All Questions
1,501 questions with no upvoted or accepted answers
6
votes
0
answers
107
views
Eigenvalues of splitting scheme
In numerical analysis it is common to approximate a solution to a PDE
$$u'(t) = (A+B) u(t), \quad u(0)=u_0$$
which is just given by $e^{t(A+B)}u_0$ by the splitting $e^{tB/2} e^{tA} e^{tB/2}u_0.$ Here,...
- 536
6
votes
0
answers
264
views
Odd Steinhaus problem for finite sets
Call a finite subset $S$ of the plane with an even number of points an odd Jackson set, if there is an $A\subset \mathbb R^2$ such that $A$ meets every congruent copy of $S$ in an odd number of points....
- 18.7k
6
votes
0
answers
445
views
Vector-valued interpolation for sublinear operators
Grafakos in his $\textit{Classical Fourier Analysis}$ formulates (see Exercise 4.5.2 therein) the following vector-valued version of the Riesz-Thorin interpolation theorem.
$\textbf{Theorem}$
Let $1\...
- 421
6
votes
0
answers
130
views
ultrametric Rademacher theorem
The classic Rademacher theorem roughly states that Lipschitz continuous functions are almost everywhere differentiable. However, there are well-known ultrametric counterexamples, see Kobliz's classic ...
- 500
6
votes
0
answers
267
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Convergence of $\sum_{n=1}^\infty x_n^k$
I thought that this question is more suitable for MSE, and asked it there. (Link to the MSE question) However, it does not get any answer despite the upvotes. It appears that I might have ...
- 1,755
6
votes
0
answers
176
views
Area-preserving map of punctured disk to itself
If $D_r = \{v\in \mathbb{R}^2 : 0 \lt |v| \lt r\}$, consider the map $f_r: D_r \to D_r$ given by:
$$f_r(x,y) = \frac{\sqrt{r^2-x^2-y^2}}{\sqrt{x^2+y^2}}\left(-y,x\right)$$
Geometrically, $f_r(v) \...
- 2,902
6
votes
0
answers
210
views
Generalized singular numbers and the Haagerup $L^p$ spaces
Let $M$ be a semi-finite von Neumann algebra with a trace $\tau$.Let $S(M)$ be the algebra of all affiliated operators measurable with respect to $M$.
The $L^p$ norm on $M$ is given by
\begin{...
- 361
6
votes
0
answers
2k
views
Interchange of integral and infimum
Can anyone please suggest how to justify widely used formula for interchange of integral and infimum:
$
\inf_{u(t)\in U}\int_{t_0}^{t_1}g(t,u(t))dt=\int_{t_0}^{t_1}\inf_{u\in U}g(t,u)dt,
$
where $ U\...
- 61
6
votes
0
answers
396
views
Recurrence Formula for Zernike polynomials
I'm not sure if this is research level, so if this result is known, please excuse the intrusion. I am trying to find a relation between solutions of the Laplacian equation in $4$ dimensions and those ...
- 263
6
votes
0
answers
206
views
Degree of Chebyshev polynomial necessary
In general, given $0<a<1$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[1-\frac{a}2,1+\frac{a}2]$ at every $x\in[1-a,1+a]$ and $f(0)=0$. What is minimum degree that is ...
- 13.9k
6
votes
0
answers
2k
views
Are planar Lipschitz curves countable unions of graphs?
More precisely:
Question:
Let $\gamma \colon [0,1] \to \mathbb{R}^2$ be Lipschitz. Do there exist Borel (or Suslin) sets $A_i \subset \mathbb{R}^2$ and directions $v_i \in \mathbb{R}^2$, for ...
- 3,270
6
votes
0
answers
223
views
Sum of product maximum
For which pairs of integers $(n,m)$ is the maximum of the following function $$f(x)=\sum_{i_1+\dots +i_n=m}\prod_{k=1}^n x^{i_k}_{k},\ \ x=(x_1,\dots,x_n), \|x\|=1$$ attained when $x_1=\dots=x_n$?
(...
- 259
6
votes
0
answers
8k
views
Dual space of continuous functions
Let $C_b(\Omega,V )=$ { $ f:\Omega\rightarrow V $ } is the Banach space of all bounded continuous functions in Banach space $V$ with a norm $\|\cdot\|$ defined as $\|f\|_\infty=\sup _{x\in\Omega}\|f(x)...
- 385
5
votes
0
answers
163
views
Does this weak omniscience principle have a name?
In constructive analysis, I'm looking at principles which follow both when there exists at least one discontinuous function from $\mathbb{R}$ to $\mathbb{R}$ (equivalent to WLPO i.e. $x > 0$ or $x \...
- 1,947
5
votes
0
answers
204
views
A proof for an $L^p$-$L^p$ inequality
This is a transfer of the question
https://math.stackexchange.com/questions/4996853/an-lp-lp-inequality
Let $a\in (0,1)$ and $1<p<\infty$ and use $L^{p}$ to denote the space $L^{p}([0,\infty))$ ...
- 852
5
votes
0
answers
156
views
What is the Hausdorff dimension of the set on which this exponential sum is bounded?
This is a direct follow up to For which rationals is this exponential sum bounded?
Given $x \in [0, 1]$, we denote by $e(x)$ the complex number $e^{2 \pi i x}$.
What is the Hausdorff dimension of the ...
- 6,155
5
votes
0
answers
285
views
How do you go about making ranges (for integer variables) independent?
Basic question: say you have a sum
$$\sum_{n_1 n_2 \dotsb n_k \leq x} f(n_1,\dotsc,n_k),$$
where $f$ decomposes in some sense (say: $f(n_1,\dotsc,n_k) = g(n_1) + \dotsb + g(n_k)$, or $f(n_1,\dotsc,n_k)...
- 20.2k
5
votes
0
answers
608
views
What is the correct $L^\infty$ limit of this strange variational problem, and what does it encode?
1. On the $L^\infty$ calculus of variations:
The field known as the $L^\infty$ calculus of variations is a relatively new field that concerns itself with minimising functionals involving the supremum ...
- 6,155
5
votes
0
answers
190
views
Number of discrete Lipschitz functions with given Lipschitz constant
Fix $T, K, N \in \mathbb Z_+$. How many distinct Lipschitz functions $f: \{0, \dots, T\} \to \mathbb Z$ are there with Lipschitz constant $K$, and supremum norm at most $N$ satisfying $f(0) = 0$?
In ...
- 6,155
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
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
5
votes
0
answers
107
views
Generalized Puiseux series for diagonal reflections of the curves $y = \frac{x}{(1-ax)(1-bx)^m}$
Reflection of the curve $y = f_m(x) = \frac{x}{(1-ax)(1-bx)^m}$ through the diagonal line $y=x$ in the $xy$-plane can be regarded as local compositional inversion of the curve $y=f_m(x)$. ($x,y,a,b$ ...
- 10.5k
5
votes
0
answers
213
views
Elliptic regularity and Sobolev spaces
Consider a linear partial differential operator $D:C^{\infty}(\mathbb{R}^{d})\to C^{\infty}(\mathbb{R}^{d})$, i.e.
$$D=\sum_{\alpha\in\mathbb{N}^{d}}a^{\alpha}(x)\partial^{\alpha}_{x}$$
where $a$ are ...
- 1,429
5
votes
0
answers
162
views
Closed formula for series $\sum_{i=1}^{\infty} \frac{1}{x^i-y^i}$
What can be said about $\sum_{i=1}^{\infty} \frac{1}{x^i-y^i}$ (for $|x|>1$ and $|y|>1$ and $x\neq y$)?
Is there a kind of closed formula for this?
By comparing to the geometric series, this sum ...
- 139
5
votes
0
answers
141
views
Maximum of a function
Let $p,q\in\Bbb N$ with $p\not=q$. Put $$M=\sup_{x\in[0,1]} \left|\cos(2 p\pi x)-\cos(2 q\pi x)\right|.$$
What is the value of $M$.
Thanks
- 501
5
votes
0
answers
417
views
All $L^pL^q$ estimates for the heat equation on $\mathbb R$ (with gain of derivatives)
I have asked this question on MSE, but this is a better place.
The heat equation and the heat kernel.
Consider the heat equation on $\mathbb R$:
$$ \left\{\begin{aligned}u_t-\Delta u&=f\\u(0,x)&...
- 1,075
5
votes
0
answers
135
views
Quantifying the degree of continuity of a function via perturbations
Let $f: \mathbb R \to \mathbb R$ be a measurable function. Define the perturbation operator $T_f$ on measurable functions $g: \mathbb R \to \mathbb R$ by
$$T_f (g)(x) := f(x + g(x)) - f(x).$$
Observe ...
- 6,155
5
votes
0
answers
132
views
Regularity of the spherical mean of a compactly-supported function
The problem
Consider a $C²$ function $f: X \to \mathbb{R}$, for some compact set $X \subset \mathbb{R}^d$ with $C^1$ boundary, say $\partial X$. I am only interested in $d\in \{2,3\}$.
Then, consider ...
- 555
5
votes
0
answers
158
views
Weaker versions of the Riemann series theorem in constructive mathematics
The classical Riemann series theorem states that given a sequence $(a_n)_{n \in \mathbb{N}}$ of real numbers such that the series $\sum_{n = 0}^\infty a_n$ is conditionally convergent, for all real ...
- 2,624
5
votes
0
answers
163
views
Is there a natural finitely additive measure for which Vitali sets have measure zero?
Vitali sets are nonmeasurable and in particular are not null sets. But all Vitali sets are in some sense small, as described below. Let $V$ be any Vitali set and let $k \in \mathbb{N}$. For each $i \...
- 303
5
votes
1
answer
630
views
Infinite dimensional involutions: infinitely large sets of multivariate polynomials self-inverse under self-substitution
Examples of infinite dimensional involutions
Edit 2/25/23, as suggested by YCOR below: (Start)
The first return on a Google search on involution--from late Latin 'a rolling up'--gives the Oxford ...
- 10.5k
5
votes
0
answers
652
views
Nature of function as $x\rightarrow\infty$
I'm studying the limits and applicability of Abel Plana summation for different test functions (class of functions). In doing so this just pops out and couldn't handle the said integral so asked here (...
- 784
5
votes
0
answers
252
views
How far can a continuous, almost everywhere differentiable function be from being a Sobolev function?
Let $\Omega$ be the open unit ball in $\mathbb R^n$. Consider the set $\mathcal D$ of continuous functions $f:\Omega \to \mathbb R$ that are differentiable a.e, and with $|\nabla f| \leq 1$ wherever $...
- 6,155
5
votes
0
answers
99
views
What is a mild sufficient condition on $X$ such that $C(X, Y)$ is sequential?
Let $X$ be a topological space, $(Y, d)$ a metric space and $C(X, Y)$ the space of continuous maps with the topology of compact convergence.
Question: What is a minimal topological condition on $X$ ...
- 315
5
votes
0
answers
233
views
Does there exist a “fat” Thomae’s function?
Definitions and some motivation:
Thomae’s function, also known as the raindrop function has several curious properties. One of which is the following - both its discontinuity set and continuity set ...
- 6,155
5
votes
0
answers
153
views
Inequality for functions on $[0,\infty)$
Let $0<q<1$ and $\varphi(q;x)=\displaystyle \prod_{j=0}^\infty (1+q^jx),\;x\geqslant 0.$ Consider the following functions:
$$l_k(x;q):=\frac{q^{k(k-1)/2} x^k}{(1-q)(1-q^2)\dots (1-q^k)\varphi(x;...
- 783
5
votes
0
answers
67
views
Minimal growth condition for a rearrangement
Let $\sigma: \mathbb{N}\to\mathbb{N}$ be bijective such that there is a sequence $(n_k)_{k\ge 0}$ in $\mathbb{N}$ satisfying $|\sigma(n_k)−n_k|\to\infty$ for $k\to\infty$.
Question: Is there a (...
- 2,160
5
votes
0
answers
113
views
Does there exist an injective Lipschitz map on the disk whose gradient switches between two given matrices?
While solving a problem in calculus of variations, I came to the following question:
Let $A,B$ be two real $2 \times 2$ matrices with positive determinants, and suppose that $\operatorname{rank}(A-B)=...
- 6,741
5
votes
0
answers
343
views
Can the inverse of the Riemann zeta function in $s > 1$ be expressed as a series?
In this post, we are interested in the Rimenann zeta function $\zeta(s)$ in $s > 1$ only where it is strictly decreasing rather than $s$ in the entire complex plane. We have the Stieltjes series ...
- 5,470
5
votes
0
answers
131
views
Is Sobolev limit of bijective maps surjective?
This is a cross-post.
Let $\Omega_1,\Omega_2 \subseteq \mathbb R^2$ be open, connected, bounded, with non-empty $C^1$ boundaries. Let $f_n:\bar\Omega_1 \to \bar\Omega_2$ be $C^1$ be bijective maps ...
- 6,741
5
votes
0
answers
140
views
Measure of the boundary of an BV-extension domain: do we have $|\nabla Eu|(\partial \Omega)=0?$
Let $\Omega\subset \Bbb R^d$ be open. The space $BV(\Omega)$ consists in functions $u\in L^1(\Omega)$ with bounded variation, i.e. $|u|_{BV(\Omega) }<\infty$ where
\begin{align}\label{eq:bounded-...
- 1,101
5
votes
0
answers
471
views
A stronger Cauchy-Schwarz in infinite dimensional Hilbert spaces?
In this MSE and question and this MO question, stronger variants of the classical Cauchy-Schwarz inequality have been suggested in finite dimensional spaces. Can we find similar results for infinite ...
- 597
5
votes
0
answers
221
views
Can we construct a computable sequence of trigonometric polynomials that converges pointwise to a given continuous function defined on the torus?
Consider any continuous function $f$ on an $m$-dimensional torus $\mathbb{T}^m$. Can we construct a sequence of band limited functions (trigonometric plynomials), with the band width (degree of the ...
- 698
5
votes
0
answers
163
views
Minimizing total variation
Let $f:\mathbb{R}\to \mathbb{R}$ be a function of bounded variation. Define $\overline{f}(x)$ by
$$\overline{f}(x) = \limsup_{\mu(I)\to 0} \frac{1}{\mu(I)} \int_I f(x) dx,$$ where $I$ ranges over ...
- 20.2k
5
votes
0
answers
192
views
Useful notion for locally convex spaces - well known?
In my current work the following property of maps between locally convex spaces showed up at several places and proved to be useful. It seems quite elementary to me, so I would like to know whether it ...
- 779
5
votes
0
answers
93
views
Points where singular sum is small
We consider $x_1,..,x_N$ points in the plane $\mathbb{R}^2.$
We define the sum
$$F(x):=\frac{1}{N^2}\sum_{i=1}^N \sum_{j \neq i} \vert x_i-x_j \vert^{-2}.$$
I am looking for a statement of the ...
- 183
5
votes
0
answers
137
views
Is finding positive integer solutions of $\zeta(a/b) = c$ equivalent to deciding the rationality of $\gamma$?
This question requires little bit of explanation of the background hence it is a bit lengthy. Note: The question was initially posted in MSE but did not get answers hence posting in MO.
For every ...
- 5,470
5
votes
0
answers
266
views
Hadamard lemma without integration
Let $I$ be the ideal of smooth germs vanishing at zero. Let $I^{k+1}$ be the ideal generated by $(k+1)$-fold product of such germs. Write $F_k$ for the ideal of $k$-flat germs at zero.
By the product ...
- 10.5k
5
votes
0
answers
205
views
Strange inequality relating Binomial pmf and cdf
I'm encountering a strange inequality I need to prove, relating the Binomial pmf and cdf.
Suppose we have $n$ coin flips, and fix an arbitrary $k \le n/2$ heads. Suppose further that we have some ...
- 101
5
votes
0
answers
89
views
Is the ratio of a number to the variance of its divisors injective?
The variance $v_n$ of a natural number $n$ is defined as the variance of its divisors. There are distinct integer whose variances are equal e,g. $v_{691} = v_{817}$. However I observed that for $n \le ...
- 5,470