All Questions
5,850 questions
2
votes
1
answer
497
views
Linear map of finite or infinite extreme points. Discuss injectivity and surjectivity
Before entering my problem, let me review some related results:
Suppose $\mathcal{S}$ is a convex hull of finite points: $\mathcal{S}=\operatorname{conv}(x_1,x_2,\ldots,x_m)$, then
By "https://...
- 345
6
votes
1
answer
314
views
Generators of a convex cone defined by a differential inequality
Consider the cone of continuously twice differentiable functions mapping positive reals to itself (i.e., $f\in C^2(\mathbb R_{++})$ and $f\colon \mathbb R_{++}\to\mathbb R_{++}$) that satisfy
\begin{...
- 61
1
vote
1
answer
192
views
Neumann-Poincare operator is in the Schatten class
Let $\Omega$ be a bounded domain in $\mathbb{R}^d$, $d\ge 3$. We define the Neumann-Poincare operator(or double layer potential) $K: L^2(\partial\Omega)\to L^2(\partial\Omega)$ by
$$(Kf)(x)=\int_{\...
- 171
4
votes
1
answer
283
views
Absolutely continuity in variation of constant formula
We are talking here about the initial value problem on some Hilbert space $H$
$$y'(t)=Ay(t)+f(t), \\ y(0)=y_0 \in D(A).$$(Problem 1.13 in the reference)
Then $y(t)=e^{At}y_0 + \int_0^t e^{A(t-s)}f(s) ...
- 43
1
vote
1
answer
211
views
Eigenvalues of the double layer potential
Consider the double layer potential $K: L^2(S^2)\to L^2(S^2)$
$$(Kf)(x)=\int_{S^2}f(y)\frac{\partial}{\partial v_y}E(x,y)dS_y,$$
where $E(x,y)=||x-y||^{-1}$ and $\frac{\partial}{\partial v_y}$ means ...
- 171
2
votes
0
answers
183
views
Fourier series and regular distribution
Assume you have a distribution $K$ on $\mathbb{T}$, the torus, such that $\sum_{n=-\infty}^{\infty} |K(e_n)|^2$ is finite, where $e_n := e^{in\cdot}$ are the Fourier basis. Does this imply that the ...
- 95
3
votes
3
answers
219
views
Asymptotic behavior of an integral transform
Given $g\in L^2(\mathbb{R}^3)$, consider the following function ( defined for $r>0$ ):
$$c(r):=\int_{\mathbb{R}^3}\frac{g(x)}{|x|^2+r}dx$$
I'm interested in the behavior of $c(r)$ for large $r$. A ...
- 943
4
votes
1
answer
215
views
On the number of repeated roots
Is there a number $c>0$ such that:
For any $n$ there is a polynomial $p(x) = a_nx^n +\cdots + a_0$ where the coefficients are $-1, 0$ or $1$ such that the number of repetition of the root $x=1$ ...
- 97
3
votes
0
answers
267
views
Link between standard convolution and Day convolution
There is a notion of convolution product between two functors called "Day convolution". (See here nlab for instance) I know that the definition of this notion is inspired by the discrete convolution $$...
- 1,017
3
votes
1
answer
187
views
Free quantum evolution operator on Sobolev space
I am not a mathematician, but would like really like to get some confirmation on the things I am doing here.
Let $-\Delta: H^2(\mathbb{R}) \subset L^2(\mathbb{R}) \rightarrow L^2(\mathbb{R})$ then ...
- 95
7
votes
0
answers
395
views
Fixed radius mean value property implies harmonicity?
Let $f$ be a continuous real-valued function on $\mathbb{R}^n$. It is well known that the following are equivalent:
$f$ is harmonic.
$f$ satisfies the ball mean value property
$$
f(x)=\frac{1}{|B(x,r)...
- 527
1
vote
1
answer
118
views
Almost periodic function and closed spaces
We denote $X_{T}$ the vector space of all $T$-periodic function with zero mean in $L^2$ ( we know that $X_{T}$ is spawn by $(e^{2i\pi nt/T})$). Let be $$X=X_{2\pi}+X_{3\pi}.$$
I think that $X_{2\pi}+...
- 75
8
votes
1
answer
617
views
Violating the Lebesgue density theorem
Can anyone exhibit a finite-dimensional metric space (preferably, $R^d$) equipped with a measure that does not satisfy the conclusions of the Lebesgue Density Theorem? Such examples exist in infinite-...
- 6,541
1
vote
0
answers
86
views
Least restrictive condition such that $-y''(x)+q(x)y(x)=\lambda y(x)$ has two solutions
Let $\lambda \in \mathbb{R}$ and consider some function $q$ on the interval $[0,1].$
What is known to be the least restrictive condition on $q$ such that there are two linearly independent $H^2$ ...
- 11
12
votes
1
answer
779
views
Is a Lebesgue measurable subgroup of $\mathbb{R}$ a Borel measurable set?
Assume that $H$ is a Lebesgue measurable additive subgroup of $\mathbb{R}$. Is $H$ necessarily a Borel subset of $\mathbb{R}$?
- 366
27
votes
7
answers
9k
views
Why are two "random" vectors in $\mathbb R^n$ approximately orthogonal for large $n$?
I saw that two random independent vectors are approximately orthogonal in high dimensional space.
How can I prove this?
And is there an intuitive explanation?
Thank you.
- 387
4
votes
1
answer
484
views
Question about normalization factors in the direct integral of operators
So the original question I wanted to ask was this one:
I'm currently a bit puzzled about the normalization for the Gelfand transform $U$:
So if we have a periodic Schrödinger operator $H$, then we ...
- 95
0
votes
0
answers
59
views
Restriction to Basis of Cadlag function
If $f \in L^2([0,T])$ then it can be written as
$$
f(t) \triangleq \sum_{i \in \mathbb{N}} c_i e_i(t),
$$
for some sequence $\{c_i\}$ of real numbers and a Schauder basis $\{e_i(t)\}$ of $L^2([0,T])$ ...
- 5,405
3
votes
2
answers
491
views
Unknown bias in a distribution related to prime numbers
If $n$ is composite then $\phi(n) < n-1$, hence there is at least one divisor $d$ of $n-1$ which does not divide $\phi(n)$. We call $d$ as the totient divisor of $n$. Trvially, if $n$ is prime then ...
- 5,470
11
votes
1
answer
1k
views
Conditional convergence of $\sum_{n\geq 1} \frac{\sin(p(n))}{n}$?
The series $\sum_{n\geq 1} \frac{\sin n}{n}$ is easily seen to be conditionally convergent, e.g. by Abel summation. But how about $\sum_{n\geq 1} \frac{\sin(n^2)}{n}$? (for which Abel summation fails)...
- 462
3
votes
0
answers
160
views
integral with simple approximation. But why?
I have the following integral
$$g(x_0) = \int_{-\infty}^{\infty} \frac{1}{(1+x^2)^{3/4}}\frac{1}{(1+(x+x_0)^2)^{3/4}}\exp\left(-\frac{2\pi i}{\lambda}\left[\sqrt{1+x^2}-\sqrt{1+(x+x_0)^2} \right] \...
4
votes
0
answers
141
views
Level sets of function of inner products of vectors on hypercube
Let $H = \{ 0, 1\}^d$ be the $d$-th Cartesian product of $\{0, 1\}$ in $\mathbb{R}^d$. Suppose $v_1, \ldots, v_k$ are $k$ vectors in $H$ in general position. We define function $F \colon H^{k}\...
- 1,127
5
votes
1
answer
2k
views
Sum of multinomial coefficients (even distribution)
By multinomial expansion formula, we know that $$ \sum_{p_1 + \cdots + p_k = r} \binom{r}{p_1,\ldots,p_k} = k^r, $$ where the multinomial coefficient is defined by $ \binom{r}{p_1, \ldots, p_k} := \...
user83150
1
vote
0
answers
448
views
Largest possible variance for log-concave distributions on a bounded interval
Let $f$ be the density of a log-concave probability distribution on the interval $[0,1]$ (with respect to Lebesgue measure). To be concrete, suppose that $f(x) = \exp( - \varphi(x))$, for some convex ...
- 251
5
votes
0
answers
235
views
Riemann theta function inequality for a class of large random matrices
The following is essentially the same question as in this previous post, but since I have completely re-formulated it (hopefully for the better ;-), I decided to post a new question instead of an edit....
- 686
2
votes
0
answers
60
views
A question about Kolmogorov Superpositions
D.A. Sprecher showed (https://www.researchgate.net/profile/David_Sprecher2/publication/243052898_A_Representation_Theorem_for_Continuous_Functions_of_Several_Variables/links/554929f20cf2ebfd8e3ad956....
- 371
1
vote
0
answers
106
views
Identifying a notion of integration
Let $f$: $I\longrightarrow\mathbb{R}$ be a (not necessarily bounded) function on an interval $I\subseteq\mathbb{R}$.
Suppose $f$ admits a function $F$: $I\longrightarrow\mathbb{R}$ such that
(1) $F$ ...
- 277
3
votes
0
answers
202
views
Difficult Gaussian-sum inequality for large random Bernoulli-Toeplitz matrices
I have come across the following problem in an attempt to prove an entropy bound for large random Bernoulli-Toeplitz matrices (Conjecture 1 on p. 16 of this preprint by Clifford et al. 2015), which is ...
- 686
15
votes
2
answers
681
views
Are Fourier transforms of L^p stable under diffeomorphisms?
Let $\xi$ be a compactly supported distribution on $\mathbb R^n$ and assume that its Fourier transform is in $L^p$. Let $\phi:\mathbb R^n\to\mathbb R^n$ be a diffeomorphism. Does the Fourier ...
- 2,649
3
votes
1
answer
146
views
Radial Kernel with Bounded Support and Norm of Gradient Bounded by a Dimension-free Constant
I was wondering if it is possible to construct a compactly supported radial kernel function in $\mathbb{R}^d$ such that the norm of the gradient is bounded by some dimension-free constant. That is, ...
- 1,127
26
votes
4
answers
2k
views
$\binom{x}{2}+\binom{x}{4}+\cdots+\binom{x}{2u}$ is a convex function on $[0,+\infty)$?
Let $f(x)=\binom{x}{2}+\binom{x}{4}+\cdots+\binom{x}{2u}$, where $u\in\mathbb{Z}^+$ and $\binom{x}{l}=\frac{x(x-1)\dots(x-l+1)}{l!}$ for all $l\in\mathbb{Z}^+$.
Then can we prove $f(x)$ is a convex ...
- 271
2
votes
1
answer
207
views
Expectation of Truncated Bivariate Gaussian Random Variables
Suppose $Z , \epsilon \sim N(0, 1)$ are independent Gaussian random variables. Let $a \ll 1$ be a small positive number. Let $W = aZ + \epsilon$. It can be show that
\begin{align}
\mathbb{E} [ W^2 (Z^...
- 1,127
7
votes
0
answers
211
views
Increasing derivatives of recursively defined polynomials
Consider recursively defined polynomials $f_0(x)=x$ and $f_{n+1}(x)=f_n(x)−f_n'(x) x (1−x)$.
These polynomials have some special properties, for example $f_n(0)=0$, $f_n(1)=1$, and all $n+1$ roots of ...
- 225
1
vote
1
answer
1k
views
Intermediate value property and continuity
We say that a function $f:\mathbb{R}\to\mathbb{R}$ has the intermediate value property (ivp) if for $a<b$ in $\mathbb{R}$ we have $$f([a,b]) \supseteq [\min\{f(a),f(b)\}, \max\{f(a), f(b)\}].$$
The ...
- 48.9k
3
votes
0
answers
848
views
Does a bounded convex domain has one smooth boundary point?
In the study of analysis and geometry of a bounded domain, its boundary regularity is important. For example, it is known that a bounded convex domain has Lipschitz bounday. This implies that a ...
- 285
4
votes
2
answers
588
views
How to prove this inequality or give a more accurate bound?
How can we prove this inequality or give a more accurate bound?
$$
1 + x + \frac{{{x^2}}}{{2!}} + ....... + \frac{{{x^n}}}{{n!}} > \frac{{{e^x}}}{2}，x \in [0,n]\
$$
I came across the equation:
$$
...
- 225
2
votes
2
answers
233
views
Is the domain of symmetric derivative borel set?
Let $\mu$ be the $n$-dimensional Lebesgue measure and $\lambda$ be a complex Borel measure on $\mathbb{R}^n$.
Let $S$ be the set of points $x\in \mathbb{R}^n$ where $\lim_{r\to 0} \frac{\lambda (B(x,...
- 337
20
votes
3
answers
1k
views
mixing convex and concave for convexity
Let $n\in\mathbb{N}$ and $0<x<1$ be a real number. Is the following a convex function of $x$?
$$G_n(x)=\log\left(\frac{(1+x^{4n+1})(1+x^{4n-1})(1+x^{2n})(1-x^{2n+1})}{(1+x^{2n+1})(1-x^{2n+2})}\...
- 43.2k
0
votes
1
answer
126
views
Comparing tails of polynomial functions
Suppose that $P(x) = a_m x^m + \dots + a_0$ and $Q(x) = b_n x^n + \dots + b_0$ are two polynomials, with $m > n > 1$ and $a_m > b_n > 0$. Suppose that $P$ has $m$ distinct real roots $y_1&...
- 225
4
votes
1
answer
1k
views
Indefinite integral of squared hypergeometric function
I am trying to compute the indefinite integral
$$
\int_0^u {}_2F_1\left(\frac{1}{4},\frac{5}{4},2,1-v^2\right)^2 dv
$$
for $0<u<1$. Using Clausen's formula for the square of the hypergeometric ...
- 165
1
vote
1
answer
1k
views
About covariance operators for probability distributions on a function space
Feel free to restrict the function space to a Hilbert space or to a RKHS. Given a probability distribution on it when can we define a ``covariance operator" for it and when would it also have a well-...
- 2,246
9
votes
3
answers
3k
views
For positive definite $A,B$ why does $AB+BA$ tend to be positive definite?
Let $A$ and $B$ be two positive definite $n \times n$ matrices. It is, of course, not true that $AB+BA$ is necessarily positive definite.
Consider, though, the results of the following numerical ...
- 117
12
votes
1
answer
520
views
Can $C^1$ mappings with derivative of low rank be approximated by smooth maps?
Asked once on SE-mathematics.
Let $U$ be an open subset in $\mathbb{R}^n$, $m\in\mathbb{N}$, $1\leq m<n$ and let
$$\mathcal{C}^k_{\leq m}(U,\mathbb{R}^n):=\lbrace g\in\mathcal{C}^k(U,\mathbb{R}^n)\...
- 123
3
votes
1
answer
290
views
Fluctuating constants
Let $p_k$ be the $k$-th prime number, $\gamma$ be the Euler-Mascheroni constant and $M$ be the Meissel–Mertens and let $m$ be the integer part of $\log p_n$. We can show that
$$
\sum_{r=1}^{m} \frac{...
- 5,470
4
votes
1
answer
370
views
Convergence of a series
Let $F(z)=\displaystyle \sum_{k=0}^\infty a_kz^k,\;|z|<R $ and $F(R)=\displaystyle \sum_{k=0}^\infty a_kR^k$ (the series converges).
Assume that $F(\alpha_j)=0,\;j=1,2,\dots ,m$, where all $|\...
- 783
-1
votes
1
answer
227
views
Solving the integral identity $ \int_{a}^{b} f(x)dx = \int_{a}^{b} f(x)g(x)dx. $ [closed]
We know that 0 is the additive identity and 1 is the multiplicative identity. In the same spirit let us define the integral identity as follows.
Definition: Let $f(x)$ be integrable in $(a,b)$. If ...
- 5,470
3
votes
1
answer
201
views
Seeking a property about Lebesgue-Stieltjes outer measure
I am a graduate student and this is not something related to my work but I was just wondering and did not find an answer on the Internet. I asked this on the other math site two weeks ago and no one ...
- 131
3
votes
0
answers
198
views
Characterizing rational functions on $\mathbb{Q}$ in terms of smooth extensions to $\mathbb{R}$ and $\mathbb{Q}_p$
Consider a function $f$ from a cofinite subset of $\mathbb{Q}$ to $\mathbb{Q}$. As established here and here $f$ extending to a smooth function on a cofinite subset of $\mathbb{R}$ is not sufficient ...
- 13.1k
0
votes
1
answer
629
views
Fourier Transform of sub-Gaussian distributions
The high level question is: Just as the Fourier transform of a Gaussian is a Gaussian, is the Fourier Transform of a sub-Gaussian also a sub-Gaussian?
Let $x \in \mathbf{R}^n$ denote some sub-...
- 445
5
votes
1
answer
337
views
Hadwiger-Nelson problem for $\ell^\infty$
Let $G=(V, E)$ be the following graph:
$V=\ell^\infty = $ set of bounded real sequences, with the norm $$\|x\|_\infty = \sup_{n\in\mathbb{N}}|x_n|,$$
$E = \big\{\{x,y\}: x,y\in \ell^\infty \text{ and ...
- 48.9k