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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-...
Amir's user avatar
  • 303
3 votes
1 answer
71 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....
D.R.'s user avatar
  • 831
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_{\...
Ali's user avatar
  • 4,135
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:\...
H A Helfgott's user avatar
  • 20.2k
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 $$ \text{BMO}(\Omega)= \big\{u\in L^1_{loc}(\Omega)\,\,:\,\, |u|_{\text{BMO}(\Omega)} <\infty \big\}, $$ ...
Guy Fsone's user avatar
  • 1,101
2 votes
0 answers
76 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 ...
H A Helfgott's user avatar
  • 20.2k
2 votes
0 answers
223 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 ...
vmist's user avatar
  • 989
2 votes
1 answer
461 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 ...
Ervand's user avatar
  • 51
1 vote
1 answer
311 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. ...
Ervand's user avatar
  • 51
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 ...
River Li's user avatar
  • 1,053
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,...
Chris Sanders's user avatar
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 ...
Alexander Wolfram's user avatar
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, $...
Gustave's user avatar
  • 617
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}...
Sam Forster's user avatar
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.
user530909's user avatar
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\...
Ali's user avatar
  • 4,135
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 $$\lim_{\vert x\vert \to \infty} \vert x \vert^2 \vert \hat{f}(x)\vert =0,$$ ...
Yizheng Yuan's user avatar
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 ...
zoran  Vicovic's user avatar
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 ...
António Borges Santos's user avatar
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 ...
Darius's user avatar
  • 21
0 votes
0 answers
118 views

Uncomplete argument in Nishioka book

In Nishioka book "Mahler functions and transcendence" in the proof of Theorem 4.2.1, Nishioka asserts the following: For a matrix $A=(a_{i,j})_{1\le i\le m}$ with coefficients in $K[z]$ ($K$ ...
joaopa's user avatar
  • 3,998
0 votes
0 answers
53 views

Spectral theory of compact operator for quasi-Banach spaces

Let $X$ be a Banach space and let $Y\subset X$ be a quasi-Banach space (with compact inclusion). Suppose $T:X\to X$ is a compact operator such that $1$ is not its eigenvalue and $T|_{Y}:Y\to Y$ is ...
Liding Yao's user avatar
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} ...
Fatima Majeed's user avatar
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 \...
H A Helfgott's user avatar
  • 20.2k
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:...
G. Blaickner's user avatar
  • 1,429
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 ...
Nate River's user avatar
  • 6,155
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 ...
Nate River's user avatar
  • 6,155
0 votes
0 answers
55 views

Compactness and Leray-Schauder degree

What's the relationship between compactness of solutions in partial differential equations (PDEs) and the Leray-Schauder degree?
Davidi Cone's user avatar
-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 ...
charlie_beck's user avatar
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 ...
Haidara's user avatar
  • 178
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 ...
Haidara's user avatar
  • 178
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 ...
asv's user avatar
  • 21.8k
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 \...
saolof's user avatar
  • 1,947
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 ...
G. Blaickner's user avatar
  • 1,429
0 votes
0 answers
22 views

Has this notion of "variation along the diagonal of a not-necessarily-smooth function" been studied before?

I am interested in knowing whether something along the lines of the "diagonal variation" defined below has been studied before. In spirit, the basic idea is that it is a kind of ...
Julian Newman's user avatar
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 ...
Shaq155's user avatar
  • 459
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 ...
miniii's user avatar
  • 71
0 votes
1 answer
71 views

Upper bound on higher order derivatives of $\frac{1}{v(t)}$

Suppose that $ v(t) >l>0$ and $$ \vert v^{(k)}(t) \vert \leq c \frac{k!}{r^k}. $$ Can we give an upper bound for $$ (\frac{1}{v(t)})^{(k)} $$ ? Attempt: We first compute the first fourth order ...
Yidong Luo's user avatar
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 \...
abcdmath's user avatar
  • 105
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 $L^{p}([0,\infty))$ ...
Medo's user avatar
  • 852
-2 votes
0 answers
64 views

A Problem using Limits of Sequences of Functions

Suppose $\{f_n\}$ is a sequence of nonnegative extended real-valued functions on $X$ and $\lim_{n\to\infty}f_n=f$. Take a simple function $0\leq\varphi\leq f$. If $X_{\infty}=\{x\in X: \varphi(x)=a>...
hunter's user avatar
  • 1
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-...
Plemath's user avatar
  • 312
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....
Darius's user avatar
  • 21
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....
Nate River's user avatar
  • 6,155
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 ...
vaoy's user avatar
  • 309
1 vote
0 answers
39 views

Hyperbolic equation without initial state

Consider the hyperbolic equation on a rectangular domain of the form $(0, L_x) \times (0, L_y)$: $$ a^2 u_{xx} - b^2 u_{yy} = f(x, y), $$ with Dirichlet boundary conditions on $u$. By using the ...
Gustave's user avatar
  • 617
1 vote
3 answers
257 views

Eigenvalues of positive matrices in $\mathrm{SL}(d,\mathbb{Z})$

Let $A\in\operatorname{SL}(d,\mathbb{Z})$ be an irreducible positive matrix, i. e. $A=(a_{i,j})_{1\leq i,j\leq d}$ with $a_{i,j}\in\mathbb{Z}_{>0}$. From the Perron-Frobenius theorem, we know that $...
Yi SHI's user avatar
  • 11
1 vote
0 answers
67 views

System of linear diophantine equations with many small solutions?

Let $n$ be positive integer, $k$,$B$ fixed positive integers. Let $f_i(x_1,x_2...x_n)$ be a system of $n-k$ linearly independent linear equations over the integers. Let $S(f_i,k,B)$ be the set of ...
joro's user avatar
  • 25.4k
3 votes
0 answers
98 views

Square Roots of Non-Negative Even Functions

I'm trying to study properties of maps between quotients of representations of compact Lie groups and I stumbled upon the following problem. Suppose you have a smooth function $f:\mathbb{R}\to\mathbb{...
Ethan Ross's user avatar
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 ...
Haidara's user avatar
  • 178

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