diff --git a/.github/workflows/buildpdf.yml b/.github/workflows/buildpdf.yml
index e0d7a55..5bd6e73 100644
--- a/.github/workflows/buildpdf.yml
+++ b/.github/workflows/buildpdf.yml
@@ -162,6 +162,11 @@ jobs:
root_file: Stochastic_processes.tex
working_directory: Mathematics/4th/Stochastic_processes/
# MATHEMATICS 5
+ - name: Compile - ADS
+ uses: xu-cheng/latex-action@v2
+ with:
+ root_file: Advanced_dynamic_systems.tex
+ working_directory: Mathematics/5th/Advanced_dynamic_systems/
- name: Compile - AP
uses: xu-cheng/latex-action@v2
with:
@@ -270,6 +275,7 @@ jobs:
Mathematics/4th/Partial_differential_equations/Partial_differential_equations.pdf
Mathematics/4th/Real_and_functional_analysis/Real_and_functional_analysis.pdf
Mathematics/4th/Stochastic_processes/Stochastic_processes.pdf
+ Mathematics/5th/Advanced_dynamic_systems/Advanced_dynamic_systems.pdf
Mathematics/5th/Advanced_probability/Advanced_probability.pdf
Mathematics/5th/Advanced_topics_in_functional_analysis_and_PDEs/Advanced_topics_in_functional_analysis_and_PDEs.pdf
Mathematics/5th/Introduction_to_evolution_PDEs/Introduction_to_evolution_PDEs.pdf
diff --git a/Mathematics/5th/Advanced_dynamical_systems/Advanced_dynamical_systems.tex b/Mathematics/5th/Advanced_dynamical_systems/Advanced_dynamical_systems.tex
new file mode 100644
index 0000000..8a80cae
--- /dev/null
+++ b/Mathematics/5th/Advanced_dynamical_systems/Advanced_dynamical_systems.tex
@@ -0,0 +1,21 @@
+\documentclass[../../../main_math.tex]{subfiles}
+
+\begin{document}
+\changecolor{ADS}
+\begin{multicols}{2}[\section{Advanced dynamical sytems}]
+ \subsection{Introduction}
+ \subsubsection{Rotations in \texorpdfstring{$\S^1$}{S1}}
+ \begin{proposition}
+ Let $\alpha=\frac{p}{q}\in\QQ$ and let $R_\alpha:\S^1\to \S^1$ be the rotation of angle $\alpha$. Then, all the points of $\S^1$ are periodic for $R_\alpha$ with period $q$.
+ \end{proposition}
+ \begin{proof}
+ We identify the elements of $\S^1$ as $\quot{\RR}{\ZZ}$. Let $x\in \S^1$. Then, ${R_\alpha}^q x=x+\alpha q=x+p=x$. And $q$ is the smallest integer such that ${R_\alpha}^q x=x$ because we assume that $p$ and $q$ are coprime.
+ \end{proof}
+ \begin{proposition}
+ Let $\alpha\in\RR\setminus\QQ$ and let $R_\alpha:\S^1\to \S^1$ be the rotation of angle $\alpha$. Then, all the points of $\S^1$ are dense in $\S^1$.
+ \end{proposition}
+ \begin{proof}
+ Let $\varepsilon>0$, $x,y\in \S^1$. Discretize $\S^1$ in intervals of length at most $\frac{1}{\varepsilon}$. Then, $\exists m,n\in \NN$ with $m< n\leq \frac{1}{\varepsilon}+1$ such that ${R_\alpha}^m x$ and ${R_\alpha}^nx$ are in the same interval. Thus, $\abs{{R_\alpha}^{n-m}x-x}<\varepsilon$. Now, concatenating ${R_\alpha}^{n-m}x$ repeatedly, we will eventually have $\abs{{R_\alpha}^{k(n-m)}x - y}<\varepsilon$ for some $k\in \NN$.
+ \end{proof}
+\end{multicols}
+\end{document}
\ No newline at end of file
diff --git a/Mathematics/5th/Limit_theorems_and_large_deviations/Limit_theorems_and_large_deviations.tex b/Mathematics/5th/Limit_theorems_and_large_deviations/Limit_theorems_and_large_deviations.tex
index f76ead4..2ff3f02 100644
--- a/Mathematics/5th/Limit_theorems_and_large_deviations/Limit_theorems_and_large_deviations.tex
+++ b/Mathematics/5th/Limit_theorems_and_large_deviations/Limit_theorems_and_large_deviations.tex
@@ -3,6 +3,60 @@
\begin{document}
\changecolor{LTLD}
\begin{multicols}{2}[\section{Limit theorem and large deviations}]
-
+ \subsection{Large deviations}
+ \subsubsection{Large deviations for \iid random variables}
+ \begin{definition}
+ Let $\alpha(\dd{x})$ be a probability distribution on $\RR$. Let $M(\lambda)=\int\exp{\lambda x}\alpha(\dd{x})$ be the moment generating function of $\alpha$. We define $Z(\lambda):= \log M(\lambda)$. Assume that $\exists \lambda^*$ such that $M(\lambda)<\infty$ if $\abs{\lambda}<\lambda^*$.
+ \end{definition}
+ \begin{remark}
+ Note that by REF JENSEN, $Z(\lambda)\geq \lambda m>-\infty$ for all $\lambda\in\RR$.
+ \end{remark}
+ \begin{definition}
+ We define $\mathcal{D}_Z:=\{\lambda\in\RR\mid Z(\lambda)<\infty\}$.
+ \end{definition}
+ \begin{remark}
+ Note that under our hypothesis, we have $0\in \Int \mathcal{D}_Z$.
+ \end{remark}
+ \begin{lemma}
+ We have:
+ \begin{enumerate}
+ \item $Z$ is convex.
+ \item $Z\in\mathcal{C}^1(\Int\mathcal{D}_Z)$ and
+ $$
+ Z'(\lambda)=\int x\exp{\lambda x-Z(\lambda)}\alpha(\dd{x})
+ $$
+ \end{enumerate}
+ \end{lemma}
+ \begin{proof}
+ \begin{enumerate}
+ \item Let $t\in[0,1]$. By \mnameref{RFA:holder} we have:
+ $$
+ M(t\lambda_1+(1-t)\lambda_2)\leq M(t\lambda_1)^tM((1-t)\lambda_2)^{1-t}
+ $$
+ Taking the logarithm we get the result.
+ \item
+ \end{enumerate}
+ \end{proof}
+ \begin{corollary}
+ We have $Z\in\mathcal{C}^\infty(\Int\mathcal{D}_Z)$ and moreover:
+ $$
+ Z''(\lambda)=\int x^2\exp{\lambda x-Z(\lambda)}\alpha(\dd{x})-{\left(\int x\exp{\lambda x-Z(\lambda)}\alpha(\dd{x})\right)}^2>0
+ $$
+ \end{corollary}
+ \begin{proof}
+ Note that $\alpha_\lambda(\dd{x}):=\exp{\lambda x-Z(\lambda)}\alpha(\dd{x})$ is a probability distribution on $\RR$. Thus, using the same technique as in the proof of the previous lemma, we get that $Z\in\mathcal{C}^\infty(\Int\mathcal{D}_Z)$ and the $Z''>0$ because it is a variance.
+ \end{proof}
+ \begin{remark}
+ Note that $Z''>0$ implies that $Z$ is strictly convex.
+ \end{remark}
+ \begin{definition}
+ We define the \emph{rate function} as the \emph{Legendre-Fenchel transform} of $Z$:
+ $$
+ I(x):=\sup_{\lambda\in\RR}(\lambda x-Z(\lambda))
+ $$
+ \end{definition}
+ \begin{remark}
+ Note that $I$ is convex (as it is the supremum of linear functions), $I(x)\geq 0$ for all $x\in\RR$ because $Z(0) = 0$, and $I(m)=0$.
+ \end{remark}
\end{multicols}
\end{document}
\ No newline at end of file
diff --git a/index.html b/index.html
index c54d744..6fa6751 100644
--- a/index.html
+++ b/index.html
@@ -70,11 +70,12 @@ Mathematics
1st year:
+ Advanced dynamic systems Advanced probability Advanced topics in functional analysis and PDEs Introduction to evolution PDEs Introduction to non linear PDEs Limit theorems and large deviations Stochastic calculus
diff --git a/main_math.idx b/main_math.idx
index 718f564..a2a6aa9 100644
--- a/main_math.idx
+++ b/main_math.idx
@@ -3045,8 +3045,8 @@
\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{Gaussian vector|hyperpage}{330}
+\indexentry{Gaussian process|hyperpage}{330}
\indexentry{mean function|hyperpage}{330}
\indexentry{covariance function|hyperpage}{330}
\indexentry{Brownian motion|hyperpage}{330}
@@ -3063,69 +3063,76 @@
\indexentry{finite-dimensional distributions|hyperpage}{333}
\indexentry{consistency condition|hyperpage}{333}
\indexentry{Kolmogorov extension theorem|hyperpage}{333}
-\indexentry{$\sigma $-algebra|hyperpage}{335}
-\indexentry{$\sigma $-algebra|hyperpage}{335}
-\indexentry{measurable space|hyperpage}{335}
-\indexentry{$\sigma $-algebra generated by $\mathcal {F}$|hyperpage}{335}
-\indexentry{measurable|hyperpage}{335}
-\indexentry{Measure|hyperpage}{335}
-\indexentry{measure|hyperpage}{335}
-\indexentry{measurable space|hyperpage}{335}
-\indexentry{integral of $f$ with respect to $\mu $|hyperpage}{335}
-\indexentry{integral of $f$ with respect to $\mu $|hyperpage}{335}
-\indexentry{Monotone convergence theorem|hyperpage}{335}
-\indexentry{Fatou's lemma|hyperpage}{335}
-\indexentry{Dominated convergence theorem|hyperpage}{335}
-\indexentry{Product measure|hyperpage}{335}
-\indexentry{product measure|hyperpage}{335}
-\indexentry{$\sigma $-finite|hyperpage}{335}
-\indexentry{Fubini|hyperpage}{335}
-\indexentry{density|hyperpage}{336}
-\indexentry{absolutely continuous|hyperpage}{336}
-\indexentry{Radon-Nikodym|hyperpage}{336}
-\indexentry{probability space|hyperpage}{336}
-\indexentry{events|hyperpage}{336}
-\indexentry{Random variable|hyperpage}{336}
-\indexentry{$E$-valued random variable|hyperpage}{336}
-\indexentry{Expectation|hyperpage}{336}
-\indexentry{expectation of $X$|hyperpage}{336}
-\indexentry{law of $X$|hyperpage}{336}
-\indexentry{$\sigma $-algebra generated by $X$|hyperpage}{336}
-\indexentry{Jensen's inequality|hyperpage}{336}
-\indexentry{conditional expectation of $X$ given $\mathcal {G}$|hyperpage}{336}
-\indexentry{Tower property|hyperpage}{337}
-\indexentry{conditional expectation of $X$ given $Y$|hyperpage}{337}
-\indexentry{probability kernel|hyperpage}{337}
-\indexentry{conditional law of $X$ given $Y$|hyperpage}{337}
-\indexentry{filtration|hyperpage}{337}
-\indexentry{filtered probability space|hyperpage}{337}
-\indexentry{adapted|hyperpage}{337}
-\indexentry{martingale|hyperpage}{337}
-\indexentry{submartingale|hyperpage}{337}
-\indexentry{supermartingale|hyperpage}{337}
-\indexentry{stopping time|hyperpage}{337}
-\indexentry{stopped process|hyperpage}{337}
-\indexentry{uniformly integrable|hyperpage}{337}
-\indexentry{converges weakly|hyperpage}{338}
-\indexentry{converges strongly|hyperpage}{338}
-\indexentry{converges weakly-*|hyperpage}{338}
-\indexentry{Banach-Alaoglu theorem|hyperpage}{338}
-\indexentry{strongly lower-semicontinuous|hyperpage}{338}
-\indexentry{weakly lower-semicontinuity|hyperpage}{338}
-\indexentry{weak-* lower-semicontinuity|hyperpage}{338}
-\indexentry{support plane|hyperpage}{338}
-\indexentry{Sobolev spaces|hyperpage}{339}
-\indexentry{Sobolev spaces|hyperpage}{339}
-\indexentry{Poincaré's inequality|hyperpage}{339}
-\indexentry{Poincaré-Wirtinger's inequality|hyperpage}{339}
-\indexentry{embedded|hyperpage}{340}
-\indexentry{compactly embedded|hyperpage}{340}
-\indexentry{Gagliardo, Nirengerg and Sobolev's inequality|hyperpage}{340}
-\indexentry{Hölder continuity|hyperpage}{340}
-\indexentry{$\mathcal {C}^{k,\theta }$-Hölder continuous|hyperpage}{340}
-\indexentry{Morrey's embedding|hyperpage}{340}
-\indexentry{extension|hyperpage}{340}
-\indexentry{extension operator|hyperpage}{340}
-\indexentry{first order reflection|hyperpage}{341}
-\indexentry{Reillich-Kondrachov's compactness theorem|hyperpage}{341}
-\indexentry{trace operator|hyperpage}{341}
+\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
index 62ab4f1..d539d3e 100644
--- a/main_math.ilg
+++ b/main_math.ilg
@@ -1,6 +1,6 @@
This is makeindex, version 2.17 [TeX Live 2023] (kpathsea + Thai support).
-Scanning input file main_math.idx.......done (3131 entries accepted, 0 rejected).
-Sorting entries..............................done (39644 comparisons).
-Generating output file main_math.ind......done (2665 lines written, 0 warnings).
+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
index bcc3526..71087f6 100644
--- a/main_math.ind
+++ b/main_math.ind
@@ -1,7 +1,7 @@
\begin{theindex}
\item $(G,\cdot )$-space, \hyperpage{211}
- \item $E$-valued random variable, \hyperpage{336}
+ \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}
@@ -23,20 +23,20 @@
\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{340}
+ \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{335}
+ \hyperpage{336}
\item $\sigma $-algebra generated, \hyperpage{172}
- \item $\sigma $-algebra generated by $X$, \hyperpage{336}
+ \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{335}
+ \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{335}
+ \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}
@@ -87,7 +87,7 @@
\item Absolute geometry, \hyperpage{64}
\item absolute value, \hyperpage{25}
\item absolutely continuous, \hyperpage{178}, \hyperpage{180},
- \hyperpage{336}
+ \hyperpage{337}
\item absolutely convergent, \hyperpage{77}, \hyperpage{106},
\hyperpage{117}, \hyperpage{300}
\item absolutely stable, \hyperpage{261}
@@ -98,7 +98,7 @@
\item Adams method, \hyperpage{262}
\item Adams-Bashforth method, \hyperpage{262}
\item Adams-Moulton method, \hyperpage{262}
- \item adapted, \hyperpage{337}
+ \item adapted, \hyperpage{338}
\item adherence, \hyperpage{54}
\item adherent point, \hyperpage{54}, \hyperpage{207}
\item adjacency matrix, \hyperpage{50}
@@ -196,7 +196,7 @@
\item ball, \hyperpage{205}
\item Banach fixed-point theorem, \hyperpage{125}
\item Banach space, \hyperpage{301}
- \item Banach-Alaoglu theorem, \hyperpage{338}
+ \item Banach-Alaoglu theorem, \hyperpage{339}
\item Banach-Steinhaus theorem, \hyperpage{310}
\item bandlimited, \hyperpage{239}
\item Barrow's law, \hyperpage{298}
@@ -420,7 +420,7 @@
\item Compact space, \hyperpage{213}
\item compact subset, \hyperpage{213}
\item compact support, \hyperpage{79}
- \item compactly embedded, \hyperpage{340}
+ \item compactly embedded, \hyperpage{341}
\item compactness, \hyperpage{213}
\item Comparison test, \hyperpage{76}, \hyperpage{80},
\hyperpage{183}
@@ -463,9 +463,9 @@
\item conditional consistency, \hyperpage{269}
\item conditional expectation, \hyperpage{186}
\item conditional expectation of $X$ given $\mathcal {G}$,
- \hyperpage{336}
- \item conditional expectation of $X$ given $Y$, \hyperpage{337}
- \item conditional law of $X$ given $Y$, \hyperpage{337}
+ \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}
@@ -540,10 +540,10 @@
\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{338}
+ \item converges strongly, \hyperpage{339}
\item converges uniformly, \hyperpage{77, 78}, \hyperpage{106, 107}
- \item converges weakly, \hyperpage{338}
- \item converges weakly-*, \hyperpage{338}
+ \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}
@@ -619,7 +619,7 @@
\item dehomogenization, \hyperpage{70}
\item Delta method, \hyperpage{196}
\item dense, \hyperpage{207}
- \item density, \hyperpage{278}, \hyperpage{336}
+ \item density, \hyperpage{278}, \hyperpage{337}
\item density function, \hyperpage{178}
\item Dependence on $\boldsymbol {\mathrm {\lambda }}$,
\hyperpage{129}
@@ -726,7 +726,8 @@
\item dominant eigenvalue, \hyperpage{100}
\item dominant eigenvector, \hyperpage{100}
\item Dominated convergence theorem, \hyperpage{184}, \hyperpage{189},
- \hyperpage{297}, \hyperpage{335}
+ \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}
@@ -762,7 +763,7 @@
\item elliptic equation, \hyperpage{277}
\item elliptic point, \hyperpage{150}
\item elliptic sector, \hyperpage{133}
- \item embedded, \hyperpage{340}
+ \item embedded, \hyperpage{341}
\item embedded methods, \hyperpage{260}
\item empirical distribution, \hyperpage{202}
\item empty set, \hyperpage{6}
@@ -821,7 +822,7 @@
\item even extension, \hyperpage{84}
\item even periodic extension, \hyperpage{283}
\item event, \hyperpage{173}
- \item events, \hyperpage{336}
+ \item events, \hyperpage{337}
\item evolute, \hyperpage{144}
\item exact, \hyperpage{156}
\item Examples of Euclidean motions, \hyperpage{73}
@@ -830,10 +831,10 @@
\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{336}
+ \item Expectation, \hyperpage{183}, \hyperpage{337}
\item expectation, \hyperpage{178}, \hyperpage{182--184},
\hyperpage{249}
- \item expectation of $X$, \hyperpage{336}
+ \item expectation of $X$, \hyperpage{337}
\item explicit, \hyperpage{258}
\item explicit Euler method, \hyperpage{258}
\item explicit form, \hyperpage{122}
@@ -848,8 +849,8 @@
\item exponential series, \hyperpage{48}
\item extended complex plane, \hyperpage{105}
\item extended real numbers, \hyperpage{105}
- \item extension, \hyperpage{126}, \hyperpage{308}, \hyperpage{340}
- \item extension operator, \hyperpage{340}
+ \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}
@@ -866,7 +867,7 @@
\item Fano configuration, \hyperpage{67}
\item fast Fourier transform, \hyperpage{240}
\item Fatou's lemma, \hyperpage{184}, \hyperpage{297},
- \hyperpage{335}
+ \hyperpage{336}
\item feasible region, \hyperpage{51}
\item feasible solution, \hyperpage{51}
\item Fejér kernel, \hyperpage{85}, \hyperpage{234}
@@ -886,8 +887,8 @@
\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{337}
- \item filtration, \hyperpage{321}, \hyperpage{337}
+ \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}
@@ -912,7 +913,7 @@
\item first integral, \hyperpage{134}, \hyperpage{154}
\item First isomorphism theorem, \hyperpage{18}, \hyperpage{39},
\hyperpage{44}
- \item first order reflection, \hyperpage{341}
+ \item first order reflection, \hyperpage{342}
\item First Sylow theorem, \hyperpage{41}
\item Fisher information, \hyperpage{195}
\item Fisher's theorem, \hyperpage{197}
@@ -954,7 +955,7 @@
\item FTBS, \hyperpage{268}
\item FTCS, \hyperpage{268}
\item FTFS, \hyperpage{268}
- \item Fubini, \hyperpage{335}
+ \item Fubini, \hyperpage{336}
\item Fubini's theorem, \hyperpage{59}, \hyperpage{175},
\hyperpage{300}
\item Fubini's theorem for elementary regions, \hyperpage{59}
@@ -979,7 +980,7 @@
\indexspace
- \item Gagliardo, Nirengerg and Sobolev's inequality, \hyperpage{340}
+ \item Gagliardo, Nirengerg and Sobolev's inequality, \hyperpage{341}
\item Galerkin approximation, \hyperpage{275}
\item Galois, \hyperpage{168}
\item Galois extension, \hyperpage{168}
@@ -1000,8 +1001,8 @@
\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 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}
@@ -1101,6 +1102,7 @@
\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}
@@ -1144,7 +1146,7 @@
\item hypothesis test, \hyperpage{199}
\item Hölder condition, \hyperpage{247}
\item Hölder conjugates, \hyperpage{302}
- \item Hölder continuity, \hyperpage{340}
+ \item Hölder continuity, \hyperpage{341}
\item Hölder's inequality, \hyperpage{302, 303}
\indexspace
@@ -1227,7 +1229,7 @@
\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{335}
+ \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}
@@ -1301,12 +1303,13 @@
\item Jacobian determinant, \hyperpage{55}
\item Jacobian matrix, \hyperpage{55}
\item Jeffrey's prior, \hyperpage{203}
- \item Jensen's inequality, \hyperpage{236}, \hyperpage{336}
+ \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}
@@ -1351,7 +1354,7 @@
\item lattice of subgroups, \hyperpage{169}
\item Laurent series, \hyperpage{117}
\item Laurent series theorem, \hyperpage{117}
- \item law of $X$, \hyperpage{336}
+ \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}
@@ -1508,8 +1511,9 @@
\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{337}
+ \item martingale, \hyperpage{338}
\item material derivative operator, \hyperpage{277}
\item Matrix, \hyperpage{13}
\item matrix, \hyperpage{13}, \hyperpage{18}
@@ -1538,10 +1542,10 @@
\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{335}
- \item measurable space, \hyperpage{175}, \hyperpage{335}
- \item Measure, \hyperpage{291}, \hyperpage{335}
- \item measure, \hyperpage{174}, \hyperpage{291}, \hyperpage{335}
+ \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}
@@ -1583,7 +1587,7 @@
\item moment-generating function, \hyperpage{186}
\item monic, \hyperpage{11}
\item Monotone convergence theorem, \hyperpage{184}, \hyperpage{296},
- \hyperpage{335}
+ \hyperpage{336}
\item monotonic, \hyperpage{26}, \hyperpage{28}
\item monotonically decreasing, \hyperpage{26}
\item monotonically increasing, \hyperpage{26}
@@ -1591,7 +1595,7 @@
\item more efficient than, \hyperpage{194}
\item Morera's theorem, \hyperpage{114}
\item morphism of field extensions, \hyperpage{165}
- \item Morrey's embedding, \hyperpage{340}
+ \item Morrey's embedding, \hyperpage{341}
\item MSE, \hyperpage{194}
\item multi-index notation, \hyperpage{57}
\item multicollinearity, \hyperpage{255}
@@ -1625,6 +1629,7 @@
\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}
@@ -1866,12 +1871,12 @@
\item Poincaré index formula, \hyperpage{139}
\item Poincaré inequality, \hyperpage{289}
\item Poincaré map, \hyperpage{136}
- \item Poincaré's inequality, \hyperpage{339}
+ \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{339}
+ \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}
@@ -1948,14 +1953,14 @@
\item prior distribution, \hyperpage{202}
\item probability, \hyperpage{173}
\item probability density function, \hyperpage{178}
- \item probability kernel, \hyperpage{337}
+ \item probability kernel, \hyperpage{338}
\item Probability mass function, \hyperpage{176}
\item probability mass function, \hyperpage{176}
- \item probability space, \hyperpage{173}, \hyperpage{336}
+ \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{335}
- \item product measure, \hyperpage{335}
+ \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}
@@ -2025,10 +2030,10 @@
\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{336}
+ \item Radon-Nikodym, \hyperpage{337}
\item Radó theorem, \hyperpage{218}
\item random sample, \hyperpage{193}
- \item Random variable, \hyperpage{336}
+ \item Random variable, \hyperpage{337}
\item random variable, \hyperpage{175}
\item random vector, \hyperpage{179}
\item Random walk, \hyperpage{319}
@@ -2077,7 +2082,7 @@
\item regular domain, \hyperpage{148}
\item regular region, \hyperpage{160}
\item regular surface, \hyperpage{147}
- \item Reillich-Kondrachov's compactness theorem, \hyperpage{341}
+ \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}
@@ -2265,7 +2270,7 @@
\item Slutsky's theorem, \hyperpage{190}
\item small change, \hyperpage{279}
\item Sobolev space, \hyperpage{289}
- \item Sobolev spaces, \hyperpage{339}
+ \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}
@@ -2335,8 +2340,8 @@
\item Stokes' theorem, \hyperpage{62}
\item Stolz-Cesàro theorem, \hyperpage{27}
\item Stone-Weierstra\ss \ theorem, \hyperpage{304}
- \item stopped process, \hyperpage{337}
- \item stopping time, \hyperpage{321}, \hyperpage{337}
+ \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}
@@ -2346,9 +2351,11 @@
\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{338}
+ \item strongly lower-semicontinuous, \hyperpage{339}
\item Strum's sequence, \hyperpage{92}
\item Student's $t$-distribution with $n$ degrees of freedom,
\hyperpage{197}
@@ -2366,7 +2373,7 @@
\item sublinear, \hyperpage{306}
\item submanifold, \hyperpage{146}
\item submanifold with boundary, \hyperpage{157}
- \item submartingale, \hyperpage{337}
+ \item submartingale, \hyperpage{338}
\item Submersion, \hyperpage{146}
\item submersion, \hyperpage{146}
\item subordinated matrix norm, \hyperpage{97}
@@ -2389,10 +2396,10 @@
\hyperpage{107}
\item sum of the series in an uniform sense, \hyperpage{78},
\hyperpage{107}
- \item supermartingale, \hyperpage{337}
+ \item supermartingale, \hyperpage{338}
\item support, \hyperpage{8}, \hyperpage{79}, \hyperpage{146},
\hyperpage{176}, \hyperpage{180}, \hyperpage{242}
- \item support plane, \hyperpage{338}
+ \item support plane, \hyperpage{339}
\item support points, \hyperpage{93}
\item supremum, \hyperpage{25}
\item Supremum axiom, \hyperpage{25}
@@ -2488,9 +2495,9 @@
\item totally isotropic, \hyperpage{74}
\item Tower formula, \hyperpage{165}
\item tower of fields, \hyperpage{165}
- \item Tower property, \hyperpage{337}
+ \item Tower property, \hyperpage{338}
\item trace, \hyperpage{141}, \hyperpage{289}
- \item trace operator, \hyperpage{341}
+ \item trace operator, \hyperpage{342}
\item Trace theorem, \hyperpage{289}
\item Traffic flow equation, \hyperpage{280}
\item trail, \hyperpage{49}
@@ -2551,7 +2558,7 @@
\item uniformly continuous, \hyperpage{33}, \hyperpage{55}
\item uniformly convergent, \hyperpage{117}
\item uniformly equicontinuous, \hyperpage{125}, \hyperpage{304}
- \item uniformly integrable, \hyperpage{337}
+ \item uniformly integrable, \hyperpage{338}
\item uniformly most powerful, \hyperpage{200}
\item union, \hyperpage{6}
\item unique factorization domain, \hyperpage{45}
@@ -2625,9 +2632,9 @@
\item Weak duality theorem, \hyperpage{51}
\item weak formulation, \hyperpage{275}
\item Weak law, \hyperpage{190}
- \item weak-* lower-semicontinuity, \hyperpage{338}
+ \item weak-* lower-semicontinuity, \hyperpage{339}
\item weakly consistent estimator, \hyperpage{194}
- \item weakly lower-semicontinuity, \hyperpage{338}
+ \item weakly lower-semicontinuity, \hyperpage{339}
\item Weierstra\ss ' theorem, \hyperpage{55}, \hyperpage{114},
\hyperpage{214}
\item Weierstra\ss \ approximation theorem, \hyperpage{79}
diff --git a/main_math.tex b/main_math.tex
index c435b75..11e3088 100644
--- a/main_math.tex
+++ b/main_math.tex
@@ -101,6 +101,9 @@ \chapter{Fourth year}
\chapter{Fifth year}
\newpage
+\subfile{Mathematics/5th/Advanced_dynamical_systems/Advanced_dynamical_systems.tex}
+\cleardoublepage
+
\subfile{Mathematics/5th/Advanced_probability/Advanced_probability.tex}
\cleardoublepage
@@ -113,8 +116,8 @@ \chapter{Fifth year}
\subfile{Mathematics/5th/Introduction_to_non_linear_elliptic_PDEs/Introduction_to_non_linear_elliptic_PDEs.tex}
\cleardoublepage
-\subfile{Mathematics/5th/Limit_theorems_and_large_deviations/Limit_theorems_and_large_deviations.tex}
-\cleardoublepage
+% \subfile{Mathematics/5th/Limit_theorems_and_large_deviations/Limit_theorems_and_large_deviations.tex}
+% \cleardoublepage
\subfile{Mathematics/5th/Stochastic_calculus/Stochastic_calculus.tex}
\cleardoublepage
diff --git a/preamble_formulas.sty b/preamble_formulas.sty
index c0fb201..9302ee2 100644
--- a/preamble_formulas.sty
+++ b/preamble_formulas.sty
@@ -89,10 +89,11 @@
{PDE}{\apl} % partial differential equations
{RFA}{\ana} % real and functional analysis
{SP}{\sta} % stochastic processes
- {ATFAPDE}{\ana} % advanced topics in functional analysis and pdes
+ {ADS}{\apl} % advanced dynamical systems
{AP}{\sta} % advanced probability
- {INLEPDE}{\ana} % introduction to non linear elliptic pdes
+ {ATFAPDE}{\ana} % advanced topics in functional analysis and pdes
{IEPDE}{\ana} % introduction to evolution pdes
+ {INLEPDE}{\ana} % introduction to non linear elliptic pdes
{LTLD}{\sta} % limit theorems and large deviations
{SC}{\sta} % stochastic calculus
}{\col}%
diff --git a/style.css b/style.css
index fe84c05..7e99b65 100644
--- a/style.css
+++ b/style.css
@@ -21,7 +21,7 @@
background-color: var(--background_color);
color: var(--general_color);
/* padding: 16px 25px; */
- text-align: center;
+ text-align: left;
text-decoration: none;
display: inline-block;
font-size: 16px;