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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5answers
503 views

Elementary inequality generalizing convexity of a function on a segment

I am looking for a proof of the following statement which is known to be true as far as I heard. Let $g\colon [a,b]\to \mathbb{R}$ be a smooth function. Assume that $$b-a< \pi.$$ Assume also $$g(a)\...
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0answers
24 views

Bounding Greens function matrix elements in terms of the diagonal elements

Consider the Hilbert space $l^2( \mathbb{Z}^2)$ and suppose that I have a unitary band matrix. I.e. $ \langle e_j , U {e_k} \rangle = 0 $ for say $\vert \vert j-k \vert \vert > 2 $ (in say taxi-cap ...
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1answer
64 views

Variational formulation of abstract Cauchy problem, possible?

Recently I have come across a method known as "variational method" in which we try to establish weak solutions of various boundary value problems involving ordinary derivatives, partial ...
2
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1answer
93 views

Functions with a Jacobian whose columns are orthogonal

I am interested in vector fields whose Jacobian has orthogonal columns; i.e. if $\mathbf{f}(\cdot):\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a function where $\mathbf{f}(\mathbf{x})=[f_1(\mathbf{x}...
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0answers
45 views

Optimal control of nonlinear harmonic oscillator

Consider the ODE $$ \begin{cases} x''(t) + \sin (x(t)) = u(t) \\ x(0)=x_0\\ x'(0)= x_1 \end{cases} $$ and the problem of minimizing $$J(u) = \int_0^T |x(t) - \bar x|^2 dt + \int_0^T u^2(t) dt$$ for $...
12
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1answer
379 views

Possible limit involving the gamma function

Does $$\lim_{n \to \infty} \int_{0}^{1} \Gamma(x)^{n/(n+1)}dx - n$$ exist? Here's some background. The integral $$\int_{0}^{1} \Gamma(x) dx$$ diverges rather slowly. Inserting the exponent $n/(n+1)$ ...
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1answer
105 views

Asymptotics for solution of transport equation and characteristics

Consider the transport equation $$u_t(t,x) + v(t,x) \cdot \nabla u(t,x) = 0.$$ Suppose that the solution of the characteristic equation $$\dot X(t) = v(t,X(t)) $$ decays to zero as $t \to \infty$. ...
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0answers
50 views

transform $ \phi '' + ( 1 +c^2/4 -|\phi |^2)\phi = 0 $ into $ \varphi '' + ( 1 - |\varphi |^2)\varphi = 0$ [closed]

Assume that $\psi: \mathbb{R}\to\mathbb{C}$ is a solution of $\psi '' + i c\psi ' + (1-\vert\psi\vert^2)\psi = 0$, where $i^2 = -1$ and $c\in (0,\sqrt{2})$. Applying the transformation $\Phi (\psi)=e^{...
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1answer
76 views

Fourier transform of a function of bounded variation

I know if $f\in L^2(\mathbb R)$ is two times continuously differentiable, then we must have that the Fourier transform is integrable. Is there any more relaxed condition than this? For example if $f$ ...
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0answers
38 views

Integrability of Fourier transform of truncated fractional power

Is the Fourier transform of the function $f$ which agrees with $1_{[-1.1]}|x|^\alpha$ on $[-1,1]$ and then decays very fast to zero to become a compactly supported continuous function, is in $L^1(\...
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1answer
144 views

Existence of entire function that yields periodicity

I have the following question: Does there exist an entire function $f(z)$ where $z=x+iy$ such that $$g(x,y) =e^{-2\pi y^2}f(z)$$ is periodic in both $x$ and $y$ direction, i.e. $$\forall x,y: g(1,y)=g(...
5
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1answer
264 views

A differential inequality involving gradient and laplacian

Let $V:\mathbb{R}^{n}\to\mathbb{R}$ smooth, such that $\lim_{|x|\to\infty}V(x)=+\infty$. What are conditions on $V$ that guarantee the existence of a function $U:\mathbb{R}^{n}\to\mathbb{R}$ such that ...
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0answers
19 views

Tightness of matrix hypergeometric bound

In Ratnarajah–Vaillancourt–Alvo (link), the authors write (on pg 3) that the following inequality for the Hypergeometric function of matrix argument holds: $${}_0F_1(b; X) < {}_0F_0(X/b)$$ where $b$...
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0answers
42 views

Brachistochrone for a rolling sphere with slippage

I was recently looking into generalisations of the brachistochrone problem: for example, in this article the authors study the brachistochrone with Amontons-Coulomb friction where a bead slides along ...
2
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0answers
91 views

Bounds for associated Legendre polynomials

I am trying to analyze the behaviour of the Associated Legendre polynomials $P_{n}^{m}$ on $[0,1]$. More specifically, I am trying to get upper bounds for $P_{n}^{m}$ on $[0,1]$. Bernstein's ...
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0answers
103 views

Sobolev extension from a discrete set of points

Let $1 > \alpha > 0$ and fix some $C > 0$. Consider $\Omega \subset \mathbb{R}^n$ a bounded domain and $Y \subset \Omega$ a discrete (finite) set of points. For $f: Y \to \mathbb{R}$ define $$...
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1answer
439 views

What fraction of fractions does Cantor's famous sequence enumerate?

Cantor's famous sequence $\frac{1}{1},\frac{1}{2},\frac{2}{1},\frac{1}{3},\frac{3}{1},\frac{1}{4}, \frac{2}{3},\frac{3}{2},\frac{4}{1}, \frac{1}{5},\frac{5}{1},\frac{1}{6}, ...$ provides a ...
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0answers
39 views

What are the weakest conditions for this inequality to be true?

Let $h:\mathbb{R}_+\to \mathbb{R}_+$ such that, for every $t\geqslant 0$, $$ h(t) \geqslant h(0) + \int_0^t \Lambda(h(s))\ ds $$ with $\Lambda(x) = a-bx - c x^2$ ($a,b,c\in\mathbb{R}-\{0\}$). a) ...
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1answer
67 views

Shortest Path finding in vector fields (2D and 3D) [closed]

Hoping someone may be able to point me in the right direction so I can research this topic further. Scenario: You have a vector field (either 2D or 3D) and you wish to find the shortest path between ...
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0answers
69 views

Intuition behind topological equivalence in dynamical systems

I have two dynamical systems defined on real line $X =\mathbb{R}$ and continuous time. They are defined by: $x' = \alpha − x^2$ $x' = \alpha − 2x^2 - 3$ When I plot the bifurcation diagrams of ...
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0answers
61 views

Reference request: sufficiently smooth functions on the plane belong to the projective tensor square of $L^2$ of the line

Let $\newcommand{\ptp}{\widehat{\otimes}}\ptp$ denote the projective tensor product of Banach spaces. Some back of the envelope calcuations, using the Fourier transform and Plancherel/Parseval, ...
3
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2answers
164 views

Morse theory for vector-valued functions

Let $f:\mathbb{R}^{m+k}\mapsto\mathbb{R}^k$ be a smooth function. I have seen quite a few books for Morse theory for $f$ when $k=1$. Is there a generalization to $k\geq2$? When $k=1$, we can define a ...
2
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1answer
93 views

Dynamical system described by coupled nonlinear differential equations

Suppose a dynamical system is described by two variables, $x$ and $y$, and they change over time according to the following two coupled nonlinear differential equations: \begin{equation} \begin{split} ...
11
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2answers
272 views

Examples of Stokes data

I'm trying to learn Stokes data but can't find an example to get my teeth into it. Background. It's well-known that on a complex manifold $X$, there is the Riemann Hilbert equivalence $$\text{regular ...
3
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0answers
64 views

Opers and global differential operators

This is a follow up question to a previous question of mine and my thought of answer to it. Given a (compact) Riemann surface $\Sigma$, a $SL(n,\mathbb{C})$-oper is a rank $n$ holomorphic vector ...
30
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3answers
2k views

What do we learn from the Wronskian in the theory of linear ODEs?

For a real interval $I$ and a continuous function $A: I \to \mathbb{R}^{d\times d}$, let $(x_1, \dots, x_d)$ denote a basis of the solution space of the non-autonomous ODE $$ \dot x(t) = A(t) x(t) \...
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1answer
78 views

Cyclic vector of holomorphic vector bundle with flat connection over compact Riemann surface

I originally posted the question on math.stackexhange, but there doesn't seem to be an answer. I apalogize in advance for cross posting. Let $E\rightarrow X$ be a holomorphic vector bundle over a ...
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1answer
79 views

Alternate proof of uniqueness of integral curves to vector fields

Let $V$ be a continuous vector field on an open set $U \subset \mathbb{R}^n$ and let $p_0 \in U$. There are many ways to construct local integral curves of $V$ through $p_0$, i.e. differentiable maps ...
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0answers
88 views

Solutions of the differential equation $f'=(f^{-1})^{[n]}$

For clarity, we use the notations $f^{[n]}=f\circ \dots\circ f$ for nesting and $f^{(n)}=\frac d{dx}\left(\frac{d}{dx}\dots\right)f$ for differentiation. After reading these two posts (here and here)...
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1answer
40 views

Computing the fractional Laplacian of power function

Is it possible to compute explicitly the fractional Laplacian (in $\mathbb R^n$) of a power function $|x|^p$?
15
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1answer
440 views

Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$

Can one show that Fourier transform of $$ f_a(x) = a^{-2} \exp(-|x|^a), \qquad a \in (0,2)$$ is decreasing in $a$? I have a solution for $a \in (0,1]$ which cannot be used for $a\in (1,2)$.
2
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1answer
33 views

Does a scalar LTV system with odd-periodic coefficients and even-periodic inputs have no periodic solutions?

Problem Setup Suppose we have the following scalar, linear time-varying (LTV) system with parameter $\mu \in [0,\pi[$: \begin{cases} \dot{x_1}(t,\mu) = a(t,\mu)x_1(t,\mu) + b(t,\mu) \\ x_1(0,\mu) = ...
1
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2answers
245 views

Linear independence of exponential functions: a reference

Is there a publication containing this obvious fact: For any real $T>0$, any natural $n$, any complex $c_1,\dots,c_n$, and any distinct complex $z_1,\dots,z_n$ such that $\sum_1^n c_k e^{tz_k}=0$ ...
0
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1answer
81 views

Analyze a complicated double summation

Let $f(x)$ be a real-valued twice continuously differentiable function, and considered the below double sum $$F(t,f(x)):=\dfrac{1}{t}\Big(\sum_{k=0}^{\infty}\sum_{m=0}^{\infty}f(x+(k-m)/\sqrt{n})\...
3
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0answers
46 views

Asymptotic behaviour of solutions to system of ODEs

Let $Y:(0,+\infty)\to\mathbb{R}^n$ be a solution to the system of ODEs $$ L[Y]=0, $$ where $L$ is a linear operator which behaves, in a neighbourhood of 0, as $$ L[Y](r)\simeq-Y''(r)-\frac{1}{r}Y'(r)...
2
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0answers
48 views

Separating a Riemann-Hilbert problem

Consider a RHP on the real line a jump is piece-wise H\"older continuous(or $L^2$), say for example the jump is $$g(x)=g_1(x)\chi_1+g_2(x)\chi_2,$$ where $g_j(x)$ are Holder continuous functions and $...
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1answer
143 views

How to solve the integral equation $f(x)=\frac{\lambda\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_1}{\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_2}$?

Recently, I have asked a question about variational analysis (First moments of uniform distribution on a curve from (0,0) to (1,1) in two-space). Such the question can be addressed in some cases by ...
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0answers
219 views

Problem with completeness of an orthogonal system

For $\nu\in (-1, \infty)$ denote by $\{\lambda_{k,\nu}\}_{k\in\mathbb{N}}$ the sequence of positive zeros of the Bessel function $J_{\nu}$. The Fourier-Bessel "Laplacean" is given by \begin{equation}...
4
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0answers
33 views

Results on: (path/initial condition)-dependent variant of exponential map generates compactly supported diffeomorphisms

Let $M$ be a connected and simply connected Riemannian manifold, non-compact, and suppose that $\{V_p\}_{p \in M}$ is a family of vector fields on $M$ indexed by $M$. Suppose moreover that the map $$ ...
2
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1answer
203 views

What is the motivation of the $L^p$ differentiability?

I was reading some papers and come up with the next definition : A function is differentiable in the $L^p$ sense at $x$ if there exists a real number $f'_p(x)$ such that $$\bigg(\frac{1}{h}∫_{-h}^...
3
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1answer
108 views

Example of a bounded function whose mean-zero mollification diverges at a point

For a Schwartz function $\psi(x)=xe^{-x^2}$ define $\varphi(x):=\psi'(x)$ and consider a family of $L^1$-dilations of $\varphi$ given by: $$ \varphi_t(x)=\frac{1}{t}\varphi(x/t), \qquad t>0. $$ $\...
2
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2answers
497 views

Monotonic and bounded sequences throughout mathematics [closed]

When I refer to the Monotone Convergence Theorem below, I refer to the very simple claim that if a non-decreasing sequence has an upper bound then it converges. I don't refer to the claim from Measure ...
5
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0answers
145 views

Minimizing total variation

Let $f:\mathbb{R}\to \mathbb{R}$ be a function of bounded variation. Define $\overline{f}(x)$ by $$\overline{f}(x) = \limsup_{\mu(I)\to 0} \frac{1}{\mu(I)} \int_I f(x) dx,$$ where $I$ ranges over ...
1
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2answers
147 views

Gradient flows: convex potential vs. contractive flow?

Take a $\mathcal C^2$ potential $V:\mathbb R^d\to \mathbb R$, and assume that it is bounded from below (say $\min V=0$ for simplicity, so that $V\geq 0$). Consider the autonomous gradient-flow $$ \dot ...
3
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0answers
41 views

Are there extensions of Hilb's and Laplace's formulas to Jacobi polynomials with $\alpha,\beta\le-1$?

In Szegő's Orthogonal Polynomials book, he gives two interesting asymptotic formulas for Jacobi polynomials with $\alpha,\beta>-1$. The first (Theorem 8.21.12, page 197 is a generalization of Hilb'...
11
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2answers
475 views

Do infinitely nested radicals have any applications?

There is a simple necessary and sufficient condition for a continued radical of the form $\sqrt{a_1 + \sqrt{a_2 + \dotsc}}$ to converge (where all terms $a_1, a_2$ etc. are nonnegative). Namely, that ...
3
votes
1answer
228 views

A second order nonlinear ODE

In my research (in differential geometry) I recently came across the following nonlinear second order ode: $$\frac{f''(x)}{f'(x)}-\frac{2}{x}+\frac{f'(x)+1}{2f(x)-x-1}+\frac{f'(x)-1}{2f(x)+x}=0$$ It ...
5
votes
1answer
1k views

Analysis of solutions to a nonlinear ODE

Consider the following ODEs: $\phi^2=\phi''\sqrt{1-\phi'^2}$, or $\phi^2=-\phi''\sqrt{1-\phi'^2}$. Is there any theory (e.g. comparison theorems) which analyzes solutions of the above ODEs? I am only ...
10
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6answers
1k views

Differentiability of eigenvalues of positive-definite symmetric matrices

Let $A\in M(n,\mathbb{R})$ be an invertible matrix. Consider the (real) eigenvalues $\lambda_1,\cdots,\lambda_n$, in increasing order, of the positive-definite symmetric matrix $A^t A$. We shall ...
41
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3answers
3k views

On which regions can Green's theorem not be applied?

In elementary calculus texts, Green's theorem is proved for regions enclosed by piecewise smooth, simple closed curves (and by extension, finite unions of such regions), including regions that are not ...

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