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.
2,198 questions
-3
votes
0answers
13 views
linear system of differential equations with variable coefficient [on hold]
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.
0
votes
0answers
19 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
votes
2answers
251 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 ...
1
vote
0answers
57 views
Algebraic Riccati and WKB [on hold]
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
votes
1answer
111 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:...
0
votes
0answers
56 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 $$\dfrac{a_1+\cdots+a_n}{n}\le \alpha (a_1\...
3
votes
1answer
156 views
+100
On convex functions which are non constant on every segment
I have been studying for the last few weeks the paper Dirichlet problem for demi-coercive functionals by Anzellotti, Buttazzo and Dal Maso. In this work the authors introduced and studied the ...
0
votes
0answers
29 views
Unique solution for a difference ODE?
Any idea how to find general solution
$$a'_{n}(t)= (n+\alpha )a_{n}(t) + \beta a_{n+1}(t) + \gamma a_{n+2}(t)$$
for some coefficients $\alpha, \beta, \gamma$?, Where $a'_{n}(t)=\frac{d}{dt} a_{n}(t)...
2
votes
1answer
75 views
Failure of Falconer distance problem in one dimension
I am told that the Falconer distance conjecture fails trivially in one dimension, but I really cannot find any reference for that. Precisely, I am asking the following question:
For a compact set $E\...
8
votes
1answer
106 views
Log-concavity of repeated convolution
Let $A = (a_0,a_1,\ldots,a_k)$ be a sequence of strictly positive numbers, and let $A^{\ast k}$ denote the $k$-fold repeated convolution (defined by $A^{\ast 1} = A$ and $A^{\ast k+1} = A^{\ast k} \...
5
votes
1answer
184 views
Dependence of a solution of a linear ODE on parameter
Is the following theorem known, or can be easily derived from known results?
Consider the differential equation
$$w''-kz^{-1}w'=(\lambda+\phi(z))w,$$
where $k>0$ is fixed, $\lambda$ is a large (...
2
votes
1answer
78 views
Discrete dynamical system and bound on norm
Let $z \in \mathbb R\backslash \left\{2 \right\}$ then I would like to understand the following:
Consider the dynamical system with $x_i \in \mathbb C^2:$
$$ x_{i} = \left(\begin{matrix} z &&...
2
votes
0answers
62 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
votes
0answers
23 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
293 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
votes
0answers
46 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
105 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
$$\...
6
votes
1answer
150 views
Eigenvalue estimates for operator perturbations
I edited the question to a general mathematical question, since I found the answer in Carlo Beenakker's reference and think that my initial question was mathematically misleading.
What was behind ...
-1
votes
0answers
135 views
what is a limit cycle?
Periodic trajectories in the plane that are stable on one side and unstable on the other side seem to be accepted or rejected as limit cycles depending on the definition.
For example,
$$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
vote
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 $k$, ...
8
votes
1answer
149 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
votes
1answer
74 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:
$$f(m,a):=\int_0^{2 \pi i m} \frac{e^{-t}}{t-a} ...
3
votes
1answer
100 views
Real part of eigenvalues and Laplacian
I am working on imaging and I am a bit puzzled by the behaviour of this matrix:
$$A:=\left(
\begin{array}{cccccc}
1 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 &...
1
vote
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
156 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
36 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 \...
-1
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
vote
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
vote
0answers
129 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
50 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
138 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
votes
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
$$
F(t,q):=\sum_{n=1}^\...
1
vote
1answer
125 views
Time ordered integral involving beta function:
Any help on unpacking integrals of the following type, would be
helpful: $$ \int_0^1 \int_0^r r^a (1-r)^b t^n (1-t)^m dr dt $$ where $a, b, n, m \in \mathbb{N}$ and $0 \le t \le 1$.
Edit/...
27
votes
2answers
815 views
Rademacher theorem
If $f:\mathbb{R}^n\to\mathbb{R}^m$ is of class $C^1$ and $\operatorname{rank} Df(x_o)=k$, then clearly $\operatorname{rank} Df\geq k$ in a neighborhood of $x_o$. It is not particularly difficult to ...
2
votes
2answers
237 views
ODE of the form $y'=\exp(-(\cos(2\pi y))$
Consider the function $h:[0,1]\to \mathbb{R}$
$$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),$$
where $a_{k},b_{k}\in\mathbb{R}$. For ...
3
votes
1answer
170 views
Local “boundary comparison principle” for harmonic functions
Let $u$ be a positive solution of the elliptic equation $\mathcal Lu = 0$ on $B^+_1 \subset \mathbb{R}^n$ such that $u$ vanishes continuously on $\{x_n = 0\}$. To fix ideas, we may take $\mathcal L = ...
2
votes
0answers
33 views
Connection Problem for the Confluent Heun Equation
Consider the Confluent Heun Equation (CHE) written in its non-symmetrical canonical form, i.e,
$$y''(z)+\left(4p+\frac{\gamma}{z}+\frac{\delta}{z-1}\right)y'(z)+\left(\frac{4p\alpha z-\sigma}{z(z-1)}\...
0
votes
0answers
52 views
Find an integral form for a target polynomial
For $x\in \mathbb{R}^d$ and multi-index $J=(j_1,\dots,j_d)$ denote $x^J=x_1^{j_1}\cdots x_d^{j_d}$.
We are given some polynomial: $p(x) = \sum_{J\leq k}\alpha_Jx^J$ where $J\leq k$ means that $j_1+\...
1
vote
1answer
134 views
Closed form expression for this infinite series?
Is there a closed-form expression for this series?
$\displaystyle\sum_{k\geq 1}^\infty \frac{\lambda^k e^{-\lambda}}{k!}\cdot[1-(1-x)^k]\cdot \frac{1}{k}$
Any answers, ideas or references would be ...
0
votes
1answer
80 views
Averaged Parseval Relation for Sampling a Function on Integers
This was asked a long time ago on math.stackexchange with no answers.
Let $f:\mathbb{Z}\rightarrow \mathbb{C}.$ Assume that the support of $f$ is finite, say it is $[1,N],$ and that $\mid f\mid$ is ...
0
votes
0answers
98 views
Asymptotic expansion in epsilon gets worse with more terms?
I'm trying to get an order bound on the following integral as $\epsilon \to 0$:
$$g(\epsilon) = \int_a^{a+\epsilon} f(\epsilon,v) dv$$ where $f$ is an ugly, but smooth, function when $0< a \leq v &...
0
votes
1answer
73 views
Alternate forms of the Bessel equation
I have a question regarding an alternate form of the Bessel equation and how that alternate form translates to the modified Bessel equation and its solution. The modified form is from:
http://...
0
votes
0answers
79 views
Search trajectory point close to line
There is a 2d mechanism:
Link AB can be rigidly tied to a point at a distance less than the radius of the circle R0 with center B to show the trajectories:
If 4 random points are rigidly tied to the ...
