All Questions
13,944 questions
11
votes
0
answers
342
views
The diagonal operators and unconditionality
The following is well-known:
Theorem: Let $X$ be a Banach space with an unconditional basis $(e_n)_n$.
Then the space of the diagonal operators with respect the basis $(e_n)_n$ endowed with
the ...
- 2,429
3
votes
0
answers
118
views
A matrix-valued analogue of a classical inequality
Let $p \geq 4$ be an even integer. In the study of variational problems in $W^{1, p}$, it is handy to know that for $a, b \in \mathbb R^d$,
$$|a - b|^p \leq 2^{p - 1} (|a|^{p - 2} + |b|^{p - 2}) |a - ...
- 6,213
4
votes
1
answer
551
views
Is there an explicit, everywhere surjective $f:\mathbb{R}\to\mathbb{R}$ whose graph has zero Hausdorff measure in its dimension?
Suppose $f:\mathbb{R}\to\mathbb{R}$ is Borel. Let $\text{dim}_{\text{H}}(\cdot)$ be the Hausdorff dimension, and $\mathcal{H}^{\text{dim}_{\text{H}}(\cdot)}(\cdot)$ be the Hausdorff measure in its ...
CommunityBot
- 1
3
votes
1
answer
175
views
Convergence rate of the sum of squares of inverse distances of random points which become dense in a region
$n$ points $\{X_i\}$ are drawn at random from a uniform distribution over a domain $\Omega\subset \mathbb{R}^m$ with a Lipschitz boundary. $D_n$ is defined as $$D_n = \sqrt{\frac{1}{\sum\limits_{1\le ...
- 10.3k
1
vote
1
answer
132
views
Can I find $n$ points on the boundary of an $n$-dimensional ball with certain properties?
My problem is the following: I want to construct $n$ rays all starting at a point $v$ that is not in the $n$-dimensional ball around $0$ such that the following is true:
The $n$-dimensional ball is a ...
2
votes
0
answers
94
views
A surprisingly simple and difficult problem on sums of upper bounds
Let $T$ be a large integer, and $C$ be a positive real constant.
Consider a sequence $\{p_t\}_{T\geq t\geq 1}$ of real numbers in $[0,1]$. The sequence $\{b_t\}_{T\geq t\geq 1}$ can be defined as ...
- 353
7
votes
1
answer
736
views
Should coffee machines be deconcentrated?
We model some region by convex and compact $E\subset \mathbb R^2$. $N\ge 1$ coffee machines are provided for the people living on $E$, of capacities $\alpha_1,\ldots, \alpha_N>0$. Assume the ...
CommunityBot
- 1
0
votes
1
answer
170
views
Summation of binomial coefficients with alternating signs
For a fixed $\alpha > 1$ and integer $n$, I want to provide some bounds or scaling results for the following summations
$$S_1(n,\alpha) = \sum_{k = 1}^{n} {n \choose k} (-1)^{k + 1} k / (\alpha k + ...
- 34.3k
6
votes
0
answers
159
views
Identification of Fock space and the $L^2$ space of tempered distributions
Let $\mathcal{S}'(\mathbb{R}^d)$ be the set of tempered distributions over $\mathbb{R}^d$ and $d\phi_C$ a Gaussian measure over $\mathcal{S}'(\mathbb{R}^d)$ with covariance operator $C$. Consider the ...
- 3,065
3
votes
0
answers
58
views
Infinitesimal generators of random evolutions
Consider two state spaces $X$ and $Y$ and infinitesimal generators of Markov processes $(A_y)_{y\in Y}$ and $B$, on $X$ and $Y$ respectively. We assume that $A_y$ share the same domain $D(A)$, and ...
- 31
4
votes
1
answer
158
views
Is the image of a complemented subspace complemented?
This question has been crossposted from mathstackexchange:
Let $X, Y$ be two Banach spaces and $T:X\to Y$ a continuous surjection. Assume $Z$ is a complemented subspace of $X$ and that $T(Z)$ is ...
- 60.5k
2
votes
0
answers
67
views
'Sublinear' and 'superlinear' moduli of continuity
Recall, given a metric space $X$, a function $f:X \rightarrow \mathbb{R}$ has (uniform) modulus of continuity $w:[0,\infty) \rightarrow [0,\infty]$ if $|f(x) - f(y)| < w(|x-y|)$ for all $x,y \in X$....
- 39.1k
5
votes
0
answers
104
views
Convolution of a bounded function and measures
Given a function $f\in L^\infty(\mathbb{R}^n)$ and a family of Radon measure $\mu_\alpha$, under what condition do we have $f*\mu_\alpha$ equi-continuous?
One condition I know is if $\mu_\alpha$ has a ...
- 375
0
votes
0
answers
59
views
Decomposition of a contractive representation into an orthogonal sum for the $n$-dimensional case. Has this been done yet?
I know that it has been done for the two-dimensional case. Marek Kosiek showed it in his work "Decomposition of operator representations of the algebra $R(K_1 \times K_2)$" and "...
- 63
3
votes
1
answer
177
views
Compactness of set of measurable functions between compact subspaces of real numbers
Let $X$ be a compact subset of $\mathbb{R}^n$ and $Y$ be a convex and compact subset of $\mathbb{R}^p$. Consider $\mathcal{F}$ the set of all measurable functions from $X$ to $Y$. Can I find ...
- 11
2
votes
0
answers
220
views
Ultraviolet divergences of entanglement entropy in QFT
I've often read that entanglement entropy in quantum field theory is ill-defined because local algebras are generally of type III, which implies that a trace doesn't exist. For a normal state $\omega_{...
- 424
3
votes
1
answer
203
views
Cohomology of the complex of differential forms with Schwartz coefficients
Let $U$ be an open manifold (say an open subset of $\mathbb{R}^n$ for simplicity). Denote by $\mathscr{S}(U)$ the space of Schwartz functions on $U$. Schwartz functions are defined as usual to be ...
- 21.5k
0
votes
1
answer
396
views
Distance function and its approximation
An easy and quick question:
Consider a function $u\in C(\Omega)$, where $\Omega$ is a bounded domain in $\mathbb{R}^n$.
Define a function $Q$ that measures the distance of a point $(x,y) \in\mathbb{...
CommunityBot
- 1
6
votes
0
answers
632
views
Generating functions in countable commutative monoids
Let $f: \mathbb{N}_0 \rightarrow \mathbb{C}$ be a function. The power series of $f$ can be viewed as the function $\mathscr{P}_f : q \mapsto \sum_{n \in \mathbb{N}_0}^{} f(n)q^n$ where $q \in \mathbb{...
- 405
6
votes
0
answers
156
views
Generalized Rademacher theorem for fractional derivatives
It is known that if $f$ is $\alpha$ Holder and $\gamma<\alpha$ then $f$ is $\gamma$ fractional differentiable. See Theorem 14 in the paper by G. H. Hardy and J. E. Littlewood, "Some properties ...
- 1,904
4
votes
1
answer
507
views
Degree four polynomials with no real roots
Consider a degree four polynomial
$$
f = a_4x^4 + a_3x^3 + a_2x^2 + a_1x+ a_0 \in \mathbb{R}[x]
$$
with real coefficients. The discriminant $\Delta_f$ of $f$ is a homogeneous polynomials of degree six ...
- 326
3
votes
0
answers
167
views
Bounding the $L^{p*}$ norm from below for functions satisfying a $p$-capacity estimate
If $1 \le p < n$, the $p$-capacity of a compact set $A \subset \mathbb{R}^n$ with respect to an open set $U$ containing it is defined as $$\text{Cap}_p(A, U) := \inf \left\{\int_U |\nabla u|^p \, ...
2
votes
1
answer
113
views
Showing $\inf _{a \neq 0} \frac{\left\|a^2\right\|}{\|a\|^2}\leq \inf _{a \neq 0} \frac{\|\hat{a}\|_{\infty}}{\|a\|}$ in a commutative banach algebra
Suppose $A$ is a commutative Banach algebra, and let $u=\inf _{a \neq 0} \frac{\left\|a^2\right\|}{\|a\|^2}$, $v=\inf _{a \neq 0} \frac{r(a)}{\|a\|}$ ($r(a)$ is the spectral radius of $a$). I need to ...
- 109k
2
votes
0
answers
126
views
Identification of Maharam extension
All definitions used in this post are from Björklund, Kosloff, and Vaes - Ergodicity and type of nonsingular Bernoulli actions. This post is inspired by the beginning of Section 2.2 in the same paper, ...
- 307
10
votes
0
answers
225
views
Can the trace be computed in any Schauder basis?
I'm cross-posting this question from Math.SE, as it didn't get much attention there.
Let $H$ be a separable Hilbert space and $T \in L(H)$ a trace-class operator. It is well known that the trace of $T$...
- 12.9k
3
votes
0
answers
116
views
On a functional equation of Mahler?
Recently, I was trying to introduce the concept of natural boundaries to a fellow math student, and what greater way to do this than using an example? In particular, I tried to use as an illustration, ...
- 131
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 ...
2
votes
1
answer
130
views
A function with one partial derivative Hölder continuous is Hölder continuous?
I'm having trouble finding a function of two variables, say $u(t,x)$, such that for some $\alpha\in ]0,1]$
1. $(t,x)\mapsto \partial_x^2 u(t,x)$ is $C^{0,\alpha}$;
2. $(t,x)\mapsto \partial_x u(t,x)...
- 4,219
6
votes
1
answer
197
views
On elliptic operators on non-compact manifolds
Let $(M,g)$ be a (connected, oriented) Riemannian manifold and $E$ some finite-rank $\mathbb{R}$- or $\mathbb{C}$-vector bundle equipped with some (positive-definite) inner product on the level of (...
- 16.2k
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
4
votes
1
answer
227
views
Continuity upgrade for nonlinear maps
Let $E,F,G$ be topological vector spaces such that $F\subset G$ with continuous embedding.
By continuity upgrade I mean the following phenomenon: In some circumstances a continuous linear map $f:E\...
- 1,075
2
votes
0
answers
142
views
A $C^*$-algebra with the bidual $B(H).$
Let $H$ be a separable Hilbert space and let $A$ be a non-unital $C^*$-subalgebra in $B(H)$ such that the second dual $A^{**} \equiv B(H).$ Does $A$ coincide with the ideal of compact operators $K(H)?$...
0
votes
0
answers
73
views
Tight tail bounds for sums of random variables
Let $X_1, X_2, \dots$ be iid uniformly on $[0,1]$. Define $Z_i^{(a)} = (X_i - a)^2$. Let $Y_n = \sum_{k=1}^n Z_k^{(1/k)}$. I am interested in matching tail bounds for $Y_n$ as $n \to \infty$. In ...
- 63.9k
4
votes
1
answer
145
views
Asymptotic decay rate of an oscillator integral
Question:
I want to evaluate the decay estimate of the integral
$I^d(t; v) = \int_0^{\sqrt{d}\pi} dr \, r^{d-2} \int_0^\pi \sin(tr) e^{i\sqrt{d}vtr\cos\theta} \sin^{d-2}\theta \, d\theta $
for ...
- 39.1k
3
votes
0
answers
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
2
votes
0
answers
58
views
An s-convex function lying between two convex functions
Let $f: \mathbb R_{+} \to \mathbb R_{+}$ be an $s$-function in the second sense, i.e.,
$$ f(\lambda x +(1-\lambda)y) \leq \lambda^s f(x) +(1-\lambda)^s f(y)$$ for every $\lambda \in (0,1)$. Assume ...
- 55
1
vote
2
answers
156
views
Numerical evaluation of monomial divided differences
Suppose $f(x)=x^{n+1}$ for some $n\in\mathbb{N}$, and define the divided difference $$f[a,b]=\frac{a^{n+1}-b^{n+1}}{a-b}.$$
I am wondering about the best way to numerically evaluate $f[a,b]$ to high ...
- 8,271
0
votes
0
answers
29
views
On constructing the canonical boundary operator for a given differential operator
Given an $n\times n$ matrix $$X=\begin{pmatrix}
x_{11} & x_{12} & \cdots & x_{1n} \\
x_{21} & x_{22} & \cdots & x_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
x_{n1}...
- 765
5
votes
0
answers
167
views
Bounding elementary symmetric polynomials away from zero
Let $2 \leq m \leq n$ be integers and let $\mathbf{x} \in \mathbb{R}^n$ (importantly, I am not assuming that the entries of $\mathbf{x}$ are non-negative). The elementary symmetric polynomials are ...
- 6,351
1
vote
2
answers
164
views
Existence of directional heat equation without uniform ellipticity
I am asking for references, or for a proof idea on how to show that weak solutions of the following problem exist: search $u$ on a bounded domain $\Omega\times (0,T]$, where $\Omega\subset\mathbb{R}^d$...
- 12.9k
3
votes
1
answer
79
views
How to deal with singularities in thin plate splines?
Follow up from this question Thin-Plate-Spline understanding and solution.
In the general case of $\mathbb{R}^N$ the following problem (interpolant which minimizes the Thin Plate Energy, specifically ...
- 346
7
votes
1
answer
253
views
Does a Banach algebra version of "the sum of a closed subspace and a finite dimensional subspace is always closed" exist?
In the setting of Banach spaces, it is well know that if $M$ is a closed subspace of a Banach space $X$ and $F$ is a finite dimensional subspace of $X$, then $M+F$ is closed.
Does a Banach algebra ...
- 2,830
1
vote
2
answers
231
views
A real root of a cubic equation for a stationary point
Let us consider the quartic polynomial in $x$
\begin{equation}
F(x) = (2 a p +2)x^4+ (6a(1-a)p^2+(6-12a)p-6)x^3
+ p(2(a-2)(a-1)a p^2 + 3(5a^2-9a+2)p +12a-18)x^2
- p^2 ((a-2)(4a^2 ...
- 371
1
vote
0
answers
88
views
Schauder estimate for $f \in L^\infty$
I was reading an article where at some point the author uses the following estimate:
Let $u$ be a solution of
$$\Delta u = f \quad \text{in } B_1$$
for $f \in L^\infty$. Then $u \in C^{1,1 - \...
- 452
0
votes
0
answers
54
views
Inequality between inverses of real functions
Let $s\geq 0$ and
$$
f(x)=-\log(x) \quad\text{an}\quad g(x)= \log(\log(1/x)+1)$$ for all $x\in(0,1)$. Is there exists $C_s>0$ such that for all $x,y\in(0,1)$,
$$
f^{-1}(s g(x)) \cdot f^{-1}(s g(y))...
- 39
54
votes
4
answers
12k
views
Everywhere differentiable function that is nowhere monotonic
It is well known that there are functions $f \colon \mathbb{R} \to \mathbb{R}$ that are everywhere continuous but nowhere monotonic (i.e. the restriction of $f$ to any non-trivial interval $[a,b]$ is ...
- 35.5k
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 ...
- 6,367
1
vote
0
answers
100
views
Difference of two completely monotonic functions
We know by the Hausdorff-Bernstein-Widder theorem that any completely monotonic function on the positive half line $[0, \infty)$ is given by the Laplace transform of a positive Borel measure on $[0, \...
CommunityBot
- 1
7
votes
5
answers
514
views
Probability of $\operatorname{Bin}(n,p)=\operatorname{Bin}(n,q)$ is decreasing when $n$ increases
$\newcommand{\Bin}{\operatorname{Bin}}$I would like to show that $\mathbb P(\operatorname{Binomial}(n,p) = \operatorname{Binomial}(n,q))$ decreases when $n$ increases for a fixed pair $(p,q)$. This ...
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,...
- 24.2k