**3**

votes

**0**answers

84 views

### What subdomains of $\mathbb{R}^2$ are diffeomorphic to $\mathbb{R}^2_+$ via rational functions?

For what subdomains $D \subset \mathbb{R}^2$ does there exist a rational diffeomorphism $h:D \to \mathbb{R}^2_+$, such that its inverse is also a rational function? (By "rational function" I mean a ...

**1**

vote

**1**answer

66 views

### Equivalent measures on algebra also equivalent on $\sigma$-algebra?

Suppose $\mu$ and $\nu$ are finite positive measures on a measurable space $(X,\mathcal A)$. Let $\mathcal G$ be an algebra of $\mathcal A$. If $\mu$ and $\nu$ are equivalent on $\mathcal G$ in the ...

**11**

votes

**4**answers

401 views

### Number of intervals needed to cross, Brownian motion

Let $B_t$ be a standard Brownian motion. Let $E_{j, n}$ denote the event$$\left\{B_t = 0 \text{ for some }{{j-1}\over{2^n}} \le t \le {j\over{2^n}}\right\},$$and let$$K_n = \sum_{j = 2^n + 1}^{2^{2n}} ...

**11**

votes

**2**answers

811 views

### On Hamkins' answer to a problem by Michael Hardy

Based on a post by Michael Hardy and Hamkins' answer to it Andreas Blass, Will Brian, Joel Hamkins, Michael Hardy and Paul Larson introduced a new cardinal characteristic of the continuum ...

**2**

votes

**0**answers

106 views

### Can one integrate around a branch-cut?

How meaningful is it to try to integrate around the branch-cut of a function?
For example lets say I have the function $\log(z^2+a^2)$ for $a>0$ and I choose my branch-cuts to be starting at $\pm ...

**6**

votes

**2**answers

115 views

### For which $r > 0$ is it the case with probability one, for all $n$ sufficiently large $M_n \le r\sqrt{\log n}$?

Let $B_t$ be a standard Brownian motion. Let$$M_n = \max\{|B_t - B_{n-1}| : n - 1 \le t \le n\}.$$For which $r > 0$ is it the case with probability one, for all $n$ sufficiently large$$M_n \le ...

**1**

vote

**0**answers

57 views

### Characterization of certain families of functions

For $R$ equal $\mathbb{R}$ or $\mathbb{Z}$, let $D^+_R:=\{(x,y)\in R^2\colon x<y\}$. For each natural $n$, let $F_{n,R}$ denote the set of all Borel-measurable functions $f\colon ...

**5**

votes

**3**answers

620 views

### A definition of the fractional derivative

I was investigating the idea of fractional derivatives and devised the following definition. WHich definition is it equivalent to and can I have a reference for it?
$$\frac{d^n}{dx^n}f(x) = \lim_{h ...

**3**

votes

**0**answers

245 views

### “Nicely” strong measure zero sets

This question is essentially an expanded version of the unanswered half of Two strengthenings of "strong measure zero".
A set $X$ of reals is strong measure zero if, for any $f: ...

**0**

votes

**0**answers

50 views

### Composition algebra of Gevrey function for $s<1$

Let $g,f$ be real-valued functions defined on the real line. Let $s$ be a real number.
Assuming that $g,f$ are both in the Gevrey class $G^{s}$, it is true that $g\circ f$ belongs to $G^{s}$ if $s\ge ...

**2**

votes

**0**answers

49 views

### Construct a sequence of i.i.d random variables with a given distribution function, diagonalization? [closed]

Assume we have a sequence of i.i.d. random variables $X_1, X_2, \dots,$ on a probability space $(\Omega, \mathcal{F}, P)$ with$$P(X_n = 1) = P(X_n = -1) = {1\over2}.$$Given a distribution function ...

**1**

vote

**0**answers

77 views

### A Question about compactness of an embedding into $L^p$ spaces

Assume $ \Omega \subset \mathbb{R}^N$ is a smooth bounded domain. There is well known Hardy inequality that says
For any $ u \in W_0^{1,2}(\Omega) $, $N\geq3$ we have
$$ \Lambda \int_{\Omega} ...

**19**

votes

**3**answers

378 views

### Which partitions of $[0,1]$ are collection of level sets of a real continuous function?

Let $f:[0,1]\to[0,1]$ be given. The level sets of $f$ (ie the collection of all sets of the form $\{x\in[0,1]:f(x)=y\}$, for each fixed $y\in[0,1]$) partition the domain of $f$. I am curious for set ...

**4**

votes

**2**answers

193 views

### Brownian motion, quadratic variation, existence of partitions?

Let $B_t$ be a standard Brownian motion. Does there with probability one exist a sequence of partitions $\{t_{k, n} : k = 0, 1, \dots, k_n\}$ $$0 = t_{0, n} < t_{1, n} < \dots < t_{k_n, n} = ...

**9**

votes

**1**answer

158 views

### Does a monotone subadditive $f: \mathcal{P}(\bf N)\to [0,1]$ admit a finite partition with values in $(0,1)$?

A function $f\colon \mathcal{P}(\mathbf{N})\to [0,1]$ is said to have the Darboux property whenever for all $X \subseteq \mathbf{N}$ and $y \in [0,f(X)]$, there exists $Y \subseteq X$ such that ...

**3**

votes

**1**answer

118 views

### Standard Brownian motion, Hölder continuous with exponent $\gamma$ for any $\gamma < 1/2$, not for any $\gamma \ge 1/2$

In some results on Hölder continuity with regards to standard Brownian motion, the following is asserted without proof.
It is not hard to see that for every $k < \infty$, and every $\epsilon ...

**5**

votes

**1**answer

64 views

### Measure of chords from a cantor set

The following problem is inspired by a problem in Pugh's Mathematical Analysis book. (Chapter 2 Problem 42).
In the problem he asks one to consider the standard Cantor set on the unit interval, and ...

**4**

votes

**1**answer

64 views

### Getting out a system of linear ODEs by knowing the Magnus expansion

Assume we are given for a transition between two time points $t_0 = 0$ and $t_1$ a matrix relationship, eventually describing the solution of a system of linear with non-constant coefficients,
...

**3**

votes

**0**answers

55 views

### Semi-continuity of the dimension of the null space

Suppose $T_n : X \rightarrow X$ is a sequence of Fredholm operators on a Banach space such that $T_k \rightarrow T$ strongly (in the induced operator norm). If $N_k$ and $N$ denote the dimensions of ...

**12**

votes

**0**answers

86 views

### A kaleidoscopic coloring of the plane

Is there a partition $\mathbb R^2=A\sqcup B$ of the Euclidean plane into two Lebesgue measurable sets such that for any disk $D$ of the unit radius we get $\lambda(A\cap D)=\lambda(B\cap ...

**7**

votes

**1**answer

180 views

### How to construct i.i.d. standard normal random variables on $\Omega = [0, 1]$ with the Lebesgue measure

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be the unit interval with Lebesgue measure on the Borel subsets. Then we can find independent random variables $X_1, X_2, X_3, \dots$ defined on $(\Omega, ...

**2**

votes

**2**answers

286 views

### Minimum of an apparently harmless function of two variables

DISCLAIMER: I already posted this question on Mathematics a month ago, here. However, since it has not been solved yet on that platform, I decided to ask it also here on mathoverflow. At a first ...

**5**

votes

**1**answer

99 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

**1**answer

101 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

**0**answers

39 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

**1**answer

284 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

**1**answer

157 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

**0**answers

109 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

**0**answers

56 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

**1**answer

63 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**

votes

**0**answers

520 views

### Relative null-ness

Here, "measure" always means Lebesgue measure on $\mathbb{R}$. This question is partly motivated by my answer ...

**3**

votes

**0**answers

45 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

**1**answer

88 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

**1**answer

86 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

**0**answers

38 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

**4**answers

473 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 ...

**3**

votes

**1**answer

72 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

**0**answers

107 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

**1**answer

134 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

**1**answer

109 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

**3**answers

215 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

**0**answers

146 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

**1**answer

101 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

**0**answers

48 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

**1**answer

183 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

**1**answer

56 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

**1**answer

204 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

**1**answer

43 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

**1**answer

50 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

**0**answers

44 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 ...