# 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.

2,175 questions

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33 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

87 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

29 views

### Integral representation of the Digamma function vs. Integral representation of the Polygamma function

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

110 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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56 views

### Spectrum of non compact operator is 0 [migrated]

Can you please help me to give an example of non compact operator which spectrum is {0}

**1**

vote

**1**answer

122 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{...

**3**

votes

**1**answer

123 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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**0**answers

85 views

### 3D Homogenous Laplace equation with integral boundary conditions

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

**3**

votes

**0**answers

48 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

49 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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**0**answers

101 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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**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**

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**1**answer

141 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

121 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/...

**23**

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**1**answer

472 views

+50

### 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

232 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 ...

**4**

votes

**0**answers

122 views

+100

### 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**

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**0**answers

32 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**

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**0**answers

51 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**

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**1**answer

127 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**

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**1**answer

72 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**

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**0**answers

96 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

69 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 ...

**4**

votes

**3**answers

151 views

### Uniqueness of minimizers in a problem in the Calculus of Variations - Part II

Take any convex set $A\subset\mathbb{R}^n$ which contains a neighborhood of the origin and let $f_A$ be the associated Minkowski functional
$$
f_A(x) = \inf\{\lambda>0\mid x\in\lambda A\},
$$
which ...

**1**

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**1**answer

67 views

### Product of independent random variables and tail deconvolution

Suppose $X, Y$ are two independent non-negative random variables. The conditions
(i) $\mathbb{P}(X > t) = \frac{C}{t^p} + o(t^{-p})$
(ii) $\mathbb{P}(Y > t) = o(t^{-q})$ for any $q > ...

**3**

votes

**2**answers

126 views

### Example of convex functions fulfilling a (strange) lower bound

I am reading a preliminary version of a paper which focuses on some minimization problems connected to a class of integral functionals. Reading the assumptions of one of the theorems I cannot convince ...

**2**

votes

**1**answer

72 views

### Uniqueness of minimizers in the Calculus of Variations

Let $f \colon \mathbb R^2 \to \mathbb R$ be the function defined by
$$
f(x,y):= (x^+)^2 + (y^+)^2
$$
where $a^+ = \max\{a,0\}$ for any real number $a$.
Given a Lipschitz regular domain $\Omega \...

**2**

votes

**1**answer

87 views

### The regularity of ODE with Zygmund coefficients

A zygmund function $f\in\mathscr C^1$ is a continuous function satisfies $|f(x+h)+f(x-h)-2f(x)|\le C|h|$ for all $x,h\in\mathbb R^n$ in the domain.
According to Markus' paper A uniqueness theorem for ...

**1**

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**0**answers

59 views

### How to prove a set of delay differential equations never converge (the delay is not constant)

Two functions $x(t)$ and $y(t)$ are coupled via:
$$\dot x(t) = a y(t)-b,y(t+x(t))=x(t)$$
where $a<0$, $b\neq 0$ is some constant.
I am mostly confused with the second equation. What is the ...

**8**

votes

**2**answers

839 views

### Differentiating an integral that grows like log asymptotically

Suppose I have a continuous function $f(x)$ that is non-increasing and always stays between $0$ and $1$, and it is known that
$$ \int_0^t f(x) dx = \log t + o(\log t), \qquad t \to \infty.$$
...

**1**

vote

**1**answer

74 views

### Interpolation of a trilinear functional

Let $f,g,h\in L^2([0,1]^2)$ and let $K:\mathbb{R}^3\to \mathbb{C}$ be some smooth kernel with support containing $[0,1]^3$. Denote by $\|f\|_2$ the $L^2([0,1]^2)$ norm of $f$, and same with $g,h.$ If ...

**3**

votes

**1**answer

201 views

### A certain generalisation of the golden ratio

Consider a real number $a \ge 1$, and let $g(a)$ be the unique positive solution $x>0$ of $x^a - x^{a-1} - 1 = 0.$
We have $g(1) = 2$, $g(2) = {1+\sqrt{5}\over2}$ (the golden ratio), and $g$ is ...

**1**

vote

**0**answers

114 views

### On the infimium of a functional

Let $(M^n,g)$ be a closed Riemannian manifold. Define
$$\lambda(g)=\inf\{\mathcal{F}(g,f),\;0<f\in C^{\infty}(M),\; \int_Mf^2\;d\nu=1\},$$
where $$\mathcal{F}(g,f)=\int_M\left(|\nabla f|^2+ af^2\...

**9**

votes

**1**answer

2k views

### Is There an Induction-Free Proof of the 'Be The Leader' Lemma?

This lemma is used in the context of online convex optimisation. It is sometimes called the 'Be the Leader Lemma'.
Lemma:
Suppose $f_1,f_2,\ldots,f_N$ are real valued functions with the same ...

**0**

votes

**1**answer

64 views

### Bringing a Heun equation into canonical form

It is a well known fact that any second order Fuchsian differential equation on the complex plane $$u''(x) + p(x)u'(x) + q(x)u(x)=0$$ with exactly $4$ regular singular points may be suitably ...

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**0**answers

60 views

### Critical growth and geodesic connectedness in Lorentz manifold

What is the deep ("heuristic") reason why the quadratic growth of $\beta$ is critical for the study of geodesic connectedness in standard static Lorentz spacetime $\mathcal M = \mathcal M_0 \times \...

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**0**answers

34 views

### Anisotropic elliptic problems

Where can I find a reference on the following anisotropic problem?
$$(\bullet) \qquad \begin{align*} \nabla_x (A(x) \nabla_x u(x,y)) + \Delta_y(K(y) \ast u(x,y))\end{align*}=0, $$
where $(x,y) \in \...

**3**

votes

**1**answer

122 views

### Lower estimate of the minimal eigenvalue of a Hamiltonian

Consider a linear operator $H\colon L^2(\mathbb{R}^3)\to L^2(\mathbb{R}^3)$ given by
$$H(\psi)(x):=-\Delta\psi(x)+V(x)\cdot \psi(x),$$
where $V\colon \mathbb{R}^3\to \mathbb{R}$ is a continuous (or ...

**2**

votes

**1**answer

67 views

### Elliptic interface problem without conditions on the interface

Consider an open domain $U$ split in two non-overlapping subdomains: $U = U_1 \cup U_2$.
For a model case, consider a ball split in a smaller ball and an anulus.
Consider the following elliptic ...

**6**

votes

**1**answer

469 views

### Famous but unavailable paper of Jan Boman

The following paper is well known, but hard to find:
J. Boman, $L^p$-estimates for very strongly elliptic systems, Report 29, Department of Mathematics, University of Stockholm, 1982.
In this paper ...

**0**

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**0**answers

30 views

### Geodesic connectedness in static Lorentz manifold vs connectedness by trajectories with potential in Riemann manifold

What is the relationship between the study of geodesic connectedness in a standard static Lorentz manifold and the connectedness of two points by trajectories with potential (i.e. solutions to $x''(t) ...

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**0**answers

56 views

### Prove that an iterative estimate implies Holder continuity

Let $u$, $w$ be nonnegative continuous functions such that $\frac{u}{w}$ is bounded on $B_{2^{-1}}$. Why the inequality
$$a_k \le \frac{u}{w} \le b_k \quad \text{ on $B_{2^{-k}}$} , \qquad b_k - a_k ...

**12**

votes

**3**answers

334 views

### (Sharp) inequality for Beta function

I am trying to prove the following inequality concerning the Beta Function:
$$
\alpha x^\alpha B(\alpha, x\alpha) \geq 1 \quad \forall 0 < \alpha \leq 1, \ x > 0,
$$
where as usual $B(a,b) = \...

**2**

votes

**1**answer

86 views

### Approximate sequence of numbers

Let $n \in \mathbb N$ and $k_n \in \left\{0,..,n \right\}$ then we define the numbers
$$x_{n,k_n} = \frac{k_n+n^2}{n^3+n^2}.$$
It is easy to see that these numbers satisfy
$$x_{n,0} = \frac{1}{n+1} ...

**0**

votes

**1**answer

65 views

### Exterior cone condition for $\mathrm{supp}\, u$ and Lebesgue points of $u$

Let $u:\mathbb{R}^n \to \mathbb{R}$ be an $L^1$ function with compact support. Let $\bar x \in \partial \mathrm{supp}\, u$ and assume that $\mathrm{supp} \, u$ satisfies the exterior cone condition at ...

**1**

vote

**1**answer

126 views

### How to recognize if a continuous vector field in the Euclidean space is a gradient

How to recognize, by "analytic" methods, if a $C^0$ vector field $v:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is the gradient of a function $h:\mathbb{R}^n \rightarrow \mathbb{R}$, given that the ...

**1**

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**0**answers

25 views

### Fredholm integral equation of third kind

Let us consider the following integral equation
$$a(x)u(x) + \int\limits_0^2 {K(s,x)u(s)ds} = f(x)$$
Let f in $L^p(0,1)$ for some $p \in [1,\infty]$ and let $K \in L^q((0,2) \times (0,1))$. Assume ...

**1**

vote

**0**answers

57 views

### On the convergence of an integral of Hardy's maximal function

Let $f:\mathbb{R}\times \mathbb{R}^N \to \mathbb{R}^N$ be an $L^1$ function.
Assume that
$$\mathcal M f(x,y) = \sup_{r< \bar r}\frac{1}{B_r(y)} \int_{B_r(y)} f(x,z)dz \to 0 $$ as $\bar r \to 0$ ...

**1**

vote

**0**answers

56 views

### Time-varying perturbations of continuous-time hyperbolic orbits

My question is the following: Assume that the flow of an autonomous ODE $\dot{x} = f(x)$ ($f$ is $C^1$) has a periodic hyperbolic orbit $\varphi^t(x_0)$, $\varphi^{t+T}(x_0) = \varphi^t(x_0)$. Then ...