All Questions
707 questions
2
votes
1
answer
324
views
Uniform estimation of an integral involving a Hölder-continuous function
Let $\Omega\subset\mathbb{R}^n$ be open and bounded, let $s\in(0,1)$, let $u\in C^{0,2s+\epsilon}(\Omega)$ bounded with $u\in C^{0,s}(\mathbb{R}^n)$ and such that: $u=0$, on $\mathbb{R}^n\setminus\...
- 339
2
votes
1
answer
451
views
Show that the absolute value of this function is twice differentiable except on a set of Lebesgue measure $0$
Let
$f\in C^3(\mathbb R)$ with $f>0$ and $$\int f(x)\:{\rm d}x=1\tag1$$
$g:=\ln f$ and assume that $g'=\frac{f'}f$ is Lipschitz continuous (note that this implies that $f'(x)\xrightarrow{|x|\to\...
- 167
2
votes
2
answers
495
views
Polynomial approximation (Weierstrass theorem) with bounds
Consider the closed interval $[0,1]$ and let $f \in C[0,1]$. Let $g$ be a real valued function on $[0,1]$ such that $g \leq f$.
Suppose $g = f$ at atmost finitely many points. Does there exist a ...
- 431
2
votes
0
answers
144
views
Does this geometric PDE have a solution?
Let $s(\theta), b(\theta)$ be two smooth non-constant real-valued functions on $\mathbb{S}^1$, and assume that $s$ never vanishes.
Does there exist a map $h:(0,1) \times \mathbb{S}^1 \to \mathbb{S}^1$,...
- 6,741
2
votes
1
answer
154
views
Grönwall-type inequality for $f(t) \le \alpha + \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(s)|^{\beta}] \, \mathrm d s$
Let $\alpha \in (0, \infty)$ and $\beta \in (0, 1]$. We assume $f : [0, 1] \to [0, \infty)$ is a measurable and bounded function such that
$$
f(t) \le \alpha + \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(...
- 825
2
votes
1
answer
154
views
Smooth conditional expectation with nonsmooth "reverse"
I am looking for a concrete example of the following: $(X,Y)$ are real-valued random variables such that:
$E[Y|X]$ is smooth
$E[X|Y]$ is discontinuous
Even better, I'd like to see an example where ...
- 23
2
votes
1
answer
389
views
Intersections of algebraic surfaces with hypercubes of a $d$-dimensional grid
This is a follow-up question, to a question I asked earlier.
See Algebraic curve intersecting square-grid.
Consider $n^d$ unit hypercubes in $d$-dimensional Euclidean space
tightly packed in the ...
- 479
2
votes
1
answer
193
views
A question on the partial sum of infinite doubly stochastic matrix
Let $A=(a_{ij})$ be an infinite doubly stochastic matrix. Is the following statement true ?
$$
\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n\sum_{j=1}^na_{ij} >0
$$
Any reference or comment on this is ...
- 171
2
votes
1
answer
677
views
Lipschitz continuity of an implicit function
Let $z=F(x,y)$ be a function from $\mathbb R^d\times \mathbb R$ to $\mathbb R$ and $z=F(x,y)$ is Lipschitz continuous. Assume that for any $x\in\mathbb R^d$, there is a unique $y$ such that $F(x,y)=0$....
- 611
2
votes
1
answer
107
views
Lower bounds on translates of a function over a compact set
Let $f\in L^p(\mathbb{R})$ and define $f_\theta(x)=f(x-\theta)$. Let $K\subset\mathbb{R}$ be a compact set. I would like to compute (or at least lower bound) the following:
$$
\inf_{\theta\ne\theta'\...
- 45
2
votes
1
answer
289
views
On semi-discrete Wasserstein distance
Let $\mu(dx)=\sum_{i=1}^np_i\delta_{x_i}(dx)$ and $\nu(dy)=\rho(y)dy$ be two probability measures on $\mathbb R^d$, where $\nu$ has a bounded support. Consider the $2-$Wasserstein distance below:
$$...
- 345
2
votes
1
answer
104
views
Limit of biggest share of the pie
A huge pie is divided among $N$ guests. The first guest gets $\frac{1}{N}$ of the pie. Guest number $k$ guest gets $\frac{k}{N}$ of what's left, for all $1\leq k\leq N$. (In particular, the last guest ...
- 48.9k
2
votes
0
answers
232
views
Is an orthogonal projection of a Lipschitz domain still a Lipschitz domain?
Let $\mathcal{X}\subseteq\mathbf{R}^n$ be a Lipschitz domain, i.e., for each $x\in\partial\mathcal{X}$, there exists a radius $r_x>0$ and a Lipschitz continuous function $F^x:\mathbf{R}^{n-1}\to\...
- 21
2
votes
1
answer
689
views
Partitions of an interval
This question asks about properties of functions which are "piecewise" polynomials. I would like to ask a specific question about the meaning of "piecewise" there.
Specifically, consider "partitions" ...
- 2,358
2
votes
1
answer
497
views
Truncated Euler products, Dirichlet eta function, and convergence issues
Can you prove that the following series does not converge if $\frac{1}{2}<\sigma<1$, no matter how close to $1$ sigma is, and no matter how large $t>0$ is? The series is defined as
$$W(\sigma,...
- 3,098
2
votes
1
answer
249
views
linear recurrence inequality
Given two real analytic functions, $g(x)$ and $f(x)$, on an open interval $I\subset \mathbb{R}$, it is obvious that $g(x) \leq f(x)$ does not imply $g_n \leq f_n$ (here $g_n = [x^n] g(x)$ denotes the $...
- 145
2
votes
1
answer
260
views
Squaring a semi-convergent series
Let $S=\sum_{n=1}^\infty a_n$, be a semi-convergent series with $T=\sum_{n=1}^\infty a_n^2 < \infty$ and $\sum_{n=1}^\infty |a_n|=\infty$. Under which conditions are the following formulas valid? ...
- 3,098
2
votes
3
answers
3k
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 ...
- 1,835
2
votes
1
answer
143
views
Proving convexity of the expected logarithm of binomial distribution
I would like to prove that the following function, for an arbitrary integer $n$:
\begin{equation}
\begin{split}
f(x) & =x\cdot E \ \log(1+\text{Binomial(n,x)}) \\
& = x \cdot \sum_{k=0}^{n} \...
- 23
2
votes
0
answers
197
views
Orthogonality relation in $L^2$ implying periodicity
Let $\theta(t)$ and $\phi(t)$ be two real $C^1$ functions $[0,2\pi]\rightarrow \mathbb{R}$. Let us assume $\theta$ has the properties
$$
\int_0^{2\pi} e^{i\theta(t)} dt=0.
$$
Geometrically this means ...
- 405
2
votes
1
answer
320
views
Fourier series but different waveform
Given a nondegenerate smooth simple closed convex curve $f: [0,2\pi]\to \mathbb C \setminus \{0\}$ with winding number (around origin) $1$, and $f$ have zero mean. Let $f_n: [0,2\pi]\to \mathbb C \...
- 807
2
votes
0
answers
190
views
What is the smallest dimension that allows finding $n$ points at distances $|x_i-x_j|^{\delta/2}$, where $0<\delta<1$, and $x \in \mathbb{R}^n$?
Let $x_1,\cdots,x_n \in \mathbb{R}$, are there $\xi_1,\cdots,\xi_n \in \mathbb{R}^s$, such that
$|x_i-x_j|^{\delta}=||\xi_i-\xi_j||^2$, $0<\delta<1$, what is the smallest $s$ to guarantee the ...
- 178
2
votes
1
answer
131
views
Uniformly Converging Metrization of Uniform Structure
This is related to trying to resolve the currently faulty second part of my answer to this question, but is by itself a purely real analysis question.
Let $X$ be a set with a uniform structure ...
- 13k
2
votes
2
answers
636
views
Continuous upper envelope of upper semicontinuous function
Let $u$ be a upper semicontinuous function on a compact set $K$ in $\mathbb R^d$. Define a space of continuous function dominating $u$ by
$$A = \{\phi \in C(K): \phi \ge u\}.$$
[Q.] Is the following ...
- 1,399
2
votes
1
answer
301
views
A question about pushforward measures and continuous Borel isomorphisms
It is fairly well known that if $\mu$ and $\nu$ are nonatomic measures on the standard Borel spaces $(X,B)$ and $(Y,C)$ such that $\mu(X)=\nu(Y)$. If $X$ and $Y$ are uncountable, then there exists a ...
- 33
2
votes
2
answers
4k
views
a limit of the laplace transform and its derivative
If $\phi(s)$ is the Laplace tranfrom of $f(t)$, then $\lim_{s\rightarrow \infty} s\phi(s) = f(0^+)$. and also $\lim_{\rightarrow \infty} s\phi'(s) = \lim_{t\rightarrow 0^+}tf(t)$ since $\phi'(s)$ is ...
2
votes
0
answers
77
views
Homomorphism of composition to additive structure
Consider the following topological groups
$\operatorname{Homeo}(\mathbb{R}^d)$ be the topological group of all homeomorphism from $\mathbb{R}^d$ onto itself; equipped with the compact-open topology (...
- 5,405
2
votes
1
answer
167
views
On a characterization of inward unit normal vector
Let $D$ be a smooth domain of $\mathbb{R}^d$. Let $\partial D$ denote the boundary of $D$. We denote by $B(x,r)=\{y \in \mathbb{R}^d \mid |y-x|<r\}$ the Euclidean ball centered at $x$ with radius $...
- 721
2
votes
1
answer
168
views
Validity of formula $u(x)=\frac{1}{4\pi}\int_G \nabla_y \frac{1}{\lvert x-y \rvert} \times \omega(y) \, d^3y +A(x)$ for periodic boundary case
I think it is better to provide context in which the previous question Any formula or estimates the Green function for the Laplacian in $3D$ periodic box? has been raised.
The motivation is the ...
- 3,477
2
votes
1
answer
61
views
$K *g_n$ converges in the topology of smooth functions, $K$ approximates $\delta(x)$ and $g_n$ is a.e convergent to $g$, then regularity of $g$?
This question is continuation from If $K *g_n$ converges in the Fréchet topology of smooth functions and $K$ approximates $\delta(x)$, is $g_n$ itself convergent? - revised.
As before, let us ...
- 3,477
2
votes
3
answers
1k
views
on the set of numbers generated by integer linear combination of two real numbers.
Let $b > a > 0$ be two real numbers. I am interested in the set of numbers
$X(p,q) = p a + q b$ with $p,q$ positive integers. Basically this is the set $a \mathbb{N} + b \mathbb{N}$.
What ...
- 29
2
votes
1
answer
437
views
If $g$ is differentiable, how can we show that $z\mapsto1\wedge e^{g(z)}$ is differentiable except on a countable set
If $g:\mathbb R\to\mathbb R$ is differentiable, how can we show that $$h(z):=\min\left(1,e^{g(z)}\right)\;\;\;\text{for }z\in\mathbb R$$ is also differentiable, except at a countable number of points, ...
- 167
2
votes
1
answer
328
views
Hausdorff dimension of the graph of a BV function (in 1 dimensional setting)
Let $u: \Omega\subset \mathbb{R} \to \mathbb{R}$ be a function of bounded variation.
Question 1.
How can we prove that the Hausdorff dimension of the essential graph of $u$ equal to $1$?
Question ...
- 839
2
votes
4
answers
3k
views
Splitting a space into positive and negative parts
Let $V$ be a vector space over $\mathbb R$. A symmetric bilinear pairing on $V$ is a linear map $a: V\otimes V \to \mathbb R$. Because $\mathbb R$ is characteristic not-two, I will freely confuse ...
- 54.6k
2
votes
2
answers
257
views
Reference request on Min-Max theorem
Consider the following min-max problem
$$\inf_{x\in M} \sup_{y\in N} F(x,y),$$
where $F: M\times N\to\mathbb R$ is Lipschitz and $y\mapsto F(x,y)$ is concave for all $x\in M$. Could we derive $\...
user128095
2
votes
1
answer
265
views
characterization of normality by selection theorem
The Urysohn's extension theorem states that a space $X$ is normal iff every continuous function $f:A \rightarrow \mathbb{R}$, with $A$ a closed subset of $X$, can be extended to a continuous function $...
- 121
1
vote
1
answer
300
views
Convergence of concave/convex function
Let assume that you have a sequence of twice differentiable functions $(f_n)_{n\in\mathbb{N}}\in\mathscr{C}^2(\mathbb{R})^{\mathbb{N}}$. Let suppose that for each $f_n$, it exists a $x_n\in\mathbb{R}$ ...
- 393
1
vote
2
answers
231
views
A real root of a cubic equation for a stationary point
Let us consider the quartic polynomial in $x$
\begin{equation}
F(x) = (2 a p +2)x^4+ (6a(1-a)p^2+(6-12a)p-6)x^3
+ p(2(a-2)(a-1)a p^2 + 3(5a^2-9a+2)p +12a-18)x^2
- p^2 ((a-2)(4a^2 ...
- 371
1
vote
0
answers
56
views
Extension of this maximisation problem : finite or not?
$\mathcal M$ is the space of real $d\times d$ matrices and $\mathcal S\subset \mathcal M$ is its subset consisting of positive semidefinite elements. We consider the distance the product space $\...
- 1,399
1
vote
1
answer
192
views
Characterization of a subset of $[0,1]$
Let $T\subseteq[0,1]$ be a subset containing $1$. Now we know that $T$ satisfies the following property:
For every $t\in [0,1)$, if there exists a decreasing sequence $\{t_n\}_{n\ge 1}\subset T$ such ...
- 1,835
1
vote
2
answers
180
views
An inequality for a real function
Let $$f(z)=(1+z)^{3/4}-\left(\frac{3}{8}+\frac{\sqrt{3}}{4}\right)^{1/4}-\frac{\left(3 z+\sqrt{6} \sqrt{-1+z^2}\right)^{3/4}}{\left(2 \left(2+\sqrt{3}\right)\right)^{3/4}}.$$ Is there a simple proof ...
- 37
1
vote
0
answers
102
views
Proving that a quantity is positive (Gaussian density and Gaussian CFD)
$\def\R{\mathbb R}$$\def\aha{{1/2}}$$\def\maha{{1/2}}$
Hi everyone,
I am interested in the following problem:
Let consider the heat equation problem:
$$\forall (t,x) \in \mathbb{R}_+\times\mathbb{R}, ~...
- 393
1
vote
1
answer
242
views
Can (how) one distinguish germs of continuous functions by a countable set of params?
Continuous functions can be distinguished by their values at say rational points of [0 1].
Germs of analytic functions can be distinguished by derivatives at a point.
So in both cases we see ...
- 24.9k
1
vote
1
answer
151
views
Monotone likelihood ratio of densities based on power function
Given $p,\phi,\theta \in \mathbb{R}$ such that $p>2$ and $0 \le \phi,\theta\le \pi/2$ define the density function:
$$f(\phi;\theta) =
\mbox{$\Large\frac{1}{p B\big(\hspace{-1pt}\frac{3}{2},\frac{p+...
- 391
1
vote
1
answer
265
views
Is there a version of dominated convergence theorem for local $L^p$ spaces?
Fix $p \in [1, \infty)$. Let $(L^p (\mathbb R^d), \|\cdot\|_{L^p})$ be the Lesbesgue space of $p$-integrable real-valued functions on $\mathbb R^d$. Let $\tilde L^p (\mathbb R^d)$ be the space of ...
- 825
1
vote
1
answer
633
views
Does sequence almost sure convergence imply almost sure convergence?
This is a cross-post of this and this questions from math.stackexchange.com since I have not received any response there. I would like to seek help here.
Suppose $x(t,\omega): [0,T]\times\Omega\...
- 2,239
1
vote
1
answer
132
views
Local maxima of the sum of Gaussian functions in *one dimension* are always strict local maxima - proof?
Motivated by this question asked earlier, I was wondering whether one can prove easily that the local maxima of the sum of Gaussians:
$$f_n(x):= \sum_{i=1}^{n}e^{-(x-x_i)^2}, \quad x_1 < x_2 < \...
- 1,467
1
vote
1
answer
401
views
linear recurrence inequality of positive terms
This is a follow up on my previous linear recurrence inequality question.
I have some matrices which satisfy a linear recurrence formula of the form
$$
A_{n+1} = \alpha A_{n} + \beta A_{n-1},\qquad n\...
- 145
1
vote
2
answers
90
views
Is the difference between $\alpha$-Hölder constants of $f*\rho$ and $g*\rho$ controlled by $\|f-g\|_\infty$?
Let $\mathcal D_1$ be the set of bounded probability density functions on $\mathbb R^d$. This means $f \in \mathcal D_1$ if and only if $f$ is non-negative measurable such that $\int_{\mathbb R^d} f (...
- 825
1
vote
1
answer
93
views
Is there $f$ such that $\int_0^t f(s)\,\mathrm d s<\infty$ and $|\partial_t g| (t, x) \le f(t)g(Ct, x)$ for all $t>0$ and $x \in \mathbb R^d$?
We consider the heat kernel
$$
g :\mathbb R_{>0} \times \mathbb R^d \to \mathbb R,\quad (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ).
$$
Then
$$
\partial_t g(t, x)...
- 657