Sobolev functions approximated by ridge functions
Let $f \in W^{k,2}(\mathbb{R}^d)$, a Sobolev space with smoothness $k$ and dimension $d$. We aim to approximate$f$ using ridge functions of the form $g(\mathbf{a}.\mathbf{x})$. Suppose the...
View ArticlePartitioning a sequence of real numbers to a finite number of convergent...
By the BolzanoβWeierstrass theorem we know that a bounded infinite sequence of real numbers has a convergent subsequence. What I'm wondering about is whether we can do more.Here's my idea, which I'm...
View ArticleTwice differentiable Lipschitz functions have Lipschitz gradient
I think I can show that for any function $f\in \mathcal{C}^2(\mathbb{R}^d)$ s.t. $f$ is Lipschitz with constant $L$, then $\nabla f$ is Lipschitz as well. This result relies on the fact that the first...
View ArticleAttempt to derive Taylor expansion using discrete time steps
I was playing around trying to check if the $n$th order Taylor approximation about $0$ can be explained and obtained by using a physical reasoning argument like below for the case $n = 2$ (that is...
View Article$I_n=\int_{0}^{\pi} e^{-n \sin x}\,dx $ [duplicate]
Study the convergence of the sequence$$I_n=\int_{0}^{\pi} e^{-n \sin x}\,dx $$ and find its limit.My idea:$I_n= \int_{0}^{\frac{\pi}{2}} e^{-n \sin x}\,dx + \int_{\frac{\pi}{2}}^{\pi} e^{-n \sin x}\,dx...
View ArticleInclusive Disjunction Problem
Given $(P:=3)$ and $x$ is an arbitrary element of the interval $S:= [1,\,5]$ except $3$, what would it be as $P \in S$?Based on the given information above, verify that $(P < x) \lor (P > x)$,...
View ArticleFinding functions $f(t)$ such that $(f(t))^{s}$ is a Bernstein function.
I'm studying Bernstein function from the book R.L. Schilling, R. Song, and Z. VondraΔek, Bernstein Functions: Theory and Applications, 2. ed (De Gruyter, Berlin, 2012).Definition. A function...
View ArticleProof of the General Change of Variables Theorem in Rn?
I was reading the theroems on site https://planetmath.org/changeofvariablesinintegralonmathbbrn and found the following general change of variables theorem:Theorem 2. Let g:XβRn be continuously...
View ArticleAre convex polytopes closed in arbitrary metric spaces?
Let $(X,d)$ be a metric space. For all points $x,y \in X$ we define the metric segment between them as the following set:$$\left [ x,y \right ] = \left \{ z \in X : d(x,z)+d(z,y)=d(x,y)\right \}$$We...
View ArticleTopology basis consisting of convex sets in metric spaces
Let $(X,d)$ be a metric space. For all points $x,y \in X$ we define the metric segment between them as the following set:$$\left [ x,y \right ] = \left \{ z \in X : d(x,z)+d(z,y)=d(x,y)\right \}$$We...
View ArticlePoints in a rectangular configuration define equal metric segments?
Let $(X,d)$ be a metric space. For points $x,y \in X$ we define the metric segment between them as the following set:$$\left [ x,y \right ] = \left \{ z \in X : d(x,z)+d(z,y)=d(x,y)\right \}$$We will...
View ArticleFamily of sets closed under arbitrary intersections and arbitrary unions of...
Let $\mathcal{U} \subseteq \wp\left ( X \right )$ be a family of sets that contains both $\emptyset$ and $X$, is closed under arbitrary intersections and is closed under arbitrary unions of chains of...
View ArticleConvex hull of open sets is an open set?
Let $(X,d)$ be a metric space. For all points $x,y \in X$ we define the metric segment between them as the following set:$$\left [ x,y \right ] = \left \{ z \in X : d(x,z)+d(z,y)=d(x,y)\right \}$$We...
View ArticleBounding the integral $\int_{-\infty}^\infty\Big|\frac{f(x)-f(0)}{x}\Big|^2\,dx$
Is there a functional inequality that allows me to bound $$\int_{-\infty}^\infty\Big|\frac{f(x)-f(0)}{x}\Big|^2\,dx$$ in terms of a Sobolev norm of $f$, say?If $f$ is equal to its Taylor series at $0$,...
View ArticleThe condition "mass preserving maps" in the motivation of optimal transportation
I am reading Monge's formulation of the optimal transportation problem. It says that one wishes to find the a transport map $T: (X,\mu) \rightarrow (Y,\nu)$ that minimizes the transport cost $$\int_X...
View ArticleProve that $\mathcal{L}^n(A(E))=|\det A|\mathcal{L}^n(E)$
Suppose I have a linear map $A:\mathbb R^n \rightarrow \mathbb R^n$I want to prove $\mathcal{L}^n(A(E))=|\det A|\mathcal{L}^n(E) $$\forall E\subset \mathbb R^n$ where $\mathcal{L}^n$ is Lebesgue...
View ArticleFind a function in the unit sphere of a function subspace with given properties
Preliminaries.$X$ is a compact Hausdorff space.$a$ is a fixed element of $X$.$C(X)$ denotes the space of real valued continuous functions on $X$$A=\{f \in C(X); f(a)=0\}$ is a Banach space respect to...
View ArticleDetermining the Least Constant π for a Summation Involving Positive Real Numbers
Given positive real numbersπ₯1,π₯2,β¦,π₯2009x1β,x2β,β¦,x2009β, consider the sum:π₯1π₯1+π₯2+π₯2π₯2+π₯3+β―+π₯2009π₯2009+π₯1x1β+x2βx1ββ+x2β+x3βx2ββ+β―+x2009β+x1βx2009ββI am trying to determine the least constantπM such...
View ArticleDecomposition of continuous function with fixed zero
Source: Harmonic Measure by Garnett & Marshall, page 95In the book, the author claims that every real $g \in C(\mathbb{C}^*)$ could be written as $g = af_1 + bf_2 + c$, where $a,b,c \in...
View ArticleIf f_n goes uniformly to 0 and its integral w.r.t the Leb. measure converges...
Assume $(f_n)_{n=1}^{\infty}$ is a sequence of functions defined on the real line such that $\lim_{n\to\infty}f_n = 0$ uniformly on $\mathbb{R}$, and assume that...
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