All Questions
18,178 questions
1
vote
0
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84
views
Percolative process distribution not equivalent to coupon collector problem distribution
I have a process where; given a $n\times 1$ matrix initially empty, an element is inserted in it at a random position, with the possibility of repeating the insertion at a filled cell. Then, after a ...
- 47
0
votes
1
answer
127
views
Continuous extensions of tangent vector fields
Let $\Omega$ be an open subset of $S^2$ with $\bar{\Omega}\neq S^2$. Suppose a continuous tangent vector field $G$ is given on $\partial \Omega$ with $|G(y)|=1$ for all $y\in \partial \Omega$. Does ...
- 434
0
votes
1
answer
80
views
Continuous modification of tangent vector fields
Let $\Omega$ be an open subset of $S^2$, and assume that there exists a continuous tangent vector field $F(x)$ defined on $\bar{\Omega}\neq S^2$ with $|F(x)|=1$ for all $x\in \bar{\Omega}$. Suppose a ...
- 434
3
votes
1
answer
249
views
Is Toeplitz operator on the Bergman space bounded iff its symbol is bounded?
It is fairly well known that if $T_\varphi$ is a Toeplitz operator on the Hardy-Hilbert space, then $\lVert T_\varphi \rVert = \lVert \varphi \rVert _{\infty}$.
Now, if $\varphi \in L^\infty (\mathbb ...
- 151
6
votes
0
answers
149
views
Running minimum of exponential random walks
Let $\{X_i\}$ be a collection of i.i.d. Exp$(1)$ random variables. For $k \in \mathbb{Z}_{>0}$, define
$$S_k = \sum_{i=1}^k X_i$$
and note that $\mathbb{E}[S_k] = k$.
I was wondering if there is ...
- 485
1
vote
1
answer
130
views
Size of epsilon net with uniformly distributed points
Consider $N$ vectors $X_1,\cdots,X_N\in \mathbb R^n$ uniformly distributed in a cube $[0,1]^n$.
What condition on $N$ such that $\{X_1,\cdots,X_N\}$ forms an $\epsilon$-net for this cube, with high ...
- 405
7
votes
2
answers
852
views
What are the Nash equilibria of the “aim for the middle” game?
Consider the following three-player game: Alice chooses an integer congruent to $0$ mod $3$, Bob chooses an integer congruent to $1$ mod $3$, and Chris chooses an integer congruent to $2$ mod $3$. (...
- 32.5k
0
votes
0
answers
81
views
White noise: a tempered distribution version of the stochastic convolution
Let $\xi$ be a space-time white noise, that is a centered Gaussian process with covariance $E[\xi_{f}\xi_h]=\int_{\mathbb{R}_+ \times \mathbb{R}^d}fh,$ for $f,h\in L^2(\mathbb{R}_+ \times \mathbb{R}^d)...
- 573
2
votes
0
answers
94
views
Concentration inequalities for functions of random binary strings
Let $(X_1,\ldots,X_n)$ be a vector in $\{0,1\}^n$ drawn uniformly at random among all vectors with exactly $k$ $1'$s. I am interested in inequalities for tail probabilities for the random variables $X,...
- 2,288
6
votes
2
answers
449
views
Asymptotic behavior of the "Cauchy square" series
$\renewcommand{\ge}{\geqslant}\renewcommand{\le}{\leqslant}$
$\newcommand{\pa}[1]{\left( #1 \right)}$
Let us take $\alpha > 0$, $x_1 := \alpha$ and for any $n \ge \mathbb{N}$,
\begin{align*}
\boxed{...
- 61
2
votes
0
answers
138
views
Is $L^2([a,b]; L^2(S^2))$ the same as $L^2([a,b] \times S^2)$?
The space $L^2([a,b];L^2(S^2))$ is a Banach space with respect to the norm
$$\left\Vert f \right\Vert_1^2 = \int_{a}^b \left\Vert f(r) \right\Vert_{L^2(S^2)}^2 dr$$
The space $L^2([a,b]\times S^2)$ ...
- 969
1
vote
0
answers
92
views
Multilinear non-commutative Khintchine inequality
Let $g_1,\ldots,g_k$ be independent standard Gaussians and for each index $(i_1,\ldots,i_k)\in [n]^k$ let $A_{i_1,\ldots,i_k}$ be a $d\times d$ symmetric matrix.
Question: Is there a known bound for ...
- 156
4
votes
0
answers
262
views
Spectrum of ring in algebraic geometry vs spectrum of Banach algebra
For a commutative unital Banach algebra $A,$ and $x\in A,$ we have $\lambda \in \sigma_A(x)$ if and only if $\phi(x) = \lambda$ for some algebra homomorphism $\phi:A \to \mathbb C.$ The set of all ...
- 1,755
3
votes
1
answer
251
views
Feynman–Kac formula for other operators
I recently came across the Feynman-Kac formula which states that given an open domain $\Omega\in\mathbb{R}^n$ and $f \in L^2(\Omega)$
where $x \in \Omega$ and $t > 0$, then
$e^{t\Delta_D}f(x) = ...
- 41
3
votes
1
answer
208
views
The signs of some mean-zero random variables
Let $X$ be a discrete random variable supported on $\{−5,\dots, 6\}$ in which the outcomes have the following respective probabilities: $$\begin{array}{rc}
n & p(n) \\ \hline −5 & 6/36 \\ −4 &...
- 19.7k
1
vote
1
answer
112
views
A bilinear estimate with a simple one-dimensional oscillatory integral kernel
Let $f\in L^{p}(\mathbb{R})$, $1\leq p\leq 2$.
I am trying to show that
$$\int_{\mathbb{R}}\int_{\mathbb{R}}
\,K(y,z)\,
\frac{f(y)f(z)}{y^{\frac{1}{2\,p^{\prime}}}\,z^{\frac{1}{2\,p^{\prime}}}}\,dy\,...
- 852
10
votes
1
answer
259
views
Sufficient condition for the graph of a measurable map to be measurable
Let $f:X \to Y$ be measurable map between measurable spaces w.r.t. to their corresponding $\sigma$-algebras $\Sigma_X$ and $\Sigma_Y$, resp.
If $(X,\Sigma_X)$ is a standard Borel space can we always ...
- 285
0
votes
0
answers
149
views
Notation $\le_{a,b,n,\ldots}$ in Analysis
In modern Analysis, especially Functional Analysis, one proves, or one uses inequalities of the form
$$F(X)\le_{a,\ldots,n}G(X).$$
The meaning of the subscripts in the inequality sign means that there ...
- 52.3k
2
votes
0
answers
28
views
Monotonicity of the rank of finitely generated closed sublattices
Let $E$ be a Banach lattice. We consider finitely generated elements in the class of all closed sublattices of $E$. Let $F$ be a closed sublattice $E$. Then a subset $G \subseteq F$ is called ...
- 467
2
votes
0
answers
78
views
What is known about $\operatorname{gap}(A) = \|A\| - r(A)$ for bounded operators on Hilbert spaces?
The gap of a bounded linear operator on a Hilbert space is defined as $$\operatorname{gap}(A) := \|A\| - r(A),$$
where $r(A)$ denotes the spectral radius of $A$. A natural question to ask is - for ...
- 165
13
votes
8
answers
1k
views
The vertices of a triangle are three random points on a unit circle. The side lengths are, in random order, $a,b,c$. Show that $P(ab>c)=\frac12$
The vertices of a triangle are three unifomly random points on a unit circle. The side lengths are, in random order, $a,b,c$.
There is a convoluted proof that $P(ab>c)=\frac12$. But since the ...
- 3,567
0
votes
0
answers
67
views
"Marginal" Three-Series Theorem?
Let $\{X_t\}_{t=1}^\infty$ be a sequence of random variables and define the partial sums by $S_t=\sum_{s=1}^t X_s$.
Kolmogorov's three-series theorem states that
Theorem (Davidson, 2021, Theorem 21.8)...
- 73
3
votes
0
answers
206
views
Explicit basis of symmetric harmonic polynomials
An orthonormal basis for the space of harmonic polynomials in $n$ variables is provided by the spherical harmonics on the $n-1$ sphere, see e.g. wiki.
From there, constructing an orthonormal basis for ...
- 31
0
votes
0
answers
73
views
Criteria giving sufficient conditions for a Borel measure to have compact support
I am interested in criteria that guarantee that a Borel probability measure has compact support.
I outline two below and I am hoping to gather more as answers (if they exist).
The first sufficient ...
- 870
0
votes
0
answers
55
views
Johnson-Lindenstrauss type result for matrix factorization
The type of result I want is: given matrix $A\in \mathbb{R}^{m\times n}$ and error tolerance $\epsilon$, what is a lower bound on $k$ such that $\|A - UV\|_{??}\le \epsilon$, where $U \in\mathbb{R}^{m\...
2
votes
0
answers
60
views
Basis vectors using anti-commuting operators?
Let $V$ be a finite-dimensional inner product space. Suppose $A_{1},...,A_{N}$ are anti-commuting operators, meaning that these are linear operators on $V$ that satisfy:
$$A_{i}A_{j}+A_{j}A_{i} = A_{i}...
- 1,305
1
vote
2
answers
66
views
Distribution of the constraint matrix conditioned on the solution of the linear system
Suppose that A is a random matrix in $R^{n\times n}$, with each component independently and identically distributed (iid) according to $\mathcal{N}(0,1)$. Additionally, b is a random vector in $R^n$, ...
- 33
0
votes
0
answers
43
views
When does the Hermite series converge pointwise and when is it uniformly bounded?
Let $\gamma$ denote the standard Gaussian measure on the real line, and consider $f \in L^2(\gamma)$. Since the Hermite polynomials $\{H_n\}_{n \geq 0}$ are a complete orthonormal system, we may ...
- 420
1
vote
1
answer
369
views
Bounding supremum norm in terms of gradient L2-norm using a Poincare-like inequality
Suppose $f$ is a Lipschitz continuous real-valued function over a bounded domain $\Omega \subset \mathbb{R}^d$ with smooth boundary, and let $\overline{f} := \frac{1}{|\Omega|}\int_\Omega f(x) dx$. Is ...
- 148
6
votes
1
answer
290
views
Analytic maps on Banach spaces: analyticity upgrade
Consider the following problem.
Let $E,F,G$ be real or complex Banach spaces, such that $F\subset G$ with continuous embedding. Let $U\subset E$ an open set and
$$ f:U\to G $$
an analytic map, such ...
- 1,075
1
vote
0
answers
128
views
Sum of upper semi continuous and lower semi continuous functions
Let $X$ be a compact metric space. Assume that $f: X \to \mathbb{R}$ is upper-semi continuous and $g:X \to \mathbb{R}$ is lower semi-continuous. Assume that $\sup \{ f(x)+g(x) : x \in X \}$ is finite. ...
- 1,043
3
votes
0
answers
281
views
Interchange limit and supremum of functionals over a bounded convex set
Let $(H, \langle\cdot,\cdot\rangle)$ be a separable real Hilbert space and $B\subset H$ be (nonempty) convex and bounded, and suppose that $(\alpha_k)\subset H$ is a sequence for which the limit $\...
- 463
2
votes
1
answer
321
views
A strange functional inequality
Let $f,g \in C([-2,2],\mathbb R_+^*)$ even and concave real functions.
Is it true that
$$
\int_0^1 f\big(\cos(x^{-1})+\sin(x^{-1})\big) \cdot g\big(\cos(x^{-1})-\sin(x^{-1})\big) \mathrm{d}x\\ \leq f(...
- 4,074
0
votes
0
answers
54
views
Reference request: "doubly empirical" measure associated to a random measure
I am considering the following type of situation. Suppose we have a random probability measure, by which I mean a probability measure on a space of probability measures atop some Polish space $X$. In ...
- 660
0
votes
0
answers
35
views
Question about the spectrum of a deformed GOE matrix
Consider a fixed real value $\sigma>0$. Let $A,Z$ be two independent $n\times n$ GOE matrices, and define $B=A+\sigma Z$. I am interested in finding a bound (possibly dependent on $n$) for the ...
- 1
0
votes
0
answers
97
views
Generator of an analytic semigroup
Perhaps I have a naive question. My question is as follows:
When we consider a Cauchy proposition of the following form:
$$
\begin{cases}
x'(t)= -Ax(t)+ F(t,x(t)) &\text{for}\ t> 0 \\
x(0)=...
- 39
0
votes
0
answers
44
views
Are there probability densities $\rho, f_n$ such that $\lim_n \frac{[\rho * f_n]_\alpha}{\|\rho * f_n\|_\infty} = \infty$?
We fix $\alpha \in (0, 1)$. Let $[f]_\alpha$ be the best $\alpha$-Hölder constant of $f: \mathbb R^d \to \mathbb R^k \otimes \mathbb R^m$, i.e., $[f]_\alpha := \sup_{x \neq y} \frac{|f(x) - f(y)|}{|x-...
- 825
1
vote
1
answer
54
views
Equivalence vs modification
Suppose that two processes have the same finite-dimensional distributions. Does there exist a coupling of them such that they are modifications of each other?
- 177
1
vote
1
answer
57
views
Lower bound the best $\alpha$-Hölder constant of a convolution
Let $\mathcal D_1$ be the set of bounded probability density functions on $\mathbb R^d$. This means $f \in \mathcal D_1$ if and only if $f$ is non-negative measurable such that $\int_{\mathbb R^d} f (...
- 825
1
vote
2
answers
90
views
Is the difference between $\alpha$-Hölder constants of $f*\rho$ and $g*\rho$ controlled by $\|f-g\|_\infty$?
Let $\mathcal D_1$ be the set of bounded probability density functions on $\mathbb R^d$. This means $f \in \mathcal D_1$ if and only if $f$ is non-negative measurable such that $\int_{\mathbb R^d} f (...
- 825
7
votes
0
answers
294
views
Applications of Banach space homology
There is a well-developed theory of Banach space homology. What are some of its useful applications to Banach space theory and which important questions can one answer using it? In other words, how ...
- 175
0
votes
0
answers
53
views
A problem about how to understand the existence of derivative of level set in Mountain-pass theorem
I'm confused about the Mountain pass theorem in Lemma4 of here.
Background :
$$
\begin{gathered}
I_\lambda(u)=\frac{1}{2} \int_M\left|\Delta_g^{\frac{m}{2}} u\right|^2 d \mu_g-\frac{\lambda}{2 m} \log ...
- 809
2
votes
1
answer
118
views
Probability that at least one of the first $k$ right-to-left maxima of $\{1, ..., n+m\}$ contains a number from $\{n+1, ..., n+m\}$
Given a permutation $\sigma$ on $[n]=\{1, ..., n\}$, we say an element $i$ is a right-to-left maximum if $\sigma(i)=\max(\sigma(1), ..., \sigma(i))$.
Suppose we sample a random permutation from $\{1, ....
7
votes
2
answers
307
views
PDE for the probability of Brownian motion staying in an area (reference request)
I am looking for a (preferably some monograph) reference on the following fact:
$$
u ( t, x ) = \mathbb{P} \{ x + B_s \in A \ \text{for all} \ s \leq t \}
$$
satisfies the heat equation
$$
\frac{\...
- 620
4
votes
1
answer
261
views
What is the convergence rate of this "infinite monkey"-type probability?
Cross-posted from Math Stack Exchange, where it hasn’t received an answer yet:
Let $S$ be a finite set and $n,m\in\mathbb N$. Consider the process $R=(R_i)_{i\in\mathbb N}$ where all $R_i$ are iid ...
- 141
1
vote
0
answers
53
views
The limit ratio of two Markov Chain Probability
Suppose there are two given SDE in $\mathbb{R}^d$:
$$
\begin{align}
\left\{
\begin{aligned}
dX_t&=\begin{bmatrix}-\nabla V(X_t)+2\beta^{-1}v_F^\theta(X_t)\end{bmatrix}dt+\sqrt{2\beta^{-1}}dW_t,&...
- 303
0
votes
0
answers
112
views
Characterization for the multipliers of Schwartz space
Is the following true?
A function $m:\mathbb R^n\to\mathbb C$ is a Schwartz multiplier (i.e. $[f\mapsto mf]:S(\mathbb R^n)\to S(\mathbb R^n)$ is bounded linear) iff the following:
For every $\alpha$ ...
- 938
4
votes
2
answers
227
views
Given an automatic set $S$ coming from a DFA $M$ when read little-endian, is $\overline{d}(S)$ at most the Büchi acceptance probability of $M$?
Note: I've entirely rewritten this question! Originally it was just the third formulation, take note of that when reading answers.
Let's say $S$ is a $b$-automatic set, and let's say $M$ is a DFA ...
- 2,585
6
votes
0
answers
197
views
What are compact manifolds such that GROWTH (of spheres volumes) is well approximated by the Gaussian normal distribution?
Consider some compact Riemannian manifold $M$. Fix some point $p$.
Consider a "sub-sphere of radius $r$" - i.e. set of points on distance $r$ from $p$.
Consider growth function $g(r)$ to be ...
- 24.9k
0
votes
0
answers
64
views
When is a symmetric block Toeplitz matrix invertible?
Let
$$
Q =
\begin{bmatrix}
Q_0 & Q_1 & Q_2 & \cdots\\
Q_{-1} & Q_{0} & Q_1 & \cdots\\
Q_{-2} & Q_{-1} & Q_0 & \cdots\\
\vdots & \vdots & \vdots & \ddots
...
