Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.

0
votes
0answers
15 views

Convergence of the asymptotic expansion solution of homogeneous linear ODE of order 2

Consider ODE $w''+pw'+qw=0$, $p$ and $q$ are functions of $z$. Denote $w_1$ and $w_2$ the ODE's two linear independent solutions. As $z\to0$, in which situation: Do both $w_1$ and $w_2$'s ...
3
votes
0answers
102 views

Integral equality of 1st intrinsic volume of spheroid

Computations suggest that $$\int_{0}^{\infty}\int_{0}^{\infty} \sqrt{x+y^2} \cdot e^{-\frac{1}{2}(\frac{x}{s}+s^2y^2)}dxdy=\frac{2}{s}+\frac{2s^2\arctan(\sqrt{s^3-1})}{\sqrt{s^3-1}}.$$ The question ...
-3
votes
0answers
51 views

Higher Derivative of Rational function [on hold]

As is well known, every rational function $R(x)$ has a partial fraction decomposition. i.e.$$ R(x) = \frac{p(x)}{q(x)} = P(x) + \sum_{i=1}^m\sum_{r=1}^{j_i} \frac{A_{ir}}{(x-a_i)^r} + \sum_{i=1}^n\...
5
votes
1answer
104 views

Finding an asymptotic solution for a first order ODE

Given strictly concave function $f(t)$ that satisfies $f'(t)>0$, $f'(t)=o(1)$ (i.e. $\lim\limits_{t\to\infty}f'(t)=0$) , and $f'(t)=\omega\left(t^{-1}\right)$ (i.e. $\lim\limits_{t\to\infty}tf'(t)=\...
2
votes
2answers
228 views

how to calculate the following integral related to Chebyshev polynomials

Chebyshev polynomials of the second kind $V_n(x)$ can be defined as $$V_n(x)=\frac{\sin(n+1)\theta}{\sin\theta}, x=2\cos\theta$$ or through the recurrence relation $$V_{n+1}=xV_n-V_{n-1}, V_0=1, V_1=...
-3
votes
0answers
34 views

show boundedness for a solution of ode [closed]

For $u(t) \ge 0 $ and $a(t)>0$ monotone increasing, and constant $c>0$ such that $$ \frac{d}{dt} u(t) + a(t)u(t)\le c $$ Show that $$\sup_t u(t) < c$$
-2
votes
3answers
159 views

A Specific Linear Homogeneous System of Differential Equations with Variable Coefficients

Is there an analytical solution satisfying these 3 equations with non-constant z? $$\frac{dx}{dt}=-z\cdot\cos(\omega t)$$ $$\frac{dy}{dt}=z\cdot\sin(\omega t)$$ $$\frac{dz}{dt}=x\cdot\cos(\omega t) - ...
9
votes
1answer
176 views

The discrete Hardy-Littlewood-Sobolev inequality

Let $p>1$, $q>1$, $0<\lambda<1$ be such that $\frac{1}{p}+\frac{1}{q}+\lambda=2$. Suppose that $(a_{k})\in \ell^{p}(\mathbb{Z})$ and $(b_{k})\in \ell^{q}(\mathbb{Z})$. It is known ([1,2,3]...
-1
votes
0answers
121 views

Integral of $C^\infty$ Analog of Unit Step Function

The function $$ f(x)\ :=\ e^{1-\frac{1}{\sqrt{1-x^2}}} $$ has the properties that $\frac{d^nf}{dx^n}(\pm 1)=0$, $\frac{d^n}{dx^n}\left(\sqrt{1-x^2}-f(x)\right) = 0$ at $x=0$, and its integral $F(...
23
votes
4answers
898 views

show this nice and hard inequality with $ \prod_{i=1}^{n}|x_{i}-y_{i}|<e^{\frac{n}{2}}$

I saw the following results in a book. The author said it was not difficult to prove how I felt it was difficult to prove, so I asked here. The result comes from a book that has no electronic version....
3
votes
3answers
169 views

ODE with Bessel decay

This is related to my previous question, but it is probably less scary and an expert in using Mathematica could figure out an answer easily. I would like to estimate the asymptotic behaviour of the ...
5
votes
0answers
269 views

Is this proof of Basel identity known?

Today, to divert myself, I tried to find a new proof of Basel identity $\boxed{\sum_{j=1}^\infty\frac{1}{j^2}=\frac{\pi^2}{6}}$. I came up with the following, which essentially interprets the identity ...
12
votes
0answers
402 views

Is it possible that the following integral is $0$?

Given any integer $n\geqslant1$, let $E,F$ be two subsets of $\{\{i,j\}:1\leqslant i<j\leqslant n\}$ such that every two sets in $F$ are disjoint. It is not difficult to see that $$\int_{1<|z|&...
1
vote
0answers
79 views

How many two-dimensional space filling Hilbert-like curves are there?

I'm interested in filling 2d square with space filling, non-self-intersecting, locality preserving, self-similar curves, like Hilbert curve. I found interesting work concerning three dimensional case ...
2
votes
0answers
72 views

Analytic continuation of an NLS soliton

The attractive nonlinear Schroedinger equation $i \partial_t \psi = - \frac {1} {2} \Delta \psi - |\psi|^{p-1} \psi$ in the $H^1$-subcritical case $1 < p$, $\frac {d} {2} + \frac {2} {p-1} < 1$ ...
0
votes
1answer
42 views

Strict positive type function on hypersurface also of positive type in neighborhood?

Let $u\in C^\infty(\mathbb{R}^n\times\mathbb{R}^n)$ be symmetric and of strictly positive type on some hypersurface $S \subset \mathbb{R}^n$ diffeomorphic to $\{0\}\times\mathbb{R}^{n-1}$. This means ...
0
votes
1answer
77 views

Chebyshev interpolation [closed]

Let's define the n-th degree Chebyshev polynomials by $$ T_{n} (x)=\cos(n\arccos(x)).$$ Find a polynomial $P$ such that $$\mid y- P (x) \mid$$ is minimal, using the first three Chebyshev ...
3
votes
0answers
204 views

Periodic orbit for certain Hamiltonian on the tangent bundle

In this question a nontrivial periodic orbit is a periodic orbit which is not a singular point. Let $p: \mathbb{R}^n \to \mathbb{R}$ be a ...
-3
votes
1answer
86 views

connected set of sum of upper semi continuous function [closed]

Let $C(X)$:space of continuous functions on a compact space.Topology $C(X)$ is generated by sup-norm($||T||=sup_{v}\frac{||T(v)||}{||v||}$). Consider $f$ and $g :C(X)\rightarrow \mathbb{R}$ are upper ...
0
votes
0answers
42 views

Function approximation via an orthonormal basis (with singular weight)

If you don't mind, please consider the eigenvalue problem $$ (1-x^2)u''+ \lambda u=0 \ \ \ \forall x\in (-1,1), $$ $$ u(\pm 1) = 0. $$ Observe that for suitable values of $\lambda$, the ODE resembles ...
1
vote
1answer
54 views

additive discrepancy under a multiplicative constraint

Consider four sequences of numbers, $0 \le a_i, b_i, c_i, d_i \le 1$, suppose they satisfy the following constraints: (1). $\sum_{i=1}^K a_i, \sum_{i=1}^K b_i, \sum_{i=1}^K c_i \ge 1/2 + \epsilon$; (...
0
votes
0answers
27 views

Existence of solutions to NLS: Local existence and boundedness

I was wondering when the following argument is valid: Consider a nonlinear Schrödinger equation $$i \partial_t \varphi = -\Delta \varphi+ N(\varphi)$$ where $N$ is a nonlinearity. Often it is ...
-2
votes
1answer
161 views

Strong estimates for the zeta function on natural numbers

Let $$\zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s}$$ be the Riemann zeta function (here we just consider real $s$). We do have a description given by $$\zeta(s) = \frac{s}{s-1}-s\int_{1}^\infty \frac{...
1
vote
1answer
86 views

Quotient with positive second derivative in the limit?

I am studying the quotient of $$f(\varepsilon) = \sum_{i=1}^{\infty} \frac{i^2}{2^{\varepsilon i^2}}$$ and $$g(\varepsilon) = \sum_{i=1}^{\infty} \frac{1}{2^{\varepsilon i^2}}$$ for some $\...
16
votes
0answers
318 views

An integral in Gradshteyn and Ryzhik

Section 3.248 of the 4th edition of the table of integrals by Gradshteyn and Ryzhik contains three entries. They are of elementary examples of the beta function. In the 5th edition there are two new ...
4
votes
0answers
87 views

Injectivity of product functions on natural number sequences

Let $M = \{ a = (a_i)_{i} : a_i \in \mathbb{N}, a_1 \geq 2, a_i > a_j \forall i>j\}$ the set of all ascending natural number sequences, with $a_1$ at least 2. We now define for each $k \geq 2$ ...
1
vote
0answers
85 views

Euclidean volumes under the matrix norm of one-parameter subsets of unit balls of $2 \times 2$ matrices

Prove that the Hilbert-Schmidt volume (normalized to equal 1 at $\varepsilon=1$) of the subset of the unit ball in operator norm (http://mathworld.wolfram.com/OperatorNorm.html) of the $2\times 2$ ...
3
votes
1answer
127 views

Is the convergence of $\dot{x}=2A(t)x$ faster than that of $\dot{x}=A(t)x$?

Let $x \in \mathbb{R}^{n}$ and $A(t) \in \mathbb{R}^{n\times n}$. If $\dot{x}=A(t)x$ and $\dot{x}=cA(t)x$ with $c>1$ are exponentially stable. Is the convergence rate of $x$ to zero of $\dot{x}=cA(...
2
votes
1answer
289 views

Functions belong to $L^{\frac{2n}{n+1}}$ whose Fourier transforms are infinite on $S^{n-1}$

I'm looking for functions $f\in L^{\frac{2n}{n+1}}$ such that $\hat{f}=\infty$ on $S^{n-1}$. Is there any explicit expression of such kind of examples? This seems to be a well-known result, but I can ...
-2
votes
1answer
118 views

Relationship between “Radial” Fourier transform and Fourier transform, especially at infinity

Let $\phi:\mathbb{R}^n \to \mathbb{R}$ be a $C^{\infty}$ function with compact support. What is the relationship between $$ \widehat{\phi}(k) = \int e^{-2\pi i x \cdot k} \phi(x) dx, \quad k \in \...
26
votes
1answer
2k views

A remarkable almost-identity

OEIS sequence A210247 gives the signs of $\text{li}(-n,-1/3) = \sum_{k=1}^\infty (-1)^k k^n/3^k$, also the signs of the Maclaurin coefficients of $4/(3 + \exp(4x))$. Mikhail Kurkov noticed that it ...
3
votes
0answers
31 views

Multi-parameter stationary phase asymptotic expansion

I am looking for an asymptotic expansion of the oscillatory integral of the form $$\int_{\mathbb{R}^n}f(x)\exp(i(\lambda_1\phi_1(x)+\dots+\lambda_k\phi_k(x))dx,$$ as $\lambda_i\to \infty$ ...
3
votes
2answers
120 views

Checking $f(x_1,y_1)f(x_2,y_2)-f(x_1,y_2)f(x_2,y_1) \ge 0$

I am working in data science and I have to deal with the following problem for which I would like to find a simplification: We call a function almost positive if $f(x_1,y_1)f(x_2,y_2)-f(x_1,y_2)f(x_2,...
4
votes
1answer
123 views

Method of characteristics beyond the Lipschitz setting

I have come across the following easy-looking problem that is driving me mad. I have a family of measures (on the real line $\mathbb R$) $\{\mu_t\}_{t>0}$ which is uniformly bounded (the measures ...
3
votes
1answer
238 views

Asymptotic solution for a first order ODE

Simplified question*: Given $f(t)$ that satisfies $f'(t)>0$, $f'(t)=\omega\left(t^{-1}\right)$, $\log\left(f'(t)\right)=o\left(f(t)\right)$ we denote $F=\exp\left(f\left(t\right)\right)$. Let $H(...
3
votes
1answer
131 views

Exponential map/ Lie derivative in variation for constant formula for ODE

In short: The question is how to go from the first equation on page 8, of this paper to the second equation. Some background I'm working in optimization and I am currently reading a paper see page ...
3
votes
1answer
187 views

Question on the definition of almost periodic function

According to Bohr, the definition of the almost periodic function is: A function $f:\mathbb{R}\rightarrow \mathbb{C}$ is called almost periodic if it is continuous and if for every positive $\epsilon$,...
12
votes
4answers
685 views

History of ODE and PDE reference request

Is there any reference (book or articles) which made the history (up to the modern times) and the conceptual development of Ordinary Differential Equations and Partial Differential Equations? It will ...
9
votes
2answers
1k views

Difficult trigonometric integral

This question was also asked here and here. I have faced some difficulties to do the following integral: $$ I=\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta~\sin\theta\int_{0}^{\infty}dr~r^2\frac{3x^2y^...
2
votes
1answer
128 views

Riemann-Stieltjes integral as a limit of Riemann integrals

Let us suppose that $f, g:(A, B)\to \mathbb{R}$ are both continuous on $(A, B)$ and for $[a, b]\subset (A, B)$, suppose that $g$ is of bounded variation on $[a, b]$ (we may add, if necessary, that ...
6
votes
2answers
280 views

asymptotic for li(x)-Ri(x)

Is it true that $$\operatorname{li}(x)-\operatorname{Ri}(x) \sim \frac{1}{2}\operatorname{li}(x^{1/2}) \ (x \to \infty),$$ where $$\operatorname{Ri}(x) = \sum_{n = 1}^\infty \frac{\mu(n)}{n} \...
7
votes
1answer
371 views

Dominated convergence 2.0?

During my research, I came across the following question. Let $(f_n)_n$ be a sequence in $C^2([0,1])$ converging pointwise to $g \in L^1([0,1])$. Assume that: $\forall n\in\mathbb N, f_n''<h$, ...
1
vote
0answers
52 views

Linear dependence of solution?

Consider the function $f_k(c):=\sum_{n=0}^{\infty} c^{n^k}$ where $k\ge 1$ is an integer. This one obviously converges for $\left\lvert c \right\rvert <1.$ In the following we want to study the ...
3
votes
2answers
284 views

Prove $\int_0^{\infty}{\frac{1}{e^{sx}\sqrt{1+s^2}}}ds < \arctan\left(\frac1x\right),\quad\forall x\ge1$

The question is to prove: $$ \int_0^{\infty}{\frac{1}{e^{sx}\sqrt{1+s^2}}}ds < \arctan\left(\frac1x\right),\quad\forall x\ge1. $$ Numerically it seems to hold true. So I have made some attempts to ...
4
votes
1answer
71 views

Convergence of a real sequence (stochastic approximation)

Let $(\gamma_n)_{n \geq 1}$ be a sequence of positive real numbers satisfying $$\sum_{n \in \mathbb{N}}\gamma_n = + \infty \text{ and }\sum_{n \in \mathbb{N}}\gamma_n^2 < + \infty$$ I would like ...
1
vote
2answers
170 views

Fourier transform of a generalized function on the plane

Is there an explicit formula for the Fourier transform of the generalized function of 2 variables $$\frac{1}{x+y^2+i0}?$$ Remark. Equivalent question: consider the Schroedinger equation one the ...
6
votes
0answers
164 views

Sobolev space under Mellin transform

The Mellin transform is known to be an isomorphism see wikipedia between $M:L^2(0, \infty) \rightarrow L^2(-\infty, \infty)$ where $$M(f):= \frac{1}{\sqrt{2\pi}}\int_0^{\infty} x^{-\frac{1}{2} + is} ...
8
votes
2answers
291 views

Hölder continuity for operators

Let $x,y$ be positive real numbers then $$|\sqrt{x}-\sqrt{y}|=\dfrac{|x-y|}{\sqrt{x}+\sqrt{y}}=\sqrt{|x-y|}\cdot \dfrac{\sqrt{|x-y|}}{\sqrt{x}+\sqrt{y}}\leq 1\cdot |x-y|^{\frac{1}{2}}$$ we obtain $1/...
4
votes
1answer
166 views

Wavefront set and Duhamel's principle

Consider the Cauchy problem: $$ \frac{\partial u}{\partial t} + \mathrm{i}\mkern1mu A(x,D_x) u = f \quad 0< t < T; \qquad u = u_0 \quad \text{when}\; t = 0, $$ where $A$ has real principal ...
7
votes
1answer
197 views

Proof of Green's formula for rectifiable Jordan curves

$\newcommand{\Ga}{\Gamma}$ I am trying to find a proof of Green's formula for rectifiable Jordan curves $\Ga$ (and the corresponding interior regions $R$). There is a proof by Ridder, followed by ...