# 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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### 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(...

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**1**answer

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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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 $n^{1/p}$.

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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}^{\...

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**3**answers

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 ...

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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),
\...

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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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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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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 ...

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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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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 ...

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**1**answer

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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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\...

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**1**answer

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 ...

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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)...

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**1**answer

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\...

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**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} \...

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**1**answer

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 (...

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**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 &&...

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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 ...

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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(...

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**1**answer

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 ...

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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 $...

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**1**answer

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,\...

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**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
$$\...

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**1**answer

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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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{...

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**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 $\...

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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 $k$, ...

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**1**answer

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 $\...

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**1**answer

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**

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**1**answer

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 &...

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**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 $z>1$....

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**3**answers

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**

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**0**answers

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^...

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**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 \...

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**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 @...

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**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 \...

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**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{...

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**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 ...

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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,...

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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 ...

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**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
$$
\inf_{x \in U} f(x) >\inf_{x \in \mathbb{R}^d} f(x),
$$
...

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**1**answer

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 ...

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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 ...

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**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}^\...

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**1**answer

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/...

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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 ...

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**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

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 = ...