Skip to main content

All Questions

Filter by
Sorted by
Tagged with
6 votes
2 answers
225 views

On a trigonometric inequality by Huygens

The following inequality, ascribed to Huygens, appeared in this post: \begin{equation*} 1-\frac43\,\frac{\sin^3\theta/2}{\theta-\sin\theta} >(1-\cos\theta/2)\Big(\frac35-\frac3{1400}\frac{\...
Iosif Pinelis's user avatar
7 votes
1 answer
580 views

Sobolev spaces are smooth? Their dual is strictly convex?

Do you know any reference which says something about the: Smoothness of the Sobolev space $W^{1,p}(\Omega)$ i.e. if the duality mapping $J\colon W^{1,p}(\Omega)\to W^{1,p}(\Omega)^*$ is a singleton. ...
Bogdan's user avatar
  • 1,759
4 votes
0 answers
284 views

Institutional approach to linear algebra

In Diaconescu's book Institution Independent Model Theory, it is mentioned on p. 37 that linear algebra can be viewed as an institution. Specifically, we have the following Definition. An institution ...
Alec Rhea's user avatar
  • 10.1k
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 ...
Ryan Hendricks's user avatar
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,...
Ryan Hendricks's user avatar
0 votes
1 answer
255 views

Carleson's theorem: proof of a lemma

I am reading the paper of Michael Lacey called "Carleson's theorem: proof, complements, variations" 1, on Carleson's theorem in Fourier analysis. At the bottom of page 20 at the beginning of ...
Alexander's user avatar
7 votes
5 answers
513 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 ...
YuiTo Cheng's user avatar
4 votes
1 answer
255 views

Asymptotic behavior and of an integral on a d-dimensional torus

I am trying to evaluate the asymptotic behavior of the following integral as $t \to \infty$: $$ I(t; \mathbf{v}) = \int_{[-\pi, \pi]^d} \frac{\sin(t f(\mathbf{k}))}{\sin(f(\mathbf{k}))} e^{i t \mathbf{...
Ko Hey's user avatar
  • 81
1 vote
1 answer
62 views

Integrability in the product space can follow from a property of the Nemytskii operator?

Let's say that $f:\Omega\times\mathbb{R}\to\mathbb{R}$ is 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 ...
Bogdan's user avatar
  • 1,759
0 votes
0 answers
115 views

Integral of a measurable function with parameter is measurable?

Say that $f:\Omega\times\mathbb{R}\to\mathbb{R}$, where $\Omega\subset\mathbb{R}^N$ is an open set, is a function such that: $f(x,\cdot)\in L^1_{\text{loc}}(\mathbb{R})$ for a.a. $x\in\Omega$ $f(\...
Bogdan's user avatar
  • 1,759
1 vote
0 answers
175 views

Solution of recurrence relation with summation

I have the following recurrence relation: $$b(n,k)=\sum _{\text{i}=0}^{2 n-1} \left(b(n-1,k-\text{i})+\frac{\text{i} (2 n-\text{i}) \binom{2 n-1}{\text{i}} \binom{(n-2)^2}{k-\text{i}}}{2 n-1} \right)$$...
Cardstdani's user avatar
0 votes
0 answers
63 views

Arrangements of fixed $k$-polyplets in a $n\times n$ matrix

Recently, I asked a question about the number of arrangements of $k$ elements inside a $n\times n$ matrix with certain restrictions. The one I´m actually interested in for this question is in its 2. ...
Cardstdani's user avatar
0 votes
0 answers
79 views

Is the Bures metric equivalent to the Euclidean one?

Let $K=\mathbb R$ (reall numbers) or $K=\mathbb C$ (complex numbers). Define $\mathcal M_n$ to be the space of $n\times n$ matrices $A=(a_{i,j})_{1\le i,j\le n}$, with $a_{i,j}\in K$. Let $\|\cdot\|$ ...
GJC20's user avatar
  • 1,334
2 votes
1 answer
246 views

Ramsey type property of the Lipschitz constant

The following problem was proposed by Pietro Majer as an extension of an earlier question of mine on Lipschitz functions. For $f$ a Lipschitz function on $\mathbb R^n$, we denote by $$\text{Lip}(f, U) ...
Nate River's user avatar
  • 6,155
4 votes
1 answer
254 views

On the Lipschitz constant outside the stretch set

Let $f: \mathbb R^n \to \mathbb R^m$ be a Lipschitz map. We define the local Lipschitz constant $Lf$ of $f$ at $x \in \mathbb R^n$ by $$Lf(x) := \lim_{r \to 0_+} \text{Lip}(f, B_r (x)),$$ where $\text{...
Nate River's user avatar
  • 6,155
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 ...
Vladimir's user avatar
  • 371
1 vote
1 answer
204 views

A question on Borel measurability

Let $(X, \mathcal{B}_{X}, \mu)$ be a measure space. Here, $\mu$ is an infinite Borel measure and $\mu$ is not $\sigma$-finite. Let $\pi$ be surjective Borel measurable map form $(X, \mathcal{B}_{X}, \...
bobscott's user avatar
5 votes
1 answer
279 views

Is there a theorem which provides conditions under which a power series satisfies the reciprocal root sum law?

Kalman - Six ways to sum a series discusses Euler's original proof for the Basel problem $\sum\limits_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6} $: $$\frac{\sin(\sqrt x)}{\sqrt x} = 1- \frac{x}{3!}+ \...
pie's user avatar
  • 541
5 votes
1 answer
229 views

Intersection between Lipschitz domains

Let $\Omega\subset\mathbb{R}^N$ be an open, bounded and connected Lipschitz domain. Is it true that we can find some $R>0$ such that any $N$-dimensional open ball $B(x,r)$ with $r\leq R$ that ...
Bogdan's user avatar
  • 1,759
3 votes
0 answers
95 views

Is it true that p-integrable function can be written as a convolution of an integrable function and p-integrable function?

We know that convolution of an integrable function with an $p$-integrable is an $p$-integrable function. This follows from Young's inequality. My question: Is it true that $L^p(\mathbb{R}^n)\subseteq ...
user531870's user avatar
7 votes
1 answer
290 views

Equivalence of omniscience principles for natural numbers and analytic omniscience principles for Cauchy real numbers

In constructive mathematics, a proposition $P$ is decidable if $P \vee \neg P$, and a proposition is stable if $\neg \neg P \implies P$. We have the following principles of omniscience for the natural ...
Madeleine Birchfield's user avatar
8 votes
0 answers
414 views

For $f$ Lipschitz with $|\nabla f| = 1$ a.e., what is the supremal Hausdorff dimension of the set on which $\varepsilon< |\nabla f| < 1-\varepsilon$?

Let $f$ be a Lipschitz function with $|\nabla f| = 1$ almost everywhere. Let $\varepsilon \geq 0$. What is the supremal Hausdorff dimension of the set on which $f$ is differentiable with $\varepsilon &...
Nate River's user avatar
  • 6,155
0 votes
0 answers
66 views

convolution of the fundamental solution with the homogeneous solution

I have a question about the convolution of the fundamental solution with the homogeneous solution. Namely if the 2 are convoluble then the homogeneous solution is necessarily zero? Let $U$ and $E$ ...
Alucard-o Ming's user avatar
2 votes
1 answer
206 views

Deriving an inequality for the integral of maximum indicator functions under measure-preserving transformations

Let's denote the measure space by $(X, \mathcal{B}, \mu)$ and the measure-preserving transformation by $T: X \to X$. Let $A \in \mathcal{B}$ be a measurable set with $0 < \mu(A) < \infty$. Let $...
abcdmath's user avatar
  • 105
1 vote
1 answer
60 views

Does Monotone (linear) convergence of iterates imply monotone (linear) convergence of function values?

I am considering a proof that would require a certain connection between convergence of iterates and corresponding function values: Consider an algorithm with iterates $\left\{{\mathbf{x}}^k\right\}_{...
AY Wer's user avatar
  • 13
1 vote
1 answer
93 views

Bound measure of difference of advected sets by norm of difference of vector fields

Consider two smooth vector fields $v$ and $u$ in $\mathbb{R}^n$, and a smooth set $\Omega$. Consider the flow of $\Omega$ via $v$ and $u$ for a time $T$, namely let $$ \Omega_v =\{x(T, x_0) | x \text{ ...
tommy1996q's user avatar
2 votes
1 answer
211 views

Hölder continuity in time of heat semigroup for regular initial distribution

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\diff}{\mathop{}\!\mathrm{d}} $ Let $(p_t)_{t>0}$ be the standard Gaussian heat kernel on $\bR^d$, i.e., $$ p_t (x) := \frac{1}{(4 \pi t)^{\frac{d}{2}}} \...
Akira's user avatar
  • 835
3 votes
1 answer
263 views

Hölder continuity in time of heat semigroup

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\diff}{\mathop{}\!\mathrm{d}} $ We fix $\alpha \in (0, 1)$ and $c>0$. Let $\ell : \bR^d \to \bR_+$ be a probability density function such that $$ \|\ell\|...
Akira's user avatar
  • 835
0 votes
0 answers
54 views

Weyl equidistribution for a periodic $L^2$ function

Let $\alpha $ be a fixed irrational number. For a function $g:\Bbb R\to\Bbb C$, define $$g^*(x)=\sup_{N\geq 1} \frac{1}{N} \sum_{n=1}^N |g(x+\alpha n)| ,$$ and assume that there is a constant $C>0$ ...
blancket's user avatar
  • 213
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 ...
Fawen90's user avatar
  • 1,389
3 votes
1 answer
248 views

Can any function in $C^\alpha$ be approximated in $C^{\alpha^-}$ by singular functions?

For every positive $\alpha < 1$, we consider the space $C^{\alpha}$ of Holder continuous functions of order $\alpha$ on $[0, 1]$, equipped with the norm $$\|f\|_{C^\alpha} := \sup|f| + \sup_{x, y \...
Nate River's user avatar
  • 6,155
1 vote
1 answer
120 views

Does Gaussian heat kernel ensure $\int_{\mathbb R^d} (1+|x|) \sqrt{\ell_{t_0} (x)} \, \mathrm d x < \infty$?

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\diff}{\mathop{}\!\mathrm{d}} $ Let $\ell : \bR^d \to \bR_+$ be a probability density function such that $$ \int_{\bR^d} (1+|x|) \sqrt{\ell (x)} \diff x < ...
Akira's user avatar
  • 835
17 votes
2 answers
1k views

Is it consistent with ZFC that the real line is approachable by sets with no accumulation points?

Let $P$ denote the following proposition: There exists a set $S$ of subsets of $\mathbb{R}$ such that $S$ is totally ordered by inclusion; each member of $S$ has no accumulation points; the union of ...
Julian Newman's user avatar
3 votes
1 answer
102 views

Literature containing basic knowledge of homogeneous functions

Let $D$ be a nonempty open subset of $\mathbb{R}\times\mathbb{R}$ and $f:D\to\mathbb{R}$ be a function of two variables. For all $(x,y)\in D$ and $t>0$ such that $(tx,ty)\in D$, if the equality $f(...
qifeng618's user avatar
  • 1,091
2 votes
1 answer
836 views

Does $\int_{\mathbb R^d} (1+|x|^{1 + \alpha}) \ell (x) \, d x < \infty$ imply $\int_{\mathbb R^d} (1+|x|) |\ell (x)|^{1-\alpha} \, d x < \infty$?

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bE}{\mathbb{E}} \newcommand{\diff}{\mathop{}\!\mathrm{d}} $ We fix $\alpha \in (0, 1)$. Let $\ell : \bR^d \to \bR_+$ be a continuous function such that $$ \|...
Akira's user avatar
  • 835
9 votes
1 answer
553 views

Does the sequence formed by Intersecting angle bisector in a pentagon converge?

I asked this question on MSE here. Given a non-regular pentagon $A_1B_1C_1D_1E_1$ with no two adjacent angle having a sum of 360 degrees, from the pentagon $A_nB_nC_nD_nE_n$ construct the pentagon $...
pie's user avatar
  • 541
1 vote
1 answer
114 views

Ensuring the measure condition $\mu(E) = \lambda$ in a lemma: need some clarification regarding the selection of $A$

I was studying a lemma from my notes on ergodic theory and encountered a difficulty. The lemma states: Let $(X, \mathcal{B}, \mu)$ be an infinite non-atomic measure space, and let $T$ be an ergodic ...
abcdmath's user avatar
  • 105
2 votes
2 answers
158 views

Does there exist a continuous field of directions in $\mathbb R^3$ tangent to every sphere?

Does there exist a nonconstant continuous map $v: \mathbb R^3 \to \mathbb S^2$ such that every sphere $S \subset \mathbb R^3$ is tangent to $v(x)$ at some $x \in S$? Bonus: I also suspect that for ...
Nate River's user avatar
  • 6,155
0 votes
1 answer
106 views

Convergence of mollified functions in weighted $L^p$ norm

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bE}{\mathbb{E}} \newcommand{\supp}{\operatorname{supp}} $ Let $(\rho_n)_{n \geq 1}$ be a sequence of mollifiers on $\bR^d$, i.e., each $\rho_n$ is a ...
Akira's user avatar
  • 835
1 vote
1 answer
39 views

Does uniform convergence of suitable functions yield pathwise convergence of their convex envelopes?

For each $k\ge 1$, let $f_k:\mathbb R\to\mathbb R_+$ be $1-$Lipschitz, increasing such that $f_k(x)\ge x^+$ for $x\in\mathbb R$, $f_k(-\infty)=0$ and $$\lim_{x\to+\infty} \big(f_k(x)-x\big)=0.$$ ...
GJC20's user avatar
  • 1,334
11 votes
1 answer
953 views

Can a differentiable function have everywhere discontinuous derivative?

For $n \geq 2$, let $f: \mathbb R^n \to \mathbb R$ be differentiable. Is it possible that $\nabla f$ is everywhere discontinuous? I believe in dimension $1$, $\nabla f$ has to be continuous on a dense ...
Nate River's user avatar
  • 6,155
4 votes
1 answer
259 views

Hausdorff dimension of the zero set of the gradient of an eikonal function

Let $f: \mathbb R^n \to \mathbb R$ be a Lipschitz function with $|\nabla f| = 1$ almost everywhere with respect to Lebesgue measure. What is the supremal Hausdorff dimension of the set on which $f$ is ...
Nate River's user avatar
  • 6,155
3 votes
0 answers
219 views

Strictly contracting solutions to the Eikonal equation on Riemannian manifolds

Given a Riemannian manifold $M$, we say $f: M \to \mathbb R$ is a strict contraction if $|f(x) - f(y)| < |x - y|$ for all distinct $x, y \in M$. Question: Does there exist, on every complete ...
Nate River's user avatar
  • 6,155
5 votes
2 answers
248 views

Hausdorff dimension of the zero set of $\nabla f$

Let $f: \mathbb R^n \to \mathbb R$ be a Lipschitz function with $\nabla f$ nonzero almost everywhere with respect to Lebesgue measure. What is the supremal Hausdorff dimension of the set on which $f$ ...
Nate River's user avatar
  • 6,155
2 votes
2 answers
151 views

Upper bound $\int_{\mathbb{R}^d \times \mathbb{R}^d} |fx)-f(y)| (1+|y|) \ell (x) p_t (x-y) \, \mathrm d x \, \mathrm d y$ in $t$

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\diff}{\mathop{}\!\mathrm{d}} $ We fix $\alpha \in (0, 1)$ and $c>0$. Let $f : \bR^d \to \bR$ and $\ell : \bR^d \to \bR_+$ be measurable such that $\ell$ ...
Akira's user avatar
  • 835
4 votes
0 answers
88 views

A question concerning regularly varying functions

In my work I need some results about regulary varying functions, which I only have a very vague understanding. A strongly related reference I found is "On the Existence of a Regularly Varying ...
Xueping's user avatar
  • 119
5 votes
1 answer
374 views

Looking for a counterexample: Conditioning increases regularity?

Let $p(x,y,z)$ be a joint density (over $\mathbb{R}^3$) under no smoothness or regularity assumptions, besides its existence. I am looking for a (counter)example where $p(y|x)$ is less regular than $p(...
user5034's user avatar
4 votes
1 answer
446 views

Is the uniform limit of "almost eikonal" maps eikonal?

Let $f_n: \mathbb R^d \to \mathbb R$ be continuously differentiable functions with $f_n \to f$ uniformly for some $f$. Suppose that $|\nabla f_n| \to 1$ uniformly. Is it true that $f$ is $C^1$ with $\...
Nate River's user avatar
  • 6,155
2 votes
1 answer
179 views

Is the average of a $\alpha$-Hölder process Hölder continuous of every order less than $\alpha$?

Let $X_t$ be a stochastic process on $[0, 1]$ that is almost surely Hölder continuous of order $\alpha > 0$, and almost surely uniformly bounded by some deterministic constant. It is not hard to ...
Nate River's user avatar
  • 6,155
3 votes
0 answers
318 views

The curse of dimensionality of the Kolmogorov–Arnold neural network

The Kolmogorov–Arnold neural networks (KAN), Ziming Liu et al., KAN: Kolmogorov–Arnold Networks is inspired by the Kolmogorov–Arnold representation theorem (KA theorem). Though it is not proved in the ...
Hans's user avatar
  • 2,239

1
3 4
5
6 7
119