# Tagged Questions

537 views

There are two classes of maps $S^1\to S^1$ for which I know how to define the winding number: • Continuous maps: Using the unique path lifting property of the universal covering map $\mathbb R\to ... 0answers 38 views ### minimum distance between sets and relation with functions [on hold] Let$A, B$be two convex and closed subsets of$\mathbb{R}^n$. We would like to the minimum distance between these two sets. i.e., we want to find a solution for the following problem. $$\min ... 1answer 129 views ### Question on Morse inequalities I want to understand why: From K.C Chang's book "Infinite Dimensional Morse Theory and Multiple Solution Problems": if i have then (4.1) is formal : it means that EDIT1: (4.1) tel us that ... 0answers 84 views ### Relationship between weak Lp and strong Lq topologies for q<p Specificaly: Does convergence in L^{\frac{1}{2}} imply weak L^2 convergence? Having a limit in L^{\frac{1}{2}} topology and a limit in weak L^2 topology whether these are always equal? If ... 1answer 155 views ### Characterization of a set in \mathbb{R}^d Let X= (X_1,\dots, X_d) be a fixed vector of random variables on the space (\Omega, \mathcal{F}, \mathbb{P}). Consider the following set. \label{main12} C= \{x\in \mathbb{R}^d ~|~ ... 2answers 140 views ### Smooth but non-analytic kernel functions Does there exist a (stationary) covariance kernel function which is C^\infty-smooth but not real analytic? If so, could you please provide an example? 0answers 346 views ### Given an even function how to obtain the most close odd function and vise versa? Given an even function f(x), how to obtain the most close to it continuous odd function g(x)? By most close I mean that \int_0^\infty |f(x)-g(x)| dx be the minimum possible and the difference ... 3answers 143 views ### Method to compute fundamental solutions which are distributions The Malgrange-Ehrenpreis theorem tells us that there is a fundamental solution for any linear differential operator of constants coefficients. The original proof was not constructive (it was based on ... 1answer 152 views ### When is the bound in Riesz-Thorin Interpolation Theorem attained? Let me recall the statement of Riesz-Thorin theorem (see also http://en.wikipedia.org/wiki/Riesz%E2%80%93Thorin_theorem). Theorem (Riesz-Thorin): Let (X,\mu) and (Y,\nu) be \sigma-finite ... 2answers 446 views ### Which smooth compactly supported functions are convolutions? If f,g are smooth functions with support in the interval [-r,r] for some r>0, then their convolution f*g is smooth with support in [-2r,2r]. My question is about the converse: Given ... 1answer 148 views ### Equivalence of negative Sobolev norm of derivative to L^2-norm Let S:=(0,1)^2 be the unit square in \mathbb{R}^2, and let M:=\{u\in L^2(S)\mid \int_S u=0\} be the space of (real-valued) L^2-functions with mean value zero. On M we can consider the ... 1answer 110 views ### Is the countably infinite product of locally convex topological vector spaces locally convex? Let (X,\tau) be a locally convex topological vector space and denote the product space$$X^{\infty}=X\times X\times X\cdots:=\big\{x=(x_i)_{i\geq 1}:~ x_i\in X\big\}$$If we endow X^{\infty} ... 1answer 162 views ### Inequality in the Sobolev space H^1 I've found the following inequality$$\int_{B_r}\vert u\vert^q\leq C \bigg(\int_{B_r}\vert\nabla u\vert^2\bigg)^{a}\bigg(\int_{B_r}\vert u\vert ... 1answer 392 views ### A generalization of a theorem of Grothendieck In this question the norm of$L^{P}[0,1]$is denoted by$\parallel . \parallel _{p}$. Let$p$and$q$be two arbitrary real numbers with$2<p<q$. Assume that$S$is a subvector space ... 2answers 107 views ### series representation of bivariate functions Given a bivariate function$f(x, y)$with$x \in [-a,a]$and$y \in [-b, b]$, what is the necessary and sufficient condition under which we can write$f(x, y) = \sum g_k(x)h_k(y)$for all$(x,y)$in ... 1answer 183 views ### Is the space of test functions separable? [closed] Consider the space$\mathcal D(\mathbb{R}^n)$of smooth functions (in the sense of having continuous derivatives of all orders) which are compactly supported. Endow it with its usual topology, i.e., ... 3answers 187 views ### dual space of a subspace of the space of bounded measures Let$\mathcal{M}=\mathcal{M}(\mathbb{R})$be the space of bounded measures. Equipped with the weak convergence, the dual space of$\mathcal{M}$is$\mathcal{C}_b(\mathbb{R})$consisting of continuous ... 0answers 107 views ### Dyadic approach to Marcinkiewicz interpolation for Lorentz Spaces In Exercise 21, in a note by professor Terrence Tao on his own blog http://terrytao.wordpress.com/2009/03/30/245c-notes-1-interpolation-of-lp-spaces/ Exercise 21 Suppose we are in the situation ... 0answers 36 views ### compactness related to some distance defined on the space of increasing functions2 Let$I=[0,1]$and denote by$C^{+}(I)$the space of continuous increasing functions. Can we find a distance$d$for$C^+(I)$such that the set of the form $$d(f,g)\rightarrow 0\Longrightarrow ... 0answers 44 views ### A fixed point problem in infinite dimensions for monotonic algebras Let A be an algebra over [0,1], whose operations are all unary monotone (increasing or decreasing) bijections, except that A also includes the infimum operation over finite or countably many ... 2answers 153 views ### Let f:J→R be an absolutely continuous and f'\in…? Let f:J\rightarrow \mathbb{R} be an absolutely continuous. Under what kind of extra condition for f', (not C) holds the following relation?$$ \Big | \frac{1}{|I_{1}|}\int_{I_{1}}f'(x)dx- ... 0answers 113 views ### Dual of the space of vector valued Borel measures What is the dual of the space of all vector valued Borel measures? 0answers 81 views ### Weak relative compactness in$L^1_{loc}$. In my work I stumbled upon a proposition (without proof, alas), which I can't really prove. Suppose we have a family of functions$\left\{\phi_\epsilon (t,x,v)\right\}_{\epsilon\in(0,1]}$, and$M(v)$... 1answer 137 views ### An Integral Functional Equation Let$f$be a non-negative function supported and integrable on the positive real axis, such that $$\int_0^\infty f(x+y)p(y) dy = c[p] f(x),$$ where$c[p]$a number (functional) dependent on function ... 1answer 129 views ### Lebesgue's integrability condition in several variables The well known Lebesgue's condition of Riemann integrability says that a bounded function in one variable$f\colon [a,b] \to \mathbb{R}$is Riemann integrable if and only if it is continuous almost ... 0answers 78 views ### Is Modulation space is close under absolute? It is clear that, for$1\leq p \leq \infty$,$f\in L^{p}$iff$| f| \in L^{p}.$By the usual modulation space$M^{p, q}(\mathbb R^{d})$,$1\leq p, q \leq \infty, d\in \mathbb N $, we mean the ... 0answers 57 views ### Fixed point theorm that does not require the hemi-continuity of the set valued map? All of the fixed point theorem I have seen (like Kakutani and Brower, Browder ) required the set valued map to be hemi-continuous (lower). Is any fixed point theorem that can assure the existence of ... 0answers 100 views ### Is reflexive Banach space valued scalarwise Lebesgue space isomorphic to the Bochner space? I first specify the setting and then formulate the question precisely. (A very long post follows.) Definitions 1. For$E$a (real Hausdorff) locally convex space, say that$E$is suitable iff there ... 2answers 182 views ### specific improper integral involving erf I have encountered an integral, and kindly ask for help with a solution. It is beyond my own capabilities, and neither Maple nor Mathematica were of any help: $$\int_{1}^{\infty} ... 1answer 182 views ### Continuous and dense embeddings and the density of sets in Hilbert space Suppose H is a Hilbert space of functions f:\Omega\to \mathbb{R}^n with \Omega\subset \mathbb{R}^n open, bounded and with Lipschitz boundary (take for example H=H_0^1(\Omega)^n) and suppose ... 1answer 117 views ### Weak continuity of Lebesgue decomposition Let X be a space with its \sigma-algebra \mathcal{B}; we are given a finite measure \mu and a sequence of finite measures \nu_n such that, for every bounded continuous function ... 0answers 66 views ### Fourier multiplier with a singularity on a convex curve Let h be a strictly convex function such that h(0) = h'(0)=0. Let \Phi: \mathbb{R}^2 \to \mathbb{R} be a C^{\infty}-function with compact support (say, \Phi is supported on ... 1answer 220 views ### Approximation of an integral over the unit ball of L_1 For every \varepsilon>0 find a piecewise continuous function q:[0,1]\rightarrow \mathbb{R} such that \int_0^1 q(x)dx=1 and$$\int_{0}^1 \int_{0}^{s} \left|\frac{q(s)q(t/s)}{s}- ... 1answer 158 views ### Does a particular iteration produce a weak solution to a non linear pde? Consider the following non linear pde in the unknown$v(x,y)$: $$\frac{\partial v(x,y)}{\partial x} + \Big(\frac{\partial v(x,y)}{\partial x} \Big)^2 = e^{2 ty}-1$$ where$t$is some fixed small ... 0answers 130 views ### Is there an appropriate weighted Sobolev space to include exponential map and projection map? Observe that given a non negative function$\omega: \mathbb{R^2} \rightarrow [0, \infty)$, we can define the weighted$L^{p}(\mathbb{R}^2, \omega) $spaces. They are measurable functions$f: ...
Let $$L(x)=Q\left(\frac{x}{2},\frac{a}{a+f(x)/\sqrt{x}}Q^{-1}\left(\frac{x}{2},1-b^{1/g(x)}\right)\right)$$ where $Q(s,x)=\frac{\Gamma(s,x)}{\Gamma(s)}$ is the upper incomplete gamma function ...