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About the recursive inequality $w_p \geq (1-\frac {\pi}n)w_{p-2n} + 2\pi + o(1)$

Suppose we have a non-decreasing sequence of positive real numbers that tend to infinity: $0<w_1\leq w_2\leq w_3\leq...$ It is known that: For every $n$ and $p\geq 2n$, we have $w_p \geq (1-\frac {...
Adrian Chu's user avatar
0 votes
1 answer
139 views

Proving negativeness of function involving $-\log t$

I have been trying to solve the following function is non-increasing with respect $\theta$ \begin{equation} h(t,\beta) = \frac{1-t-\frac{\beta(-\log t)^{\theta}}{\theta(-\log \beta)^{\theta -1}}}{1-\...
MSquared's user avatar
2 votes
0 answers
99 views

Closed form for $\int_0^{+\infty} \ln^p(t) \frac{\sin^q(t)}{t^r}dt$

Do you know if there exists a closed form for the integral : $$I_{p,q,r} = \int_0^{+ \infty} \ln^p(t) \frac{ \sin^q (t)}{t^r} dt$$ where $p$, $q$, $r$ are natural integers such as this integral ...
Azoth's user avatar
  • 69
0 votes
0 answers
73 views

An example of a groupoid that satisfy the following hypothesis

In the paper titled, 'Tannaka–Krein duality for compact groupoids I, Representation theory', the author proves the Peter Weyl theorem on compact groupoids. In the statement, he gives the hypothesis ...
K N SRIDHARAN NAMBOODIRI's user avatar
2 votes
1 answer
133 views

Points of differentiability of convex functions

Let $U$ be an open neighbourhood of $0 \in \mathbb{R}^2$ and $f\colon U \to \mathbb{R}$ a convex (and bounded) function. Denote by $D \subset U$ the set of points on which $f$ is totally ...
AlexE's user avatar
  • 2,998
0 votes
0 answers
60 views

Criteria for log-absolute-monotonicity

Consider a function $f: [0,1] \rightarrow \mathbb R$ defined by a power series $f(x) = a_0 + a_1 x + a_2 x^2 + \dots$, where all $a_i$ are positive. Is there are any criterion in terms of the ...
David Harris's user avatar
  • 3,475
7 votes
1 answer
270 views

Sequential continuity and the Axiom of Choice

It is well-known that ZF cannot prove the following: "for a function $f$ from reals to reals and any real $x$, $f$ is continuous at $x$ if and only if $f$ is sequentially continuous at $x$."...
Sam Sanders's user avatar
  • 4,359
23 votes
4 answers
2k views

Identity for an infinite product

Here is an experimental "result" exhibiting the difference of two (formal) infinite products that "almost factorizes". QUESTION. Is this true? $$\prod_{n\geq1}(1+x^{2n-1})^{24} - \...
T. Amdeberhan's user avatar
2 votes
0 answers
85 views

Higher cohomology groups for the trivial action of the reals on themselves

For a freely generated countable abelian group $A$ with the trivial action on itself ($a\cdot b = b$) the resulting cohomology groups are well-known and eventually vanish (see e.g. here). Coming from ...
Ollie's user avatar
  • 1,411
8 votes
1 answer
361 views

Invertible matrix with group ring coefficient

Before asking the question I do need some notations. $G$ a (torsion-free) group, $\mathbb{Z}^{´}=\mathbb{Z}[\frac{1}{2}]$ $R:= \mathbb{Z}[G]$, $R^{´}=\mathbb{Z}^{´}[G]$ group rings. $Mat_{n}(R)$ the ...
GSM's user avatar
  • 223
-1 votes
1 answer
61 views

Asking for some references on correlations of joint optimization problems

Here are two problems that I am trying to understand, and it would be nice if someone could provide references on whether there is some structure theorem for these problems that have been studied in ...
Aaradhya Pandey's user avatar
7 votes
1 answer
346 views

Mean Cauchy sequences

Let $X$ be a complete metric space. Suppose a sequence of elements $x_n$ is Cauchy in mean, in the sense that $$\lim_{K \to \infty} \limsup_{N, M \to \infty} \frac{1}{NM} \sum_{i = K+1}^{K + N} \sum_{...
Nate River's user avatar
  • 6,155
2 votes
1 answer
315 views

Are surjective homogeneous maps open at zero?

I'm asking this question as a follow-up inspired by this one: An open mapping theorem for homogeneous functions? I'm actually wondering whether there exists an homogeneous map $f:\mathbb R^n\to\mathbb ...
Gil Sanders's user avatar
7 votes
1 answer
224 views

Does the decomposability of $\mathbb{R}$ imply analytic LLPO?

By "BISH" I mean constructive mathematics without axiom of countable choice. By $\mathbb{R}^f$ I mean real numbers as fundamental sequences of rational numbers and by $\mathbb{R}^d$ I mean ...
Mohammad Tahmasbi's user avatar
3 votes
1 answer
240 views

Solutions and asymptotics of the ODE $ f''=f^{-\alpha} $

Consider the ODE $ f''=f^{-\alpha} $, where $ \alpha>1 $ and $ f>0 $ in $ \mathbb{R} $. Assume that for $ [f]_{\frac{2}{\alpha+1}}\leq A $, where $ A>0 $ is a constant and $$ [f]_{\frac{2}{\...
Luis Yanka Annalisc's user avatar
0 votes
1 answer
153 views

Lebesgue measure of the level set of sum of two nonnegative functions

Let $f, g:\mathbb{R}^n\to \mathbb{R}$ be nonnegative functions such that $g$ is a strictly positive homogeneous function. As commented by Fedor Petrov below, one may not have that for any $\lambda>...
Ribhu's user avatar
  • 407
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 ...
Nate River's user avatar
  • 6,155
4 votes
1 answer
249 views

Does this functional admit an absolute minimizer?

This is a close relative of the following problem. Let $\Omega$ be an open, bounded subdomain of $\mathbb R^n$ with smooth boundary, and $f_i \in W^{1, \infty} (\Omega)$ a sequence of functions ...
Nate River's user avatar
  • 6,155
3 votes
2 answers
614 views

Should coffee machines be placed at the region's boundary?

This is a continuation of Should coffee machines be deconcentrated? Recall that some region is denoted by convex and compact $E\subset \mathbb R^2$. $N\ge 1$ coffee machines are provided for the ...
Fawen90's user avatar
  • 1,389
-1 votes
1 answer
122 views

Divergent summation [closed]

Let $(x_i)_{i=0}^\infty$ be a sequence such that $0<x_i<1\ \forall i \in \mathbb{N} \cup {0}$.Consider the following series: $$\sum_{i=1}^\infty \frac{x_i}{\left(\sum_{k=0}^{i-1} x_k \right)^2}.$...
Paul Deerock's user avatar
7 votes
1 answer
179 views

More on the Gram matrix of $6$ unit vectors in $\Bbb R^3$

Let $G=(g_{ij}\colon i,j=1,\dots,6)$ be the $6\times6$ Gram matrix of $6$ unit vectors in $\Bbb R^3$. Let $$u:=\sum_{1\le i<j\le 6}g_{ij}^2,\quad v:=\sum_{1\le i<j<k\le 6}g_{ij}g_{ik}g_{jk}.$$...
Iosif Pinelis's user avatar
2 votes
1 answer
231 views

Is Boltzmann entropy well-defined for arbitrary probability density function?

$\newcommand{\bR}{\mathbb{R}}\newcommand{\diff}{\mathop{}\!\mathrm{d}}$ We define a continuous function $\varphi : \bR_+ \to \bR$ by $$ \varphi (s) := \begin{cases} 0 &\text{if} \quad s =0 , \\ s \...
Akira's user avatar
  • 835
4 votes
1 answer
96 views

On the Gram matrix of $6$ unit vectors in $\Bbb R^3$

Let $G$ be the $6\times6$ Gram matrix of $6$ unit vectors in $\Bbb R^3$. Can the mean of the squares of the off-diagonal entries of $G$ be $<1/5$? Remark 1: A numerical experiment suggests that $...
Iosif Pinelis's user avatar
11 votes
2 answers
425 views

Maximization of a cubic form over the $14$-dimensional sphere

For any integers $i$ and $j$ such as $1\le i<j\le6$, let $x_{ij}$ be a nonnegative real number. Is it true that, given the condition $$\sum_{1\le i<j\le6}x_{ij}^2=1,$$ the sum $$\sum_{1\le i<...
Iosif Pinelis's user avatar
6 votes
2 answers
492 views

Does this polynomial have a real zero less than or equal to $1/2$?

Is the smallest root $x$ of $$ 10x^{3}-30x^{2}+\left(30-2\sum_{1\le i<j\le6}\cos^{2}\alpha_{ij}\right)x\\ +2\sum_{1\le i<j\le6}\cos^{2}\alpha_{ij}-\sum_{1\le i<j<k\le6}\cos\alpha_{ij}\cos\...
user avatar
4 votes
1 answer
110 views

Scaling of stopped Hölder norm of Brownian motion

I'm interested in the behaviour of the stopped $\alpha$-Hölder norm of a one-dimensional real-valued Brownian motion $(B_t)_{t \geq 0}$ for $\alpha < 1/2$. For fixed $T>0$, self similarity ...
user2103480's user avatar
-4 votes
1 answer
302 views

A Question in Fourier Analysis proposing a conjecture

Let $f$ be a $2\pi$ periodic BV function whose derivative is also BV.Let the amount of jump at a point $x$ is denoted as $\lfloor f \rfloor (x) = f(x+0)-f(x-0)$ Define function $J:\mathbb{R} \to\...
Rajesh D's user avatar
  • 698
2 votes
0 answers
58 views

$L^2$ approximation of delta functions on real algebraic varieties and asymptotic bounds

Let $X$ be a smooth projective variety over $\mathbb{C}$ of dimension $n$. Consider a probability measure $\mu$ on $X(\mathbb{R})$, absolutely continuous with respect to the Lebesgue measure induced ...
Raphael Riviera's user avatar
1 vote
0 answers
125 views

Relating singular homology of function spaces: a natural transformation from $C(\mathbb{R}, -)$ to $L^p(\mathbb{R}, -)$

Consider the category $\mathcal{Top}_*$ of pointed topological spaces and continuous basepoint-preserving maps. Let $C(\mathbb{R}, X)$ denote the space of continuous maps from the real line $\mathbb{R}...
user avatar
0 votes
0 answers
100 views

Construct a bi-Lipschitz mapping that maps a cube to a ball which has the same center with the cube

A mapping $f: \mathbb{R}^n\to \mathbb{R}^n$ is said to be $K$-bi-Lipschitz, $K>1$, if \begin{equation*} \dfrac{1}{K}\leqslant \dfrac{|f(x)-f(y)|}{|x-y|}\leqslant K, \end{equation*} for any $x,y\in \...
Javier's user avatar
  • 69
9 votes
3 answers
2k views

Smallest root of a degree 3 polynomial

Is it true that the smallest root $t$ of the polynomial $$ 20 t^3 - 30 t^2 + (12 - 4 \cos^2 \alpha - 4 \cos^2 \beta - 4 \cos^2 \gamma) t + \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma - 2 \cos \alpha \...
Venus's user avatar
  • 171
0 votes
0 answers
71 views

Nearest integer to fractional power series

Let $k$ be a positive integer. Let $$\displaystyle f_0(x) = a_n x^{\frac{n}{k}} + \cdots + a_1 x^{\frac{1}{k}} + a_0 + \sum_{h \geq 1} a_{-h} x^{-\frac{h}{k}}$$ be a Laurent series in the variable $x^{...
Stanley Yao Xiao's user avatar
8 votes
0 answers
103 views

Sobolev embedding theorems in vector bundles on non-compact manifolds

Let $(M,g)$ be a smooth (not necessarily compact) Riemannian $n$-manifold. It is well-known that dealing with Sobolev spaces in the general non-compact case becomes tricky, since for instance, there ...
G. Blaickner's user avatar
  • 1,429
1 vote
0 answers
46 views

Optimal transport and the geometry of singular measures on fractal Sets

Let $K$ be a self-similar fractal set in $\mathbb{R}^n$ with Hausdorff dimension $d < n$, equipped with a self-similar measure $\mu$ supported on $K$. Let $\mathcal{P}(K)$ denote the space of ...
danyerdos's user avatar
7 votes
0 answers
313 views

Did Lebesgue like non-measurable set or not?

I was surprised by the following paragraph in Bressoud's A radical approach to Lebesgue's theory of integration, quoted by Caicedo's in his comment to this question: Vitali's nonmeasurable set, ...
new account's user avatar
3 votes
0 answers
45 views

Small deviation asymptotics for sub-gaussian diffusions in dirichlet spaces

Let $(X,d,\mu)$ be a metric measure space equipped with a strongly local, regular Dirichlet form $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ on $L^2(X,\mu)$. Assume that the associated heat kernel $p_t(...
Thomas Frenkel's user avatar
2 votes
0 answers
65 views

Construct a differentiable function whose gradient has a prescribed modulus of continuity

$\newcommand{\bR}{\mathbb{R}}$ Let $\alpha := e^{-(1 + \sqrt{2})}$. We define the following modulus $\psi : \bR_+ \to \bR_+$ of continuity $$ \psi (x) := \begin{cases} 0 &\text{if} \quad x =0 , \\ ...
Akira's user avatar
  • 835
2 votes
0 answers
65 views

Generalized Fourier transforms associated to Schroedinger operators

Let $n\geq 1$. Let $q\in C^{\infty}_0(\mathbb R^n)$ be compactly supported and consider the operator $P= -\Delta+q(x)$ on $\mathbb R^n$. We will assume that $q$ is sufficiently small so that the ...
Ali's user avatar
  • 4,135
7 votes
0 answers
249 views

Proving this function is convex

Let $C$ be a symmetric positive definite matrix such that $0\leq c_{ij} \leq 1$, $c_{ii}=1$, and define $f$ as $$f(x)=\sum_{i}x_{i}\log(\sum_{j}c_{ij}x_{j})$$ for positive vectors $x$ (in fact let's ...
Tom Solberg's user avatar
  • 4,049
1 vote
0 answers
42 views

Approximation of the function $f(z)=z^2/|z|$ by $C^1$ immersions

Let $D$ denote the unit disk in $\mathbb C=\mathbb R^2$. We consider the function $f:D\rightarrow\mathbb C $ defined by $$f(z):=\frac{z^2}{|z|}.$$ Then as proved in Global invertibility (p324 Remark 4)...
Tian LAN's user avatar
  • 435
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 ...
Nate River's user avatar
  • 6,155
9 votes
1 answer
366 views

Can the canonical Eudoxus-real representatives be defined easily?

(See e.g. here for background on the Eudoxus reals, which motivates this question.) Let $\mathcal{Z}=(\mathbb{Z};+,<)$. Say that a Eudoxus function is an $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such ...
Noah Schweber's user avatar
0 votes
0 answers
101 views

A special Hamel basis and a special additive function

On mathstackexchange I recently asked whether for an irrational number $a$ a special Hamel basis of type $\bigcup_{i\in I}\{x_i,y_i,ay_i\}$ exists, where $x_i, y_i$ and $ay_i$ are $\mathbb Q$-...
ray's user avatar
  • 687
2 votes
1 answer
127 views

Density of smooth functions in weighted Sobolev space

Let $\rho(x)=e^{-\phi(x)}$, where $\phi$ is an even polynomial with positive leading coefficient. I am interested in a proof of the fact that the space of smooth compactly supported functions $\...
Bastien's user avatar
  • 23
20 votes
1 answer
2k views

How rich is the richest person in a society satisfying the Pareto principle?

The Pareto Principle roughly states that in many societies, the top 20% of people hold over 80% of the wealth. Suppose we had a society that satisfied this principle in every stratum of society - how ...
Nate River's user avatar
  • 6,155
0 votes
0 answers
52 views

References on a variant of Geometric Calculus

Geometric algebra and (standard) calculus, when synthesized, give rise to geometric calculus, a very powerful formalism. I have read a bit about fractional calculus and time-scale calculus, both very ...
user avatar
0 votes
1 answer
128 views

Characterizing the integral as a function of $n$

Let $\alpha \in [0,3], \beta \geq 1, \lambda \geq 1$ and fix $n \in \mathbb{N}$. Consider the function $f(x;\alpha, \beta, \lambda) = x^{\alpha}\exp(-\lambda x^\beta)$. Let $I(n; \alpha,\beta,\lambda) ...
yfful's user avatar
  • 25
2 votes
0 answers
120 views

On mollifiers acting between $L^2$ and Sobolev spaces

(I'm reposting here this question from MSE as it didn't receive any answer for two weeks.) Consider a sequence of finite lattices in $\mathbb{R}^n$ defined by $$L_k= [-k,k]^n \cap 2^{-k}\cdot \mathbb{...
S.Z.'s user avatar
  • 505
3 votes
1 answer
224 views

Extension of Sobolev function defined on unit cube

Im wondering about theorems concerning extending Sobolev functions defined on the $d$-dimensional unit cube to all of $\mathbb{R}^d$. More precisely, given $f:[0,1]^d \to \mathbb{R}$ with $f\in H^k([0,...
Jjj's user avatar
  • 93
10 votes
1 answer
1k views

A strange Lipschitz function

Let $n \geq 3$. Does there exist a Lipschitz function $f: \mathbb R^n \to \mathbb R$ such that the following conditions hold? The origin is a weak Lebesgue point of $\nabla f$, in the sense that the ...
Nate River's user avatar
  • 6,155