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Properties of solution to Schrödinger equation

Given a Schrödinger equation with, let's say continuous, periodic potential $$-y''(x)+V(x)y(x)=\lambda y(x)$$ where $V(x+1)=V(x)$ and $V$ is even, i.e. for $x \in (0,\frac{1}{2})$ we have $V(x+\frac{...
Zinkin's user avatar
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0 answers
122 views

Does there exist curve (for example, in $\mathbb R^2$) that either touches itself or intersects itself at every one of its points?

I really do not even know how to constructively think about this question that I wanted to post before, but delayed. I know that there are space-filling curves and curves of positive area and those ...
user avatar
4 votes
0 answers
144 views

Asymptotic expansion of a Gaussian integral and heat kernel

When considering the heat kernel of a Schr\"odinger operator $$- \Delta + V(x) $$ where $\Delta$ is the standard Laplacian on ${\mathbb R}^n$ and $V$ is a nonnegative potential function that has ...
Guangbo Xu's user avatar
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4 votes
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633 views

Problem with an integral equation taken from a paper

I am reading a paper (the 2015 paper by A. Falkowski and L. Slominski Stochastic Differential Equation with Constraints Driven by Processes with Bounded $p-$variation, page 353, proof of the Lemma 3.1)...
Joe's user avatar
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4 votes
0 answers
261 views

Is the following integral positive or not?

Let $n$ be a given even positive integer. We have the following integral \begin{eqnarray} &&\int_0^1\cdots\int_0^1\prod\limits_{i=1}^n\prod\limits_{j=1}^n(x_i-y_j)dx_1\cdots dx_ndy_1\cdots ...
user173856's user avatar
  • 1,997
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349 views

Fractional integral inequality (Hardy-Littlewood-Sobolev)

I am investigating the following integral \begin{equation} I^*(x) = \int_{\mathbb{R}} \frac{f(y) \ln |y-x| }{|y - x|^{\mu}} \, dy \end{equation} where $f \in L_p(\mathbb{R})$, $ 1 < p < q <...
Narek Margaryan's user avatar
4 votes
0 answers
139 views

The class of all iterated antiderivatives of rational functions

Consider the following property of a function $f$: There exists a non-negative integer $n$ such that the $n$'th derivative of $f$ is a rational function. Question 1: Is there a name in the ...
Andreas Holmstrom's user avatar
4 votes
0 answers
96 views

Bessel in matrix?

Let $M_n$ be the matrix $$M_n=\begin{pmatrix} 1&\binom{1}{1}\binom{1-1}{1-1} &0 &0\qquad \qquad \dots &0\\ 1&\binom{2}{1}\binom{2-1}{1-1} &\binom{2}{2}\binom{2-1}{2-1} &0 \...
T. Amdeberhan's user avatar
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298 views

Operator topologies

Let $L(H)$ be the space of bounded operators on some Hilbert space. We can endow this space with the operator norm topology, the strong operator topology (SOT) and the weak operator topology (WOT). ...
Zwars's user avatar
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672 views

Proofs of the second fundamental theorem of calculus

I am referring to the following version of the theorem, in the setting of the Lebesgue integral. Theorem Let $f: [a,b] \rightarrow \bf R$ be an everywhere differentiable function whose derivative is ...
coudy's user avatar
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147 views

The asymptotic behavior of the ratio between the largest two of $n$ i.i.d. chi-square random variables

My question is about the asymptotic behavior of the ratio between the largest and second largest values of $n$ independent chi-square random variables. Let $X_1, \ldots, X_n$ be $n$ independent and ...
Steve's user avatar
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0 answers
95 views

Approximating martingales given marginal distributions

Let $(\mu_0,\mu_1)$ be a vector of probability measures on $\mathbb R$ that are of finite first moment, i.e. $$\int_{\mathbb{R}}|x|\mu_i(dx)~<~+\infty \mbox{ for } i=0,1$$ and increasing in ...
CodeGolf's user avatar
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Level sets of function of inner products of vectors on hypercube

Let $H = \{ 0, 1\}^d$ be the $d$-th Cartesian product of $\{0, 1\}$ in $\mathbb{R}^d$. Suppose $v_1, \ldots, v_k$ are $k$ vectors in $H$ in general position. We define function $F \colon H^{k}\...
Steve's user avatar
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0 answers
136 views

Classifying countable sets of weighted dots on a real line

Each dot is located on the real line and assigned a weight that can be positive or negative. A dot is equivalent to two(or more) dots located at the same place whose weights sum is equal to that of ...
Anixx's user avatar
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131 views

Well-definedness on $C_{0}^{\infty}(\mathbb{R}^{n})$

Let $T$ be a Calderon-Zygmund operator associated to a Calderon-Zygmund kernel $K\in CZK_{\alpha}$ of order $\alpha>0$ and $b\in BMO(\mathbb{R}^{n})$. Then for $f\in C_{0}^{\infty}(\mathbb{R}^{n})$ ...
Timothy's user avatar
  • 355
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0 answers
500 views

Properties of the solution of the heat equation

Note 1: the following question has been post on Math Stackexchange here but receive no respond. So I post it here to get more attention. Note 2: This is my research problem, but the original problem ...
JumpJump's user avatar
  • 679
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0 answers
716 views

Can one integrate around a branch-cut?

How meaningful is it to try to integrate around the branch-cut of a function? For example lets say I have the function $\log(z^2+a^2)$ for $a>0$ and I choose my branch-cuts to be starting at $\pm ...
user6818's user avatar
  • 1,893
4 votes
0 answers
197 views

Dynamics of an inequality

The dynamics $D\ni(r_i,r_{i+1})\mapsto(r_{i+1},r_{i+2})\in D$ on the set $D:=\{(x,y)\in\mathbb{R}^2\colon x>0,y>x^2/2\}$ is given by the recurrence $$r_{i+2}=\frac{r_{i+1}^2}2+\frac1{r_{i+1}^3} ...
Iosif Pinelis's user avatar
4 votes
0 answers
121 views

The best constant in Poincare-liked inequality in $BV$ and $BD$ space

This question has been posted on Math Stack exchange for a while and received no response. So I decide to move it here to get more attention. Let $\Omega\subset \mathbb R^N$ be open, bounded and with ...
JumpJump's user avatar
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4 votes
0 answers
188 views

Evaluate a multiple integral

I want to compute this integral and I would appreciate any help: $N\geq 1$ is fixed. $$I_N=\int_{0\le r_n\le r_{n-1}\le\cdots\le r_1} e^{-(r_1^2+\cdots+r_n^2)} \prod_{i<j} \sinh(r_i-r_j) dr_1\...
Hatem's user avatar
  • 41
4 votes
0 answers
684 views

A difficult integral which the Risch algorithm shows is not elementary

For reasons which aren't conceptually related to the problem a few of my colleagues and I are in need of finding an expression for the following integral in terms of $a$ and $\delta$: $$\int_{\delta}^...
Samuel Reid's user avatar
  • 1,441
4 votes
0 answers
453 views

Under what conditions do time averages of ergodic transformations satisfy a central limit theorem?

Let $(X, \mu)$ be a probability space and $T:X\rightarrow X $ an ergodic transformation, i.e. $T$ is measure preserving and the only $T$ invariant subspaces have either measure $0$ or measure $1$ (...
Ritwik's user avatar
  • 3,245
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\...
Joseph Van Name's user avatar
4 votes
0 answers
896 views

A strong form of implicit function theorem (what happens when the derivative is degenerate?)

(this can be considered as some ad) Consider the system of equations $F(x,y)=0$. (Here $x$, $y$ are multi-variables. The equations are over a local ring. e.g. polynomial/analytic/formal/$C^\infty$ ...
Dmitry Kerner's user avatar
4 votes
0 answers
428 views

Inverse of matrix-valued function

Given $c>0$. Let $\gamma_c:{\cal M}_{k \times k}^+\mapsto {\cal M}_{k \times k}^+$ is a function defined by \begin{equation} \gamma_c(\Omega)=\frac1{\sqrt{(2\pi)^{k}|\Omega|}}\int_{\mathbb{R}^k}\{(-...
Jlamprong's user avatar
  • 133
4 votes
0 answers
340 views

Viscosity solution of the PDE

Let $\Omega$ be bounded domain, $u=0$ on $\delta\Omega$ and $$|Du|-f(x,u)=0$$ where $f\ge 0$ and $f$ is strictly monotone for fixed $x.$ I am looking for the reference to show that it has unique ...
nick's user avatar
  • 61
4 votes
0 answers
462 views

System of Equations Upper Bound

I asked a related question on math.stackexchange here but would now like to obtain a better bound. This question comes from a graph theory problem. I'll restate the new question here: For $i=1,2,\...
Alex R.'s user avatar
  • 4,952
4 votes
0 answers
213 views

The ring generated by measures

Suppose $X$ is a space equipped with a $\sigma$-algebra $\mathcal{M}_X$. Then the set of measures on $X$ is closed under addition and scalar multiplication by elements of ${\mathbb R}$. Formally ...
David Spivak's user avatar
  • 8,659
4 votes
0 answers
273 views

Real Analytic Function and nth Prime

It is trivial that there are no polynomial function $P$ with integer coefficients that has the property $P(n)=p_n$ where $p_n$ is the $n$th prime.While it is true that can always construct a smooth ...
Marcus's user avatar
  • 153
4 votes
0 answers
109 views

rank of a C^1 map

I saw this three star problem in Hirsch .. If we have open sets $U \subset R^3$ ,$V \subset R^2$ and $f:U \to V$ is $C^1$ and onto...Prove there is at least one point in $U$ where $f$ has full rank ...
Marcus's user avatar
  • 153
4 votes
0 answers
162 views

Symmetric functions and regularity (II)

My previous question (where $n=2$) was a bit too naive. I think that this one, which is the one being of genuine interest to me, is more involved. Let $f=\mathbb R^n\rightarrow\mathbb R$ be a ...
Denis Serre's user avatar
  • 52.3k
4 votes
0 answers
939 views

Proofs of Baire category theorem

I would like to have a list of proofs of the fact that the real line is not meager (also very useful would be a reference to such a list, if it already exists somewhere). My motivation is the ...
Antongiulio's user avatar
3 votes
0 answers
15 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
  • 4,145
3 votes
0 answers
91 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
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
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
3 votes
0 answers
100 views

How to compute the partial derivatives of this function?

For any probability measure $\mu$ on $\mathbb R^2$ and $\theta\in [0,2\pi]$, denote by $\mu_\theta$ its projection along $v:=(\cos\theta,\sin\theta)$. Namely, if $X$ is a random variable distributed ...
Fawen90's user avatar
  • 1,399
3 votes
0 answers
45 views

Small deviation asymptotics for sub-gaussian diffusions in dirichlet spaces

Let $(X,d,\mu)$ be a metric measure space equipped with a strongly local, regular Dirichlet form $(\mathcal{E}, \mathcal{D}(\mathcal{E}))$ on $L^2(X,\mu)$. Assume that the associated heat kernel $p_t(...
Thomas Frenkel's user avatar
3 votes
0 answers
118 views

A matrix-valued analogue of a classical inequality

Let $p \geq 4$ be an even integer. In the study of variational problems in $W^{1, p}$, it is handy to know that for $a, b \in \mathbb R^d$, $$|a - b|^p \leq 2^{p - 1} (|a|^{p - 2} + |b|^{p - 2}) |a - ...
Aidan Backus's user avatar
3 votes
0 answers
167 views

Bounding the $L^{p*}$ norm from below for functions satisfying a $p$-capacity estimate

If $1 \le p < n$, the $p$-capacity of a compact set $A \subset \mathbb{R}^n$ with respect to an open set $U$ containing it is defined as $$\text{Cap}_p(A, U) := \inf \left\{\int_U |\nabla u|^p \, ...
Cauchy's Sequence's user avatar
3 votes
0 answers
95 views

Is it true that p-integrable function can be written as a convolution of an integrable function and p-integrable function?

We know that convolution of an integrable function with an $p$-integrable is an $p$-integrable function. This follows from Young's inequality. My question: Is it true that $L^p(\mathbb{R}^n)\subseteq ...
user531870's user avatar
3 votes
0 answers
219 views

Strictly contracting solutions to the Eikonal equation on Riemannian manifolds

Given a Riemannian manifold $M$, we say $f: M \to \mathbb R$ is a strict contraction if $|f(x) - f(y)| < |x - y|$ for all distinct $x, y \in M$. Question: Does there exist, on every complete ...
Nate River's user avatar
  • 6,215
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 ...
Hans's user avatar
  • 2,239
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 ...
Nilotpal Kanti Sinha's user avatar
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: ...
Jan Bohr's user avatar
  • 779
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?
Ali Taghavi's user avatar
3 votes
0 answers
141 views

Existence of very weak solution to the elliptic equation $\partial_i(a^{ij}\partial_j u)=\partial_k\partial_l f$

Let $a^{ij}\in W^{1,n}\cap L^\infty (B^1)$ be uniformly elliptic, i.e. $\lambda|\xi|^2\le a_{ij}(x)\xi_i\xi_j\le \Lambda |\xi|^2$ for a.e. $x\in B^1$, $\xi\in\mathbb R^n$, where $B_1\subset \mathbb R^...
Tian LAN's user avatar
  • 435
3 votes
0 answers
68 views

How powerful are sequences of Steiner symmetrizations?

I was studying geometric analysis and have encountered something called Steiner symmetrization method. Intuitively I understand how it's made to be applied and used, but Wikipedia pages do not give ...
cnikbesku's user avatar
  • 171
3 votes
0 answers
84 views

About the naturality of Krasnoselskii genus on Variational Methods

I have recently watched a seminar about Variational Methods from Mónica Clapp and she gave a very interesting motivation of why the Lusternik–Schnirelmann category (click on the link for the ...
Pitbull's user avatar
  • 131
3 votes
0 answers
86 views

Finite dimensional distribution of a stochastic process Lipschitz on every relatively compact set

Let $X_t$ be a Markovian Itô diffusion process, defined by an SDE \begin{equation} dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t\,. \end{equation} Let $f(x,t|x_0,0)$ denote its transition density function. ...
Luís Ferreira's user avatar

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