updated week

.github/workflows/buildpdf.yml

@@ -185,8 +185,8 @@ jobs:
       - name: Compile - INEPDE
         uses: xu-cheng/latex-action@v2
         with:
-          root_file: Introduction_to_non_linear_elliptic_PDEs.tex
-          working_directory: Mathematics/5th/Introduction_to_non_linear_elliptic_PDEs/
+          root_file: Introduction_to_nonlinear_elliptic_PDEs.tex
+          working_directory: Mathematics/5th/Introduction_to_nonlinear_elliptic_PDEs/
       - name: Compile - LTLD
         uses: xu-cheng/latex-action@v2
         with:
@@ -279,7 +279,7 @@ jobs:
             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/Introduction_to_non_linear_elliptic_PDEs/Introduction_to_non_linear_elliptic_PDEs.pdf
+            Mathematics/5th/Introduction_to_nonlinear_elliptic_PDEs/Introduction_to_nonlinear_elliptic_PDEs.pdf
             Mathematics/5th/Limit_theorems_and_large_deviations/Limit_theorems_and_large_deviations.pdf
             Mathematics/5th/Stochastic_calculus/Stochastic_calculus.pdf
             main_physics.pdf

Mathematics/3rd/Probability/Probability.tex

@@ -217,7 +217,7 @@
       \item There exists $A\in\mathcal{E}$ such that $\mu(A)<\infty$.
       \item If $\{A_n\in\mathcal{E}:n\in\NN\}$ is a collection of pairwise disjoint sets, then: $$\mu\left(\bigcup_{n=1}^\infty A_n\right)=\sum_{n=1}^\infty \mu(A_n)$$
     \end{enumerate}
-    The triplet $(E,\mathcal{E},\mu)$ is called a \emph{measurable space}.
+    The triplet $(E,\mathcal{E},\mu)$ is called a \emph{measure space}.
   \end{definition}
   \begin{definition}
     The \emph{$\sigma$-algebra of all Lebesgue measurable sets in $\RR^n$}, $\mathcal{L}_n\subset\mathcal{P}(\RR^n)$, is defined as:
@@ -230,10 +230,10 @@
     We can extend the concept of volume on rectangles in $\RR^n$ to all the elements in $\mathcal{L}_n$. This extension is called \emph{Lebesgue measure} (or simply \emph{volume}) in $\RR^n$.
   \end{theorem}
   \begin{definition}
-    Let $(E,\mathcal{E},\mu)$ be a measurable space and $f:E\rightarrow\RR$ be a function. We say that $f$ is \emph{measurable} if $\forall B\in\mathcal{B}(\RR)$ we have $f^{-1}(B)\in\mathcal{E}$. The \emph{Lebesgue integral} of $f$ over $E$ with respect to $\mu$ is denoted by: $$\int_Ef\dd\mu$$
+    Let $(E,\mathcal{E},\mu)$ be a measure space and $f:E\rightarrow\RR$ be a function. We say that $f$ is \emph{measurable} if $\forall B\in\mathcal{B}(\RR)$ we have $f^{-1}(B)\in\mathcal{E}$. The \emph{Lebesgue integral} of $f$ over $E$ with respect to $\mu$ is denoted by: $$\int_Ef\dd\mu$$
   \end{definition}
   \begin{proposition}
-    Let $(E,\mathcal{E},\mu)$ be a measurable space and $f:E\rightarrow\RR$ be a measurable function such that $f(x)\geq 0$ $\forall x\in E$. Then, we can always define the integral $$\int_Ef\dd\mu$$ taking into account that may be $+\infty$.
+    Let $(E,\mathcal{E},\mu)$ be a measure space and $f:E\rightarrow\RR$ be a measurable function such that $f(x)\geq 0$ $\forall x\in E$. Then, we can always define the integral $$\int_Ef\dd\mu$$ taking into account that may be $+\infty$.
   \end{proposition}
   \begin{definition}
     Let $(E,\mathcal{E},\mu)$ be a measurable space and $f:E\rightarrow\RR$ be a measurable function. We say that $f$ is \emph{Lebesgue integrable} with respect to $\mu$ if: $$\int_E|f|\dd\mu<\infty$$

Mathematics/4th/Dynamical_systems/Dynamical_systems.tex

@@ -201,15 +201,15 @@
     The \emph{codimension} of a bifurcation is the number of parameters which must be varied for the bifurcation to occur.
   \end{definition}
   \begin{definition}[Saddle-node bifurcation]
-    The normal form of the codimension-one \emph{saddle-node bifurcation} is: $$x'=x^2+\mu$$
+    The normal form of the codimension-one \emph{saddle-node bifurcation} is: $$x'=\mu+x^2$$
     \mcref{DS:sn} shows the qualitative behavior of that system\footnote{In these images, the red lies means that the point $(\mu,x)$ is repelling. The blue lines means that the point $(\mu,x)$ is attracting.}.
   \end{definition}
   \begin{definition}[Transcritical bifurcation]
-    The normal form of the codimension-one \emph{transcritical bifurcation} is: $$x'=x^2+\mu x$$
+    The normal form of the codimension-one \emph{transcritical bifurcation} is: $$x'=\mu x+x^2$$
     \mcref{DS:trans} shows a qualitative behavior of the stability of the equilibria.
   \end{definition}
   \begin{definition}[Pitchfork bifurcation]
-    The normal form of the codimension-one \emph{pitchfork bifurcation} is: $$x'=x^3+\mu x$$
+    The normal form of the codimension-one \emph{pitchfork bifurcation} is: $$x'=\mu x+x^3$$
     \mcref{DS:fork} shows a qualitative behavior of the stability of the equilibria.
   \end{definition}
   \begin{figure}[H]

Mathematics/5th/Advanced_probability/Advanced_probability.tex

@@ -32,7 +32,7 @@
               \mu\left(\bigcup_{n\in\NN}{A_n}\right)=\sum_{n\in\NN}{\mu(A_n)}
             $$
     \end{enumerate}
-    The triple $(E,\mathcal{E},\mu)$ is called a \emph{measurable space}.
+    The triple $(E,\mathcal{E},\mu)$ is called a \emph{measure space}.
   \end{definition}
   \begin{definition}
     Let $(E,\mathcal{E},\mu)$ be a measurable space and $f:E\to [0,\infty]$ be a measurable function. We define the \emph{integral of $f$ with respect to $\mu$} as:

Mathematics/5th/Advanced_topics_in_functional_analysis_and_PDEs/Advanced_topics_in_functional_analysis_and_PDEs.tex

@@ -331,6 +331,17 @@
   \begin{theorem}
     Let $1\leq p<\infty$ and $u\in W^{1,p}(\RR_{\geq 0}^d)$. Then, $Tu=0$ if and only if $u\in W_0^{1,p}(\RR_{\geq 0}^d)$.
   \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
+    $$
+      \function{T}{W^{1,p}(\Omega)}{L^p(\Fr{\Omega})}{u}{u|_{\Fr{\Omega}}}
+    $$
+    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.
+    \end{itemize}
+  \end{theorem}
   \begin{lemma}
     Let $\Omega\subseteq \RR^d$ be an open set and $u\in W^{1,p}(\Omega)$ with $1\leq p\leq \infty$. Then, $\norm{\grad \abs{u}}\almoste{\leq}\norm{\grad u}$.
   \end{lemma}

Mathematics/5th/Instabilities_and_nonlinear_phenomena/Instabilities_and_nonlinear_phenomena.tex

@@ -0,0 +1,8 @@
+\documentclass[../../../main_math.tex]{subfiles}
+
+\begin{document}
+\changecolor{INLP}
+\begin{multicols}{2}[\section{Instabilities and nonlinear phenomena}]
+
+\end{multicols}
+\end{document}

Mathematics/5th/Introduction_to_non_linear_elliptic_PDEs/Introduction_to_non_linear_elliptic_PDEs.tex → Mathematics/5th/Introduction_to_nonlinear_elliptic_PDEs/Introduction_to_nonlinear_elliptic_PDEs.tex

@@ -2,40 +2,48 @@
 
 \begin{document}
 \changecolor{INLEPDE}
-\begin{multicols}{2}[\section{Introduction to non linear elliptic PDEs}]
+\begin{multicols}{2}[\section{Introduction to nonlinear elliptic PDEs}]
   \subsection{Introduction}
   \begin{definition}
-    Let $a_{ij}$, $b_j$, $c$, $f$ be known functions on $\Omega\subseteq \RR^d$. Usually we will denote $\vf{A}=(a_{ij})$ and $\vf{b}=(b_j)$ A \emph{linear second-order PDE} is an equation of the form:
+    Let $a_{ij}$, $b_j$, $c$, $f$ be known scalar functions defined on $\Omega\subseteq \RR^d$. Usually we will denote $\vf{A}=(a_{ij})$ and $\vf{b}=(b_j)$. A \emph{linear second-order PDE} is an equation of the form:
     \begin{equation*}
-      -\sum_{i,j=1}^da_{ij}(x){\partial_{ij}}^2u(x)+\sum_{j=1}^db_j(x)\partial_ju(x)+c(x)u(x)=f(x)
+      -\sum_{i,j=1}^da_{ij}(\vf{x})\partial_{ij}^2u(\vf{x})+\sum_{j=1}^db_j(\vf{x})\partial_ju(\vf{x})+c(\vf{x})u(\vf{x})=f(\vf{x})
     \end{equation*}
     where $u:\Omega\to \RR$ is the unknown function. This form is called \emph{non-divergence form}. If we write the equation in the form:
     \begin{multline*}
-      -\sum_{i=1}^d\pdv{}{x_i}\left(\sum_{j=1}^da_{ij}(x)\partial_ju(x)\right)+\sum_{j=1}^db_j(x)\partial_ju(x)+\\+c(x)u(x)=f(x)
+      -\sum_{i=1}^d\pdv{}{x_i}\left(\sum_{j=1}^da_{ij}(\vf{x})\partial_ju(\vf{x})\right)+\sum_{j=1}^db_j(\vf{x})\partial_ju(\vf{x})+\\+c(\vf{x})u(\vf{x})=f(\vf{x})
     \end{multline*}
     then we say that the equation is in \emph{divergence form}. Together with the PDE we usually impose boundary conditions on $\partial\Omega$. The \emph{Dirichlet boundary condition} is:
     $$
       u|_{\partial\Omega}=g
     $$
     and it is called \emph{homogeneous} if $g=0$. The \emph{Neumann boundary condition} is:
     $$
-      \langle \vf{n},\vf{A} \nabla u\rangle|_{\partial\Omega}=g
+      \langle \vf{n},\vf{A} \grad u\rangle|_{\partial\Omega}=g
     $$
     where we have assumed that the boundary of $\Omega$ is smooth enough to define the normal vector $\vf{n}$. The condition is called \emph{homogeneous} if $g=0$. Note that if $\vf{A}=\vf{I}_d$, then the Neumann boundary condition is just $\partial_{\vf{n}} u=g$.
   \end{definition}
   \begin{definition}
-    Let $a_{ij},b_j,c$ be known functions on $\Omega\subseteq \RR^d$. We say that the operator $$L=-\sum_{i,j=1}^da_{ij}{\partial_{ij}}^2 + \sum_{j=1}^d b_j\partial_j+c$$ is \emph{uniformly elliptic} if there exists $\theta>0$ such that for all $x\in \Omega$ and all $p\in \RR^d$ we have:
+    Let $a_{ij},b_j,c$ be known functions on $\Omega\subseteq \RR^d$. We say that the operator $$L=-\sum_{i,j=1}^da_{ij}\partial_{ij}^2 + \sum_{j=1}^d b_j\partial_j+c$$ is \emph{uniformly elliptic} if there exists $\theta>0$ such that for all $x\in \Omega$ and all $p\in \RR^d$ we have:
     \begin{equation}
-      Q_x(p)=\sum_{i,j=1}^da_{ij}(x)p_ip_j\geq \theta \sum_{i=1}^{d} {p_i}^2
+      Q_x(\vf{p})=\sum_{i,j=1}^da_{ij}(\vf{x})p_ip_j\geq \theta \sum_{i=1}^{d} {p_i}^2
     \end{equation}
   \end{definition}
   \begin{remark}
     Geometrically speaking, this implies that the sets
     $$
-      \xi_{x,h}=\{ p\in \RR^d: Q_x(p)=h\}
+      \xi_{x,h}=\{ \vf{p}\in \RR^d: Q_x(\vf{p})=h\}
     $$
     are ellipsoids.
   \end{remark}
+  \begin{proposition}
+    Let $H$ be Hilbert and $K:H\to H$ be a continuous linear operator. Then, the following are equivalent:
+    \begin{enumerate}
+      \item $K$ is compact.
+      \item For any bounded sequence $(u_n)\in H$, the sequence $(Ku_n)$ has a convergent subsequence.
+      \item For any sequence $(u_n)\in H$ such that $u_n\rightharpoonup u$, we have $Ku_n\to Ku$.
+    \end{enumerate}
+  \end{proposition}
   \subsection{Hilbert space methods for divergence form linear PDEs}
   In this section, we will assume that $\Omega\subset\RR^d$ is an open, bounded subset, $a_{ij}=a_{ji}$ and $a_{ij},b_j,c\in L^\infty(\Omega)$.
   \subsubsection{Fredholm alternative}

Mathematics/5th/Stochastic_calculus/Stochastic_calculus.tex

@@ -36,6 +36,22 @@
     \end{multline*}
   \end{proposition}
   \subsubsection{Brownian motion}
+  \begin{definition}
+    A \emph{Brownian motion} is a stochastic process ${(B_t)}_{t\geq 0}$ such that:
+    \begin{enumerate}
+      \item $B$ is Gaussian with $\Exp(B_t)=0$ and $\cov(B_s,B_t)=s\wedge t$.
+      \item $B$ has continuous paths.
+    \end{enumerate}
+  \end{definition}
+  \begin{proposition}
+    Let $B$ be a Brownian motion. Then:
+    \begin{enumerate}
+      \item $B_0=0$ a.s.
+      \item $B$ has independent increments.
+      \item $B$ has stationary increments.
+    \end{enumerate}
+    Conversely, any stochastic process with these properties has the law of a Brownian motion.
+  \end{proposition}
   \begin{theorem}[Strong law of large numbers for Brownian motion]
     Let ${(B_t)}_{t\geq 0}$ be a Brownian motion. Then:
     $$
@@ -93,7 +109,7 @@
     Now, $\left\{d(X_s,A)\leq\frac{1}{k}\right\}\in \mathcal{F}_s\subseteq \mathcal{F}_t$ because $X$ is adapted and $z\mapsto d(z,A)$ is measurable.
     Thus, $\{T_A \leq t\}\in \mathcal{F}_t$ because it is a countable union and intersection of events in $\mathcal{F}_t$.
   \end{proof}
-  \begin{theorem}[Doob's optional sampling theorem]
+  \begin{theorem}[Doob's optional sampling theorem]\label{SC:doob_sampling}
     Let ${(M_t)}_{t\geq 0}$ be a continuous martingale and $T$ be a stopping time. Then, the \emph{stopped process} $M^T:={(M_{t\wedge T})}_{t\geq 0}$ is a continuous martingale. In particular, $\forall t\geq 0$, $\Exp(M_{t\wedge T})=\Exp(M_0)$. If $M^T$ is uniformly integrable and $T\overset{\text{a.s.}}{\leq}\infty$, then taking $t\to\infty$ we have $\Exp(M_T)=\Exp(M_0)$.
   \end{theorem}
   \begin{lemma}[Orthogonality of martingales]\label{SC:orthogonality_martingales}
@@ -205,7 +221,7 @@
   \end{proof}
   \subsubsection{Local martingales}
   \begin{definition}
-    A stochastic process ${(M_t)}_{t\geq 0}$ is a \emph{local martingale} if there exists a sequence of stopping times ${(T_n)}_{n\in\NN}$ (called \emph{localizing sequence}) such that:
+    A stochastic process ${(M_t)}_{t\geq 0}$ is a \emph{continuous local martingale} if there exists a sequence of stopping times ${(T_n)}_{n\in\NN}$ (called \emph{localizing sequence}) such that:
     \begin{enumerate}
       \item $T_n\nearrow \infty$ a.s.
       \item $M^{T_n}:={(M_{t\wedge T_n})}_{t\geq 0}$ is a martingale for all $n\in\NN$.
@@ -235,14 +251,49 @@
     $$
     And using the \mnameref{P:dominated} with $M_{t\wedge T_n}\leq \sup_{0\leq s\leq t}\abs{M_s}$ we conclude the result.
   \end{proof}
-
+  \begin{remark}
+    Note that if $M$ is a continuous local martingale with $M_0=0$, then we can always take $T_n=\inf\{t\geq 0:\abs{M_t}\geq n\}$ as a localizing sequence.
+  \end{remark}
+  \begin{theorem}[Doob's optional sampling theorem for local martingales]
+    Let $M={(M_t)}_{t\geq 0}$ be a continuous local martingale and $T$ be a stopping time. Then, the stopped process $M^T:={(M_{t\wedge T})}_{t\geq 0}$ is a continuous local martingale.
+  \end{theorem}
+  \begin{proof}
+    Let ${(T_n)}_{n\in\NN}$ be a localizing sequence for $M$. Since $M^{T_n}$ is a continuous martingale, by \mnameref{SC:doob_sampling} we have that $M^{T_n\wedge T}$ is a continuous martingale. Thus, $M^T$ is a local martingale with localizing sequence ${(T_n)}_{n\in\NN}$.
+  \end{proof}
+  \begin{proposition}
+    Continuous local martingales form a vector space.
+  \end{proposition}
+  \begin{proof}
+    Let $M$, $\tilde{M}$ be continuous local martingales with localizing sequences ${(T_n)}_{n\in\NN}$ and ${(\tilde{T}_n)}_{n\in\NN}$ respectively and $\lambda,\tilde{\lambda}\in\RR$. Then, ${(T_n\wedge \tilde{T}_n)}_{n\in\NN}$ is a localizing sequence for both $M$ and $\tilde{M}$ and so $\lambda M^{T_n\wedge \tilde{T}_n}+\tilde{\lambda}\tilde{M}^{T_n\wedge \tilde{T}_n}$ is a martingale.
+  \end{proof}
+  \begin{proposition}
+    If $M$ is a continuous local martingale which has finite variation a.s., then:
+    $$
+      \Prob(\forall t\geq 0,\ M_t=M_0)=1
+    $$
+  \end{proposition}
+  \begin{proof}
+    Let ${(T_n)}_{n\in\NN}$ be a localizing sequence for $M$. Then, $M^{T_n}$ is a martingale and $V(M^{T_n},0,t)=V(M,0,t\wedge T_n)<\infty$. Thus, by \mcref{SC:corollary_finite_variation} we have that $M^{T_n}_t=M^{T_n}_0$ $\forall t\geq 0$ and $n\in\NN$. Taking $n\to\infty$ we get the result.
+  \end{proof}
+  \begin{proposition}
+    Let $M$ be a continuous local martingale. Then, the limit
+    $$
+      {\langle M\rangle}_t:=\lim_{n\to\infty}\sum_{k=1}^{n}\abs{M_{t_k^n}-M_{t_{k-1}^n}}^2
+    $$
+    exists in probability for any $t\geq 0$ and does not depend on the partition ${(t_k^n)}_{0\leq k\leq n}\in \mathrm{P}([0,t])$ chosen as long as $\Delta_n\to 0$. Moreover, $\langle M\rangle=({\langle M\rangle}_t)_{t\geq 0}$ is the unique process (up to modification) such that:
+    \begin{enumerate}
+      \item ${\langle M\rangle}_0=0$
+      \item $t\mapsto {\langle M\rangle}_t$ is a.s.\ continuous.
+      \item $\langle M\rangle$ is a.s.\ non-decreasing.
+      \item ${({M_t}^2-{\langle M\rangle}_t)}_{t\geq 0}$ is a continuous local martingale.
+    \end{enumerate}
+  \end{proposition}
   \begin{theorem}[Levy's characterization of Brownian motion]
-    Let $M={(M_t)}_{t\geq 0}$ be a  stochastic process. Then, the following are equivalent:
+    Let $M={(M_t)}_{t\geq 0}$ be a stochastic process. Then, the following are equivalent:
     \begin{enumerate}
       \item $M$ is a continuous local square-integrable martingale with $M_0=0$ and ${\langle M\rangle}_t=t$.
       \item $M$ is a ${(\mathcal{F}_t)}_{t\geq 0}$-Brownian motion.
     \end{enumerate}
   \end{theorem}
-
 \end{multicols}
 \end{document}

index.html

@@ -74,7 +74,7 @@ <h2 class="special-color">Mathematics</h2>
           <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/Introduction_to_nonlinear_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/Stochastic_calculus.pdf';" target="_top">Stochastic calculus</button></li>
         </ul>

main_math.tex

@@ -113,7 +113,7 @@ \chapter{Fifth year}
 \subfile{Mathematics/5th/Introduction_to_evolution_PDEs/Introduction_to_evolution_PDEs.tex}
 \cleardoublepage
 
-\subfile{Mathematics/5th/Introduction_to_non_linear_elliptic_PDEs/Introduction_to_non_linear_elliptic_PDEs.tex}
+\subfile{Mathematics/5th/Introduction_to_nonlinear_elliptic_PDEs/Introduction_to_nonlinear_elliptic_PDEs.tex}
 \cleardoublepage
 
 % \subfile{Mathematics/5th/Limit_theorems_and_large_deviations/Limit_theorems_and_large_deviations.tex}