From c752fb6cecdbaf6c1ffeae3694d91865f737a0eb Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?V=C3=ADctor?= Date: Sat, 7 Oct 2023 14:00:42 +0200 Subject: [PATCH] updated stochastic calc + elliptic pdes --- ...topics_in_functional_analysis_and_PDEs.tex | 10 +- ...ntroduction_to_nonlinear_elliptic_PDEs.tex | 105 +- .../Stochastic_calculus.tex | 21 + main_math.idx | 3138 ----------------- main_math.ilg | 6 - main_math.ind | 2672 -------------- preamble_formulas.sty | 2 +- 7 files changed, 131 insertions(+), 5823 deletions(-) delete mode 100644 main_math.idx delete mode 100644 main_math.ilg delete mode 100644 main_math.ind diff --git a/Mathematics/5th/Advanced_topics_in_functional_analysis_and_PDEs/Advanced_topics_in_functional_analysis_and_PDEs.tex b/Mathematics/5th/Advanced_topics_in_functional_analysis_and_PDEs/Advanced_topics_in_functional_analysis_and_PDEs.tex index 7c4953b..5620a50 100644 --- a/Mathematics/5th/Advanced_topics_in_functional_analysis_and_PDEs/Advanced_topics_in_functional_analysis_and_PDEs.tex +++ b/Mathematics/5th/Advanced_topics_in_functional_analysis_and_PDEs/Advanced_topics_in_functional_analysis_and_PDEs.tex @@ -325,21 +325,21 @@ \begin{definition} We define the \emph{trace operator} as the map: \begin{align*} - \function{T}{W^{1,p}(\RR_{\geq 0}^d)}{L^p(\RR^{d-1})}{u}{u|_{\RR_0^d}} + \function{\Tr}{W^{1,p}(\RR_{\geq 0}^d)}{L^p(\RR^{d-1})}{u}{u|_{\RR_0^d}} \end{align*} \end{definition} \begin{theorem} - Let $1\leq p<\infty$ and $u\in W^{1,p}(\RR_{\geq 0}^d)$. Then, $Tu=0$ if and only if $u\in W_0^{1,p}(\RR_{\geq 0}^d)$. + Let $1\leq p<\infty$ and $u\in W^{1,p}(\RR_{\geq 0}^d)$. Then, $\Tr u=0$ if and only if $u\in W_0^{1,p}(\RR_{\geq 0}^d)$. \end{theorem} \begin{theorem} Let $1\leq p<\infty$ and $\Omega\subset\RR^d$ be a bounded domain with $\mathcal{C}^1$ boundary. Then, the trace operator $$ - \function{T}{W^{1,p}(\Omega)}{L^p(\Fr{\Omega})}{u}{u|_{\Fr{\Omega}}} + \function{\Tr}{W^{1,p}(\Omega)}{L^p(\Fr{\Omega})}{u}{u|_{\Fr{\Omega}}} $$ is bounded. Here we are taking the norm of $L^p(\Fr{\Omega})$ as ${\norm{u}_{L^p(\Fr{\Omega})}}^p:=\int_{\Fr{\Omega}}{\abs{u}^p}$. In addition: \begin{itemize} - \item $\forall u\in W^{1,p}(\Omega)$, $Tu=0$ if and only if $u\in W_0^{1,p}(\Omega)$. - \item For $p=2$, $T$ is surjective. + \item $\forall u\in W^{1,p}(\Omega)$, $\Tr u=0$ if and only if $u\in W_0^{1,p}(\Omega)$. + \item For $p=2$, $\Tr$ is surjective. \end{itemize} \end{theorem} \begin{lemma} diff --git a/Mathematics/5th/Introduction_to_nonlinear_elliptic_PDEs/Introduction_to_nonlinear_elliptic_PDEs.tex b/Mathematics/5th/Introduction_to_nonlinear_elliptic_PDEs/Introduction_to_nonlinear_elliptic_PDEs.tex index d10ecbb..8f78029 100644 --- a/Mathematics/5th/Introduction_to_nonlinear_elliptic_PDEs/Introduction_to_nonlinear_elliptic_PDEs.tex +++ b/Mathematics/5th/Introduction_to_nonlinear_elliptic_PDEs/Introduction_to_nonlinear_elliptic_PDEs.tex @@ -101,7 +101,7 @@ \begin{proposition} $\mathcal{N}_f$ has at least one solution if and only if for any weak solution $v$ of $\mathcal{N}_0$ we have $\langle f,v\rangle=0$. \end{proposition} - \subsection{Spectrum of compact operators} + \subsubsection{Spectrum of compact operators} In this section $\KK$ will denote either $\RR$ or $\CC$. \begin{definition} Let $H$ be a $\KK$-Hilbert space and $K:H\to H$ be a compact operator. We define the \emph{resolvent set} of $K$ as: @@ -120,5 +120,108 @@ $$ If $\sigma(K)\cap\RR^*$ is infinite, then it is of the form $\{\lambda_n\}_{n\in \NN}$ with $\lambda_n\to 0$. \end{theorem} + \begin{lemma} + Let $H$ be a Hilbert space and $K:H\to H$ be a continuous self-adjoint operator. Then: + $$ + \norm{K}=\sup_{\norm{x}=1}\langle x,Kx\rangle + $$ + \end{lemma} + \begin{lemma} + Let $H\ne\{ 0\}$ be Hilbert and $K:H\to H$ be a compact and self-adjoint operator. Then: + $$ + \sup_{\norm{x}=1}\langle x,Kx\rangle=\lambda + $$ + where $\lambda$ is the largest eigenvalue of $K$. + \end{lemma} + \subsubsection{Regularity theorems for weak solutions of divergence-form elliptic PDEs} + \begin{theorem}[Inner regularity] + Assume, in addition to the usual assumptions, that $a_{ij}\in \mathcal{C}^1(\Omega)$. Let $f\in L^2(\Omega)$ and $u$ be a weak solution of $Lu=f$. Then, $u\in H^2_{\text{loc}}(\Omega)$ and for any compact $\omega\subset\subset \Omega$, meaning that $\overline{\omega}\subset\Omega$ compact, we have $u\in H^2(\omega)$ and: + $$ + \norm{u}_{H^2(\omega)}\leq C\left(\norm{f}_{L^2(\Omega)}+\norm{u}_{L^2(\Omega)}\right) + $$ + \end{theorem} + \begin{corollary} + Assume that $a_{ij}\in\mathcal{C}^m(\Omega)$ for some $m\geq 2$, and $b_j,c\in \mathcal{C}^{m-1}(\Omega)$. Let $f\in H^{m-1}(\Omega)$ and $u\in H^1_{\text{loc}}(\Omega)\cap L^2(\Omega)$ be a weak solution of $Lu=f$. Then, $u\in H^{m+1}_{\text{loc}}(\Omega)$ and for any $\omega\subset\subset \Omega$ we have $u\in H^{m+2}(\omega)$ and: + $$ + \norm{u}_{H^{m+2}(\omega)}\leq C\left(\norm{f}_{H^{m-1}(\Omega)}+\norm{u}_{L^2(\Omega)}\right) + $$ + \end{corollary} + \begin{theorem}[Regularity up to the boundary] + Assume that $\Fr{\Omega}$ is $\mathcal{C}^2$ and that $a_{ij}\in \mathcal{C}^1(\overline{\Omega})$, $b_j,c\in L^\infty(\Omega)$. Let $f\in L^2(\Omega)$ and $u\in H^1_0(\Omega)$ be a weak solution of $\mathcal{D}_f$. Then, $u\in H^2(\Omega)$ and: + $$ + \norm{u}_{H^2(\Omega)}\leq C\left(\norm{f}_{L^2(\Omega)}+\norm{u}_{L^2(\Omega)}\right) + $$ + \end{theorem} + \begin{corollary} + Assume that $\Fr{\Omega}$ is $\mathcal{C}^{m+1}$, $m\geq 2$, and that $a_{ij}\in \mathcal{C}^m(\overline{\Omega})$, $b_j,c\in \mathcal{C}^{m-1}(\overline{\Omega})$. Let $f\in H^{m-1}(\Omega)$ and $u\in H^1_0(\Omega)$ be a weak solution of $\mathcal{D}_f$. Then, $u\in H^{m+2}(\Omega)$ and: + $$ + \norm{u}_{H^{m+2}(\Omega)}\leq C\left(\norm{f}_{H^{m-1}(\Omega)}+\norm{u}_{L^2(\Omega)}\right) + $$ + \end{corollary} + \begin{corollary} + Assume that $\Fr{\Omega}$ is $\mathcal{C}^\infty$ and that $a_{ij},b_j,c,f\in \mathcal{C}^\infty(\overline{\Omega})$. Let $u\in H^1_0(\Omega)$ be a weak solution of $\mathcal{D}_f$. Then, $u\in \mathcal{C}^\infty(\Omega)$ and $\forall m\geq 1$: + $$ + \norm{u}_{H^{m+1}(\Omega)}\leq C\left(\norm{f}_{H^{m-1}(\Omega)}+\norm{u}_{L^2(\Omega)}\right) + $$ + \end{corollary} + \subsubsection{Weak maximum principle for weak solutions of divergence-form elliptic PDEs} + \begin{lemma} + Let $\Omega\subseteq\RR^d$ open and $u\in H^1(\Omega)$. Then: + $$ + u^{+}:=\begin{cases} + u & \text{if }u\geq 0 \\ + 0 & \text{if }u<0 + \end{cases}\qquad + u^{-}:=\begin{cases} + -u & \text{if }u\leq 0 \\ + 0 & \text{if }u>0 + \end{cases} + $$ + are also in $H^1(\Omega)$ and: + $$ + \grad(u^+)\almoste{=}\begin{cases} + \grad u & \text{if }u>0 \\ + 0 & \text{if }u\leq 0 + \end{cases}\quad + \grad(u^-)\almoste{=}\begin{cases} + -\grad u & \text{if }u<0 \\ + 0 & \text{if }u\geq 0 + \end{cases} + $$ + \end{lemma} + \begin{corollary} + Let $\Omega\subseteq\RR^d$ open and $u\in H^1(\Omega)$. Then, $\abs{u}\in H^1(\Omega)$ and $\grad{\abs{u}}=\sign\grad{u}$. + \end{corollary} + \begin{lemma} + Let $(u_n)\in H^1(\Omega)$ be such that $u_n\overset{H^1(\Omega)}{\longrightarrow} u$. Then, $u_n^\pm\overset{H^1(\Omega)}{\longrightarrow} u^\pm$. + \end{lemma} + \begin{corollary} + Let $u\in H^1(\Omega)$. Then, $\Tr_{\partial\Omega}(u^\pm)={(\Tr_{\partial\Omega}u)}^\pm$. + \end{corollary} + \begin{lemma} + Let $\Omega\subseteq\RR^d$ open with $\mathcal{C}^1$ boundary, $u\in H^1(\Omega)$ and $\Tr_{\partial\Omega}u\almoste{\leq}0$. Then, $u^+\in H^1_0(\Omega)$. + \end{lemma} + \begin{theorem}[Weak maximum principle] + Let $\Omega\subseteq\RR^d$ open and bounded with $\mathcal{C}^1$ boundary, $a_{ij}=a_{ji},c\in L^\infty(\Omega)$, $c\almoste{\geq}0$, $L=-\sum_{i,j=1}^d\partial_i(a_{ij}\partial_j)+c$ be elliptic and $f\in L^2(\Omega)$ with $f\almoste{\leq} 0$. Let $u\in H^1(\Omega)$ be such that: + \begin{itemize} + \item $\displaystyle \int_\Omega\left[\sum_{i,j=1}^da_{ij}\partial_iu\partial_jv+cuv\right]=\int_\Omega fv$ $\forall v\in H^1_0(\Omega)$ + \item $\Tr_{\partial\Omega}u\almoste{\leq}0$ + \end{itemize} + Then, $u\almoste{\leq}0$. + \end{theorem} + \begin{theorem}[Weak maximum principle] + Let $\Omega\subseteq\RR^d$ open and bounded with $\mathcal{C}^1$ boundary, $a_{ij}=a_{ji},b_j,c\in L^\infty(\Omega)$, $L=-\sum_{i,j=1}^d\partial_i(a_{ij}\partial_j)+\sum_{j=1}^db_j\partial_j+c$ be elliptic and $f\in L^2(\Omega)$ with $f\almoste{\leq} 0$. Let $u\in H^1(\Omega)$ be such that: + \begin{itemize} + \item $\displaystyle \int_\Omega\left[\sum_{i,j=1}^da_{ij}\partial_iu\partial_jv+cuv\right]=\int_\Omega fv$ $\forall v\in H^1_0(\Omega)$ + \item $\Tr_{\partial\Omega}u\almoste{\leq}0$ + \end{itemize} + Then, $u\almoste{\leq}0$. + \end{theorem} + \begin{corollary} + For each $f\in L^2(\Omega)$, the problem $\mathcal{D}_f$ has a unique weak solution $u_f$. Moreover, if $\Fr{\Omega}\in\mathcal{C}^1$, then $u_f\in H^2(\Omega)$ and $f\mapsto u_f$ is a bounded linear operator from $L^2(\Omega)$ to $H^2(\Omega)$. If $\Fr{\Omega}\in\mathcal{C}^{m+1}$, $b_j\in\mathcal{C}^{m-1}$ and $f\in H^{m-1}(\Omega)$, then $u_f\in H^{m+1}(Omega)$ and $f\mapsto u_f$ is a bounded linear operator from $H^{m-1}(\Omega)$ to $H^{m+1}(\Omega)$. + \end{corollary} + \begin{definition} + If the weak solution $u_f$ of the problem $\mathcal{D}_f$ is in $H^1_0(\Omega)\cap W^{2,p}(\Omega)$ for some $p\in [1,\infty)$, then we say that $u_f$ is called a \emph{strong solution} of $\mathcal{D}_f$. If $u_f\in \mathcal{C}^2(\Omega)\cap H^1_0(\Omega)$, then we say that $u_f$ is a \emph{classical solution} of $\mathcal{D}_f$. + \end{definition} \end{multicols} \end{document} \ No newline at end of file diff --git a/Mathematics/5th/Stochastic_calculus/Stochastic_calculus.tex b/Mathematics/5th/Stochastic_calculus/Stochastic_calculus.tex index 690be2d..b53e07b 100644 --- a/Mathematics/5th/Stochastic_calculus/Stochastic_calculus.tex +++ b/Mathematics/5th/Stochastic_calculus/Stochastic_calculus.tex @@ -667,6 +667,26 @@ \dd{f(X_t)}=f'(X_t)\dd{X_t}+\frac{1}{2}f''(X_t)\dd{{\langle X\rangle}_t} $$ \end{theorem} + \begin{proof} + Let $t\geq 0$ and ${(t_k^n)}_{0\leq k\leq n}\in \mathrm{P}([0,t])$ such that $\Delta_n\to 0$. Then, using the Taylor expansion of $f$ we have: + \begin{align*} + f(X_t)-f(X_0) & =\sum_{k=0}^{n-1}f(X_{t_{k+1}^n})-f(X_{t_k^n})= \\&=\sum_{k=0}^{n-1}f'(X_{t_k^n})(X_{t_{k+1}^n}-X_{t_k^n})+\\ + & \qquad\quad+\frac{1}{2}\sum_{k=0}^{n-1}f''(X_{u_k^n}){(X_{t_{k+1}^n}-X_{t_k^n})}^2 + \end{align*} + with $u_k^n\in [t_k^n,t_{k+1}^n]$. By a previous remark we have that: + $$ + \sum_{k=0}^{n-1}f'(X_{t_k^n})(X_{t_{k+1}^n}-X_{t_k^n})\overset{\Prob}{\longrightarrow} \int_0^t f'(X_u)\dd{X_u} + $$ + Now, by \mcref{SC:ito_quadratic_variation} we have that: + $$ + \sum_{k=0}^{n-1}Y_{t_k^n}{(X_{t_{k+1}^n}-X_{t_k^n})}^2 \overset{\Prob}{\longrightarrow} \int_0^t Y_u\dd{{\langle X\rangle}_u} + $$ + in the elementary case where $Y_u=\indi{(0,s]}(u)$, $s\geq 0$. By linearity, this immediately extends to the case where $Y$ is a random step function. By density, it further extends to the case where $Y$ is any continuous and adapted process. In particular, we may take + $Y_u=f''(X_u)$, and the formula holds by uniform continuity: + $$ + \max_{0\leq k\leq n}\abs{f''(X_{t_k^n})-f''(X_{u_k^n})}\almoste{\longrightarrow} 0 + $$ + \end{proof} \begin{theorem} Let $X^1,\ldots,X^d$ be Itô processes and $f\in C^2(\RR^d)$. Then, ${(f(X^1_t,\ldots,X^d_t))}_{t\geq 0}$ is an Itô process and: $$ @@ -674,5 +694,6 @@ $$ where $\vf{X}:=(X^1,\ldots,X^d)$. \end{theorem} + \subsubsection{Exponential martingales} \end{multicols} \end{document} \ No newline at end of file diff --git a/main_math.idx b/main_math.idx deleted file mode 100644 index a2a6aa9..0000000 --- a/main_math.idx +++ /dev/null @@ -1,3138 +0,0 @@ -\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} -\indexentry{predicate|hyperpage}{6} -\indexentry{Cartesian product|hyperpage}{7} -\indexentry{function|hyperpage}{7} -\indexentry{composition|hyperpage}{7} -\indexentry{image|hyperpage}{7} -\indexentry{image|hyperpage}{7} -\indexentry{preimage|hyperpage}{7} -\indexentry{preimage|hyperpage}{7} -\indexentry{injective|hyperpage}{7} -\indexentry{surjective|hyperpage}{7} -\indexentry{bijective|hyperpage}{7} -\indexentry{inverse function|hyperpage}{7} -\indexentry{invertible|hyperpage}{7} -\indexentry{negation|hyperpage}{8} -\indexentry{$P$ and $Q$ are both true|hyperpage}{8} -\indexentry{either $P$ or $Q$ are true|hyperpage}{8} -\indexentry{$Q$ is true whenever $P$ is true|hyperpage}{8} -\indexentry{$P$ and $Q$ have the same truth-value|hyperpage}{8} -\indexentry{symmetric group|hyperpage}{8} -\indexentry{order|hyperpage}{8} -\indexentry{support|hyperpage}{8} -\indexentry{orbit|hyperpage}{8} -\indexentry{Orbit structure|hyperpage}{8} -\indexentry{Orbit linear structure|hyperpage}{8} -\indexentry{cycle|hyperpage}{8} -\indexentry{transposition|hyperpage}{8} -\indexentry{sign|hyperpage}{8} -\indexentry{equivalence relation|hyperpage}{8} -\indexentry{equivalence class|hyperpage}{8} -\indexentry{partial order relation|hyperpage}{9} -\indexentry{partially ordered set|hyperpage}{9} -\indexentry{poset|hyperpage}{9} -\indexentry{minimal element|hyperpage}{9} -\indexentry{least element|hyperpage}{9} -\indexentry{maximal element|hyperpage}{9} -\indexentry{total order relation|hyperpage}{9} -\indexentry{well-order relation|hyperpage}{9} -\indexentry{well-ordered set|hyperpage}{9} -\indexentry{Cantor-Bernstein theorem|hyperpage}{9} -\indexentry{Cantor's theorem|hyperpage}{9} -\indexentry{characteristic function|hyperpage}{9} -\indexentry{indicator function|hyperpage}{9} -\indexentry{Binomial coefficient formulas|hyperpage}{9} -\indexentry{Pigeonhole principle|hyperpage}{10} -\indexentry{Combinations without repetition|hyperpage}{10} -\indexentry{Combinations with 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-\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{$\sigma $-algebra|hyperpage}{336} -\indexentry{$\sigma $-algebra|hyperpage}{336} -\indexentry{measurable space|hyperpage}{336} -\indexentry{$\sigma $-algebra generated by $\mathcal {F}$|hyperpage}{336} -\indexentry{measurable|hyperpage}{336} -\indexentry{Measure|hyperpage}{336} -\indexentry{measure|hyperpage}{336} -\indexentry{measurable space|hyperpage}{336} -\indexentry{integral of $f$ with respect to $\mu $|hyperpage}{336} -\indexentry{integral of $f$ with respect to $\mu $|hyperpage}{336} -\indexentry{Monotone convergence theorem|hyperpage}{336} -\indexentry{Fatou's lemma|hyperpage}{336} -\indexentry{Dominated convergence theorem|hyperpage}{336} -\indexentry{Product measure|hyperpage}{336} -\indexentry{product measure|hyperpage}{336} -\indexentry{$\sigma $-finite|hyperpage}{336} -\indexentry{Fubini|hyperpage}{336} -\indexentry{density|hyperpage}{337} -\indexentry{absolutely continuous|hyperpage}{337} -\indexentry{Radon-Nikodym|hyperpage}{337} -\indexentry{probability space|hyperpage}{337} -\indexentry{events|hyperpage}{337} -\indexentry{Random variable|hyperpage}{337} -\indexentry{$E$-valued random variable|hyperpage}{337} -\indexentry{Expectation|hyperpage}{337} -\indexentry{expectation of $X$|hyperpage}{337} -\indexentry{law of $X$|hyperpage}{337} -\indexentry{$\sigma $-algebra generated by $X$|hyperpage}{337} -\indexentry{Jensen's inequality|hyperpage}{337} -\indexentry{conditional expectation of $X$ given $\mathcal {G}$|hyperpage}{337} -\indexentry{Tower property|hyperpage}{338} -\indexentry{conditional expectation of $X$ given $Y$|hyperpage}{338} -\indexentry{probability kernel|hyperpage}{338} -\indexentry{conditional law of $X$ given $Y$|hyperpage}{338} -\indexentry{filtration|hyperpage}{338} -\indexentry{filtered probability space|hyperpage}{338} -\indexentry{adapted|hyperpage}{338} -\indexentry{martingale|hyperpage}{338} -\indexentry{submartingale|hyperpage}{338} -\indexentry{supermartingale|hyperpage}{338} -\indexentry{stopping time|hyperpage}{338} -\indexentry{stopped process|hyperpage}{338} -\indexentry{uniformly integrable|hyperpage}{338} -\indexentry{converges weakly|hyperpage}{339} -\indexentry{converges strongly|hyperpage}{339} -\indexentry{converges weakly-*|hyperpage}{339} -\indexentry{Banach-Alaoglu theorem|hyperpage}{339} -\indexentry{strongly lower-semicontinuous|hyperpage}{339} -\indexentry{weakly lower-semicontinuity|hyperpage}{339} -\indexentry{weak-* lower-semicontinuity|hyperpage}{339} -\indexentry{support plane|hyperpage}{339} -\indexentry{Sobolev spaces|hyperpage}{340} -\indexentry{Sobolev spaces|hyperpage}{340} -\indexentry{Poincaré's inequality|hyperpage}{340} -\indexentry{Poincaré-Wirtinger's inequality|hyperpage}{340} -\indexentry{embedded|hyperpage}{341} -\indexentry{compactly embedded|hyperpage}{341} -\indexentry{Gagliardo, Nirengerg and Sobolev's inequality|hyperpage}{341} -\indexentry{Hölder continuity|hyperpage}{341} -\indexentry{$\mathcal {C}^{k,\theta }$-Hölder continuous|hyperpage}{341} -\indexentry{Morrey's embedding|hyperpage}{341} -\indexentry{extension|hyperpage}{341} -\indexentry{extension operator|hyperpage}{341} -\indexentry{first order reflection|hyperpage}{342} -\indexentry{Reillich-Kondrachov's compactness theorem|hyperpage}{342} -\indexentry{trace operator|hyperpage}{342} -\indexentry{jointly Gaussian|hyperpage}{345} -\indexentry{Strong law of large numbers for Brownian motion|hyperpage}{345} -\indexentry{Markov property for Brownian motion|hyperpage}{345} -\indexentry{natural filtration|hyperpage}{345} -\indexentry{hitting time|hyperpage}{345} -\indexentry{Doob's optional sampling theorem|hyperpage}{345} -\indexentry{stopped process|hyperpage}{345} diff --git a/main_math.ilg b/main_math.ilg deleted file mode 100644 index d539d3e..0000000 --- a/main_math.ilg +++ /dev/null @@ -1,6 +0,0 @@ -This is makeindex, version 2.17 [TeX Live 2023] (kpathsea + Thai support). -Scanning input file main_math.idx.......done (3138 entries accepted, 0 rejected). -Sorting entries.............................done (38728 comparisons). -Generating output file main_math.ind......done (2672 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 deleted file mode 100644 index 71087f6..0000000 --- a/main_math.ind +++ /dev/null @@ -1,2672 +0,0 @@ -\begin{theindex} - - \item $(G,\cdot )$-space, \hyperpage{211} - \item $E$-valued random variable, \hyperpage{337} - \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 $N$-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 $\mathcal {C}^{k,\theta }$-Hölder continuous, \hyperpage{341} - \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{336} - \item $\sigma $-algebra generated, \hyperpage{172} - \item $\sigma $-algebra generated by $X$, \hyperpage{337} - \item $\sigma $-algebra generated by $\boldsymbol {\mathrm {X}}$, - \hyperpage{321} - \item $\sigma $-algebra generated by $\mathcal {F}$, \hyperpage{336} - \item $\sigma $-algebra of all Lebesgue measurable sets in $\ensuremath {\mathbb {R}}^n$, - \hyperpage{175} - \item $\sigma $-finite, \hyperpage{336} - \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{337} - \item absolutely convergent, \hyperpage{77}, \hyperpage{106}, - \hyperpage{117}, \hyperpage{300} - \item absolutely stable, \hyperpage{261} - \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{338} - \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 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 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 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 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 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{339} - \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 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 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 bridge, \hyperpage{50} - \item Brouwer's fixed-point theorem, \hyperpage{215} - \item Brownian motion, \hyperpage{330} - \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} - \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 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} - \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 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{341} - \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} - \item conditional consistency, \hyperpage{269} - \item conditional expectation, \hyperpage{186} - \item conditional expectation of $X$ given $\mathcal {G}$, - \hyperpage{337} - \item conditional expectation of $X$ given $Y$, \hyperpage{338} - \item conditional law of $X$ given $Y$, \hyperpage{338} - \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 Congruence axioms, \hyperpage{63} - \item congruence relation, \hyperpage{63} - \item conic, \hyperpage{73} - \item conjugacy, \hyperpage{132} - \item conjugate, \hyperpage{104}, \hyperpage{132}, \hyperpage{202} - \item conjugate dynamical systems, \hyperpage{131} - \item Conjugate gradient method, \hyperpage{264} - \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 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 function, \hyperpage{208} - \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 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{339} - \item converges uniformly, \hyperpage{77, 78}, \hyperpage{106, 107} - \item converges weakly, \hyperpage{339} - \item converges weakly-*, \hyperpage{339} - \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} - - \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{337} - \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 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 coefficient, \hyperpage{278} - \item Diffusion equation, \hyperpage{278} - \item diffusion flux, \hyperpage{278} - \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 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 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 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{336} - \item Doob's optional sampling theorem, \hyperpage{345} - \item dot product, \hyperpage{53} - \item double, \hyperpage{90} - \item Double dual space, \hyperpage{19} - \item double dual space, \hyperpage{19} - \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{341} - \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} - \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} - \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{337} - \item evolute, \hyperpage{144} - \item exact, \hyperpage{156} - \item Examples of Euclidean motions, \hyperpage{73} - \item Excluded point topology, \hyperpage{206} - \item Existence of orthogonal polynomials, \hyperpage{96} - \item Existence of the splitting field, \hyperpage{167} - \item exists and it is finite, \hyperpage{298} - \item Expansive fixed point theorem, \hyperpage{229} - \item Expectation, \hyperpage{183}, \hyperpage{337} - \item expectation, \hyperpage{178}, \hyperpage{182--184}, - \hyperpage{249} - \item expectation of $X$, \hyperpage{337} - \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{341} - \item extension operator, \hyperpage{341} - \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{336} - \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 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{338} - \item filtration, \hyperpage{321}, \hyperpage{338} - \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 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-dimensional distributions, \hyperpage{333} - \item finite-dimensional distributions, \hyperpage{333} - \item finite-rank operator, \hyperpage{307} - \item finitely generated, \hyperpage{165} - \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{342} - \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 linear ode, \hyperpage{126} - \item flux, \hyperpage{62}, \hyperpage{159} - \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 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{336} - \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{341} - \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 linear model, \hyperpage{250} - \item general solution of the ode, \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 solution, \hyperpage{245} - \item generating set, \hyperpage{16} - \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 distribution, \hyperpage{177} - \item geometric distribution, \hyperpage{177} - \item geometric multiplicity, \hyperpage{20} - \item Gershgorin circle theorem, \hyperpage{92} - \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 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 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 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 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 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{345} - \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} - \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 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-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 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{341} - \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 Euler method, \hyperpage{258} - \item implicit form, \hyperpage{122} - \item Implicit function theorem, \hyperpage{57} - \item Implicit scheme in finite differences, \hyperpage{286} - \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 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{336} - \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 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 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} - \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 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{337} - \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{345} - \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 forward equation, \hyperpage{328} - \item Kolmogorov's strong law, \hyperpage{191} - \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 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{337} - \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 lexicographic degree, \hyperpage{163} - \item lexicographic order, \hyperpage{163} - \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 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 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 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 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{345} - \item Markov's inequality, \hyperpage{185} - \item martingale, \hyperpage{338} - \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{336} - \item measurable space, \hyperpage{175}, \hyperpage{336} - \item Measure, \hyperpage{291}, \hyperpage{336} - \item measure, \hyperpage{174}, \hyperpage{291}, \hyperpage{336} - \item Melnikov's method, \hyperpage{227} - \item memoryless, \hyperpage{177, 178} - \item meromorphic, \hyperpage{118} - \item Mesh, \hyperpage{275} - \item mesh, \hyperpage{275} - \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 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{336} - \item monotonic, \hyperpage{26}, \hyperpage{28} - \item monotonically decreasing, \hyperpage{26} - \item monotonically increasing, \hyperpage{26} - \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{341} - \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 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 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{345} - \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 Neumann, \hyperpage{274} - \item Neumann series, \hyperpage{308} - \item Neville's algorithm, \hyperpage{93} - \item Newton method, \hyperpage{262}, \hyperpage{264} - \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-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 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 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 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 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 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{340} - \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{340} - \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 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 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{338} - \item Probability mass function, \hyperpage{176} - \item probability mass function, \hyperpage{176} - \item probability space, \hyperpage{173}, \hyperpage{337} - \item probability-generating function, \hyperpage{316} - \item product, \hyperpage{11}, \hyperpage{13}, \hyperpage{106} - \item Product measure, \hyperpage{336} - \item product measure, \hyperpage{336} - \item product topology, \hyperpage{209} - \item products of group subsets, \hyperpage{40} - \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 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{337} - \item Radó theorem, \hyperpage{218} - \item random sample, \hyperpage{193} - \item Random variable, \hyperpage{337} - \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 linear ode 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} - \item regular distributions, \hyperpage{241} - \item regular domain, \hyperpage{148} - \item regular region, \hyperpage{160} - \item regular surface, \hyperpage{147} - \item Reillich-Kondrachov's compactness theorem, \hyperpage{342} - \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 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 Rolle's theorem, \hyperpage{30} - \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 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 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 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} - \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} - \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{340} - \item solution, \hyperpage{90}, \hyperpage{139} - \item solution of a system of equations, \hyperpage{13} - \item solution of the ode, \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} - \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 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{338}, \hyperpage{345} - \item stopping time, \hyperpage{321}, \hyperpage{338} - \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 law, \hyperpage{191} - \item Strong law of large numbers for Brownian motion, - \hyperpage{345} - \item Strong Markov property, \hyperpage{321, 322} - \item strongly consistent estimator, \hyperpage{194} - \item strongly lower-semicontinuous, \hyperpage{339} - \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-multiplicativity, \hyperpage{97} - \item subalgebra, \hyperpage{303} - \item subbasis, \hyperpage{206} - \item subcover, \hyperpage{213} - \item subfield generated, \hyperpage{163} - \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{338} - \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 supermartingale, \hyperpage{338} - \item support, \hyperpage{8}, \hyperpage{79}, \hyperpage{146}, - \hyperpage{176}, \hyperpage{180}, \hyperpage{242} - \item support plane, \hyperpage{339} - \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{338} - \item trace, \hyperpage{141}, \hyperpage{289} - \item trace operator, \hyperpage{342} - \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 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 equicontinuous, \hyperpage{125}, \hyperpage{304} - \item uniformly integrable, \hyperpage{338} - \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 formulation, \hyperpage{275} - \item Weak law, \hyperpage{190} - \item weak-* lower-semicontinuity, \hyperpage{339} - \item weakly consistent estimator, \hyperpage{194} - \item weakly lower-semicontinuity, \hyperpage{339} - \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 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/preamble_formulas.sty b/preamble_formulas.sty index 69b9a45..14411ec 100644 --- a/preamble_formulas.sty +++ b/preamble_formulas.sty @@ -372,7 +372,7 @@ %%% PARTIAL DIFFERENTIAL EQUATIONS \newcommand{\matdv}[2]{\frac{\mathrm{D}#1}{\mathrm{D}#2}} % material derivative - +% \newcommand{\Tr}{\mathrm{Tr}} % trace of an operator (already defined and it is equal to Tr, so perfect) %%% LINEAR MODELS \DeclareMathOperator{\MST}{MST} % Total Mean Square