All Questions
714 questions
4
votes
2
answers
846
views
Generalized Jordan theorem and winding number
By the generalized Jordan theorem any continuous injective map
$S^{n-1} \hookrightarrow R^n$ splits $R^n$ into two regions, one being bounded (interior) and the other one unbounded (exterior). It ...
- 1,875
4
votes
1
answer
321
views
Can planar set contain even many vertices of every unit equilateral triangle?
Is there a nonempty planar set that contains $0$ or $2$ vertices from each unit equilateral triangle?
I know that such a set cannot be measurable. In fact, my motivation is to extend a Falconer-Croft ...
- 19k
4
votes
1
answer
150
views
Quantitative analytic continuation estimate for functions small except on a small set
This question arises as a variation of this question, which was helpfully answered in the negative. It turns out that for my application, a substantially weaker conjecture suffices, which fails to be ...
- 324
4
votes
1
answer
155
views
How do the balls maximizing the maximal function depend on their centers?
Let $\mu$ be a finite Borel measure on $\mathbb R^N$ and let $f\in L^1(\mu)$ be a non-negative function. Let $M_\mu f$ denote the maximal function of $f$ relative to $\mu$, i.e. $(M_\mu f)(x)=0$ if $\...
- 1,277
4
votes
1
answer
642
views
Explicit and fast error bounds for approximating continuous functions
Main Question
This question is about finding explicit, calculable, and fast error bounds (no hidden constants) when approximating continuous functions with polynomials or simpler functions to a user-...
- 697
4
votes
1
answer
668
views
Optimal Transport: how is this transport map Borel measurable?
I'm reading Theorem 1.17. and its proof at page 14 of Santambrogio's Optimal transport for applied mathematicians. The content is not hard but a little bit long (because of related detail). Please ...
- 825
4
votes
1
answer
109
views
Dividing a spherical cap into three equal wedges
Background: Optimal ways to cut an orange.
In this problem, we have a spherical orange, and we do not wish to eat its central column which is modelled as a cylinder. Part of the procedure involves an ...
- 1,459
4
votes
1
answer
287
views
Local maxima of the sum of Gaussian functions in *multiple dimensions* are always strict local maxima - prove/disprove/prove conditionally?
This is a follow up of the question in one dimension, that asked to show that the all the maxima of the sum of Gaussian
$$f_n(x):= \sum_{i=1}^{n}e^{-(x-x_i)^2}, x_1 < x_2 < \dots < x_n$$
are ...
- 1,467
4
votes
0
answers
219
views
Is every supersmooth function a local polynomial?
This question is a follow up question to this question that I recently asked.
A $C^{\infty}$ function $f:(c,d)\rightarrow\mathbb{R}$ shall be called a local polynomial if whenever $f:(c,d)\rightarrow\...
4
votes
1
answer
149
views
An algebraic inequality in three real variables
Is it true that
$$(v-u)^2+(w-u)^2+(w-v)^2 \\
+\left(\sqrt{\frac{1+u^2}{1+v^2}}
+\sqrt{\frac{1+v^2}{1+u^2}}\right) (w-u)(w-v) \\
-\left(\sqrt{\frac{1+u^2}{1+w^2}}+\sqrt{\frac{1+w^2}{1+u^2}}\right) (w-...
- 128k
4
votes
1
answer
204
views
Is there a density theorem for Banach measure?
Fix a finitely additive measure $\mu$ on $\mathbb R^2$ that is consistent with the Lebesgue measure. Does Lebesgue's density theorem hold for such a $\mu$, i.e., is it true that for every $A$ we have $...
- 19k
4
votes
1
answer
367
views
Inequality with decreasing rearrangement function
Let $f^{*}$ be the usual decreasing rearrangement function of a measurable function $f$ on a measure space $(X, \mu)$. Let $1<p<n$ and set $$p'=\frac{pn}{n-p}.$$ Also, let $g$ be a positive ...
- 459
4
votes
1
answer
222
views
A continuous bi-Lipschitz shrinking of a domain into a compact subset
Let $\Omega \subset \mathbb{R}^n$ be a bounded domain. My main problem/question is:
(1) Show there exist a sequence of bi-Lipschitz (i.e injective Lipschitz function with Lipschitz inverse) maps $F_n ...
- 401
4
votes
1
answer
134
views
On partial absolute continuity
$\newcommand\B{\mathscr B}\newcommand\A{\mathscr A}\newcommand\si{\sigma}$Let $I:=[0,1]$, and let $\B$ and $\B^2$ denote the Borel $\si$-algebras over $I$ and $I^2$, respectively. Let $\A$ stand for ...
- 128k
3
votes
0
answers
689
views
"Nicely" strong measure zero sets
This question is essentially an expanded version of the unanswered half of Two strengthenings of "strong measure zero".
A set $X$ of reals is strong measure zero if, for any $f: \omega\...
- 20.5k
3
votes
1
answer
250
views
Characterization of a subset of [0,1] $II$
My question follows the previous one
Characterization of a subset of $[0,1]$
But I don't know whether it is correct to ask again with a new title.
Thanks a lot for pointing the mistake and I ...
- 1,835
3
votes
0
answers
144
views
Noncrossing partitions in Hopf algebras/monoids via compositional inversion
Partition polynomials constructed from the face structures of the associahedra (OEIS A133437) and permutahedra (A133314) comprise the antipodes/compositional inverses in a Faa-di-Bruno-type Hopf ...
- 10.5k
3
votes
1
answer
624
views
Forwards Feynman–Kac formula
This might be a simple question, but I'm having trouble with it.
Consider the Cauchy problem with final condition.
\begin{equation}
\begin{cases}
\frac{\partial u}{\partial t}(t,x) + \mathcal{L}u(t,x) ...
- 33
3
votes
2
answers
2k
views
Expected gradient vs. gradient of expectation
Suppose a function $f(x): \mathbb R^d \mapsto \mathbb R^D$, and its stochastic approximator, $g(x; W): \mathbb R^d \mapsto \mathbb R^D$. Here $W$ is some random variable. Then $g(x; W)$ is unbiased in ...
- 205
3
votes
1
answer
299
views
Lipschitz functions that saturate the Lipschitz inequality on the average (part 1)
Consider a 1-Lipschitz function $f: \mathbb R^n \to \mathbb R$ satisfying the inequality
\begin{align*}
|f(x) - f(y)| \le \|x-y\|_2, \;\forall x,y \in \mathbb R^n.
\end{align*}
For $n \ge 2$, can we ...
- 1,731
3
votes
0
answers
306
views
Metric analogues of bounded variation
A function $f:[a,b]\to\mathbb{R}$ is said to be of bounded variation if
$$ \sup_I \sum_{i=1}^n |f(x_i)-f(x_{i-1})| \le V $$
for some finite $V>0$, where the supremum is over all finite partitions
$...
- 6,541
3
votes
1
answer
220
views
Looking for non-polynomial functions: with the growth condition: $\phi\big(\theta \frac{s}{t}\big) \leq \frac{\phi(s)}{\phi(t)}$
I am for example(s) of an invertible Convex or concave function $\phi: [0,\infty)\to [0, \infty)$ such that $\phi(0)=0$ and there exists $\theta>0$ and for all $s\leq t$ we have
\begin{align}\label{...
- 1,101
3
votes
1
answer
296
views
Does this condition characterise intervals, among subsets of the real line?
For a real number, $c\in \left]0,1\right[$, consider the following property $\mathbf(\mathbf P_c\mathbf)$ of subsets $A$ of $\mathbb R$:
$\mathbf(\mathbf P_c\mathbf)$ For every bounded set $B\subset \...
- 60.5k
3
votes
1
answer
255
views
Is this constraint convex?
I have an optimization problem where the following constraint causes DCP Rule Error.
$$e^{x_n} \leq B \log _2\left(1+\frac{e^{\rho_n} g_n^2}{\sum_{i=1}^{n-1} e^{\...
- 31
3
votes
0
answers
290
views
Does there exist a supersmooth non-polynomial function?
Let's call a $C^{\infty}$-function $f:\mathbb{R}\rightarrow\mathbb{R}$ Lebesgue supersmooth if whenever $a_{n}\in\mathbb{R}$ for all $n$, then $\lim_{n\rightarrow\infty}a_{n}f^{(n)}(x)\rightarrow 0$ ...
3
votes
1
answer
322
views
Special version of Tonelli’s theorem
I am trying to prove this theorem. I have not found anything similar to it on the internet.
Special version of Tonelli’s theorem
Assume that the functions $f(x,u): [a,b] \times \mathbb{R} \to \mathbb{...
- 159
3
votes
1
answer
304
views
Question abouth Skorokhod representation of random variables
It is known that for any two probability measures $\mu$ and $\nu$ on $\mathbb R$ that are close in the Prokhorov metric $\rho$, i.e.
$$\rho(\mu,\nu)<\varepsilon,$$
then there exist two random ...
- 1,835
3
votes
2
answers
487
views
Integrating over the Intersection of Convex Regions
Is there a way to integrate over the intersection of a finite collection of convex regions, using only the definition of the regions (i.e. without actually calculating the intersections)?
The ...
- 13.2k
3
votes
1
answer
233
views
A special approximation of the Heaviside function
Is there a $C^m$ approximation $f_\epsilon$ of the Heaviside function such that
$$f_\epsilon(x) = f_1(x/\epsilon) = \begin{cases} 0 & \text{ if } x < 0 \\
1 & \text{ if } x/\epsilon \ge 1
\...
- 131
3
votes
2
answers
309
views
Seeking proof to an asymptotics of a recursion or functional equation
My question on math.stackexchange.com and the continuation by an answer to it gives the two summation expressions for the recursion
$$a_n = 1+\frac1{2^n}\sum_{k=0}^n {n\choose k}a_k,\, \forall n\in\...
- 2,239
3
votes
2
answers
472
views
Regularity of lipschitz and derivable function
Let be lipschitz $f$ on $[0,1]$ and everywhere derivable. Is it true that $f\in C^1([0,1])$ ?
- 4,074
3
votes
0
answers
1k
views
On new (purely analytic) perspective towards theory of prime numbers
[I'm going to ask this question very carefully as a question similar to this received a critical response on this platform.
I myself am very skeptical about this but I want to know, from the experts' ...
- 375
3
votes
3
answers
427
views
Quantitative analytic continuation estimate for a function small on a set of positive measure
The following conjecture about analytic functions arose as a way to show the asymptotic growth for certain PDE solutions. As I am unfamiliar with any results of this type, I thought I'd ask here.
In ...
- 324
3
votes
2
answers
447
views
Algebraic curve intersecting square-grid
Let us subdivide the unit square into square-grid cells with sidelength $w$. This will give us roughly $w^{-2}$ cells.
Formally
$$ g_{ij} = \{(wi, wj) + (x,y) : 0\leq x,y\leq w \},$$
for $i,j = 0,\...
- 479
3
votes
1
answer
557
views
Is there real or complex analytic function whose positive real zeros are the primes?
Related to this question
Q1 Is there real or complex analytic function $f(x)$ such
that its positive real zeros are the primes and it is
given in closed form of compositions of already named ...
- 25.4k
3
votes
1
answer
411
views
Continuation of a smooth function, whose every derivative is strictly monotonic
Let $f$ be a function defined on $(-\infty, a]$ such that every derivative of $f$ is strictly monotonic. Does it guarantee uniqueness of a smooth continuation $g$ of $f$ to the whole real line, where ...
3
votes
1
answer
248
views
"Lagrange inversion" around an extremum
Cross-posted from Math Stackexchange.
In an older question to which I provided an answer it was asked how to compute a particular limit involving the roots of a transcedental function around its ...
- 151
3
votes
0
answers
106
views
The behavior of an integral related to the inward normal vector near a point of the boundary of a domain
Inspired by this Q&A, I am asking for what kind of non-smooth domains $D$ the following limit
$$
\lim_{r \to 0}\frac{1}{m(D \cap B(x,r))}\int\limits_{D \cap B(x,r)}\frac{z-x}{r}\,m(dz)
$$
where
$...
- 6,367
3
votes
1
answer
84
views
Existence and uniqueness of an Euler-type ODE with varying parameters part 2
I am working on some non-local differential equations that appear in geometric analysis.
One of which I posted here and was answered by @WillieWong and @losifPinelis.
Consider this non-local ...
- 969
3
votes
1
answer
158
views
Explicit eigenvalues of matrix?
Consider the matrix-valued operator
$$A = \begin{pmatrix} x & -\partial_x \\ \partial_x & -x \end{pmatrix}.$$
I am wondering if one can explicitly compute the eigenfunctions of that object on ...
- 536
3
votes
0
answers
232
views
When polynomial f(t+1/t) can be factored as g(t)·g(1/t)?
In venue of my old question When polynomial f(x^2) can be factored as g(x)·g(-x)? and this recent answer to a different question, I wonder:
How to characterize polynomials $f(x)$ with rational ...
- 34.3k
3
votes
1
answer
496
views
"Square root" of multiplication operator on Sobolev space
Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a non-negative, smooth, uniformly bounded function with uniformly bounded first derivative. Then $f$ defines a bounded operator on $L^2(\mathbb{R}^n)$ as ...
- 1,903
3
votes
1
answer
415
views
Inverse of block matrix
Let $V$ be a finite-dimensional vector space and consider the space $X=V\times V\times V\times V.$
Consider the block matrix
$$A = \begin{pmatrix} A_1 & A_2 \\ A_2^* & -A_1\end{pmatrix}$$
...
- 192
3
votes
0
answers
238
views
Move one element of finite set out from A in plane
Suppose we are given two sets, $S$ and $A$ in the plane, such that $S$ is finite, with a special point, $s_0$, while neither $A$ nor its complement is a null-set, i.e., the outer Lebesgue measure of $...
- 19k
3
votes
1
answer
496
views
Prove that these two definitions of "natural" integration constant coincide when both converge
These are two possible definitions of antiderivative (integral) incorporating a supposedly natural choice of an integration constant (see this question for further details).
The first one is based on ...
- 10.1k
3
votes
1
answer
1k
views
Reference request: interpolation of Hölder spaces
On the Wikipedia page on interpolation space, it is written that the space $C^\theta([0, 1])$ is the (real) interpolation of $C^0([0, 1])$ and $C^1([0, 1])$, where $C^\theta([0, 1])$ denotes the space ...
- 1,263
3
votes
1
answer
340
views
Shrinking a disk with fixed differential
Consider mappings $f$ from $\mathbb{R}^2$ to $\mathbb{R}^2$ with differential
\begin{align}
\mathsf{d} f= \begin{pmatrix}
\cos\psi(x) &\cos\phi(y) \\
\sin \psi(x)& \sin\phi(y)
\end{...
- 134
3
votes
1
answer
186
views
packing with special sets in high dimensional Euclidean space
Let $\lambda$ be Lebesgue measure on $[0,1]$. For $\mathbf{x}=(x_1,x_2,..,x_k)\in[0,1]^k$, define $$A(\mathbf{x}):=\{(y_1,\dots,y_k)\in [0,1]^k: \text{there exist intervals }I_1,\dots,I_k \text{ in }[...
- 75
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
3
votes
1
answer
99
views
A bound on an oscillatory solution of an ODE
This question was restated as follows:
Let $V\colon[a,b]\to\mathbb{R}$ be smooth, strictly decreasing and
$V(b) = 0$. Suppose that $f\colon[a,b]\to\mathbb{R}$ is smooth and
satisfies $f''(x)+V(x) f(x)...
- 128k