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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5
votes
5answers
368 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)\...
2
votes
1answer
360 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 &...
2
votes
1answer
91 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}...
1
vote
2answers
113 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 ...
1
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1answer
62 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 ...
0
votes
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 ...
1
vote
1answer
210 views

Space derivative of flow of ODE with monotone source

Consider the ODE $$ \begin{cases} \partial_t\Phi(t,x) = f(t,\Phi(t,x)), &\ t>0, \ x \in \mathbb R \\ \Phi(0,x) = x, & x \in \mathbb R \end{cases} $$ where $f$ is function which is a non-...
7
votes
2answers
871 views

Is there a differentiable but nonsmooth version of the continuous Implicit Function Theorem?

From the result discussed in Does the inverse function theorem hold for everywhere differentiable maps? (which I'll call the differentiable nonsmooth Inverse Function Theorem) one can obtain a ...
110
votes
5answers
20k views

Does the inverse function theorem hold for everywhere differentiable maps?

(This question was posed to me by a colleague; I was unable to answer it, so am posing it here instead.) Let $f: {\bf R}^n \to {\bf R}^n$ be an everywhere differentiable map, and suppose that at each ...
4
votes
3answers
2k views

How to compute $\prod_{n=1}^{\infty} (1-p^{-n})$

We know it converges for any prime $p$. I just want to know how to compute its exact value: $$\prod_{n=1}^{\infty} (1-p^{-n})$$
5
votes
1answer
263 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 ...
1
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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 $...
11
votes
6answers
4k views

Does the derivative of log have a Dirac delta term?

Dirac writes down the following formula on page 61 of his "Principles of quantum mechanics": $\frac{d}{dx}\log x = \frac{1}{x} -i\pi\delta(x)$, see http://adsabs.harvard.edu/abs/1947pqm..book.....D ...
1
vote
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(...
12
votes
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 $n/(n+1)$ ...
0
votes
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 $t \to \infty$. ...
1
vote
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 ...
-1
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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^{...
2
votes
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 $f$ ...
71
votes
4answers
7k views

Nonexistence of boundary between convergent and divergent series?

The following is a FAQ that I sometimes get asked, and it occurred to me that I do not have an answer that I am completely satisfied with. In Rudin's Principles of Mathematical Analysis, following ...
1
vote
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(\...
0
votes
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 $b$...
8
votes
1answer
831 views

On the convergence of the function series $\sum_{n=0}^\infty(-1)^n\frac{f^{(n)}(x)}{n!}x^n$

Let $f$ be a smooth real function defined around origin. If we differentiate term by term the series $\hat{f}(x):=\sum_{n=0}^\infty(-1)^n\frac{f^{(n)}(x)}{n!}x^n$, we get $\frac{d}{dx}\hat{f}(x)=0$. \...
5
votes
1answer
437 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 ...
94
votes
4answers
24k views

Is the analysis as taught in universities in fact the analysis of definable numbers?

Ten years ago, when I studied in university, I had no idea about definable numbers, but I came to this concept myself. My thoughts were as follows: All numbers are divided into two classes: those ...
2
votes
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 ...
1
vote
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 ...
15
votes
1answer
434 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 $a\in (1,2)$.
5
votes
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 $$...
0
votes
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) ...
3
votes
0answers
106 views

Relationship between three different definitions of solutions for ODE with irregular coefficient

What is the difference between the notions of Regular Lagrangian flow Filippov solution Caratheodory solution of an ODE $\dot \Phi(t,x) = b(t,\Phi(t,x))$, with initial condition $\Phi(0,x) = x$, ...
0
votes
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 ...
5
votes
1answer
107 views

When is the log-permanent concave?

Let $\operatorname{PSD}_n$ be the cone of $n\times n$ semidefinite positive matrices. For any $X\in \operatorname{PSD}_n$, define $$f(X)=\log(\det(X)).$$ Then $f$ is a concave function on $\...
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 ...
-2
votes
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 ...
20
votes
8answers
3k views

Euclidean volume of the unit ball of matrices under the matrix norm

The matrix norm for an $n$-by-$n$ matrix $A$ is defined as $$|A| := \max_{|x|=1} |Ax|$$ where the vector norm is the usual Euclidean one. This is also called the induced (matrix) norm, the operator ...
11
votes
2answers
271 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 ...
0
votes
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 ...
5
votes
0answers
60 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
votes
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
votes
1answer
92 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} ...
3
votes
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
votes
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) \...
1
vote
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 ...
2
votes
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)...
3
votes
1answer
109 views

Strict inequality in decoupling inequality

I am working on the decoupling inequality developed by Bourgain and Demeter: https://arxiv.org/abs/1604.06032. Is there an example where we have strict inequality in Theorem 1.1, say in the case $n=...
1
vote
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 $|x|^p$?
6
votes
6answers
4k views

Fourier transform of (real) exponential

Is it possible to make sense, in distributional sense, of the Fourier transform of the exponential function (defined over the whole real line)?
2
votes
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) = ...
22
votes
3answers
1k views

Integral $\int_0^1 \int_0^1 \cdots \int_0^1\frac{x_{1}^2+x_{2}^2+\cdots+x_{n}^2}{x_{1}+x_{2}+\cdots+x_{n}}dx_{1}\, dx_{2}\cdots \, dx_{n}=?$

How to evaluate this integral: $$\int_0^1 \int_0^1 \cdots \int_0^1\frac{x_{1}^2+x_{2}^2+\cdots+x_{n}^2}{x_{1}+x_{2}+\cdots+x_{n}}dx_{1}\, dx_{2}\cdots \, dx_{n}=?$$ I'm making use of the integral ...

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