Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

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5
votes
1answer
114 views

On the zero set of a $C^2$ function on $[0,1]^2$

Let $f:[0,1]^2\rightarrow \mathbb{R}$ be a twice continuously differentiable function with the property that for all $x\in [0,1]$, there is an interval $I_x\subset [0,1]$ such that $f(x,y)=0$ for all ...
1
vote
1answer
114 views

Are solutions of the Beltrami Equations necessarily smooth?

Let $ a $, $ b $ and $ c $ be real constants such that $ \Delta \stackrel{\text{df}}{=} a c - b^{2} > 0 $. The Beltrami Equations are defined as the following system of PDE’s on the domain $ ...
1
vote
0answers
41 views

Average - Map - Infinite number of points [closed]

I have a problem to solve in the context of the preparation of the PUTNAM competition. I am asked to find the average of a certain map of $S \subset \mathbb{R^3}$ (domain $S$ is uncountable) into ...
8
votes
1answer
285 views

Why should the map $-\Delta^{-1}$ be continuous?

I'm reading an article by Wei-Ming Ni about the existence of solutions for the elliptic problem $$\Delta u +|x|^\lambda |u|^\tau =0,$$ in the unit ball $\Omega$ in dimension $>2$. I'm looking for ...
5
votes
1answer
163 views

Order between two completely monotone functions?

I am wondering if the following assertion is true: Let $f,g:\mathbb{R}_+\rightarrow [0,1]$ be completely monotone functions on $\mathbb{R}_+^*$, that is, $(-1)^n f^{(n)}(x)\geq 0$ and $(-1)^n ...
3
votes
0answers
121 views

Reference request: Darboux properties of real-valued set functions (measures, densities, etc.)

Fix a set $S$ and let $f: \mathcal P(S) \rightharpoonup \mathbf R$ be a real-valued partial function on the power set of $S$; denote by $\mathcal D$ the domain of $f$. We say that $f$ has: (i) the ...
2
votes
0answers
59 views

Optimal condition for the weak convergence of the jacobian determinant

Whenever $n<q,$ it is known that given a sequence $\{ u_{k} \}$ which is weakly convergent in $W^{1,q}(U)$ one has that the Jacobian determinants $\text{det} Du_{k}$ converge weakly in ...
2
votes
1answer
68 views

interpret of Picone inequality for non-regular functions

Assume $\Omega \subset \mathbb{R}^N$, $ N>4 $ is open set. There is a well-known picone identity that says Let $u,v \in C^2(\Omega)$ satisfy $v>0$ and $-\Delta v \geq 0$ in $\Omega$. The ...
21
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0answers
533 views

Relative null-ness

Here, "measure" always means Lebesgue measure on $\mathbb{R}$. This question is partly motivated by my answer ...
3
votes
0answers
46 views

$X_t = B_t^q$, $X_t = (\sin B_t)^q$, $X_t = B_t^q (\sin B_t)^r$, $dM_t = R_t\,M_t\,dB_t$ [closed]

What are the SDE's satisfied by the following processes? $X_t = B_t^q$ $X_t = (\sin B_t)^q$ $X_t = B_t^q (\sin B_t)^r$ Assume $B_t$ is a standard Brownian motion with $B_0 > 0$ and the ...
3
votes
1answer
102 views

$\int_0^t f(s)\,dB_s$ normally distributed, mean and variance

Suppose that $f(t)$ is a (non-random) continuous function on $[0, \infty)$. Let$$Z_t = \int_0^t f(s)\,dB_s.$$ How do I see that $Z_t$ is normally distributed? What is the mean and variance? I need ...
4
votes
1answer
91 views

On continuous perturbations of functions of the first Baire class on the Cantor set

Is it true that for any function of the first Baire class $f:X\to\mathbb R$ on the Cantor cube $X=2^\omega$ there is a continuous function $g:X\to[0,1]$ such that the image $(f+g)(X)$ is disjoint with ...
1
vote
0answers
53 views

Derivatives of $O$-regular varying functions are $O$-regular varying functions?

The Monotone Density Theorem for regularly varying functions says, in essence: Theorem (Monotone Density Theorem). Let $f$ be a differentiable regularly varying real-function of index $\rho$ ...
10
votes
4answers
504 views

Probability that planar Brownian motion doesn't “encircle” 0

Suppose $B_t$ is a standard Brownian motion in $\mathbb{R}^2$ and $T = \text{inf}\{t : |B_t| = 1\}$. Let $E$ denote the event that $0$ is contained in the unbounded component of $\mathbb{R}^2 ...
4
votes
1answer
79 views

$M_t = f(B_{t \wedge \tau}) + (t \wedge \tau)$ local martingale, $\textbf{E}^x[\tau] = f(x)?$

Suppose $D \subset \mathbb{R}^d$ is a domain and $f: \overline{D} \to \mathbb{R}$ is a continuous function, $C^2$ in $D$, satisfying$$f(x) = 0\text{ for }x\in \partial D,$$$${1\over2} \Delta f(x) = -1 ...
5
votes
0answers
127 views

Uniform approximation of separately continuous functions on zero-dimensional spaces

For topological spaces $X,Y,Z$ а function $f:X\times Y\to Z$ is called separately continuous if for any $(x,y)\in X\times Y$ the restrictions of $f$ to the sets $\{x\}\times Y$ and $X\times \{y\}$ are ...
1
vote
1answer
136 views

Is this function concave or convex? [closed]

let $g_{n,\gamma}(\sigma)$ be the function defined as the following $$ g_{n,\gamma}(\sigma)= \left(\frac{(\sigma-1)^2 +\gamma^2}{\sigma^2 +\gamma^2} \right)^{n/2} T_n\left( \frac{\sigma(\sigma-1) ...
4
votes
1answer
116 views

Can the integral of a “generic” bounded measurable function be determined by its values on the rationals?

[This question is an extension of my question Does a positive-measure subset of the unit interval almost surely intersect a random translation of some countable subgroup of $\mathbb{R}$?. I'm asking ...
6
votes
3answers
227 views

Quantum Mechanics and bilinear optimal control theory

I was wondering whether there are any rigorous results about the optimal controllability of Schrödinger operators. So my question is something like this: Let $i \partial_t \psi(x,t) = ...
3
votes
0answers
149 views

Dynamics of an inequality

The dynamics $D\ni(r_i,r_{i+1})\mapsto(r_{i+1},r_{i+2})\in D$ on the set $D:=\{(x,y)\in\mathbb{R}^2\colon x>0,y>x^2/2\}$ is given by the recurrence $$r_{i+2}=\frac{r_{i+1}^2}2+\frac1{r_{i+1}^3} ...
2
votes
1answer
114 views

Monotonicity of the integral

Let $R(x)$ be the residual function associated to the normal probability density, i.e. $$R(x)~=~\int_x^{+\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{y^2}{2}}dy, \mbox{ for all } x\in R.$$ Define ...
1
vote
0answers
50 views

concavity of a vector function

I'm given a function $g:\mathbb{R}^n \mapsto \mathbb{R}$, $g(y) = \prod_{i\in[n]} (1+y_i\cdot c_i)$, where $c_i>0$. Let $e_a,e_b$ be two arbitrary standard vectors. It is easy to show that for any ...
0
votes
1answer
217 views

About the generating structure of Borel field

This is a graduate-level measure theory problem. I have thought throught it and asked on math.SE but received no satisfying answer. On P.32 of [P.Billingsley] Probability and Measure, 3ed, 1993, the ...
0
votes
1answer
59 views

Introducton books for ‎$\frak{E}_p(I)$

Are there any good books different from abstract harmonic analysis by hewitt to study ‎$\frak{E}_p(I)$. where ‎$\frak{E}_p(I)$ is: ‎Let $I$ be an arbitrary index set‎. ‎For each $i\in I$ let $H_i$ ...
-1
votes
1answer
208 views

An infinite set in a compact space

Let $X$ be a topological space. Is there any characterization for the property that says "for every infinit subset $A$ of $X$ there exists $a\in A$ such that if $f$ be an arbitrary real continuous ...
2
votes
1answer
50 views

Bounding the difference in the value of a strongly convex function at its integer minimum and other integer points

I am currently working on a problem where I have to minimize a $m$-strongly convex function $$f ~: ~\mathbb{R}^n \rightarrow \mathbb{R}^+$$ over a bounded integer lattice, $$L = \mathbb{Z}^n \cap ...
1
vote
1answer
73 views

floating point representation via the perspective of TTE/computable analysis

Floating point numbers are not compatible with the usual theory of type 2 theory of effectivity (TTE), and not even the real-RAM model; there are functions that are computable in one model but not ...
0
votes
0answers
46 views

Dimension of the set of the polynomial growth harmonic function on the hyperbolic plane

We consider the hyperbolic plane and the harmonic function there. Pick any point $p$. Let $H_n, n \in\mathbb N$ be the set of the harmonic functions $f$ such that $|f(x)|\leq c(1+ d(x,p))^n$. What is ...
14
votes
3answers
910 views

Can integration spoil real-analyticity?

Is there an example of a function $f:(a,b)\times(c,d)\to\mathbb{R}$, which is real analytic in its domain, integrable in the second variable, and such that the function $$ g:(a,b)\to\mathbb{R},\qquad ...
5
votes
1answer
320 views

Improper integral $\int_0^1 \frac{\exp(ctx)}{\sqrt{(\exp(bt)-1)(1-\exp(atx))-(1-\exp(at))(\exp(btx)-1)}} dx$ with $-a$ and $b$ positive

Is the following function real analytic in $t>0$: $$F(t)=\int_0^1\frac{\exp(ctx)}{\sqrt{(\exp(bt)-1)(1-\exp(atx))-(1-\exp(at))(\exp(btx)-1)}} dx,$$ where $-a$ and $b$ are positive, and $c\not=a$? ...
12
votes
0answers
111 views

Interval arithmetic with different definitions of intervals

Interval arithmetic normally deals with intervals defined as $[a,b]$ with rules like $$[a,b]+[c,d]=[a+c,b+d]$$ I am interested in interval arithmetic with different interval definitions such as ...
18
votes
0answers
571 views

A Linear Order from AP Calculus

In teaching my calculus students about limits and function domination, we ran into the class of functions $$\Theta=\{x^\alpha (\ln{x})^\beta\}_{(\alpha,\beta)\in\mathbb{R}^2}$$ Suppose we say that ...
2
votes
1answer
156 views

Hardy space, Lebesgue space for $p<1$,

We denote $\mathcal D'(\mathbb R^n)$ the space of distributions, and $\mathcal D(\mathbb R^n)$ the space of smooth, compactly supported functions. Let $\rho\in \mathcal D'(\mathbb R^n)$ such that ...
0
votes
1answer
113 views

For a given $n$, under what condition(s) there exists (at least) two different $c$ and $c′$ such that $X_n^c=X_n^{c'}$

Let $X_n^c=\{\cos\left((4k-c)\frac{\pi}{2n}+\frac{\pi}{4}\right): k=0, 1, \dots, n-1\}$ where $c\in\{0, 1, \ldots, \lfloor\frac{n}{2}\rfloor\}$ and $n$ is any positive integer greater than 3. I want ...
3
votes
0answers
122 views

Involutions on $[0,1]$ given by power series (related to probability generating functions)

Let $A$ be a function from $[0,1]$ to $[0,1]$. $A$ is an involution if $A(A(x))=x$ for all $x\in[0,1]$. Which involutions $A$ exist such that $A(x)=\sum_{k=0}^\infty a_k x^k$ with $a_0=1$ and ...
3
votes
0answers
67 views

Can Mumford-Shah functional be adapted to lower $L^1$ space?

The well know Mumford-Shah functional functional $$ F(u)=\int_\Omega|\nabla u|^2+\mathcal H^{N-1}(S_u) \tag 1 $$ where $u\in SBV(\Omega)$ and $\nabla u$ is the absolutely continuous part of ...
3
votes
1answer
154 views

Scaling properties of the Hölder estimate for heat equation

Lately, I have been interested in scaling properties of parabolic equations, and this question is related to an earlier one I asked about Harnack constants. Let $Q(R) := Q(R^2,R) = B(0, R) \times ...
0
votes
0answers
54 views

Absolute continuity and the Luzin N-Property for functions of two variables

It is a well known fact that absolutely continuous functions of one real variable have the so-called Luzin N-property. That is, if $E\subset\operatorname{Domain}(f)$ has zero measure, then $f(E)$ has ...
10
votes
5answers
578 views

Identities and inequalities in analysis and probability

Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of ...
5
votes
1answer
193 views

Smoothening a measure, II

There is an almost invisible, but significant difference between the question below and that recently answered by Boris Bukh. Given a probability measure $\mu$ supported on a finite set ...
54
votes
6answers
2k views

Why does $d^n \exp(-x-x^{-1})/(dx)^n$ only have $n$ positive real zeroes?

Set $f(x) = \exp(-x-x^{-1})$. An easy induction shows that $$\frac{d^n}{(dx)^n} f(x) = \phi_n(x^{-1}) f(x)$$ for $\phi_n$ a polynomial of degree $2n$. Clearly, the roots of $\phi_n(x^{-1})$ are the ...
1
vote
0answers
61 views

Monotone version of one-dimensional Whitney extension theorem

Is there a version of the Whitney extension theorem that would extend a monotone $C^\infty$ function on a compact subset of $\mathbb R$ (satisfying the usual Whitney's compatibility conditions) to a ...
3
votes
1answer
131 views

Smoothening a probability measure

Given a probability measure $\mu$ supported on a finite set $S\subset{\mathbb R}^2$, define $$ f(z):=\max\left\{\frac{\mu(x)+\mu(y)}2\colon \frac{x+y}2=z,\ x,y\in S \right\}, \ z\in{\mathbb ...
2
votes
1answer
113 views

Heat equation: impact of the diffusion coefficient on the Harnack constant

Consider the heat equation $$ u_t - div[a(x,t) \nabla u] =0,\quad (x,t) \in B(r) \times [-r^2, 0] \subset \mathbb R^{d+1} $$ for a Hölder continuous coefficient $a(x,t)$ satisfying $$ 0<C_o \le ...
1
vote
0answers
172 views

Density of subspace with nonlocal/Wentzell boundary condition

Given the space $F$ defined by: $$F=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(z,x)g(z)dz, x>0\right\},$$ I want to prove that the subspace $E$ of $F$ defined by ...
0
votes
0answers
46 views

The union of weighted compact supported continuous function

Let $\Omega\subset \mathbb R^N$ be open. Given a weight function $v\geq 1$ such that $v\in L^1_{\text{loc}}(\Omega)$ and $l.s.c$. Also supposethere exists a Lipschitz continuous sequence $v_n$ such ...
0
votes
0answers
35 views

The Best Korn's constant for bounded deformation

I am studying the following version of Korn's inequality. For $u\in BD(\Omega)$, $BD$ denotes the bounded deformation space, we have, there exists a $r(u)\in \operatorname{ker}\mathcal E$ such that ...
0
votes
1answer
103 views

Discussion for the sign of a specific sum

Given a modular form $f$ of an even weight $k$ for the full modular group. Let $\lambda_f(n)$ the $n$-th normalized Fourier coefficient of $f.$ For a fixed positive integers $a$ and $b,$ I want to ...
3
votes
1answer
134 views

Real analysis on vector-valued spaces, $L^{p}(\mathbb{R}^N,E)$ ,$H^{s}(\mathbb{R}^N,E)$

I am dealing with some vector-valued Sobolev spaces $H^{s}(\mathbb{R}^N,E)$ where $E$ is a Banach space. I am looking for references about results for the scalar case ...
2
votes
2answers
119 views

The convolution between weighted $L^1$ space and normal $L^1$ space

Let $\omega\in L^1_{\text{loc}}(\mathbb R^N$) be given. We assume that $\omega\geq 1$, l.s.c, and satisfies, for a constant $C>0$, $$ \frac{1}{|B(x,r)|}\int_{B(x,r)}\omega(y)dy\leq C\omega(x) $$ ...