updated dynamical systems

.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

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}

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}

index.html

@@ -70,11 +70,12 @@ <h2 class="special-color">Mathematics</h2>
       <ul>
         <li>1st year:</li>
         <ul>
+          <li><button class="button" onclick="window.location.href='https://github.com/victorballester7/Complete-summaries/releases/latest/download/Advanced_dynamic_systems.pdf';" target="_top">Advanced dynamic systems</button></li>
           <li><button class="button" onclick="window.location.href='https://github.com/victorballester7/Complete-summaries/releases/latest/download/Advanced_probability.pdf';" target="_top">Advanced probability</button></li>
           <li><button class="button" onclick="window.location.href='https://github.com/victorballester7/Complete-summaries/releases/latest/download/Advanced_topics_in_functional_analysis_and_PDEs.pdf';" target="_top">Advanced topics in functional analysis and PDEs</button></li>
           <li><button class="button" onclick="window.location.href='https://github.com/victorballester7/Complete-summaries/releases/latest/download/Introduction_to_evolution_PDEs.pdf';" target="_top">Introduction to evolution PDEs</button></li>
           <li><button class="button" onclick="window.location.href='https://github.com/victorballester7/Complete-summaries/releases/latest/download/Introduction_to_non_linear_PDEs.pdf';" target="_top">Introduction to non linear PDEs</button></li>
-          <li><button class="button" onclick="window.location.href='https://github.com/victorballester7/Complete-summaries/releases/latest/download/Limit_theorems_and_large_deviations.pdf';" target="_top">Limit theorems and large deviations</button></li>
+          <!-- <li><button class="button" onclick="window.location.href='https://github.com/victorballester7/Complete-summaries/releases/latest/download/Limit_theorems_and_large_deviations.pdf';" target="_top">Limit theorems and large deviations</button></li> -->
           <li><button class="button" onclick="window.location.href='https://github.com/victorballester7/Complete-summaries/releases/latest/download/Stochastic_calculus.pdf';" target="_top">Stochastic calculus</button></li>
         </ul>
       </ul>

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}

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.