All Questions
5,788 questions
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 ...
- 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(...
- 51
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 $\...
- 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 ...
- 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 ...
- 2,239
7
votes
1
answer
561
views
How are real numbers defined in elementary recursive arithmetic?
I am currently reading about elementary function arithmetic and Harvey Friedman's grand conjecture.
In Number theory and elementary arithmetic, Jeremy Avigad expressed Fermat's last theorem, ...
user507115
2
votes
0
answers
88
views
Dependence and $L^2$ projections of functions
tl;dr: Is it possible that the best approximation to a nonnegative function of three variables with a bivariate function is no better than the best univariate function?
Let $w$ be a density on $\...
- 21
6
votes
1
answer
413
views
Analyticity of $f*g$ with $f$ and $g$ smooth on $\mathbb{R}$ and analytic on $\mathbb{R}^*$
Suppose that we have two real functions $f$ and $g$ both belonging to $\mathcal{C}^\infty(\mathbb{R},\mathbb{R})$ analytic on $\mathbb{R}\setminus\{0\}$ but non-analytic at $x=0$. Is the convolution (...
- 393
4
votes
1
answer
217
views
$2$ continuous, commuting functions doesn't always have a common fixed point
The question is as such:
If two continuous mappings $f$ and $g$ of a closed interval into itself commute, that is, $f\circ g=g\circ f$, then they do not always have a common fixed point.
-- Zorich ...
- 183
0
votes
0
answers
63
views
A maximisation problem : finite or not?
Let $\mathcal M_2$ be the space of real $2\times 2$ matrices and $\mathcal S_2\subset \mathcal M_2$ be its subset consisting of positive semidefinite elements, i.e. $A\in \mathcal S_2$ iff $A$ is ...
- 1,389
3
votes
0
answers
138
views
What is the probability that the absolute value of the root of a polynomial is greater than $x$?
Note: This question was unanswered in MSE for a month so posting it in MO.
Let $f(x) = 0$ be an equation of degree $n$. WLOG we can assume that the its coefficients are in $(-1,1)$. This is because we ...
- 5,470
0
votes
0
answers
21
views
Unimodality of distribution from Lévy symbol
Also posted in MSE.
Assume that one want to study a distribution $f$ on $\mathbb{R}$ for which the Lévy symboln, i.e.:
$$
\forall u\in\mathbb{R},\quad\psi(u) := \log \mathbb{E}\left[e^{iuX}\right]
$$
...
- 393
2
votes
1
answer
110
views
Equivalence among these functions
Let $\Phi$ be the CDF of a standard Gaussian distribution, i.e.
$$\Phi(x):=\int_{-\infty}^x \frac{1}{\sqrt{2\pi}} e^{-y^2/2}dy,\quad \forall~ x\in \mathbb R.$$
Denote by $\Phi^{-1}$ its inverse ...
- 1,389
3
votes
0
answers
212
views
Differentiability along hyperplanes for rational functions
This is a follow up to my previous question.
Let $f\colon \mathbb R^3\to \mathbb R$ be a continuous function that is rational and differentiable along all planes through $0$, that is, we assume:
...
- 779
9
votes
1
answer
492
views
Dispersion points of Lipschitz functions
For a function $f: \mathbb R^n \to \mathbb R^m$ with $m < n$, we say that $x \in \mathbb R^n$ is a dispersion point of $f$ if
$$\liminf_{y \to x} \frac{|f(y) - f(x)|}{|y - x|} > 0.$$
Question: ...
- 6,155
2
votes
0
answers
160
views
An "almost" true inequality for Hermitian matrices
Let $A$ be an $N\times N$ Hermitian matrix. For $p+q$ even, consider the following inequality:
$$\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} (A^q)_{ii} \geq \Big(\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} \Big) \Big(\...
- 593
3
votes
1
answer
193
views
Differentiability along hyperplanes
Definition. Let us say that a function $f\colon \mathbb R^d\to \mathbb R$ is differentiable along hyperplanes in the point $0\in \mathbb R^d$, if $f\circ \varphi\colon \mathbb R^{d-1}\to \mathbb R$ is ...
- 779
1
vote
1
answer
161
views
An "almost" geodesic dome
A regular $ n$-gon is inscribed in the unit circle centered in $0$.
We want to build an "almost" geodesic dome upon it this way: on each side of the $n$-gon we build an equilateral triangle ...
- 387
1
vote
0
answers
56
views
Extension of this maximisation problem : finite or not?
$\mathcal M$ is the space of real $d\times d$ matrices and $\mathcal S\subset \mathcal M$ is its subset consisting of positive semidefinite elements. We consider the distance the product space $\...
- 1,389
7
votes
1
answer
271
views
Can a differentiable function be nowhere locally $\alpha$-Hölder for all $\alpha > 0$?
Does there exist a real valued function on $[0, 1]$ that is differentiable everywhere, but for every $\alpha > 0$ is nowhere locally $\alpha$-Hölder continuous? That is, it is not $\alpha$-Hölder ...
- 6,155
0
votes
0
answers
56
views
How explicit the optimiser of this optimisation problem can be?
Provided the given parameters as follows :
$\mu\in\mathbb R, \sigma\in\mathbb R_+$ are constant, $\kappa, r, \alpha, \beta: \mathbb R_+\to\mathbb R_+ $ are measurable functions such that $\kappa(y)\...
- 1,334
6
votes
3
answers
851
views
Almost everywhere-periodic functions with many periods
Let $f : \mathbb{R} \to \mathbb{R}$ be a Lebesgue measurable function and $D$ be a countable dense subset of $\mathbb{R}$.
Suppose that for a.e. $x \in \mathbb{R}$ we have
\begin{equation*}
f(x + d) = ...
2
votes
1
answer
360
views
Asymptotics of an oscillatory integral
For $a > 0$ and $n \in \mathbb Z_+$, consider the oscillatory integral
$$\int_{0}^1 f(x) f(ax) \dots f(a^n x) \, dx,$$
where $f$ is an integrable function on $[0, 1]$, which we extend by ...
- 6,155
7
votes
2
answers
178
views
Separating domains in $\mathbb{R}^{2n}$ by a real algebraic variety
Suppose $\Omega_1$ and $\Omega_2$ are two disjoint unbounded domains in $\mathbb{R}^{2n}$, $n \in \mathbb{N}$. Can there be conditions on $\Omega_1$ and $\Omega_2$ so that these two domains can be ...
- 171
4
votes
1
answer
128
views
Lower bound of mean curvature implies that the set is subset of a given ball
If a simply connected set $\Omega\subset\mathbb{R}^n$ has $C^2$ boundary such that the mean curvature $H$ of $\partial \Omega$ satisfies:
$$H\geq 1$$
Does this imply that $\Omega\subset B_1$ after ...
- 421
2
votes
0
answers
80
views
Stability of Hölder constants of frozen Itô stochastic integrals
$
\newcommand{\RR}{\mathbb{R}}
\newcommand{\TT}{\mathbb{T}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\EE}{\mathbb{E}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\PPP}{\...
- 835
20
votes
1
answer
2k
views
Is $1/F$ Schwartz if $F$ is "reverse Schwartz"?
Let's call a positive function $F:\mathbb{R}\to\mathbb{R}$ "reverse Schwartz" if $F$ is smooth and
$$\forall n \forall k,\quad\lim_{x\to\infty}\frac{|x|^n}{|\partial_x^k F(x)|}=0\quad .$$
In ...
- 23.3k
4
votes
1
answer
205
views
Existence of an $\alpha$-Hölder continuous function whose graph has positive Hausdorff measure of maximal dimension
It is standard that if $f:[0,1] \rightarrow \mathbb{R}$ is $\alpha$-Hölder continuous, then its graph has Hausdorff dimension at most $2-\alpha$. My naive expectation was that "most" graphs ...
- 45
1
vote
1
answer
118
views
Is there $\varepsilon \in (0, 1)$ such that $\sup_{t \in [0, \varepsilon]} [\ell_t]_\beta < \infty$?
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bT}{\mathbb{T}}
\newcommand{\bN}{\mathbb{N}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\bD}{\mathbb{...
- 835
6
votes
1
answer
816
views
Is the $L^\infty$ norm of the derivative the same under the Hausdorff and Lebesgue measure?
Note: Here $\mathcal H^k$ denotes the $k$-dimensional Hausdorff measure, and $\|f\|_{L^\infty (\mathcal H^k)}$ denotes the $L^\infty$ norm of a function $f$ with respect to $\mathcal H^k$.
Let $\Omega$...
- 6,155
0
votes
0
answers
143
views
A Poincaré inequality holds for $p>2$ but fails for $p\leqslant 2$
I am confused with the following example taken from page 6 of Sobolev Met Poincaré, by Hajłasz and Koskela (MR1683160, Zbl 0954.46022).
Let $(X,d,\mu)$ be a metric measure space and let $\Omega\subset ...
- 69
3
votes
0
answers
146
views
Two algebraically independent irrational numbers $\alpha,\beta$ s.t. $\alpha^\beta$ is a rational number
Are there two algebraically independent irrational numbers $\alpha,\beta$ s.t. $\alpha^\beta$ is a rational number?
- 356
5
votes
1
answer
245
views
Are singular functions dense in the space of Hölder continuous functions?
We say a non-constant function $f$ on $[0, 1]$ is singular if it is continuous, and in addition differentiable almost everywhere with $f' = 0$ a.e.
For every positive $\alpha < 1$, is the set of ...
- 6,155
0
votes
0
answers
125
views
Has anyone seen such a function/quantity?
I am dealing with a problem wherein I encounter the following quantity-
$$
Q_{d, \epsilon}(t_0) = \sup_{t' \notin B(t_0, \epsilon)} \inf_{t \in B(t_0, \epsilon)} \frac{d(t') - d(t)}{t'-t}.
$$
Here,...
- 11
4
votes
1
answer
100
views
Limits along lines for the gradient of a convex function
It is easy to see that if a function $f: \mathbb{R} \to \mathbb{R}$ is strictly convex, $C^1$ and $f'$ has bounded image, then as $t\to \infty$ the limit
$$
\lim_{t\to\infty} f'(t) = \lim_{t\to\infty} ...
- 89
8
votes
2
answers
509
views
Condition to guarantee that an inhabited and bounded set of reals has a supremum
This question is about constructive mathematics (without Choice), such as in the internal logic of a topos with natural numbers object, or in IZF. The “reals” (and the symbol $\mathbb{R}$) refer to ...
- 32.5k
-1
votes
1
answer
223
views
Centroid of $\Omega$ and $\partial\Omega$ concides then $\Omega$ must be a ball
Hi I just happened to have a small question. If we have
$$\frac{\int_\Omega x}{|\Omega|}=\frac{\int_{\partial\Omega} x}{|\partial\Omega|}$$
for a simply connected set $\Omega$ with analytic boundary. ...
- 421
1
vote
1
answer
101
views
Unique continuation property of the equation $ -\Delta u=|u|^{p-1}u $ with $ p>2 $
Assume that $ \{u_i\}_{i=1}^{2} $ satisfies $ -\Delta u_i=|u_i|^{p-1}u_i $ in $ B_1 $ with $ p>2 $ and $ u_1=u_2 $ in an open set $ A\subset B_1 $. I want to ask that if $ u_1=u_2 $ in $ B_1 $.
...
- 1,219
0
votes
0
answers
86
views
Solve equation with three square roots
I am trying to solve a more general question and I have the following subproblem:
Find $x>0$ that satisfies for fixed $ i \geq 3$,
$$\left(1 + \frac{1}{b^2}\right) x = \frac{\sum_i a_i^2} {b^2} + \...
- 49
2
votes
1
answer
106
views
Lower bounds for the expectation of log ratio between the posterior and prior Beta densities
The quantity I'm interested in is expressed as follows:
$$
I = \mathbb{E}_{k\sim \text{Binom}(n,p)} \left[\ln \frac{\text{Beta}(p;a+k,b+n-k)}{\text{Beta}(p;a,b)}\right]
$$
The term inside the ...
- 63
2
votes
1
answer
103
views
Sufficient conditions for the space of Radon measure to be a Banach space
Let $\mathcal{X}$ be a Hausdorff space and consider the space of Radon measures with bounded total variation $M(\mathcal{X})$ on $\mathcal{X}$.
Usually, the additional assumptions on $\mathcal{X}$ are ...
- 171
2
votes
0
answers
43
views
Good Polynomial lower estimates for beta function
I'm looking for polynomial lower estimates for beta function, and what I've found so far is this, which can be found in proposition 2.3 in this paper
Proposition 2.3 1. If $0<𝑞<1$ and $𝑝 \geq ...
- 677
9
votes
1
answer
845
views
Convergence of sequences formed by orthocenters, incenters, and centroids in repeated triangle constructions
I asked this question on MSE here.
Given a scalene triangle $A_1B_1C_1$ , construct a triangle $A_{n+1}B_{n+1}C_{n+1}$ from the triangle $A_nB_nC_n$ where $A_{n+1}$ is the orthocenter of $A_nB_nC_n$, ...
- 541
2
votes
1
answer
246
views
Inequality with Hermite polynomials
Consider the (physicist's) Hermite polynomials $H_n(x)$ which are divided by
$$\sqrt{\sqrt{\pi} 2^n n!}$$
for the purpose of normalization.
These are orthogonal with respect to the weight function $e^{...
- 43.2k
2
votes
0
answers
97
views
On the second order analog of the upper 1-Lipschitz envelope of a function
Let $u: \mathbb R \to \mathbb R$ be a given function. Then we can consider its upper 1-Lip envelope
$$
\hat u(x) \doteq \inf\{g(x) \, \mid\, g \, \text{has Lipschitz constant 1 and}\, g(y) \geq u(y) \,...
- 355
0
votes
0
answers
109
views
A Lipschitz function induced by the infimum of the length of curves
Recently I have read a paper, Quasiconformal Images of Hölder Domains, written by S. M. Buckley in 2004, published by Annales Academiæ Scientiarum Fennicæ Mathematica. I am confused about page 33 of ...
- 69
5
votes
1
answer
281
views
Is there a singular function that is Hölder continuous of every order less than $1$?
We say a non-constant function $f$ on $[0, 1]$ is singular if it is continuous, and in addition differentiable almost everywhere with $f' = 0$ a.e.
Does there exist a singular function that is Hölder ...
- 6,155
16
votes
4
answers
2k
views
Is the $W^{1, \infty}$ limit of differentiable functions also differentiable?
Let $f_n$ be a sequence of differentiable functions on $[0, 1]$ with
$f_n \to f$ uniformly for some (necessarily) continuous $f$.
$f'_n - g \to 0$ in $L^{\infty}$ for some measurable $g$.
Is it true ...
- 6,155
5
votes
2
answers
297
views
Is the $W^{1, \infty}$ limit of differentiable a.e. functions also differentiable a.e.?
Let $f_n$ be a sequence of continuous, differentiable a.e. functions on $[0, 1]$ with
$f_n \to f$ uniformly for some continuous $f$.
$f'_n - g \to 0$ in $L^\infty$ for some measurable $g$,
where we ...
- 6,155
0
votes
0
answers
95
views
Functions representing all strings somewhere
Do there exist "nice" (maybe analytic?) functions $f_0,f_1:\mathbb R \to \mathbb R$ such that
$\forall n\in\mathbb N,\forall \sigma\in\{0,1\}^n,\exists x\in\mathbb R, \forall \tau\in\{0,1\}^...
- 24.8k