# 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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**3**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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 ...