All Questions
5,871 questions
0
votes
0
answers
55
views
A system of nonlinear Diophantine equations whose positive solutions are not coprime
Consider the following system of Diophantine equations:
$$v_1k_1=k_1^3-k_2^3+k_3^3 \\
v_2k_2=k_1^3+k_2^3-k_3^3 \\
v_3k_3=-k_1^3+k_2^3+k_3^3 \tag{1}$$
where $v_1,v_2,v_3$ and $k_1,k_2,k_3$ are integer-...
- 303
3
votes
1
answer
73
views
How irregular can the set of points of non-differentiability for an L1 function's primitive F get, before the FTC fails?
A Fundamental Theorem of Calculus for Lebesgue Integration, J. J. Koliha begins with the passage
Lebesgue proved a number of remarkable results on the relation between integration and differentiation....
- 831
4
votes
1
answer
273
views
Eigenvalue of a convolution and a restriction?
Let $\epsilon>0$ be small. Let $\eta(t) = \frac{2\epsilon}{\epsilon^2+(2\pi t)^2}$ (the Fourier transform of $x\mapsto e^{-\epsilon |x|}$). Let $V$ be the space of integrable, bounded functions $f:\...
- 20.2k
-3
votes
0
answers
145
views
A presentation for the group $GL(n,\mathbb{Z}_p)$
Let $n\ge 2$. Let $p$ be a prime and $\mathbb{Z}_p$ denote the finite field with $p$ elements.
I want to know about the presentation for the group $GL(n,\mathbb{Z}_p)$ consisting of its generators and ...
- 103
2
votes
1
answer
462
views
About the number of critical points of a function
Suppose that $f$ is a totally monotone function on $(0,\infty)$, so that $(-1)^n f^{(n)}\ge0$ for all $n=0,1,\dots$, $f(0+)\in(0,\infty)$, and $f(t)\sim\frac{1}{t^{\frac{3}{2}}}$ as $t\to\infty$. Can ...
- 51
2
votes
0
answers
13
views
On compact embeddings in weighted Riesz potential spaces
I wonder if there is any references for the study of the following type of spaces
$$ X_{\delta,\alpha}=\{ u\in L^2_\delta(\mathbb{R}^n):\, u= (-\Delta)^\alpha f \quad\text{for some}\quad f\in L^2_{\...
- 4,135
3
votes
0
answers
90
views
About BMO space on smooth open bounded domain
Let $\Omega$ be any open domain in $\Bbb R^d$.
Define the $\text{BMO}(\Omega)$ space as
$$ \text{BMO}(\Omega)= \big\{u\in L^1_{loc}(\Omega)\,\,:\,\, |u|_{\text{BMO}(\Omega)} <\infty \big\},
$$
...
- 1,101
15
votes
1
answer
764
views
Does there exist a nowhere smooth function, that has arbitrary many derivatives?
I'm sorry if my title sounds misleading, I don't know a better way to word my question briefly. But I have the following question about functions.
First, as long as $A$ is a dense subset of $\mathbb{R}...
- 700
2
votes
0
answers
224
views
A deceptively simple regularity problem for functions on the plane
By various meanderings and toying with simpler problems, my current research has lead me to the following quite straightforward question, which I am wholly unable to answer:
Consider a twice ...
- 989
2
votes
0
answers
77
views
Function that is (essentially) a self-convolution but not a multiple of a self-convolution
Call a function $F:\mathbb{R}\to C$ nice if it is of the form $F = f\ast \tilde{f}$, where $\tilde{f}(x) = \overline{f(-x)}$. (Of course nice functions are precisely those whose Fourier transform is ...
- 20.2k
1
vote
1
answer
315
views
The number of intersection points of two completely monotonic functions
Is there a way to show that two completely bounded decreasing monotonic function which are also differentiable over $[0,\infty)$ intersect at most at one point? or showing some counterexamples.
...
- 51
2
votes
1
answer
155
views
Does the support of a smooth compactly supported function have a smooth boundary?
Let $U$ be a bounded domain in $\mathbb{R}^n$ and $f:U\to \mathbb{R}^n$ an infinitely differentiable function with compact support. My question is whether $\operatorname{supp}(f)$ has a smooth ...
3
votes
1
answer
144
views
Jordan plane curve such that $\frac{d(g(x),g(y))}{d(x,y)}\to0$?
Write $g$ as the inverse of $f$.
Is there a continuous injective $f:S^1\to C\subset\mathbb{R}^2$ such that
$$
\displaystyle\sup_{d(x,y)<r}\dfrac{d(g(x),g(y))}{d(x,y)}\to0
$$ as $r\to0$?
If you like,...
- 211
6
votes
3
answers
748
views
Clarification and Proof of Inequality (8.11) in Analytic Number Theory by Iwaniec and Kowalski
I am studying inequality (8.11) from Analytic Number Theory by Iwaniec and Kowalski. It is found on top of page 200. In bottom of page 199, the authors prove that
$$
|S_f(N)|^2 \leq N + \frac{2N^2}{q} ...
- 111
2
votes
0
answers
101
views
An inequality related to Problem 10210 AMM 1992 No. 3
Problem. Let $A$ be a $N \times N$ real matrix whose $(i,j)$ entry is $a_{ij} \ge 0, \forall i, j$. Let $1$ denote $N\times 1$ all-ones vector. Prove that
$$N^2 1^\top A^\top A A^\top 1 \ge (1^\top A ...
- 1,053
0
votes
1
answer
142
views
Why are the homeomorphisms from the unit circle to the unit circle preserving measure affine? [closed]
Why are the homeomorphisms from the unit circle to the unit circle preserving measure affine? The affine is composition of rotation and continue automorphism.
- 83
5
votes
2
answers
355
views
Can one show that $(-1)^{n-1} {(1/\zeta)}^{(n)}(x) >0$ for all real $x>1$?
Is it true that $(-1)^{n-1} {(1/\zeta)}^{(n)}(x) >0$ for all real $x>1$ ?
Or in other words can you show that the higher order derivatives of the reciprocal of the Riemann zeta function ...
- 178
0
votes
0
answers
87
views
Curl-Div equation with singular matrix
I want to solve the equation:
$$
\begin{cases}
\nabla \times (A \mathbf v)=f, \quad x\in \Omega \\
\operatorname{div}(\mathbf v)=0,
\end{cases}
$$
where $\Omega \subset\mathbb{R}^n$, is an open set, $...
- 617
7
votes
1
answer
554
views
Example of continuous function which is not differentiable everywhere in a strong sense
Is there a continuous function
$$u\colon (0,1)\to \mathbb{R}$$
such that at every point $x\in (0,1)$ one has
$$\lim\sup_{y\to x+0}\frac{u(y)-u(x)}{y-x}=+\infty?$$
In particular $u$ is not ...
- 21.8k
0
votes
1
answer
115
views
Fourier transform of exponential over torus
I found the following formula for the Fourier transform on a flat 2-torus, but I don't quite know how to derive it. We have a variable $q=(q_x,q_y) \in [0,2\pi)^2$ and by considering it in polar ...
0
votes
1
answer
124
views
Holomorphic functions of certain blow up at origin
Suppose that $D=\{z\in \mathbb C\,:\, |z|\leq 1\}$ and let $f$ be holomorphic on $D\setminus\{0\}$ such that $|f(z)|\leq e^{\frac{1}{|z|}}$ for all $0<|z|\leq 1$ and assume additionally that $\lim\...
- 4,135
1
vote
0
answers
146
views
integral over the unit sphere of $\Bbb C^n$
Please, is there a way to calculate this integral
$$\int_{S_{2n-1}} \frac{e^{a \langle z, \zeta \rangle}}{|z - \zeta|^{\beta}} \, d\sigma(\zeta)$$
where $ z $ is a fixed point in the complex unit ball ...
- 511
0
votes
0
answers
71
views
Fourier decay implies what kind of regularity
We consider a function $f:\mathbb R^2 \to \mathbb C$ that is compactly supported and bounded. In addition, we know that
$$\lim_{\vert x\vert \to \infty} \vert x \vert^2 \vert \hat{f}(x)\vert =0,$$
...
2
votes
0
answers
194
views
Functions such that the *integral* of the Fourier transform is non-negative?
Let $f:\mathbb{R}\to \mathbb{R}$ be in $L^1$, with its Fourier transform $\widehat{f}$ also in $L^1$. What is a necessary and sufficient condition on $f$ so that
$$\int_{-\infty}^x \widehat{f}(t) dt \...
- 20.2k
-3
votes
1
answer
194
views
Bounding a number-theoretic integral
Find a good upper bound on $$\int_1^T\frac{\zeta'(s)}{\zeta(s)\zeta(1-s)}X^sdt,$$ where $s=c+it$ for a constant $c>1$ and $X>0$ is a parameter. If needed, we can assume RH.
My attempt here is ...
- 107
6
votes
0
answers
130
views
Do there exist strictly contracting eikonal functions on $\mathbb R^n$?
A function $f: \mathbb R^n \to \mathbb R$ is said to be a strict contraction if
$$|f(x) - f(y)| < |x - y|$$
for all $x \neq y$.
A function $f$ is said to be eikonal if it is differentiable ...
- 6,165
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?
$$\prod_{n\geq1}(1+x^{2n-1})^{24} - \...
- 43.2k
2
votes
0
answers
52
views
On distributions and kernels
Let $U\subset\mathbb{R}^{d}$ be an open set and consider $X=\mathbb{R}\times U$. Now, lets consider a smooth (regular) kernel $k_{A}\in C^{\infty}(X\times X)$ and corresponding continuous operator $A:...
- 1,429
1
vote
0
answers
59
views
Asymptotic behavior of the Hermite functions
I would like to understand the asymptotic behavior of the Hermite function :
$$\psi_k(x) = \frac{1}{\sqrt{2^k k!}}H_k(x) e^{-\frac{x^2}{2}},$$
where $H_k(x)$ is the $k-$th Hermite polynomial. For ...
- 21
9
votes
3
answers
1k
views
Examples of combinatorial problems where the only known solutions, or most "natural" solutions, use representation theory?
In Solution of two difficult combinatorial problems with linear algebra, Robert Proctor presents two simply stated combinatorial problems, and gives solutions to them using a linear algebraic approach ...
5
votes
0
answers
163
views
Does this weak omniscience principle have a name?
In constructive analysis, I'm looking at principles which follow both when there exists at least one discontinuous function from $\mathbb{R}$ to $\mathbb{R}$ (equivalent to WLPO i.e. $x > 0$ or $x \...
- 1,947
0
votes
1
answer
66
views
Does convergence in probability of iid samples imply convergence in measure of the sampled functions?
Let $g_i: [0, 1] \to \mathbb R$ be $L^1$ functions, equibounded in $L^1$ norm. Let $X_i$ a sequence of iid uniform random variables on $[0, 1]$. Suppose that
$$\frac{1}{n} \sum_{i = 1}^n g_i (X_i) \to ...
- 6,165
1
vote
0
answers
100
views
Prove or disprove that $|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$
$|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$.
I had this conjecture for a long time. I tried various methods and techniques but they all failed. It might also be wrong ...
- 178
7
votes
2
answers
331
views
Does every subset of $\mathbb N$ with full natural density contain arbitrarily long geometric progressions?
We use the standard definition of natural density. We say a subset of $\mathbb N$ has full natural density if it has natural density $1$.
Question: Does every subset of the naturals with full natural ...
- 6,165
6
votes
2
answers
773
views
Finiteness of an integral
In a paper I am reading, the following seems to be claimed:
Let $f:[0,\infty)\to [2,\infty)$ be a continuous, monotonically increasing function with $\lim_{x\to\infty}f(x)=\infty$ and let $\alpha>3/...
- 460
1
vote
2
answers
117
views
If $f\in C([0,\infty))$, does $\delta>0$ and $g\in C^1((0,\delta))\cap C([0,\delta])$ s.t. $g\geq f$ on $[0,\delta]$ and $g(0)=f(0)$ exist?
The question is the following:
Suppose $f : [0,\infty) \rightarrow \mathbb{R}$ is a continuous function. Can I find $\delta \in (0,\infty)$ and a function $g : [0,\delta] \rightarrow \mathbb{R}$ such ...
- 309
1
vote
1
answer
90
views
Sobolev inequality with weight in the case $1<n\leq p$
Assume that $1<n\leq p$. Does there exist a (non-negative) measure $\mu$ (preferably with some positive density function with respect to the Lebesue measure $dx$) and $q>p$ so that for all $f\in ...
- 459
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
0
votes
2
answers
148
views
Asymptotic behavior of the integral of Hermite functions/polynomials on half-lines
I would like to understand the asymptotic behaviour of the following integrals with fixed $x_0>0$:
$$J_m=\int^{+\infty}_{x_0}|H_m(x)|^2 e^{-x^2}dx,$$
where $H_m(x)$ is the $m-$th Hermite polynomial....
- 21
2
votes
0
answers
43
views
Distributions and time-kernels
Let $U\subset\mathbb{R}^{d}$ be an open subset and set $M:=I\times U$, where $I=(a,b)\subset\mathbb{R}$ is some open subset. Lets consider a linear operator $B:C^{\infty}_{c}(M)\to C^{\infty}(M)$ that ...
- 1,429
1
vote
1
answer
69
views
Exhausting sequences contain a $\pi$ lift of a subset with a $(1-\delta)$ factor
Let $\pi : Y \to X$ be a measurable map between the $\sigma$-finite measure spaces $(Y, \mathcal{B}, \nu)$ and $(X, \mathcal{A}, \mu)$. Suppose there exists $c \in (0, \infty)$ such that for all $A \...
- 105
1
vote
1
answer
188
views
Can one show $h(x)=|2(\zeta'(x))^2-\zeta''(x)\zeta(x)|$ is a decreasing function for $x\in\mathbb{R}\cap [1,\infty)$?
This question is related to This question.
When I tried to approach it I couldn't even proof that the LHS is a decreasing function on the given domain using regular methods. I have tried to write the ...
- 178
4
votes
1
answer
256
views
Approximating an $L^1$ function with Riemann sums
Note: Here all functions are genuine functions, i.e. pointwise defined measurable functions instead of defined only a.e.
Let $f: [0, 1] \to \mathbb R$ be an arbitrary $L^1$ function. Of course, $f$ is ...
- 6,165
0
votes
2
answers
364
views
Can one show $\left|\frac{2(\zeta'(x))^2-\zeta''(x)\zeta(x)}{\zeta^3(x)}\right|\leq \frac{2}{(x-\frac{1}{2})^2}$ for $x\in\mathbb{R}\cap [1,\infty)$?
I have found that $\left|\frac{2(\zeta'(x))^2-\zeta''(x)\zeta(x)}{\zeta^3(x)}\right|\leq \frac{2}{(x-\frac{1}{2})^2}$ for all real $x$ such that $x>1$ seems to be true. I have plotted the ...
- 178
6
votes
1
answer
568
views
Can one show that $|\zeta'(x) / \zeta^2(x)| \leq 1/(x-.5)$ for $x\in\mathbb{R}\cap [1,\infty)$?
I have found that $\left|\frac{\zeta'(x)}{\zeta^2(x)}\right|\leq \frac{1}{x-\frac{1}{2}}$ for all real $x$ such that $x>1$ seems to be true. I have plotted the inequality and got this inequality ...
- 178
3
votes
0
answers
95
views
Deeper reason for why classical orthogonal polynomials have simple generating functions?
Is there a known reason why all classical families of orthogonal polynomials have simple generating functions? I was wondering whether one could get an explanation using the connection with Sturm-...
- 312
3
votes
2
answers
153
views
On nowhere differentiability of functions that just barely fail to be Lipschitz
By Rademacher’s theorem, Lipschitz functions are differentiable almost everywhere. I am wondering how badly this pointwise differentiability fails for functions that “just barely” fail to be Lipschitz....
- 6,165
12
votes
2
answers
866
views
Sets that project to zero measure on all lines except one
It is a (difficult) exercise to show that there exists a measurable set $E \subset [0,1]^2$ (necessarily with zero 2-dimensional Lebesgue measure) such that the projection on every line passing ...
- 355
0
votes
1
answer
93
views
A question on finite Fourier series
Let $\mathcal F(N)$ denote the space of finite Fourier series up to frequency $N > 0$, i.e. $f\in \mathcal F(N)$ if and only if it can be written as
$$f(x) = \sum_{k=0}^N a_k\cos(kx+\theta_k)$$
for ...
- 71
3
votes
1
answer
212
views
$\nabla \times (F\times \mathbf v)=g$, $\operatorname{div}(\mathbf v)=0$
I want to solve the equation:
$$
\begin{cases}
\nabla \times (F\times\mathbf v)=g, \\
\operatorname{div}(\mathbf v)=0,
\end{cases}\label{1}\tag{1}
$$ where $F$ and $g$ are given vector fields. The ...
- 617