# Tagged Questions

**14**

votes

**3**answers

489 views

### Infinitely many $k$ such that $[a_k,a_{k+1}]>ck^2$

Let $a_n\in \mathbf{N}$ be an infinite sequence such that $\forall i\neq j, a_i\neq a_j$.
I have the following theorem:
For $0<c<\frac{3}{2}$, there are infinitely many $k$ for which ...

**3**

votes

**1**answer

54 views

### Injectivity of vector functions: Numerical Verification

Problem Setup
Let $f:A\rightarrow B$, be a continuous function, $A\subset\Re^{n}$,$B\subset\Re^{m}$, $m\geq n$ and $A, B$ compact.
The function $f(\cdot)$ can only be evaluated numerically.
...

**1**

vote

**1**answer

157 views

### Dense subspaces of $L^p(0,T;X)$

Given a Banach space $X$ and $1\leq p<\infty$, let's define the space $L^p(0,T;X)$ as the set of all strongly measurable functions $f:(0,T)\mapsto X$ such that
$$\int_0^T\Vert ...

**2**

votes

**1**answer

85 views

### Scaling of distributions

Suppose we have a sequence of $L^1(\mathbb{R})$ functions $p_\epsilon$ with $\|p_\epsilon\|_{L^1} \leq 1$ for all $n$. Suppose we know that $p_\epsilon \to 0$ in distributions. Is it obvious that ...

**2**

votes

**1**answer

109 views

### Does the Abel transform preserve analyticity?

Let $I=(0,1]$ and $T=\{(x,y)\in I^2;x\geq y\}$.
If functions $f:I\to\mathbb R$ and $w:T\to\mathbb R$ are analytic, is the function $A_wf:I\to\mathbb R$,
$$
...

**1**

vote

**1**answer

115 views

### Question abouth Skorokhod representation of random variables (II)

This is a continuation of
Question abouth Skorokhod representation of random variables
Let $\mu$ and $\nu$ be two probability measures on $\mathbb R$ such that
$$\int_{\mathbb R}|x|^pd\mu(x),~ ...

**4**

votes

**1**answer

108 views

### Hellinger integral for the Student/Cauchy family

Let $p$ and $q$ be probability densities on $\mathbb R$, with respect to the Lebesgue measure $dx$. The corresponding Hellinger integral is
$H(p,q):=\int_{\mathbb R}\sqrt{pq}\,dx$.
Let now $p$ be ...

**4**

votes

**1**answer

134 views

### Compact, not local uniform convergence of sequences of functions on the rationals

I stumbled upon the following elementary problem while trying to come up with a certain counterexample in category theory. (Basically, I am interested in the constant sheaf of $\mathbb F_2$-vector ...

**2**

votes

**2**answers

255 views

### Are the closed and unbounded subsets of $\mathbb{R}$ known up to homeomorphism?

I am currently working on a problem for which this knowledge could greatly reduce the number of cases, but I have yet to find anything after searching online. Are the closed unbounded subsets of ...

**0**

votes

**1**answer

71 views

### A $W^{1,2}_{loc}$ function with uniformly bounded integrals on compact subsets $W^{1,2}$?

Let $M$ be a Riemannian manifold, $\Omega\subset M$ is an open subset, let $f\in W^{1,2}_{loc}(\Omega)$ with uniformly bounded integrals on compact subset, i.e. there exists a $C>0$, such that for ...

**3**

votes

**0**answers

101 views

### Completion of $C_{0,rad}^{\infty}(\Omega)$ with respect to the norm $\|u\|= \Bigg(\int_{\Omega} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $

I have a question that it seems simple but I can not solve it.
Let $\Omega$ be the unit ball centered at zero in $\mathbb{R}^N$, $N>4$. Assume that $C_{0,rad}^{\infty}(\Omega)$ is the space of all ...

**1**

vote

**0**answers

81 views

### monotonicity of a function

I want to know if the function below is monotonically decreasing for all $a,b >0, a\neq b $
\begin{equation}
x\rightarrow \frac{\sinh^2((a-b)x)}{\sinh(2ax)\sinh(2bx)} \text{, $x >0. $}
...

**1**

vote

**0**answers

91 views

### Interchange of integral and infimum

Can anyone please suggest how to justify widely used formula for interchange of integral and infimum:
$
\inf_{u(t)\in U}\int_{t_0}^{t_1}g(t,u(t))dt=\int_{t_0}^{t_1}\inf_{u\in U}g(t,u)dt,
$
where $ ...

**-1**

votes

**1**answer

54 views

### Idempotent solutions to the implict function theorem other than the identity?

I am interested in the following problem. Assume that an (anti)symmetric function $g:\mathbb{R}^2 \to \mathbb{R}$ satisfies the implicit function theorem. That is, $g(x,y) = \pm g(y,x)$ and $g(x,y)=0$ ...

**3**

votes

**1**answer

155 views

### Question abouth Skorokhod representation of random variables

It is known that for any two probability measures $\mu$ and $\nu$ on $\mathbb R$ that are close in the Prokhorov metric $\rho$, i.e.
$$\rho(\mu,\nu)<\varepsilon,$$
then there exist two random ...

**6**

votes

**1**answer

149 views

### Can the potential of a complete Kahler metric be bounded?

Let $X$ be a complex manifold and $\omega$ a Kahler form on $X$. A smooth function $\rho$ is called a potential of $\omega$ if $i\partial\bar\partial\rho=\omega$. By intuition, it seems that $\rho$ ...

**1**

vote

**0**answers

99 views

### Monotonicity of the Hellinger integral/distance

Let $p$ and $q$ be probability densities on $\mathbb R$, with respect to the Lebesgue measure $dx$. The corresponding Hellinger integral and distance are
$H(p,q):=\int_{\mathbb R}\sqrt{pq}\,dx$ and ...

**2**

votes

**1**answer

81 views

### Majorization of cyclic products

Let $k,m,n\in\mathbb N$ such that $n>k$. For a partition $\alpha=(\alpha_1,\dots,\alpha_k)\vdash m$ with $\alpha_1\ge\dots\ge \alpha_k>0$ and nonnegative $ x_1,\dots,x_n$ define $x^\alpha ...

**2**

votes

**0**answers

32 views

### Constant periodic Sobolev embedding

Dear mathoverflowers,
I would like to have a reference regarding the optimal constant in the Sobolev embedding
$$
\|u\|_{L^q}\leq C_{s,q}\|u\|_{\dot{H}^s},
$$
($H^s$ denotes the standard L^2 ...

**4**

votes

**1**answer

128 views

### Mean value of a function associated with continued fractions

Suppose that an irrational $x$ in $(0,1)$ has convergents $c(k,x)$, and let
$$d(x) = \sum_{k=0}^{\infty} \mid x - c(k,x)\mid.$$
What is the mean value of $d$?

**4**

votes

**1**answer

121 views

### Superadditivity of the lower density

Let $\mu^\star$ be a real-valued function defined on the power set of the positive integers $\mathbf{N}^+$ such that for all $X,Y\subseteq \mathbf{N}^+$ the following axioms hold:
(F1) ...

**0**

votes

**1**answer

189 views

### Is the limsup or liminf of n-wise independent events independent?

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space.
Consider events indexed by $m, n \in \mathbb N$:
$ \ \ \ \ \ \ \ \ \ \ \ A_{1,n}, A_{2,n}, A_{3,n} ...$ are n-wise independent.
...

**8**

votes

**1**answer

570 views

### Sort-of Converse of Kolmogorov Zero-One Theorem

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. The Kolmogorov Zero-One Theorem states that
Suppose we have independent random variables $X_1, X_2, ...$. Then $\forall \ A \in ...

**0**

votes

**0**answers

183 views

### Can we find an upper bound?

Let $f\in C^1(\mathbb R)$ with $f(0)=0$ and $|f'(x)|\le m$, where $m\in (1,2]$.
Let $x(0)\in\mathbb R$ be arbitrary, and define $x(n),y(n)$ recursively by
$$
x(n+1)=f(x(n)) , \quad\quad
...

**1**

vote

**2**answers

129 views

### Convergence of Sobolev functions near the boundary

Let $B_0(1)$ be the unit ball in $\mathbb R^n$, $n\geq2$. Let $f\in W_0^{1,2}(B_0(1))$, and $W^{1,2}(B_0(1))\ni f_i\to f$ in the sense of $L^2(B_0(1))$-norm, as $i\to \infty$.
Question 1: Can we ...

**2**

votes

**2**answers

353 views

### A logarithmic cotangent inequality

I must be a terrible googling searcher but I cannot find a reference to the following inequality:
$$ \forall_{\phi\in(0;\frac \pi 4)}\ \ln(\cot(\phi)))\, <\, \cot(2\!\cdot\!\phi) $$
I have just ...

**2**

votes

**1**answer

71 views

### Convergence of energy of Sobolev functions near the boundary

Let $B_0(1)$ be the unit ball in $\mathbb R^n$, $n\geq2$. $h\in W_0^{1,2}(B_0(1))$. For $r\in (0,1)$, define a function $f_r(x):[0,1]\rightarrow \mathbb R$ by
\begin{equation}
f_r(x):=
\begin{cases}
...

**20**

votes

**3**answers

531 views

### Evaluating an integral using real methods

This is a bit of recreational integration. The following, rather attractive integral is quite straightforward via residues:
$$\int_0^1 x^{-x}(1-x)^{x-1}\sin \pi x\,\mathrm{d}x=\frac{\pi}{e}$$
...

**2**

votes

**1**answer

153 views

### Proofs of inequalities used by Erdos-Renyi in their Random Graphs Paper 1

Please refer to this, it is Erdos-Renyi 1959 paper 1 on Random Graphs. I am currently working on this, but I am stuck on the fifth page, where they use two estimates. More specifically, here's the ...

**2**

votes

**0**answers

83 views

### Error term for Euler-MacLaurin summation formula when applied to infinitely smooth functions?

A function $f(z,x)$ is tempered if all of the following are true:
$f(z, x)$ is infinitely differentiable in $z$
$f(z,x)$ is defined for all $z,x \in \mathbb{R}$
Every derivative of $f(z,x)$ is ...

**7**

votes

**1**answer

261 views

### An interesting integration

For any positive integer $n$, let
$$A_n=\idotsint\limits_{\substack{x_1+\cdots+x_n+y_1+\cdots+y_n\leq1\\x_1,\cdots,x_n,y_1,\cdots,y_n\geq0}}\prod_{i,j=1}^n(x_i-y_j)dx_1\cdots dx_ndy_1\cdots dy_n.$$
It ...

**12**

votes

**1**answer

257 views

### An interesting inequality

Let $\mathbb{R}$ be the real field. For any homogeneous polynomial $f(X_1,\cdots,X_n)$ in $\mathbb{R}[X_1,\cdots,X_n]$, we use $S_f(X_1,\cdots,X_n)$ to denote the following homogeneous symmetric ...

**9**

votes

**2**answers

420 views

### When does $\nabla\times(\nabla\times F)=0$ imply $\nabla \times F=0$

On a (simply connected) domain $\Omega$ for a smooth vector field $F\colon \Omega \to \mathbb{R}^3$, when does $\nabla\times(\nabla\times F)=0$ imply $\nabla \times F=0$. I know that ...

**3**

votes

**1**answer

43 views

### Determine a sign of the limitation of a certain integral

I can't determine a sign of an integral written below and it has hit a dead end.
My setting is rather special.
Let $a\in(0,1)$ be a given constant and ...

**0**

votes

**1**answer

184 views

### the double dual of “little l one” sequence space

I remember a professor remarking a while back that the double dual of the sequence space $l_1^{\infty}(\mathbb{R})$ is a very complicated space. I understand it must contain a copy of the original ...

**9**

votes

**3**answers

292 views

### Decay of real continuous algebraic functions at infinity

Let $f$ be a real valued continuous algebraic function on $\mathbb R^n$. Suppose the zero set of $f$ is bounded, i.e., if $|x|$ is large enough, $f(x)\neq 0$. Is there any estimate of the sort ...

**1**

vote

**1**answer

101 views

### An analytic family of in fact non-existent improper Riemann integrals

Question:
Are there any useful interpretations or "applications" of the formula
$$\intop_0^\infty e^{-ax}\frac{\sin x}{x}\,dx=\frac{\pi}{2}-\arctan(a)\qquad \forall\;a\in \mathbb{R},
$$
in which the ...

**1**

vote

**0**answers

139 views

### On the differentiability of a certain map from $ (0,\infty) $ to $ \Bbb{R} $

This problem arose from my study of energy-conservation for non-linear Schrödinger equations. Suppose that we have the following data:
$ u \in C^{1} \! \left( (0,\infty),{L^{2}}(\Bbb{R}^{n}) \right) ...

**1**

vote

**0**answers

46 views

### Derivatives of Minkowski function?

Let $A\subset \mathbb R^n$ and $M$ be the convex hull of the set $A$, e.g., $M:=Conv(A)$. The Minkowski function on $M$ is defined as follows
\begin{align*}
&f: \mathbb R^n \to \mathbb R\\
...

**2**

votes

**0**answers

58 views

### Minimal condition $\sum \rho_{ij} \Psi_{ij} s_i s_j < \sum s_i s_j $, $\mid \rho_{ij} \mid \leq 1$, $s_i \in \mathbb R$ and $\Psi_{ij} \in \{0,1\}$

Consider a sequence of real number $\{s_i\}_{i\leq n}$. Now consider the real numbers $F$, $G$ and $\alpha$ defined below
$$F= \sqrt{ \left( \sum ~\rho_{ij} ~\Psi_{ij}~ s_i ~s_j \right)^+}, $$
$$G = ...

**0**

votes

**3**answers

82 views

### Bounds on derivative of integrable, monotonically decreasing, differentiable functions on $\mathbb R_+$

The following three conditions have shown up as hypotheses in some recent work, and despite not having been able to find an example, we assume the third is not implied by the former two. We're hoping ...

**2**

votes

**0**answers

90 views

### Distributive law

I was wondering whether there is any reference that deals with the distributive law for infinitely many elements, i.e.
$$
\prod_{i\in \mathbb N} \sum_{k\in \mathbb N} \alpha_{i,k} = \sum_{(k_i)_{i\in ...

**12**

votes

**3**answers

289 views

### Is a certain subset of the disc a convex set?

Some one asked me this question and I thought about it and I don't have any good idea to solve that. Can some one help me and give me an idea to start solve that?
Draw a Cantor set $C$ on the circle ...

**4**

votes

**1**answer

382 views

### A continuous path between two Sobolev functions

Let $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. Let $u_1$, $u_2\in H^{1}(\Omega)$ such that $T[u_1]=T[u_2]=T[\omega]$ where $T$ stands for the trace operator and $\omega\in ...

**-1**

votes

**1**answer

139 views

### separable BV space for PDE's, Whats stopping us? [closed]

Consider the metric space BV(0,1) with the following metric
$$ d(u,v) = \|u-v\|_{L^1} + |TV(u)-TV(v)| $$. It is separable, compact, uniformly bounded and complete. So What is the really obvious thing ...

**1**

vote

**2**answers

237 views

### Fourier transform localisation (still unanswered, but apparently off-topic?) [closed]

In the context of Pólya's theorem I was reading these notes here on p. 19. In the last paragraph the authors claim (it is the sentence starting like "standard Fourier theory shows...") that the ...

**-2**

votes

**1**answer

59 views

### Suppose a real differentiable function with its derivative not infinity, it is sure that its second symmetric derivative should exist? [closed]

Suppose a real differentiable function $h(x)$ with its derivative not infinity, it is sure that its second symmetric derivative ...

**0**

votes

**0**answers

70 views

### Derivatives of Mollified functions

I'm reading Controlled Diffusion Process by N.V. Krylov. On page 87-88, in the proof of theorem II.6.1, it says the following:
Let $\sigma(t,x)$ be a matrix of dimension $d\times d$, and let $b(t,x)$ ...

**7**

votes

**0**answers

150 views

### Improving Baumgartner's result?

Q1: Is it consistent with the failure of CH to have an $\aleph_1$-dense subset $A \subseteq \mathbb{R}$ such that for every $X \subseteq \mathbb{R}$ of size $\aleph_1$, there is a $C^{\infty}$ map $F: ...

**0**

votes

**1**answer

82 views

### Complement of a finite union of convex sets

Question. Let $V_1,\ldots,V_n$ be open, bounded and convex subsets of $\mathbb R^2$. Show that $F=\mathbb R^2\smallsetminus\bigcup_{i=1}^n V_i$ possesses only finitely many connected components.
I ...