All Questions
18,178 questions
3
votes
0
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69
views
Perturbation of one-parameter groups of unitary operators
Let $H$ be a Hilbert space and let $h$ be a fixed, densely defined, possibly unbounded, self-adjoint operator on $H$. Letting $B(H)$ denote the space of all bounded operators on $H$, it is well ...
- 2,263
0
votes
0
answers
29
views
On constructing the canonical boundary operator for a given differential operator
Given an $n\times n$ matrix $$X=\begin{pmatrix}
x_{11} & x_{12} & \cdots & x_{1n} \\
x_{21} & x_{22} & \cdots & x_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
x_{n1}...
- 765
2
votes
0
answers
95
views
On analytic functions on the complement of a curve without jump across the curve almost everywhere
Question. Suppose $f$ is an analytic function on $\mathbb C\setminus\mathbb R$ and assume that the boundary values of $f$ from above and below the real axis (denoted $f_\pm$ respectively) exist almost ...
3
votes
1
answer
175
views
Convergence rate of the sum of squares of inverse distances of random points which become dense in a region
$n$ points $\{X_i\}$ are drawn at random from a uniform distribution over a domain $\Omega\subset \mathbb{R}^m$ with a Lipschitz boundary. $D_n$ is defined as $$D_n = \sqrt{\frac{1}{\sum\limits_{1\le ...
- 698
2
votes
1
answer
105
views
Inequality for Gaussian measures
Let $\mu$ denote a centered Gaussian measure on $\mathbb{R}^k$, $K=(-\infty, a] \times \mathbb{R}^{k-1}$ ($a\ge 0$) and $L=\mathbb{R}\times C$ where $C$ is a convex set in $\mathbb{R}^{k-1}$, ...
- 197
0
votes
0
answers
31
views
What is the Fisher information matrix of the von Mises-Fisher distribution?
Assuming the von Mises-Fisher distribution as
$$f_{p}(\mathbf{x}; \boldsymbol{\mu}, \kappa) = C_{p}(\kappa) \exp \left( {\kappa \boldsymbol{\mu}^\mathsf{T} \mathbf{x} } \right),$$
where $\kappa \ge 0$,...
- 287
3
votes
1
answer
79
views
How to deal with singularities in thin plate splines?
Follow up from this question Thin-Plate-Spline understanding and solution.
In the general case of $\mathbb{R}^N$ the following problem (interpolant which minimizes the Thin Plate Energy, specifically ...
- 115
2
votes
1
answer
179
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Reference request: Parabolic Schauder estimates for the heat equation with $f \in L^\infty$
Let us consider the heat equation
$$\partial_t u - \Delta u = f(x, t) \quad \text{in }Q_R $$
where $Q_R = B_R \times (-R^2,0].$ I would like to know the kind of regularity we should expect of $u$ if ...
- 452
1
vote
2
answers
164
views
Existence of directional heat equation without uniform ellipticity
I am asking for references, or for a proof idea on how to show that weak solutions of the following problem exist: search $u$ on a bounded domain $\Omega\times (0,T]$, where $\Omega\subset\mathbb{R}^d$...
- 35
1
vote
0
answers
88
views
Schauder estimate for $f \in L^\infty$
I was reading an article where at some point the author uses the following estimate:
Let $u$ be a solution of
$$\Delta u = f \quad \text{in } B_1$$
for $f \in L^\infty$. Then $u \in C^{1,1 - \...
- 452
19
votes
2
answers
2k
views
Higher or lower?
Consider the following game - I draw a number from $[0, 1]$ uniformly, and show it to you. I tell you I am going to draw another $1000$ numbers in sequence, independently and uniformly. Your task is ...
- 6,215
5
votes
1
answer
192
views
Non-equivalent definitions of Markov process
As far as I know, there are three definitions of Markov processes (or of Markov chains).
DEFINITION 1 (WEAKER). A process $(X_t)_{t\in[0,\infty)}$ on $(\Omega,\mathcal{F},\mathbb{P})$ with values in ...
- 1,149
8
votes
1
answer
522
views
One step forward, one step back
$N \geq 2$ players play a cooperative game on the integers $\mathbb Z$. All of them start from $0$. At each turn, they are simultanously given the same yes or no question to answer.
The questions ...
- 6,215
3
votes
0
answers
167
views
Bounding the $L^{p*}$ norm from below for functions satisfying a $p$-capacity estimate
If $1 \le p < n$, the $p$-capacity of a compact set $A \subset \mathbb{R}^n$ with respect to an open set $U$ containing it is defined as $$\text{Cap}_p(A, U) := \inf \left\{\int_U |\nabla u|^p \, ...
1
vote
0
answers
80
views
Moments from characteristic function for matrices
When $x$ is a random variable with the smooth characteristic function $\phi_x(t) = \mathbb{E}e^{itx}$, we can easily compute the moments as $\mathbb{E}[x^k] = i^{-n}\phi_x^{(n)}(0)$. There is no magic ...
- 193
2
votes
0
answers
56
views
Sum of independent Wisharts
Suppose random vectors $y_1,y_2,\ldots,y_m$ are independent and the distribution of each $y_i$ is a $d$-dimensional complex Gaussian with mean $0$ and covariance $\Gamma_i$, that is $y_i \sim \mathcal{...
- 193
2
votes
0
answers
29
views
Steiner symmetrization of smooth function on non-simply connected regions
Given a smooth function $u$ defined on $\mathbb{R}^2$, restrict $u$ to a subset $\Omega \subset \mathbb{R}^2$ (possibly not simply connected) foliated by level sets of a smooth function $\psi: \Omega \...
- 434
1
vote
2
answers
209
views
Approximate simple function $f$ by a sequence of continuous functions on $\mathbb{R}^d$ such that $\|f_n\|_\infty\leq \|f\|_\infty$
Let $f=\sum_{i=1}^n c_i 1_{\Delta_i}$ be a simple function on $\mathbb{R}^d$, where $c_i\in\mathbb{C}$. Then we can find sequnces of continuous functions $\{f_k^{(i)}\}$ for each $i=1,\ldots,n$ such ...
- 29
1
vote
1
answer
152
views
SOT and WOT convergence of Toeplitz operators
For the Hardy space $H^2$, every $\phi \in L^\infty (\mathbb T)$ induces a bounded Toeplitz operator $T_\phi$ on the Hardy space and $\lVert T_\phi \rVert = \lVert \phi \rVert _{\infty}$. Consequently,...
- 151
1
vote
0
answers
248
views
Solving functional analysis problems by using Algebraic geometry
I am thinking about some open problems in nonlinear functional analysis and I just wanted to know if there are any problems that have been solved by using Algebraic geometry techniques in these fields....
1
vote
0
answers
66
views
Random lattice always has trivial automorphism group?
In example 2.5 of a paper [LS17] written by Lenstra and Silverberg, it is written that “Random” lattices have $Aut(L) = \{ \pm 1 \}$, I guess the 'Random' here refers to the distribution in Siegel's ...
- 253
1
vote
0
answers
55
views
Limit process of a sequence of Gaussian variables on mesh grid going to zero
Consider the interval $[0,1]$ and a partition $\mathscr{P}_n = \{ [t_i,t_{i+1}), \, i=1,\ldots,N_n \, : \, 0=t_0 < \ldots < t_{N_n} = 1\}$. Suppose that for all $i$ and $t \in [t_i,t_{i+1})$, we ...
- 141
2
votes
0
answers
82
views
What is Lipschitz constant of the radial renormalization $(X,\|\cdot\|_a) \rightarrow (X,\|\cdot\|_b)$ on a normed vector space $X$
Suppose that $X$ is a vector space with two norms $\|\cdot\|_a$ and $\|\cdot\|_b$. The mapping
$$
f(x) = \frac{\|x\|_{a}}{\|x\|_{b}} x, \qquad \forall x \in X,
$$
with $f(0)=0$
is a radial and maps ...
- 5,327
3
votes
1
answer
194
views
Dynamics of a random stretch map
Notation: Here $S^1$ denotes the circle, which we view as the unit sphere in $\mathbb C$. We equip the circle with its natural length metric.
Let $\{\epsilon_n\}_{n \geq 1}$ be iid uniformly ...
- 6,215
6
votes
1
answer
257
views
Example/Existence of Positive Linear Functional which is NOT Hermitian
We know that if $\mathcal{A}$ is a unital $C^*$-algebra and if $f:\mathcal{A}\to\mathbb{C}$ is a positive linear functional then it is Hermitian. It simply follows from the fact that in $\mathcal{A}$ ...
- 173
34
votes
7
answers
3k
views
A hat puzzle question—how to prove the standard solution is optimal?
I am currently writing an essay on hat puzzles, and for the warm-up section I introduce some of the standard finite hat puzzles. One of these proceeds as follows:
You and two friends are each given a ...
- 236k
0
votes
0
answers
93
views
Orthogonalization of symmetric non-degenerate bilinear forms
It is well-known that given a field $k$ with characteristic different from $2$, every symmetric non-degenerate bilinear form $B$ over a finite-dimensional space can be orthogonalized. This means that ...
- 1,485
2
votes
1
answer
99
views
A question on Bloch functions
Let $\mathcal{B}(\Delta)$ be the space of Bloch functions in the unit disk $\Delta$. For any $f\in \mathcal{B}(\Delta)$, we define the Bloch norm by
$$
\|f\|_{\mathcal{B}}=\sup_{|z|<1}|f'(z)|(1-|z|^...
- 1,706
0
votes
0
answers
56
views
What is the maximum of $ \frac{\sin(n(x+a))}{\sin(x+a)} + \frac{\sin(n(x-a))}{\sin(x-a)}$?
I have asked this here. Due to inactivity and no satisfying answers, I am asking here. Hope that's okay.
We know the global maxima of the function $\frac{\sin(nx)}{\sin(x)}$
is $n$ (thanks to this ...
- 113
9
votes
1
answer
155
views
How to sample exactly k indices given the inclusion probabilities of all indices?
Let $k<d$ two positive integers, and $\{p_i\}_{i=1}^d$ a series of probabilities, with $p_i \in (0,1)$ and $\sum_{i=1}^d p_i = k$.
We wish to sample exactly $k$ distinct indices $\mathcal{I}\...
- 968
1
vote
1
answer
151
views
Some operators on spheres
Let $S_2$ be the unit sphere in $\mathbb R^3$ equipped with normalized Haar measure. For a continuous function f and $\delta\in (-1,1)$ define $T_\delta f(x):=\int_{\{y:<x,y>=\delta\}}f(y)d_\...
- 1,879
2
votes
1
answer
89
views
The contractivity of the time derivative of the heat semigroup in $L^p$ spaces
Let $M$ be a complete manifold. The heat semigroup $e^{-tL}$ is bounded on $L^p(M)$, for any $1 \leq p \leq \infty$;
see this for instance.
It seems that we can deduce the time derivative of the heat ...
- 121
2
votes
1
answer
276
views
Probability of visible permutations
A visible permutation $\sigma$ of $[1,2, ...,n]$ has a permutation matrix such that all "1" locations are visible from the origin $(0,0)$.
Two "1" locations are visible if the two ...
- 2,993
5
votes
2
answers
258
views
Boundary value of Sobolev space
Let $D$ be a regular domain in $\mathbb R^2$. Suppose that $u \in H_0^1(D) \cap C(D)$. Does this imply $u \in C(\overline D)$ and $u|_{\partial D} = 0$?
- 177
0
votes
0
answers
118
views
Find the maximum of an expression under the logconcave assumption
Let $F(v)$ be a cdf over $\left[0,v_{max}\right]$, $1-F(v)$ is logconcave. The corresponding density function is $f(v)$. Let $p^m$ solve $1-F(v)-f(v)v=0$ (it is a FOC of a profit maximization problem)....
1
vote
0
answers
30
views
Generalization of subadditivity analogous to quasiconvexity, and variants
I am curious if there are natural generalizations of subadditivity which have been studied in the past or have been stated in the literature? I (and people that I have talked to) have not had much ...
- 21
2
votes
1
answer
118
views
Proving that a polynomial $f(x,y)$ that is unbounded in every direction is bounded below by $1$ outside of a disc of finite radius
This is a follow up from this question.
I have a polynomial function $f(x,y)$ that is unbounded in every direction. In other words, if we choose a direction $(a,b)\in S^1$ and keep moving along the ...
- 765
5
votes
2
answers
364
views
Euler-Lagrange equations for minimizer of energy with indicator function
I'm looking for a modern explanation/proof of the derivation of Euler-Lagrange (or first-order or the "first variation") conditions for
$$\min_{u \in H^1_0(\Omega), u \geq 0} \int_\Omega |\...
- 93
1
vote
1
answer
142
views
Operator norm of some type of discrete Fourier matrix
Let $N$ be a natural number and let $w$ be a complex number.
We define the $N\times N$ matrix $C_w=(a_{k,l})_{k,l=1}^N$ as follows,
$$
a_{k,l}=\begin{cases}1 & l=k+1\\
w &...
- 4,058
1
vote
1
answer
76
views
Proving that a function $f(x,y)$, that is unbounded in every direction, is uniformly bounded below by $1$ outside some disc of large enough radius
I have a smooth function $f(x,y)$ that is unbounded in every direction. In other words, if we choose a direction $(a,b)\in S^1$ and keep moving along the curve $(ta,tb)$, then $$\lim_{t\to\infty}f(ta,...
- 765
0
votes
0
answers
30
views
Why is the $\alpha$-divergence unique in positive measure space $\mathcal{M}$?
In this article https://bsi-ni.brain.riken.jp/database/file/298/303.pdf (S. Amari 2009), it is said that a $f$-divergence (eq. 17) which can be written by a decomposable Bregman divergence (eq. 53) ...
- 1
1
vote
1
answer
115
views
Block-diagonal embedding of $U(n)$ into $U(mn)$
What is known about the subgroup $U(n)\subset U(mn)$ for $m,n\in\mathbb{N}$ given by the diagonal embedding
$$ \alpha\mapsto \text{diag}(\alpha,\cdots, \alpha),$$
for $\alpha$ appearing $m$ times?
For ...
4
votes
0
answers
127
views
A "resampling identity" for the Bessel(3) process
I've come across the following resampling identity and was wondering if this is known since it seems rather natural. Take $X$ a two-sided Brownian motion conditioned to always stay below $1$. (So if ...
- 10.3k
2
votes
0
answers
94
views
Nemytskij operator for Lebesgue variable UNBOUNDED exponent spaces
Let $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Caratheodory function (i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is measurable for all $t\in\mathbb{R}$), where $\Omega\...
- 1,759
3
votes
1
answer
232
views
Bounds on relative entropy for MLE in Bernoulli coin tosses
In the context of estimating the parameter $p$ from a dataset of $n$ i.i.d Bernoulli coin tosses, we often use the relative entropy $D(p \parallel \hat{p})$ to measure the performance of an estimator $...
- 63
4
votes
2
answers
312
views
What is the expected size of the smallest hitting set?
Suppose we pick $n$ subsets of size $j$ of an $N$-element set $S$ uniformly at random. A hitting set is a subset of $S$ that intersects all our subsets. I am interested in the smallest size of an ...
- 11.1k
7
votes
1
answer
253
views
Does a Banach algebra version of "the sum of a closed subspace and a finite dimensional subspace is always closed" exist?
In the setting of Banach spaces, it is well know that if $M$ is a closed subspace of a Banach space $X$ and $F$ is a finite dimensional subspace of $X$, then $M+F$ is closed.
Does a Banach algebra ...
- 515
1
vote
0
answers
43
views
Moments on the Stiefel manifold
Let $S_{n, k} = \{V \in \mathbb{R}^{n \times k} : V^T V = I_k\}$ denote the Stiefel manifold, $1 \leq k \leq n$.
Let $P \in \mathbb{R}^{n \times n}$ denote a symmetric real, positive definite matrix, ...
- 420
8
votes
0
answers
177
views
Understanding spaces of negative regularity
I apologize if this question is too basic for this site, but I posted it on mathSE and did not get any responses (link can be found here) so I'm crossposting it here.
Let $C^k(\mathbb{R}^n$) be the ...
- 721
2
votes
1
answer
202
views
Strong Liouville property of virtually abelian groups
Let $G$ be a finitely generated group and let $\mu$ be a symmetric non-degenerate measure on $G$. By strong Liouville property for $(G, \mu)$, we mean that every positive $\mu$-harmonic function on $G$...
- 1,407