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Questions tagged [ca.classical-analysis-and-odes]

Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.

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0answers
33 views

Approximation of functions by tensor products

Given a function $f(x,y)\in L^p(R^d;L^\infty(B_R))$ with $1<p<\infty$, where $B_R:=\{y\in R^d: |y|\le R\}$, can we find a sequence of functions $f_n$ of the form $f_n(x,y)=\sum_{i=1}^ng_i(x)h_i(...
2
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1answer
44 views

Reverse Loomis-Whitney Inequality for funcctions

I was wondering if the reverse Loomis-Whitney inequality holds for general functions: Let $n\geq 2$. Let $(X_i,\mu_i)$, $1\leq i\leq n$ be measure spaces. Write $x=(x_1,\dots,x_n)$ and for each $1\...
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0answers
55 views

How to see the divergence of a series is not faster than some order? [on hold]

$$ \sum_{m=1}^{n} m^{-1+1/p} \leq Cn^{1/p} $$ For $1<p<2$, I know the LHS is divergent but I can't see its speed of divergence is not faster than $n^{1/p}$.
4
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2answers
197 views

Sums of entire surjective functions

Suppose $(f_n)_n$ is a countable family of entire, surjective functions, each $f_n:\mathbb{C}\to\mathbb{C}$. Can one always find complex scalars $(a_n)_n$, not all zero, such that $\sum_{n=1}^{\...
3
votes
3answers
140 views

Existence of solution to linear fractional equation

We consider the equation $$ \sum_{j=1}^n \frac{\lambda_j}{x-x_j} =i$$ where $\lambda_j>0$ and $x_j$ are real distinct numbers. I want to show that if $\lambda_k$ is small compared to the ...
1
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0answers
155 views

Bounding the $L^2$ norm of a polynomial from below

Let $\sigma >0$ be fixed. For even $k \in \mathbb{N} \cup \{0\}$, we consider the polynomial \begin{equation} \varphi_k(x) = \sum_{j=0}^{k} (-1)^j {k \choose j} b_j \, x^{2j} \quad x \in (-1,1), \...
4
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2answers
257 views

Can we use Ramanujan's parameterization of Klein's quartic to solve Klein's septic?

I. Klein In "On the Order-Seven Transformation of Elliptic Functions" (pp. 287-331), he discusses in p. 298 what we now call the Klein quartic, $$\lambda^3\mu+\mu^3\nu+\nu^3\lambda= 0\tag1$$ and in ...
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0answers
18 views

linear system of differential equations with variable coefficient [closed]

Any application about linear system of differential equation with variable coefficient which is related to modelling or other fields and mehods to solve them as well.
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0answers
20 views

Can time-scale calculus be used to derive a counterpart theorem of discrete-time dynamic systems directly from continuos-time dynamics systems?

From what I read of time-scale calculus literature, most results of continuous-time and discrete-time systems can be generalized to arbitrary time-scales by considering the generalized derivative ...
6
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2answers
272 views

Properties of heat equation

** I simplified the question: ** On bounded domains, the maximum principle implies that the solution to the heat equation is (strictly) positive, if the initial and boundary data is positive. I ...
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0answers
57 views

Algebraic Riccati and WKB [closed]

It's a one-liner to show that the algebraic Riccati equation (ARE) and the lowest order form of WKB for a linear ode are the same. But I've looked all over the web and there does not seem to be a ...
4
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1answer
121 views

Brascamp-Lieb inequalities on the sphere

In the paper [CLL], Carlen, Lieb, and Loss demonstrate a version of the Young inequality on the sphere $S^{N-1}$ in $\mathbb{R}^N$. For positive functions $f_j$ on $[-1,1]$, the following bound holds:...
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0answers
57 views

Does there exist $\alpha>0, \beta\in (0,1)$ such that $\dfrac{\sum_{k=1}^n a_k}{n}\le \alpha (a_1\cdots a_n)^{1/n} + \beta \max_i(a_i)$ holds?

Let $a_1\ge a_2\ge \cdots\ge a_n\ge 0$ be given non-negative numbers. My question is the following: Is there any $\beta \in(0,1),\ \alpha>0$, such that $$\dfrac{a_1+\cdots+a_n}{n}\le \alpha (a_1\...
3
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1answer
168 views

On convex functions which are non constant on every segment

I have been studying for the last few weeks the paper Dirichlet problem for demi-coercive functionals by Anzellotti, Buttazzo and Dal Maso. In this work the authors introduced and studied the ...
0
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0answers
29 views

Unique solution for a difference ODE?

Any idea how to find general solution $$a'_{n}(t)= (n+\alpha )a_{n}(t) + \beta a_{n+1}(t) + \gamma a_{n+2}(t)$$ for some coefficients $\alpha, \beta, \gamma$?, Where $a'_{n}(t)=\frac{d}{dt} a_{n}(t)...
2
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1answer
77 views

Failure of Falconer distance problem in one dimension

I am told that the Falconer distance conjecture fails trivially in one dimension, but I really cannot find any reference for that. Precisely, I am asking the following question: For a compact set $E\...
8
votes
1answer
106 views

Log-concavity of repeated convolution

Let $A = (a_0,a_1,\ldots,a_k)$ be a sequence of strictly positive numbers, and let $A^{\ast k}$ denote the $k$-fold repeated convolution (defined by $A^{\ast 1} = A$ and $A^{\ast k+1} = A^{\ast k} \...
5
votes
1answer
197 views

Dependence of a solution of a linear ODE on parameter

Is the following theorem known, or can be easily derived from known results? Consider the differential equation $$w''-kz^{-1}w'=(\lambda+\phi(z))w,$$ where $k>0$ is fixed, $\lambda$ is a large (...
2
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1answer
78 views

Discrete dynamical system and bound on norm

Let $z \in \mathbb R\backslash \left\{2 \right\}$ then I would like to understand the following: Consider the dynamical system with $x_i \in \mathbb C^2:$ $$ x_{i} = \left(\begin{matrix} z &&...
2
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0answers
64 views

Off-diagonal estimates for Poisson kernels on manifolds

Let $(M,g)$ be a complete Riemannian manifold, $\Delta$ its Laplace-Beltrami operator and $T_t = (e^{t \Delta})_{t \geq 0}$ the associated heat semigroup. We can define the subordinated Poisson ...
0
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0answers
24 views

Independence of eigenfunction evaluations at random locations

My question is based on this one. Let the Gaussian kernel be $k(x,y) = \exp(-(x-y)^2/2)$ for $x, y \in R$. Then the Hilbert-Schmidt integral operator is defined as $T_k: (T_k f)(x) = \int_y k(x,y) f(...
8
votes
1answer
297 views

Constants of motion for Droop equation

There is an important ODE system in biochemistry, Droop's equations: $$s'=1-s-\frac{sx}{a_1+s}$$ $$x'=a_2\big(1-\frac{1}{q}\big)x-x$$ $$q'=\frac{a_3s}{a_1+s}-a_2(q-1)$$ Relatively easy one finds a ...
2
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0answers
47 views

Traceless sobolev forms on compact manifolds with boundary

Let $(M,g)$ be a smooth, compact, oriented Riemannian manifold with smooth, oriented boundary $\partial M$. Further, let $\Omega^p(M)$ and $\Omega^p(\partial M)$ be the spaces of smooth differential $...
2
votes
1answer
109 views

Eigenvalues Sturm-Liouville Operator

Is the eigenvalue decomposition of the Sturm-Liouville operator $$ Lf(x)=-f''(x)+h\sin(x)f'(x),\quad h>0, $$ with Neumann boundary conditions $f'(-\pi)=f'(\pi)=0$ on the Hilbert space $L^2([-\pi,\...
2
votes
1answer
86 views

How to get asymptotic expansion of the sum of modified Bessel function $\sum_{n=1}^\infty K_0(s\, n)$ as $s\to 0^+$?

I guess for the modified Bessel funcion $K_0(z)$, $$\sum_{n=1}^\infty K_0(s\, n) \sim \frac{-2\log 2 - \log \pi + \gamma}{2} + \frac{\log s}{2} + \frac{\pi}{2\, s}, \quad s\to 0^+,$$ if taking $$\...
6
votes
1answer
155 views

Eigenvalue estimates for operator perturbations

I edited the question to a general mathematical question, since I found the answer in Carlo Beenakker's reference and think that my initial question was mathematically misleading. What was behind ...
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0answers
135 views

what is a limit cycle?

Periodic trajectories in the plane that are stable on one side and unstable on the other side seem to be accepted or rejected as limit cycles depending on the definition. For example, $$f=\begin{...
5
votes
1answer
163 views

Perturbing a normal matrix

Let $N$ be a normal matrix. Now I consider a perturbation of the matrix by another matrix $A.$ The perturbed matrix shall be called $M=N+A.$ Now assume there is a normalized vector $u$ such that $\...
1
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0answers
69 views

Uniform $L_\infty$ bound on eigenfunctions of HS integral operator (Mercer's Theorem)

My question extends this one: 1. In Hermann Koenig's book: Eigenvalue distribution of compact operators (page 145), the Mercer's theorem is stated with an extra claim. Given a Mercer's kernel $k$, ...
11
votes
1answer
182 views

Poincaré on analytic dependence on parameters of solutions of linear differential equations

There is the following important General Principle: if a parameter enters in a linear differential equation additively, for example $$\frac{d^2w}{dx^2}+(q(x)+\lambda)w=0,$$ where the parameter is $\...
0
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1answer
75 views

Solving or bounding the real part of the integral $\int_0^{2 \pi i m} \frac{e^{-t}}{t-a} dt$

I would be interested in finding a closed form or, at least, bounding (in terms of $m$ as it becomes larger) the real part of the following itnegral: $$f(m,a):=\int_0^{2 \pi i m} \frac{e^{-t}}{t-a} ...
3
votes
1answer
131 views

Real part of eigenvalues and Laplacian

I am working on imaging and I am a bit puzzled by the behaviour of this matrix: $$A:=\left( \begin{array}{cccccc} 1 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 &...
1
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2answers
86 views

Asymptotic expansion of hypergeometric function near $z=1$

Given the hypergeometric function $_2F_1[a,b,c,z]$ in the interval $z\in(1,\infty)$. What is the proper asymptotic expansion of the aforesaid function near $z=1$, when one is approaching from $z>1$....
4
votes
3answers
158 views

A Riccati type integral inequality

Let $x(t),t\in [1,\infty)$ be a nondecreasing positive function satisfying the following inequality: $$ x'(t) \le \int_t^{+\infty} x(s)\frac{k(s)}{s^2}\,ds, $$ for any $t \ge 1$, where $k(t),t\in [1,\...
0
votes
0answers
37 views

Ellipticity-type condition

An elliptic operator $L=\mathrm{div}(A(x)\nabla u)$, is called uniformly elliptic if $$C^{-1}\mathrm{Id} \le A(x) \le C \mathrm{Id}$$ If $A$ depends also on $u$, what is the condition $$C^{-1} + C^...
4
votes
1answer
104 views

Oscillation and Holder continuity

Where can I find a proof of the follwing fact? If $$w(u,x_{0},r)=\sup _{B_{r}(x_{0})}u-\inf _{B_{r}(x_{0})}u$$ for some function $u(x)$ satisfies $$ w\left(u,x_{0},{\tfrac {r}{2}}\right)\leq \...
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votes
1answer
30 views

Integral representation of the Digamma function vs. Integral representation of the Polygamma function [closed]

Yesterday I asked for the derivation of the Integral representation of the Digamma-Function: https://math.stackexchange.com/questions/3113119/integral-representation-of-digamma-function Thanks again @...
2
votes
1answer
119 views

Must $q$ be analytic?

I have a continuous function $q:\mathbb{R}^+ \to \mathbb{R}^+$. An interesting property of this function is that $$F(s) = \frac{e^{-q(s)}q(s+1)}{1-e^{s-q(s)}}$$ which also takes $\mathbb{R}^+ \to \...
1
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1answer
133 views

Why is this series summable?

Let $\delta, \epsilon \in \mathbb{R}$, $\delta >0$, $\epsilon >0$. Let $\{ a_k\}^\infty$,$\{ b_k\}^\infty$ be sequences of positive integers such that $\lim \sup_{k \rightarrow \infty} \frac{...
4
votes
1answer
156 views

ODE with a measurable vector field

Suppose we have a bounded Borel measurable vector field $F:\mathbb{R}^n\to\mathbb{R}^n$. To make the question non-trivial, assume that $F\neq 0$ eveywhere. Question. Does there exist at least one ...
1
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0answers
133 views

3D Homogenous Laplace equation with integral boundary conditions

I have the 3D Laplace equation: $$\nabla^{2} T_w = 0$$ where $\nabla^{2}=(\frac{\partial^{2}}{\partial x^2}+\frac{\partial^{2}}{\partial y^2}+\frac{\partial^{2}}{\partial z^2})$ defined on $x \in [0,...
3
votes
0answers
51 views

system of Euler like ode's

I am interested in solving some linear elliptic system like $$ -\Delta \phi(x) + \frac{C_1 \psi(x)}{|x|^\beta} =f(x)$$ $$ -\Delta \psi(x) + \frac{C_2 \phi(x)}{|x|^\alpha} =g(x)$$ in $B_1$ (the ...
0
votes
1answer
51 views

Empty interior lack of minima

Suppose that $U \subseteq \mathbb{R}^d$, and satsifies $U$ is dense in $\mathbb{R}^d$, U has empty interior, Then is it possible that $$ \inf_{x \in U} f(x) >\inf_{x \in \mathbb{R}^d} f(x), $$ ...
1
vote
1answer
141 views

Laplace equation with integral source terms

I am not specifically asking for a solution, but any reference on any method i could read about would be a big help. This I clarify as I am aware of the fact that MathOverflow only deals with research ...
0
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0answers
32 views

Feller semigroups and fractional operators

Have Feller semigroups been used to investigate the properties of the Cauchy problem associated with the fractional Laplacian (just like they have been used to study local degenerate second order ...
5
votes
1answer
149 views

“One half of a theta-function” - is there something in the literature about it?

In an answer to my question Enumeration of lattice paths of a specific type (which was in turn caused by an older one, "Special" meanders) I encountered the series $$ F(t,q):=\sum_{n=1}^\...
1
vote
1answer
126 views

Time ordered integral involving beta function:

Any help on unpacking integrals of the following type, would be helpful: $$ \int_0^1 \int_0^r r^a (1-r)^b t^n (1-t)^m dr dt $$ where $a, b, n, m \in \mathbb{N}$ and $0 \le t \le 1$. Edit/...
27
votes
2answers
895 views

Rademacher theorem

If $f:\mathbb{R}^n\to\mathbb{R}^m$ is of class $C^1$ and $\operatorname{rank} Df(x_o)=k$, then clearly $\operatorname{rank} Df\geq k$ in a neighborhood of $x_o$. It is not particularly difficult to ...
2
votes
2answers
237 views

ODE of the form $y'=\exp(-(\cos(2\pi y))$

Consider the function $h:[0,1]\to \mathbb{R}$ $$h(\theta):=\sum_{k\geq 1}\frac{a_{k}}{\sqrt{k}}\cos(2\pi k \theta)+\frac{b_{k}}{\sqrt{k}}\sin(2\pi k \theta),$$ where $a_{k},b_{k}\in\mathbb{R}$. For ...
3
votes
1answer
179 views

Local “boundary comparison principle” for harmonic functions

Let $u$ be a positive solution of the elliptic equation $\mathcal Lu = 0$ on $B^+_1 \subset \mathbb{R}^n$ such that $u$ vanishes continuously on $\{x_n = 0\}$. To fix ideas, we may take $\mathcal L = ...