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Does weak continuity of Jacobians hold for non nondegenerate maps?

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ Let $\M,\N$ be two-dimensional smooth, compact, connected, oriented Riemannian manifolds. (with or without boundaries). Let $f_n \...
Asaf Shachar's user avatar
  • 6,741
1 vote
1 answer
471 views

k-th largest root in common interlacing polynomials

In their proof of the celebrated Kadison-Singer conjecture, Marcus, Spielman and Srivastava exploited so-called interlacing families which are originally defined for their work on Ramanujan graphs. ...
Federico Magallanez's user avatar
1 vote
1 answer
171 views

Superharmonic extension 2

This question is a simplified version of the one in the MO post Superharmonic extension. Suppose $K$ is a compact of $\mathbb{R}^m$ ($m\geq2$), and $U(x)=\log\frac{1}{|x|}$ if $m=2$, and $=|x|^{2-m}$ ...
M. Rahmat's user avatar
  • 411
1 vote
1 answer
368 views

Does the almost sure convergence of absolutely continuous r.v.'s imply the weak convergence of the pdf's in $(L^\infty)^*$?

The following question was asked in a comment at Almost sure convergence vs convergence of probability density functions : Suppose that $(X_n)$ is a sequence of random variables (r.v.'s) converging ...
Iosif Pinelis's user avatar
1 vote
1 answer
348 views

Baire class 1 and (uncountably many) discontinuities

Consider a function $f:[0,1]\to[0,1]$ which is continuous on a co-meager set $C\subset[0,1]$ and discontinuous on $D=[0,1]\setminus C$. Suppose that $D\cap I$ is uncountable for every open interval $I\...
Alessandro Della Corte's user avatar
1 vote
1 answer
224 views

Sum of negative roots of a $5^{th}$ degree monic polynomial

Let $f(x)$ be a $4^{th}$ degree monic polynomial say $f(x) = x^4 + a_1x^3+a_2x^2+a_3x+a_4$ with the property that $a_1<0, a_4>0$ and $a_2<a_3$. They by Descartes' rule of signs we can ...
User8976's user avatar
  • 199
1 vote
1 answer
307 views

Convexity of discrete Fourier transform

Let $f : [0,2\pi] \to \mathbb{R}$ be a continuous convex function on $(0,2\pi)$ which is singular about $0$ and $2\pi$ but finite when evaluated at the boundaries. Assume also that $f$ is symmetric ...
spaceman's user avatar
  • 595
1 vote
0 answers
47 views

Another uniform estimation of an integral involving an Hölder function with derivative that is Hölder

Let $\Omega\subset\mathbb{R}^n$, let $s\in [1/2,1)$, let $u\in C^{1,2s-1+\epsilon}(\Omega)$ such that: $u=0$ on $\mathbb{R}^n\setminus\Omega$, and: $u\in C^{0,s}(\mathbb{R}^n)$, is true that there ...
inoc's user avatar
  • 339
1 vote
1 answer
117 views

Shrinking subset with disjoint unions

Given a segment and a value $c$ less than the segment length, let $A_1,\dots,A_n$ be disjoint finite unions of intervals on the segment. We choose a finite union of intervals $B$ with $|B|=c$ that ...
pi66's user avatar
  • 1,209
1 vote
0 answers
123 views

Generalization of concave envelope

Let $g:\mathbb R_+\to\mathbb R$ be a measurable function (which could be supposed to be bounded and Lipschitz if required). Let $\mathcal P$ be the collection of probability measures $\mu$ on $\mathbb ...
CodeGolf's user avatar
  • 1,835
1 vote
1 answer
319 views

Is $(f \ast K)'' \in L^1(\mathbb R)$ for $f \in L^1 \cap L^\infty(\mathbb R)$ and $K \in BV(\mathbb R)$?

Is it possible to deduce that $$(f \ast K)'' \in L^1(\mathbb R)$$ if $f \in L^1 \cap L^\infty(\mathbb R)$ and $K \in BV(\mathbb R)$? What I can prove is that $(f \ast K)' \in L^1 \cap L^\infty$. Is ...
Hiro's user avatar
  • 131
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1 answer
2k views

About the coefficients of Taylor series for the complex Riemann Zeta function $\zeta(s)$

The following real-valued functions are closely related to the zeros of $\zeta(s)$ in the critical strip $\frac{1}{2}<\Re(s) < 1$. $$\phi_1(\sigma, t) = \sum_{n=1}^\infty (-1)^{n+1}\frac{\cos(t\...
Vincent Granville's user avatar
1 vote
1 answer
110 views

Prove that $\dfrac{g(x,u_{n})}{\left\Vert u_{n}\right\Vert ^{p-1}}\rightarrow g_{0}$ weakly in $L^{\overline{p}}$

Let $\Omega \subset \mathbb{R}^{N}$ be a smooth bounded domain , $g:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Caratheodory function such that $g(x,t)=0$ for $t\leq0$ . Suppose that ...
user109584's user avatar
1 vote
0 answers
186 views

Lipschitz continuity of an implicit function generated by a monotonic and Lipschitz multivariate function

Let $z=F(x,y)$ be a function from $\mathbb R^d\times \mathbb R$ to $\mathbb R$ satisfying the following conditions: $z=F(x,y)$ is Lipschitz continuous w.r.t. $(x,y)$; Given $x$, $F(x,y)$ is non-...
zbh2047's user avatar
  • 601
1 vote
1 answer
114 views

question about $TGV^2$ space

Let us just stay in $\mathbb R^1$. The space $TGV^k$ is defined as the function $u\in L^1(I)$ and $$ TGV^k(u,I):=\sup\left\{\int_I u\,\phi^{(k)}\,d\mu, \,\phi\in C_c^\infty(I),\,\|\phi\|_{L^{\infty}(...
JumpJump's user avatar
  • 679
1 vote
1 answer
87 views

Oscillating sums

Let $\left\{a_k\right\}_{0\leqslant k\leqslant n}$ and $\left\{b_k\right\}_{0\leqslant k\leqslant n}$ be two positive sequences such that $c_1a_k\leqslant b_k\leqslant c_2 a_k$ for all $k$ for some ...
coco's user avatar
  • 539
1 vote
1 answer
119 views

A non-polynomial Young function satisfying a power-like condition

This post asked, essentially, for an example of a "non-polynomial" invertible increasing function $f\colon[0,\infty)\to[0,\infty)$ such that $f(0)=0$ and \begin{equation} f(cu)f(t)\le f(...
Iosif Pinelis's user avatar
0 votes
0 answers
123 views

Reference for the Hardy maximal function on the torus

I am searching for a reference for the (sharp) Hardy maximal function on the torus $\mathbb{T}^2:=\mathbb{R^2}/\mathbb{Z}^2$, for instance I would need result result of the following type : if $g\in H^...
Ayman Moussa's user avatar
  • 3,425
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
127 views

Continuous extensions of tangent vector fields

Let $\Omega$ be an open subset of $S^2$ with $\bar{\Omega}\neq S^2$. Suppose a continuous tangent vector field $G$ is given on $\partial \Omega$ with $|G(y)|=1$ for all $y\in \partial \Omega$. Does ...
MathLearner's user avatar
0 votes
1 answer
347 views

Asymptotic behaviour of fixed points in permutations

For any $n\in\mathbb{N}$ let $S_n$ denote the set of all permutations (bijective maps) $\pi:\{1,\ldots, n\} \to \{1,\ldots,n\}$. For $\pi \in S_n$ we set $$\text{fix}(\pi) = \{x\in \{1,\ldots, n\}: \...
Dominic van der Zypen's user avatar
0 votes
1 answer
297 views

Approximating characteristic functions by cutting the real axis into smaller and smaller pieces

Let $\Lambda_r^*=\frac{1}{2\pi r} \mathbb{Z} \subset\mathbb{R} (r>0)$, let $E\subset\mathbb{R}$ be a Lebesgue measurable set with finite measure $|E|$, define $J_r=(-\frac{1}{4\pi r}, \frac{1}{4\pi ...
Lao-tzu's user avatar
  • 1,906
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 ...
Fawen90's user avatar
  • 1,389
0 votes
1 answer
53 views

Rate of convergence of the minimum point over a product space

Let $f(\theta, \epsilon)$ be smooth on $[0,2\pi] \times [0,\infty)$ such that $f(\theta, \epsilon)$ converges to $f(\theta, 0)$ uniformly as $\epsilon \rightarrow 0$. $f(\theta, \epsilon) > 0$ for ...
MathLearner's user avatar
0 votes
1 answer
77 views

Decay rate of minimum point over a product space

Let $f(\theta, \epsilon)$ be smooth on $[0,2\pi] \times [0,\infty)$ such that $f(\theta, \epsilon)$ converges to $f(\theta, 0)$ uniformly as $\epsilon \rightarrow 0$. $f(\theta, \epsilon) > 0$ for ...
MathLearner's user avatar
0 votes
1 answer
697 views

How much do we know about this "local" Hardy-Littlewood maximal function?

The "local" Hardy-Littlewood maximal function is given by $$(M_\phi f)(x)= \sup_{0<\epsilon<1}|\phi_\epsilon \ast f|(x),$$ which is similar to the classical Hardy-Littlewood maximal function : $$...
Mr.right's user avatar
  • 171
0 votes
1 answer
285 views

Infinite products for linear combinations of sines or cosines

There is a well known infinite product both for $\phi(x)=\sin x$ and $\phi(x)=\cos x$. These are particular cases of the Weierstrass factorization theorem. What about $\phi(x)=a_1\cos b_1 x + a_2\cos ...
Vincent Granville's user avatar
0 votes
1 answer
169 views

Unimodality of a certain parametric integral

Suppose $f: [0,1] \to [0,\infty)$ is a smooth, concave and strictly increasing function satisfying $f(0)=0$. Is it true that the map $$ F(y) = \int_0^1 \frac{y^{3/2}}{(y+f(x))^2} dx $$ has exactly one ...
thomas's user avatar
  • 191
0 votes
1 answer
110 views

Functions for which $|f^{(k)}|_{C^{0,\alpha}(0,1)} \le \Vert f \Vert_{L^1(0,1)}$

Let $f \in C^k(0,1)$ and assume that the $k$-th derivative is $\alpha$-Hölder continuous. Assume that $f(x) = 0$ in a fixed interval $(a,b) \subset (0,1)$. Can we characterize (or at least find some ...
Hiro's user avatar
  • 131
0 votes
0 answers
161 views

Superharmonic extension

We know the following classic result. If $K\subset\mathbb{R}^m$ ($m>1$) is a compact set and $u$ is superharmonic on a neighborhood of $K$, then we can extend $u$ to a superharmonic function $\...
M. Rahmat's user avatar
  • 411
0 votes
1 answer
245 views

Riemann-Liouville integral of $f$ is zero implies $f =0$ a.e

The Riemann-Liouville integral is defined by $$ I^\alpha f(x)=\frac{1}{\Gamma(\alpha)} \int_a^x f(t)(x-t)^{\alpha-1} d t $$ where $\Gamma$ is the gamma function and $a$ is an arbitrary but fixed base ...
Grandes Jorasses's user avatar
0 votes
1 answer
124 views

Uniform estimation of an integral

Let $\Omega\subset\mathbb{R}^n$ be open and bounded, let $s\in(0,1)$, let $u\in C^{0,2s+\epsilon}(\Omega)$ bounded and such that: $u=0$, on $\mathbb{R}^n\setminus\Omega$, is true that there exist a ...
inoc's user avatar
  • 339
0 votes
1 answer
230 views

Can we construct a sequence of trigonometric polynomials that converges pointwise to a given continuous function on the torus?

Consider any continuous function $f$ on an $m$-dimensional Torus $\mathbb{T}^m$. Can we construct a sequence of band limited functions (trigonometric polynomials), with the band width (degree of the ...
Rajesh D's user avatar
  • 698
0 votes
1 answer
175 views

Asymptotic of ratio between l1 / l2 norm of a structured vector

As suggested in this discussion, I would like to inquire about the following question: Consider a matrix B of size $n\times n$ defined as: $$B_{ij}(\pmb{\theta})=(\theta_i-\theta_j)\sin(\theta_i-\...
tony's user avatar
  • 405
0 votes
1 answer
127 views

asymptotic of ratio between two summations (l1 / l2 norm)

Let $B$ as a $n\times n$ matrix where $$B_{ij}(\pmb{\theta})=(\theta_i-\theta_j)\sin(\theta_i-\theta_j), 1\leq i<j\leq n$$ and other entries equals to $0$, and $$\theta=[\theta_1,\cdots,\theta_n]\...
tony's user avatar
  • 405
0 votes
1 answer
275 views

Estimate for computing the $L^2$-norm of a function from its data

Let $f:\mathbb{T}^m \to \mathbb{R}$ is a function of bounded variation(BV). Let $D=\{\boldsymbol{p}_i,i=1,2,3\ldots\}$ be a countable dense subset of $(0,1)^m$. Let $E_n, n = 1,2,3\ldots$ be a ...
user102868's user avatar
0 votes
1 answer
246 views

can there be a function $f:\mathbb Q_{+}^{*}\longmapsto\mathbb Q_{+}^{*}$ such that $f(xf(y))=\frac{f(f(x))}{y}$?

Problem: Can an $f$ function be created where:$$f\colon\mathbb Q_{+}^{*}\to \mathbb Q_{+}^{*}$$ The function is defined on the set of fully positive rational numbers and is achieved: $\forall(x,y)\in \...
Bachamohamed's user avatar
0 votes
0 answers
173 views

Is this has anything to do with Riesz representation?

The Riesz representation is very useful in study BV space. There is a lot of version of it and one of the good one can be found in this book, page 49. Here I come up with a question which has similar ...
JumpJump's user avatar
  • 679
0 votes
1 answer
143 views

Existence of smooth functions $f$ satisfying $\sup_{x \in \mathbb{R}} \lvert x^k f^{(q)}(x) \rvert \leq C B^q k^{1/8} q^{q/2}$

$\mathcal{S}^{1/2}_{1/2}(\mathbb{R})$ is defined to be the collection of $C^\infty$ functions $f$ on $\mathbb{R}$ such that \begin{equation} \sup_{x \in \mathbb{R}} \lvert x^k f^{(q)}(x) \rvert \leq ...
Isaac's user avatar
  • 3,477
0 votes
1 answer
78 views

The conditions used to prove upper semicontinuous of generalized directional derivative (in Clarke sense)

Let $X$ be a reflexive Banach space, $z, x, v \in X$. $\{z_i\}, \{x_i\}$ and $\{v_i\}$ are arbitrary sequences converging to $z, x$ and $v$, respectively. I would like to know under which conditions ...
superlit's user avatar
0 votes
0 answers
112 views

Fixed point of a contraction map

This question is a continuation of Is this a contraction mapping for small $T$? Set, for $T, m>0$, $H^m_T:=\{h:[0,T]\to [0,m]:~ h,~h' \mbox{ are both continuous on } [0,T]\}$ endowed with the norm $...
GJC20's user avatar
  • 1,334
0 votes
1 answer
236 views

Is this a contraction mapping for small $T$?

Let $G$ be the heat kernal, i.e. for $0\le t<s$ and $x,y\in\mathbb R$ $$G(t,x;s,y):=\frac{1}{\sqrt{4\pi(s-t)}}\exp\left(-\frac{(y-x)^2}{4(s-t)}\right).$$ For $T>0$, let $\mathcal H_T:=\{h:[0,T]\...
GJC20's user avatar
  • 1,334
0 votes
0 answers
80 views

Alternative to the Sampling Theorem / Invertible transform with sampling criteria

I seek a transform $T$ that operates on real-valued $x(t)$, that Is perfectly invertible Has discrete counterpart with continuous reconstructor Provides conditional reconstruction guarantees ...
OverLordGoldDragon's user avatar
0 votes
1 answer
124 views

Bounding integral expression with Sobolev norm of integrand

Consider the following integral expression: $$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$ for $\epsilon>0$, $f \in L^\...
user avatar
0 votes
1 answer
137 views

Zeros of entire functions with parameter

Let $f_w:\mathbb C \to \mathbb C$ be an entire function with $f_w(0)=1$ and at least one root for any choice of $w \in (0,1)$. Assume further that for a dense set of $w$ the function $f_w$ has ...
Kung Yao's user avatar
  • 192
0 votes
1 answer
227 views

Laplace transform injectivity for different values of $p$

Let $y\in L^{2}(0,1)$ and let $\widetilde{y}$ be its extension on $(0,\infty ).$ Assume that there exist $p_{0},p_{1}\in %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion ,$ $p_{0}\neq ...
Gustave's user avatar
  • 617
0 votes
1 answer
375 views

Bringing a Heun equation into canonical form

It is a well known fact that any second order Fuchsian differential equation on the complex plane $$u''(x) + p(x)u'(x) + q(x)u(x)=0$$ with exactly $4$ regular singular points may be suitably ...
Max's user avatar
  • 213
0 votes
1 answer
116 views

Integrable function [closed]

Suppose that $a, b, c_1$ and $c_2$ are real constant. Is there the necessary and sufficient conditions of $a ,b, c_1,c_2 $ for the following integration is integrable? i.e. $$\int_1^{\infty}\int_1^{\...
Xiaopai Song's user avatar
0 votes
1 answer
491 views

Is this set of function belongs to $L^\infty$?

Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Let $u\in SBV\cap L^\infty(\Omega)$ be given. We write $$ Du = \nabla u\lfloor \mathcal L^N + (u^+-u^-)\otimes \nu_u\mathcal H^{N-...
JumpJump's user avatar
  • 679
0 votes
2 answers
387 views

Derivative of fractional Laplacian is the fractional Laplacian of the derivative

Is it true that $$\partial_x ((-\Delta)^s u(x)) = ((-\Delta)^s \partial_x u(x))?$$
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