From db02704dbac935585861e09f02c255ae8eecf7df Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?V=C3=ADctor?= Date: Fri, 27 Oct 2023 17:01:20 +0200 Subject: [PATCH] created stochastic control --- .../Stochastic_calculus.tex | 4 +- .../Stochastic_control/Stochastic_control.tex | 142 + main_math.idx | 3301 +++++++++++++++++ main_math.ilg | 6 + main_math.ind | 2816 ++++++++++++++ main_math.tex | 2 + preamble_formulas.sty | 1 + 7 files changed, 6270 insertions(+), 2 deletions(-) create mode 100644 Mathematics/5th/Stochastic_control/Stochastic_control.tex create mode 100644 main_math.idx create mode 100644 main_math.ilg create mode 100644 main_math.ind diff --git a/Mathematics/5th/Stochastic_calculus/Stochastic_calculus.tex b/Mathematics/5th/Stochastic_calculus/Stochastic_calculus.tex index 154d02d..1b5848f 100644 --- a/Mathematics/5th/Stochastic_calculus/Stochastic_calculus.tex +++ b/Mathematics/5th/Stochastic_calculus/Stochastic_calculus.tex @@ -137,7 +137,7 @@ \begin{proof} By \mnameref{SC:doob_maximal} applied to $M^n-M^m$ we have that for fixed $t\geq 0$ and $k\in\NN$: $$ - \Prob\left(\sup_{0\leq s\leq t}\abs{M_s^n-M_s^m}\geq \frac{1}{kmn }\right)\leq k^2\Exp({(M_t^n-M_t^m)}^2)\leq \frac{1}{k^2} + \Prob\left(\sup_{0\leq s\leq t}\abs{M_s^n-M_s^m}\geq \frac{1}{k^2}\right)\leq k^2\Exp({(M_t^n-M_t^m)}^2)\leq \frac{1}{k^2} $$ where in the last inequality we have used that ${(M^n)}$ converges in $L^2$ and so we have chosen $n,m$ large enough so that the inequality holds. Thus, there is an increasing sequence $(n_k)$ such that: $$ @@ -808,7 +808,7 @@ $$ \end{definition} \subsubsection{Existence and uniqueness of solutions} - \begin{lemma}[Gronwall's lemma] + \begin{lemma}[Gronwall's lemma]\label{SC:gronwall} Let ${(x_t)}_{t\in[0,T]}$ be a non-negative function in $L^1([0,T])$ satisfying that $\forall t\in[0,T]$: $$ x_t\leq \alpha+\beta\int_0^t x_s\dd{s} diff --git a/Mathematics/5th/Stochastic_control/Stochastic_control.tex b/Mathematics/5th/Stochastic_control/Stochastic_control.tex new file mode 100644 index 0000000..306d978 --- /dev/null +++ b/Mathematics/5th/Stochastic_control/Stochastic_control.tex @@ -0,0 +1,142 @@ +\documentclass[../../../main_math.tex]{subfiles} + +\begin{document} +\changecolor{SCO} +\begin{multicols}{2}[\section{Stochastic control}] + \subsection{SDEs and Feynman-Kac formula} + In this section we will work in a filtered space $(\Omega, \mathcal{F}, \Prob, {(\mathcal{F}_t)}_{t\geq 0})$. We will denote by $\vf{X}={(\vf{X}_t)}_{t\geq 0}$ a $d$-dimensional Itô process + $$ + \vf{X}_t=\vf{X}_0+\int_0^t \vf{b}(s,\vf{X}_s)ds+\int_0^t \vf{\sigma}(s,\vf{X}_s)d\vf{B}_s + $$ + with $\vf{b}:[0,T]\times\RR^d\to \RR^d$, $\vf{\sigma}:[0,T]\times\RR^d\to \RR^{d\times m}$ be progressively measurable with respect to ${(\mathcal{F}_t)}_{t\geq 0}$ and $\vf{b}\in\MM^1_{\text{loc}}(\RR^d)$, $\vf{\sigma}\in\MM^2_{\text{loc}}(\RR^d)$. + \subsubsection{Itô's formula} + \begin{theorem}[Itô's formula]\label{SCO:ito_formula} + For all $\varphi:\RR_{\geq 0}\times\RR^d\to\RR$ of class $\mathcal{C}^{1,2}$, we have: + \begin{multline*} + \varphi(t,\vf{X}_t)=\varphi(0,\vf{X}_0)+\\+\int_0^t \left(\pdv{\varphi}{s}+\vf{b}\cdot\grad\varphi+\frac{1}{2}\tr\left[\vf{\sigma}\transpose{\vf\sigma}\vf{D}\varphi\right]\right)\dd{s}+\\+\int_0^t \grad\varphi\cdot\vf{\sigma}\dd{\vf{B}_s} + \end{multline*} + \end{theorem} + \subsubsection{SDEs} + Suppose now that $\vf{b}$ and $\vf\sigma$ are Lipschitz-continuous in the second variable and consider the following SDE: + \begin{equation}\label{SCO:SDE} + \begin{cases} + d\vf{X}_t=\vf{b}(t,\vf{X}_t)\dd{t}+\vf{\sigma}(t,\vf{X}_t)\dd{\vf{B}_t} \\ + \vf{X}_{t_0}=\vf{x}_0 + \end{cases} + \end{equation} + with $t_0\geq 0$. + \begin{proposition} + Let ${(\vf{X}_t^{t_0,\vf{x}_0})}_{t\geq t_0}$ be the solution of \eqref{SCO:SDE}. Then, for all $T>0$ $\exists C_T>0$ such that: + \begin{enumerate} + \item + $$ + \Exp\left(\sup_{s\in [t_0,T]}\abs{\vf{X}_s^{t_0,\vf{x}_0}-\vf{X}_s^{t_0,\vf{y}_0}}^2\right)\leq C_T\abs{\vf{x}_0-\vf{y}_0}^2 + $$ + \item $$ + \Exp\left(\sup_{s\in [t_0,T]}\abs{\vf{X}_s^{t_0,\vf{x}_0}}^2\right)\leq C_T(1+\abs{\vf{x}_0}^2) + $$ + \item For all $t_0\leq t_10$ and $g:\RR^d\to\RR$ be continuous and bounded. We define $u(t_0,\vf{x}_0)=\Exp(g(\vf{X}_T^{t_0,\vf{x}_0}))$. + \end{definition} + \begin{proposition} + Let ${(\mathcal{F}_t)}_{t\geq 0}$ be the filtration generated by the Brownian motion. If $t_0\leq t_1\leq T$, then: + $$ + u(t_0,\vf{x}_0)=\Exp\left(g\left(\vf{X}_T^{t_1,\vf{X}_{t_1}^{t_0,\vf{x}_0}}\right)\mid \mathcal{F}_{t_1}\right) + $$ + \end{proposition} + \begin{theorem}[Dynamic programming principle]\label{SCO:dynamic_programming} + If $t_0\leq t_1\leq T$, we have that $u:[0,T]\times\RR^d\to\RR$ satisfies: + $$ + u(t_0,\vf{x}_0)=\Exp\left(u(t_1,\vf{X}_{t_1}^{t_0,\vf{x}_0})\right) + $$ + \end{theorem} + \begin{corollary} + Let $\tau\geq t_0$ be a stopping time. Then, $$u(t_0,\vf{x}_0)=\Exp\left(g(\vf{X}_\tau^{t_0,\vf{x}_0})\right)$$ + \end{corollary} + \begin{theorem}\hfill + \begin{enumerate} + \item If $u\in\mathcal{C}^{1,2}$ and $\partial_t u,\grad u, \vf{D}^2u,\vf\sigma,\vf{b}$ are bounded, then $u$ solves the \emph{Fokker-Planck equation}: + $$ + \begin{cases} + \partial_tu+\frac{1}{2}\tr(\vf{\sigma}\transpose{\vf\sigma}\vf{D}^2u)-\vf{b}\cdot\grad u=0 & \!\text{in }(0,T)\!\times\!\RR^d \\ + u(T,\vf{x})=g(\vf{x}) & \!\text{in }\RR^d + \end{cases} + $$ + \item If $v\in \mathcal{C}^{1,2}$ solves the Fokker-Planck equation and it is bounded, then $u=v$. + \end{enumerate} + \end{theorem} + \begin{proof}\hfill + \begin{enumerate} + \item Fix $(t,\vf{x})\in (0,T)\times\RR^d$. By the \mnameref{SCO:dynamic_programming} we have that $\forall h\in (0,T)$ we have: + \begin{multline*} + u(t,\vf{x})=\Exp\left(u(t+h,\vf{X}_{t+h}^{t,\vf{x}})\right)=\\=\Exp\Bigg(u(t,\vf{x})+\\\left.\int_t^{t+h}\left[\partial_t u + \vf{b}\cdot\grad u + \frac{1}{2}\tr(\vf{\sigma}\transpose{\vf\sigma}\vf{D}^2u)\right]\!(r,\vf{X}_r^{t,\vf{x}})\dd{r}\!\right)\!+\\+\Exp\left(\int_t^{t+h}\grad u(r,\vf{X}_r^{t,\vf{x}})\cdot\vf{\sigma}(r,\vf{X}_r^{t,\vf{x}})\dd{\vf{B}_r}\right) + \end{multline*} + where we have used \mnameref{SCO:ito_formula}. Note that the last term is zero because it is a martingale. Now dividing by $h$ we have: + $$ + \Exp\left(\frac{1}{h}\int_t^{t+h}\partial_t u + \vf{b}\cdot\grad u + \frac{1}{2}\tr(\vf{\sigma}\transpose{\vf\sigma}\vf{D}^2u) \dd{r}\right)=0 + $$ + And now use \mnameref{P:dominated}. + \item Assume $v$ solves the Fokker-Planck equation. We have: + \begin{equation*} + g(\vf{X}_T^{t,\vf{x}})=v(T,\vf{X}_T^{t,\vf{x}})=v(t,\vf{X}_t^{t,\vf{x}})+\int_t^T(\transpose{\vf\sigma} \grad v) \dd{\vf{B}_r} + \end{equation*} + the second term is a local a bounded (by hypothesis) local martingale. Hence, it is a martingale, and taking expectations we get the result. + \end{enumerate} + \end{proof} + \begin{remark} + Note that the Fokker-Planck equation becomes the backward in time heat equation if $\vf{b}=0$ and $\vf{\sigma}=\sqrt{2}\vf{I}$. + \end{remark} + \subsection{Dynamic programming} + \subsubsection{Optimal control problem} + \begin{definition} + Consider the following SDE: + \begin{equation}\label{SCO:SDE_control} + \begin{cases} + \dd{\vf{X}_t}=\vf{b}_t(\vf{X}_t,\alpha_t)\dd{t}+\vf{\sigma}_t(\vf{X}_t,\alpha_t)\dd{\vf{B}_t} \\ + \vf{X}_{t_0}=\vf{x}_0 + \end{cases} + \end{equation} + where $\vf{b}:\RR_{\geq 0}\times\RR^d\times A\to\RR^d$, $\vf{\sigma}:\RR_{\geq 0}\times\RR^d\times A\to\RR^{d\times m}$ are continuous, $A$ is a compact metric space and $\alpha_t\in \text{ct}_t:=\{\rho:[0,t]\times\Omega\to A:\rho\text{ is progressively measurable}\}$ is a \emph{control parameter}. + \end{definition} + From here on we will assume that both $\vf{b}$ and $\vf{\sigma}$ are uniformly Lipschitz-continuous in the second variable. + \begin{theorem} + For all $\alpha\in \text{ct}_{t_0}$ and all $(t_0,\vf{x}_0)\in [0,T]\times\RR^d$ there exists a unique solution ${(\vf{X}_t^{t_0,\vf{x}_0,\alpha})}_{t\geq t_0}$ of \mcref{SCO:SDE_control}. + \end{theorem} + \begin{lemma} + Let ${(\vf{X}_t^{t_0,\vf{x}_0,\alpha})}_{t\geq t_0}$ be the solution of \mcref{SCO:SDE_control}. Then, for all $t_2>t_1\geq t_0$ and all $\alpha\in \text{ct}_{t_0}$ we have: + $$ + \vf{X}_{t_2}^{t_0,\vf{x}_0,\alpha}=\vf{X}_{t_2}^{t_1,\vf{X}_{t_1}^{t_0,\vf{x}_0,\alpha},\alpha} + $$ + \end{lemma} + \begin{definition}[Finite horizon problem] + Let $T>0$, $g:\RR^d\to\RR$ be continuous and bounded and $\ell:[0,T]\times\RR^d\times A\to\RR$ be continuous and bounded. We define the following problem: + $$ + \inf_{\alpha\in \text{ct}_{t_0}}\Exp\left(\int_{t_0}^T \ell(r,\vf{X}_r^{t_0,\vf{x}_0,\alpha},\alpha_r)\dd{r}+g(\vf{X}_T^{t_0,\vf{x}_0,\alpha})\right) + $$ + The first term in the expectation is called \emph{running cost} and the second one \emph{terminal cost}. + \end{definition} + \begin{definition}[Infinte horizon problem] + Let $g:\RR^d\to\RR$ be continuous and bounded and $\ell:[0,\infty)\times\RR^d\times A\to\RR$ be continuous and bounded. We define the following problem: + $$ + \inf_{\alpha\in \text{ct}_{t_0}}\Exp\left(\int_{t_0}^\infty \exp{-\tau r}\ell(r,\vf{X}_r^{t_0,\vf{x}_0,\alpha},\alpha_r)\dd{r}\right) + $$ + \end{definition} +\end{multicols} +\end{document} \ No newline at end of file diff --git a/main_math.idx b/main_math.idx new file mode 100644 index 0000000..1dcb1b5 --- /dev/null +++ b/main_math.idx @@ -0,0 +1,3301 @@ +\indexentry{Peano axioms|hyperpage}{6} +\indexentry{Induction axiom|hyperpage}{6} +\indexentry{Induction axiom|hyperpage}{6} +\indexentry{set|hyperpage}{6} +\indexentry{cardinal|hyperpage}{6} +\indexentry{subset|hyperpage}{6} +\indexentry{Axiom of extensionality|hyperpage}{6} +\indexentry{equal|hyperpage}{6} +\indexentry{power set|hyperpage}{6} +\indexentry{empty set|hyperpage}{6} +\indexentry{intersection|hyperpage}{6} +\indexentry{union|hyperpage}{6} +\indexentry{complement|hyperpage}{6} +\indexentry{De Morgan's laws|hyperpage}{6} +\indexentry{set difference|hyperpage}{6} 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morphism|hyperpage}{38} +\indexentry{group morphism|hyperpage}{38} +\indexentry{normal|hyperpage}{38} +\indexentry{kernel|hyperpage}{38} +\indexentry{image|hyperpage}{38} +\indexentry{isomorphism|hyperpage}{38} +\indexentry{isomorphic|hyperpage}{38} +\indexentry{Classification of cyclic groups|hyperpage}{38} +\indexentry{symmetric group|hyperpage}{38} +\indexentry{Cayley's theorem|hyperpage}{39} +\indexentry{right cosets|hyperpage}{39} +\indexentry{left cosets|hyperpage}{39} +\indexentry{set of right cosets|hyperpage}{39} +\indexentry{set of left cosets|hyperpage}{39} +\indexentry{Lagrange's theorem|hyperpage}{39} +\indexentry{index|hyperpage}{39} +\indexentry{Quotient group|hyperpage}{39} +\indexentry{quotient group|hyperpage}{39} +\indexentry{First isomorphism theorem|hyperpage}{39} +\indexentry{products of group subsets|hyperpage}{40} +\indexentry{Second isomorphism theorem|hyperpage}{40} +\indexentry{Correspondence theorem|hyperpage}{40} +\indexentry{Third isomorphism theorem|hyperpage}{40} +\indexentry{(left) group action|hyperpage}{40} +\indexentry{(left) $G$-set|hyperpage}{40} +\indexentry{orbit|hyperpage}{40} +\indexentry{stabilizer|hyperpage}{40} +\indexentry{Orbit-stabilizer theorem|hyperpage}{40} +\indexentry{Orbits formula|hyperpage}{40} +\indexentry{Cauchy's theorem|hyperpage}{41} +\indexentry{conjugation action|hyperpage}{41} +\indexentry{Center of a group|hyperpage}{41} +\indexentry{center|hyperpage}{41} +\indexentry{normalizer|hyperpage}{41} +\indexentry{First Sylow theorem|hyperpage}{41} +\indexentry{$p$-group|hyperpage}{41} +\indexentry{Sylow $p$-subgroup|hyperpage}{41} +\indexentry{solvable|hyperpage}{41} +\indexentry{Second Sylow theorem|hyperpage}{41} +\indexentry{Third Sylow theorem|hyperpage}{41} +\indexentry{alternating group|hyperpage}{42} +\indexentry{dihedral group|hyperpage}{42} +\indexentry{quaternion group|hyperpage}{42} +\indexentry{dicyclic group|hyperpage}{42} +\indexentry{Ring|hyperpage}{42} 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minimize|hyperpage}{51} +\indexentry{Strong duality theorem|hyperpage}{52} +\indexentry{Complementary property|hyperpage}{52} +\indexentry{distance|hyperpage}{53} +\indexentry{triangular inequality|hyperpage}{53} +\indexentry{metric space|hyperpage}{53} +\indexentry{norm|hyperpage}{53} +\indexentry{triangular inequality|hyperpage}{53} +\indexentry{normed vector space|hyperpage}{53} +\indexentry{dot product|hyperpage}{53} +\indexentry{Euclidean space|hyperpage}{53} +\indexentry{Cauchy-Schwarz inequality|hyperpage}{53} +\indexentry{Parallelogram law|hyperpage}{53} +\indexentry{norm|hyperpage}{53} +\indexentry{sphere|hyperpage}{54} +\indexentry{open ball|hyperpage}{54} +\indexentry{closed ball|hyperpage}{54} +\indexentry{bounded set|hyperpage}{54} +\indexentry{neighbourhood|hyperpage}{54} +\indexentry{interior point|hyperpage}{54} +\indexentry{interior|hyperpage}{54} +\indexentry{exterior point|hyperpage}{54} +\indexentry{exterior|hyperpage}{54} +\indexentry{adherent point|hyperpage}{54} 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continuous|hyperpage}{55} +\indexentry{contraction|hyperpage}{55} +\indexentry{modulus of continuity|hyperpage}{55} +\indexentry{differentiable|hyperpage}{55} +\indexentry{differential|hyperpage}{55} +\indexentry{directional derivative|hyperpage}{55} +\indexentry{partial derivative|hyperpage}{55} +\indexentry{Jacobian matrix|hyperpage}{55} +\indexentry{Jacobian determinant|hyperpage}{55} +\indexentry{gradient|hyperpage}{56} +\indexentry{Chain rule|hyperpage}{56} +\indexentry{class|hyperpage}{56} +\indexentry{class|hyperpage}{56} +\indexentry{Differentiability criterion|hyperpage}{56} +\indexentry{Mean value theorem|hyperpage}{56} +\indexentry{Mean value theorem for vector-valued functions|hyperpage}{56} +\indexentry{partial derivative of order $k$|hyperpage}{56} +\indexentry{Hessian matrix|hyperpage}{56} +\indexentry{Schwarz's theorem|hyperpage}{57} +\indexentry{Inverse function theorem|hyperpage}{57} +\indexentry{diffeomorphism|hyperpage}{57} +\indexentry{Implicit function theorem|hyperpage}{57} +\indexentry{Taylor's theorem|hyperpage}{57} +\indexentry{multi-index notation|hyperpage}{57} +\indexentry{local maximum|hyperpage}{57} +\indexentry{local minimum|hyperpage}{57} +\indexentry{local extremum|hyperpage}{57} +\indexentry{global maximum|hyperpage}{57} +\indexentry{global minimum|hyperpage}{57} +\indexentry{critical point|hyperpage}{57} +\indexentry{saddle point|hyperpage}{57} +\indexentry{Sylvester's criterion|hyperpage}{58} +\indexentry{Lagrange multipliers theorem|hyperpage}{58} +\indexentry{rectangle|hyperpage}{58} +\indexentry{$n$-dimensional volume|hyperpage}{58} +\indexentry{length|hyperpage}{58} +\indexentry{surface|hyperpage}{58} +\indexentry{partition|hyperpage}{58} +\indexentry{regular|hyperpage}{58} +\indexentry{finer than|hyperpage}{58} +\indexentry{lower sum|hyperpage}{58} +\indexentry{upper sum|hyperpage}{58} +\indexentry{lower integral|hyperpage}{58} +\indexentry{upper integral|hyperpage}{58} +\indexentry{Riemann-integrable|hyperpage}{58} +\indexentry{Riemann sum|hyperpage}{58} +\indexentry{Fubini's theorem|hyperpage}{59} +\indexentry{elementary region|hyperpage}{59} +\indexentry{$x$-simple|hyperpage}{59} +\indexentry{$xy$-simple|hyperpage}{59} +\indexentry{Fubini's theorem for elementary regions|hyperpage}{59} +\indexentry{$n$-dimensional volume|hyperpage}{59} +\indexentry{Cavalieri's principle|hyperpage}{59} +\indexentry{Center of mass|hyperpage}{59} +\indexentry{center of mass|hyperpage}{59} +\indexentry{Moment of inertia|hyperpage}{59} +\indexentry{moment of inertia|hyperpage}{59} +\indexentry{integrable function|hyperpage}{59} +\indexentry{Riemann-integrable function|hyperpage}{59} +\indexentry{$y$-simple|hyperpage}{59} +\indexentry{$yz$-simple|hyperpage}{59} +\indexentry{$xz$-simple|hyperpage}{59} +\indexentry{Change of variable theorem|hyperpage}{60} +\indexentry{Integral in polar coordinates|hyperpage}{60} +\indexentry{Integral in cylindrical coordinates|hyperpage}{60} +\indexentry{Integral in spherical coordinates|hyperpage}{60} +\indexentry{length of the polygonal|hyperpage}{60} +\indexentry{arc length|hyperpage}{60} +\indexentry{rectifiable|hyperpage}{60} +\indexentry{reparametrization|hyperpage}{60} +\indexentry{arc length parameter|hyperpage}{60} +\indexentry{vector field|hyperpage}{60} +\indexentry{curvature|hyperpage}{61} +\indexentry{line integral|hyperpage}{61} +\indexentry{line integral|hyperpage}{61} +\indexentry{circulation|hyperpage}{61} +\indexentry{Jordan arc|hyperpage}{61} +\indexentry{Jordan closed curve|hyperpage}{61} +\indexentry{conservative|hyperpage}{61} +\indexentry{gradient vector field|hyperpage}{61} +\indexentry{potential|hyperpage}{61} +\indexentry{divergence|hyperpage}{61} +\indexentry{curl|hyperpage}{61} +\indexentry{Laplacian|hyperpage}{61} +\indexentry{parametrized surface|hyperpage}{61} +\indexentry{surface integral|hyperpage}{61} +\indexentry{surface integral|hyperpage}{62} +\indexentry{flux|hyperpage}{62} +\indexentry{differential 1-form|hyperpage}{62} +\indexentry{Green's theorem|hyperpage}{62} +\indexentry{Divergence theorem on $\ensuremath {\mathbb {R}}^2$|hyperpage}{62} +\indexentry{Stokes' theorem|hyperpage}{62} +\indexentry{symmetric|hyperpage}{62} +\indexentry{Divergence theorem on $\ensuremath {\mathbb {R}}^3$|hyperpage}{62} +\indexentry{2-forms|hyperpage}{62} +\indexentry{3-forms|hyperpage}{62} +\indexentry{Euclid's axioms|hyperpage}{63} +\indexentry{Playfair's axiom|hyperpage}{63} +\indexentry{points|hyperpage}{63} +\indexentry{lines|hyperpage}{63} +\indexentry{incidence relation|hyperpage}{63} +\indexentry{order relation|hyperpage}{63} +\indexentry{congruence relation|hyperpage}{63} +\indexentry{Incidence axioms|hyperpage}{63} +\indexentry{Order axioms|hyperpage}{63} +\indexentry{Pasch's axiom|hyperpage}{63} +\indexentry{ray|hyperpage}{63} +\indexentry{half-line|hyperpage}{63} +\indexentry{half-plane|hyperpage}{63} +\indexentry{angle|hyperpage}{63} +\indexentry{Congruence axioms|hyperpage}{63} +\indexentry{SAS criterion|hyperpage}{63} +\indexentry{Continuity axioms|hyperpage}{63} +\indexentry{Axiom of Archimedes|hyperpage}{63} +\indexentry{Axiom of completeness|hyperpage}{63} +\indexentry{RC|hyperpage}{64} +\indexentry{CC|hyperpage}{64} +\indexentry{Axiom of Parallels|hyperpage}{64} +\indexentry{Hilbert plane|hyperpage}{64} +\indexentry{Pythagorean plane|hyperpage}{64} +\indexentry{Euclidean plane|hyperpage}{64} +\indexentry{Cartesian geometry|hyperpage}{64} +\indexentry{Absolute geometry|hyperpage}{64} +\indexentry{SAS criterion|hyperpage}{64} +\indexentry{SSS criterion|hyperpage}{64} +\indexentry{Exterior angle theorem|hyperpage}{64} +\indexentry{ASA criterion|hyperpage}{64} +\indexentry{SAA criterion|hyperpage}{64} +\indexentry{Saccheri-Legendre theorem|hyperpage}{64} +\indexentry{ordered field|hyperpage}{64} +\indexentry{Pythagorean|hyperpage}{64} +\indexentry{Archimedean|hyperpage}{64} +\indexentry{Euclidean|hyperpage}{64} +\indexentry{Hilbert field|hyperpage}{64} +\indexentry{constructible field|hyperpage}{64} +\indexentry{Hyperbolic geometry|hyperpage}{65} +\indexentry{Hyperbolic geometry|hyperpage}{65} +\indexentry{Non-Paschian geometry|hyperpage}{65} +\indexentry{Non-Paschian geometry|hyperpage}{65} +\indexentry{Construction of a non-Paschian geometry|hyperpage}{65} +\indexentry{Non-SAS geometry|hyperpage}{66} +\indexentry{Non-SAS geometry|hyperpage}{66} +\indexentry{Construction of a non-SAS geometry|hyperpage}{66} +\indexentry{Non-Archimedean geometry|hyperpage}{66} +\indexentry{Non-Archimedean geometry|hyperpage}{66} +\indexentry{axiomatic projective space|hyperpage}{66} +\indexentry{Projective axiom|hyperpage}{66} +\indexentry{projective subvariety|hyperpage}{66} +\indexentry{collineation|hyperpage}{66} +\indexentry{dimension|hyperpage}{66} +\indexentry{projective plane|hyperpage}{66} +\indexentry{Duality principle|hyperpage}{66} +\indexentry{affine plane|hyperpage}{66} +\indexentry{Passage from the projective plane to the affine plane|hyperpage}{66} +\indexentry{Passage from the affine plane to the projective plane|hyperpage}{66} +\indexentry{$n$-dimensional projective space|hyperpage}{67} +\indexentry{homography|hyperpage}{67} +\indexentry{semilinear isomorphism|hyperpage}{67} +\indexentry{isomorphism between projective spaces|hyperpage}{67} +\indexentry{$m$-dimensional projective subvariety|hyperpage}{67} +\indexentry{line|hyperpage}{67} +\indexentry{hyperplane|hyperpage}{67} +\indexentry{homogeneous coordinates|hyperpage}{67} +\indexentry{projective frame|hyperpage}{67} +\indexentry{equation of the hyperplane|hyperpage}{67} +\indexentry{Gra\ss mann formula|hyperpage}{67} +\indexentry{configuration|hyperpage}{67} +\indexentry{realizable|hyperpage}{67} +\indexentry{theorem|hyperpage}{67} +\indexentry{Fano configuration|hyperpage}{67} +\indexentry{Pappus configuration|hyperpage}{68} +\indexentry{in perspective with respect to a point|hyperpage}{68} +\indexentry{centre of perspectivity|hyperpage}{68} +\indexentry{in perspective with respect to a line|hyperpage}{68} +\indexentry{axis of perspectivity|hyperpage}{68} +\indexentry{Desargues' theorem|hyperpage}{68} +\indexentry{Desargues configuration|hyperpage}{68} +\indexentry{non-Desarguesian planes|hyperpage}{68} +\indexentry{Coordination theorem|hyperpage}{68} +\indexentry{Fundamental theorem of projective geometry|hyperpage}{68} +\indexentry{Cross ratio|hyperpage}{69} +\indexentry{cross ratio|hyperpage}{69} +\indexentry{harmonic ratio|hyperpage}{69} +\indexentry{cross ratio|hyperpage}{69} +\indexentry{perspectivity|hyperpage}{69} +\indexentry{projectivity|hyperpage}{69} +\indexentry{Plücker coordinates|hyperpage}{69} +\indexentry{affine space|hyperpage}{69} +\indexentry{dimension|hyperpage}{69} +\indexentry{affine combination|hyperpage}{69} +\indexentry{affinely independents|hyperpage}{69} +\indexentry{barycenter|hyperpage}{69} +\indexentry{affine subvariety|hyperpage}{70} +\indexentry{director subspace|hyperpage}{70} +\indexentry{line|hyperpage}{70} +\indexentry{hyperplane|hyperpage}{70} +\indexentry{parallel|hyperpage}{70} +\indexentry{sum|hyperpage}{70} +\indexentry{Affine Gra\ss mann formulas|hyperpage}{70} +\indexentry{origin|hyperpage}{70} +\indexentry{affine coordinates|hyperpage}{70} +\indexentry{homogenization|hyperpage}{70} +\indexentry{dehomogenization|hyperpage}{70} +\indexentry{parametric equations|hyperpage}{70} +\indexentry{Cartesian equations|hyperpage}{70} +\indexentry{affinity|hyperpage}{70} +\indexentry{differential|hyperpage}{70} +\indexentry{semiaffinity|hyperpage}{70} +\indexentry{similar|hyperpage}{71} +\indexentry{fixed point|hyperpage}{71} +\indexentry{invariant under an affinity|hyperpage}{71} +\indexentry{invariance level|hyperpage}{71} +\indexentry{Translations|hyperpage}{71} +\indexentry{translation|hyperpage}{71} +\indexentry{Properties of translations|hyperpage}{71} +\indexentry{Reflections|hyperpage}{71} +\indexentry{reflection|hyperpage}{71} +\indexentry{mirror of the reflection|hyperpage}{71} +\indexentry{root of the reflection|hyperpage}{71} +\indexentry{Properties of reflections|hyperpage}{71} +\indexentry{Projections|hyperpage}{71} +\indexentry{projection|hyperpage}{71} +\indexentry{Properties of projections|hyperpage}{71} +\indexentry{Homotheties|hyperpage}{72} +\indexentry{homothety|hyperpage}{72} +\indexentry{similitude ratio of the homothety|hyperpage}{72} +\indexentry{Properties of homotheties|hyperpage}{72} +\indexentry{glide reflection|hyperpage}{72} +\indexentry{Simple ratio|hyperpage}{72} +\indexentry{simple ratio|hyperpage}{72} +\indexentry{Fundamental theorem of affine geometry|hyperpage}{72} +\indexentry{Euclidean affine space|hyperpage}{72} +\indexentry{distance between two points|hyperpage}{72} +\indexentry{segment|hyperpage}{72} +\indexentry{Triangular inequality|hyperpage}{72} +\indexentry{Pythagorean theorem|hyperpage}{72} +\indexentry{orthogonal|hyperpage}{72} +\indexentry{distance between two affine subvarieties|hyperpage}{72} +\indexentry{Euclidean motion|hyperpage}{73} +\indexentry{Examples of Euclidean motions|hyperpage}{73} +\indexentry{orthogonal reflections|hyperpage}{73} +\indexentry{Glide orthogonal reflections|hyperpage}{73} +\indexentry{Classification of isometries|hyperpage}{73} +\indexentry{glide vector|hyperpage}{73} +\indexentry{Classification of Euclidean motions|hyperpage}{73} +\indexentry{quadric|hyperpage}{73} +\indexentry{points of the quadric|hyperpage}{73} +\indexentry{conic|hyperpage}{73} +\indexentry{equivalent|hyperpage}{73} +\indexentry{quadric|hyperpage}{73} +\indexentry{points of the quadric|hyperpage}{73} +\indexentry{equivalent|hyperpage}{73} +\indexentry{Homogenization|hyperpage}{74} +\indexentry{Dehomogenization|hyperpage}{74} +\indexentry{anisotropic|hyperpage}{74} +\indexentry{elliptic|hyperpage}{74} +\indexentry{hyperbolic plane|hyperpage}{74} +\indexentry{quadratic form|hyperpage}{74} +\indexentry{non-degenerate|hyperpage}{74} +\indexentry{quadratic space|hyperpage}{74} +\indexentry{isometry|hyperpage}{74} +\indexentry{totally isotropic|hyperpage}{74} +\indexentry{rank|hyperpage}{74} +\indexentry{Witt's theorem|hyperpage}{74} +\indexentry{index|hyperpage}{74} +\indexentry{Classification of quadratic forms in $\ensuremath {\mathbb {C}}$|hyperpage}{74} +\indexentry{Classification of quadratic forms in $\ensuremath {\mathbb {F}}_q$|hyperpage}{74} +\indexentry{Classification of quadratic forms in $\ensuremath {\mathbb {R}}$|hyperpage}{74} +\indexentry{Classification of projective qua\-drics in $\ensuremath {\mathbb {C}}$|hyperpage}{74} +\indexentry{Classification of projective qua\-drics in $\ensuremath {\mathbb {F}}_q$|hyperpage}{75} +\indexentry{Classification of projective qua\-drics in $\ensuremath {\mathbb {R}}$|hyperpage}{75} +\indexentry{Classification of affine quadrics|hyperpage}{75} +\indexentry{numeric series|hyperpage}{76} 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+\indexentry{rearrangement of the sequence|hyperpage}{77} +\indexentry{rearrangement of the series|hyperpage}{77} +\indexentry{positive part|hyperpage}{77} +\indexentry{negative part|hyperpage}{77} +\indexentry{Riemann's theorem|hyperpage}{77} +\indexentry{sequence of real functions|hyperpage}{77} +\indexentry{converges pointwise|hyperpage}{77} +\indexentry{converges uniformly|hyperpage}{77} +\indexentry{Cauchy's test|hyperpage}{78} +\indexentry{series of functions|hyperpage}{78} +\indexentry{converges pointwise|hyperpage}{78} +\indexentry{sum of the series in a pointwise sense|hyperpage}{78} +\indexentry{converges uniformly|hyperpage}{78} +\indexentry{sum of the series in an uniform sense|hyperpage}{78} +\indexentry{Cauchy's test|hyperpage}{78} +\indexentry{Weierstra\ss \ M-test|hyperpage}{78} +\indexentry{Dirichlet's test|hyperpage}{78} +\indexentry{Abel's test|hyperpage}{78} +\indexentry{power series|hyperpage}{78} +\indexentry{radius of convergence of the power series|hyperpage}{79} +\indexentry{Abel's theorem|hyperpage}{79} +\indexentry{analytic|hyperpage}{79} +\indexentry{compact support|hyperpage}{79} +\indexentry{convolution|hyperpage}{79} +\indexentry{kernel function|hyperpage}{79} +\indexentry{approximation of identity|hyperpage}{79} +\indexentry{Weierstra\ss \ approximation theorem|hyperpage}{79} +\indexentry{locally integrable|hyperpage}{79} +\indexentry{improper integral|hyperpage}{79} +\indexentry{convergent|hyperpage}{79} +\indexentry{support|hyperpage}{79} +\indexentry{Cauchy's test|hyperpage}{80} +\indexentry{Comparison test|hyperpage}{80} +\indexentry{Integral test|hyperpage}{80} +\indexentry{converges absolutely|hyperpage}{80} +\indexentry{Dirichlet's test|hyperpage}{80} +\indexentry{Abel's test|hyperpage}{80} +\indexentry{Gamma function|hyperpage}{81} +\indexentry{Stirling's formula|hyperpage}{81} +\indexentry{$T$-periodic|hyperpage}{81} +\indexentry{periodic with period $T$|hyperpage}{81} +\indexentry{inner product|hyperpage}{82} +\indexentry{distance|hyperpage}{82} +\indexentry{Cauchy-Schwarz inequality|hyperpage}{82} +\indexentry{Minkowski inequality|hyperpage}{82} +\indexentry{orthogonal|hyperpage}{82} +\indexentry{orthonormal|hyperpage}{82} +\indexentry{orthonormal system|hyperpage}{82} +\indexentry{linearly dependent|hyperpage}{82} +\indexentry{linearly independent|hyperpage}{82} +\indexentry{$n$-th Fourier coefficient|hyperpage}{83} +\indexentry{Fourier series|hyperpage}{83} +\indexentry{$N$-th partial sum of $Sf$|hyperpage}{83} +\indexentry{even extension|hyperpage}{84} +\indexentry{odd extension|hyperpage}{84} +\indexentry{Dirichlet kernel|hyperpage}{84} +\indexentry{Dirichlet kernel|hyperpage}{84} +\indexentry{Riemann-Lebesgue lemma|hyperpage}{85} +\indexentry{Dini's theorem|hyperpage}{85} +\indexentry{Lipschitz's theorem|hyperpage}{85} +\indexentry{Cesàro summable|hyperpage}{85} +\indexentry{Fejér kernel|hyperpage}{85} +\indexentry{Fejér kernel|hyperpage}{85} +\indexentry{Fejér means|hyperpage}{86} +\indexentry{Fejér's theorem|hyperpage}{86} +\indexentry{Fejér's theorem|hyperpage}{86} +\indexentry{converge in norm $L^p$|hyperpage}{87} +\indexentry{Bessel's inequality|hyperpage}{87} +\indexentry{Parseval's identity|hyperpage}{87} +\indexentry{Wirtinger's inequality|hyperpage}{87} +\indexentry{Wirtinger's inequality|hyperpage}{88} +\indexentry{Isoperimetric inequality|hyperpage}{88} +\indexentry{Floating-point representation|hyperpage}{89} +\indexentry{floating-point representation|hyperpage}{89} +\indexentry{sign|hyperpage}{89} +\indexentry{significant|hyperpage}{89} +\indexentry{mantissa|hyperpage}{89} +\indexentry{exponent|hyperpage}{89} +\indexentry{normalized|hyperpage}{89} +\indexentry{floating-point representation by truncation|hyperpage}{89} +\indexentry{floating-point representation by rounding|hyperpage}{89} +\indexentry{absolute error|hyperpage}{89} +\indexentry{relative error|hyperpage}{89} +\indexentry{has $t$ correct decimal digits|hyperpage}{89} +\indexentry{has $u$ significant digits|hyperpage}{89} +\indexentry{machine epsilon $\epsilon $|hyperpage}{89} +\indexentry{Propagation of absolute errors|hyperpage}{89} +\indexentry{absolute condition numbers|hyperpage}{89} +\indexentry{Propagation of relative errors|hyperpage}{89} +\indexentry{relative condition numbers|hyperpage}{90} +\indexentry{numerically stable|hyperpage}{90} +\indexentry{numerically unstable|hyperpage}{90} +\indexentry{well-conditioned|hyperpage}{90} +\indexentry{ill-conditioned|hyperpage}{90} +\indexentry{zero|hyperpage}{90} +\indexentry{solution|hyperpage}{90} +\indexentry{zero of multiplicity $m\in \ensuremath {\mathbb {N}}$|hyperpage}{90} +\indexentry{simple|hyperpage}{90} +\indexentry{double|hyperpage}{90} +\indexentry{triple|hyperpage}{90} +\indexentry{Bisection method|hyperpage}{90} +\indexentry{Regula falsi method|hyperpage}{90} +\indexentry{Secant method|hyperpage}{90} +\indexentry{Newton-Raphson method|hyperpage}{90} +\indexentry{Newton-Raphson modified method|hyperpage}{90} +\indexentry{Chebyshev method|hyperpage}{90} +\indexentry{$n$-periodic|hyperpage}{90} +\indexentry{contraction|hyperpage}{90} +\indexentry{Fixed-point theorem|hyperpage}{90} +\indexentry{attractor fixed point|hyperpage}{91} +\indexentry{repulsor fixed point|hyperpage}{91} +\indexentry{chaotic behavior|hyperpage}{91} +\indexentry{Order of convergence|hyperpage}{91} +\indexentry{order of convergence|hyperpage}{91} +\indexentry{asymptotic error constant|hyperpage}{91} +\indexentry{linear convergence|hyperpage}{91} +\indexentry{quadratic convergence|hyperpage}{91} +\indexentry{cubic convergence|hyperpage}{91} +\indexentry{order of convergence at least $p$|hyperpage}{91} +\indexentry{computational efficiency|hyperpage}{91} +\indexentry{Aitken's $\Delta ^2$ method|hyperpage}{91} +\indexentry{Aitken's $\Delta ^2$ method|hyperpage}{91} +\indexentry{Steffensen's method|hyperpage}{92} +\indexentry{Strum's sequence|hyperpage}{92} +\indexentry{Sturm's sequence|hyperpage}{92} +\indexentry{Sturm's theorem|hyperpage}{92} +\indexentry{Budan-Fourier theorem|hyperpage}{92} +\indexentry{Descartes' rule of signs|hyperpage}{92} +\indexentry{Gershgorin circle theorem|hyperpage}{92} +\indexentry{vector space of polynomials with real coefficients and degree less than or equal to $n$|hyperpage}{93} +\indexentry{support points|hyperpage}{93} +\indexentry{interpolation problem|hyperpage}{93} +\indexentry{Lagrange's interpolation problem|hyperpage}{93} +\indexentry{Lagrange basis polynomials|hyperpage}{93} +\indexentry{Neville's algorithm|hyperpage}{93} +\indexentry{divided difference|hyperpage}{93} +\indexentry{Newton's divided differences\\method|hyperpage}{93} +\indexentry{equally-spaced|hyperpage}{93} +\indexentry{Hermite interpolation problem|hyperpage}{94} +\indexentry{Spline|hyperpage}{94} +\indexentry{spline|hyperpage}{94} +\indexentry{cubic splines|hyperpage}{94} +\indexentry{Natural cubic spline|hyperpage}{94} +\indexentry{Cubic Hermite spline|hyperpage}{94} +\indexentry{Cubic periodic spline|hyperpage}{94} +\indexentry{seminorm|hyperpage}{94} +\indexentry{Intermediate value theorem|hyperpage}{94} +\indexentry{Forward and backward difference formula of order 1|hyperpage}{94} +\indexentry{seminorm|hyperpage}{94} +\indexentry{norm|hyperpage}{94} +\indexentry{Symmetric difference formula of order 1|hyperpage}{95} +\indexentry{Symmetric difference formula of order 2|hyperpage}{95} +\indexentry{Richardson extrapolation|hyperpage}{95} +\indexentry{quadrature formula|hyperpage}{95} +\indexentry{degree of precision|hyperpage}{95} +\indexentry{Mean value theorem for integrals|hyperpage}{95} +\indexentry{Closed Newton-Cotes Formulas|hyperpage}{95} +\indexentry{Trapezoidal rule|hyperpage}{95} +\indexentry{Simpson's rule|hyperpage}{95} +\indexentry{Composite Trapezoidal rule|hyperpage}{95} +\indexentry{Composite Simpson's rule|hyperpage}{96} +\indexentry{Bernoulli polynomials|hyperpage}{96} +\indexentry{Bernoulli numbers|hyperpage}{96} +\indexentry{Euler-Maclaurin formula|hyperpage}{96} +\indexentry{Romberg method|hyperpage}{96} +\indexentry{Orthogonal polynomials|hyperpage}{96} +\indexentry{orthogonal with respect to the weight $\omega (x)$|hyperpage}{96} +\indexentry{orthogonal basis|hyperpage}{96} +\indexentry{Existence of orthogonal polynomials|hyperpage}{96} +\indexentry{Chebyshev polynomials|hyperpage}{96} +\indexentry{Chebyshev polynomials|hyperpage}{96} +\indexentry{Laguerre polynomials|hyperpage}{96} +\indexentry{Laguerre polynomials|hyperpage}{96} +\indexentry{Legendre polynomials|hyperpage}{97} +\indexentry{Legendre polynomials|hyperpage}{97} +\indexentry{quadrature formula with weight $\omega (x)$|hyperpage}{97} +\indexentry{upper triangular|hyperpage}{97} +\indexentry{lower triangular|hyperpage}{97} +\indexentry{triangular system|hyperpage}{97} +\indexentry{matrix norm|hyperpage}{97} +\indexentry{sub-multiplicativity|hyperpage}{97} +\indexentry{compatible with|hyperpage}{97} +\indexentry{subordinated matrix norm|hyperpage}{97} +\indexentry{spectrum|hyperpage}{97} +\indexentry{spectral radius|hyperpage}{98} +\indexentry{Properties of matrix norms|hyperpage}{98} +\indexentry{convergent|hyperpage}{98} +\indexentry{condition number|hyperpage}{99} +\indexentry{iteration matrix|hyperpage}{99} +\indexentry{rate of convergence|hyperpage}{99} +\indexentry{strictly diagonally dominant by rows|hyperpage}{99} +\indexentry{strictly diagonally dominant by columns|hyperpage}{99} +\indexentry{Jacobi method|hyperpage}{100} +\indexentry{Jacobi method|hyperpage}{100} +\indexentry{Gau\ss -Seidel method|hyperpage}{100} +\indexentry{Gau\ss -Seidel method|hyperpage}{100} +\indexentry{Over-relaxation methods|hyperpage}{100} +\indexentry{relaxation factor|hyperpage}{100} +\indexentry{Over-relaxation methods|hyperpage}{100} +\indexentry{Successive over-relaxation \\method|hyperpage}{100} +\indexentry{Successive over-relaxation method|hyperpage}{100} +\indexentry{SOR|hyperpage}{100} +\indexentry{dominant eigenvalue|hyperpage}{100} +\indexentry{dominant eigenvector|hyperpage}{100} +\indexentry{reducible|hyperpage}{100} +\indexentry{irreducible|hyperpage}{100} +\indexentry{Perron-Frobenius theorem|hyperpage}{101} +\indexentry{Power method|hyperpage}{101} +\indexentry{power method|hyperpage}{101} +\indexentry{Normalized power method|hyperpage}{101} +\indexentry{normalized power method|hyperpage}{101} +\indexentry{Rayleigh quotient|hyperpage}{101} +\indexentry{Rayleigh quotient|hyperpage}{101} +\indexentry{Inverse power method|hyperpage}{101} +\indexentry{inverse power method|hyperpage}{101} +\indexentry{Gaussian elimination|hyperpage}{101} +\indexentry{Partial pivoting|hyperpage}{102} +\indexentry{Complete pivoting|hyperpage}{102} +\indexentry{LU descompostion|hyperpage}{102} +\indexentry{LU decomposition|hyperpage}{102} +\indexentry{permutation matrix|hyperpage}{102} +\indexentry{field of complex numbers|hyperpage}{104} +\indexentry{complex conjugate|hyperpage}{104} +\indexentry{conjugate|hyperpage}{104} +\indexentry{real part|hyperpage}{104} +\indexentry{imaginary part|hyperpage}{104} +\indexentry{modulus|hyperpage}{104} +\indexentry{argument|hyperpage}{104} +\indexentry{multivalued function|hyperpage}{104} +\indexentry{determination of the argument of $f$|hyperpage}{104} +\indexentry{principal argument|hyperpage}{104} +\indexentry{Cartesian form|hyperpage}{104} +\indexentry{extended complex plane|hyperpage}{105} +\indexentry{extended real numbers|hyperpage}{105} +\indexentry{Stereographic projection|hyperpage}{105} +\indexentry{stereographic projection|hyperpage}{105} +\indexentry{sequence of complex numbers|hyperpage}{105} +\indexentry{limit|hyperpage}{105} +\indexentry{converges|hyperpage}{105} +\indexentry{convergent|hyperpage}{105} +\indexentry{divergent|hyperpage}{105} +\indexentry{bounded|hyperpage}{105} +\indexentry{Cauchy|hyperpage}{105} +\indexentry{numeric series of complex numbers|hyperpage}{105} +\indexentry{general term of the series|hyperpage}{105} +\indexentry{$N$-th partial sum of the series|hyperpage}{105} +\indexentry{convergent|hyperpage}{106} +\indexentry{sum of the series|hyperpage}{106} +\indexentry{divergent|hyperpage}{106} +\indexentry{reordering|hyperpage}{106} +\indexentry{Abel's summation formula|hyperpage}{106} +\indexentry{absolutely convergent|hyperpage}{106} +\indexentry{Cauchy product|hyperpage}{106} +\indexentry{product|hyperpage}{106} +\indexentry{complex function|hyperpage}{106} +\indexentry{continuous|hyperpage}{106} +\indexentry{sequence of complex functions|hyperpage}{106} +\indexentry{converges pointwise|hyperpage}{106} +\indexentry{converges uniformly|hyperpage}{106} +\indexentry{Cauchy's test|hyperpage}{107} +\indexentry{series of functions|hyperpage}{107} +\indexentry{converges pointwise|hyperpage}{107} +\indexentry{sum of the series in a pointwise sense|hyperpage}{107} +\indexentry{converges uniformly|hyperpage}{107} +\indexentry{sum of the series in an uniform sense|hyperpage}{107} +\indexentry{Cauchy's test|hyperpage}{107} +\indexentry{Weierstra\ss \ M-test|hyperpage}{107} +\indexentry{Dirichlet's test|hyperpage}{107} +\indexentry{Abel's test|hyperpage}{107} +\indexentry{Dedekind's test|hyperpage}{107} +\indexentry{Du Bois-Reymond's test|hyperpage}{107} +\indexentry{complex power series|hyperpage}{107} +\indexentry{Cauchy-Hadamard theorem|hyperpage}{107} +\indexentry{radius of convergence|hyperpage}{108} +\indexentry{Abel's theorem|hyperpage}{108} +\indexentry{cone|hyperpage}{108} +\indexentry{Abel's theorem|hyperpage}{108} +\indexentry{complex exponential function|hyperpage}{108} +\indexentry{Euler's formula|hyperpage}{108} +\indexentry{Polar form|hyperpage}{108} +\indexentry{polar form|hyperpage}{108} +\indexentry{De Moivre's formula|hyperpage}{108} +\indexentry{complex natural logarithm|hyperpage}{108} +\indexentry{principal value|hyperpage}{108} +\indexentry{complex sine|hyperpage}{108} +\indexentry{complex cosine|hyperpage}{108} +\indexentry{complex tangent|hyperpage}{109} +\indexentry{complex secant|hyperpage}{109} +\indexentry{complex cosecant|hyperpage}{109} +\indexentry{complex cotangent|hyperpage}{109} +\indexentry{complex hyperbolic sine|hyperpage}{109} +\indexentry{complex hyperbolic cosine|hyperpage}{109} +\indexentry{complex hyperbolic tangent|hyperpage}{109} +\indexentry{complex hyperbolic secant|hyperpage}{109} +\indexentry{complex hyperbolic cosecant|hyperpage}{109} +\indexentry{complex hyperbolic cotangent|hyperpage}{109} +\indexentry{$\ensuremath {\mathbb {C}}$-differentiable|hyperpage}{109} +\indexentry{derivative of $f$ at $z_0$|hyperpage}{109} +\indexentry{Chain rule|hyperpage}{109} +\indexentry{holomorphic|hyperpage}{109} +\indexentry{entire function|hyperpage}{109} +\indexentry{determination|hyperpage}{110} +\indexentry{determination of the logarithm|hyperpage}{110} +\indexentry{determination|hyperpage}{110} +\indexentry{associated multivalued function|hyperpage}{110} +\indexentry{$\ensuremath {\mathbb {R}}$-differentiable|hyperpage}{110} +\indexentry{differential|hyperpage}{110} +\indexentry{Cauchy-Riemann theorem|hyperpage}{111} +\indexentry{Cauchy-Riemann equations|hyperpage}{111} +\indexentry{Wirtinger operators|hyperpage}{111} +\indexentry{integral|hyperpage}{111} +\indexentry{line integral|hyperpage}{112} +\indexentry{Goursat's theorem|hyperpage}{112} +\indexentry{Local Cauchy's integral theorem|hyperpage}{112} +\indexentry{Fresnel integrals|hyperpage}{113} +\indexentry{index|hyperpage}{113} +\indexentry{winding number|hyperpage}{113} +\indexentry{Local Cauchy's integral formula|hyperpage}{113} +\indexentry{analytic|hyperpage}{113} +\indexentry{Local Cauchy's integral formula for derivatives|hyperpage}{113} +\indexentry{Cauchy's inequality|hyperpage}{114} +\indexentry{Liouville's theorem|hyperpage}{114} +\indexentry{Fundamental theorem of algebra|hyperpage}{114} +\indexentry{Cardano-Vieta's formulas|hyperpage}{114} +\indexentry{Morera's theorem|hyperpage}{114} +\indexentry{Weierstra\ss ' theorem|hyperpage}{114} +\indexentry{multiplicity|hyperpage}{114} +\indexentry{Analytic continuation theorem|hyperpage}{114} +\indexentry{Maximum modulus principle|hyperpage}{114} +\indexentry{Minimum modulus principle|hyperpage}{114} +\indexentry{harmonic|hyperpage}{115} +\indexentry{domain|hyperpage}{115} +\indexentry{region|hyperpage}{115} +\indexentry{star domain|hyperpage}{115} +\indexentry{convex|hyperpage}{115} +\indexentry{Mean value property|hyperpage}{115} +\indexentry{chain of paths|hyperpage}{115} +\indexentry{image|hyperpage}{115} +\indexentry{cycle|hyperpage}{115} +\indexentry{index|hyperpage}{115} +\indexentry{homologous to zero|hyperpage}{115} +\indexentry{General Cauchy's integral formula|hyperpage}{115} +\indexentry{Cauchy's integral theorem|hyperpage}{116} +\indexentry{General Cauchy's integral formula for derivatives|hyperpage}{116} +\indexentry{Local behaviour of a holomorphic function|hyperpage}{116} +\indexentry{Open mapping theorem|hyperpage}{116} +\indexentry{simply connected|hyperpage}{116} +\indexentry{simply connected|hyperpage}{116} +\indexentry{isolated singularity|hyperpage}{116} +\indexentry{Removable singularity|hyperpage}{116} +\indexentry{removable singularity|hyperpage}{116} +\indexentry{Riemann's theorem on removable singularities|hyperpage}{116} +\indexentry{pole|hyperpage}{116} +\indexentry{Pole|hyperpage}{116} +\indexentry{order|hyperpage}{116} +\indexentry{simple|hyperpage}{116} +\indexentry{residue|hyperpage}{116} +\indexentry{Essential singularity|hyperpage}{117} +\indexentry{essential singularity|hyperpage}{117} +\indexentry{convergent|hyperpage}{117} +\indexentry{absolutely convergent|hyperpage}{117} +\indexentry{uniformly convergent|hyperpage}{117} +\indexentry{Laurent series theorem|hyperpage}{117} +\indexentry{Laurent series|hyperpage}{117} +\indexentry{Partial fraction decomposition theorem|hyperpage}{117} +\indexentry{Residues theorem|hyperpage}{117} +\indexentry{Calculation of integrals|hyperpage}{117} +\indexentry{Casorati-Weierstra\ss \ theorem|hyperpage}{118} +\indexentry{Little Picard's theorem|hyperpage}{118} +\indexentry{Great Picard's theorem|hyperpage}{118} +\indexentry{meromorphic|hyperpage}{118} +\indexentry{Argument principle|hyperpage}{118} +\indexentry{Generalized argument principle|hyperpage}{118} +\indexentry{Rouché's theorem|hyperpage}{119} +\indexentry{Binomial theorem|hyperpage}{119} +\indexentry{Hurwitz's theorem|hyperpage}{119} +\indexentry{conformal|hyperpage}{119} +\indexentry{conformal|hyperpage}{119} +\indexentry{conformal representation|hyperpage}{119} +\indexentry{Möbius transformations|hyperpage}{119} +\indexentry{Möbius transformation|hyperpage}{119} +\indexentry{translation|hyperpage}{119} +\indexentry{rotation|hyperpage}{119} +\indexentry{dilatation|hyperpage}{119} +\indexentry{inversion|hyperpage}{119} +\indexentry{cross ratio|hyperpage}{120} +\indexentry{circle|hyperpage}{120} +\indexentry{orientation|hyperpage}{120} +\indexentry{Right side|hyperpage}{120} +\indexentry{Left side|hyperpage}{120} +\indexentry{Center|hyperpage}{120} +\indexentry{Orientation principle|hyperpage}{120} +\indexentry{reflection|hyperpage}{120} +\indexentry{Symmetry principle|hyperpage}{120} +\indexentry{Schwarz lemma|hyperpage}{120} +\indexentry{Schwarz-Pick lemma|hyperpage}{121} +\indexentry{surjective|hyperpage}{121} +\indexentry{relatively compact|hyperpage}{121} +\indexentry{normal|hyperpage}{121} +\indexentry{equicontinuous|hyperpage}{121} +\indexentry{Arzelà-Ascoli theorem|hyperpage}{121} +\indexentry{locally bounded|hyperpage}{121} +\indexentry{bounded by compact sets|hyperpage}{121} +\indexentry{Montel's theorem|hyperpage}{121} +\indexentry{Riemann conformal representation theorem|hyperpage}{121} +\indexentry{ordinary differential equation|hyperpage}{122} +\indexentry{ode|hyperpage}{122} +\indexentry{implicit form|hyperpage}{122} +\indexentry{independent variable|hyperpage}{122} +\indexentry{explicit form|hyperpage}{122} +\indexentry{solution of theODE|hyperpage}{122} +\indexentry{general solution of theODE|hyperpage}{122} +\indexentry{system of ordinary differential equations|hyperpage}{122} +\indexentry{differential system|hyperpage}{122} +\indexentry{autonomous|hyperpage}{122} +\indexentry{non-autonomous|hyperpage}{122} +\indexentry{linear|hyperpage}{122} +\indexentry{homogeneous|hyperpage}{122} +\indexentry{constant coefficients|hyperpage}{122} +\indexentry{Initial value problem|hyperpage}{122} +\indexentry{initial value problem|hyperpage}{122} +\indexentry{ivp|hyperpage}{122} +\indexentry{Cauchy problem|hyperpage}{122} +\indexentry{initial conditions|hyperpage}{122} +\indexentry{Separation of variables|hyperpage}{123} +\indexentry{Variation of constants|hyperpage}{123} +\indexentry{Characteristic equation|hyperpage}{123} +\indexentry{characteristic equation|hyperpage}{123} +\indexentry{Reducible linearODE of second order|hyperpage}{123} +\indexentry{Bernoulli differential equation|hyperpage}{123} +\indexentry{Bernoulli differential equation|hyperpage}{123} +\indexentry{Riccati differential equation|hyperpage}{123} +\indexentry{Riccati differential equation|hyperpage}{123} +\indexentry{Integrating factor|hyperpage}{124} +\indexentry{integrating factor|hyperpage}{124} +\indexentry{Lipschitz continuous with respect to the second variable|hyperpage}{124} +\indexentry{locally Lipschitz continuous with respect to the second variable|hyperpage}{124} +\indexentry{operator|hyperpage}{124} +\indexentry{Banach fixed-point theorem|hyperpage}{125} +\indexentry{Picard theorem|hyperpage}{125} +\indexentry{Picard iteration process|hyperpage}{125} +\indexentry{pointwise bounded|hyperpage}{125} +\indexentry{uniformly bounded|hyperpage}{125} +\indexentry{equicontinuous at a point|hyperpage}{125} +\indexentry{pointwise equicontinuous|hyperpage}{125} +\indexentry{uniformly equicontinuous|hyperpage}{125} +\indexentry{Arzelà-Ascoli theorem|hyperpage}{125} +\indexentry{Peano theorem|hyperpage}{125} +\indexentry{extension|hyperpage}{126} +\indexentry{maximal element|hyperpage}{126} +\indexentry{maximal|hyperpage}{126} +\indexentry{chain|hyperpage}{126} +\indexentry{upper bound|hyperpage}{126} +\indexentry{greatest element|hyperpage}{126} +\indexentry{Zorn's lemma|hyperpage}{126} +\indexentry{Wintner lemma|hyperpage}{126} +\indexentry{tends to the boundary|hyperpage}{126} +\indexentry{system of linear differential equations|hyperpage}{126} +\indexentry{homogeneous|hyperpage}{126} +\indexentry{constant coefficients|hyperpage}{126} +\indexentry{flow of the linearODE|hyperpage}{126} +\indexentry{matrix solution|hyperpage}{127} +\indexentry{fundamental matrix solution|hyperpage}{127} +\indexentry{Liouville's formula|hyperpage}{127} +\indexentry{matrix exponential|hyperpage}{127} +\indexentry{generalized eigenvector|hyperpage}{128} +\indexentry{generalized eigenspace|hyperpage}{128} +\indexentry{flow|hyperpage}{128} +\indexentry{Grönwall's lemma|hyperpage}{129} +\indexentry{Dependence on $\boldsymbol {\mathrm {x}}_0$|hyperpage}{129} +\indexentry{variational equations|hyperpage}{129} +\indexentry{Dependence on $t_0$|hyperpage}{129} +\indexentry{Dependence on $\boldsymbol {\mathrm {\lambda }}$|hyperpage}{129} +\indexentry{dynamical system|hyperpage}{130} +\indexentry{discrete|hyperpage}{130} +\indexentry{continuous|hyperpage}{130} +\indexentry{semidynamical system|hyperpage}{130} +\indexentry{orbit|hyperpage}{130} +\indexentry{positive semi-orbit|hyperpage}{130} +\indexentry{negative semi-orbit|hyperpage}{130} +\indexentry{phase portrait|hyperpage}{130} +\indexentry{phase space|hyperpage}{130} +\indexentry{equivalent|hyperpage}{130} +\indexentry{critical point|hyperpage}{131} +\indexentry{singular point|hyperpage}{131} +\indexentry{non-singular|hyperpage}{131} +\indexentry{regular|hyperpage}{131} +\indexentry{periodic|hyperpage}{131} +\indexentry{$\alpha $-limit point|hyperpage}{131} +\indexentry{$\alpha $-limit set|hyperpage}{131} +\indexentry{$\omega $-limit point|hyperpage}{131} +\indexentry{$\omega $-limit point|hyperpage}{131} +\indexentry{$\omega $-limit set|hyperpage}{131} +\indexentry{$\omega $-limit point|hyperpage}{131} +\indexentry{invariant|hyperpage}{131} +\indexentry{positively invariant|hyperpage}{131} +\indexentry{negatively invariant|hyperpage}{131} +\indexentry{positively stable|hyperpage}{131} +\indexentry{negatively stable|hyperpage}{131} +\indexentry{attracting|hyperpage}{131} +\indexentry{repelling|hyperpage}{131} +\indexentry{asymptotically stable|hyperpage}{131} +\indexentry{center|hyperpage}{131} +\indexentry{basin|hyperpage}{131} +\indexentry{equivalent dynamical systems|hyperpage}{131} +\indexentry{topologically equivalent|hyperpage}{131} +\indexentry{equivalence|hyperpage}{131} +\indexentry{conjugate dynamical systems|hyperpage}{131} +\indexentry{topologically conjugate|hyperpage}{132} 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focus|hyperpage}{133} +\indexentry{source|hyperpage}{133} +\indexentry{center|hyperpage}{133} +\indexentry{saddle separatrices|hyperpage}{133} +\indexentry{homoclinic orbit|hyperpage}{133} +\indexentry{heteroclinic orbit|hyperpage}{133} +\indexentry{elliptic sector|hyperpage}{133} +\indexentry{hyperbolic sector|hyperpage}{133} +\indexentry{attracting parabolic sector|hyperpage}{133} +\indexentry{repelling parabolic sector|hyperpage}{133} +\indexentry{sectorial decomposition|hyperpage}{133} +\indexentry{first integral|hyperpage}{134} +\indexentry{functionally independent|hyperpage}{134} +\indexentry{independent|hyperpage}{134} +\indexentry{Hamiltonian system|hyperpage}{134} +\indexentry{Hamiltonian|hyperpage}{134} +\indexentry{hyperbolic critical point|hyperpage}{134} +\indexentry{Hartman-Grobman theorem|hyperpage}{134} +\indexentry{hyperbolic matrix|hyperpage}{134} +\indexentry{stability number|hyperpage}{134} +\indexentry{polynomial system|hyperpage}{135} +\indexentry{polynomial vector field|hyperpage}{135} +\indexentry{degree|hyperpage}{135} +\indexentry{algebraic curve|hyperpage}{135} +\indexentry{invariant algebraic curve|hyperpage}{135} +\indexentry{cofactor|hyperpage}{136} +\indexentry{Poincaré map|hyperpage}{136} +\indexentry{Poincaré map|hyperpage}{136} +\indexentry{limit cycle|hyperpage}{136} +\indexentry{stable|hyperpage}{136} +\indexentry{unstable|hyperpage}{136} +\indexentry{semi-stable|hyperpage}{136} +\indexentry{hyperbolic periodic orbit|hyperpage}{136} +\indexentry{Poincaré-Bendixson theorem|hyperpage}{136} +\indexentry{Poincaré-Bendixson theorem|hyperpage}{136} +\indexentry{Lyapunov stable|hyperpage}{137} +\indexentry{Lyapunov function|hyperpage}{137} +\indexentry{strict Lyapunov function|hyperpage}{137} +\indexentry{Lyapunov's theorem|hyperpage}{137} +\indexentry{Bendixson's theorem|hyperpage}{137} +\indexentry{Bendixson-Dulac theorem|hyperpage}{137} +\indexentry{Generalized Bendixson-Dulac theorem|hyperpage}{137} +\indexentry{Poincaré compactification|hyperpage}{138} +\indexentry{Poincaré disk|hyperpage}{138} +\indexentry{integrating factor|hyperpage}{138} +\indexentry{exponential factor|hyperpage}{138} +\indexentry{Darboux theorem|hyperpage}{138} +\indexentry{index|hyperpage}{138} +\indexentry{homotopic|hyperpage}{139} +\indexentry{homotopy|hyperpage}{139} +\indexentry{contractible|hyperpage}{139} +\indexentry{index|hyperpage}{139} +\indexentry{index|hyperpage}{139} +\indexentry{tangent vector field|hyperpage}{139} +\indexentry{Poincaré-Hopf theorem|hyperpage}{139} +\indexentry{Poincaré-Hopf theorem on $S^2$|hyperpage}{139} +\indexentry{Poincaré index formula|hyperpage}{139} +\indexentry{partial differential equation|hyperpage}{139} +\indexentry{PDE|hyperpage}{139} +\indexentry{solution|hyperpage}{139} +\indexentry{quasilinear PDE|hyperpage}{140} +\indexentry{Heat equation|hyperpage}{140} +\indexentry{heat equation|hyperpage}{140} +\indexentry{Wave equation|hyperpage}{140} +\indexentry{wave equation|hyperpage}{140} 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plane|hyperpage}{143} +\indexentry{General Frenet-Serret formulas|hyperpage}{143} +\indexentry{contact|hyperpage}{144} +\indexentry{contact|hyperpage}{144} +\indexentry{tangent line|hyperpage}{144} +\indexentry{osculating circle|hyperpage}{144} +\indexentry{radius of curvature|hyperpage}{144} +\indexentry{osculating sphere|hyperpage}{144} +\indexentry{envelope|hyperpage}{144} +\indexentry{evolute|hyperpage}{144} +\indexentry{involute|hyperpage}{144} +\indexentry{signed curvature|hyperpage}{145} +\indexentry{local canonical form|hyperpage}{145} +\indexentry{dextrorotation|hyperpage}{145} +\indexentry{levorotation|hyperpage}{145} +\indexentry{orthogonal group|hyperpage}{145} +\indexentry{special orthogonal group|hyperpage}{145} +\indexentry{Fundamental theorem of curves|hyperpage}{146} +\indexentry{support|hyperpage}{146} +\indexentry{Immersion|hyperpage}{146} +\indexentry{immersion|hyperpage}{146} +\indexentry{immersion|hyperpage}{146} +\indexentry{Submersion|hyperpage}{146} +\indexentry{submersion|hyperpage}{146} +\indexentry{submersion|hyperpage}{146} +\indexentry{Local structure of immersions|hyperpage}{146} +\indexentry{Local structure of submersions|hyperpage}{146} +\indexentry{submanifold|hyperpage}{146} +\indexentry{parametrization|hyperpage}{147} +\indexentry{coordinate chart|hyperpage}{147} +\indexentry{regular surface|hyperpage}{147} +\indexentry{surface|hyperpage}{147} +\indexentry{surface of revolution|hyperpage}{147} +\indexentry{generatrix|hyperpage}{147} +\indexentry{differentiable|hyperpage}{147} +\indexentry{differentiable|hyperpage}{147} +\indexentry{tangent vector|hyperpage}{147} +\indexentry{tangent space|hyperpage}{147} +\indexentry{tangent plane|hyperpage}{147} +\indexentry{tangent function|hyperpage}{148} +\indexentry{differential|hyperpage}{148} +\indexentry{Inverse function theorem for surfaces|hyperpage}{148} +\indexentry{first fundamental form|hyperpage}{148} +\indexentry{orthogonal|hyperpage}{148} +\indexentry{orthogonal coordinates|hyperpage}{148} +\indexentry{regular domain|hyperpage}{148} +\indexentry{domain|hyperpage}{148} +\indexentry{region|hyperpage}{148} +\indexentry{integral of $f$ over the region $R$|hyperpage}{148} +\indexentry{local isometry|hyperpage}{148} +\indexentry{isometry|hyperpage}{148} +\indexentry{conformal|hyperpage}{149} +\indexentry{angle-preserving|hyperpage}{149} +\indexentry{conformal|hyperpage}{149} +\indexentry{orientable|hyperpage}{149} +\indexentry{normal unit field|hyperpage}{149} +\indexentry{Gau\ss \ map|hyperpage}{149} +\indexentry{orientation|hyperpage}{149} +\indexentry{compatible|hyperpage}{149} +\indexentry{Weingarten endomorphsim|hyperpage}{149} +\indexentry{principal directons|hyperpage}{149} +\indexentry{principal curvatures|hyperpage}{149} +\indexentry{umbilic point|hyperpage}{149} +\indexentry{Gau\ss \ curvature|hyperpage}{149} +\indexentry{mean curvature|hyperpage}{149} +\indexentry{minimal surface|hyperpage}{149} +\indexentry{second fundamental form|hyperpage}{150} +\indexentry{normal curvature|hyperpage}{150} +\indexentry{Meusnier's theorem|hyperpage}{150} +\indexentry{normal curvature|hyperpage}{150} +\indexentry{asymptotic direction|hyperpage}{150} +\indexentry{line of curvature|hyperpage}{150} +\indexentry{asymptotic line|hyperpage}{150} +\indexentry{Olinde Rodrigues' theorem|hyperpage}{150} +\indexentry{Euler's formula|hyperpage}{150} +\indexentry{elliptic point|hyperpage}{150} +\indexentry{hyperbolic point|hyperpage}{150} +\indexentry{parabolic point|hyperpage}{150} +\indexentry{plane point|hyperpage}{150} +\indexentry{ruled surface|hyperpage}{151} +\indexentry{developable|hyperpage}{151} +\indexentry{regression axis|hyperpage}{151} +\indexentry{Christoffel symbols|hyperpage}{151} +\indexentry{Gau\ss \ equations|hyperpage}{152} +\indexentry{Codazzi-Mainardi equations|hyperpage}{152} +\indexentry{Gau\ss ' Theorema Egregium|hyperpage}{152} +\indexentry{Bonnet's theorem|hyperpage}{152} +\indexentry{vector field|hyperpage}{152} +\indexentry{differentiable|hyperpage}{152} +\indexentry{differentiable|hyperpage}{152} +\indexentry{covariant derivative|hyperpage}{152} +\indexentry{vector field|hyperpage}{152} +\indexentry{differentiable|hyperpage}{152} +\indexentry{parallel|hyperpage}{152} +\indexentry{parallel transport|hyperpage}{152} +\indexentry{geodesic|hyperpage}{153} +\indexentry{algebraic value|hyperpage}{153} +\indexentry{geodesic curvature|hyperpage}{153} +\indexentry{Liouville's formula|hyperpage}{153} +\indexentry{vector field|hyperpage}{154} +\indexentry{differentiable|hyperpage}{154} +\indexentry{differentiable|hyperpage}{154} +\indexentry{integral curve|hyperpage}{154} +\indexentry{first integral|hyperpage}{154} +\indexentry{vector field|hyperpage}{154} +\indexentry{differentiable|hyperpage}{154} +\indexentry{tangent|hyperpage}{154} +\indexentry{multilinear map|hyperpage}{155} +\indexentry{$k$-linear map|hyperpage}{155} +\indexentry{degree|hyperpage}{155} 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boundary|hyperpage}{157} +\indexentry{interior point|hyperpage}{157} +\indexentry{boundary|hyperpage}{157} +\indexentry{tangent space|hyperpage}{157} +\indexentry{set of interior vectors|hyperpage}{157} +\indexentry{exterior vectors|hyperpage}{157} +\indexentry{orientable|hyperpage}{158} +\indexentry{compatible|hyperpage}{158} +\indexentry{volume element|hyperpage}{158} +\indexentry{unit normal exterior vector field|hyperpage}{158} +\indexentry{positive|hyperpage}{158} +\indexentry{orientation induced by $M$|hyperpage}{158} +\indexentry{integral|hyperpage}{158} +\indexentry{integral|hyperpage}{158} +\indexentry{integral|hyperpage}{158} +\indexentry{partition of unity|hyperpage}{158} +\indexentry{integral|hyperpage}{159} +\indexentry{Change of variables|hyperpage}{159} +\indexentry{Stokes theorem|hyperpage}{159} +\indexentry{Green's formula|hyperpage}{159} +\indexentry{line intergal|hyperpage}{159} +\indexentry{circulation|hyperpage}{159} +\indexentry{surface intergal|hyperpage}{159} +\indexentry{flux|hyperpage}{159} +\indexentry{Curl theorem|hyperpage}{159} +\indexentry{Divergence theorem on $\ensuremath {\mathbb {R}}^3$|hyperpage}{160} +\indexentry{Divergence theorem|hyperpage}{160} +\indexentry{simple region|hyperpage}{160} +\indexentry{edges|hyperpage}{160} +\indexentry{vertices|hyperpage}{160} +\indexentry{exterior angle|hyperpage}{160} +\indexentry{Local Gau\ss -Bonnet theorem|hyperpage}{160} +\indexentry{triangle|hyperpage}{160} +\indexentry{regular region|hyperpage}{160} +\indexentry{Global Gau\ss -Bonnet theorem|hyperpage}{160} +\indexentry{singular point|hyperpage}{160} +\indexentry{isolated singular point|hyperpage}{160} +\indexentry{isolated singularity|hyperpage}{160} +\indexentry{index|hyperpage}{160} +\indexentry{Poincaré-Hopf theorem|hyperpage}{161} +\indexentry{quadratic polynomial|hyperpage}{162} +\indexentry{cubic polynomial|hyperpage}{162} +\indexentry{quartic polynomial|hyperpage}{162} +\indexentry{depressed equation|hyperpage}{162} +\indexentry{field morphism|hyperpage}{162} +\indexentry{characteristic|hyperpage}{162} +\indexentry{polynomial ring|hyperpage}{162} +\indexentry{Universal property of polynomials in several variables|hyperpage}{162} +\indexentry{evaluation|hyperpage}{162} +\indexentry{Universal property of the field of fractions|hyperpage}{163} +\indexentry{field of fractions|hyperpage}{163} +\indexentry{field of rational functions|hyperpage}{163} +\indexentry{subring generated|hyperpage}{163} +\indexentry{subfield generated|hyperpage}{163} +\indexentry{Symmetric polynomials|hyperpage}{163} +\indexentry{symmetric polynomial|hyperpage}{163} +\indexentry{elementary symmetric polynomials|hyperpage}{163} +\indexentry{lexicographic order|hyperpage}{163} +\indexentry{lexicographic degree|hyperpage}{163} +\indexentry{Waring's method|hyperpage}{164} +\indexentry{Fundamental theorem of symmetric polynomials|hyperpage}{164} +\indexentry{$n$-th cyclotomic polynomial|hyperpage}{164} +\indexentry{field extension|hyperpage}{164} +\indexentry{degree|hyperpage}{164} +\indexentry{finite|hyperpage}{164} +\indexentry{infinite|hyperpage}{164} +\indexentry{Kronecker's lemma|hyperpage}{164} +\indexentry{algebraic|hyperpage}{164} +\indexentry{irreducible polynomial|hyperpage}{164} +\indexentry{transcendental|hyperpage}{164} +\indexentry{simple|hyperpage}{164} +\indexentry{primitive element|hyperpage}{164} +\indexentry{Steinitz's theorem|hyperpage}{164} +\indexentry{Tower formula|hyperpage}{165} +\indexentry{tower of fields|hyperpage}{165} +\indexentry{finitely generated|hyperpage}{165} +\indexentry{compositum|hyperpage}{165} +\indexentry{algebraic|hyperpage}{165} +\indexentry{purely transcendental|hyperpage}{165} +\indexentry{morphism of field extensions|hyperpage}{165} +\indexentry{$K$-field morphism|hyperpage}{165} +\indexentry{Galois group|hyperpage}{166} +\indexentry{Finite field|hyperpage}{166} +\indexentry{finite field|hyperpage}{166} +\indexentry{Frobenius endomorphism|hyperpage}{166} +\indexentry{Perfect fields|hyperpage}{166} +\indexentry{perfect|hyperpage}{166} +\indexentry{splits into linear factors|hyperpage}{167} +\indexentry{Kronecker's theorem|hyperpage}{167} +\indexentry{Splitting field|hyperpage}{167} +\indexentry{splitting field|hyperpage}{167} +\indexentry{Existence of the splitting field|hyperpage}{167} +\indexentry{Unicity of the splitting field|hyperpage}{167} +\indexentry{normal|hyperpage}{167} +\indexentry{normal closure|hyperpage}{167} +\indexentry{Formal derivative|hyperpage}{167} +\indexentry{formal derivative|hyperpage}{167} +\indexentry{separable|hyperpage}{168} +\indexentry{Separable extension|hyperpage}{168} +\indexentry{separable|hyperpage}{168} +\indexentry{separable|hyperpage}{168} +\indexentry{Separability theorem|hyperpage}{168} +\indexentry{purely inseparable|hyperpage}{168} +\indexentry{Primitive element theorem|hyperpage}{168} +\indexentry{Galois extension|hyperpage}{168} +\indexentry{Galois|hyperpage}{168} +\indexentry{fixed field|hyperpage}{168} +\indexentry{Artin's lemma|hyperpage}{169} +\indexentry{Fundamental theorem of Galois theory|hyperpage}{169} +\indexentry{lattice of subgroups|hyperpage}{169} +\indexentry{transitive|hyperpage}{169} +\indexentry{discriminant|hyperpage}{169} +\indexentry{algebraically closed|hyperpage}{170} +\indexentry{Fundamental theorem of algebra|hyperpage}{170} +\indexentry{algebraic closure|hyperpage}{170} +\indexentry{solvable|hyperpage}{170} +\indexentry{simple|hyperpage}{170} +\indexentry{radical|hyperpage}{170} +\indexentry{cyclotomic|hyperpage}{170} +\indexentry{radical tower|hyperpage}{170} +\indexentry{solvable by radicals|hyperpage}{170} +\indexentry{solvable by radicals|hyperpage}{170} +\indexentry{$n$-th root of unity|hyperpage}{170} +\indexentry{$n$-th primitive root of unity|hyperpage}{170} +\indexentry{abelian|hyperpage}{170} +\indexentry{cyclic|hyperpage}{170} +\indexentry{Dedekind's lemma|hyperpage}{170} +\indexentry{$F$-linearly independent|hyperpage}{170} +\indexentry{Abel-Ruffini theorem|hyperpage}{171} +\indexentry{Nart-Vila theorem|hyperpage}{171} +\indexentry{Algebra|hyperpage}{172} +\indexentry{algebra|hyperpage}{172} +\indexentry{$\sigma $-algebra|hyperpage}{172} +\indexentry{$\sigma $-algebra|hyperpage}{172} +\indexentry{trivial $\sigma $-algebra|hyperpage}{172} +\indexentry{discrete $\sigma $-algebra|hyperpage}{172} +\indexentry{$\sigma $-algebra generated|hyperpage}{172} +\indexentry{$\sigma $-algebra generated|hyperpage}{172} +\indexentry{Borel $\sigma $-algebra|hyperpage}{172} +\indexentry{Sample space|hyperpage}{173} +\indexentry{sample space|hyperpage}{173} +\indexentry{Kolmogorov axioms|hyperpage}{173} +\indexentry{probability|hyperpage}{173} +\indexentry{$\sigma $-additivity|hyperpage}{173} +\indexentry{event|hyperpage}{173} +\indexentry{probability space|hyperpage}{173} +\indexentry{Finite subadditivity|hyperpage}{173} +\indexentry{Continuity from below|hyperpage}{173} +\indexentry{Continuity from above|hyperpage}{173} +\indexentry{Countable subadditivity|hyperpage}{173} +\indexentry{conditional probability|hyperpage}{174} +\indexentry{Compound probability formula|hyperpage}{174} +\indexentry{Generalized compound probability formula|hyperpage}{174} +\indexentry{partition|hyperpage}{174} +\indexentry{Law of total probability|hyperpage}{174} +\indexentry{Bayes' formula|hyperpage}{174} +\indexentry{independent events|hyperpage}{174} +\indexentry{independent events|hyperpage}{174} +\indexentry{independent events|hyperpage}{174} +\indexentry{null set|hyperpage}{174} +\indexentry{set of zero-content|hyperpage}{174} +\indexentry{measure|hyperpage}{174} +\indexentry{measure space|hyperpage}{175} +\indexentry{$\sigma $-algebra of all Lebesgue measurable sets in $\ensuremath {\mathbb {R}}^n$|hyperpage}{175} +\indexentry{Lebesgue measure|hyperpage}{175} +\indexentry{volume|hyperpage}{175} +\indexentry{measurable|hyperpage}{175} +\indexentry{Lebesgue integral|hyperpage}{175} +\indexentry{Lebesgue integrable|hyperpage}{175} +\indexentry{Tonelli's theorem|hyperpage}{175} +\indexentry{Fubini's theorem|hyperpage}{175} +\indexentry{real random variable|hyperpage}{175} +\indexentry{random variable|hyperpage}{175} +\indexentry{distribution|hyperpage}{176} +\indexentry{equal in distribution|hyperpage}{176} +\indexentry{equal almost surely|hyperpage}{176} +\indexentry{Cumulative distribution function|hyperpage}{176} +\indexentry{cumulative distribution function|hyperpage}{176} +\indexentry{cdf|hyperpage}{176} +\indexentry{càdlàg|hyperpage}{176} +\indexentry{discrete|hyperpage}{176} +\indexentry{support|hyperpage}{176} +\indexentry{Probability mass function|hyperpage}{176} +\indexentry{probability mass function|hyperpage}{176} +\indexentry{pmf|hyperpage}{176} +\indexentry{distribution functions|hyperpage}{176} +\indexentry{Degenerated distribution|hyperpage}{177} +\indexentry{degenerated distribution|hyperpage}{177} +\indexentry{Bernoulli distribution|hyperpage}{177} +\indexentry{Bernoulli distribution|hyperpage}{177} +\indexentry{Discrete uniform distribution|hyperpage}{177} +\indexentry{discrete uniform distribution|hyperpage}{177} +\indexentry{equiprobable space|hyperpage}{177} +\indexentry{Binomial distribution|hyperpage}{177} +\indexentry{binomial distribution|hyperpage}{177} +\indexentry{Poisson distribution|hyperpage}{177} +\indexentry{Poisson distribution|hyperpage}{177} +\indexentry{Geometric distribution|hyperpage}{177} +\indexentry{geometric distribution|hyperpage}{177} +\indexentry{Discrete memorylessness property|hyperpage}{177} +\indexentry{memoryless|hyperpage}{177} +\indexentry{Hypergeometric distribution|hyperpage}{177} +\indexentry{hypergeometric distribution|hyperpage}{177} +\indexentry{success|hyperpage}{177} +\indexentry{true|hyperpage}{177} +\indexentry{failure|hyperpage}{177} +\indexentry{false|hyperpage}{177} +\indexentry{Negative binomial distribution|hyperpage}{178} +\indexentry{negative binomial distribution|hyperpage}{178} +\indexentry{absolutely continuous|hyperpage}{178} +\indexentry{probability density function|hyperpage}{178} +\indexentry{pdf|hyperpage}{178} +\indexentry{density function|hyperpage}{178} +\indexentry{Continuous uniform distribution|hyperpage}{178} +\indexentry{continuous uniform distribution|hyperpage}{178} +\indexentry{Exponential distribution|hyperpage}{178} +\indexentry{exponential distribution|hyperpage}{178} +\indexentry{Continuous memorylessness property|hyperpage}{178} +\indexentry{memoryless|hyperpage}{178} +\indexentry{Standard normal distribution|hyperpage}{178} +\indexentry{standard normal distribution|hyperpage}{178} +\indexentry{Normal distribution|hyperpage}{178} +\indexentry{normal distribution|hyperpage}{178} +\indexentry{mean|hyperpage}{178} +\indexentry{expectation|hyperpage}{178} +\indexentry{variance|hyperpage}{178} +\indexentry{standard deviation|hyperpage}{178} +\indexentry{standardized form|hyperpage}{179} +\indexentry{Gamma distribution|hyperpage}{179} +\indexentry{gamma distribution|hyperpage}{179} +\indexentry{shape|hyperpage}{179} +\indexentry{rate|hyperpage}{179} +\indexentry{scale|hyperpage}{179} +\indexentry{beta function|hyperpage}{179} +\indexentry{Beta distribution|hyperpage}{179} +\indexentry{beta distribution|hyperpage}{179} +\indexentry{Cauchy distribution|hyperpage}{179} +\indexentry{Cauchy distribution|hyperpage}{179} +\indexentry{mixed random variable|hyperpage}{179} +\indexentry{random vector|hyperpage}{179} +\indexentry{distribution|hyperpage}{180} +\indexentry{discrete|hyperpage}{180} +\indexentry{support|hyperpage}{180} +\indexentry{Joint probability mass function|hyperpage}{180} +\indexentry{joint probability mass function|hyperpage}{180} +\indexentry{joint pmf|hyperpage}{180} +\indexentry{Marginal probability mass functions|hyperpage}{180} +\indexentry{marginal probability mass functions|hyperpage}{180} +\indexentry{marginal pmf|hyperpage}{180} +\indexentry{Multinomial distrbution|hyperpage}{180} +\indexentry{multinomial distribution|hyperpage}{180} +\indexentry{absolutely continuous|hyperpage}{180} +\indexentry{joint probability density function|hyperpage}{180} +\indexentry{joint pdf|hyperpage}{180} +\indexentry{Marginal probability density functions|hyperpage}{180} +\indexentry{marginal probability density functions|hyperpage}{180} +\indexentry{marginal pdf|hyperpage}{180} +\indexentry{Multivariate standard normal distribution|hyperpage}{180} +\indexentry{multivariate normal distribution|hyperpage}{180} +\indexentry{Multivariate uniform distribution|hyperpage}{180} +\indexentry{multivariate uniform distribution|hyperpage}{180} +\indexentry{multivariate cumulative distribution function|hyperpage}{181} +\indexentry{multivariate cdf|hyperpage}{181} +\indexentry{Borel measurable|hyperpage}{181} +\indexentry{Multivariate normal distribution|hyperpage}{181} +\indexentry{mean vector|hyperpage}{181} +\indexentry{covariance matrix|hyperpage}{181} +\indexentry{independent|hyperpage}{181} +\indexentry{independent and identically distributed|hyperpage}{181} +\indexentry{i.i.d.|hyperpage}{181} +\indexentry{increment|hyperpage}{181} +\indexentry{increasing|hyperpage}{181} +\indexentry{conditional probability mass function|hyperpage}{182} +\indexentry{conditional probability density function|hyperpage}{182} +\indexentry{simple random variable|hyperpage}{182} +\indexentry{finite expectation|hyperpage}{182} +\indexentry{integrable|hyperpage}{182} +\indexentry{expectation|hyperpage}{182} +\indexentry{not integrable|hyperpage}{182} +\indexentry{Expectation|hyperpage}{183} +\indexentry{finite expectation|hyperpage}{183} +\indexentry{integrable|hyperpage}{183} +\indexentry{expectation|hyperpage}{183} +\indexentry{Comparison test|hyperpage}{183} +\indexentry{dyadic partition of order $n$|hyperpage}{183} +\indexentry{Monotone convergence theorem|hyperpage}{184} +\indexentry{Dominated convergence theorem|hyperpage}{184} +\indexentry{Fatou's lemma|hyperpage}{184} +\indexentry{finite expectation|hyperpage}{184} +\indexentry{expectation|hyperpage}{184} +\indexentry{Moment|hyperpage}{185} +\indexentry{finite moment of order $k$|hyperpage}{185} +\indexentry{finite $k$-th moment|hyperpage}{185} +\indexentry{Cauchy-Schwarz inequality|hyperpage}{185} +\indexentry{Variance|hyperpage}{185} +\indexentry{variance|hyperpage}{185} +\indexentry{Standard deviation|hyperpage}{185} +\indexentry{standard deviation|hyperpage}{185} +\indexentry{standard error|hyperpage}{185} +\indexentry{Covariance|hyperpage}{185} +\indexentry{covariance|hyperpage}{185} +\indexentry{uncorrelated|hyperpage}{185} +\indexentry{Pearson correlation coefficient|hyperpage}{185} +\indexentry{Pearson correlation coefficient|hyperpage}{185} +\indexentry{correlation coefficient|hyperpage}{185} +\indexentry{Markov's inequality|hyperpage}{185} +\indexentry{Chebyshev's inequality|hyperpage}{186} +\indexentry{Moment-generating function|hyperpage}{186} +\indexentry{moment-generating function|hyperpage}{186} +\indexentry{conditional expectation|hyperpage}{186} +\indexentry{Law of total expectation|hyperpage}{186} +\indexentry{conditional expectation|hyperpage}{186} +\indexentry{Law of total expectation|hyperpage}{187} +\indexentry{converges in probability|hyperpage}{187} +\indexentry{metrizable|hyperpage}{187} +\indexentry{Cauchy condition in probability|hyperpage}{187} +\indexentry{Cauchy in probability|hyperpage}{187} +\indexentry{converges almost surely|hyperpage}{188} +\indexentry{limit superior|hyperpage}{188} +\indexentry{limit inferior|hyperpage}{188} +\indexentry{limit|hyperpage}{188} +\indexentry{First Borel-Cantelli lemma|hyperpage}{188} +\indexentry{Second Borel-Cantelli lemma|hyperpage}{189} +\indexentry{converges in the $p$-th mean|hyperpage}{189} +\indexentry{Dominated convergence theorem|hyperpage}{189} +\indexentry{Cauchy condition in $p$-th mean|hyperpage}{189} +\indexentry{Cauchy in $p$-th mean|hyperpage}{189} +\indexentry{converges in distribution|hyperpage}{190} +\indexentry{Skorokhod's representation theorem|hyperpage}{190} +\indexentry{Slutsky's theorem|hyperpage}{190} +\indexentry{Weak law of large numbers|hyperpage}{190} +\indexentry{Weak law of large numbers|hyperpage}{190} +\indexentry{Kolmogorov's strong law of large numbers|hyperpage}{191} +\indexentry{Strong law of large numbers|hyperpage}{191} +\indexentry{simply normal|hyperpage}{191} +\indexentry{normal|hyperpage}{191} +\indexentry{Borel's theorem|hyperpage}{191} +\indexentry{complex random variable|hyperpage}{191} +\indexentry{Characteristic function|hyperpage}{191} +\indexentry{characteristic function|hyperpage}{191} +\indexentry{Lévy-Lindeberg central limit theorem|hyperpage}{192} +\indexentry{Lyapunov central limit theorem|hyperpage}{192} +\indexentry{sample mean|hyperpage}{192} +\indexentry{De Moivre-Laplace theorem|hyperpage}{192} +\indexentry{Continuity correction|hyperpage}{192} +\indexentry{continuity correction|hyperpage}{192} +\indexentry{statistical model|hyperpage}{193} +\indexentry{sample space|hyperpage}{193} +\indexentry{parameter space|hyperpage}{193} +\indexentry{random sample|hyperpage}{193} +\indexentry{simple random sample|hyperpage}{193} +\indexentry{realization|hyperpage}{193} +\indexentry{identifiable|hyperpage}{193} +\indexentry{parametric|hyperpage}{193} +\indexentry{Statistic|hyperpage}{193} +\indexentry{statistic|hyperpage}{193} +\indexentry{dimension|hyperpage}{193} +\indexentry{sample mean|hyperpage}{193} +\indexentry{sample variance|hyperpage}{193} +\indexentry{corrected sample variance|hyperpage}{193} +\indexentry{estimator|hyperpage}{193} +\indexentry{Bias|hyperpage}{193} +\indexentry{bias|hyperpage}{193} +\indexentry{unbiased estimator|hyperpage}{193} +\indexentry{biased estimator|hyperpage}{193} +\indexentry{mean squared error|hyperpage}{194} +\indexentry{MSE|hyperpage}{194} +\indexentry{more efficient than|hyperpage}{194} +\indexentry{minimum-variance unbiased estimator|hyperpage}{194} +\indexentry{MVUE|hyperpage}{194} +\indexentry{sufficient|hyperpage}{194} +\indexentry{Fisher-Neyman factorization theorem|hyperpage}{194} +\indexentry{weakly consistent estimator|hyperpage}{194} +\indexentry{strongly consistent estimator|hyperpage}{194} +\indexentry{consistent estimator in $L^2$|hyperpage}{194} +\indexentry{asymptotically unbiased estimator|hyperpage}{194} +\indexentry{an asymptotically normal estimator|hyperpage}{194} +\indexentry{Method of moments|hyperpage}{194} +\indexentry{Likelihood|hyperpage}{195} +\indexentry{likelihood function|hyperpage}{195} +\indexentry{likelihood function|hyperpage}{195} +\indexentry{Maximum likelihood method|hyperpage}{195} +\indexentry{maximum likelihood estimator|hyperpage}{195} +\indexentry{MLE|hyperpage}{195} +\indexentry{log-likelihood function|hyperpage}{195} +\indexentry{score function|hyperpage}{195} +\indexentry{Invariance of the MLE|hyperpage}{195} +\indexentry{regular|hyperpage}{195} +\indexentry{observed information|hyperpage}{195} +\indexentry{Fisher information|hyperpage}{195} +\indexentry{regular|hyperpage}{196} +\indexentry{Cramér-Rao bound|hyperpage}{196} +\indexentry{efficient estimator|hyperpage}{196} +\indexentry{asymptotic efficient estimator|hyperpage}{196} +\indexentry{Delta method|hyperpage}{196} +\indexentry{order statistic|hyperpage}{196} +\indexentry{order statistics|hyperpage}{196} +\indexentry{Quantile|hyperpage}{196} +\indexentry{multivariate normal distribution|hyperpage}{196} +\indexentry{mean vector|hyperpage}{196} +\indexentry{covariance matrix|hyperpage}{196} +\indexentry{multivariate standard deviation|hyperpage}{196} +\indexentry{chi-squared distribution with $n$ degrees of freedom|hyperpage}{197} +\indexentry{Student's $t$-distribution with $n$ degrees of freedom|hyperpage}{197} +\indexentry{Fisher's theorem|hyperpage}{197} +\indexentry{confidence region|hyperpage}{198} +\indexentry{confidence level|hyperpage}{198} +\indexentry{confidence intervals|hyperpage}{198} +\indexentry{significance level|hyperpage}{198} +\indexentry{pivot|hyperpage}{198} +\indexentry{Interval for $\mu $ with $\sigma $ known|hyperpage}{198} +\indexentry{Intervals for $\mu $ and $\sigma ^2$|hyperpage}{198} +\indexentry{Independent samples with known variances|hyperpage}{198} +\indexentry{Independent samples with unknown equal variances|hyperpage}{198} +\indexentry{Independent samples with unknown variances|hyperpage}{198} +\indexentry{Related samples with unknown variances|hyperpage}{199} +\indexentry{hypothesis|hyperpage}{199} +\indexentry{null hypothesis|hyperpage}{199} +\indexentry{alternative hypothesis|hyperpage}{199} +\indexentry{simple|hyperpage}{199} +\indexentry{Hypothesis test|hyperpage}{199} +\indexentry{hypothesis test|hyperpage}{199} +\indexentry{acceptation region|hyperpage}{199} +\indexentry{critical region|hyperpage}{199} +\indexentry{error of type I|hyperpage}{199} +\indexentry{error of type II|hyperpage}{199} +\indexentry{power|hyperpage}{199} +\indexentry{size|hyperpage}{199} +\indexentry{significance level|hyperpage}{199} +\indexentry{power function|hyperpage}{199} +\indexentry{test statistic|hyperpage}{199} +\indexentry{$p$-value|hyperpage}{199} +\indexentry{one-sided right tail test|hyperpage}{199} +\indexentry{one-sided left tail test|hyperpage}{199} +\indexentry{two-sided test|hyperpage}{200} +\indexentry{uniformly most powerful|hyperpage}{200} +\indexentry{UMP|hyperpage}{200} +\indexentry{test|hyperpage}{200} +\indexentry{Neymann-Pearson test|hyperpage}{200} +\indexentry{Neyman-Pearson test|hyperpage}{200} +\indexentry{Neyman-Pearson lemma|hyperpage}{200} +\indexentry{Likelihood-ratio test|hyperpage}{200} +\indexentry{likelihood ratio test|hyperpage}{200} +\indexentry{LRT|hyperpage}{200} +\indexentry{Asymptotic behaviour of the LRT|hyperpage}{200} +\indexentry{LRT test statistic|hyperpage}{200} +\indexentry{Goodness of fit|hyperpage}{200} +\indexentry{Test of homogenity|hyperpage}{201} +\indexentry{Test of independence|hyperpage}{201} +\indexentry{$t$-test|hyperpage}{201} +\indexentry{$t$-test|hyperpage}{201} +\indexentry{Wald test|hyperpage}{201} +\indexentry{Wald test|hyperpage}{201} +\indexentry{contrast matrix|hyperpage}{201} +\indexentry{Score test|hyperpage}{202} +\indexentry{score test|hyperpage}{202} +\indexentry{Non-parametric bootstrap|hyperpage}{202} +\indexentry{empirical distribution|hyperpage}{202} +\indexentry{bootstrap distribution|hyperpage}{202} +\indexentry{Parametric bootstrap|hyperpage}{202} +\indexentry{parametric bootstrap distribution|hyperpage}{202} +\indexentry{Normal confidence interval|hyperpage}{202} +\indexentry{normal confidence interval|hyperpage}{202} +\indexentry{Basic bootstrap confidence interval|hyperpage}{202} +\indexentry{basic bootstrap confidence interval|hyperpage}{202} +\indexentry{Bootstrap-t confidence interval|hyperpage}{202} +\indexentry{bootstrap-t confidence interval|hyperpage}{202} +\indexentry{Percentile confidence interval|hyperpage}{202} +\indexentry{percentile confidence interval|hyperpage}{202} +\indexentry{prior distribution|hyperpage}{202} +\indexentry{prior|hyperpage}{202} +\indexentry{posterior distribution|hyperpage}{202} +\indexentry{posterior|hyperpage}{202} +\indexentry{conjugate|hyperpage}{202} +\indexentry{Bayesian point estimates|hyperpage}{203} +\indexentry{posterior mean|hyperpage}{203} +\indexentry{posterior mode|hyperpage}{203} +\indexentry{posterior median|hyperpage}{203} +\indexentry{flat|hyperpage}{203} +\indexentry{improper|hyperpage}{203} +\indexentry{Jeffrey's prior|hyperpage}{203} +\indexentry{Jeffrey's prior|hyperpage}{203} +\indexentry{loss function|hyperpage}{203} +\indexentry{Quadratic loss function|hyperpage}{203} +\indexentry{Linear loss function|hyperpage}{203} +\indexentry{Zero-one loss function|hyperpage}{203} +\indexentry{Bayes estimate|hyperpage}{203} +\indexentry{credible region|hyperpage}{203} +\indexentry{credible interval|hyperpage}{203} +\indexentry{highest posterior density|hyperpage}{204} +\indexentry{region|hyperpage}{204} +\indexentry{Q-Q plots|hyperpage}{204} +\indexentry{Quantile-Quantile plot|hyperpage}{204} +\indexentry{Q-Q plots|hyperpage}{204} +\indexentry{theoretical quantiles|hyperpage}{204} +\indexentry{sample quantiles|hyperpage}{204} +\indexentry{Normal Q-Q plots|hyperpage}{204} +\indexentry{distance|hyperpage}{205} +\indexentry{metric|hyperpage}{205} +\indexentry{triangular inequality|hyperpage}{205} +\indexentry{metric space|hyperpage}{205} +\indexentry{Euclidean metric|hyperpage}{205} +\indexentry{Taxicab metric|hyperpage}{205} +\indexentry{Maximum metric|hyperpage}{205} +\indexentry{discrete metric|hyperpage}{205} +\indexentry{ball|hyperpage}{205} +\indexentry{open|hyperpage}{205} +\indexentry{Topological space|hyperpage}{205} +\indexentry{topology|hyperpage}{205} +\indexentry{topological space|hyperpage}{205} +\indexentry{points|hyperpage}{205} +\indexentry{open sets|hyperpage}{205} +\indexentry{finer|hyperpage}{205} +\indexentry{Topology induced from the metric|hyperpage}{205} +\indexentry{Trivial topology|hyperpage}{206} +\indexentry{Discrete topology|hyperpage}{206} +\indexentry{Cofinite topology|hyperpage}{206} +\indexentry{Cocountable topology|hyperpage}{206} +\indexentry{Particular point topology|hyperpage}{206} +\indexentry{Excluded point topology|hyperpage}{206} +\indexentry{Sierpiński topology|hyperpage}{206} +\indexentry{closed|hyperpage}{206} +\indexentry{clopen|hyperpage}{206} +\indexentry{basis|hyperpage}{206} +\indexentry{topology generated|hyperpage}{206} +\indexentry{lower limit topology|hyperpage}{206} +\indexentry{digital topology|hyperpage}{206} +\indexentry{subbasis|hyperpage}{206} +\indexentry{Interior|hyperpage}{206} +\indexentry{interior|hyperpage}{206} +\indexentry{Closure|hyperpage}{206} +\indexentry{closure|hyperpage}{206} +\indexentry{dense|hyperpage}{207} +\indexentry{Boundary|hyperpage}{207} +\indexentry{boundary|hyperpage}{207} +\indexentry{neighbourhood|hyperpage}{207} +\indexentry{interior point|hyperpage}{207} +\indexentry{adherent point|hyperpage}{207} +\indexentry{Kuratowski's problem|hyperpage}{207} +\indexentry{separated|hyperpage}{207} +\indexentry{separated by closed neighbourhoods|hyperpage}{207} +\indexentry{Continuous function|hyperpage}{208} +\indexentry{homeomorphism|hyperpage}{208} +\indexentry{homeomorphic|hyperpage}{208} +\indexentry{Open function|hyperpage}{208} +\indexentry{open|hyperpage}{208} +\indexentry{Closed function|hyperpage}{208} +\indexentry{closed|hyperpage}{208} +\indexentry{topological subspace|hyperpage}{208} +\indexentry{subspace topology|hyperpage}{208} +\indexentry{Cantor set|hyperpage}{209} +\indexentry{product topology|hyperpage}{209} +\indexentry{box topology|hyperpage}{209} +\indexentry{infinite product topology|hyperpage}{209} +\indexentry{$n-1$-th sphere|hyperpage}{210} +\indexentry{$n$-th ball|hyperpage}{210} +\indexentry{Torus|hyperpage}{210} +\indexentry{torus|hyperpage}{210} +\indexentry{quotient topology|hyperpage}{210} +\indexentry{quotient map|hyperpage}{210} +\indexentry{quotient space|hyperpage}{210} +\indexentry{quotient space of collapsing a set to a point|hyperpage}{211} +\indexentry{action|hyperpage}{211} +\indexentry{$(G,\cdot )$-space|hyperpage}{211} +\indexentry{Möbius band|hyperpage}{211} +\indexentry{Klein bottle|hyperpage}{211} +\indexentry{$T_0$ space|hyperpage}{212} +\indexentry{Kolmogorov|hyperpage}{212} +\indexentry{$T_1$ space|hyperpage}{212} +\indexentry{Fréchet|hyperpage}{212} +\indexentry{$T_2$ space|hyperpage}{212} +\indexentry{Hausdorff|hyperpage}{212} +\indexentry{$T_{2\frac {1}{2}}$ space|hyperpage}{212} +\indexentry{regular|hyperpage}{212} +\indexentry{$T_3$ space|hyperpage}{212} +\indexentry{normal|hyperpage}{212} +\indexentry{$T_4$ space|hyperpage}{212} +\indexentry{Urysohn's lemma|hyperpage}{212} +\indexentry{Tietze extension theorem|hyperpage}{212} +\indexentry{metrizable|hyperpage}{212} +\indexentry{Urysohn's metrization theorem|hyperpage}{212} +\indexentry{topological property|hyperpage}{213} +\indexentry{Cover|hyperpage}{213} +\indexentry{cover|hyperpage}{213} +\indexentry{finite|hyperpage}{213} +\indexentry{countable|hyperpage}{213} +\indexentry{open cover|hyperpage}{213} +\indexentry{cover|hyperpage}{213} +\indexentry{subcover|hyperpage}{213} +\indexentry{Compact space|hyperpage}{213} +\indexentry{compact|hyperpage}{213} +\indexentry{compactness|hyperpage}{213} +\indexentry{compact subset|hyperpage}{213} +\indexentry{Tychonoff's theorem|hyperpage}{213} +\indexentry{Axiom of choice|hyperpage}{213} +\indexentry{Kelley's theorem|hyperpage}{213} +\indexentry{topological embedding|hyperpage}{214} +\indexentry{One-point compactification|hyperpage}{214} +\indexentry{one-point compactification|hyperpage}{214} +\indexentry{locally|hyperpage}{214} +\indexentry{locally compact|hyperpage}{214} +\indexentry{Heine-Borel theorem|hyperpage}{214} +\indexentry{Heine-Borel theorem|hyperpage}{214} +\indexentry{Weierstra\ss ' theorem|hyperpage}{214} +\indexentry{Connected space|hyperpage}{214} +\indexentry{connected|hyperpage}{214} +\indexentry{disconnected|hyperpage}{214} +\indexentry{connected subset|hyperpage}{214} +\indexentry{totally disconnected|hyperpage}{215} +\indexentry{Intermediate value theorem|hyperpage}{215} +\indexentry{Bolzano's theorem|hyperpage}{215} +\indexentry{Brouwer's fixed-point theorem|hyperpage}{215} +\indexentry{Borsuk-Ulam theorem|hyperpage}{215} +\indexentry{connected components|hyperpage}{215} +\indexentry{locally connected|hyperpage}{215} +\indexentry{path|hyperpage}{215} +\indexentry{initial point|hyperpage}{215} +\indexentry{terminal point|hyperpage}{215} +\indexentry{path from $x$ to $y$|hyperpage}{215} +\indexentry{loop|hyperpage}{215} +\indexentry{path-connected|hyperpage}{215} +\indexentry{path-connected|hyperpage}{216} +\indexentry{path-connected components|hyperpage}{216} +\indexentry{locally path-connected|hyperpage}{216} +\indexentry{simply connected|hyperpage}{216} +\indexentry{Topological manifold|hyperpage}{216} +\indexentry{topological manifold|hyperpage}{216} +\indexentry{coordinate chart|hyperpage}{216} +\indexentry{atlas|hyperpage}{216} +\indexentry{transition functions|hyperpage}{216} +\indexentry{piecewise linear manifold|hyperpage}{216} +\indexentry{differentiable manifold|hyperpage}{216} +\indexentry{holes|hyperpage}{216} +\indexentry{Connected sum|hyperpage}{217} +\indexentry{connected sum|hyperpage}{217} +\indexentry{same orientation|hyperpage}{217} +\indexentry{opposite orientations|hyperpage}{217} +\indexentry{orientation|hyperpage}{217} +\indexentry{oriented vector space|hyperpage}{217} +\indexentry{unoriented vector space|hyperpage}{217} +\indexentry{orientation-preserving|hyperpage}{217} +\indexentry{not orientation-preserving|hyperpage}{217} +\indexentry{orientation-preserving|hyperpage}{217} +\indexentry{not orientation-preserving|hyperpage}{217} +\indexentry{orientable|hyperpage}{217} +\indexentry{Classification of connected 1-manifolds|hyperpage}{217} +\indexentry{surface|hyperpage}{217} +\indexentry{Connected sum of surfaces|hyperpage}{217} +\indexentry{genus $g$ orientable surface|hyperpage}{217} +\indexentry{genus $h$ non-orientable surface|hyperpage}{217} +\indexentry{standard $n$-simplex|hyperpage}{218} +\indexentry{triangularization|hyperpage}{218} +\indexentry{simple curve|hyperpage}{218} +\indexentry{Jordan curve theorem|hyperpage}{218} +\indexentry{Radó theorem|hyperpage}{218} +\indexentry{fundamental polygon of the surface|hyperpage}{218} +\indexentry{normalized|hyperpage}{218} +\indexentry{Euler characteristic|hyperpage}{218} +\indexentry{Classification of compact connected surfaces|hyperpage}{219} +\indexentry{equilibrium point|hyperpage}{221} +\indexentry{fixed point|hyperpage}{221} +\indexentry{periodic orbit of period $T$|hyperpage}{221} +\indexentry{symmetric with respect to the $x$-axis|hyperpage}{221} +\indexentry{symmetric with respect to the $y$-axis|hyperpage}{221} +\indexentry{holomorphic differential equation|hyperpage}{221} +\indexentry{linearization|hyperpage}{221} +\indexentry{minimal|hyperpage}{221} +\indexentry{Kolmogorov system|hyperpage}{221} +\indexentry{homogeneous of degree $k$|hyperpage}{221} +\indexentry{homogeneous of degree $k$|hyperpage}{221} +\indexentry{discrete map of order $m$|hyperpage}{222} +\indexentry{Characteristic equation|hyperpage}{222} +\indexentry{linear discrete map|hyperpage}{222} +\indexentry{Hartman-Grobman theorem|hyperpage}{222} +\indexentry{The stable manifold theorem|hyperpage}{222} +\indexentry{stable manifold|hyperpage}{222} +\indexentry{unstable manifold|hyperpage}{222} +\indexentry{global stable manifold|hyperpage}{222} +\indexentry{global unstable manifold|hyperpage}{222} +\indexentry{The center manifold theorem|hyperpage}{223} +\indexentry{central manifolds|hyperpage}{223} +\indexentry{local bifurcation|hyperpage}{223} +\indexentry{normal form|hyperpage}{223} +\indexentry{codimension|hyperpage}{223} +\indexentry{Saddle-node bifurcation|hyperpage}{223} +\indexentry{saddle-node bifurcation|hyperpage}{223} +\indexentry{Transcritical bifurcation|hyperpage}{223} +\indexentry{transcritical bifurcation|hyperpage}{223} +\indexentry{Pitchfork bifurcation|hyperpage}{223} +\indexentry{pitchfork bifurcation|hyperpage}{223} +\indexentry{Lyapunov's theorem|hyperpage}{223} +\indexentry{Lyapunov's theorem|hyperpage}{223} +\indexentry{Semi-hyperbolic singular points classification theorem|hyperpage}{223} +\indexentry{saddle-node|hyperpage}{224} +\indexentry{nilpotent|hyperpage}{224} +\indexentry{Nilpotent singular points classification theorem|hyperpage}{224} +\indexentry{cusp|hyperpage}{224} +\indexentry{Blow-up in polar coordinates|hyperpage}{225} +\indexentry{blowing-up|hyperpage}{225} +\indexentry{blow-down|hyperpage}{225} +\indexentry{Blow-up in generalized polar coordinates|hyperpage}{225} +\indexentry{Blow-up in cartesian coordinates|hyperpage}{226} +\indexentry{Lyapunov's method|hyperpage}{226} +\indexentry{$n$-th Lyapunov constant|hyperpage}{226} +\indexentry{Hopf bifurcation theorem|hyperpage}{226} +\indexentry{Hopf-bifurcation|hyperpage}{226} +\indexentry{Poincaré's method|hyperpage}{226} +\indexentry{Bautin's theorem|hyperpage}{226} +\indexentry{Homoclinic bifurcation|hyperpage}{226} +\indexentry{homoclinic bifurcation|hyperpage}{226} +\indexentry{Bogdanov-Takens bifurcation|hyperpage}{227} +\indexentry{Bogdanov-Takens bifurcation|hyperpage}{227} +\indexentry{Routh-Hurwitz stability criterion|hyperpage}{227} +\indexentry{Routh-Hurwitz stability criterion|hyperpage}{227} +\indexentry{one-parameter family of rotated vector fields|hyperpage}{227} +\indexentry{positively rotated|hyperpage}{227} +\indexentry{negatively rotated|hyperpage}{227} +\indexentry{Melnikov's method|hyperpage}{227} +\indexentry{Melnikov's method|hyperpage}{227} +\indexentry{graph|hyperpage}{227} +\indexentry{non-degenerated|hyperpage}{227} +\indexentry{linear saddles|hyperpage}{227} +\indexentry{Liénard system|hyperpage}{228} +\indexentry{Liénard's theorem|hyperpage}{228} +\indexentry{Van der Pol oscillator|hyperpage}{228} +\indexentry{Lorenz system|hyperpage}{228} +\indexentry{periodic point|hyperpage}{228} +\indexentry{prime period|hyperpage}{228} +\indexentry{set of periodic points|hyperpage}{228} +\indexentry{set of periods|hyperpage}{228} +\indexentry{logistic map|hyperpage}{228} +\indexentry{Period-doubling bifurcation|hyperpage}{229} +\indexentry{period-doubling bifurcation|hyperpage}{229} +\indexentry{period-halving bifurcation|hyperpage}{229} +\indexentry{period-doubling cascade|hyperpage}{229} +\indexentry{Sharkovskii's order|hyperpage}{229} +\indexentry{Expansive fixed point theorem|hyperpage}{229} +\indexentry{Itinerary lemma|hyperpage}{229} +\indexentry{Sharkovskii's theorem|hyperpage}{229} +\indexentry{Period three theorem|hyperpage}{229} +\indexentry{Proximity theorem|hyperpage}{229} +\indexentry{Shift map|hyperpage}{229} +\indexentry{shift map|hyperpage}{229} +\indexentry{itinerary|hyperpage}{230} +\indexentry{topologically transitive|hyperpage}{230} +\indexentry{sensitive dependence on initial conditions|hyperpage}{230} +\indexentry{Chaos|hyperpage}{230} +\indexentry{chaotic|hyperpage}{230} +\indexentry{Poisson kernel|hyperpage}{231} +\indexentry{Poisson kernel|hyperpage}{231} +\indexentry{Fourier transform|hyperpage}{231} +\indexentry{inverse Fourier transform|hyperpage}{231} +\indexentry{Fourier transform operator|hyperpage}{232} +\indexentry{Riemann-Lebesgue lemma|hyperpage}{232} +\indexentry{Inversion theorem|hyperpage}{233} +\indexentry{partial inverse Fourier transform|hyperpage}{233} +\indexentry{Dirichlet kernel|hyperpage}{233} +\indexentry{Dirichlet kernel|hyperpage}{233} +\indexentry{Dini's theorem|hyperpage}{234} +\indexentry{Fejér mean|hyperpage}{234} +\indexentry{Fejér kernel|hyperpage}{234} +\indexentry{Poisson kernel|hyperpage}{234} +\indexentry{Weierstra\ss \ kernel|hyperpage}{234} +\indexentry{approximations of the identity|hyperpage}{235} +\indexentry{Minkowski's integral inequality|hyperpage}{235} +\indexentry{Plancherel theorem|hyperpage}{236} +\indexentry{Jensen's inequality|hyperpage}{236} +\indexentry{Generalized Hölder's inequality|hyperpage}{237} +\indexentry{Young's convolution inequality|hyperpage}{237} +\indexentry{Fourier transform|hyperpage}{237} +\indexentry{Fourier transform|hyperpage}{237} +\indexentry{inverse Fourier transform|hyperpage}{237} +\indexentry{Riemann-Lebesgue lemma|hyperpage}{238} +\indexentry{Plancherel theorem|hyperpage}{238} +\indexentry{Uncertainty principle|hyperpage}{238} +\indexentry{Uncertainty principle in $\ensuremath {\mathbb {R}}^n$|hyperpage}{238} +\indexentry{Poisson summation formula|hyperpage}{238} +\indexentry{bandlimited|hyperpage}{239} +\indexentry{Nyquist-Shannon sampling theorem|hyperpage}{239} +\indexentry{discrete Fourier transform|hyperpage}{239} +\indexentry{DFT|hyperpage}{239} +\indexentry{inverse discrete Fourier transform|hyperpage}{239} +\indexentry{Plancherel theorem|hyperpage}{240} +\indexentry{convolution|hyperpage}{240} +\indexentry{Poisson summation formula|hyperpage}{240} +\indexentry{fast Fourier transform|hyperpage}{240} +\indexentry{FFT|hyperpage}{240} +\indexentry{Radix-2 DIT Cooley-Tukey FFT algorithm|hyperpage}{240} +\indexentry{radix-2 decimation-in-time (DIT) FFT|hyperpage}{240} +\indexentry{bump functions|hyperpage}{240} +\indexentry{test functions|hyperpage}{240} +\indexentry{Distribution|hyperpage}{240} +\indexentry{distribution|hyperpage}{240} +\indexentry{Fundamental lemma of calculus of variations|hyperpage}{240} +\indexentry{regular distributions|hyperpage}{241} +\indexentry{Dirac's $\delta $ distribution|hyperpage}{241} +\indexentry{Dirac's $\delta $ distribution|hyperpage}{241} +\indexentry{Cauchy principal value|hyperpage}{241} +\indexentry{Cauchy principal value|hyperpage}{241} +\indexentry{distribution of order $N\in \ensuremath {\mathbb {N}}\cup \{0\}$|hyperpage}{241} +\indexentry{infinite order|hyperpage}{241} +\indexentry{approximation of identity|hyperpage}{241} +\indexentry{support|hyperpage}{242} +\indexentry{distributional derivative|hyperpage}{242} +\indexentry{Heaviside step function|hyperpage}{242} +\indexentry{Schwarz theorem|hyperpage}{243} +\indexentry{Leibnitz rule|hyperpage}{243} +\indexentry{Schwartz space|hyperpage}{243} +\indexentry{space of rapidly decreasing functions|hyperpage}{243} +\indexentry{tempered distribution|hyperpage}{244} +\indexentry{convolution|hyperpage}{244} +\indexentry{Fourier transform|hyperpage}{244} +\indexentry{inverse Fourier transform|hyperpage}{244} +\indexentry{direct product|hyperpage}{245} +\indexentry{Homogeneous distribution|hyperpage}{245} +\indexentry{homogeneous of degree $r\in \ensuremath {\mathbb {R}}$|hyperpage}{245} +\indexentry{differential operator over distributions|hyperpage}{245} +\indexentry{generalized solution|hyperpage}{245} +\indexentry{fundamental solution|hyperpage}{245} +\indexentry{fundamental solution|hyperpage}{246} +\indexentry{Malgrange-Ehrenpreis theorem|hyperpage}{247} +\indexentry{truncated Hilbert transform|hyperpage}{247} +\indexentry{Hilbert transform|hyperpage}{247} +\indexentry{Hölder condition|hyperpage}{247} +\indexentry{Riesz theorem|hyperpage}{248} +\indexentry{conjugate Poisson kernel|hyperpage}{248} +\indexentry{Riesz transform|hyperpage}{248} +\indexentry{Sample variance and covariance|hyperpage}{249} +\indexentry{sample covariance|hyperpage}{249} +\indexentry{sample variance|hyperpage}{249} +\indexentry{sample correlation coefficient|hyperpage}{249} +\indexentry{expectation|hyperpage}{249} +\indexentry{covariance matrix|hyperpage}{249} +\indexentry{covariance|hyperpage}{249} +\indexentry{Simple model|hyperpage}{250} +\indexentry{preditions|hyperpage}{250} +\indexentry{Least-squares method|hyperpage}{250} +\indexentry{least-squares method|hyperpage}{250} +\indexentry{residuals|hyperpage}{250} +\indexentry{General linear model|hyperpage}{250} +\indexentry{design matrix|hyperpage}{250} +\indexentry{regressor coefficients|hyperpage}{250} +\indexentry{regression coefficients|hyperpage}{250} +\indexentry{preditions|hyperpage}{250} +\indexentry{Least-squares method|hyperpage}{250} +\indexentry{MLE method|hyperpage}{251} +\indexentry{mean square error|hyperpage}{251} +\indexentry{Gau\ss -Markov|hyperpage}{251} +\indexentry{BLUE|hyperpage}{251} +\indexentry{the Best Linear Unbiased Estimator|hyperpage}{251} +\indexentry{no-intercept|hyperpage}{252} +\indexentry{coefficient of determination|hyperpage}{252} +\indexentry{regression sum of squares|hyperpage}{252} +\indexentry{total sum of squares|hyperpage}{252} +\indexentry{error sum of squares|hyperpage}{252} +\indexentry{adjusted coefficient of determination|hyperpage}{252} +\indexentry{total mean of squares|hyperpage}{252} +\indexentry{$F$-distribution with degrees of freedom $d_1$ and $d_2$|hyperpage}{252} +\indexentry{Cochran's theorem|hyperpage}{252} +\indexentry{Bonferroni's method|hyperpage}{252} +\indexentry{Model without interaction|hyperpage}{253} +\indexentry{dummy variable|hyperpage}{253} +\indexentry{Model with interaction|hyperpage}{253} +\indexentry{model with interaction|hyperpage}{253} +\indexentry{Segmented regression|hyperpage}{253} +\indexentry{response|hyperpage}{253} +\indexentry{prediction|hyperpage}{253} +\indexentry{average response|hyperpage}{253} +\indexentry{estimated average response|hyperpage}{253} +\indexentry{estimation of the prediction|hyperpage}{253} +\indexentry{Confidence interval for the average response|hyperpage}{253} +\indexentry{leverage|hyperpage}{253} +\indexentry{confidence band|hyperpage}{254} +\indexentry{Confidence interval for the predicted value|hyperpage}{254} +\indexentry{prediction band|hyperpage}{254} +\indexentry{LOF test|hyperpage}{254} +\indexentry{lack of fit test|hyperpage}{254} +\indexentry{LOF test|hyperpage}{254} +\indexentry{internally studentized residuals|hyperpage}{254} +\indexentry{externally studentized residuals|hyperpage}{254} +\indexentry{high-leverage points|hyperpage}{255} +\indexentry{outliars|hyperpage}{255} +\indexentry{high-leverage point|hyperpage}{255} +\indexentry{Cook's distance|hyperpage}{255} +\indexentry{multicollinearity|hyperpage}{255} +\indexentry{variance inflation factor|hyperpage}{255} +\indexentry{VIF|hyperpage}{255} +\indexentry{tolerance|hyperpage}{255} +\indexentry{Bias on the estimations|hyperpage}{255} +\indexentry{Bias on the predictions|hyperpage}{255} +\indexentry{Mallow's $C_p$ statistic|hyperpage}{256} +\indexentry{Mallow's $C_p$ statistic|hyperpage}{256} +\indexentry{Information|hyperpage}{256} +\indexentry{information of an event|hyperpage}{256} +\indexentry{Cauchy's functional equation|hyperpage}{256} +\indexentry{Entropy|hyperpage}{256} +\indexentry{entropy|hyperpage}{256} +\indexentry{Kullback-Leibler divergence|hyperpage}{256} +\indexentry{Akaike information criterion|hyperpage}{257} +\indexentry{Akaike information criterion|hyperpage}{257} +\indexentry{AIC|hyperpage}{257} +\indexentry{Box-Cox transformation|hyperpage}{257} +\indexentry{exponential|hyperpage}{257} +\indexentry{odds|hyperpage}{257} +\indexentry{log-odds|hyperpage}{257} +\indexentry{logit|hyperpage}{257} +\indexentry{well-posed in the Hadamard sense|hyperpage}{258} +\indexentry{well-posed|hyperpage}{258} +\indexentry{step size|hyperpage}{258} +\indexentry{mesh-points|hyperpage}{258} +\indexentry{explicit|hyperpage}{258} +\indexentry{implicit|hyperpage}{258} +\indexentry{one-step method|hyperpage}{258} +\indexentry{incremental function|hyperpage}{258} +\indexentry{local truncation errors|hyperpage}{258} +\indexentry{global truncation error|hyperpage}{258} +\indexentry{Euler method|hyperpage}{258} +\indexentry{forward Euler method|hyperpage}{258} +\indexentry{explicit Euler method|hyperpage}{258} +\indexentry{backward Euler method|hyperpage}{258} +\indexentry{implicit Euler method|hyperpage}{258} +\indexentry{implicit|hyperpage}{258} +\indexentry{Trapezoidal method|hyperpage}{259} +\indexentry{Trapezoidal method|hyperpage}{259} +\indexentry{Heun method|hyperpage}{259} +\indexentry{Heun method|hyperpage}{259} +\indexentry{Taylor method|hyperpage}{259} +\indexentry{Taylor method of order $r$|hyperpage}{259} +\indexentry{consistent|hyperpage}{259} +\indexentry{order of consistency|hyperpage}{259} +\indexentry{order of accuracy|hyperpage}{259} +\indexentry{order|hyperpage}{259} +\indexentry{convergent|hyperpage}{259} +\indexentry{order of convergence|hyperpage}{259} +\indexentry{$s$-stage Runge-Kutta methods|hyperpage}{260} +\indexentry{RK methods|hyperpage}{260} +\indexentry{Butcher tableau|hyperpage}{260} +\indexentry{Richardson extrapolation|hyperpage}{260} +\indexentry{Runge-Kutta-Fehlberg method|hyperpage}{260} +\indexentry{embedded methods|hyperpage}{260} +\indexentry{stability function|hyperpage}{261} +\indexentry{stable|hyperpage}{261} +\indexentry{absolutely stable|hyperpage}{261} +\indexentry{stability region|hyperpage}{261} +\indexentry{A-stable|hyperpage}{261} +\indexentry{unconditionally absolutely stable|hyperpage}{261} +\indexentry{conditionally absolutely stable|hyperpage}{261} +\indexentry{stiff equations|hyperpage}{261} +\indexentry{Lax theorem|hyperpage}{261} +\indexentry{test problem|hyperpage}{261} +\indexentry{Linear multistep method|hyperpage}{261} +\indexentry{linear $k$-step method|hyperpage}{261} +\indexentry{first and second characteristic polynomials|hyperpage}{261} +\indexentry{zero-stable|hyperpage}{262} +\indexentry{root condition|hyperpage}{262} +\indexentry{Adams method|hyperpage}{262} +\indexentry{Adams method|hyperpage}{262} +\indexentry{Adams-Bashforth method|hyperpage}{262} +\indexentry{Adams-Moulton method|hyperpage}{262} +\indexentry{local truncation error|hyperpage}{262} +\indexentry{consistent|hyperpage}{262} +\indexentry{order of consistency|hyperpage}{262} +\indexentry{order of accuracy|hyperpage}{262} +\indexentry{Dahlquist's theorem|hyperpage}{262} +\indexentry{Newton method|hyperpage}{262} +\indexentry{Newton method|hyperpage}{262} +\indexentry{quasi-Newton method|hyperpage}{263} +\indexentry{damping parameter|hyperpage}{263} +\indexentry{Broyden's method|hyperpage}{263} +\indexentry{Broyden's method|hyperpage}{263} +\indexentry{secant-like method|hyperpage}{263} +\indexentry{Descent method|hyperpage}{264} +\indexentry{descent method|hyperpage}{264} +\indexentry{step size|hyperpage}{264} +\indexentry{descent direction|hyperpage}{264} +\indexentry{Newton method|hyperpage}{264} +\indexentry{Inexact Newton method|hyperpage}{264} +\indexentry{Steepest descent|hyperpage}{264} +\indexentry{Gradient descent|hyperpage}{264} +\indexentry{Conjugate gradient method|hyperpage}{264} +\indexentry{line search method|hyperpage}{264} +\indexentry{Broyden-Fletcher-Goldfarb-Shanno method|hyperpage}{264} +\indexentry{Broyden-Fletcher-Goldfarb-Shanno method|hyperpage}{264} +\indexentry{BFGS method|hyperpage}{264} +\indexentry{Shooting method|hyperpage}{264} +\indexentry{shooting method|hyperpage}{264} +\indexentry{Multiple shooting method|hyperpage}{264} +\indexentry{multiple shooting method|hyperpage}{264} +\indexentry{finite difference method|hyperpage}{265} +\indexentry{Singular value|hyperpage}{265} +\indexentry{singular values|hyperpage}{265} +\indexentry{Singular value decomposition|hyperpage}{265} +\indexentry{singular value decomposition|hyperpage}{265} +\indexentry{SVD|hyperpage}{265} +\indexentry{left singular vectors|hyperpage}{265} +\indexentry{right singular vectors|hyperpage}{265} +\indexentry{truncated|hyperpage}{266} +\indexentry{truncated singular value decomposition|hyperpage}{266} +\indexentry{TSVD|hyperpage}{266} +\indexentry{QR descompostion|hyperpage}{267} +\indexentry{QR decomposition|hyperpage}{267} +\indexentry{thin QR decomposition|hyperpage}{267} +\indexentry{full QR decomposition|hyperpage}{267} +\indexentry{hyperbolic|hyperpage}{268} +\indexentry{grid|hyperpage}{268} +\indexentry{finite difference scheme|hyperpage}{268} +\indexentry{number of steps|hyperpage}{268} +\indexentry{one-step|hyperpage}{268} +\indexentry{multistep|hyperpage}{268} +\indexentry{Forward-time forward-space|hyperpage}{268} +\indexentry{FTFS|hyperpage}{268} +\indexentry{Forward-time backward-space|hyperpage}{268} +\indexentry{FTBS|hyperpage}{268} +\indexentry{Forward-time central-space|hyperpage}{268} +\indexentry{FTCS|hyperpage}{268} +\indexentry{Backward-time central-space|hyperpage}{268} +\indexentry{BTCS|hyperpage}{268} +\indexentry{Leapfrog scheme|hyperpage}{268} +\indexentry{Lax-Friedrichs scheme|hyperpage}{268} +\indexentry{stability region|hyperpage}{268} +\indexentry{unconditionally convergent|hyperpage}{268} +\indexentry{conditionally convergent|hyperpage}{269} +\indexentry{consistent|hyperpage}{269} +\indexentry{conditional consistency|hyperpage}{269} +\indexentry{upwind condition|hyperpage}{269} +\indexentry{stable|hyperpage}{269} +\indexentry{Courant-Friedrichs-Lewy condition|hyperpage}{270} +\indexentry{Semidiscrete Fourier transform|hyperpage}{270} +\indexentry{semidiscrete Fourier transform|hyperpage}{270} +\indexentry{inverse semidiscrete Fourier transform|hyperpage}{270} +\indexentry{Semidiscrete Parseval identity|hyperpage}{270} +\indexentry{amplification factor|hyperpage}{270} +\indexentry{Lax-Wendroff|hyperpage}{271} +\indexentry{Lax-Wendroff scheme|hyperpage}{271} +\indexentry{Crank-Nicolson|hyperpage}{272} +\indexentry{Crank-Nicolson scheme|hyperpage}{272} +\indexentry{numerical boundary condition|hyperpage}{272} +\indexentry{Lax-Richtmyer equivalence theorem|hyperpage}{273} +\indexentry{amplification polynomial|hyperpage}{273} +\indexentry{compatibility condition|hyperpage}{273} +\indexentry{elliptic|hyperpage}{273} +\indexentry{hyperbolic|hyperpage}{273} +\indexentry{parabolic|hyperpage}{273} +\indexentry{stable|hyperpage}{274} +\indexentry{Forward-time central-space|hyperpage}{274} +\indexentry{Backward-time central-space|hyperpage}{274} +\indexentry{Crank-Nicolson scheme|hyperpage}{274} +\indexentry{Leapfrog scheme|hyperpage}{274} +\indexentry{Du-Fort-Frankel scheme|hyperpage}{274} +\indexentry{Dirichlet|hyperpage}{274} +\indexentry{Neumann|hyperpage}{274} +\indexentry{Robin|hyperpage}{274} +\indexentry{discrete laplacian|hyperpage}{274} +\indexentry{Discrete maximum principle|hyperpage}{274} +\indexentry{Discrete minimum principle|hyperpage}{274} +\indexentry{finite element method|hyperpage}{275} +\indexentry{variational formulation|hyperpage}{275} +\indexentry{weak formulation|hyperpage}{275} +\indexentry{Galerkin approximation|hyperpage}{275} +\indexentry{Galerkin approximation|hyperpage}{275} +\indexentry{Mesh|hyperpage}{275} 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+\indexentry{potential|hyperpage}{277} +\indexentry{Poisson equation|hyperpage}{277} +\indexentry{elliptic equation|hyperpage}{277} +\indexentry{Fick's law of diffusion|hyperpage}{278} +\indexentry{diffusivity|hyperpage}{278} +\indexentry{diffusion coefficient|hyperpage}{278} +\indexentry{diffusion flux|hyperpage}{278} +\indexentry{Fick's law|hyperpage}{278} +\indexentry{Diffusion equation|hyperpage}{278} +\indexentry{Fourier's law|hyperpage}{278} +\indexentry{thermal conductivity|hyperpage}{278} +\indexentry{heat flux|hyperpage}{278} +\indexentry{Fourier's law|hyperpage}{278} +\indexentry{Heat equation|hyperpage}{278} +\indexentry{specific heat capacity|hyperpage}{278} +\indexentry{density|hyperpage}{278} +\indexentry{thermal diffusivity|hyperpage}{278} +\indexentry{Gau\ss ' law|hyperpage}{278} +\indexentry{Gau\ss ' law|hyperpage}{278} +\indexentry{Gau\ss ' law for magnetism|hyperpage}{278} +\indexentry{Gau\ss ' law for magnetism|hyperpage}{278} +\indexentry{Maxwell-Faraday equation|hyperpage}{278} +\indexentry{Maxwell-Faraday equation|hyperpage}{278} +\indexentry{Ampère-Maxwell circuital law|hyperpage}{278} +\indexentry{Ampère's law|hyperpage}{278} +\indexentry{displacement current|hyperpage}{278} +\indexentry{refractive index|hyperpage}{278} +\indexentry{Fermat's principle|hyperpage}{278} +\indexentry{Fermat's principle|hyperpage}{278} +\indexentry{Eikonal equation|hyperpage}{278} +\indexentry{action|hyperpage}{278} +\indexentry{Principle of least action|hyperpage}{279} +\indexentry{small change|hyperpage}{279} +\indexentry{Hamilton-Jacobi equation|hyperpage}{279} +\indexentry{Schrödinger equation|hyperpage}{279} +\indexentry{Schrödinger equation|hyperpage}{279} +\indexentry{wave function|hyperpage}{279} +\indexentry{Fundamental lemma of calculus of variations|hyperpage}{279} +\indexentry{Green identities|hyperpage}{279} +\indexentry{Method of characteristics|hyperpage}{279} +\indexentry{characteristic curves|hyperpage}{279} +\indexentry{Traffic flow equation|hyperpage}{280} +\indexentry{Rankine-Hugoniot equation|hyperpage}{280} +\indexentry{Entropy condition|hyperpage}{280} +\indexentry{D'Alembert formula|hyperpage}{281} +\indexentry{Conservation of energy|hyperpage}{282} +\indexentry{even periodic extension|hyperpage}{283} +\indexentry{odd periodic extension|hyperpage}{283} +\indexentry{Separation of variables|hyperpage}{283} +\indexentry{separation of variables|hyperpage}{283} +\indexentry{Sturm-Picone comparison theorem|hyperpage}{283} +\indexentry{self-similar|hyperpage}{284} +\indexentry{Distribution|hyperpage}{285} +\indexentry{distribution|hyperpage}{285} +\indexentry{Dirac's $\delta $ distribution|hyperpage}{285} +\indexentry{differentiation operator|hyperpage}{285} +\indexentry{distributional derivative|hyperpage}{285} +\indexentry{Heaviside step function|hyperpage}{285} +\indexentry{fundamental solution|hyperpage}{285} +\indexentry{heat kernel|hyperpage}{285} +\indexentry{generalized heat kernel|hyperpage}{286} +\indexentry{Explicit scheme in finite differences|hyperpage}{286} +\indexentry{Implicit scheme in finite differences|hyperpage}{286} +\indexentry{Duhamel principle|hyperpage}{287} +\indexentry{parabolic cylinder|hyperpage}{287} +\indexentry{parabolic boundary|hyperpage}{287} +\indexentry{Maximum principle|hyperpage}{287} +\indexentry{Minimum principle|hyperpage}{287} +\indexentry{Uniqueness of the heat equation|hyperpage}{287} +\indexentry{Maximum principle on unbounded domains|hyperpage}{287} +\indexentry{Minimum principle on unbounded domains|hyperpage}{288} +\indexentry{Uniqueness of the heat equation on the unbounded domains|hyperpage}{288} +\indexentry{Laplace equation|hyperpage}{288} +\indexentry{Laplace equation|hyperpage}{288} +\indexentry{Dirichlet problem in the disc|hyperpage}{288} +\indexentry{Dirichlet problem|hyperpage}{288} +\indexentry{Dirichlet problem|hyperpage}{288} +\indexentry{Uniqueness of Dirichlet problem|hyperpage}{288} +\indexentry{energy functional|hyperpage}{288} +\indexentry{Dirichlet's principle|hyperpage}{288} +\indexentry{Sobolev space|hyperpage}{289} +\indexentry{average|hyperpage}{289} +\indexentry{Trace theorem|hyperpage}{289} +\indexentry{trace|hyperpage}{289} +\indexentry{Poincaré inequality|hyperpage}{289} +\indexentry{$\sigma $-algebra|hyperpage}{291} +\indexentry{$\sigma $-algebra|hyperpage}{291} +\indexentry{Measure|hyperpage}{291} +\indexentry{measure|hyperpage}{291} +\indexentry{$\sigma $-additivity|hyperpage}{291} +\indexentry{interval|hyperpage}{291} +\indexentry{volume|hyperpage}{291} +\indexentry{$m$-th dyadic cube|hyperpage}{291} +\indexentry{Outer measure|hyperpage}{292} +\indexentry{outer measure|hyperpage}{292} +\indexentry{null set|hyperpage}{293} +\indexentry{almost everywhere|hyperpage}{293} +\indexentry{a.e.|hyperpage}{293} +\indexentry{Lebesgue measure|hyperpage}{293} +\indexentry{Lebesgue measurable|hyperpage}{293} +\indexentry{measurable|hyperpage}{293} +\indexentry{Lebesgue measure|hyperpage}{293} +\indexentry{real function|hyperpage}{294} +\indexentry{finite|hyperpage}{294} +\indexentry{Lebesgue measurable|hyperpage}{294} +\indexentry{measurable|hyperpage}{294} +\indexentry{Borel $\sigma $-algebra|hyperpage}{295} +\indexentry{Borel measurable|hyperpage}{295} +\indexentry{simple function|hyperpage}{295} +\indexentry{integral of $s$ over $\ensuremath {\mathbb {R}}^n$|hyperpage}{295} +\indexentry{integral of $s$ over a measurable set $E$|hyperpage}{295} +\indexentry{integral of $f$ over $\ensuremath {\mathbb {R}}^n$|hyperpage}{296} +\indexentry{integral of $f$ over a measurable set $E\subseteq \ensuremath {\mathbb {R}}^n$|hyperpage}{296} +\indexentry{Monotone convergence theorem|hyperpage}{296} +\indexentry{Chebyshev's inequality|hyperpage}{297} +\indexentry{Fatou's lemma|hyperpage}{297} +\indexentry{integral of $f$ over $E$|hyperpage}{297} +\indexentry{integrable function over $E$|hyperpage}{297} +\indexentry{Dominated convergence theorem|hyperpage}{297} +\indexentry{converge in mean|hyperpage}{298} +\indexentry{exists and it is finite|hyperpage}{298} +\indexentry{Mean value theorem for integrals|hyperpage}{298} +\indexentry{Barrow's law|hyperpage}{298} +\indexentry{Fundamental theorem of calculus|hyperpage}{298} +\indexentry{Integration by parts|hyperpage}{298} +\indexentry{Hardy-Littlewood maximal function|hyperpage}{298} +\indexentry{Hardy-Littlewood maximal function|hyperpage}{298} +\indexentry{Lebesgue differentiation theorem|hyperpage}{299} +\indexentry{section|hyperpage}{299} +\indexentry{Tonelli's theorem|hyperpage}{299} +\indexentry{Fubini's theorem|hyperpage}{300} +\indexentry{change of variables|hyperpage}{300} +\indexentry{Change of variables|hyperpage}{300} +\indexentry{distance|hyperpage}{300} +\indexentry{triangular inequality|hyperpage}{300} +\indexentry{metric space|hyperpage}{300} +\indexentry{complete|hyperpage}{300} +\indexentry{norm|hyperpage}{300} +\indexentry{triangular inequality|hyperpage}{300} +\indexentry{normed vector space|hyperpage}{300} +\indexentry{convergent series|hyperpage}{300} +\indexentry{absolutely convergent|hyperpage}{300} +\indexentry{Banach space|hyperpage}{301} +\indexentry{Banach space|hyperpage}{301} +\indexentry{uniform norm|hyperpage}{301} +\indexentry{total subset|hyperpage}{301} +\indexentry{separable|hyperpage}{301} +\indexentry{quotient space|hyperpage}{302} +\indexentry{Young's inequality for products|hyperpage}{302} +\indexentry{Hölder conjugates|hyperpage}{302} +\indexentry{Hölder's inequality|hyperpage}{302} +\indexentry{Hölder's inequality|hyperpage}{303} +\indexentry{Interpolation inequality|hyperpage}{303} +\indexentry{Minkowski inequality|hyperpage}{303} +\indexentry{uniform norm|hyperpage}{303} +\indexentry{uniform norm|hyperpage}{303} +\indexentry{supremum norm|hyperpage}{303} +\indexentry{subalgebra|hyperpage}{303} +\indexentry{separating set|hyperpage}{303} +\indexentry{separate the points|hyperpage}{303} +\indexentry{vanishes nowhere|hyperpage}{304} +\indexentry{self-conjugate|hyperpage}{304} +\indexentry{Stone-Weierstra\ss \ theorem|hyperpage}{304} +\indexentry{pointwise bounded|hyperpage}{304} +\indexentry{locally bounded|hyperpage}{304} +\indexentry{uniformly bounded|hyperpage}{304} +\indexentry{equicontinuous at a point|hyperpage}{304} +\indexentry{pointwise equicontinuous|hyperpage}{304} +\indexentry{uniformly equicontinuous|hyperpage}{304} +\indexentry{relatively compact|hyperpage}{304} +\indexentry{Arzelà-Ascoli theorem|hyperpage}{304} +\indexentry{operator|hyperpage}{305} +\indexentry{norm|hyperpage}{305} +\indexentry{sublinear|hyperpage}{306} +\indexentry{Marcinkiewicz interpolation theorem|hyperpage}{306} +\indexentry{topological homeomorphism|hyperpage}{306} +\indexentry{isomorphic|hyperpage}{306} +\indexentry{is finer than|hyperpage}{306} +\indexentry{equivalent|hyperpage}{306} +\indexentry{Almost orthogonality lemma|hyperpage}{306} +\indexentry{Riesz's theorem|hyperpage}{306} +\indexentry{normed algebra|hyperpage}{307} +\indexentry{dual space|hyperpage}{307} +\indexentry{bidual space|hyperpage}{307} +\indexentry{reflexive|hyperpage}{307} +\indexentry{compact operator|hyperpage}{307} +\indexentry{Fredholm operator with kernel $K$|hyperpage}{307} +\indexentry{Volterra operator with kernel $K$|hyperpage}{307} +\indexentry{Hilbert-Schmidt operator with kernel $K$|hyperpage}{307} +\indexentry{finite-rank operator|hyperpage}{307} +\indexentry{Neumann series|hyperpage}{308} +\indexentry{convex functional|hyperpage}{308} +\indexentry{Hahn-Banach theorem|hyperpage}{308} +\indexentry{extension|hyperpage}{308} +\indexentry{Seminorm|hyperpage}{308} +\indexentry{seminorm|hyperpage}{308} +\indexentry{Hahn-Banach theorem|hyperpage}{308} +\indexentry{Hahn-Banach theorem|hyperpage}{308} +\indexentry{reflexive|hyperpage}{308} +\indexentry{dual map|hyperpage}{308} +\indexentry{proper subspace of $T$|hyperpage}{309} +\indexentry{eigenvectors|hyperpage}{309} +\indexentry{eigenvalue|hyperpage}{309} +\indexentry{spectrum|hyperpage}{309} +\indexentry{spectral values|hyperpage}{309} +\indexentry{Baire's theorem|hyperpage}{309} +\indexentry{Open mapping theorem|hyperpage}{309} +\indexentry{Closed graph theorem|hyperpage}{309} +\indexentry{Banach-Steinhaus theorem|hyperpage}{310} +\indexentry{semilinear|hyperpage}{310} +\indexentry{inner product|hyperpage}{310} +\indexentry{pre-Hilbert space|hyperpage}{310} +\indexentry{Cauchy-Schwarz inequality|hyperpage}{310} +\indexentry{Minkowski inequality|hyperpage}{310} +\indexentry{Polarization identity|hyperpage}{310} +\indexentry{orthogonal|hyperpage}{310} +\indexentry{orthogonal complement|hyperpage}{310} +\indexentry{Pythagorean theorem|hyperpage}{310} +\indexentry{Parallelogram law|hyperpage}{311} +\indexentry{Hilbert space|hyperpage}{311} +\indexentry{Hilbert space|hyperpage}{311} +\indexentry{minimizer|hyperpage}{311} +\indexentry{Projection theorem|hyperpage}{311} +\indexentry{orthogonal projection on $F$|hyperpage}{311} +\indexentry{Riesz representation theorem|hyperpage}{312} +\indexentry{adjoint operator|hyperpage}{312} +\indexentry{self-adjoint|hyperpage}{312} +\indexentry{continuous|hyperpage}{313} +\indexentry{coercive|hyperpage}{313} +\indexentry{symmetric|hyperpage}{313} +\indexentry{Lax-Milgram theorem|hyperpage}{313} +\indexentry{orthogonal system|hyperpage}{313} +\indexentry{orthonormal system|hyperpage}{313} +\indexentry{complete|hyperpage}{313} +\indexentry{Hilbert basis|hyperpage}{313} +\indexentry{Fourier coefficients|hyperpage}{313} +\indexentry{Fourier series|hyperpage}{313} +\indexentry{Gram-Schmidt process|hyperpage}{313} +\indexentry{orthonormalization|hyperpage}{313} +\indexentry{Bessel's inequality|hyperpage}{313} +\indexentry{Fourier transform|hyperpage}{313} +\indexentry{Riesz-Fischer theorem|hyperpage}{313} +\indexentry{Parseval identity|hyperpage}{313} +\indexentry{Spectral theorem|hyperpage}{314} +\indexentry{Hilbert-Schmidt spectral representation theorem|hyperpage}{314} +\indexentry{Fredholm alternative|hyperpage}{314} +\indexentry{Law of total probability|hyperpage}{315} +\indexentry{Substitution principle|hyperpage}{315} +\indexentry{Law of total expectation|hyperpage}{315} +\indexentry{Wald theorem|hyperpage}{315} +\indexentry{probability-generating function|hyperpage}{316} +\indexentry{pgf|hyperpage}{316} +\indexentry{Stochastic process|hyperpage}{316} +\indexentry{stochastic process|hyperpage}{316} +\indexentry{parameter set|hyperpage}{316} +\indexentry{state space|hyperpage}{316} +\indexentry{independent|hyperpage}{316} +\indexentry{Gambler's ruin problem|hyperpage}{318} +\indexentry{Markov chain|hyperpage}{318} +\indexentry{Markov property|hyperpage}{318} +\indexentry{time-homogeneous Markov chain|hyperpage}{318} +\indexentry{state space|hyperpage}{318} +\indexentry{states|hyperpage}{318} +\indexentry{Stochastic matrix|hyperpage}{318} +\indexentry{stochastic matrix|hyperpage}{318} +\indexentry{transition probabilities|hyperpage}{318} +\indexentry{transition matrix|hyperpage}{318} +\indexentry{initial distribution|hyperpage}{318} +\indexentry{Random walk|hyperpage}{319} +\indexentry{random walk|hyperpage}{319} +\indexentry{simple random walk|hyperpage}{319} +\indexentry{$n$-step transition probabilities|hyperpage}{319} +\indexentry{$n$-step transition matrix|hyperpage}{319} +\indexentry{$n$-step distribution|hyperpage}{319} +\indexentry{Chapman-Kolmogorov equation|hyperpage}{320} +\indexentry{reachable|hyperpage}{320} +\indexentry{communicate|hyperpage}{320} +\indexentry{irreducible class|hyperpage}{320} +\indexentry{irreducible chain|hyperpage}{320} +\indexentry{period|hyperpage}{320} +\indexentry{aperiodic|hyperpage}{320} +\indexentry{aperiodic|hyperpage}{320} +\indexentry{filtration|hyperpage}{321} +\indexentry{filtration space|hyperpage}{321} +\indexentry{$\sigma $-algebra generated by $\boldsymbol {\mathrm {X}}$|hyperpage}{321} +\indexentry{stopping time|hyperpage}{321} +\indexentry{Strong Markov property|hyperpage}{321} +\indexentry{Strong Markov property|hyperpage}{322} +\indexentry{transient|hyperpage}{322} +\indexentry{recurrent|hyperpage}{322} +\indexentry{$k$-th hitting time|hyperpage}{322} +\indexentry{recurrent|hyperpage}{324} +\indexentry{transient|hyperpage}{324} +\indexentry{Polya's theorem on $\ensuremath {\mathbb {Z}}$|hyperpage}{324} +\indexentry{Polya's theorem on $\ensuremath {\mathbb {Z}}^2$|hyperpage}{324} +\indexentry{Polya's theorem on $\ensuremath {\mathbb {Z}}^3$|hyperpage}{324} +\indexentry{positive recurrent|hyperpage}{324} +\indexentry{null recurrent|hyperpage}{324} +\indexentry{Ergotic theorem|hyperpage}{324} +\indexentry{Ergotic theorem|hyperpage}{325} +\indexentry{stationary distribution|hyperpage}{325} +\indexentry{continuous-time Markov chain|hyperpage}{326} +\indexentry{transition probabilities|hyperpage}{326} +\indexentry{Chapman-Kolmogorov equation|hyperpage}{326} +\indexentry{Poisson process|hyperpage}{326} +\indexentry{independent increments|hyperpage}{326} +\indexentry{stationary increments|hyperpage}{326} +\indexentry{trajectories|hyperpage}{326} +\indexentry{càd|hyperpage}{326} +\indexentry{holding times|hyperpage}{327} +\indexentry{inter-arrival times|hyperpage}{327} +\indexentry{infinitesimal generator|hyperpage}{328} +\indexentry{infinitesimal transition scheme|hyperpage}{328} +\indexentry{Kolmogorov's backward equation|hyperpage}{328} +\indexentry{Kolmogorov's forward equation|hyperpage}{328} +\indexentry{jump process|hyperpage}{328} +\indexentry{regular|hyperpage}{328} +\indexentry{stable|hyperpage}{328} +\indexentry{conservative|hyperpage}{328} +\indexentry{stationary distribution|hyperpage}{329} +\indexentry{balance equation|hyperpage}{329} +\indexentry{irreducible|hyperpage}{329} +\indexentry{limit distribution|hyperpage}{329} +\indexentry{birth and death process|hyperpage}{329} +\indexentry{pure death process|hyperpage}{329} +\indexentry{pure birth process|hyperpage}{329} +\indexentry{Reuter criterion|hyperpage}{330} +\indexentry{Gaussian vector|hyperpage}{330} +\indexentry{Gaussian process|hyperpage}{330} +\indexentry{mean function|hyperpage}{330} +\indexentry{covariance function|hyperpage}{330} +\indexentry{Brownian motion|hyperpage}{330} +\indexentry{Wiener process|hyperpage}{330} +\indexentry{standard|hyperpage}{330} +\indexentry{stochastically equivalent|hyperpage}{331} +\indexentry{version|hyperpage}{331} +\indexentry{indistinguishable|hyperpage}{331} +\indexentry{Kolmogorov's continuity theorem|hyperpage}{331} +\indexentry{Paley-Wiener-Zygmund theorem|hyperpage}{331} +\indexentry{$d$-dimensional standard Brownian motion|hyperpage}{332} +\indexentry{Law of the iterated logarithm|hyperpage}{333} +\indexentry{Finite-dimensional distributions|hyperpage}{333} +\indexentry{finite-dimensional distributions|hyperpage}{333} +\indexentry{consistency condition|hyperpage}{333} +\indexentry{Kolmogorov extension theorem|hyperpage}{333} +\indexentry{shift map|hyperpage}{335} +\indexentry{expansion map|hyperpage}{335} +\indexentry{hyperbolicity|hyperpage}{336} +\indexentry{mixing|hyperpage}{336} +\indexentry{$\mathcal {C}^0$-close|hyperpage}{336} +\indexentry{$\mathcal {C}^1$-close|hyperpage}{336} +\indexentry{Structal stability|hyperpage}{336} +\indexentry{conjugacy equation|hyperpage}{336} +\indexentry{sensitive dependence on initial conditions|hyperpage}{336} +\indexentry{Lyapunov exponent|hyperpage}{336} +\indexentry{chaotic|hyperpage}{337} +\indexentry{Hamiltonian vector field|hyperpage}{337} +\indexentry{circle|hyperpage}{337} +\indexentry{Existence of a lift|hyperpage}{337} +\indexentry{lift|hyperpage}{337} +\indexentry{preserves orientation|hyperpage}{337} +\indexentry{Arnold family|hyperpage}{337} +\indexentry{subadditive|hyperpage}{338} +\indexentry{superadditive|hyperpage}{338} +\indexentry{Existence of the rotation number|hyperpage}{338} +\indexentry{rotation number|hyperpage}{338} +\indexentry{$\sigma $-algebra|hyperpage}{339} +\indexentry{$\sigma $-algebra|hyperpage}{339} +\indexentry{measurable space|hyperpage}{339} +\indexentry{$\sigma $-algebra generated by $\mathcal {F}$|hyperpage}{339} +\indexentry{measurable|hyperpage}{339} +\indexentry{Measure|hyperpage}{339} +\indexentry{measure|hyperpage}{339} +\indexentry{measure space|hyperpage}{339} +\indexentry{integral of $f$ with respect to $\mu $|hyperpage}{339} +\indexentry{integral of $f$ with respect to $\mu $|hyperpage}{339} +\indexentry{Monotone convergence theorem|hyperpage}{339} +\indexentry{Fatou's lemma|hyperpage}{339} +\indexentry{Dominated convergence theorem|hyperpage}{339} +\indexentry{Product measure|hyperpage}{339} +\indexentry{product measure|hyperpage}{339} +\indexentry{$\sigma $-finite|hyperpage}{339} +\indexentry{Fubini|hyperpage}{339} +\indexentry{density|hyperpage}{340} +\indexentry{absolutely continuous|hyperpage}{340} +\indexentry{Radon-Nikodym|hyperpage}{340} +\indexentry{probability space|hyperpage}{340} +\indexentry{events|hyperpage}{340} +\indexentry{Random variable|hyperpage}{340} +\indexentry{$E$-valued random variable|hyperpage}{340} +\indexentry{Expectation|hyperpage}{340} +\indexentry{expectation of $X$|hyperpage}{340} +\indexentry{law of $X$|hyperpage}{340} +\indexentry{$\sigma $-algebra generated by $X$|hyperpage}{340} +\indexentry{Jensen's inequality|hyperpage}{340} +\indexentry{conditional expectation of $X$ given $\mathcal {G}$|hyperpage}{340} +\indexentry{Tower property|hyperpage}{341} +\indexentry{conditional expectation of $X$ given $Y$|hyperpage}{341} +\indexentry{probability kernel|hyperpage}{341} +\indexentry{conditional law of $X$ given $Y$|hyperpage}{341} +\indexentry{filtration|hyperpage}{341} +\indexentry{filtered probability space|hyperpage}{341} +\indexentry{adapted|hyperpage}{341} +\indexentry{martingale|hyperpage}{341} +\indexentry{submartingale|hyperpage}{341} +\indexentry{supermartingale|hyperpage}{341} +\indexentry{stopping time|hyperpage}{341} +\indexentry{stopped process|hyperpage}{341} +\indexentry{uniformly integrable|hyperpage}{341} +\indexentry{converges weakly|hyperpage}{342} +\indexentry{converges strongly|hyperpage}{342} +\indexentry{converges weakly-*|hyperpage}{342} +\indexentry{Banach-Alaoglu theorem|hyperpage}{342} +\indexentry{strongly lower-semicontinuous|hyperpage}{342} +\indexentry{weakly lower-semicontinuity|hyperpage}{342} +\indexentry{weak-* lower-semicontinuity|hyperpage}{342} +\indexentry{support plane|hyperpage}{342} +\indexentry{Sobolev spaces|hyperpage}{343} +\indexentry{Sobolev spaces|hyperpage}{343} +\indexentry{Poincaré's inequality|hyperpage}{343} +\indexentry{Poincaré-Wirtinger's inequality|hyperpage}{343} +\indexentry{embedded|hyperpage}{344} +\indexentry{compactly embedded|hyperpage}{344} +\indexentry{Gagliardo, Nirengerg and Sobolev's inequality|hyperpage}{344} +\indexentry{Hölder continuity|hyperpage}{344} +\indexentry{$\mathcal {C}^{k,\theta }$-Hölder continuous|hyperpage}{344} +\indexentry{Morrey's embedding|hyperpage}{344} +\indexentry{extension|hyperpage}{344} +\indexentry{extension operator|hyperpage}{344} +\indexentry{first order reflection|hyperpage}{345} +\indexentry{Reillich-Kondrachov's compactness theorem|hyperpage}{345} +\indexentry{trace operator|hyperpage}{345} +\indexentry{Gradient descent algorithm|hyperpage}{346} +\indexentry{gradient descent algorithm|hyperpage}{346} +\indexentry{Armijo-type rule|hyperpage}{346} +\indexentry{Frank-Wolfe-type method|hyperpage}{346} +\indexentry{$L$-co-coercive|hyperpage}{346} +\indexentry{firmly non-expansive|hyperpage}{346} +\indexentry{Baillon-Haddad|hyperpage}{346} +\indexentry{strongly convex|hyperpage}{346} +\indexentry{$\gamma $-convex|hyperpage}{346} +\indexentry{Newton method|hyperpage}{347} +\indexentry{Newton method|hyperpage}{347} +\indexentry{multistep method|hyperpage}{347} +\indexentry{Heavy ball method|hyperpage}{347} +\indexentry{Conjugate gradient|hyperpage}{347} +\indexentry{conjugate gradient method|hyperpage}{347} +\indexentry{Nesterov's accelerated gradient method|hyperpage}{347} +\indexentry{Nesterov's method|hyperpage}{347} +\indexentry{subgradient descent method|hyperpage}{347} +\indexentry{implicit descent|hyperpage}{347} +\indexentry{nonexpansive|hyperpage}{347} +\indexentry{avergaed|hyperpage}{347} +\indexentry{Opial's lemma|hyperpage}{347} +\indexentry{Krasnoselskii-Mann's convergence theorem|hyperpage}{348} +\indexentry{linear second-order PDE|hyperpage}{349} +\indexentry{non-divergence form|hyperpage}{349} +\indexentry{divergence form|hyperpage}{349} +\indexentry{Dirichlet boundary condition|hyperpage}{349} +\indexentry{homogeneous|hyperpage}{349} +\indexentry{Neumann boundary condition|hyperpage}{349} +\indexentry{homogeneous|hyperpage}{349} +\indexentry{uniformly elliptic|hyperpage}{349} +\indexentry{Abstract Fredholm alternative|hyperpage}{349} +\indexentry{weak formulation|hyperpage}{349} +\indexentry{formal adjoint|hyperpage}{349} +\indexentry{homogeneous adjoint problem|hyperpage}{349} +\indexentry{resolvent set|hyperpage}{350} +\indexentry{spectrum|hyperpage}{350} +\indexentry{Inner regularity|hyperpage}{350} +\indexentry{Regularity up to the boundary|hyperpage}{350} +\indexentry{Weak maximum principle|hyperpage}{350} +\indexentry{Weak maximum principle|hyperpage}{350} +\indexentry{strong solution|hyperpage}{351} +\indexentry{classical solution|hyperpage}{351} +\indexentry{Weak maximum principle|hyperpage}{351} +\indexentry{Hopf's lemma|hyperpage}{351} +\indexentry{interior ball condition|hyperpage}{351} +\indexentry{Strong maximum principle|hyperpage}{351} +\indexentry{Continuation method|hyperpage}{351} +\indexentry{Montecarlo estimator|hyperpage}{352} +\indexentry{seed|hyperpage}{352} +\indexentry{Mersenne Twister algorithm|hyperpage}{352} +\indexentry{Acceptance-rejection method|hyperpage}{352} +\indexentry{Box-Muller method|hyperpage}{353} +\indexentry{Polar method|hyperpage}{353} +\indexentry{antithetic method|hyperpage}{353} +\indexentry{control variate|hyperpage}{353} +\indexentry{multiple control variate estimator|hyperpage}{354} +\indexentry{equivalent|hyperpage}{354} +\indexentry{importance sampling estimator|hyperpage}{354} +\indexentry{Euler method|hyperpage}{355} +\indexentry{continuous Euler scheme|hyperpage}{355} +\indexentry{Strong error of the Euler scheme|hyperpage}{355} +\indexentry{Weak error of the Euler scheme|hyperpage}{355} +\indexentry{Romberg Extrapolation|hyperpage}{355} +\indexentry{Romberg Extrapolation|hyperpage}{355} +\indexentry{Dirichlet boundary conditions|hyperpage}{356} +\indexentry{Neumann boundary conditions|hyperpage}{356} +\indexentry{Robin boundary conditions|hyperpage}{356} +\indexentry{conforming Galerkin method|hyperpage}{356} +\indexentry{Céa's lemma|hyperpage}{356} +\indexentry{finite element|hyperpage}{356} +\indexentry{basis functions|hyperpage}{357} +\indexentry{local interpolant|hyperpage}{357} +\indexentry{subdivision|hyperpage}{357} +\indexentry{global interpolant|hyperpage}{357} +\indexentry{triangulation|hyperpage}{357} +\indexentry{affinely equivalent|hyperpage}{357} +\indexentry{Bramble-Hilbert lemma|hyperpage}{357} +\indexentry{diameter|hyperpage}{357} +\indexentry{insphere diameter|hyperpage}{357} +\indexentry{condition number|hyperpage}{357} +\indexentry{Local interpolation error|hyperpage}{358} +\indexentry{regular|hyperpage}{358} +\indexentry{Global interpolation error|hyperpage}{358} +\indexentry{jointly Gaussian|hyperpage}{359} +\indexentry{Brownian motion|hyperpage}{359} +\indexentry{Strong law of large numbers for Brownian motion|hyperpage}{359} +\indexentry{Markov property for Brownian motion|hyperpage}{359} +\indexentry{natural filtration|hyperpage}{359} +\indexentry{Martingale|hyperpage}{359} +\indexentry{martingale|hyperpage}{359} +\indexentry{adapted|hyperpage}{359} +\indexentry{sub-martingale|hyperpage}{359} +\indexentry{super-martingale|hyperpage}{359} +\indexentry{hitting time|hyperpage}{360} +\indexentry{Doob's optional sampling theorem|hyperpage}{360} +\indexentry{stopped process|hyperpage}{360} +\indexentry{Orthogonality of martingales|hyperpage}{360} +\indexentry{Doob's maximal inequality|hyperpage}{360} +\indexentry{absoulte variation|hyperpage}{360} +\indexentry{finite variation|hyperpage}{360} +\indexentry{Quadratic variation|hyperpage}{360} +\indexentry{mesh|hyperpage}{360} +\indexentry{continuous local martingale|hyperpage}{361} +\indexentry{localizing sequence|hyperpage}{361} +\indexentry{Doob's optional sampling theorem for local martingales|hyperpage}{361} +\indexentry{Levy's characterization of Brownian motion|hyperpage}{362} +\indexentry{isometry|hyperpage}{362} +\indexentry{partial isometry|hyperpage}{362} +\indexentry{Wiener integral|hyperpage}{362} +\indexentry{Wiener isometry|hyperpage}{362} +\indexentry{Wiener integral|hyperpage}{362} +\indexentry{Wiener integral|hyperpage}{362} +\indexentry{Chasles relation|hyperpage}{362} +\indexentry{progressive|hyperpage}{363} +\indexentry{Itô integral|hyperpage}{364} +\indexentry{Itô isometry|hyperpage}{364} +\indexentry{Itô integral|hyperpage}{364} +\indexentry{generalized Itô integral|hyperpage}{364} +\indexentry{Stochastic dominated convergence theorem|hyperpage}{364} +\indexentry{Itô process|hyperpage}{365} +\indexentry{martingale term|hyperpage}{365} +\indexentry{drift term|hyperpage}{365} +\indexentry{stochastic differential|hyperpage}{365} +\indexentry{quadratic variation|hyperpage}{365} +\indexentry{Stochastic integration by parts|hyperpage}{366} +\indexentry{Itô term|hyperpage}{366} +\indexentry{Itô's formula|hyperpage}{366} +\indexentry{Doléans-Dade exponential|hyperpage}{366} +\indexentry{Novikov's condition|hyperpage}{366} +\indexentry{Giranov's theorem|hyperpage}{367} +\indexentry{drift|hyperpage}{367} +\indexentry{diffusion|hyperpage}{367} +\indexentry{stochastic differential equation|hyperpage}{367} +\indexentry{SDE|hyperpage}{367} +\indexentry{solution of the SDE|hyperpage}{367} +\indexentry{Gronwall's lemma|hyperpage}{367} +\indexentry{Existence and uniqueness of solutions of SDEs|hyperpage}{367} +\indexentry{Langevin equation|hyperpage}{367} +\indexentry{Geometric Brownian motion|hyperpage}{368} +\indexentry{Black-Scholes process|hyperpage}{368} +\indexentry{homogeneous SDE|hyperpage}{368} +\indexentry{diffusions|hyperpage}{368} +\indexentry{Invariance under time shift|hyperpage}{368} +\indexentry{Generator|hyperpage}{368} +\indexentry{generator|hyperpage}{368} +\indexentry{Kolmogorov's equation|hyperpage}{368} +\indexentry{Feynman-Kac's formula|hyperpage}{369} +\indexentry{General Itô's formula|hyperpage}{370} diff --git a/main_math.ilg b/main_math.ilg new file mode 100644 index 0000000..fca4743 --- /dev/null +++ b/main_math.ilg @@ -0,0 +1,6 @@ +This is makeindex, version 2.17 [TeX Live 2023] (kpathsea + Thai support). +Scanning input file main_math.idx.......done (3301 entries accepted, 0 rejected). +Sorting entries................................done (42230 comparisons). +Generating output file main_math.ind.......done (2816 lines written, 0 warnings). +Output written in main_math.ind. +Transcript written in main_math.ilg. diff --git a/main_math.ind b/main_math.ind new file mode 100644 index 0000000..747ebec --- /dev/null +++ b/main_math.ind @@ -0,0 +1,2816 @@ +\begin{theindex} + + \item $(G,\cdot )$-space, \hyperpage{211} + \item $E$-valued random variable, \hyperpage{340} + \item $F$-distribution with degrees of freedom $d_1$ and $d_2$, + \hyperpage{252} + \item $F$-linearly independent, \hyperpage{170} + \item $K$-field morphism, \hyperpage{165} + \item $L$-co-coercive, \hyperpage{346} + \item $N$-th partial sum of $Sf$, \hyperpage{83} + \item $N$-th partial sum of the series, \hyperpage{76}, + \hyperpage{105} + \item $P$ and $Q$ are both true, \hyperpage{8} + \item $P$ and $Q$ have the same truth-value, \hyperpage{8} + \item $Q$ is true whenever $P$ is true, \hyperpage{8} + \item $T$-periodic, \hyperpage{81} + \item $T_0$ space, \hyperpage{212} + \item $T_1$ space, \hyperpage{212} + \item $T_2$ space, \hyperpage{212} + \item $T_3$ space, \hyperpage{212} + \item $T_4$ space, \hyperpage{212} + \item $T_{2\frac {1}{2}}$ space, \hyperpage{212} + \item $\alpha $-limit point, \hyperpage{131} + \item $\alpha $-limit set, \hyperpage{131} + \item $\ensuremath {\mathbb {C}}$-differentiable, \hyperpage{109} + \item $\ensuremath {\mathbb {R}}$-differentiable, \hyperpage{110} + \item $\gamma $-convex, \hyperpage{346} + \item $\mathcal {C}^0$-close, \hyperpage{336} + \item $\mathcal {C}^1$-close, \hyperpage{336} + \item $\mathcal {C}^{k,\theta }$-Hölder continuous, \hyperpage{344} + \item $\omega $-limit point, \hyperpage{131} + \item $\omega $-limit set, \hyperpage{131} + \item $\sigma $-additivity, \hyperpage{173}, \hyperpage{291} + \item $\sigma $-algebra, \hyperpage{172}, \hyperpage{291}, + \hyperpage{339} + \item $\sigma $-algebra generated, \hyperpage{172} + \item $\sigma $-algebra generated by $X$, \hyperpage{340} + \item $\sigma $-algebra generated by $\boldsymbol {\mathrm {X}}$, + \hyperpage{321} + \item $\sigma $-algebra generated by $\mathcal {F}$, \hyperpage{339} + \item $\sigma $-algebra of all Lebesgue measurable sets in $\ensuremath {\mathbb {R}}^n$, + \hyperpage{175} + \item $\sigma $-finite, \hyperpage{339} + \item $d$-dimensional standard Brownian motion, \hyperpage{332} + \item $i$-th pivot, \hyperpage{14} + \item $k$-linear map, \hyperpage{155} + \item $k$-th hitting time, \hyperpage{322} + \item $m$-dimensional projective subvariety, \hyperpage{67} + \item $m$-th dyadic cube, \hyperpage{291} + \item $n$-dimensional projective space, \hyperpage{67} + \item $n$-dimensional volume, \hyperpage{58, 59} + \item $n$-periodic, \hyperpage{90} + \item $n$-step distribution, \hyperpage{319} + \item $n$-step transition matrix, \hyperpage{319} + \item $n$-step transition probabilities, \hyperpage{319} + \item $n$-th Fourier coefficient, \hyperpage{83} + \item $n$-th Lyapunov constant, \hyperpage{226} + \item $n$-th ball, \hyperpage{210} + \item $n$-th cyclotomic polynomial, \hyperpage{164} + \item $n$-th derivative of $f$ at the point $a\in \ensuremath {\mathbb {R}}$, + \hyperpage{31} + \item $n$-th primitive root of unity, \hyperpage{170} + \item $n$-th root of unity, \hyperpage{170} + \item $n-1$-th sphere, \hyperpage{210} + \item $p$-group, \hyperpage{41} + \item $p$-value, \hyperpage{199} + \item $s$-stage Runge-Kutta methods, \hyperpage{260} + \item $t$-test, \hyperpage{201} + \item $x$-simple, \hyperpage{59} + \item $xy$-simple, \hyperpage{59} + \item $xz$-simple, \hyperpage{59} + \item $y$-simple, \hyperpage{59} + \item $yz$-simple, \hyperpage{59} + \item (left) $G$-set, \hyperpage{40} + \item (left) group action, \hyperpage{40} + \item 2-forms, \hyperpage{62} + \item 3-forms, \hyperpage{62} + + \indexspace + + \item A-stable, \hyperpage{261} + \item a.e., \hyperpage{293} + \item Abel's summation formula, \hyperpage{77}, \hyperpage{106} + \item Abel's test, \hyperpage{77, 78}, \hyperpage{80}, + \hyperpage{107} + \item Abel's theorem, \hyperpage{79}, \hyperpage{108} + \item Abel-Ruffini theorem, \hyperpage{171} + \item abelian, \hyperpage{37}, \hyperpage{170} + \item absolute condition numbers, \hyperpage{89} + \item absolute error, \hyperpage{89} + \item Absolute geometry, \hyperpage{64} + \item absolute value, \hyperpage{25} + \item absolutely continuous, \hyperpage{178}, \hyperpage{180}, + \hyperpage{340} + \item absolutely convergent, \hyperpage{77}, \hyperpage{106}, + \hyperpage{117}, \hyperpage{300} + \item absolutely stable, \hyperpage{261} + \item absoulte variation, \hyperpage{360} + \item Abstract Fredholm alternative, \hyperpage{349} + \item Acceptance-rejection method, \hyperpage{352} + \item acceptation region, \hyperpage{199} + \item ACCP, \hyperpage{46} + \item accumulation point, \hyperpage{26}, \hyperpage{54} + \item action, \hyperpage{211}, \hyperpage{278} + \item Adams method, \hyperpage{262} + \item Adams-Bashforth method, \hyperpage{262} + \item Adams-Moulton method, \hyperpage{262} + \item adapted, \hyperpage{341}, \hyperpage{359} + \item adherence, \hyperpage{54} + \item adherent point, \hyperpage{54}, \hyperpage{207} + \item adjacency matrix, \hyperpage{50} + \item adjacent, \hyperpage{49} + \item adjoint, \hyperpage{24} + \item adjoint operator, \hyperpage{312} + \item adjugate matrix, \hyperpage{15} + \item adjusted coefficient of determination, \hyperpage{252} + \item affine combination, \hyperpage{69} + \item affine coordinates, \hyperpage{70} + \item Affine Gra\ss mann formulas, \hyperpage{70} + \item affine plane, \hyperpage{66} + \item affine space, \hyperpage{69} + \item affine subvariety, \hyperpage{70} + \item affinely equivalent, \hyperpage{357} + \item affinely independents, \hyperpage{69} + \item affinity, \hyperpage{70} + \item AIC, \hyperpage{257} + \item Aitken's $\Delta ^2$ method, \hyperpage{91} + \item Akaike information criterion, \hyperpage{257} + \item Algebra, \hyperpage{172} + \item algebra, \hyperpage{172} + \item algebraic, \hyperpage{164, 165} + \item algebraic closure, \hyperpage{170} + \item algebraic curve, \hyperpage{135} + \item algebraic multiplicity, \hyperpage{20} + \item algebraic value, \hyperpage{153} + \item algebraically closed, \hyperpage{170} + \item algebraically closed field, \hyperpage{21} + \item almost everywhere, \hyperpage{293} + \item Almost orthogonality lemma, \hyperpage{306} + \item alternating, \hyperpage{155} + \item alternating group, \hyperpage{42} + \item alternating series, \hyperpage{77} + \item alternative hypothesis, \hyperpage{199} + \item amplification factor, \hyperpage{270} + \item amplification polynomial, \hyperpage{273} + \item Ampère's law, \hyperpage{278} + \item Ampère-Maxwell circuital law, \hyperpage{278} + \item an asymptotically normal estimator, \hyperpage{194} + \item analytic, \hyperpage{79}, \hyperpage{113} + \item analytic at $a$, \hyperpage{32} + \item Analytic continuation theorem, \hyperpage{114} + \item angle, \hyperpage{24}, \hyperpage{63} + \item angle-preserving, \hyperpage{149} + \item anisotropic, \hyperpage{74} + \item annihilator, \hyperpage{19} + \item antithetic method, \hyperpage{353} + \item aperiodic, \hyperpage{320} + \item approximation of identity, \hyperpage{79}, \hyperpage{241} + \item approximations of the identity, \hyperpage{235} + \item arc length, \hyperpage{60} + \item arc length parameter, \hyperpage{60} + \item arc-length function, \hyperpage{142} + \item arc-length parameter, \hyperpage{141} + \item Archimedean, \hyperpage{64} + \item Archimedean property, \hyperpage{25} + \item argument, \hyperpage{104} + \item Argument principle, \hyperpage{118} + \item Armijo-type rule, \hyperpage{346} + \item Arnold family, \hyperpage{337} + \item Artin's lemma, \hyperpage{169} + \item Arzelà-Ascoli theorem, \hyperpage{121}, \hyperpage{125}, + \hyperpage{304} + \item ASA criterion, \hyperpage{64} + \item ascending chain condition on principal ideals, \hyperpage{46} + \item associated, \hyperpage{45} + \item associated multivalued function, \hyperpage{110} + \item Associative property, \hyperpage{76} + \item Asymptotic behaviour of the LRT, \hyperpage{200} + \item asymptotic direction, \hyperpage{150} + \item asymptotic efficient estimator, \hyperpage{196} + \item asymptotic error constant, \hyperpage{91} + \item asymptotic line, \hyperpage{150} + \item asymptotically stable, \hyperpage{131} + \item asymptotically unbiased estimator, \hyperpage{194} + \item atlas, \hyperpage{216} + \item attracting, \hyperpage{131} + \item attracting parabolic sector, \hyperpage{133} + \item attractor fixed point, \hyperpage{91} + \item auto-adjoint, \hyperpage{24} + \item autonomous, \hyperpage{122} + \item average, \hyperpage{289} + \item average response, \hyperpage{253} + \item avergaed, \hyperpage{347} + \item Axiom of Archimedes, \hyperpage{63} + \item Axiom of choice, \hyperpage{213} + \item Axiom of completeness, \hyperpage{63} + \item Axiom of extensionality, \hyperpage{6} + \item Axiom of Parallels, \hyperpage{64} + \item axiomatic projective space, \hyperpage{66} + \item axis of perspectivity, \hyperpage{68} + + \indexspace + + \item backward Euler method, \hyperpage{258} + \item Backward-time central-space, \hyperpage{268}, \hyperpage{274} + \item Baillon-Haddad, \hyperpage{346} + \item Baire's theorem, \hyperpage{309} + \item balance equation, \hyperpage{329} + \item ball, \hyperpage{205} + \item Banach fixed-point theorem, \hyperpage{125} + \item Banach space, \hyperpage{301} + \item Banach-Alaoglu theorem, \hyperpage{342} + \item Banach-Steinhaus theorem, \hyperpage{310} + \item bandlimited, \hyperpage{239} + \item Barrow's law, \hyperpage{298} + \item barycenter, \hyperpage{69} + \item Basic bootstrap confidence interval, \hyperpage{202} + \item basic bootstrap confidence interval, \hyperpage{202} + \item basic feasible solutions, \hyperpage{51} + \item basin, \hyperpage{131} + \item basis, \hyperpage{16}, \hyperpage{206} + \item basis functions, \hyperpage{357} + \item Bautin's theorem, \hyperpage{226} + \item Bayes estimate, \hyperpage{203} + \item Bayes' formula, \hyperpage{174} + \item Bayesian point estimates, \hyperpage{203} + \item Bendixson's theorem, \hyperpage{137} + \item Bendixson-Dulac theorem, \hyperpage{137} + \item Bernoulli differential equation, \hyperpage{123} + \item Bernoulli distribution, \hyperpage{177} + \item Bernoulli numbers, \hyperpage{96} + \item Bernoulli polynomials, \hyperpage{96} + \item Bessel's inequality, \hyperpage{87}, \hyperpage{313} + \item Beta distribution, \hyperpage{179} + \item beta distribution, \hyperpage{179} + \item beta function, \hyperpage{179} + \item BFGS method, \hyperpage{264} + \item Bias, \hyperpage{193} + \item bias, \hyperpage{193} + \item Bias on the estimations, \hyperpage{255} + \item Bias on the predictions, \hyperpage{255} + \item biased estimator, \hyperpage{193} + \item bidual space, \hyperpage{307} + \item bijective, \hyperpage{7} + \item bilinear, \hyperpage{22} + \item bilinear form, \hyperpage{22} + \item Binomial coefficient formulas, \hyperpage{9} + \item Binomial distribution, \hyperpage{177} + \item binomial distribution, \hyperpage{177} + \item Binomial theorem, \hyperpage{119} + \item binormal vector, \hyperpage{143} + \item bipartite, \hyperpage{50} + \item birth and death process, \hyperpage{329} + \item Bisection method, \hyperpage{90} + \item Black-Scholes process, \hyperpage{368} + \item block matrix, \hyperpage{15} + \item blow-down, \hyperpage{225} + \item Blow-up in cartesian coordinates, \hyperpage{226} + \item Blow-up in generalized polar coordinates, \hyperpage{225} + \item Blow-up in polar coordinates, \hyperpage{225} + \item blowing-up, \hyperpage{225} + \item BLUE, \hyperpage{251} + \item Bogdanov-Takens bifurcation, \hyperpage{227} + \item Bolzano's theorem, \hyperpage{28}, \hyperpage{215} + \item Bolzano-Weierstra\ss \ theorem, \hyperpage{27}, \hyperpage{54} + \item Bonferroni's method, \hyperpage{252} + \item Bonnet's theorem, \hyperpage{152} + \item Boolean ring, \hyperpage{42} + \item bootstrap distribution, \hyperpage{202} + \item Bootstrap-t confidence interval, \hyperpage{202} + \item bootstrap-t confidence interval, \hyperpage{202} + \item Borel $\sigma $-algebra, \hyperpage{172}, \hyperpage{295} + \item Borel measurable, \hyperpage{181}, \hyperpage{295} + \item Borel's theorem, \hyperpage{191} + \item Borsuk-Ulam theorem, \hyperpage{215} + \item Boundary, \hyperpage{207} + \item boundary, \hyperpage{54}, \hyperpage{157}, \hyperpage{207} + \item boundary point, \hyperpage{54} + \item bounded, \hyperpage{26, 27}, \hyperpage{105} + \item bounded by compact sets, \hyperpage{121} + \item bounded from above, \hyperpage{25, 26} + \item bounded from below, \hyperpage{25, 26} + \item bounded set, \hyperpage{54} + \item box topology, \hyperpage{209} + \item Box-Cox transformation, \hyperpage{257} + \item Box-Muller method, \hyperpage{353} + \item Bramble-Hilbert lemma, \hyperpage{357} + \item bridge, \hyperpage{50} + \item Brouwer's fixed-point theorem, \hyperpage{215} + \item Brownian motion, \hyperpage{330}, \hyperpage{359} + \item Broyden's method, \hyperpage{263} + \item Broyden-Fletcher-Goldfarb-Shanno method, \hyperpage{264} + \item BTCS, \hyperpage{268} + \item Budan-Fourier theorem, \hyperpage{92} + \item bump functions, \hyperpage{240} + \item Butcher tableau, \hyperpage{260} + \item Bézout's theorem, \hyperpage{10, 11} + + \indexspace + + \item Calculation of integrals, \hyperpage{117} + \item canonical form of a linear programming to maximize, + \hyperpage{51} + \item canonical form of a linear programming to minimize, + \hyperpage{51} + \item Cantor set, \hyperpage{209} + \item Cantor's theorem, \hyperpage{9} + \item Cantor-Bernstein theorem, \hyperpage{9} + \item Cardano-Vieta's formulas, \hyperpage{114} + \item cardinal, \hyperpage{6} + \item Cartesian equations, \hyperpage{70} + \item Cartesian form, \hyperpage{104} + \item Cartesian geometry, \hyperpage{64} + \item Cartesian product, \hyperpage{7} + \item Casorati-Weierstra\ss \ theorem, \hyperpage{118} + \item Cauchy, \hyperpage{105} + \item Cauchy condition in $p$-th mean, \hyperpage{189} + \item Cauchy condition in probability, \hyperpage{187} + \item Cauchy distribution, \hyperpage{179} + \item Cauchy in $p$-th mean, \hyperpage{189} + \item Cauchy in probability, \hyperpage{187} + \item Cauchy momentum equation, \hyperpage{277} + \item Cauchy principal value, \hyperpage{241} + \item Cauchy problem, \hyperpage{122} + \item Cauchy product, \hyperpage{106} + \item Cauchy sequence, \hyperpage{27}, \hyperpage{54} + \item Cauchy's functional equation, \hyperpage{256} + \item Cauchy's inequality, \hyperpage{114} + \item Cauchy's integral theorem, \hyperpage{116} + \item Cauchy's mean value theorem, \hyperpage{30} + \item Cauchy's test, \hyperpage{76}, \hyperpage{78}, \hyperpage{80}, + \hyperpage{107} + \item Cauchy's theorem, \hyperpage{41} + \item Cauchy-Hadamard theorem, \hyperpage{107} + \item Cauchy-Riemann equations, \hyperpage{111} + \item Cauchy-Riemann theorem, \hyperpage{111} + \item Cauchy-Schwarz inequality, \hyperpage{24}, \hyperpage{53}, + \hyperpage{82}, \hyperpage{185}, \hyperpage{310} + \item Cavalieri's principle, \hyperpage{59} + \item Cayley's theorem, \hyperpage{39} + \item Cayley-Hamilton theorem, \hyperpage{21} + \item CC, \hyperpage{64} + \item cdf, \hyperpage{176} + \item cells, \hyperpage{275} + \item Center, \hyperpage{120} + \item center, \hyperpage{41}, \hyperpage{131}, \hyperpage{133} + \item Center of a group, \hyperpage{41} + \item Center of mass, \hyperpage{59} + \item center of mass, \hyperpage{59} + \item Center of masses, \hyperpage{35} + \item center of masses, \hyperpage{35} + \item central manifolds, \hyperpage{223} + \item centre of perspectivity, \hyperpage{68} + \item Cesàro summable, \hyperpage{85} + \item chain, \hyperpage{126} + \item chain of paths, \hyperpage{115} + \item Chain rule, \hyperpage{29}, \hyperpage{56}, \hyperpage{109} + \item Change of basis formula, \hyperpage{18} + \item change of parameter, \hyperpage{141} + \item Change of variable theorem, \hyperpage{60} + \item Change of variables, \hyperpage{159}, \hyperpage{300} + \item change of variables, \hyperpage{300} + \item change-of-basis matrix, \hyperpage{18} + \item Chaos, \hyperpage{230} + \item chaotic, \hyperpage{230}, \hyperpage{337} + \item chaotic behavior, \hyperpage{91} + \item Chapman-Kolmogorov equation, \hyperpage{320}, \hyperpage{326} + \item characteristic, \hyperpage{162} + \item characteristic curves, \hyperpage{279} + \item Characteristic equation, \hyperpage{123}, \hyperpage{222} + \item characteristic equation, \hyperpage{123} + \item Characteristic function, \hyperpage{191} + \item characteristic function, \hyperpage{9}, \hyperpage{191} + \item characteristic polynomial, \hyperpage{20}, \hyperpage{49} + \item Chasles relation, \hyperpage{362} + \item Chebyshev method, \hyperpage{90} + \item Chebyshev polynomials, \hyperpage{96} + \item Chebyshev's inequality, \hyperpage{186}, \hyperpage{297} + \item chi-squared distribution with $n$ degrees of freedom, + \hyperpage{197} + \item Chinese remainder theorem, \hyperpage{11} + \item Christoffel symbols, \hyperpage{151} + \item circle, \hyperpage{120}, \hyperpage{337} + \item circulation, \hyperpage{61}, \hyperpage{159} + \item class, \hyperpage{56} + \item class $\mathcal {C}^\infty $ at $a$, \hyperpage{31} + \item class $\mathcal {C}^n$ at a point $a\in \ensuremath {\mathbb {R}}$, + \hyperpage{31} + \item class $\mathcal {C}^p$, \hyperpage{31} + \item classical solution, \hyperpage{351} + \item Classification of affine quadrics, \hyperpage{75} + \item Classification of compact connected surfaces, \hyperpage{219} + \item Classification of connected 1-manifolds, \hyperpage{217} + \item Classification of cyclic groups, \hyperpage{38} + \item Classification of Euclidean motions, \hyperpage{73} + \item Classification of isometries, \hyperpage{73} + \item Classification of projective qua\-drics in $\ensuremath {\mathbb {C}}$, + \hyperpage{74} + \item Classification of projective qua\-drics in $\ensuremath {\mathbb {F}}_q$, + \hyperpage{75} + \item Classification of projective qua\-drics in $\ensuremath {\mathbb {R}}$, + \hyperpage{75} + \item Classification of quadratic forms in $\ensuremath {\mathbb {C}}$, + \hyperpage{74} + \item Classification of quadratic forms in $\ensuremath {\mathbb {F}}_q$, + \hyperpage{74} + \item Classification of quadratic forms in $\ensuremath {\mathbb {R}}$, + \hyperpage{74} + \item clopen, \hyperpage{206} + \item closed, \hyperpage{49}, \hyperpage{54}, \hyperpage{156}, + \hyperpage{206}, \hyperpage{208} + \item closed ball, \hyperpage{54} + \item Closed forms, \hyperpage{48} + \item Closed function, \hyperpage{208} + \item Closed graph theorem, \hyperpage{309} + \item Closed interval, \hyperpage{25} + \item Closed Newton-Cotes Formulas, \hyperpage{95} + \item closed trail, \hyperpage{49} + \item Closure, \hyperpage{206} + \item closure, \hyperpage{206} + \item Cochran's theorem, \hyperpage{252} + \item Cocountable topology, \hyperpage{206} + \item Codazzi-Mainardi equations, \hyperpage{152} + \item codimension, \hyperpage{223} + \item coefficient of determination, \hyperpage{252} + \item coefficients, \hyperpage{11} + \item coercive, \hyperpage{313} + \item cofactor, \hyperpage{136} + \item cofactor matrix, \hyperpage{15} + \item Cofinite topology, \hyperpage{206} + \item collineation, \hyperpage{66} + \item column rank, \hyperpage{16} + \item Combinations with repetition, \hyperpage{10} + \item Combinations without repetition, \hyperpage{10} + \item communicate, \hyperpage{320} + \item commutative, \hyperpage{37} + \item compact, \hyperpage{54}, \hyperpage{213} + \item compact operator, \hyperpage{307} + \item Compact space, \hyperpage{213} + \item compact subset, \hyperpage{213} + \item compact support, \hyperpage{79} + \item compactly embedded, \hyperpage{344} + \item compactness, \hyperpage{213} + \item Comparison test, \hyperpage{76}, \hyperpage{80}, + \hyperpage{183} + \item compatibility condition, \hyperpage{273} + \item compatible, \hyperpage{149}, \hyperpage{158} + \item compatible with, \hyperpage{97} + \item complement, \hyperpage{6} + \item Complementary property, \hyperpage{52} + \item complementary subspace, \hyperpage{17} + \item complete, \hyperpage{54}, \hyperpage{300}, \hyperpage{313} + \item complete graph, \hyperpage{49} + \item Complete pivoting, \hyperpage{102} + \item complex conjugate, \hyperpage{24}, \hyperpage{104} + \item complex cosecant, \hyperpage{109} + \item complex cosine, \hyperpage{108} + \item complex cotangent, \hyperpage{109} + \item complex exponential function, \hyperpage{108} + \item complex function, \hyperpage{106} + \item complex hyperbolic cosecant, \hyperpage{109} + \item complex hyperbolic cosine, \hyperpage{109} + \item complex hyperbolic cotangent, \hyperpage{109} + \item complex hyperbolic secant, \hyperpage{109} + \item complex hyperbolic sine, \hyperpage{109} + \item complex hyperbolic tangent, \hyperpage{109} + \item complex natural logarithm, \hyperpage{108} + \item complex power series, \hyperpage{107} + \item complex random variable, \hyperpage{191} + \item complex secant, \hyperpage{109} + \item complex sine, \hyperpage{108} + \item complex tangent, \hyperpage{109} + \item Composite Simpson's rule, \hyperpage{96} + \item Composite Trapezoidal rule, \hyperpage{95} + \item composition, \hyperpage{7} + \item compositum, \hyperpage{165} + \item Compound probability formula, \hyperpage{174} + \item computational efficiency, \hyperpage{91} + \item concave, \hyperpage{30} + \item Condensation test, \hyperpage{76} + \item condition number, \hyperpage{99}, \hyperpage{357} + \item conditional consistency, \hyperpage{269} + \item conditional expectation, \hyperpage{186} + \item conditional expectation of $X$ given $\mathcal {G}$, + \hyperpage{340} + \item conditional expectation of $X$ given $Y$, \hyperpage{341} + \item conditional law of $X$ given $Y$, \hyperpage{341} + \item conditional probability, \hyperpage{174} + \item conditional probability density function, \hyperpage{182} + \item conditional probability mass function, \hyperpage{182} + \item conditionally absolutely stable, \hyperpage{261} + \item conditionally convergent, \hyperpage{269} + \item cone, \hyperpage{108} + \item confidence band, \hyperpage{254} + \item Confidence interval for the average response, \hyperpage{253} + \item Confidence interval for the predicted value, \hyperpage{254} + \item confidence intervals, \hyperpage{198} + \item confidence level, \hyperpage{198} + \item confidence region, \hyperpage{198} + \item configuration, \hyperpage{67} + \item conformal, \hyperpage{119}, \hyperpage{149} + \item conformal representation, \hyperpage{119} + \item conforming Galerkin method, \hyperpage{356} + \item Congruence axioms, \hyperpage{63} + \item congruence relation, \hyperpage{63} + \item conic, \hyperpage{73} + \item conjugacy, \hyperpage{132} + \item conjugacy equation, \hyperpage{336} + \item conjugate, \hyperpage{104}, \hyperpage{132}, \hyperpage{202} + \item conjugate dynamical systems, \hyperpage{131} + \item Conjugate gradient, \hyperpage{347} + \item Conjugate gradient method, \hyperpage{264} + \item conjugate gradient method, \hyperpage{347} + \item conjugate Poisson kernel, \hyperpage{248} + \item conjugation action, \hyperpage{41} + \item connected, \hyperpage{50}, \hyperpage{54}, \hyperpage{214} + \item connected components, \hyperpage{50}, \hyperpage{215} + \item Connected space, \hyperpage{214} + \item connected subset, \hyperpage{214} + \item Connected sum, \hyperpage{217} + \item connected sum, \hyperpage{217} + \item Connected sum of surfaces, \hyperpage{217} + \item Conservation of energy, \hyperpage{282} + \item conservative, \hyperpage{61}, \hyperpage{328} + \item consistency condition, \hyperpage{333} + \item consistent, \hyperpage{259}, \hyperpage{262}, \hyperpage{269} + \item consistent estimator in $L^2$, \hyperpage{194} + \item constant coefficients, \hyperpage{122}, \hyperpage{126} + \item constant term, \hyperpage{11}, \hyperpage{13} + \item constructible field, \hyperpage{64} + \item Construction of a non-Paschian geometry, \hyperpage{65} + \item Construction of a non-SAS geometry, \hyperpage{66} + \item contact, \hyperpage{144} + \item contact of order $\geq n$ at $a$, \hyperpage{31} + \item Continuation method, \hyperpage{351} + \item Continuity axioms, \hyperpage{63} + \item Continuity correction, \hyperpage{192} + \item continuity correction, \hyperpage{192} + \item Continuity from above, \hyperpage{173} + \item Continuity from below, \hyperpage{173} + \item continuous, \hyperpage{28}, \hyperpage{55}, \hyperpage{106}, + \hyperpage{130}, \hyperpage{313} + \item continuous at $x_0$, \hyperpage{28} + \item Continuous equation, \hyperpage{277} + \item continuous equation, \hyperpage{277} + \item continuous Euler scheme, \hyperpage{355} + \item Continuous function, \hyperpage{208} + \item continuous local martingale, \hyperpage{361} + \item Continuous memorylessness property, \hyperpage{178} + \item Continuous uniform distribution, \hyperpage{178} + \item continuous uniform distribution, \hyperpage{178} + \item continuous-time Markov chain, \hyperpage{326} + \item contractible, \hyperpage{139} + \item contraction, \hyperpage{55}, \hyperpage{90} + \item contrast matrix, \hyperpage{201} + \item control variate, \hyperpage{353} + \item converge in mean, \hyperpage{298} + \item converge in norm $L^p$, \hyperpage{87} + \item convergent, \hyperpage{26}, \hyperpage{54}, \hyperpage{76}, + \hyperpage{79}, \hyperpage{98}, \hyperpage{105, 106}, + \hyperpage{117}, \hyperpage{259} + \item convergent series, \hyperpage{300} + \item converges, \hyperpage{105} + \item converges absolutely, \hyperpage{80} + \item converges almost surely, \hyperpage{188} + \item converges in distribution, \hyperpage{190} + \item converges in probability, \hyperpage{187} + \item converges in the $p$-th mean, \hyperpage{189} + \item converges pointwise, \hyperpage{77, 78}, \hyperpage{106, 107} + \item converges strongly, \hyperpage{342} + \item converges uniformly, \hyperpage{77, 78}, \hyperpage{106, 107} + \item converges weakly, \hyperpage{342} + \item converges weakly-*, \hyperpage{342} + \item convex, \hyperpage{30}, \hyperpage{115} + \item convex functional, \hyperpage{308} + \item convolution, \hyperpage{79}, \hyperpage{240}, \hyperpage{244} + \item Cook's distance, \hyperpage{255} + \item coordinate chart, \hyperpage{147}, \hyperpage{216} + \item coordinates, \hyperpage{16} + \item Coordination theorem, \hyperpage{17}, \hyperpage{68} + \item coprime, \hyperpage{10, 11} + \item corrected sample variance, \hyperpage{193} + \item correlation coefficient, \hyperpage{185} + \item Correspondence theorem, \hyperpage{40}, \hyperpage{44} + \item countable, \hyperpage{25}, \hyperpage{213} + \item Countable subadditivity, \hyperpage{173} + \item Courant-Friedrichs-Lewy condition, \hyperpage{270} + \item Covariance, \hyperpage{185} + \item covariance, \hyperpage{185}, \hyperpage{249} + \item covariance function, \hyperpage{330} + \item covariance matrix, \hyperpage{181}, \hyperpage{196}, + \hyperpage{249} + \item covariant derivative, \hyperpage{152} + \item Cover, \hyperpage{213} + \item cover, \hyperpage{213} + \item Cramér-Rao bound, \hyperpage{196} + \item Crank-Nicolson, \hyperpage{272} + \item Crank-Nicolson scheme, \hyperpage{272}, \hyperpage{274} + \item credible interval, \hyperpage{203} + \item credible region, \hyperpage{203} + \item critical point, \hyperpage{57}, \hyperpage{131} + \item critical region, \hyperpage{199} + \item cross product, \hyperpage{142} + \item Cross ratio, \hyperpage{69} + \item cross ratio, \hyperpage{69}, \hyperpage{120} + \item cubic convergence, \hyperpage{91} + \item Cubic Hermite spline, \hyperpage{94} + \item Cubic periodic spline, \hyperpage{94} + \item cubic polynomial, \hyperpage{162} + \item cubic splines, \hyperpage{94} + \item Cumulative distribution function, \hyperpage{176} + \item cumulative distribution function, \hyperpage{176} + \item curl, \hyperpage{61} + \item Curl theorem, \hyperpage{159} + \item curvature, \hyperpage{61}, \hyperpage{143} + \item cusp, \hyperpage{224} + \item cycle, \hyperpage{8}, \hyperpage{49}, \hyperpage{115} + \item cyclic, \hyperpage{170} + \item cyclic group, \hyperpage{37} + \item cyclotomic, \hyperpage{170} + \item càd, \hyperpage{326} + \item càdlàg, \hyperpage{176} + \item Céa's lemma, \hyperpage{356} + + \indexspace + + \item D'Alembert formula, \hyperpage{140}, \hyperpage{281} + \item D'Alembert theorem, \hyperpage{12} + \item Dahlquist's theorem, \hyperpage{262} + \item damping parameter, \hyperpage{263} + \item Darboux theorem, \hyperpage{138} + \item Darboux's theorem, \hyperpage{30} + \item De Moivre's formula, \hyperpage{108} + \item De Moivre-Laplace theorem, \hyperpage{192} + \item De Morgan's laws, \hyperpage{6} + \item decreasing, \hyperpage{28} + \item Dedekind's lemma, \hyperpage{170} + \item Dedekind's test, \hyperpage{107} + \item Degenerated distribution, \hyperpage{177} + \item degenerated distribution, \hyperpage{177} + \item degree, \hyperpage{11}, \hyperpage{45}, \hyperpage{135}, + \hyperpage{155}, \hyperpage{164} + \item degree of a vertex, \hyperpage{49} + \item degree of precision, \hyperpage{95} + \item degree sequence, \hyperpage{49} + \item Dehomogenization, \hyperpage{74} + \item dehomogenization, \hyperpage{70} + \item Delta method, \hyperpage{196} + \item dense, \hyperpage{207} + \item density, \hyperpage{278}, \hyperpage{340} + \item density function, \hyperpage{178} + \item Dependence on $\boldsymbol {\mathrm {\lambda }}$, + \hyperpage{129} + \item Dependence on $\boldsymbol {\mathrm {x}}_0$, \hyperpage{129} + \item Dependence on $t_0$, \hyperpage{129} + \item depressed equation, \hyperpage{162} + \item derivative of $f$ at $x_0$, \hyperpage{29} + \item derivative of $f$ at $z_0$, \hyperpage{109} + \item Desargues configuration, \hyperpage{68} + \item Desargues' theorem, \hyperpage{68} + \item Descartes' rule of signs, \hyperpage{24}, \hyperpage{92} + \item descent direction, \hyperpage{264} + \item Descent method, \hyperpage{264} + \item descent method, \hyperpage{264} + \item design matrix, \hyperpage{250} + \item Determinant, \hyperpage{14} + \item determinant, \hyperpage{142} + \item determination, \hyperpage{110} + \item determination of the argument of $f$, \hyperpage{104} + \item determination of the logarithm, \hyperpage{110} + \item determined and consistent, \hyperpage{14} + \item developable, \hyperpage{151} + \item dextrorotation, \hyperpage{145} + \item DFT, \hyperpage{239} + \item diagonal, \hyperpage{20} + \item diagonalizable, \hyperpage{20} + \item Diagonalization theorem, \hyperpage{20} + \item diameter, \hyperpage{357} + \item dicyclic group, \hyperpage{42} + \item diffeomorphism, \hyperpage{57} + \item Differentiability criterion, \hyperpage{56} + \item differentiable, \hyperpage{29}, \hyperpage{55}, \hyperpage{147}, + \hyperpage{152}, \hyperpage{154}, \hyperpage{156, 157} + \item differentiable at $x_0\in (a,b)$, \hyperpage{29} + \item differentiable manifold, \hyperpage{216} + \item differential, \hyperpage{55}, \hyperpage{70}, \hyperpage{110}, + \hyperpage{148}, \hyperpage{155}, \hyperpage{157} + \item differential $\ell $-form, \hyperpage{156} + \item differential $k$-form, \hyperpage{155} + \item differential 1-form, \hyperpage{62} + \item differential operator over distributions, \hyperpage{245} + \item differential system, \hyperpage{122} + \item differentiation operator, \hyperpage{285} + \item diffusion, \hyperpage{367} + \item diffusion coefficient, \hyperpage{278} + \item Diffusion equation, \hyperpage{278} + \item diffusion flux, \hyperpage{278} + \item diffusions, \hyperpage{368} + \item diffusivity, \hyperpage{278} + \item digital topology, \hyperpage{206} + \item dihedral group, \hyperpage{42} + \item dilatation, \hyperpage{119} + \item dimension, \hyperpage{16}, \hyperpage{66}, \hyperpage{69}, + \hyperpage{193} + \item Dini's theorem, \hyperpage{85}, \hyperpage{234} + \item Dirac's $\delta $ distribution, \hyperpage{241}, + \hyperpage{285} + \item direct, \hyperpage{16} + \item direct product, \hyperpage{245} + \item Direct sum, \hyperpage{16} + \item directional derivative, \hyperpage{55} + \item director subspace, \hyperpage{70} + \item Dirichlet, \hyperpage{274} + \item Dirichlet boundary condition, \hyperpage{349} + \item Dirichlet boundary conditions, \hyperpage{356} + \item Dirichlet kernel, \hyperpage{84}, \hyperpage{233} + \item Dirichlet problem, \hyperpage{140}, \hyperpage{288} + \item Dirichlet problem in the disc, \hyperpage{288} + \item Dirichlet's principle, \hyperpage{288} + \item Dirichlet's test, \hyperpage{77, 78}, \hyperpage{80}, + \hyperpage{107} + \item disconnected, \hyperpage{214} + \item Discontinuity of the first kind, \hyperpage{28} + \item Discontinuity of the second kind, \hyperpage{28} + \item discrete, \hyperpage{130}, \hyperpage{176}, \hyperpage{180} + \item discrete $\sigma $-algebra, \hyperpage{172} + \item discrete Fourier transform, \hyperpage{239} + \item discrete laplacian, \hyperpage{274} + \item discrete map of order $m$, \hyperpage{222} + \item Discrete maximum principle, \hyperpage{274} + \item Discrete memorylessness property, \hyperpage{177} + \item discrete metric, \hyperpage{205} + \item Discrete minimum principle, \hyperpage{274} + \item Discrete topology, \hyperpage{206} + \item Discrete uniform distribution, \hyperpage{177} + \item discrete uniform distribution, \hyperpage{177} + \item discriminant, \hyperpage{169} + \item displacement current, \hyperpage{278} + \item distance, \hyperpage{53}, \hyperpage{82}, \hyperpage{205}, + \hyperpage{300} + \item distance between two affine subvarieties, \hyperpage{72} + \item distance between two points, \hyperpage{72} + \item Distribution, \hyperpage{240}, \hyperpage{285} + \item distribution, \hyperpage{176}, \hyperpage{180}, \hyperpage{240}, + \hyperpage{285} + \item distribution functions, \hyperpage{176} + \item distribution of order $N\in \ensuremath {\mathbb {N}}\cup \{0\}$, + \hyperpage{241} + \item distributional derivative, \hyperpage{242}, \hyperpage{285} + \item divergence, \hyperpage{61} + \item divergence form, \hyperpage{349} + \item Divergence theorem, \hyperpage{160} + \item Divergence theorem on $\ensuremath {\mathbb {R}}^2$, + \hyperpage{62} + \item Divergence theorem on $\ensuremath {\mathbb {R}}^3$, + \hyperpage{62}, \hyperpage{160} + \item divergent, \hyperpage{26}, \hyperpage{76}, \hyperpage{105, 106} + \item divided difference, \hyperpage{93} + \item Doléans-Dade exponential, \hyperpage{366} + \item domain, \hyperpage{115}, \hyperpage{148} + \item dominant eigenvalue, \hyperpage{100} + \item dominant eigenvector, \hyperpage{100} + \item Dominated convergence theorem, \hyperpage{184}, \hyperpage{189}, + \hyperpage{297}, \hyperpage{339} + \item Doob's maximal inequality, \hyperpage{360} + \item Doob's optional sampling theorem, \hyperpage{360} + \item Doob's optional sampling theorem for local martingales, + \hyperpage{361} + \item dot product, \hyperpage{53} + \item double, \hyperpage{90} + \item Double dual space, \hyperpage{19} + \item double dual space, \hyperpage{19} + \item drift, \hyperpage{367} + \item drift term, \hyperpage{365} + \item Du Bois-Reymond's test, \hyperpage{107} + \item Du-Fort-Frankel scheme, \hyperpage{274} + \item dual basis, \hyperpage{19} + \item Dual map, \hyperpage{19} + \item dual map, \hyperpage{19}, \hyperpage{308} + \item dual program, \hyperpage{51} + \item dual space, \hyperpage{19}, \hyperpage{307} + \item Duality principle, \hyperpage{66} + \item Duhamel principle, \hyperpage{287} + \item dummy variable, \hyperpage{253} + \item dyadic partition of order $n$, \hyperpage{183} + \item dynamical system, \hyperpage{130} + + \indexspace + + \item edges, \hyperpage{160} + \item efficient estimator, \hyperpage{196} + \item eigenspace, \hyperpage{20} + \item eigenvalue, \hyperpage{309} + \item eigenvector, \hyperpage{20} + \item eigenvectors, \hyperpage{309} + \item Eikonal equation, \hyperpage{278} + \item Eisenstein's criterion, \hyperpage{45}, \hyperpage{47} + \item either $P$ or $Q$ are true, \hyperpage{8} + \item elastodynamics, \hyperpage{277} + \item elementary region, \hyperpage{59} + \item elementary symmetric polynomials, \hyperpage{163} + \item elliptic, \hyperpage{74}, \hyperpage{273} + \item elliptic equation, \hyperpage{277} + \item elliptic point, \hyperpage{150} + \item elliptic sector, \hyperpage{133} + \item embedded, \hyperpage{344} + \item embedded methods, \hyperpage{260} + \item empirical distribution, \hyperpage{202} + \item empty set, \hyperpage{6} + \item energy functional, \hyperpage{288} + \item entire function, \hyperpage{109} + \item Entropy, \hyperpage{256} + \item entropy, \hyperpage{256} + \item Entropy condition, \hyperpage{280} + \item envelope, \hyperpage{144} + \item equal, \hyperpage{6} + \item equal almost surely, \hyperpage{176} + \item equal in distribution, \hyperpage{176} + \item equally-spaced, \hyperpage{93} + \item equation of the hyperplane, \hyperpage{67} + \item equicontinuous, \hyperpage{121} + \item equicontinuous at a point, \hyperpage{125}, \hyperpage{304} + \item equilibrium point, \hyperpage{221} + \item equiprobable space, \hyperpage{177} + \item equivalence, \hyperpage{131} + \item equivalence class, \hyperpage{8}, \hyperpage{17} + \item equivalence relation, \hyperpage{8} + \item equivalent, \hyperpage{13}, \hyperpage{17}, \hyperpage{23}, + \hyperpage{73}, \hyperpage{130}, \hyperpage{132}, + \hyperpage{306}, \hyperpage{354} + \item equivalent dynamical systems, \hyperpage{131} + \item Ergotic theorem, \hyperpage{324, 325} + \item error of type I, \hyperpage{199} + \item error of type II, \hyperpage{199} + \item error sum of squares, \hyperpage{252} + \item Essential singularity, \hyperpage{117} + \item essential singularity, \hyperpage{117} + \item estimated average response, \hyperpage{253} + \item estimation of the prediction, \hyperpage{253} + \item estimator, \hyperpage{193} + \item Euclid's axioms, \hyperpage{63} + \item Euclid's theorem, \hyperpage{10} + \item Euclidean, \hyperpage{64} + \item Euclidean affine space, \hyperpage{72} + \item Euclidean metric, \hyperpage{205} + \item Euclidean motion, \hyperpage{73} + \item Euclidean plane, \hyperpage{64} + \item Euclidean space, \hyperpage{53} + \item Euclidean vector space, \hyperpage{24} + \item Euclidian division, \hyperpage{11} + \item Euler characteristic, \hyperpage{218} + \item Euler method, \hyperpage{258}, \hyperpage{355} + \item Euler theorem, \hyperpage{50} + \item Euler's formula, \hyperpage{108}, \hyperpage{150} + \item Euler's theorem, \hyperpage{11} + \item Euler's totient function, \hyperpage{11} + \item Euler-Maclaurin formula, \hyperpage{96} + \item Eulerian, \hyperpage{50} + \item Eulerian circuit, \hyperpage{50} + \item Eulerian trail, \hyperpage{50} + \item evaluation, \hyperpage{12}, \hyperpage{162} + \item even extension, \hyperpage{84} + \item even periodic extension, \hyperpage{283} + \item event, \hyperpage{173} + \item events, \hyperpage{340} + \item evolute, \hyperpage{144} + \item exact, \hyperpage{156} + \item Examples of Euclidean motions, \hyperpage{73} + \item Excluded point topology, \hyperpage{206} + \item Existence and uniqueness of solutions of SDEs, \hyperpage{367} + \item Existence of a lift, \hyperpage{337} + \item Existence of orthogonal polynomials, \hyperpage{96} + \item Existence of the rotation number, \hyperpage{338} + \item Existence of the splitting field, \hyperpage{167} + \item exists and it is finite, \hyperpage{298} + \item expansion map, \hyperpage{335} + \item Expansive fixed point theorem, \hyperpage{229} + \item Expectation, \hyperpage{183}, \hyperpage{340} + \item expectation, \hyperpage{178}, \hyperpage{182--184}, + \hyperpage{249} + \item expectation of $X$, \hyperpage{340} + \item explicit, \hyperpage{258} + \item explicit Euler method, \hyperpage{258} + \item explicit form, \hyperpage{122} + \item Explicit scheme in finite differences, \hyperpage{286} + \item exponent, \hyperpage{89} + \item exponential, \hyperpage{257} + \item Exponential distribution, \hyperpage{178} + \item exponential distribution, \hyperpage{178} + \item exponential factor, \hyperpage{138} + \item exponential function with base $a$, \hyperpage{29} + \item exponential generating function, \hyperpage{48} + \item exponential series, \hyperpage{48} + \item extended complex plane, \hyperpage{105} + \item extended real numbers, \hyperpage{105} + \item extension, \hyperpage{126}, \hyperpage{308}, \hyperpage{344} + \item extension operator, \hyperpage{344} + \item exterior, \hyperpage{54} + \item exterior angle, \hyperpage{160} + \item Exterior angle theorem, \hyperpage{64} + \item exterior differential, \hyperpage{156, 157} + \item exterior point, \hyperpage{54} + \item exterior product, \hyperpage{155} + \item exterior vectors, \hyperpage{157} + \item externally studentized residuals, \hyperpage{254} + + \indexspace + + \item failure, \hyperpage{177} + \item false, \hyperpage{177} + \item Fano configuration, \hyperpage{67} + \item fast Fourier transform, \hyperpage{240} + \item Fatou's lemma, \hyperpage{184}, \hyperpage{297}, + \hyperpage{339} + \item feasible region, \hyperpage{51} + \item feasible solution, \hyperpage{51} + \item Fejér kernel, \hyperpage{85}, \hyperpage{234} + \item Fejér mean, \hyperpage{234} + \item Fejér means, \hyperpage{86} + \item Fejér's theorem, \hyperpage{86} + \item FEM, \hyperpage{275} + \item Fermat's little theorem, \hyperpage{11} + \item Fermat's principle, \hyperpage{278} + \item Feynman-Kac's formula, \hyperpage{369} + \item FFT, \hyperpage{240} + \item Fick's law, \hyperpage{278} + \item Fick's law of diffusion, \hyperpage{278} + \item Field, \hyperpage{42} + \item field, \hyperpage{42} + \item field extension, \hyperpage{164} + \item field morphism, \hyperpage{162} + \item field of complex numbers, \hyperpage{104} + \item field of fractions, \hyperpage{46}, \hyperpage{163} + \item field of rational functions, \hyperpage{163} + \item filtered probability space, \hyperpage{341} + \item filtration, \hyperpage{321}, \hyperpage{341} + \item filtration space, \hyperpage{321} + \item finer, \hyperpage{205} + \item finer than, \hyperpage{32}, \hyperpage{58} + \item finite, \hyperpage{25}, \hyperpage{49}, \hyperpage{164}, + \hyperpage{213}, \hyperpage{294} + \item finite $k$-th moment, \hyperpage{185} + \item finite difference method, \hyperpage{265} + \item finite difference scheme, \hyperpage{268} + \item finite element, \hyperpage{356} + \item finite element method, \hyperpage{275} + \item finite expectation, \hyperpage{182--184} + \item Finite field, \hyperpage{166} + \item finite field, \hyperpage{166} + \item finite moment of order $k$, \hyperpage{185} + \item Finite subadditivity, \hyperpage{173} + \item finite variation, \hyperpage{360} + \item Finite-dimensional distributions, \hyperpage{333} + \item finite-dimensional distributions, \hyperpage{333} + \item finite-rank operator, \hyperpage{307} + \item finitely generated, \hyperpage{165} + \item firmly non-expansive, \hyperpage{346} + \item first and second characteristic polynomials, \hyperpage{261} + \item First Borel-Cantelli lemma, \hyperpage{188} + \item first fundamental form, \hyperpage{148} + \item first integral, \hyperpage{134}, \hyperpage{154} + \item First isomorphism theorem, \hyperpage{18}, \hyperpage{39}, + \hyperpage{44} + \item first order reflection, \hyperpage{345} + \item First Sylow theorem, \hyperpage{41} + \item Fisher information, \hyperpage{195} + \item Fisher's theorem, \hyperpage{197} + \item Fisher-Neyman factorization theorem, \hyperpage{194} + \item fixed field, \hyperpage{168} + \item fixed point, \hyperpage{71}, \hyperpage{221} + \item Fixed-point theorem, \hyperpage{90} + \item flat, \hyperpage{203} + \item Floating-point representation, \hyperpage{89} + \item floating-point representation, \hyperpage{89} + \item floating-point representation by rounding, \hyperpage{89} + \item floating-point representation by truncation, \hyperpage{89} + \item flow, \hyperpage{128}, \hyperpage{132} + \item Flow box theorem, \hyperpage{132} + \item flow of the linearODE, \hyperpage{126} + \item flux, \hyperpage{62}, \hyperpage{159} + \item formal adjoint, \hyperpage{349} + \item Formal derivative, \hyperpage{167} + \item formal derivative, \hyperpage{167} + \item Forward and backward difference formula of order 1, + \hyperpage{94} + \item forward Euler method, \hyperpage{258} + \item Forward-time backward-space, \hyperpage{268} + \item Forward-time central-space, \hyperpage{268}, \hyperpage{274} + \item Forward-time forward-space, \hyperpage{268} + \item Fourier coefficients, \hyperpage{313} + \item Fourier series, \hyperpage{83}, \hyperpage{313} + \item Fourier transform, \hyperpage{231}, \hyperpage{237}, + \hyperpage{244}, \hyperpage{313} + \item Fourier transform operator, \hyperpage{232} + \item Fourier's law, \hyperpage{278} + \item Frank-Wolfe-type method, \hyperpage{346} + \item Fredholm alternative, \hyperpage{314} + \item Fredholm operator with kernel $K$, \hyperpage{307} + \item free variables, \hyperpage{14} + \item Frenet-Serret formulas, \hyperpage{143} + \item Frenet-Serret frame, \hyperpage{143} + \item Fresnel integrals, \hyperpage{113} + \item Frobenius endomorphism, \hyperpage{166} + \item Fréchet, \hyperpage{212} + \item FTBS, \hyperpage{268} + \item FTCS, \hyperpage{268} + \item FTFS, \hyperpage{268} + \item Fubini, \hyperpage{339} + \item Fubini's theorem, \hyperpage{59}, \hyperpage{175}, + \hyperpage{300} + \item Fubini's theorem for elementary regions, \hyperpage{59} + \item full QR decomposition, \hyperpage{267} + \item function, \hyperpage{7} + \item functionally independent, \hyperpage{134} + \item Fundamental lemma of calculus of variations, \hyperpage{240}, + \hyperpage{279} + \item fundamental matrix solution, \hyperpage{127} + \item fundamental polygon of the surface, \hyperpage{218} + \item fundamental solution, \hyperpage{245, 246}, \hyperpage{285} + \item Fundamental theorem of affine geometry, \hyperpage{72} + \item Fundamental theorem of algebra, \hyperpage{45}, \hyperpage{114}, + \hyperpage{170} + \item Fundamental theorem of arithmetic, \hyperpage{10} + \item Fundamental theorem of calculus, \hyperpage{33}, + \hyperpage{298} + \item Fundamental theorem of curves, \hyperpage{146} + \item Fundamental theorem of Galois theory, \hyperpage{169} + \item Fundamental theorem of projective geometry, \hyperpage{68} + \item Fundamental theorem of symmetric polynomials, \hyperpage{164} + + \indexspace + + \item Gagliardo, Nirengerg and Sobolev's inequality, \hyperpage{344} + \item Galerkin approximation, \hyperpage{275} + \item Galois, \hyperpage{168} + \item Galois extension, \hyperpage{168} + \item Galois group, \hyperpage{166} + \item Gambler's ruin problem, \hyperpage{318} + \item Gamma distribution, \hyperpage{179} + \item gamma distribution, \hyperpage{179} + \item Gamma function, \hyperpage{81} + \item Gau\ss ' law, \hyperpage{278} + \item Gau\ss ' law for magnetism, \hyperpage{278} + \item Gau\ss ' lemma, \hyperpage{47} + \item Gau\ss ' theorem, \hyperpage{10, 11}, \hyperpage{14} + \item Gau\ss ' Theorema Egregium, \hyperpage{152} + \item Gau\ss -Markov, \hyperpage{251} + \item Gau\ss -Seidel method, \hyperpage{100} + \item Gau\ss \ curvature, \hyperpage{149} + \item Gau\ss \ equations, \hyperpage{152} + \item Gau\ss \ map, \hyperpage{149} + \item Gau\ss ian integers, \hyperpage{47} + \item Gaussian elimination, \hyperpage{101} + \item Gaussian process, \hyperpage{330} + \item Gaussian vector, \hyperpage{330} + \item General Cauchy's integral formula, \hyperpage{115} + \item General Cauchy's integral formula for derivatives, + \hyperpage{116} + \item General Eisenstein's criterion, \hyperpage{45} + \item General Frenet-Serret formulas, \hyperpage{143} + \item General Itô's formula, \hyperpage{370} + \item General linear model, \hyperpage{250} + \item general solution of theODE, \hyperpage{122} + \item general term of the series, \hyperpage{76}, \hyperpage{105} + \item Generalized argument principle, \hyperpage{118} + \item Generalized Bendixson-Dulac theorem, \hyperpage{137} + \item Generalized compound probability formula, \hyperpage{174} + \item generalized eigenspace, \hyperpage{128} + \item generalized eigenvector, \hyperpage{128} + \item generalized heat kernel, \hyperpage{286} + \item Generalized Hölder's inequality, \hyperpage{237} + \item generalized Itô integral, \hyperpage{364} + \item generalized solution, \hyperpage{245} + \item generating set, \hyperpage{16} + \item Generator, \hyperpage{368} + \item generator, \hyperpage{368} + \item generator tree, \hyperpage{50} + \item generatrix, \hyperpage{147} + \item genus $g$ orientable surface, \hyperpage{217} + \item genus $h$ non-orientable surface, \hyperpage{217} + \item geodesic, \hyperpage{153} + \item geodesic curvature, \hyperpage{153} + \item Geometric Brownian motion, \hyperpage{368} + \item Geometric distribution, \hyperpage{177} + \item geometric distribution, \hyperpage{177} + \item geometric multiplicity, \hyperpage{20} + \item Gershgorin circle theorem, \hyperpage{92} + \item Giranov's theorem, \hyperpage{367} + \item Glide orthogonal reflections, \hyperpage{73} + \item glide reflection, \hyperpage{72} + \item glide vector, \hyperpage{73} + \item Global Gau\ss -Bonnet theorem, \hyperpage{160} + \item global interpolant, \hyperpage{357} + \item Global interpolation error, \hyperpage{358} + \item global maximum, \hyperpage{57} + \item global minimum, \hyperpage{57} + \item global stable manifold, \hyperpage{222} + \item global truncation error, \hyperpage{258} + \item global unstable manifold, \hyperpage{222} + \item Goodness of fit, \hyperpage{200} + \item Goursat's theorem, \hyperpage{112} + \item Gra\ss mann formula, \hyperpage{16}, \hyperpage{67} + \item gradient, \hyperpage{56} + \item Gradient descent, \hyperpage{264} + \item Gradient descent algorithm, \hyperpage{346} + \item gradient descent algorithm, \hyperpage{346} + \item gradient vector field, \hyperpage{61} + \item Gram-Schmidt process, \hyperpage{23}, \hyperpage{313} + \item graph, \hyperpage{49}, \hyperpage{55}, \hyperpage{227} + \item Graph of a function, \hyperpage{55} + \item Great Picard's theorem, \hyperpage{118} + \item greatest common divisor, \hyperpage{10}, \hyperpage{46} + \item greatest element, \hyperpage{126} + \item Green identities, \hyperpage{279} + \item Green's formula, \hyperpage{159} + \item Green's theorem, \hyperpage{62} + \item grid, \hyperpage{268} + \item Gronwall's lemma, \hyperpage{367} + \item Group, \hyperpage{37} + \item group, \hyperpage{37} + \item Group morphism, \hyperpage{38} + \item group morphism, \hyperpage{38} + \item Grönwall's lemma, \hyperpage{129} + + \indexspace + + \item Hahn-Banach theorem, \hyperpage{308} + \item half-line, \hyperpage{63} + \item half-plane, \hyperpage{63} + \item Hamilton-Jacobi equation, \hyperpage{279} + \item Hamiltonian, \hyperpage{134} + \item Hamiltonian system, \hyperpage{134} + \item Hamiltonian vector field, \hyperpage{337} + \item Handshaking lemma, \hyperpage{49} + \item Hardy-Littlewood maximal function, \hyperpage{298} + \item harmonic, \hyperpage{115} + \item harmonic ratio, \hyperpage{69} + \item Hartman-Grobman theorem, \hyperpage{134}, \hyperpage{222} + \item has $t$ correct decimal digits, \hyperpage{89} + \item has $u$ significant digits, \hyperpage{89} + \item Hausdorff, \hyperpage{212} + \item Heat equation, \hyperpage{140}, \hyperpage{278} + \item heat equation, \hyperpage{140} + \item heat flux, \hyperpage{278} + \item heat kernel, \hyperpage{285} + \item Heaviside step function, \hyperpage{242}, \hyperpage{285} + \item Heavy ball method, \hyperpage{347} + \item Heine's theorem, \hyperpage{55} + \item Heine-Borel theorem, \hyperpage{214} + \item Hermite interpolation problem, \hyperpage{94} + \item Hermite reduction method, \hyperpage{35} + \item Hessian matrix, \hyperpage{56} + \item heteroclinic orbit, \hyperpage{133} + \item Heun method, \hyperpage{259} + \item high-leverage point, \hyperpage{255} + \item high-leverage points, \hyperpage{255} + \item highest posterior density, \hyperpage{204} + \item Hilbert basis, \hyperpage{313} + \item Hilbert field, \hyperpage{64} + \item Hilbert plane, \hyperpage{64} + \item Hilbert space, \hyperpage{311} + \item Hilbert transform, \hyperpage{247} + \item Hilbert's basis theorem, \hyperpage{43} + \item Hilbert's Nullstellensatz, \hyperpage{45} + \item Hilbert-Schmidt operator with kernel $K$, \hyperpage{307} + \item Hilbert-Schmidt spectral representation theorem, + \hyperpage{314} + \item hitting time, \hyperpage{360} + \item holding times, \hyperpage{327} + \item holes, \hyperpage{216} + \item holomorphic, \hyperpage{109} + \item holomorphic differential equation, \hyperpage{221} + \item homeomorphic, \hyperpage{208} + \item homeomorphism, \hyperpage{208} + \item Homoclinic bifurcation, \hyperpage{226} + \item homoclinic bifurcation, \hyperpage{226} + \item homoclinic orbit, \hyperpage{133} + \item homogeneous, \hyperpage{49}, \hyperpage{122}, \hyperpage{126}, + \hyperpage{349} + \item homogeneous adjoint problem, \hyperpage{349} + \item homogeneous coordinates, \hyperpage{67} + \item Homogeneous distribution, \hyperpage{245} + \item homogeneous of degree $k$, \hyperpage{221} + \item homogeneous of degree $r\in \ensuremath {\mathbb {R}}$, + \hyperpage{245} + \item homogeneous SDE, \hyperpage{368} + \item Homogenization, \hyperpage{74} + \item homogenization, \hyperpage{70} + \item homography, \hyperpage{67} + \item homologous to zero, \hyperpage{115} + \item Homotheties, \hyperpage{72} + \item homothety, \hyperpage{19}, \hyperpage{72} + \item homotopic, \hyperpage{139} + \item homotopy, \hyperpage{139} + \item Hopf bifurcation theorem, \hyperpage{226} + \item Hopf's lemma, \hyperpage{351} + \item Hopf-bifurcation, \hyperpage{226} + \item Hurwitz's theorem, \hyperpage{119} + \item hyperbolic, \hyperpage{268}, \hyperpage{273} + \item hyperbolic critical point, \hyperpage{134} + \item hyperbolic equations, \hyperpage{277} + \item Hyperbolic geometry, \hyperpage{65} + \item hyperbolic matrix, \hyperpage{134} + \item hyperbolic periodic orbit, \hyperpage{136} + \item hyperbolic plane, \hyperpage{74} + \item hyperbolic point, \hyperpage{150} + \item hyperbolic sector, \hyperpage{133} + \item hyperbolicity, \hyperpage{336} + \item Hypergeometric distribution, \hyperpage{177} + \item hypergeometric distribution, \hyperpage{177} + \item hyperplane, \hyperpage{67}, \hyperpage{70} + \item hypothesis, \hyperpage{199} + \item Hypothesis test, \hyperpage{199} + \item hypothesis test, \hyperpage{199} + \item Hölder condition, \hyperpage{247} + \item Hölder conjugates, \hyperpage{302} + \item Hölder continuity, \hyperpage{344} + \item Hölder's inequality, \hyperpage{302, 303} + + \indexspace + + \item i.i.d., \hyperpage{181} + \item Ideal, \hyperpage{43} + \item ideal, \hyperpage{43} + \item idempotent, \hyperpage{14} + \item identifiable, \hyperpage{193} + \item ill-conditioned, \hyperpage{90} + \item image, \hyperpage{7}, \hyperpage{17}, \hyperpage{38}, + \hyperpage{115} + \item imaginary part, \hyperpage{104} + \item Immersion, \hyperpage{146} + \item immersion, \hyperpage{146} + \item implicit, \hyperpage{258} + \item implicit descent, \hyperpage{347} + \item implicit Euler method, \hyperpage{258} + \item implicit form, \hyperpage{122} + \item Implicit function theorem, \hyperpage{57} + \item Implicit scheme in finite differences, \hyperpage{286} + \item importance sampling estimator, \hyperpage{354} + \item improper, \hyperpage{37}, \hyperpage{203} + \item improper integral, \hyperpage{79} + \item in perspective with respect to a line, \hyperpage{68} + \item in perspective with respect to a point, \hyperpage{68} + \item Incidence axioms, \hyperpage{63} + \item incidence relation, \hyperpage{63} + \item incident, \hyperpage{49} + \item incompressible flow, \hyperpage{277} + \item inconsistent, \hyperpage{14} + \item increasing, \hyperpage{28}, \hyperpage{181} + \item increment, \hyperpage{181} + \item incremental function, \hyperpage{258} + \item independent, \hyperpage{134}, \hyperpage{181}, \hyperpage{316} + \item independent and identically distributed, \hyperpage{181} + \item independent events, \hyperpage{174} + \item independent increments, \hyperpage{326} + \item Independent samples with known variances, \hyperpage{198} + \item Independent samples with unknown equal variances, + \hyperpage{198} + \item Independent samples with unknown variances, \hyperpage{198} + \item independent variable, \hyperpage{122} + \item indeterminate, \hyperpage{11}, \hyperpage{14} + \item index, \hyperpage{14}, \hyperpage{39}, \hyperpage{74}, + \hyperpage{113}, \hyperpage{115}, \hyperpage{138, 139}, + \hyperpage{160} + \item indicator function, \hyperpage{9} + \item indistinguishable, \hyperpage{331} + \item induced subgraph, \hyperpage{49} + \item Induction axiom, \hyperpage{6} + \item Inexact Newton method, \hyperpage{264} + \item infimum, \hyperpage{25} + \item infinite, \hyperpage{25}, \hyperpage{164} + \item infinite order, \hyperpage{241} + \item infinite product topology, \hyperpage{209} + \item infinitesimal generator, \hyperpage{328} + \item infinitesimal transition scheme, \hyperpage{328} + \item inflection point, \hyperpage{31} + \item Information, \hyperpage{256} + \item information of an event, \hyperpage{256} + \item initial conditions, \hyperpage{122} + \item initial distribution, \hyperpage{318} + \item initial point, \hyperpage{215} + \item Initial value problem, \hyperpage{122} + \item initial value problem, \hyperpage{122} + \item initial values, \hyperpage{48} + \item injective, \hyperpage{7} + \item inner product, \hyperpage{24}, \hyperpage{82}, \hyperpage{310} + \item Inner regularity, \hyperpage{350} + \item insphere diameter, \hyperpage{357} + \item integrable, \hyperpage{32}, \hyperpage{182, 183} + \item integrable function, \hyperpage{59} + \item integrable function over $E$, \hyperpage{297} + \item integral, \hyperpage{111}, \hyperpage{158, 159} + \item integral curve, \hyperpage{154} + \item integral domain, \hyperpage{44} + \item Integral in cylindrical coordinates, \hyperpage{60} + \item Integral in polar coordinates, \hyperpage{60} + \item Integral in spherical coordinates, \hyperpage{60} + \item integral of $f$ over $\ensuremath {\mathbb {R}}^n$, + \hyperpage{296} + \item integral of $f$ over $E$, \hyperpage{297} + \item integral of $f$ over a measurable set $E\subseteq \ensuremath {\mathbb {R}}^n$, + \hyperpage{296} + \item integral of $f$ over the region $R$, \hyperpage{148} + \item integral of $f$ with respect to $\mu $, \hyperpage{339} + \item integral of $s$ over $\ensuremath {\mathbb {R}}^n$, + \hyperpage{295} + \item integral of $s$ over a measurable set $E$, \hyperpage{295} + \item Integral test, \hyperpage{77}, \hyperpage{80} + \item Integrating factor, \hyperpage{124} + \item integrating factor, \hyperpage{124}, \hyperpage{138} + \item Integration by parts, \hyperpage{33}, \hyperpage{298} + \item Integration by substitution, \hyperpage{33} + \item inter-arrival times, \hyperpage{327} + \item Interior, \hyperpage{206} + \item interior, \hyperpage{54}, \hyperpage{206} + \item interior ball condition, \hyperpage{351} + \item interior point, \hyperpage{54}, \hyperpage{157}, + \hyperpage{207} + \item interior product, \hyperpage{156} + \item Intermediate value theorem, \hyperpage{28}, \hyperpage{55}, + \hyperpage{94}, \hyperpage{215} + \item internally studentized residuals, \hyperpage{254} + \item Interpolation inequality, \hyperpage{303} + \item interpolation problem, \hyperpage{93} + \item intersection, \hyperpage{6} + \item interval, \hyperpage{291} + \item Interval for $\mu $ with $\sigma $ known, \hyperpage{198} + \item Intervals for $\mu $ and $\sigma ^2$, \hyperpage{198} + \item invariance level, \hyperpage{71} + \item Invariance of the MLE, \hyperpage{195} + \item Invariance under time shift, \hyperpage{368} + \item invariant, \hyperpage{131} + \item invariant algebraic curve, \hyperpage{135} + \item invariant subspace, \hyperpage{21} + \item invariant under an affinity, \hyperpage{71} + \item inverse discrete Fourier transform, \hyperpage{239} + \item inverse Fourier transform, \hyperpage{231}, \hyperpage{237}, + \hyperpage{244} + \item inverse function, \hyperpage{7} + \item Inverse function rule, \hyperpage{30} + \item Inverse function theorem, \hyperpage{57} + \item Inverse function theorem for surfaces, \hyperpage{148} + \item Inverse power method, \hyperpage{101} + \item inverse power method, \hyperpage{101} + \item inverse semidiscrete Fourier transform, \hyperpage{270} + \item inversion, \hyperpage{119} + \item Inversion theorem, \hyperpage{233} + \item invertible, \hyperpage{7}, \hyperpage{14} + \item Inviscid flow, \hyperpage{277} + \item involute, \hyperpage{144} + \item irreducible, \hyperpage{12}, \hyperpage{44}, \hyperpage{100}, + \hyperpage{329} + \item irreducible chain, \hyperpage{320} + \item irreducible class, \hyperpage{320} + \item irreducible polynomial, \hyperpage{164} + \item is finer than, \hyperpage{306} + \item isolated point, \hyperpage{54} + \item isolated singular point, \hyperpage{160} + \item isolated singularity, \hyperpage{116}, \hyperpage{160} + \item isometric, \hyperpage{23} + \item isometry, \hyperpage{23}, \hyperpage{74}, \hyperpage{148}, + \hyperpage{362} + \item isomorphic, \hyperpage{17}, \hyperpage{38}, \hyperpage{50}, + \hyperpage{306} + \item isomorphism, \hyperpage{38} + \item isomorphism between projective spaces, \hyperpage{67} + \item Isoperimetric inequality, \hyperpage{88} + \item isotropic, \hyperpage{22} + \item iteration matrix, \hyperpage{99} + \item itinerary, \hyperpage{230} + \item Itinerary lemma, \hyperpage{229} + \item Itô integral, \hyperpage{364} + \item Itô isometry, \hyperpage{364} + \item Itô process, \hyperpage{365} + \item Itô term, \hyperpage{366} + \item Itô's formula, \hyperpage{366} + \item ivp, \hyperpage{122} + + \indexspace + + \item Jacobi identity, \hyperpage{142} + \item Jacobi method, \hyperpage{100} + \item Jacobian determinant, \hyperpage{55} + \item Jacobian matrix, \hyperpage{55} + \item Jeffrey's prior, \hyperpage{203} + \item Jensen's inequality, \hyperpage{236}, \hyperpage{340} + \item joint pdf, \hyperpage{180} + \item joint pmf, \hyperpage{180} + \item joint probability density function, \hyperpage{180} + \item Joint probability mass function, \hyperpage{180} + \item joint probability mass function, \hyperpage{180} + \item jointly Gaussian, \hyperpage{359} + \item Jordan arc, \hyperpage{61} + \item Jordan block, \hyperpage{21} + \item Jordan closed curve, \hyperpage{61} + \item Jordan curve theorem, \hyperpage{218} + \item Jordan form, \hyperpage{21} + \item Jordan matrix, \hyperpage{21} + \item Jump discontinuity, \hyperpage{28} + \item jump process, \hyperpage{328} + + \indexspace + + \item Kelley's theorem, \hyperpage{213} + \item kernel, \hyperpage{17}, \hyperpage{38} + \item kernel function, \hyperpage{79} + \item Klein bottle, \hyperpage{211} + \item Kolmogorov, \hyperpage{212} + \item Kolmogorov axioms, \hyperpage{173} + \item Kolmogorov extension theorem, \hyperpage{333} + \item Kolmogorov system, \hyperpage{221} + \item Kolmogorov's backward equation, \hyperpage{328} + \item Kolmogorov's continuity theorem, \hyperpage{331} + \item Kolmogorov's equation, \hyperpage{368} + \item Kolmogorov's forward equation, \hyperpage{328} + \item Kolmogorov's strong law of large numbers, \hyperpage{191} + \item Krasnoselskii-Mann's convergence theorem, \hyperpage{348} + \item Kronecker delta, \hyperpage{19} + \item Kronecker's lemma, \hyperpage{164} + \item Kronecker's theorem, \hyperpage{167} + \item Kullback-Leibler divergence, \hyperpage{256} + \item Kuratowski's problem, \hyperpage{207} + + \indexspace + + \item L'H\^opital's rule, \hyperpage{30} + \item lack of fit test, \hyperpage{254} + \item Lagrange basis polynomials, \hyperpage{93} + \item Lagrange multipliers theorem, \hyperpage{58} + \item Lagrange's interpolation problem, \hyperpage{93} + \item Lagrange's theorem, \hyperpage{39} + \item Laguerre polynomials, \hyperpage{96} + \item Lamé coefficients, \hyperpage{277} + \item Langevin equation, \hyperpage{367} + \item Laplace equation, \hyperpage{140}, \hyperpage{288} + \item Laplacian, \hyperpage{61} + \item lattice of subgroups, \hyperpage{169} + \item Laurent series, \hyperpage{117} + \item Laurent series theorem, \hyperpage{117} + \item law of $X$, \hyperpage{340} + \item Law of the iterated logarithm, \hyperpage{333} + \item Law of total expectation, \hyperpage{186, 187}, \hyperpage{315} + \item Law of total probability, \hyperpage{174}, \hyperpage{315} + \item Lax theorem, \hyperpage{261} + \item Lax-Friedrichs scheme, \hyperpage{268} + \item Lax-Milgram theorem, \hyperpage{313} + \item Lax-Richtmyer equivalence theorem, \hyperpage{273} + \item Lax-Wendroff, \hyperpage{271} + \item Lax-Wendroff scheme, \hyperpage{271} + \item leading coefficient, \hyperpage{11} + \item Leapfrog scheme, \hyperpage{268}, \hyperpage{274} + \item least common multiple, \hyperpage{10} + \item least element, \hyperpage{9} + \item Least-squares method, \hyperpage{250} + \item least-squares method, \hyperpage{250} + \item leave, \hyperpage{50} + \item Lebesgue differentiation theorem, \hyperpage{299} + \item Lebesgue integrable, \hyperpage{175} + \item Lebesgue integral, \hyperpage{175} + \item Lebesgue measurable, \hyperpage{293, 294} + \item Lebesgue measure, \hyperpage{175}, \hyperpage{293} + \item left cosets, \hyperpage{39} + \item Left side, \hyperpage{120} + \item left singular vectors, \hyperpage{265} + \item Left-open interval, \hyperpage{25} + \item left-sided limit of $f$ at $x_0$, \hyperpage{27} + \item Legendre polynomials, \hyperpage{97} + \item Leibnitz rule, \hyperpage{243} + \item Leibnitz's test, \hyperpage{77} + \item length, \hyperpage{58}, \hyperpage{141} + \item length of the polygonal, \hyperpage{34}, \hyperpage{60}, + \hyperpage{141} + \item level set, \hyperpage{55} + \item leverage, \hyperpage{253} + \item levorotation, \hyperpage{145} + \item Levy's characterization of Brownian motion, \hyperpage{362} + \item lexicographic degree, \hyperpage{163} + \item lexicographic order, \hyperpage{163} + \item lift, \hyperpage{337} + \item Likelihood, \hyperpage{195} + \item likelihood function, \hyperpage{195} + \item likelihood ratio test, \hyperpage{200} + \item Likelihood-ratio test, \hyperpage{200} + \item Limit, \hyperpage{26} + \item limit, \hyperpage{105}, \hyperpage{188} + \item Limit comparison test, \hyperpage{76} + \item limit cycle, \hyperpage{136} + \item limit distribution, \hyperpage{329} + \item limit inferior, \hyperpage{27}, \hyperpage{188} + \item limit of $f$ at infinity, \hyperpage{28} + \item limit of the function $f$ at the point $x_0$, \hyperpage{27} + \item limit set, \hyperpage{54} + \item limit superior, \hyperpage{27}, \hyperpage{188} + \item line, \hyperpage{67}, \hyperpage{70} + \item line integral, \hyperpage{61}, \hyperpage{112} + \item line intergal, \hyperpage{159} + \item line of curvature, \hyperpage{150} + \item line search method, \hyperpage{264} + \item linear, \hyperpage{122} + \item linear $k$-step method, \hyperpage{261} + \item linear combination, \hyperpage{16} + \item linear convergence, \hyperpage{91} + \item linear discrete map, \hyperpage{222} + \item linear equation, \hyperpage{13} + \item Linear loss function, \hyperpage{203} + \item linear map, \hyperpage{17} + \item Linear multistep method, \hyperpage{261} + \item linear programming to maximize, \hyperpage{51} + \item linear programming to minimize, \hyperpage{51} + \item linear recurrence relation of order $k$, \hyperpage{48} + \item linear saddles, \hyperpage{227} + \item linear second-order PDE, \hyperpage{349} + \item linear span, \hyperpage{16} + \item Linearity, \hyperpage{76} + \item linearization, \hyperpage{221} + \item linearly dependent, \hyperpage{16}, \hyperpage{82} + \item linearly independent, \hyperpage{16}, \hyperpage{82} + \item lines, \hyperpage{63} + \item Liouville's formula, \hyperpage{127}, \hyperpage{153} + \item Liouville's theorem, \hyperpage{114} + \item Lipschitz continuous, \hyperpage{55} + \item Lipschitz continuous with respect to the second variable, + \hyperpage{124} + \item Lipschitz's theorem, \hyperpage{85} + \item Little Picard's theorem, \hyperpage{118} + \item Liénard system, \hyperpage{228} + \item Liénard's theorem, \hyperpage{228} + \item load vector, \hyperpage{276} + \item Local behaviour of a holomorphic function, \hyperpage{116} + \item local bifurcation, \hyperpage{223} + \item local canonical form, \hyperpage{145} + \item Local Cauchy's integral formula, \hyperpage{113} + \item Local Cauchy's integral formula for derivatives, + \hyperpage{113} + \item Local Cauchy's integral theorem, \hyperpage{112} + \item local conjugacy, \hyperpage{132} + \item local diffeomorphism, \hyperpage{141} + \item local equivalence, \hyperpage{132} + \item local extremum, \hyperpage{30}, \hyperpage{57} + \item Local Gau\ss -Bonnet theorem, \hyperpage{160} + \item local interpolant, \hyperpage{357} + \item Local interpolation error, \hyperpage{358} + \item local isometry, \hyperpage{148} + \item local maximum, \hyperpage{30}, \hyperpage{57} + \item local minimum, \hyperpage{30}, \hyperpage{57} + \item Local structure of immersions, \hyperpage{146} + \item Local structure of submersions, \hyperpage{146} + \item local transversal section, \hyperpage{132} + \item local truncation error, \hyperpage{262} + \item local truncation errors, \hyperpage{258} + \item localizing sequence, \hyperpage{361} + \item locally, \hyperpage{214} + \item locally bounded, \hyperpage{121}, \hyperpage{304} + \item locally compact, \hyperpage{214} + \item locally connected, \hyperpage{215} + \item locally integrable, \hyperpage{79} + \item locally Lipschitz continuous with respect to the second variable, + \hyperpage{124} + \item locally path-connected, \hyperpage{216} + \item locally topologically conjugate, \hyperpage{132} + \item locally topologically equivalent, \hyperpage{132} + \item LOF test, \hyperpage{254} + \item log-likelihood function, \hyperpage{195} + \item log-odds, \hyperpage{257} + \item logarithm with base $a$, \hyperpage{29} + \item Logarithmic test, \hyperpage{76} + \item logistic map, \hyperpage{228} + \item logit, \hyperpage{257} + \item loop, \hyperpage{49}, \hyperpage{215} + \item Lorenz system, \hyperpage{228} + \item loss function, \hyperpage{203} + \item lower bound, \hyperpage{25} + \item lower integral, \hyperpage{32}, \hyperpage{58} + \item lower limit topology, \hyperpage{206} + \item lower sum, \hyperpage{32}, \hyperpage{58} + \item lower triangular, \hyperpage{97} + \item LRT, \hyperpage{200} + \item LRT test statistic, \hyperpage{200} + \item LU decomposition, \hyperpage{102} + \item LU descompostion, \hyperpage{102} + \item Lyapunov central limit theorem, \hyperpage{192} + \item Lyapunov exponent, \hyperpage{336} + \item Lyapunov function, \hyperpage{137} + \item Lyapunov stable, \hyperpage{137} + \item Lyapunov's method, \hyperpage{226} + \item Lyapunov's theorem, \hyperpage{137}, \hyperpage{223} + \item Lévy-Lindeberg central limit theorem, \hyperpage{192} + + \indexspace + + \item machine epsilon $\epsilon $, \hyperpage{89} + \item Malgrange-Ehrenpreis theorem, \hyperpage{247} + \item Mallow's $C_p$ statistic, \hyperpage{256} + \item mantissa, \hyperpage{89} + \item Marcinkiewicz interpolation theorem, \hyperpage{306} + \item marginal pdf, \hyperpage{180} + \item marginal pmf, \hyperpage{180} + \item Marginal probability density functions, \hyperpage{180} + \item marginal probability density functions, \hyperpage{180} + \item Marginal probability mass functions, \hyperpage{180} + \item marginal probability mass functions, \hyperpage{180} + \item Markov chain, \hyperpage{318} + \item Markov property, \hyperpage{318} + \item Markov property for Brownian motion, \hyperpage{359} + \item Markov's inequality, \hyperpage{185} + \item Martingale, \hyperpage{359} + \item martingale, \hyperpage{341}, \hyperpage{359} + \item martingale term, \hyperpage{365} + \item material derivative operator, \hyperpage{277} + \item Matrix, \hyperpage{13} + \item matrix, \hyperpage{13}, \hyperpage{18} + \item matrix exponential, \hyperpage{127} + \item matrix norm, \hyperpage{97} + \item matrix of the bilinear form, \hyperpage{22} + \item matrix solution, \hyperpage{127} + \item maximal, \hyperpage{44}, \hyperpage{126} + \item maximal element, \hyperpage{9}, \hyperpage{126} + \item maximum likelihood estimator, \hyperpage{195} + \item Maximum likelihood method, \hyperpage{195} + \item Maximum metric, \hyperpage{205} + \item Maximum modulus principle, \hyperpage{114} + \item Maximum principle, \hyperpage{287} + \item Maximum principle on unbounded domains, \hyperpage{287} + \item Maxwell-Faraday equation, \hyperpage{278} + \item mean, \hyperpage{178} + \item mean curvature, \hyperpage{149} + \item mean function, \hyperpage{330} + \item mean square error, \hyperpage{251} + \item mean squared error, \hyperpage{194} + \item Mean value property, \hyperpage{115} + \item Mean value theorem, \hyperpage{30}, \hyperpage{56} + \item Mean value theorem for integrals, \hyperpage{95}, + \hyperpage{298} + \item Mean value theorem for vector-valued functions, \hyperpage{56} + \item mean vector, \hyperpage{181}, \hyperpage{196} + \item measurable, \hyperpage{175}, \hyperpage{293, 294}, + \hyperpage{339} + \item measurable space, \hyperpage{339} + \item Measure, \hyperpage{291}, \hyperpage{339} + \item measure, \hyperpage{174}, \hyperpage{291}, \hyperpage{339} + \item measure space, \hyperpage{175}, \hyperpage{339} + \item Melnikov's method, \hyperpage{227} + \item memoryless, \hyperpage{177, 178} + \item meromorphic, \hyperpage{118} + \item Mersenne Twister algorithm, \hyperpage{352} + \item Mesh, \hyperpage{275} + \item mesh, \hyperpage{275}, \hyperpage{360} + \item mesh-points, \hyperpage{258} + \item Method of characteristics, \hyperpage{279} + \item Method of moments, \hyperpage{194} + \item metric, \hyperpage{205} + \item metric space, \hyperpage{53}, \hyperpage{205}, \hyperpage{300} + \item metrizable, \hyperpage{187}, \hyperpage{212} + \item Meusnier's theorem, \hyperpage{150} + \item minimal, \hyperpage{221} + \item minimal element, \hyperpage{9} + \item minimal polynomial, \hyperpage{20} + \item minimal surface, \hyperpage{149} + \item minimizer, \hyperpage{311} + \item Minimum modulus principle, \hyperpage{114} + \item Minimum principle, \hyperpage{287} + \item Minimum principle on unbounded domains, \hyperpage{288} + \item minimum-variance unbiased estimator, \hyperpage{194} + \item Minkowski inequality, \hyperpage{82}, \hyperpage{303}, + \hyperpage{310} + \item Minkowski's integral inequality, \hyperpage{235} + \item minor, \hyperpage{16} + \item mirror of the reflection, \hyperpage{71} + \item mixed random variable, \hyperpage{179} + \item mixing, \hyperpage{336} + \item MLE, \hyperpage{195} + \item MLE method, \hyperpage{251} + \item Model with interaction, \hyperpage{253} + \item model with interaction, \hyperpage{253} + \item Model without interaction, \hyperpage{253} + \item modulus, \hyperpage{104} + \item modulus of continuity, \hyperpage{55} + \item Moment, \hyperpage{185} + \item Moment of inertia, \hyperpage{59} + \item moment of inertia, \hyperpage{59} + \item Moment-generating function, \hyperpage{186} + \item moment-generating function, \hyperpage{186} + \item monic, \hyperpage{11} + \item Monotone convergence theorem, \hyperpage{184}, \hyperpage{296}, + \hyperpage{339} + \item monotonic, \hyperpage{26}, \hyperpage{28} + \item monotonically decreasing, \hyperpage{26} + \item monotonically increasing, \hyperpage{26} + \item Montecarlo estimator, \hyperpage{352} + \item Montel's theorem, \hyperpage{121} + \item more efficient than, \hyperpage{194} + \item Morera's theorem, \hyperpage{114} + \item morphism of field extensions, \hyperpage{165} + \item Morrey's embedding, \hyperpage{344} + \item MSE, \hyperpage{194} + \item multi-index notation, \hyperpage{57} + \item multicollinearity, \hyperpage{255} + \item multilinear forms, \hyperpage{155} + \item multilinear map, \hyperpage{155} + \item Multinomial distrbution, \hyperpage{180} + \item multinomial distribution, \hyperpage{180} + \item multiple, \hyperpage{10} + \item multiple control variate estimator, \hyperpage{354} + \item multiple edges, \hyperpage{49} + \item Multiple shooting method, \hyperpage{264} + \item multiple shooting method, \hyperpage{264} + \item multiplicative group, \hyperpage{43} + \item multiplicity, \hyperpage{114} + \item multistep, \hyperpage{268} + \item multistep method, \hyperpage{347} + \item multivalued function, \hyperpage{104} + \item multivariate cdf, \hyperpage{181} + \item multivariate cumulative distribution function, \hyperpage{181} + \item Multivariate normal distribution, \hyperpage{181} + \item multivariate normal distribution, \hyperpage{180}, + \hyperpage{196} + \item multivariate standard deviation, \hyperpage{196} + \item Multivariate standard normal distribution, \hyperpage{180} + \item Multivariate uniform distribution, \hyperpage{180} + \item multivariate uniform distribution, \hyperpage{180} + \item MVUE, \hyperpage{194} + \item Möbius band, \hyperpage{211} + \item Möbius transformation, \hyperpage{119} + \item Möbius transformations, \hyperpage{119} + + \indexspace + + \item Nart-Vila theorem, \hyperpage{171} + \item Natural cubic spline, \hyperpage{94} + \item natural filtration, \hyperpage{359} + \item Navier-Cauchy equation, \hyperpage{277} + \item negation, \hyperpage{8} + \item negative, \hyperpage{141} + \item negative basis, \hyperpage{142} + \item Negative binomial distribution, \hyperpage{178} + \item negative binomial distribution, \hyperpage{178} + \item negative part, \hyperpage{77} + \item negative semi-orbit, \hyperpage{130} + \item negative-definite, \hyperpage{23} + \item negative-semidefinite, \hyperpage{23} + \item negatively invariant, \hyperpage{131} + \item negatively rotated, \hyperpage{227} + \item negatively stable, \hyperpage{131} + \item negatively-oriented, \hyperpage{142} + \item neighbourhood, \hyperpage{25}, \hyperpage{54}, \hyperpage{207} + \item Nesterov's accelerated gradient method, \hyperpage{347} + \item Nesterov's method, \hyperpage{347} + \item Neumann, \hyperpage{274} + \item Neumann boundary condition, \hyperpage{349} + \item Neumann boundary conditions, \hyperpage{356} + \item Neumann series, \hyperpage{308} + \item Neville's algorithm, \hyperpage{93} + \item Newton method, \hyperpage{262}, \hyperpage{264}, + \hyperpage{347} + \item Newton's divided differences\\method, \hyperpage{93} + \item Newton-Raphson method, \hyperpage{90} + \item Newton-Raphson modified method, \hyperpage{90} + \item Neyman-Pearson lemma, \hyperpage{200} + \item Neyman-Pearson test, \hyperpage{200} + \item Neymann-Pearson test, \hyperpage{200} + \item nilpotent, \hyperpage{14}, \hyperpage{224} + \item Nilpotent singular points classification theorem, + \hyperpage{224} + \item no-intercept, \hyperpage{252} + \item nodal basis, \hyperpage{275} + \item nodes, \hyperpage{275} + \item Noetherian, \hyperpage{43} + \item Non-Archimedean geometry, \hyperpage{66} + \item non-autonomous, \hyperpage{122} + \item non-degenerate, \hyperpage{74} + \item non-degenerated, \hyperpage{227} + \item non-Desarguesian planes, \hyperpage{68} + \item non-divergence form, \hyperpage{349} + \item Non-parametric bootstrap, \hyperpage{202} + \item Non-Paschian geometry, \hyperpage{65} + \item Non-SAS geometry, \hyperpage{66} + \item non-singular, \hyperpage{131} + \item noncommutative ring, \hyperpage{42} + \item nonexpansive, \hyperpage{347} + \item nonsingular, \hyperpage{22} + \item norm, \hyperpage{24}, \hyperpage{53}, \hyperpage{94}, + \hyperpage{300}, \hyperpage{305} + \item norm associated with the inner product, \hyperpage{24} + \item normal, \hyperpage{38}, \hyperpage{121}, \hyperpage{167}, + \hyperpage{191}, \hyperpage{212} + \item normal closure, \hyperpage{167} + \item Normal confidence interval, \hyperpage{202} + \item normal confidence interval, \hyperpage{202} + \item normal curvature, \hyperpage{150} + \item Normal distribution, \hyperpage{178} + \item normal distribution, \hyperpage{178} + \item normal form, \hyperpage{223} + \item Normal plane, \hyperpage{143} + \item Normal Q-Q plots, \hyperpage{204} + \item normal unit field, \hyperpage{149} + \item normalized, \hyperpage{89}, \hyperpage{218} + \item Normalized power method, \hyperpage{101} + \item normalized power method, \hyperpage{101} + \item normalizer, \hyperpage{41} + \item normed algebra, \hyperpage{307} + \item normed vector space, \hyperpage{53}, \hyperpage{300} + \item not integrable, \hyperpage{182} + \item not orientation-preserving, \hyperpage{217} + \item Novikov's condition, \hyperpage{366} + \item null hypothesis, \hyperpage{199} + \item null recurrent, \hyperpage{324} + \item null set, \hyperpage{174}, \hyperpage{293} + \item number of steps, \hyperpage{268} + \item numeric series, \hyperpage{76} + \item numeric series of complex numbers, \hyperpage{105} + \item numerical boundary condition, \hyperpage{272} + \item numerically stable, \hyperpage{90} + \item numerically unstable, \hyperpage{90} + \item Nyquist-Shannon sampling theorem, \hyperpage{239} + + \indexspace + + \item objective function, \hyperpage{51} + \item observed information, \hyperpage{195} + \item odd extension, \hyperpage{84} + \item odd periodic extension, \hyperpage{283} + \item odds, \hyperpage{257} + \item ode, \hyperpage{122} + \item Olinde Rodrigues' theorem, \hyperpage{150} + \item one-parameter family of rotated vector fields, \hyperpage{227} + \item One-point compactification, \hyperpage{214} + \item one-point compactification, \hyperpage{214} + \item one-sided left tail test, \hyperpage{199} + \item one-sided right tail test, \hyperpage{199} + \item one-step, \hyperpage{268} + \item one-step method, \hyperpage{258} + \item open, \hyperpage{54}, \hyperpage{205}, \hyperpage{208} + \item open ball, \hyperpage{54} + \item open cover, \hyperpage{213} + \item Open function, \hyperpage{208} + \item Open interval, \hyperpage{25} + \item Open mapping theorem, \hyperpage{116}, \hyperpage{309} + \item open sets, \hyperpage{205} + \item operator, \hyperpage{124}, \hyperpage{305} + \item Opial's lemma, \hyperpage{347} + \item opposite orientations, \hyperpage{142}, \hyperpage{217} + \item orbit, \hyperpage{8}, \hyperpage{40}, \hyperpage{130} + \item Orbit linear structure, \hyperpage{8} + \item Orbit structure, \hyperpage{8} + \item Orbit-stabilizer theorem, \hyperpage{40} + \item Orbits formula, \hyperpage{40} + \item order, \hyperpage{8}, \hyperpage{37}, \hyperpage{49}, + \hyperpage{116}, \hyperpage{259} + \item Order axioms, \hyperpage{63} + \item order of accuracy, \hyperpage{259}, \hyperpage{262} + \item order of consistency, \hyperpage{259}, \hyperpage{262} + \item Order of convergence, \hyperpage{91} + \item order of convergence, \hyperpage{91}, \hyperpage{259} + \item order of convergence at least $p$, \hyperpage{91} + \item order relation, \hyperpage{63} + \item order statistic, \hyperpage{196} + \item order statistics, \hyperpage{196} + \item ordered field, \hyperpage{25}, \hyperpage{64} + \item ordinary differential equation, \hyperpage{122} + \item ordinary generating function, \hyperpage{48} + \item orientable, \hyperpage{149}, \hyperpage{158}, \hyperpage{217} + \item orientation, \hyperpage{120}, \hyperpage{142}, \hyperpage{149}, + \hyperpage{217} + \item orientation induced by $M$, \hyperpage{158} + \item Orientation principle, \hyperpage{120} + \item orientation-preserving, \hyperpage{142}, \hyperpage{217} + \item oriented vector space, \hyperpage{142}, \hyperpage{217} + \item origin, \hyperpage{70} + \item orthogonal, \hyperpage{22--24}, \hyperpage{72}, \hyperpage{82}, + \hyperpage{148}, \hyperpage{310} + \item orthogonal basis, \hyperpage{96} + \item orthogonal complement, \hyperpage{22}, \hyperpage{310} + \item orthogonal coordinates, \hyperpage{148} + \item orthogonal geometry, \hyperpage{23} + \item orthogonal group, \hyperpage{145} + \item Orthogonal polynomials, \hyperpage{96} + \item orthogonal projection, \hyperpage{23} + \item orthogonal projection on $F$, \hyperpage{311} + \item orthogonal reflections, \hyperpage{73} + \item orthogonal system, \hyperpage{313} + \item orthogonal with respect to the weight $\omega (x)$, + \hyperpage{96} + \item Orthogonality of martingales, \hyperpage{360} + \item orthonormal, \hyperpage{22}, \hyperpage{82} + \item orthonormal system, \hyperpage{82}, \hyperpage{313} + \item orthonormalization, \hyperpage{313} + \item osculating circle, \hyperpage{144} + \item Osculating plane, \hyperpage{143} + \item osculating sphere, \hyperpage{144} + \item Outer measure, \hyperpage{292} + \item outer measure, \hyperpage{292} + \item outliars, \hyperpage{255} + \item Over-relaxation methods, \hyperpage{100} + + \indexspace + + \item Paley-Wiener-Zygmund theorem, \hyperpage{331} + \item Pappus configuration, \hyperpage{68} + \item PAQ reduction theorem, \hyperpage{14} + \item parabolic, \hyperpage{273} + \item parabolic boundary, \hyperpage{287} + \item parabolic cylinder, \hyperpage{287} + \item parabolic point, \hyperpage{150} + \item parallel, \hyperpage{70}, \hyperpage{152} + \item parallel transport, \hyperpage{152} + \item Parallelogram law, \hyperpage{53}, \hyperpage{311} + \item parameter set, \hyperpage{316} + \item parameter space, \hyperpage{193} + \item parametric, \hyperpage{193} + \item Parametric bootstrap, \hyperpage{202} + \item parametric bootstrap distribution, \hyperpage{202} + \item parametric equations, \hyperpage{70} + \item parametrization, \hyperpage{141}, \hyperpage{147} + \item parametrized surface, \hyperpage{61} + \item Parseval identity, \hyperpage{313} + \item Parseval's identity, \hyperpage{87} + \item partial derivative, \hyperpage{55} + \item partial derivative of order $k$, \hyperpage{56} + \item partial differential equation, \hyperpage{139} + \item Partial fraction decomposition theorem, \hyperpage{117} + \item partial inverse Fourier transform, \hyperpage{233} + \item partial isometry, \hyperpage{362} + \item partial order relation, \hyperpage{9} + \item Partial pivoting, \hyperpage{102} + \item partially ordered set, \hyperpage{9} + \item Particular point topology, \hyperpage{206} + \item partition, \hyperpage{32}, \hyperpage{58}, \hyperpage{174} + \item partition of unity, \hyperpage{158} + \item Pasch's axiom, \hyperpage{63} + \item Passage from the affine plane to the projective plane, + \hyperpage{66} + \item Passage from the projective plane to the affine plane, + \hyperpage{66} + \item path, \hyperpage{49}, \hyperpage{215} + \item path from $x$ to $y$, \hyperpage{215} + \item path-connected, \hyperpage{215, 216} + \item path-connected components, \hyperpage{216} + \item PDE, \hyperpage{139} + \item pdf, \hyperpage{178} + \item Peano axioms, \hyperpage{6} + \item Peano theorem, \hyperpage{125} + \item Pearson correlation coefficient, \hyperpage{185} + \item Percentile confidence interval, \hyperpage{202} + \item percentile confidence interval, \hyperpage{202} + \item perfect, \hyperpage{166} + \item Perfect fields, \hyperpage{166} + \item period, \hyperpage{320} + \item Period three theorem, \hyperpage{229} + \item Period-doubling bifurcation, \hyperpage{229} + \item period-doubling bifurcation, \hyperpage{229} + \item period-doubling cascade, \hyperpage{229} + \item period-halving bifurcation, \hyperpage{229} + \item periodic, \hyperpage{131} + \item periodic orbit of period $T$, \hyperpage{221} + \item periodic point, \hyperpage{228} + \item periodic with period $T$, \hyperpage{81} + \item permutation matrix, \hyperpage{102} + \item Perron-Frobenius theorem, \hyperpage{101} + \item perspectivity, \hyperpage{69} + \item pgf, \hyperpage{316} + \item phase portrait, \hyperpage{130} + \item phase space, \hyperpage{130} + \item Picard iteration process, \hyperpage{125} + \item Picard theorem, \hyperpage{125} + \item PID, \hyperpage{44} + \item piecewise linear manifold, \hyperpage{216} + \item Pigeonhole principle, \hyperpage{10} + \item Pitchfork bifurcation, \hyperpage{223} + \item pitchfork bifurcation, \hyperpage{223} + \item pivot, \hyperpage{198} + \item Plancherel theorem, \hyperpage{236}, \hyperpage{238}, + \hyperpage{240} + \item plane curve, \hyperpage{141} + \item plane point, \hyperpage{150} + \item Playfair's axiom, \hyperpage{63} + \item Plücker coordinates, \hyperpage{69} + \item pmf, \hyperpage{176} + \item Poincaré compactification, \hyperpage{138} + \item Poincaré disk, \hyperpage{138} + \item Poincaré index formula, \hyperpage{139} + \item Poincaré inequality, \hyperpage{289} + \item Poincaré map, \hyperpage{136} + \item Poincaré's inequality, \hyperpage{343} + \item Poincaré's method, \hyperpage{226} + \item Poincaré-Bendixson theorem, \hyperpage{136} + \item Poincaré-Hopf theorem, \hyperpage{139}, \hyperpage{161} + \item Poincaré-Hopf theorem on $S^2$, \hyperpage{139} + \item Poincaré-Wirtinger's inequality, \hyperpage{343} + \item points, \hyperpage{63}, \hyperpage{205} + \item points of the quadric, \hyperpage{73} + \item pointwise bounded, \hyperpage{125}, \hyperpage{304} + \item pointwise equicontinuous, \hyperpage{125}, \hyperpage{304} + \item Poisson distribution, \hyperpage{177} + \item Poisson equation, \hyperpage{277} + \item Poisson kernel, \hyperpage{231}, \hyperpage{234} + \item Poisson process, \hyperpage{326} + \item Poisson summation formula, \hyperpage{238}, \hyperpage{240} + \item Polar form, \hyperpage{108} + \item polar form, \hyperpage{108} + \item Polar method, \hyperpage{353} + \item Polarization identity, \hyperpage{310} + \item Pole, \hyperpage{116} + \item pole, \hyperpage{116} + \item Polya's theorem on $\ensuremath {\mathbb {Z}}$, + \hyperpage{324} + \item Polya's theorem on $\ensuremath {\mathbb {Z}}^2$, + \hyperpage{324} + \item Polya's theorem on $\ensuremath {\mathbb {Z}}^3$, + \hyperpage{324} + \item polyhedron, \hyperpage{51} + \item polynomial, \hyperpage{11} + \item polynomial ring, \hyperpage{162} + \item polynomial system, \hyperpage{135} + \item polynomial vector field, \hyperpage{135} + \item polytope, \hyperpage{51} + \item poset, \hyperpage{9} + \item positive, \hyperpage{141}, \hyperpage{158} + \item positive basis, \hyperpage{142} + \item positive part, \hyperpage{77} + \item positive recurrent, \hyperpage{324} + \item positive semi-orbit, \hyperpage{130} + \item positive-definite, \hyperpage{23} + \item positive-semidefinite, \hyperpage{23} + \item positively invariant, \hyperpage{131} + \item positively oriented, \hyperpage{142} + \item positively rotated, \hyperpage{227} + \item positively stable, \hyperpage{131} + \item posterior, \hyperpage{202} + \item posterior distribution, \hyperpage{202} + \item posterior mean, \hyperpage{203} + \item posterior median, \hyperpage{203} + \item posterior mode, \hyperpage{203} + \item potential, \hyperpage{61}, \hyperpage{277} + \item power, \hyperpage{199} + \item power function, \hyperpage{199} + \item Power method, \hyperpage{101} + \item power method, \hyperpage{101} + \item power series, \hyperpage{78} + \item power set, \hyperpage{6} + \item pre-Hilbert space, \hyperpage{310} + \item predicate, \hyperpage{6} + \item prediction, \hyperpage{253} + \item prediction band, \hyperpage{254} + \item preditions, \hyperpage{250} + \item preimage, \hyperpage{7} + \item preserves orientation, \hyperpage{337} + \item primal, \hyperpage{51} + \item prime, \hyperpage{10}, \hyperpage{12}, \hyperpage{44} + \item Prime number theorem, \hyperpage{10} + \item prime period, \hyperpage{228} + \item primitive element, \hyperpage{164} + \item Primitive element theorem, \hyperpage{168} + \item primitive function, \hyperpage{33} + \item primitive polynomial, \hyperpage{47} + \item principal argument, \hyperpage{104} + \item principal curvatures, \hyperpage{149} + \item principal directons, \hyperpage{149} + \item Principal ideal, \hyperpage{43} + \item principal ideal, \hyperpage{43} + \item principal ideal domain, \hyperpage{44} + \item principal value, \hyperpage{108} + \item Principle of least action, \hyperpage{279} + \item prior, \hyperpage{202} + \item prior distribution, \hyperpage{202} + \item probability, \hyperpage{173} + \item probability density function, \hyperpage{178} + \item probability kernel, \hyperpage{341} + \item Probability mass function, \hyperpage{176} + \item probability mass function, \hyperpage{176} + \item probability space, \hyperpage{173}, \hyperpage{340} + \item probability-generating function, \hyperpage{316} + \item product, \hyperpage{11}, \hyperpage{13}, \hyperpage{106} + \item Product measure, \hyperpage{339} + \item product measure, \hyperpage{339} + \item product topology, \hyperpage{209} + \item products of group subsets, \hyperpage{40} + \item progressive, \hyperpage{363} + \item projection, \hyperpage{71} + \item Projection theorem, \hyperpage{311} + \item Projections, \hyperpage{71} + \item Projective axiom, \hyperpage{66} + \item projective frame, \hyperpage{67} + \item projective plane, \hyperpage{66} + \item projective subvariety, \hyperpage{66} + \item projectivity, \hyperpage{69} + \item Propagation of absolute errors, \hyperpage{89} + \item Propagation of relative errors, \hyperpage{89} + \item proper, \hyperpage{37} + \item proper subspace of $T$, \hyperpage{309} + \item Properties of addition and scalar multiplication of matrices, + \hyperpage{13} + \item Properties of homotheties, \hyperpage{72} + \item Properties of matrix norms, \hyperpage{98} + \item Properties of matrix product, \hyperpage{13} + \item Properties of projections, \hyperpage{71} + \item Properties of reflections, \hyperpage{71} + \item Properties of translations, \hyperpage{71} + \item Proximity theorem, \hyperpage{229} + \item pull-back, \hyperpage{156} + \item pure birth process, \hyperpage{329} + \item pure death process, \hyperpage{329} + \item purely inseparable, \hyperpage{168} + \item purely transcendental, \hyperpage{165} + \item Pythagorean, \hyperpage{64} + \item Pythagorean plane, \hyperpage{64} + \item Pythagorean theorem, \hyperpage{72}, \hyperpage{310} + + \indexspace + + \item Q-Q plots, \hyperpage{204} + \item QR decomposition, \hyperpage{267} + \item QR descompostion, \hyperpage{267} + \item quadratic convergence, \hyperpage{91} + \item quadratic form, \hyperpage{74} + \item Quadratic loss function, \hyperpage{203} + \item quadratic polynomial, \hyperpage{162} + \item quadratic space, \hyperpage{74} + \item Quadratic variation, \hyperpage{360} + \item quadratic variation, \hyperpage{365} + \item quadrature formula, \hyperpage{95} + \item quadrature formula with weight $\omega (x)$, \hyperpage{97} + \item quadric, \hyperpage{73} + \item Quantile, \hyperpage{196} + \item Quantile-Quantile plot, \hyperpage{204} + \item quartic polynomial, \hyperpage{162} + \item quasi-Newton method, \hyperpage{263} + \item quasilinear PDE, \hyperpage{140} + \item quaternion group, \hyperpage{42} + \item Quotient group, \hyperpage{39} + \item quotient group, \hyperpage{39} + \item quotient map, \hyperpage{210} + \item quotient space, \hyperpage{17}, \hyperpage{210}, + \hyperpage{302} + \item quotient space of collapsing a set to a point, \hyperpage{211} + \item quotient topology, \hyperpage{210} + + \indexspace + + \item Raabe's test, \hyperpage{76} + \item radical, \hyperpage{22}, \hyperpage{170} + \item radical tower, \hyperpage{170} + \item radius of convergence, \hyperpage{108} + \item radius of convergence of the power series, \hyperpage{79} + \item radius of curvature, \hyperpage{144} + \item radix-2 decimation-in-time (DIT) FFT, \hyperpage{240} + \item Radix-2 DIT Cooley-Tukey FFT algorithm, \hyperpage{240} + \item Radon-Nikodym, \hyperpage{340} + \item Radó theorem, \hyperpage{218} + \item random sample, \hyperpage{193} + \item Random variable, \hyperpage{340} + \item random variable, \hyperpage{175} + \item random vector, \hyperpage{179} + \item Random walk, \hyperpage{319} + \item random walk, \hyperpage{319} + \item Rank, \hyperpage{14} + \item rank, \hyperpage{14}, \hyperpage{16}, \hyperpage{23}, + \hyperpage{74} + \item Rankine-Hugoniot equation, \hyperpage{280} + \item rate, \hyperpage{179} + \item rate of convergence, \hyperpage{99} + \item Ratio test, \hyperpage{26}, \hyperpage{76} + \item ray, \hyperpage{63} + \item Rayleigh quotient, \hyperpage{101} + \item RC, \hyperpage{64} + \item reachable, \hyperpage{320} + \item real function, \hyperpage{294} + \item real part, \hyperpage{104} + \item real random variable, \hyperpage{175} + \item realizable, \hyperpage{67} + \item realization, \hyperpage{193} + \item rearrangement of the sequence, \hyperpage{77} + \item rearrangement of the series, \hyperpage{77} + \item rectangle, \hyperpage{58} + \item rectifiable, \hyperpage{34}, \hyperpage{60}, \hyperpage{141} + \item Rectifying plane, \hyperpage{143} + \item recurrence relation of order $k$, \hyperpage{48} + \item recurrent, \hyperpage{48}, \hyperpage{322}, \hyperpage{324} + \item Reduced row echelon form, \hyperpage{14} + \item reduced row echelon form, \hyperpage{14} + \item reducible, \hyperpage{100} + \item Reducible linearODE of second order, \hyperpage{123} + \item reflection, \hyperpage{71}, \hyperpage{120} + \item Reflections, \hyperpage{71} + \item reflexive, \hyperpage{307, 308} + \item refractive index, \hyperpage{278} + \item region, \hyperpage{115}, \hyperpage{148}, \hyperpage{204} + \item regression axis, \hyperpage{151} + \item regression coefficients, \hyperpage{250} + \item regression sum of squares, \hyperpage{252} + \item regressor coefficients, \hyperpage{250} + \item Regula falsi method, \hyperpage{90} + \item regular, \hyperpage{49}, \hyperpage{58}, \hyperpage{131}, + \hyperpage{141}, \hyperpage{195, 196}, \hyperpage{212}, + \hyperpage{328}, \hyperpage{358} + \item regular distributions, \hyperpage{241} + \item regular domain, \hyperpage{148} + \item regular region, \hyperpage{160} + \item regular surface, \hyperpage{147} + \item Regularity up to the boundary, \hyperpage{350} + \item Reillich-Kondrachov's compactness theorem, \hyperpage{345} + \item Related samples with unknown variances, \hyperpage{199} + \item relative condition numbers, \hyperpage{90} + \item relative error, \hyperpage{89} + \item relatively compact, \hyperpage{121}, \hyperpage{304} + \item relatively prime, \hyperpage{10, 11} + \item relaxation factor, \hyperpage{100} + \item Removable discontinuity, \hyperpage{28} + \item Removable singularity, \hyperpage{116} + \item removable singularity, \hyperpage{116} + \item reordering, \hyperpage{106} + \item reparametrization, \hyperpage{60}, \hyperpage{141} + \item repelling, \hyperpage{131} + \item repelling parabolic sector, \hyperpage{133} + \item repulsor fixed point, \hyperpage{91} + \item residuals, \hyperpage{250} + \item residue, \hyperpage{116} + \item Residues theorem, \hyperpage{117} + \item resolvent set, \hyperpage{350} + \item response, \hyperpage{253} + \item restrictions, \hyperpage{51} + \item Reuter criterion, \hyperpage{330} + \item Riccati differential equation, \hyperpage{123} + \item Richardson extrapolation, \hyperpage{95}, \hyperpage{260} + \item Riemann conformal representation theorem, \hyperpage{121} + \item Riemann sum, \hyperpage{34}, \hyperpage{58} + \item Riemann's theorem, \hyperpage{77} + \item Riemann's theorem on removable singularities, \hyperpage{116} + \item Riemann-integrable, \hyperpage{58} + \item Riemann-integrable function, \hyperpage{59} + \item Riemann-Lebesgue lemma, \hyperpage{85}, \hyperpage{232}, + \hyperpage{238} + \item Riesz representation theorem, \hyperpage{312} + \item Riesz theorem, \hyperpage{248} + \item Riesz transform, \hyperpage{248} + \item Riesz's theorem, \hyperpage{306} + \item Riesz-Fischer theorem, \hyperpage{313} + \item right cosets, \hyperpage{39} + \item Right side, \hyperpage{120} + \item right singular vectors, \hyperpage{265} + \item Right-open interval, \hyperpage{25} + \item right-sided limit of $f$ at $x_0$, \hyperpage{27} + \item Ring, \hyperpage{42} + \item ring, \hyperpage{42} + \item Ring morphism, \hyperpage{43} + \item ring morphism, \hyperpage{43} + \item RK methods, \hyperpage{260} + \item Robin, \hyperpage{274} + \item Robin boundary conditions, \hyperpage{356} + \item Rolle's theorem, \hyperpage{30} + \item Romberg Extrapolation, \hyperpage{355} + \item Romberg method, \hyperpage{96} + \item root, \hyperpage{12} + \item root condition, \hyperpage{262} + \item root of the reflection, \hyperpage{71} + \item Root test, \hyperpage{26}, \hyperpage{76} + \item rotation, \hyperpage{119} + \item rotation number, \hyperpage{338} + \item Rouché's theorem, \hyperpage{119} + \item Rouché-Frobenius theorem, \hyperpage{14} + \item Routh-Hurwitz stability criterion, \hyperpage{227} + \item Row echelon form, \hyperpage{14} + \item row echelon form, \hyperpage{14} + \item row rank, \hyperpage{16} + \item Ruffini's rule, \hyperpage{12} + \item ruled surface, \hyperpage{151} + \item Runge-Kutta-Fehlberg method, \hyperpage{260} + + \indexspace + + \item SAA criterion, \hyperpage{64} + \item Saccheri-Legendre theorem, \hyperpage{64} + \item saddle point, \hyperpage{57}, \hyperpage{133} + \item saddle separatrices, \hyperpage{133} + \item saddle-node, \hyperpage{224} + \item Saddle-node bifurcation, \hyperpage{223} + \item saddle-node bifurcation, \hyperpage{223} + \item same orientation, \hyperpage{142}, \hyperpage{217} + \item sample correlation coefficient, \hyperpage{249} + \item sample covariance, \hyperpage{249} + \item sample mean, \hyperpage{192, 193} + \item sample quantiles, \hyperpage{204} + \item Sample space, \hyperpage{173} + \item sample space, \hyperpage{173}, \hyperpage{193} + \item sample variance, \hyperpage{193}, \hyperpage{249} + \item Sample variance and covariance, \hyperpage{249} + \item SAS criterion, \hyperpage{63, 64} + \item scalars, \hyperpage{15} + \item scale, \hyperpage{179} + \item Schrödinger equation, \hyperpage{279} + \item Schwartz space, \hyperpage{243} + \item Schwarz lemma, \hyperpage{120} + \item Schwarz theorem, \hyperpage{243} + \item Schwarz's theorem, \hyperpage{57} + \item Schwarz-Pick lemma, \hyperpage{121} + \item score function, \hyperpage{195} + \item Score test, \hyperpage{202} + \item score test, \hyperpage{202} + \item SDE, \hyperpage{367} + \item Secant method, \hyperpage{90} + \item secant-like method, \hyperpage{263} + \item Second Borel-Cantelli lemma, \hyperpage{189} + \item second derivative of $f$ at $a$, \hyperpage{31} + \item second fundamental form, \hyperpage{150} + \item Second isomorphism theorem, \hyperpage{18}, \hyperpage{40}, + \hyperpage{44} + \item Second Sylow theorem, \hyperpage{41} + \item section, \hyperpage{299} + \item sectorial decomposition, \hyperpage{133} + \item seed, \hyperpage{352} + \item segment, \hyperpage{72} + \item Segmented regression, \hyperpage{253} + \item self-adjoint, \hyperpage{312} + \item self-conjugate, \hyperpage{304} + \item self-similar, \hyperpage{284} + \item Semi-hyperbolic singular points classification theorem, + \hyperpage{223} + \item semi-stable, \hyperpage{136} + \item semiaffinity, \hyperpage{70} + \item Semidiscrete Fourier transform, \hyperpage{270} + \item semidiscrete Fourier transform, \hyperpage{270} + \item Semidiscrete Parseval identity, \hyperpage{270} + \item semidynamical system, \hyperpage{130} + \item semilinear, \hyperpage{310} + \item semilinear isomorphism, \hyperpage{67} + \item Seminorm, \hyperpage{308} + \item seminorm, \hyperpage{94}, \hyperpage{308} + \item sensitive dependence on initial conditions, \hyperpage{230}, + \hyperpage{336} + \item Separability theorem, \hyperpage{168} + \item separable, \hyperpage{168}, \hyperpage{301} + \item Separable extension, \hyperpage{168} + \item separate the points, \hyperpage{303} + \item separated, \hyperpage{207} + \item separated by closed neighbourhoods, \hyperpage{207} + \item separating set, \hyperpage{303} + \item Separation of variables, \hyperpage{123}, \hyperpage{283} + \item separation of variables, \hyperpage{283} + \item sequence, \hyperpage{54} + \item sequence of complex functions, \hyperpage{106} + \item sequence of complex numbers, \hyperpage{105} + \item sequence of real functions, \hyperpage{77} + \item sequence of real numbers, \hyperpage{26} + \item series of functions, \hyperpage{78}, \hyperpage{107} + \item set, \hyperpage{6} + \item set difference, \hyperpage{6} + \item set of interior vectors, \hyperpage{157} + \item set of left cosets, \hyperpage{39} + \item set of periodic points, \hyperpage{228} + \item set of periods, \hyperpage{228} + \item set of polynomials, \hyperpage{42} + \item set of right cosets, \hyperpage{39} + \item set of zero-content, \hyperpage{174} + \item shape, \hyperpage{179} + \item Sharkovskii's order, \hyperpage{229} + \item Sharkovskii's theorem, \hyperpage{229} + \item Shift map, \hyperpage{229} + \item shift map, \hyperpage{229}, \hyperpage{335} + \item Shooting method, \hyperpage{264} + \item shooting method, \hyperpage{264} + \item Sierpiński topology, \hyperpage{206} + \item sign, \hyperpage{8}, \hyperpage{89} + \item signature, \hyperpage{23} + \item signed curvature, \hyperpage{145} + \item significance level, \hyperpage{198, 199} + \item significant, \hyperpage{89} + \item similar, \hyperpage{20}, \hyperpage{71} + \item similitude ratio of the homothety, \hyperpage{72} + \item simple, \hyperpage{49}, \hyperpage{90}, \hyperpage{116}, + \hyperpage{164}, \hyperpage{170}, \hyperpage{199} + \item simple curve, \hyperpage{218} + \item simple function, \hyperpage{295} + \item Simple model, \hyperpage{250} + \item simple random sample, \hyperpage{193} + \item simple random variable, \hyperpage{182} + \item simple random walk, \hyperpage{319} + \item Simple ratio, \hyperpage{72} + \item simple ratio, \hyperpage{72} + \item simple region, \hyperpage{160} + \item Simplex method: Phase I, \hyperpage{51} + \item Simplex method: Phase II, \hyperpage{51} + \item simply connected, \hyperpage{116}, \hyperpage{216} + \item simply normal, \hyperpage{191} + \item Simpson's rule, \hyperpage{95} + \item singular point, \hyperpage{131}, \hyperpage{160} + \item Singular value, \hyperpage{265} + \item Singular value decomposition, \hyperpage{265} + \item singular value decomposition, \hyperpage{265} + \item singular values, \hyperpage{265} + \item sink, \hyperpage{133} + \item size, \hyperpage{49}, \hyperpage{199} + \item Skorokhod's representation theorem, \hyperpage{190} + \item Slutsky's theorem, \hyperpage{190} + \item small change, \hyperpage{279} + \item Sobolev space, \hyperpage{289} + \item Sobolev spaces, \hyperpage{343} + \item solution, \hyperpage{90}, \hyperpage{139} + \item solution of a system of equations, \hyperpage{13} + \item solution of the SDE, \hyperpage{367} + \item solution of theODE, \hyperpage{122} + \item solvable, \hyperpage{41}, \hyperpage{170} + \item solvable by radicals, \hyperpage{170} + \item SOR, \hyperpage{100} + \item source, \hyperpage{133} + \item space of rapidly decreasing functions, \hyperpage{243} + \item special orthogonal group, \hyperpage{145} + \item specific heat capacity, \hyperpage{278} + \item spectral radius, \hyperpage{98} + \item Spectral theorem, \hyperpage{24}, \hyperpage{314} + \item spectral values, \hyperpage{309} + \item spectrum, \hyperpage{97}, \hyperpage{309}, \hyperpage{350} + \item sphere, \hyperpage{54} + \item Spline, \hyperpage{94} + \item spline, \hyperpage{94} + \item splits into linear factors, \hyperpage{167} + \item Splitting field, \hyperpage{167} + \item splitting field, \hyperpage{167} + \item Squeeze theorem, \hyperpage{26} + \item SSS criterion, \hyperpage{64} + \item stability function, \hyperpage{261} + \item stability number, \hyperpage{134} + \item stability region, \hyperpage{261}, \hyperpage{268} + \item stabilizer, \hyperpage{40} + \item stable, \hyperpage{136}, \hyperpage{261}, \hyperpage{269}, + \hyperpage{274}, \hyperpage{328} + \item stable degenerated node, \hyperpage{133} + \item stable focus, \hyperpage{133} + \item stable manifold, \hyperpage{222} + \item stable node, \hyperpage{133} + \item stable star, \hyperpage{133} + \item standard, \hyperpage{330} + \item standard $n$-simplex, \hyperpage{218} + \item Standard deviation, \hyperpage{185} + \item standard deviation, \hyperpage{178}, \hyperpage{185} + \item standard error, \hyperpage{185} + \item standard form, \hyperpage{51} + \item Standard normal distribution, \hyperpage{178} + \item standard normal distribution, \hyperpage{178} + \item standardized form, \hyperpage{179} + \item star domain, \hyperpage{115} + \item state space, \hyperpage{316}, \hyperpage{318} + \item states, \hyperpage{318} + \item stationary distribution, \hyperpage{325}, \hyperpage{329} + \item stationary increments, \hyperpage{326} + \item Statistic, \hyperpage{193} + \item statistic, \hyperpage{193} + \item statistical model, \hyperpage{193} + \item Steepest descent, \hyperpage{264} + \item Steffensen's method, \hyperpage{92} + \item Steinitz exchange lemma, \hyperpage{16} + \item Steinitz's theorem, \hyperpage{164} + \item step size, \hyperpage{258}, \hyperpage{264} + \item Stereographic projection, \hyperpage{105} + \item stereographic projection, \hyperpage{105} + \item stiff equations, \hyperpage{261} + \item stiffness matrix, \hyperpage{276} + \item Stirling's formula, \hyperpage{81} + \item stochastic differential, \hyperpage{365} + \item stochastic differential equation, \hyperpage{367} + \item Stochastic dominated convergence theorem, \hyperpage{364} + \item Stochastic integration by parts, \hyperpage{366} + \item Stochastic matrix, \hyperpage{318} + \item stochastic matrix, \hyperpage{318} + \item Stochastic process, \hyperpage{316} + \item stochastic process, \hyperpage{316} + \item stochastically equivalent, \hyperpage{331} + \item Stokes theorem, \hyperpage{159} + \item Stokes' theorem, \hyperpage{62} + \item Stolz-Cesàro theorem, \hyperpage{27} + \item Stone-Weierstra\ss \ theorem, \hyperpage{304} + \item stopped process, \hyperpage{341}, \hyperpage{360} + \item stopping time, \hyperpage{321}, \hyperpage{341} + \item strict Lyapunov function, \hyperpage{137} + \item strictly concave, \hyperpage{30} + \item strictly convex, \hyperpage{30} + \item strictly decreasing, \hyperpage{26}, \hyperpage{28} + \item strictly diagonally dominant by columns, \hyperpage{99} + \item strictly diagonally dominant by rows, \hyperpage{99} + \item strictly increasing, \hyperpage{26}, \hyperpage{28} + \item Strong duality theorem, \hyperpage{52} + \item Strong error of the Euler scheme, \hyperpage{355} + \item Strong law of large numbers, \hyperpage{191} + \item Strong law of large numbers for Brownian motion, + \hyperpage{359} + \item Strong Markov property, \hyperpage{321, 322} + \item Strong maximum principle, \hyperpage{351} + \item strong solution, \hyperpage{351} + \item strongly consistent estimator, \hyperpage{194} + \item strongly convex, \hyperpage{346} + \item strongly lower-semicontinuous, \hyperpage{342} + \item Structal stability, \hyperpage{336} + \item Strum's sequence, \hyperpage{92} + \item Student's $t$-distribution with $n$ degrees of freedom, + \hyperpage{197} + \item Sturm's sequence, \hyperpage{92} + \item Sturm's theorem, \hyperpage{92} + \item Sturm-Picone comparison theorem, \hyperpage{283} + \item sub-martingale, \hyperpage{359} + \item sub-multiplicativity, \hyperpage{97} + \item subadditive, \hyperpage{338} + \item subalgebra, \hyperpage{303} + \item subbasis, \hyperpage{206} + \item subcover, \hyperpage{213} + \item subdivision, \hyperpage{357} + \item subfield generated, \hyperpage{163} + \item subgradient descent method, \hyperpage{347} + \item Subgroup, \hyperpage{37} + \item subgroup, \hyperpage{37} + \item subgroup generated, \hyperpage{37} + \item sublinear, \hyperpage{306} + \item submanifold, \hyperpage{146} + \item submanifold with boundary, \hyperpage{157} + \item submartingale, \hyperpage{341} + \item Submersion, \hyperpage{146} + \item submersion, \hyperpage{146} + \item subordinated matrix norm, \hyperpage{97} + \item Subring, \hyperpage{42} + \item subring, \hyperpage{42} + \item subring generated, \hyperpage{163} + \item Subsequence, \hyperpage{26} + \item subsequence, \hyperpage{26} + \item subset, \hyperpage{6} + \item subspace topology, \hyperpage{208} + \item Substitution principle, \hyperpage{315} + \item success, \hyperpage{177} + \item Successive over-relaxation \\method, \hyperpage{100} + \item Successive over-relaxation method, \hyperpage{100} + \item sufficient, \hyperpage{194} + \item sum, \hyperpage{11}, \hyperpage{13}, \hyperpage{16}, + \hyperpage{70} + \item sum of the series, \hyperpage{76}, \hyperpage{106} + \item sum of the series in a pointwise sense, \hyperpage{78}, + \hyperpage{107} + \item sum of the series in an uniform sense, \hyperpage{78}, + \hyperpage{107} + \item super-martingale, \hyperpage{359} + \item superadditive, \hyperpage{338} + \item supermartingale, \hyperpage{341} + \item support, \hyperpage{8}, \hyperpage{79}, \hyperpage{146}, + \hyperpage{176}, \hyperpage{180}, \hyperpage{242} + \item support plane, \hyperpage{342} + \item support points, \hyperpage{93} + \item supremum, \hyperpage{25} + \item Supremum axiom, \hyperpage{25} + \item supremum norm, \hyperpage{303} + \item surface, \hyperpage{58}, \hyperpage{147}, \hyperpage{217} + \item surface integral, \hyperpage{61, 62} + \item surface intergal, \hyperpage{159} + \item Surface of revolution, \hyperpage{34} + \item surface of revolution, \hyperpage{147} + \item surjective, \hyperpage{7}, \hyperpage{121} + \item SVD, \hyperpage{265} + \item Sylow $p$-subgroup, \hyperpage{41} + \item Sylvester's criterion, \hyperpage{58} + \item Sylvester's law of inertia, \hyperpage{23} + \item symmetric, \hyperpage{22}, \hyperpage{62}, \hyperpage{313} + \item Symmetric difference formula of order 1, \hyperpage{95} + \item Symmetric difference formula of order 2, \hyperpage{95} + \item symmetric group, \hyperpage{8}, \hyperpage{38} + \item symmetric polynomial, \hyperpage{163} + \item Symmetric polynomials, \hyperpage{163} + \item symmetric with respect to the $x$-axis, \hyperpage{221} + \item symmetric with respect to the $y$-axis, \hyperpage{221} + \item Symmetry principle, \hyperpage{120} + \item system of linear differential equations, \hyperpage{126} + \item system of linear equations, \hyperpage{13} + \item system of ordinary differential equations, \hyperpage{122} + + \indexspace + + \item tangent, \hyperpage{154}, \hyperpage{156} + \item tangent function, \hyperpage{148} + \item tangent line, \hyperpage{141}, \hyperpage{144} + \item tangent line to the graph at the point $(x_0,f(x_0))$, + \hyperpage{29} + \item tangent plane, \hyperpage{147} + \item tangent space, \hyperpage{147}, \hyperpage{156, 157} + \item tangent vector, \hyperpage{141}, \hyperpage{147}, + \hyperpage{156} + \item tangent vector field, \hyperpage{139} + \item Taxicab metric, \hyperpage{205} + \item Taylor method, \hyperpage{259} + \item Taylor method of order $r$, \hyperpage{259} + \item Taylor polynomial of order $n$ of $f$ centered at $a$, + \hyperpage{31} + \item Taylor's theorem, \hyperpage{57} + \item tempered distribution, \hyperpage{244} + \item tends to the boundary, \hyperpage{126} + \item terminal point, \hyperpage{215} + \item test, \hyperpage{200} + \item test functions, \hyperpage{240} + \item Test of homogenity, \hyperpage{201} + \item Test of independence, \hyperpage{201} + \item test problem, \hyperpage{261} + \item test statistic, \hyperpage{199} + \item the Best Linear Unbiased Estimator, \hyperpage{251} + \item The center manifold theorem, \hyperpage{223} + \item The stable manifold theorem, \hyperpage{222} + \item theorem, \hyperpage{67} + \item theoretical quantiles, \hyperpage{204} + \item thermal conductivity, \hyperpage{278} + \item thermal diffusivity, \hyperpage{278} + \item thin QR decomposition, \hyperpage{267} + \item Third isomorphism theorem, \hyperpage{18}, \hyperpage{40}, + \hyperpage{44} + \item Third Sylow theorem, \hyperpage{41} + \item Tietze extension theorem, \hyperpage{212} + \item time-homogeneous Markov chain, \hyperpage{318} + \item TNB frame, \hyperpage{143} + \item tolerance, \hyperpage{255} + \item Tonelli's theorem, \hyperpage{175}, \hyperpage{299} + \item topological embedding, \hyperpage{214} + \item topological homeomorphism, \hyperpage{306} + \item Topological manifold, \hyperpage{216} + \item topological manifold, \hyperpage{216} + \item topological property, \hyperpage{213} + \item Topological space, \hyperpage{205} + \item topological space, \hyperpage{205} + \item topological subspace, \hyperpage{208} + \item topologically conjugate, \hyperpage{132} + \item topologically equivalent, \hyperpage{131} + \item topologically transitive, \hyperpage{230} + \item topology, \hyperpage{205} + \item topology generated, \hyperpage{206} + \item Topology induced from the metric, \hyperpage{205} + \item torsion, \hyperpage{143} + \item Torus, \hyperpage{210} + \item torus, \hyperpage{210} + \item total mean of squares, \hyperpage{252} + \item total order relation, \hyperpage{9} + \item total subset, \hyperpage{301} + \item total sum of squares, \hyperpage{252} + \item totally disconnected, \hyperpage{215} + \item totally isotropic, \hyperpage{74} + \item Tower formula, \hyperpage{165} + \item tower of fields, \hyperpage{165} + \item Tower property, \hyperpage{341} + \item trace, \hyperpage{141}, \hyperpage{289} + \item trace operator, \hyperpage{345} + \item Trace theorem, \hyperpage{289} + \item Traffic flow equation, \hyperpage{280} + \item trail, \hyperpage{49} + \item trajectories, \hyperpage{326} + \item transcendental, \hyperpage{164} + \item Transcritical bifurcation, \hyperpage{223} + \item transcritical bifurcation, \hyperpage{223} + \item transient, \hyperpage{322}, \hyperpage{324} + \item transition functions, \hyperpage{216} + \item transition matrix, \hyperpage{318} + \item transition probabilities, \hyperpage{318}, \hyperpage{326} + \item transitive, \hyperpage{169} + \item translation, \hyperpage{71}, \hyperpage{119} + \item Translations, \hyperpage{71} + \item transpose, \hyperpage{14} + \item Transposition, \hyperpage{14} + \item transposition, \hyperpage{8} + \item transversal section, \hyperpage{132} + \item Trapezoidal method, \hyperpage{259} + \item Trapezoidal rule, \hyperpage{95} + \item traversable, \hyperpage{50} + \item tree, \hyperpage{50} + \item triangle, \hyperpage{160} + \item Triangular inequality, \hyperpage{25}, \hyperpage{72} + \item triangular inequality, \hyperpage{53}, \hyperpage{205}, + \hyperpage{300} + \item triangular system, \hyperpage{97} + \item triangularization, \hyperpage{218} + \item triangulation, \hyperpage{357} + \item triple, \hyperpage{90} + \item trivial $\sigma $-algebra, \hyperpage{172} + \item Trivial topology, \hyperpage{206} + \item true, \hyperpage{177} + \item truncated, \hyperpage{266} + \item truncated Hilbert transform, \hyperpage{247} + \item truncated singular value decomposition, \hyperpage{266} + \item TSVD, \hyperpage{266} + \item two times differentiable, \hyperpage{31} + \item two times differentiable at $a$, \hyperpage{31} + \item two-sided test, \hyperpage{200} + \item Tychonoff's theorem, \hyperpage{213} + + \indexspace + + \item UFD, \hyperpage{45} + \item umbilic point, \hyperpage{149} + \item UMP, \hyperpage{200} + \item unbiased estimator, \hyperpage{193} + \item Uncertainty principle, \hyperpage{238} + \item Uncertainty principle in $\ensuremath {\mathbb {R}}^n$, + \hyperpage{238} + \item unconditionally absolutely stable, \hyperpage{261} + \item unconditionally convergent, \hyperpage{268} + \item uncorrelated, \hyperpage{185} + \item uncountable, \hyperpage{25} + \item Unicity of the splitting field, \hyperpage{167} + \item uniform norm, \hyperpage{301}, \hyperpage{303} + \item uniformly bounded, \hyperpage{125}, \hyperpage{304} + \item uniformly continuous, \hyperpage{33}, \hyperpage{55} + \item uniformly convergent, \hyperpage{117} + \item uniformly elliptic, \hyperpage{349} + \item uniformly equicontinuous, \hyperpage{125}, \hyperpage{304} + \item uniformly integrable, \hyperpage{341} + \item uniformly most powerful, \hyperpage{200} + \item union, \hyperpage{6} + \item unique factorization domain, \hyperpage{45} + \item Uniqueness of Dirichlet problem, \hyperpage{288} + \item Uniqueness of the heat equation, \hyperpage{287} + \item Uniqueness of the heat equation on the unbounded domains, + \hyperpage{288} + \item unit, \hyperpage{43} + \item unit normal exterior vector field, \hyperpage{158} + \item unit normal vector, \hyperpage{143} + \item unit-speed parametrization, \hyperpage{141} + \item Universal property of polynomials, \hyperpage{45} + \item Universal property of polynomials in several variables, + \hyperpage{45}, \hyperpage{162} + \item Universal property of the field of fractions, \hyperpage{163} + \item unknowns, \hyperpage{13} + \item unoriented vector space, \hyperpage{142}, \hyperpage{217} + \item unstable, \hyperpage{136} + \item unstable degenerated node, \hyperpage{133} + \item unstable focus, \hyperpage{133} + \item unstable manifold, \hyperpage{222} + \item unstable node, \hyperpage{133} + \item unstable star, \hyperpage{133} + \item upper bound, \hyperpage{25}, \hyperpage{126} + \item upper integral, \hyperpage{32}, \hyperpage{58} + \item upper sum, \hyperpage{32}, \hyperpage{58} + \item upper triangular, \hyperpage{97} + \item upwind condition, \hyperpage{269} + \item Urysohn's lemma, \hyperpage{212} + \item Urysohn's metrization theorem, \hyperpage{212} + + \indexspace + + \item Van der Pol oscillator, \hyperpage{228} + \item vanishes nowhere, \hyperpage{304} + \item variable, \hyperpage{11} + \item variables, \hyperpage{13} + \item Variacions with repetition, \hyperpage{10} + \item Variance, \hyperpage{185} + \item variance, \hyperpage{178}, \hyperpage{185} + \item variance inflation factor, \hyperpage{255} + \item Variation of constants, \hyperpage{123} + \item variational equations, \hyperpage{129} + \item variational formulation, \hyperpage{275} + \item Variations without repetition, \hyperpage{10} + \item vector field, \hyperpage{60}, \hyperpage{152}, \hyperpage{154}, + \hyperpage{156} + \item vector space, \hyperpage{15} + \item vector space of polynomials with real coefficients and degree less than or equal to $n$, + \hyperpage{93} + \item vector subspace, \hyperpage{16} + \item vectors, \hyperpage{15} + \item version, \hyperpage{331} + \item vertices, \hyperpage{160} + \item VIF, \hyperpage{255} + \item Viscid flow, \hyperpage{277} + \item Volterra operator with kernel $K$, \hyperpage{307} + \item volume, \hyperpage{175}, \hyperpage{291} + \item volume element, \hyperpage{156}, \hyperpage{158} + \item Volume of revolution, \hyperpage{34} + + \indexspace + + \item Wald test, \hyperpage{201} + \item Wald theorem, \hyperpage{315} + \item walk, \hyperpage{49} + \item Waring's method, \hyperpage{164} + \item Wave equation, \hyperpage{140}, \hyperpage{277} + \item wave equation, \hyperpage{140} + \item wave function, \hyperpage{279} + \item Weak duality theorem, \hyperpage{51} + \item Weak error of the Euler scheme, \hyperpage{355} + \item weak formulation, \hyperpage{275}, \hyperpage{349} + \item Weak law of large numbers, \hyperpage{190} + \item Weak maximum principle, \hyperpage{350, 351} + \item weak-* lower-semicontinuity, \hyperpage{342} + \item weakly consistent estimator, \hyperpage{194} + \item weakly lower-semicontinuity, \hyperpage{342} + \item Weierstra\ss ' theorem, \hyperpage{55}, \hyperpage{114}, + \hyperpage{214} + \item Weierstra\ss \ approximation theorem, \hyperpage{79} + \item Weierstra\ss \ kernel, \hyperpage{234} + \item Weierstra\ss \ M-test, \hyperpage{78}, \hyperpage{107} + \item Weierstra\ss \ theorem, \hyperpage{28} + \item Weingarten endomorphsim, \hyperpage{149} + \item well-conditioned, \hyperpage{90} + \item well-order relation, \hyperpage{9} + \item well-ordered set, \hyperpage{9} + \item well-posed, \hyperpage{258} + \item well-posed in the Hadamard sense, \hyperpage{258} + \item Wiener integral, \hyperpage{362} + \item Wiener isometry, \hyperpage{362} + \item Wiener process, \hyperpage{330} + \item winding number, \hyperpage{113} + \item Wintner lemma, \hyperpage{126} + \item Wirtinger operators, \hyperpage{111} + \item Wirtinger's inequality, \hyperpage{87, 88} + \item Witt's theorem, \hyperpage{74} + + \indexspace + + \item Young's convolution inequality, \hyperpage{237} + \item Young's inequality for products, \hyperpage{302} + + \indexspace + + \item zero, \hyperpage{90} + \item zero divisor, \hyperpage{44} + \item zero of multiplicity $m\in \ensuremath {\mathbb {N}}$, + \hyperpage{90} + \item Zero-one loss function, \hyperpage{203} + \item zero-stable, \hyperpage{262} + \item Zorn's lemma, \hyperpage{126} + +\end{theindex} diff --git a/main_math.tex b/main_math.tex index fb48fd8..99f1fe4 100644 --- a/main_math.tex +++ b/main_math.tex @@ -125,6 +125,8 @@ \chapter{Fifth year} \subfile{Mathematics/5th/Stochastic_calculus/Stochastic_calculus.tex} \cleardoublepage +\subfile{Mathematics/5th/Stochastic_control/Stochastic_control.tex} +\cleardoublepage \printindex diff --git a/preamble_formulas.sty b/preamble_formulas.sty index 6bb2e41..385da8d 100644 --- a/preamble_formulas.sty +++ b/preamble_formulas.sty @@ -100,6 +100,7 @@ {MM}{\sta} % Montecarlo methods {CO}{\apl} % Continuous optimization {NMPDE}{\apl} % Numerical methods for pdes + {SCO}{\sta} % Stochastic control }{\col}% } \ExplSyntaxOff