**5**

votes

**1**answer

167 views

### Unicity of additive, $(-1)$-homogeneous, and shift invariant probability measures on $\mathbf N^+$

Let $\mathcal D$ be the set of all (finitely) additive probability measures $\mu^\ast: \mathcal P(\mathbf N^+) \to [0,\infty[$ such that $\mu^\ast(k \cdot X + h) = \frac{1}{k} \mu^\ast(X)$ for all $X ...

**2**

votes

**0**answers

66 views

### Does there exist $k\ge2$ s.t. $X \subseteq \mathbf N^+$ has positive upper Banach density if the counting function of $X$ is $\gg n/\log^{[k]}(n)$?

Does there exist an integer $k \ge 1$ such that ${\sf bd}^\ast(X) > 0$ whenever $X \subseteq \mathbf N^+$ and $\pi_X(n) \gg \frac{n}{\log^{[k]}(n)}$ as $n \to \infty$? Here, ${\sf bd}^\ast$ is the ...

**0**

votes

**0**answers

85 views

### The following ODE global existence theorem reference?

There is an ODE existence theorem of the form:
Let $f:[a,b]\times \mathbb{R}^n \to \mathbb{R}^n$ be a Caratheodory function.
Suppose that there is a constant $c$ such that if $y$ is a ...

**2**

votes

**2**answers

136 views

### Are all mixtures of these unimodal functions unimodal?

Let us say that a function $F\colon(0,\infty)\to\mathbb{R}$ is increasing-decreasing if, for some $c\in[0,\infty]$, $F$ is non-decreasing on $(0,c]$ and non-increasing on $[c,\infty)$. Is it true that ...

**4**

votes

**1**answer

250 views

### Ref. request: Additive probability measure on $\mathcal P({\bf N})$ supplies subset of $\mathbf R$ without Baire property

ZFC proves, among the other things, the existence of a (finitely) additive probability measure $\theta: \mathcal P(\mathbf N) \to \mathbf R$ on the power set of $\mathbf N$ such that $\theta(X) = 0$ ...

**2**

votes

**1**answer

85 views

### Numerical solution of singular ODE

Consider the singular ODE
$y''+\frac{y'}{r}+p(r)y=0 \ \ with \ \ y(0)=1 \ \ and \ \ y'(0)=0$.
Theoretically such solution exists and is unique if $p$ is nice. Is there a method to numerically ...

**1**

vote

**0**answers

177 views

### p-adic asymptotic analysis

There is a huge field of asymptotic expansions and such over the real and complex fields (see Bender and Orszag, or Copson, or Whittaker and Watson). How different is the theory over p-adic fields? ...

**4**

votes

**0**answers

164 views

### Evaluate a multiple integral

I want to compute this integral and I would appreciate any help: $N\geq 1$ is fixed.
$$I_N=\int_{0\le r_n\le r_{n-1}\le\cdots\le r_1} e^{-(r_1^2+\cdots+r_n^2)} \prod_{i<j} \sinh(r_i-r_j) ...

**1**

vote

**0**answers

128 views

### First to note the relation between Stasheff polytopes (associahedra) and compositional inversion?

In my answer to MO-Q: Enumerative geometry and nonlinear waves, I outline the relation between the refined face polynomials of the Stasheff polytopes (associahedra) and the partition polynomials for ...

**8**

votes

**1**answer

395 views

### The space of polynomials with all real roots

The question stems from an attempt to answer a question of David Speyer. Let $R \subseteq \mathbb{R}^n$ be the space of coefficients of all polynomials of degree $n$ whose all roots are real, i.e.
$R ...

**49**

votes

**3**answers

5k views

### Is this differential identity known?

Recently I discovered the differential identity
$$ \frac{d^{k+1}}{dx^{k+1}} (1+x^2)^{k/2} = \frac{(1 \times 3 \times \dots \times k)^2}{(1+x^2)^{(k+2)/2}}$$
valid for any odd natural number $k$; for ...

**2**

votes

**1**answer

136 views

### Construct smooth functions with prescribed derivatives

To be specific, suppose we are given a sequence of smooth functions $\{f_k\}_{k\geq 0}$ on flat torus $\mathbf{T}^2$(you may think of it as doubly periodic functions on $\mathbf{R}^2$ and smooth).
...

**5**

votes

**1**answer

248 views

### Dominated convergence to characteristic function

Let $\phi_m(x):=\chi_{[0,1]} * \chi_{[0,1]} *...* \chi_{[0,1]}$ be the m -times convolution (so $m+1$ characteristic functions are involved).
Then the Fourier transform of this function is given by
...

**3**

votes

**2**answers

371 views

### Who needs a symmetric upper asymptotic density on the integers?

The upper asymptotic density on $\mathbf Z$, viz. the function
$$
{\sf d}^\ast: \mathcal P(\mathbf Z) \to [0,1]: X \mapsto \limsup_{n \to \infty} \frac{|X \cap [1,n]|}{n},
$$
has a ''symmetric ...

**5**

votes

**0**answers

51 views

### Value of prolate speroidal wave function at 0

I have a very basic question about prolate speroidal wave functions that can be defined as eigenfunctions to the following integral equation:
$$
\lambda\cdot \psi(x) = \int_{-1}^{1}\frac{\sin ...

**24**

votes

**0**answers

496 views

### Finding a path through real rooted polynomials

This is a lemma I wanted in order to solve Patrizio Neff's conjecture. It turned out to be the wrong way to think about it, but I am still curious if it is true.
Let $z^n+a_{n-1} z^{n-1} + \cdots + ...

**7**

votes

**2**answers

301 views

### Radial limit does not exist almost everywhere

Problem 4 in Chapter 4 of Stein's book "Real Analysis" says
$\sum_{n\geqslant 0}z^{2^n}$
doesn't have radial limit as $z$ approaches the unit circle from inside almost everywhere. It's fairly easy ...

**3**

votes

**0**answers

100 views

### Reference for a lemma on the asymptotic upper density of special sets with large gaps and intervals

Update. Based on Anthony Quas' comment below, the proof can be made sensibly shorter and the lemma can be slightly generalized by weakening the old assumption (iii).
In a joint paper that I am ...

**3**

votes

**1**answer

172 views

### A question on the Frechet derivative

Suppose the derivative of a functional is given by
\begin{equation*}
\int_{\Omega}(\vec{v}.\nabla u)|\nabla u|^{p-2} \phi=\int_{\Omega}\nabla.(u\vec{v})|\nabla u|^{p-2} \phi,~\phi\in ...

**3**

votes

**1**answer

84 views

### Hyperelliptic generalization of Euler's formula

Are there any hyperelliptic generalizations of the following formula, first proved by Euler in 1782,
...

**0**

votes

**0**answers

51 views

### Compact embedding for Bochner spaces

I was reading some topics about the Bochner integral, and this question arises:
Assume we have a sequence $u_n \in H^s(0,2\pi,L^2(\mathbb{T}))$ converging weakly to $v$ for all $s \in (\frac12, ...

**10**

votes

**1**answer

309 views

### On the independence of lower and upper asymptotic and Banach densities

Given a set $X \subseteq \mathbf N^+$, denote by $\mathsf{d}_\ast(X)$ and $\mathsf{d}^\ast(X)$, respectively, the lower and upper asymptotic (or natural) density of $X$, viz. $$\mathsf{d}_\ast(X) := ...

**9**

votes

**1**answer

317 views

### Sard's Theorem For Banach Spaces

Given a smooth map from $\phi: B \rightarrow M$ where $B$ is a Banach Space and $M$ is a finite dimensional smooth manifold (for example, the end point map for a control system), what is the strongest ...

**0**

votes

**1**answer

114 views

### Quadratic stability of linear time varying system

(This question was originally asked at Math.SE, where it didn't receive any answers.)
Consider the linear time-varying system
$$ \dot{x} = A(t) x, $$
where $x \in \mathbb{R}^n$ and $A: [0,+\infty) ...

**0**

votes

**0**answers

138 views

### Closed form for convolution of two-dimensional Gaussian with characteristic function of a disk

Is there a closed form expression for the convolution of a two-dimensional (elliptical) Gaussian function with the characteristic function of the interior of an ellipse?
The motivation is that I have ...

**3**

votes

**1**answer

227 views

### An integral equation

I have a Fredholm integral equation of second kind
$$\frac{-1}{2\pi \omega'^2}+\int_{-\infty}^{\infty}\frac{1}{\pi ...

**0**

votes

**0**answers

67 views

### Derivative of a conjugation of matrices

Let $\mathcal{M}_n$ be the space of complex $n\times n$ matrices. Let $\Phi\colon \mathbb{D}\to \mathcal{M}_n$ and $\psi \colon \mathbb{D}\to \mathcal{M}_n$ be holomorphic functions. Consider the ...

**5**

votes

**1**answer

106 views

### Gradient flow in simple settings

I would like to solve an equation of the type $\partial_t X = \nabla U (X)$ in $\Omega \subset\mathbb{R}^2$ with an initial condition.
I know that under some conditions, $X(t)$ will converge to a ...

**1**

vote

**0**answers

51 views

### help with an asymptotic estimate for a certain product

(I apologize in advance if this question is not suitable for Math Overflow, but it came up in a research problem and thought perhaps I could find some help here.)
I'm having difficulty finding an ...

**6**

votes

**2**answers

763 views

### A (likely) positivity property of the Lerch zeta-function

The problem is to show that $\Re L(b/2,1/2,p+1)>0$ for all real $b\ne0$ and all real $p>-1$, where
$$L(\lambda,c,s):=\sum_{k=0}^\infty\frac{\exp(2\pi i\lambda k)}{(k+c)^s}$$
is the Lerch ...

**2**

votes

**1**answer

130 views

### On nonlinear fractional field equations

In a paper by Vazquez I read that an interesting variant of fractional diffusion (where the fractional operator is usually $(-\Delta)^s$) could be $(I-\Delta)^s$. Therefore I am wondering if there are ...

**4**

votes

**1**answer

109 views

### Uniform boundedness of eigenfunctions of an elementary differential operator

Consider a differential operator
$L=(-1)^ma(x)\frac{d^{2m}}{dx^{2m}}$, with boundary value conditions
$u^{(j)}(0)=u^{(j)}(1)=0, j=m,m+1,\ldots,2m-1$,
where $m\ge1$ is an integer and $a(x)>0$ is ...

**2**

votes

**1**answer

307 views

### Does the following type of Gronwall inequality hold?

Let $I=[0,b)$, $b< \infty$. Suppose $u$ is a positive bounded measurable function on $I$. $v(s)$ is a positive, smooth function on $I$. Note that $u(b),v(b)$ may be $0$.
Suppose that
$$
u(t) ...

**0**

votes

**0**answers

94 views

### On isolated points of the approximate point spectrum of a bounded operator

Let $X$ be a complex Banach space and $T$ be a bounded operator acting on $X$.
Let $\sigma(T)$ and $\sigma_{ap}(T)$ denote the spectrum and approximate point spectrum of $T$, respectively.
Let ...

**1**

vote

**1**answer

97 views

### Derivative of a time evolution operator w.r.t. a parameter

Let $N\geq1$ be an integer and let $H:[0,1]^2\to\mathbb C^{N\times N}$ be a pointwise hermitean matrix valued function.
For $y\in[0,1]$ and $0\leq a\leq b\leq 1$, let $U_y(b,a)$ be the time evolution ...

**5**

votes

**0**answers

114 views

### behaviour of first eigenfunction near the boundary

Consider $\Omega$ a smooth bounded domain in $R^N$ and suppose $ \phi_1(x)>0$ is the first eigenfunction of $ -\Delta$ in $H_0^1(\Omega)$ normalized however one chooses.
My interest is in how $ ...

**6**

votes

**2**answers

353 views

### Existence of a measure-preserving bijection

Let $f, g \, \colon \mathbb{R}^n \rightarrow \mathbb{R}$ be two Borel-measurable functions such that $f$ is non negative and
g is radially symmetric,
the function $ (0, \infty )\ni t \mapsto g ...

**2**

votes

**0**answers

92 views

### Motivation for the existence of periodic solutions [closed]

I have been reading the book Critical Point Theory and Hamiltonian System by Mawhin and Willem, as well as several other papers on the existence of periodic solutions for equations of the form
...

**1**

vote

**0**answers

58 views

### Characterization of the maximizer of a function based on a parameter's value

Consider a smooth, continuously differentiable, and jointly concave function $f(x,y,z;a)$, where $x,y$ and $z$ are decision variables and $a$ is a problem parameter.
I have two optimization problems. ...

**5**

votes

**3**answers

555 views

### Is there an analytic solution for this partial differential equation?

The Fokker-Planck equation for a probability distribution $P(\theta,t)$:
\begin{align}
\frac{\partial P(\theta,t)}{\partial ...

**3**

votes

**1**answer

216 views

### methods for situations where well-posedness criteria hold but global solutions do not exist

I have been learning PDEs (more specifically, nonlinear dispersive equations (Schrödinger/wave/ Klein-Gordan equations etc...)) through the harmonic analysis methods. And I have read a couple of ...

**0**

votes

**0**answers

61 views

### Orthogonality relation for associated Legendre functions

The associated Legendre Polynomials $P_l^m(x)$ obey orthogonality relations for fixed $m$ and fixed $l$:
$$
\begin{align}
\int_0^\pi P_k^m(\cos\theta)P_l^m(\cos\theta)\sin\theta d\theta ...

**1**

vote

**0**answers

44 views

### A fundamental lemma involving a certain exponential kernel

Let $h \in L^1(\mathbb R^n, \mathbb R)$ be a scalar field and let $\Psi_t: \mathbb R^n \to \mathbb R$ be smooth mappings, parameterized by $t \in \mathbb R$.
Suppose that we are given data $$D(v,t) = ...

**9**

votes

**1**answer

140 views

### Nonconventional ergodic averages for commuting transformations

Let $S$ and $T$ be commuting measure-preserving transformations of a standard probability space $(X,\mu)$, so $S$ and $T$ define an action of $\mathbb{Z}^2$ on $(X,\mu)$. I am wondering about ...

**1**

vote

**0**answers

59 views

### persistence of regularity for nonlinear Klein-Gordon equation

I have been reading the paper on nonlinear Klein-Gordon equation(NLKG) for initial data in modulation space: For detail please see the paper "Klein-Gordon Equations on Modulation Spaces (2014)" ...

**2**

votes

**1**answer

125 views

### Is every set of small measure contained in an open set of small measure with null boundary?

Let $\lambda( \cdot )$ denote Lebesgue measure on $[0,1]$. Let $(A_n)_{n=1}^\infty$ be a decreasing sequence of Borel subsets of $[0,1]$ such that $\bigcap_{n=1}^\infty A_n = \emptyset$. Given ...

**2**

votes

**2**answers

190 views

### “C^0 estimate for solutions to $\Delta(u)+e^{-u} \geq 0$”

Let $u: \mathbb{R}^2 \to \mathbb{R}.$ Suppose I have a solution to the equation
$$\Delta(u)+e^{-u} \geq 0$$ on $\mathbb{R}^2$. Let r be the radial coordinate on $\mathbb{R}^2$. Suppose that $$lim_{r ...

**6**

votes

**2**answers

392 views

### Texts about Dwork's work

I want to ask about references to papers, that probably exist, which explain the articles of Bernard Dwork starting from "The rationality of the zeta function of an algebraic variety" to "On the ...

**2**

votes

**0**answers

101 views

### Oscillation aspects of two-way infinite alternating series (a followup from the MO-question “functions that eventually oscillate”)

In the recent question on "eventually oscillating function" I had a heuristic for the function $d(x)$ that its amplitude is constant, but could not further describe that function. I just found a ...

**0**

votes

**0**answers

52 views

### What can we say about variational energies here?

Let $V_{ij}^{lk}$ be any $nm \times nm$ real symmetric matrix, $\forall i,j,k,l$
\begin{equation}
V_{ij}^{kl}=V_{ji}^{lk}
\end{equation}
(So for the indices we have $1 \leq k,l \leq m$ and $1 \leq i,j ...