\indexentry {*conventions@[{[\texttt {xx}]}] see reference number \texttt {xx} in \hyperref [ch:bibliography]{the bibliography}|nopage}{100}
\indexentry {Ageneral.0@[$=$] \index {mathematics"!equality ($=$)"|indexglo}is equal to|nopage}{100}
\indexentry {Ageneral.0@[$\triangleq $] \index {mathematics"!definition ($\triangleq $)"|indexglo}defined as|nopage}{100}
\indexentry {Csets.0@[$\set {X}$] a set $\set {X}$|nopage}{100}
\indexentry {Csets.1a@[$\{a,b,c\}$] a set of objects $a$, $b$, and $c$|nopage}{100}
\indexentry {Csets.1aa@[$\dots $] continue the established pattern \adinfinitum {} (\eg {}, the infinite set $\{1,2,3,\dots \}$)|nopage}{100}
\indexentry {Csets.1b@[$\emptyset $] \index {mathematics"!sets"!empty set ($\emptyset $)"|indexglo}the empty set (\ie {}, $\{\}$)|nopage}{100}
\indexentry {Csets.1b@[$\in $] is an element of (\ie {}, \index {mathematics"!sets"!set element ($\in $)"|indexglo}set inclusion)|nopage}{100}
\indexentry {Csets.1b@[$\notin $] is not an element of (\ie {}, \index {mathematics"!sets"!exclusion ($\notin $)"|indexglo}set exclusion)|nopage}{100}
\indexentry {Csets.1c@[$\subseteq $ ($\supseteq $)] is a \index {mathematics"!sets"!subset ($\subseteq $ or $\subset $)"|indexglo}subset (superset) of|nopage}{100}
\indexentry {Csets.1d@[$\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}
\indexentry {Csets.1d@[$\set {X} \neq \set {Y}$] set $\set {X}$ is not equal to set $\set {Y}$|nopage}{100}
\indexentry {Csets.1c@[$\subset $ ($\supset $)] is a \index {mathematics"!sets"!superset ($\supseteq $ or $\supset $)"|indexglo}proper/strict subset (superset) of|nopage}{100}
\indexentry {Csets.1ab@[$\{ u : p \}$] set of all elements of $u$ such that $p$|nopage}{100}
\indexentry {Csets.1ab@[$\{ u : p, q, r \}$] set of all elements of $u$ such that $p$, $q$, and $r$|nopage}{100}
\indexentry {Dseq.0@[$x(i)$~or~$x_i$~or~$x^i$] alternate notations for an index $i$ on a symbol $x$|nopage}{100}
\indexentry {Csets.2cart0@[$(a,b)$] \index {mathematics"!ordered pair"|indexglo}ordered pair of objects $a$ and $b$ (\ie {}, $(a,b) \triangleq \{\{a\},\{a,b\}\}$)|nopage}{100}
\indexentry {Csets.2cart01@[$(x_1,x_2,\dots ,x_n)$] $n$-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}
\indexentry {Csets.2cart1@[$\set {X} \times \set {Y}$] \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}
\indexentry {Csets.2cart10@[$\set {X}_1 \times \cdots \times \set {X}_n$] \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}
\indexentry {Csets.2cart11@[$\set {X}^n$] \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 $)"|)indexglo}|nopage}{100}
\indexentry {Ganalysis.0011@[$f: \set {X} \mapsto \set {Y}$] a \index {mathematics"!functions"|indexglo}function $f$ with domain $\set {X}$ and codomain $\set {Y}$|nopage}{100}
\indexentry {Dseq.1@[$(x_i:i \in \set {I})$] an indexed family with index set $\set {I}$ (also $(x_i)_{i \in \set {I}}$)|nopage}{100}
\indexentry {Dseq.2@[$(x(t):t \geq 0)$] an ordered indexed family with a directed index set $\set {T}$ where $0 \in \set {T}$|nopage}{100}
\indexentry {Csets.1zz@[$\pipe \set {X}\pipe $] cardinality of set $\set {X}$|nopage}{100}
\indexentry {Csets.1z@[$\Pow (\set {U})$] \index {mathematics"!sets"!power set ($\Pow $)"|indexglo}power set of set $\set {U}$ (\ie {}, the set of all subsets of $\set {U}$)|nopage}{100}
\indexentry {Csets.207@[$\set {X}^c$] \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}
\indexentry {Csets.202@[$\set {X} \cup \set {Y}$] \index {mathematics"!sets"!set union ($\cup $)"|indexglo}set union (or join) of sets $\set {X}$ and $\set {Y}$|nopage}{100}
\indexentry {Csets.2@[$\bigcup $] union of many sets (compare to $\sum $)|nopage}{100}
\indexentry {Csets.201@[$\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}
\indexentry {Csets.2@[$\bigcap $] intersection of many sets (compare to $\sum $)|nopage}{100}
\indexentry {Csets.203@[$\set {X} \setdiff \set {Y}$] \index {mathematics"!sets"!set difference ($\setdiff $)"|indexglo}difference of sets $\set {X}$ and $\set {Y}$|nopage}{100}
\indexentry {Elogic@[$\implies $] \index {mathematics"!logic"!implication ($\implies $)"|indexglo}logical implication|nopage}{100}
\indexentry {Elogic@[$\iff $] \index {mathematics"!logic"!equivalence ($\iff $)"|indexglo}logical equivalence|nopage}{100}
\indexentry {Ageneral.5@[$\leq $ ($\geq $)] less (greater) than or equal to|nopage}{100}
\indexentry {Ageneral.5@[$<$ ($>$)] strictly less (greater) than|nopage}{100}
\indexentry {Csets.2intervals1@[${[a,b]}$] \index {mathematics"!numbers"!real number intervals"|(indexglo}interval $[a,b] \triangleq \{ x \in \set {X} : a \leq x \leq b \}$|nopage}{100}
\indexentry {Csets.2intervals2@[${(a,b]}$] interval $(a,b] \triangleq \{ x \in \set {X} : a < x \leq b \}$|nopage}{100}
\indexentry {Csets.2intervals3@[${[a,b)}$] interval $[a,b) \triangleq \{ x \in \set {X} : a \leq x < b \}$|nopage}{100}
\indexentry {Csets.2intervals4@[${(a,b)}$] interval $(a,b) \triangleq \{ x \in \set {X} : a < x < b \}$\index {mathematics"!numbers"!real number intervals"|)indexglo}|nopage}{100}
\indexentry {Forder.201@[$\sup $] \index {mathematics"!order"!supremum ($\sup $)"|indexglo}supremum (\ie {}, lowest upper bound or join)|nopage}{100}
\indexentry {Forder.202@[$\max $] \index {mathematics"!order"!maximum ($\max $)"|indexglo}maximum element|nopage}{100}
\indexentry {Forder.201@[$\inf $] \index {mathematics"!order"!infimum ($\inf $)"|indexglo}infimum (\ie {}, greatest lower bound or meet)|nopage}{100}
\indexentry {Forder.202@[$\min $] \index {mathematics"!order"!minimum ($\min $)"|indexglo}minimum element|nopage}{100}
\indexentry {Dseq.3@[$(x_\alpha )$] a net (\ie {}, an ordered indexed family $(x_\alpha : \alpha \in \set {A})$ with directed index set $\set {A}$)|nopage}{100}
\indexentry {Dseq.3@[$(x_n)$] a sequence (\ie {}, an ordered indexed family $(x_n : n \in \N )$ with totally ordered index set $\N $)|nopage}{100}
\indexentry {Ageneral.541@[$x + y$] sum of $x$ and $y$|nopage}{100}
\indexentry {Ageneral.542@[$x \times y$] product of $x$ and $y$ (also denoted $xy$)|nopage}{100}
\indexentry {Ageneral.543@[$-x$] additive inverse of $x$|nopage}{100}
\indexentry {Ageneral.5431@[$x - y$] difference of $x$ and $y$ (\ie {}, $x - y \triangleq x + -y$)|nopage}{100}
\indexentry {Ageneral.5432@[$\sgn (x)$] sign function of $x$|nopage}{100}
\indexentry {Ageneral.5433@[$\pipe x \pipe $] \index {mathematics\"!numbers\"!absolute value\"|indexglo}absolute value of $x$ (\ie {}, $x = \sgn (x) \pipe x \pipe $)|nopage}{100}
\indexentry {Ageneral.z@[$\sum $] sum of elements of a set|nopage}{100}
\indexentry {Ageneral.z@[$\prod $] product of elements of a set|nopage}{100}
\indexentry {Bnumbers.2@[$\W $] the set of the \index {mathematics"!numbers"!whole numbers ($\W $)"|indexglo}whole numbers (\ie {}, $\{0,1,2,3,\dots \}$)|nopage}{100}
\indexentry {Bnumbers.1@[$\N $] the set of the \index {mathematics"!numbers"!natural numbers ($\N $)"|indexglo}natural numbers (\ie {}, $\{1,2,3,\dots \}$)|nopage}{100}
\indexentry {Bnumbers.3@[$\Z $] the set of the \index {mathematics"!numbers"!integers ($\Z $)"|indexglo}integers (\ie {}, $\{\dots ,-3,-2,-1,0,1,2,3,\dots \}$)|nopage}{100}
\indexentry {Bnumbers.4@[$\Q $] the set of the \index {mathematics"!numbers"!rationals ($\Q $)"|indexglo}rationals (\ie {}, ratios of integers)|nopage}{100}
\indexentry {Bnumbers.50@[$\R $] the set of the \index {mathematics"!numbers"!real numbers ($\R $)"|indexglo}real numbers|nopage}{100}
\indexentry {Bnumbers.510@[$\R _{>0}$] the set of the \index {mathematics"!numbers"!real positive numbers ($\R _{>0}$)"|indexglo}strictly positive real numbers|nopage}{100}
\indexentry {Bnumbers.511@[$\R _{\geq 0}$] the set of the \index {mathematics"!numbers"!real non-negative numbers ($\R _{\geq 0}$)"|indexglo}non-negative real numbers|nopage}{100}
\indexentry {Bnumbers.520@[$\R _{<0}$] the set of the \index {mathematics"!numbers"!real negative numbers ($\R _{<0}$)"|indexglo}strictly negative real numbers|nopage}{100}
\indexentry {Bnumbers.521@[$\R _{\leq 0}$] the set of the \index {mathematics"!numbers"!real non-positive numbers ($\R _{\leq 0}$)"|indexglo}non-positive real numbers|nopage}{100}
\indexentry {Bnumbers.53@[$\R _{\neq 0}$] the set of the \index {mathematics"!numbers"!real non-zero numbers ($\R _{\neq 0}$)"|indexglo}non-zero real numbers|nopage}{100}
\indexentry {Bnumbers.61@[$\lfloor x \rfloor $] the floor of real number $x$ (\ie {}, the greatest integer not greater than $x$)|nopage}{100}
\indexentry {Bnumbers.60@[$\lceil x \rceil $] the ceiling of real number $x$ (\ie {}, the least integer not less than $x$)|nopage}{100}
\indexentry {Bnumbers.54@[$\extR $] the set of the \index {mathematics"!numbers"!extended real numbers ($\extR $)"|indexglo}extended real numbers (\ie {}, $\R \cup \{-\infty ,+\infty \}$)|nopage}{100}
\indexentry {Ganalysis.120@[$\to $] a limit|nopage}{100}
\indexentry {Ganalysis.10@[$\lim $] \index {mathematics"!limit ($\lim $ or $\to $)"|indexglo}limit (\eg {}, unique limit of filter base, function, net, or sequence)|nopage}{100}
\indexentry {Ganalysis.122@[$f(x) \to q$] limit of function $f$ (\eg {}, as $x \to p$)|nopage}{100}
\indexentry {Ganalysis.121@[$p_n \to p$] limit of sequence $(p_n)$|nopage}{100}
\indexentry {Ganalysis.2b@[$f'(x_0)$] the \index {mathematics"!functions"!total derivative ($\total $)"|indexglo}first (total) derivative of function $f$ at point $x_0$|nopage}{100}
\indexentry {Ganalysis.2a@[$f'(x_0+)$] the right-hand derivative of function $f$ at point $x_0$|nopage}{100}
\indexentry {Ganalysis.2a@[$f'(x_0-)$] the left-hand derivative of function $f$ at point $x_0$|nopage}{100}
\indexentry {Ganalysis.2b2@[$f''(x_0)$] the second (ordinary) derivative of function $f$ at point $x_0$|nopage}{100}
\indexentry {Ganalysis.2b3@[$f'''(x_0)$] the third (ordinary) derivative of function $f$ at point $x_0$|nopage}{100}
\indexentry {Ganalysis.2b4@[$f^{(n)}(x_0)$] the $n\th $ (ordinary) derivative of function $f$ at point $x_0$ where $n \in \{4,5,6,\dots \}$|nopage}{100}
\indexentry {Ganalysis.2y@[$\total f/\total t$] total derivative of function $f$ at point $t$|nopage}{100}
\indexentry {Ganalysis.2z@[$\partial f/\partial x$] \index {mathematics"!functions"!partial derivative ($\partial $)"|indexglo}partial derivative of function $f$ with respect to $x$|nopage}{100}
\indexentry {Ganalysis.2y2@[$\total ^2 f/{\total t}^2$] second total derivative of function $f$ (\ie {}, $f''$)|nopage}{100}
\indexentry {Ganalysis.2y3@[$\total ^3 f/{\total t}^3$] third total derivative of function $f$ (\ie {}, $f'''$)|nopage}{100}
\indexentry {Ganalysis.2yn@[$\total ^n f/{\total t}^n$] $n\th $ total derivative of function $f$ (\ie {}, $f^{(n)}$)|nopage}{100}
\indexentry {Ganalysis.2zxy@[$\partial ^2 f/\partial x \partial y$] partial derivative of function $\partial f/\partial x$ with respect to $y$|nopage}{100}
\indexentry {Bnumbers.55@[$e$] \aimention {Leonhard Euler}Euler's number (\ie {}, constant $e \approx 2.71828182845904523536$)|nopage}{100}
\indexentry {Ganalysis.001@[${n\bang }$] factorial of $n$ (\ie {}, ${n\bang }=1\times 2\times \cdots \times n$ with ${0\bang }=1$)|nopage}{100}
\indexentry {Ageneral.1@[$\approx $] is approximately equal to|nopage}{100}
\indexentry {Bnumbers.595@[$\exp (x)$] exponential function (\ie {}, $\exp (x) \triangleq e^x$)|nopage}{100}
\indexentry {Bnumbers.58@[$\ln (x)$] natural logarithm of positive real number $x$ (\ie {}, $e^{\ln (x)} = x$)|nopage}{100}
\indexentry {Bnumbers.57@[$\log (x)$] common logarithm of positive real number $x$ (\ie {}, $10^{\log (x)} = x$)|nopage}{100}
\indexentry {Bnumbers.56@[$\log _b(x)$] logarithm of positive real number $x$ in base $b$ (\ie {}, $b^{\log _b(x)} = x$)|nopage}{100}
\indexentry {Hvectors.2@[$y_i$] the $i\th $ coordinate of vector $\v {y}$|nopage}{100}
\indexentry {Hvectors.42@[$\v {e}_i$] the $i\th $ elementary (or standard) basis vector|nopage}{100}
\indexentry {Hvectors.3@[$\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}
\indexentry {Bnumbers.545@[$\R ^n$] the \index {mathematics"!numbers"!Euclidean $n$-space ($\R ^n$)"|indexglo}\aimention {Euclid}Euclidean $n$-space|nopage}{100}
\indexentry {Bnumbers.5451@[$\R ^{n \times m}$] space of $n$-by-$m$ real matrices|nopage}{100}
\indexentry {Hvectors.31@[$\mat {A}^\T $] the \index {mathematics"!vector spaces"!matrix transpose (${}^\T $)"|indexglo}transpose of matrix $\mat {A}$|nopage}{100}
\indexentry {Hvectors.45@[$\I $] the identity matrix|nopage}{100}
\indexentry {Bnumbers.5452@[$\R ^{n \times n}$] the unitary associative real algebra|nopage}{100}
\indexentry {Hvectors.5@[$\nabla _{\v {x}} f(\v {x})$] the \index {mathematics"!functions"!gradient ($\nabla $)"|indexglo}gradient vector of function $f$ at $\v {x}$|nopage}{100}
\indexentry {Hvectors.51@[$\nabla ^2_{\v {x}\v {x}} f(\v {x})$] the \index {mathematics"!functions"!Hessian ($\nabla ^2$)"|indexglo}\aimention {Ludwig Otto Hesse}Hessian matrix of function $f$ at point $\v {x}$|nopage}{100}
\indexentry {Iprob.3@[$\Borel (\set {U})$] 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}
\indexentry {Iprob.4@[$\int _a^b f(x) \total x$] 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}
\indexentry {Iprob.5@[$\delta _a(\set {E})$] 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}
\indexentry {Iprob.50@[$\delta (x-p)$] Simplified Dirac delta measure notation (\ie {}, $\delta (x-p) \triangleq \delta _p(\{x\})$)|nopage}{100}
\indexentry {Iprob.41@[$f * g$] \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 $)|nopage}{100}
\indexentry {Iprob.541@[$(\set {U},\Sigma ,\Pr )$] \index {stochasticity"!probability space"|indexglo}Probability space with outcomes $\set {U}$, $\sigma $-field of events $\Sigma $, and probability measure $\Pr $|nopage}{100}
\indexentry {Iprob.540@[$\Pr $] \index {stochasticity"!probability measure"|indexglo}Probability measure|nopage}{100}
\indexentry {Iprob.545@[$\{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}
\indexentry {Iprob.546@[$\Pr (X \leq a)$] Probability induced by preimage of random variable $X$ (\ie {}, \linebreak [3] $\Pr (\{ \zeta \in \set {U} : X(\zeta ) \leq a \})$)|nopage}{100}
\indexentry {Iprob.55@[$F_X(x)$] \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}
\indexentry {Iprob.56@[$f_X(x)$] \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}
\indexentry {Iprob.61@[$\E (g(X))$] \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}
\indexentry {Iprob.60@[$\E (X)$] \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}
\indexentry {Iprob.62@[$\var (X)$] \index {stochasticity"!statistics"!variance ($\var $)|indexglo}Variance of random variable $X$ (\ie {}, $\var (X) = \E (X^2) - \E (X)^2$)|nopage}{100}
\indexentry {Iprob.63@[$\cov (X,Y)$] Covariance of random variables $X$ and $Y$ (\ie {}, $\cov (X,Y) = \E (XY) - \E (X)\E (Y)$)|nopage}{100}
\indexentry {Iprob.65@[$F_{XY}(x,y)$] Joint distribution function for random variables $X$ and $Y$ (\ie {}, $F_{XY}(a,b) \triangleq \Pr (X \leq a, Y \leq b)$)|nopage}{100}
\indexentry {Iprob.66@[$f_{XY}(x,y)$] Joint density function for random variables $X$ and $Y$|nopage}{100}
\indexentry {Iprob.670@[$f_{Y \pipe X}(y \pipe x)$] Conditional density function for random variable $Y$ given $X=x$|nopage}{100}
\indexentry {Iprob.671@[$F_{Y \pipe X}(y \pipe x)$] Conditional distribution function for random variable $Y$ given $X=x$|nopage}{100}
\indexentry {Iprob.68@[$\E (Y \pipe X)$] \index {stochasticity"!statistics"!conditional expectation"|indexglo}Conditional expectation of $Y$ given $X$|nopage}{100}
\indexentry {Iprob.70@[$( \v {N}(t) : t \in \R _{\geq 0})$] \index {stochasticity"!random process"|indexglo}Random process (\ie {}, $\v {N}(t)$ is a random vector for all $t \in \R _{>0}$)|nopage}{100}
\indexentry {Iprob.7301@[$Y(t) \xto {a.s.} Y$] \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}
\indexentry {Iprob.7303@[$\aslim \limits _{t \to \infty } Y(t) = Y$] Random process $Y(t)$ converges almost surely (\ie {}, with probability 1) to $Y$|nopage}{100}