A very complete proof on the separability of $L^p$.
I have to prove the following important result.Theorem. Let $(X,\mathcal{A},\mu)$ be a measure space such that:$(1)\;$ the measurable space $(X,\mathcal{A})$ is separable.$(2)\;$$\mu$ is sigma...
View ArticleProof of Kolmogorov's Existence Theorem in Dudley's Real Analysis and...
The proof is written allowing for a state space that is an arbitrary measurable space, but I am interested in the case of real valued stochastic processes.I am working through the proof of the theorem,...
View ArticleConvergence of series $\sum_{n=1}^{\infty}x_n^{\alpha}$ subjected to...
Suppose $x_1\in(0,1),x_{n+1}=x_n-\frac{x_n^2}{\sqrt{n}}$, find the values of $\alpha\in\mathbb{R}$ for which the series $\sum_{n=1}^{\infty}x_n^{\alpha}$ is convergent.The following is my solution,...
View ArticleDiscrete product rule for $D_x^c(u\bar{u})$
For $(u_i)_{i\in I}$ define the central gradient by$$(D_x^c u)_i = \frac{u_{i+1}-u_{i-1}}{2\Delta x}\textrm{ for all }i\in I.$$If, additionally, I have some $(\bar{u}_i)_{i\in I}$, I am wondering...
View ArticleHow to check the compactness of following sets?
(1) Let $K \subset M_n(\mathbb{R})$ be defined by $$K = \{A \in M_n(\mathbb{R})\mid A = A^T, \ \operatorname{tr}(A) = 1, x^TAx \geq 0 \text{ for all } x \in \mathbb{R}\}$$Then $K$ is compact.(2) Let $K...
View ArticleCentral gradient, product, product rule
For $(u_i)_{i\in I}$ define the central gradient by$$(D_x^c u)_i = \frac{u_{i+1}-u_{i-1}}{2\Delta x}\textrm{ for all }i\in I.$$Now, for $u=(u_i)_{i\in I}$ and $\bar{u}=(\bar{u}_i)_{i\in I}$, I am...
View ArticleDistance module of tangent vector equal metric norm
Let $(M,g)$ be a Riemannian manifold, and let $d$ denotes the distance induced by $g$.Let $\gamma : I \to M$, where $I$ is an interval, be an absolutely continuous curve.We know that$$ |\dot\gamma_t|...
View ArticleLet $f$ be a continuous function mapping the unit interval to the unit...
Let $f$ be a continuous function mapping the unit interval to the unit interval such that $\int_{0}^{x}f = \int_{x}^{1}f$. Prove that $f=0$.I´ve tried doing a proof by contradiction. Suppose that...
View ArticleLe Gall Exercise 1.3
I am trying to do exercise 1.3 from Le Gall's Measure theory, probability and stochastic processes: Let $C([0,1],\mathbb{R}^d)$ be the space of all continuous functions from $[0,1]$ into...
View Articlecontraction mapping theorem for 'non-strict' contraction
Given a map $F:X \to X$ on a complete metric space $(X,d)$, and let $K<1$ such that:$$d(F(x), F(y)) \le K d(x,y), \quad \forall x,y \in X$$then the contraction mapping theorem tells us that $F$ has...
View ArticleAre all continuous functions that send conics to conics in $\mathbb{R}^2$ a...
Motivation:Consider an arbitrary conic section in $\mathbb{R}^2$ given by$$ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 $$Now consider the map $$\phi: \mathbb{R}^2 \rightarrow \mathbb{R}^2,...
View ArticleCan the rate or order of convergence of Gradient Descent change when we...
For example if the algorithm needs $O(1/\varepsilon)$ iterations to get to a point where $\|\nabla f(x)\| \leq \varepsilon$ can this Big-O complexity change when we instead use $\|x_{k+1} - x_{k}\|...
View ArticleFubini for Hausdorff measures
Let $f \in L^2(S^{n-1})$, where $S^{n-1}$ denotes the $n$ dimensional sphere. Now, by using Fubini we can write$$\int_{S^{n-1}}\int_{S^{n-1}} 1_{x \cdot y = 0} |f(x)|^2 d\mathcal{H}^{n-2}(x)...
View ArticleApplication of mean value property of harmonic functions?
let u be the eigenfunction of $-\Delta$ with eigenvalue $\lambda$ on domain $D$. Suppose $u < c_0r$ in $B_r(0)$, show that for every $y \in B_(r/2)$, $r<r_0$,$$u(y) \leq...
View ArticleGradient of softmax function with the inner product argument
Suppose that $x_i$ and $y_j$ are vectors in $\mathbb{R}^d$, where $i,j\in\{1,2,\dots,N\}$. Let the loss function be defined as $$\mathcal{L} = -\frac{1}{N}\ln\left(\frac{\exp({\frac{x_i.y_i}{\tau\lVert...
View ArticleFind the limit points of the sequence $a_n \ = \sin(\frac{n \pi} {4}) $
I know that I have to find some sub-sequences for each point, but how do I obtain a sub-sequence that does not alternate?
View ArticleRadius of convergence of $\sum\limits_{n=0}^{\infty}a_nx^{2n}$ and...
The question:If $\lim\limits_{n\to\infty}\sqrt[n]{|a_n|}=L$, prove that the power series $\sum\limits_{n=0}^{\infty}a_nx^{2n}$ and $\sum\limits_{n=0}^{\infty}a_nx^{2n+1}$ have both radius of...
View ArticleUnderstanding Proof: Simple Functions In...
I have difficulties understanding the proof of the following proposition:Proposition$\quad$ Let $(X,\mathscr{A},\mu)$ be a measure space, and let $p$ satisfy $1\leq p$. Then the simple functions in...
View ArticleLet $S$ be a non empty subset of $\Bbb R$ and bounded above and $M=\sup...
I tried the math but I am stuck at the point that how to show that $M \in S$. I did the part that $M$ is a limit point of $S$.$M=SupS$.Let $M$ does not belong to $S$. $M=sup s$So $x<M$ for all x...
View ArticleA random variable is symmetric if and only if its characteristic function is...
Quick summary: I am stuck on the implication: $\phi_X$ real-valued $\rightarrow$ $X$ symmetric.Assume you have a probability space $(\Omega, \mathcal{F},P)$, and a random varaiable $X: \Omega...
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