$] interval $(a,b) \triangleq \{ x \in \set {X} : a < x < b \}$\index {mathematics!numbers!real number intervals|)indexglo}
		\nopage{100}

  \indexspace
\item[\symheader{D}\hfill] 

  \item [$x(i)$~or~$x_i$~or~$x^i$] alternate notations for an index $i$ on a symbol $x$
		\nopage{100}
  \item [$(x_i:i \in \set {I})$] an indexed family with index set $\set {I}$ (also $(x_i)_{i \in \set {I}}$)
		\nopage{100}
  \item [$(x(t):t \geq 0)$] an ordered indexed family with a directed index set $\set {T}$ where $0 \in \set {T}$
		\nopage{100}
  \item [$(x_\alpha )$] a net (\ie {}, an ordered indexed family $(x_\alpha : \alpha \in \set {A})$ with directed index set $\set {A}$)
		\nopage{100}
  \item [$(x_n)$] a sequence (\ie {}, an ordered indexed family $(x_n : n \in \N )$ with totally ordered index set $\N $)
		\nopage{100}

  \indexspace
\item[\symheader{E}\hfill] 

  \item [$\iff $] \index {mathematics!logic!equivalence ($\iff $)|indexglo}logical equivalence
		\nopage{100}
  \item [$\implies $] \index {mathematics!logic!implication ($\implies $)|indexglo}logical implication
		\nopage{100}

  \indexspace
\item[\symheader{F}\hfill] 

  \item [$\inf $] \index {mathematics!order!infimum ($\inf $)|indexglo}infimum (\ie {}, greatest lower bound or meet)
		\nopage{100}
  \item [$\sup $] \index {mathematics!order!supremum ($\sup $)|indexglo}supremum (\ie {}, lowest upper bound or join)
		\nopage{100}
  \item [$\max $] \index {mathematics!order!maximum ($\max $)|indexglo}maximum element
		\nopage{100}
  \item [$\min $] \index {mathematics!order!minimum ($\min $)|indexglo}minimum element
		\nopage{100}

  \indexspace
\item[\symheader{G}\hfill] 

  \item [${n\bang }$] factorial of $n$ (\ie {}, ${n\bang }=1\times 2\times \cdots \times n$ with ${0\bang }=1$)
		\nopage{100}
  \item [$f: \set {X} \mapsto \set {Y}$] a \index {mathematics!functions|indexglo}function $f$ with domain $\set {X}$ and codomain $\set {Y}$
		\nopage{100}
  \item [$\lim $] \index {mathematics!limit ($\lim $ or $\to $)|indexglo}limit (\eg {}, unique limit of filter base, function, net, or sequence)
		\nopage{100}
  \item [$\to $] a limit\nopage{100}
  \item [$p_n \to p$] limit of sequence $(p_n)$\nopage{100}
  \item [$f(x) \to q$] limit of function $f$ (\eg {}, as $x \to p$)
		\nopage{100}
  \item [$f'(x_0+)$] the right-hand derivative of function $f$ at point $x_0$
		\nopage{100}
  \item [$f'(x_0-)$] the left-hand derivative of function $f$ at point $x_0$
		\nopage{100}
  \item [$f'(x_0)$] the \index {mathematics!functions!total derivative ($\total $)|indexglo}first (total) derivative of function $f$ at point $x_0$
		\nopage{100}
  \item [$f''(x_0)$] the second (ordinary) derivative of function $f$ at point $x_0$
		\nopage{100}
  \item [$f'''(x_0)$] the third (ordinary) derivative of function $f$ at point $x_0$
		\nopage{100}
  \item [$f^{(n)}(x_0)$] the $n\th $ (ordinary) derivative of function $f$ at point $x_0$ where $n \in \{4,5,6,\dots \}$
		\nopage{100}
  \item [$\total f/\total t$] total derivative of function $f$ at point $t$
		\nopage{100}
  \item [$\total ^2 f/{\total t}^2$] second total derivative of function $f$ (\ie {}, $f''$)
		\nopage{100}
  \item [$\total ^3 f/{\total t}^3$] third total derivative of function $f$ (\ie {}, $f'''$)
		\nopage{100}
  \item [$\total ^n f/{\total t}^n$] $n\th $ total derivative of function $f$ (\ie {}, $f^{(n)}$)
		\nopage{100}
  \item [$\partial f/\partial x$] \index {mathematics!functions!partial derivative ($\partial $)|indexglo}partial derivative of function $f$ with respect to $x$
		\nopage{100}
  \item [$\partial ^2 f/\partial x \partial y$] partial derivative of function $\partial f/\partial x$ with respect to $y$
		\nopage{100}

  \indexspace
\item[\symheader{H}\hfill] 

  \item [$y_i$] the $i\th $ coordinate of vector $\v {y}$\nopage{100}
  \item [$\v {x}^\T $] the \index {mathematics!vector spaces!vector transpose (${}^\T $)|indexglo}transpose of vector or covector $\v {x}$ (\ie {}, if $\v {x}$ is an $n$-vector then $\v {x} = [x_1, x_2, \dots , x_n]^\T )$
		\nopage{100}
  \item [$\mat {A}^\T $] the \index {mathematics!vector spaces!matrix transpose (${}^\T $)|indexglo}transpose of matrix $\mat {A}$
		\nopage{100}
  \item [$\v {e}_i$] the $i\th $ elementary (or standard) basis vector
		\nopage{100}
  \item [$\I $] the identity matrix\nopage{100}
  \item [$\nabla _{\v {x}} f(\v {x})$] the \index {mathematics!functions!gradient ($\nabla $)|indexglo}gradient vector of function $f$ at $\v {x}$
		\nopage{100}
  \item [$\nabla ^2_{\v {x}\v {x}} f(\v {x})$] the \index {mathematics!functions!Hessian ($\nabla ^2$)|indexglo}\aimention {Ludwig Otto Hesse}Hessian matrix of function $f$ at point $\v {x}$
		\nopage{100}

  \indexspace
\item[\symheader{I}\hfill] 

  \item [$\Borel (\set {U})$] the \index {mathematics!sets!Borel algebra ($\Borel $)|indexglo}\aimention {\'{E}mile Borel}Borel algebra of set $\set {U}$ (\ie , $\Borel (\set {U})$ is the minimal a $\sigma $-algebra containing the open sets; elements of $\Borel (\set {U})$ are called \emph {\aimention {\'{E}mile Borel}Borel sets} and are subsets of $\set {U}$, so $\Borel (\set {U} \in \Pow (\set {U})$
		\nopage{100}
  \item [$\int _a^b f(x) \total x$] the \index {mathematics!functions!integral ($\int $)|indexglo}\aimention {Henri L. Lebesgue}Lebesgue integral of function $f$ over interval $[a,b] \subset \extR $ with respect to the \aimention {Henri L. Lebesgue}Lebesgue measure
		\nopage{100}
  \item [$f * g$] \index {mathematics!functions!convolution ($*$)|indexglo}convolution of function $f$ with function $g$ (\ie {}, $(f * g)(t) \triangleq \int _{-\infty }^\infty f(\tau ) g(t-\tau ) \total \tau $)
		\nopage{100}
  \item [$\delta _a(\set {E})$] Dirac delta measure of set $\set {E}$ at point $a$ (\eg {}, $f(0) = \linebreak [4] \int _{-1}^1 f(x) \delta _0(\{x\}) \total x$)
		\nopage{100}
  \item [$\delta (x-p)$] Simplified Dirac delta measure notation (\ie {}, $\delta (x-p) \triangleq \delta _p(\{x\})$)
		\nopage{100}
  \item [$\Pr $] \index {stochasticity!probability measure|indexglo}Probability measure
		\nopage{100}
  \item [$(\set {U},\Sigma ,\Pr )$] \index {stochasticity!probability space|indexglo}Probability space with outcomes $\set {U}$, $\sigma $-field of events $\Sigma $, and probability measure $\Pr $
		\nopage{100}
  \item [$\{X \leq a\}$] Measurable set induced by preimage of random variable $X$ (\ie {}, \linebreak [3] $\{ \zeta \in \set {U} : X(\zeta ) \leq a \}$)
		\nopage{100}
  \item [$\Pr (X \leq a)$] Probability induced by preimage of random variable $X$ (\ie {}, \linebreak [3] $\Pr (\{ \zeta \in \set {U} : X(\zeta ) \leq a \})$)
		\nopage{100}
  \item [$F_X(x)$] \index {stochasticity!random variable!cumulative distribution function ($F$)|indexglo}Cumulative distribution function for random variable $X$ (\ie {}, $F_X(a) \triangleq \Pr (X \leq a)$)
		\nopage{100}
  \item [$f_X(x)$] \index {stochasticity!random variable!probability density function ($f$)|indexglo}Probability density function for random variable $X$ (\ie {}, $F_X(a) = \int _{-\infty }^a f_X(x) \total x$)
		\nopage{100}
  \item [$\E (X)$] \index {stochasticity!statistics!expectation ($\E $)|indexglo}Expectation of random variable $X$ (\ie {}, \linebreak [4] $\int _{-\infty }^\infty x f_X(x) \total x$)
		\nopage{100}
  \item [$\E (g(X))$] \index {stochasticity!statistics!expectation of function|indexglo}Expectation of function $g$ of random variable $X$ (\ie {}, \linebreak [4] $\int _{-\infty }^\infty g(x) f_X(x) \total x$)
		\nopage{100}
  \item [$\cov (X,Y)$] Covariance of random variables $X$ and $Y$ (\ie {}, $\cov (X,Y) = \E (XY) - \E (X)\E (Y)$)
		\nopage{100}
  \item [$F_{XY}(x,y)$] Joint distribution function for random variables $X$ and $Y$ (\ie {}, $F_{XY}(a,b) \triangleq \Pr (X \leq a, Y \leq b)$)
		\nopage{100}
  \item [$f_{XY}(x,y)$] Joint density function for random variables $X$ and $Y$
		\nopage{100}
  \item [$f_{Y \pipe X}(y \pipe x)$] Conditional density function for random variable $Y$ given $X=x$
		\nopage{100}
  \item [$F_{Y \pipe X}(y \pipe x)$] Conditional distribution function for random variable $Y$ given $X=x$
		\nopage{100}
  \item [$\E (Y \pipe X)$] \index {stochasticity!statistics!conditional expectation|indexglo}Conditional expectation of $Y$ given $X$
		\nopage{100}
  \item [$( \v {N}(t) : t \in \R _{\geq 0})$] \index {stochasticity!random process|indexglo}Random process (\ie {}, $\v {N}(t)$ is a random vector for all $t \in \R _{>0}$)
		\nopage{100}
  \item [$Y(t) \xto {a.s.} Y$] \index {stochasticity!random process!almost sure limit ($\aslim $ or $\xto {a.s.}$)|indexglo}Random process $Y(t)$ converges almost surely (\ie {}, $\Pr (\lim _{t \to \infty } Y(t) = Y) = 1$) to $Y$
		\nopage{100}
  \item [$\aslim \limits _{t \to \infty } Y(t) = Y$] Random process $Y(t)$ converges almost surely (\ie {}, with probability 1) to $Y$
		\nopage{100}\index{stochasticity|)indexglo}\index{mathematics|)indexglo}

\end{theglossary}