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.
2,461
questions
5
votes
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 $$g(a)\...
0
votes
0answers
24 views
Bounding Greens function matrix elements in terms of the diagonal elements
Consider the Hilbert space $l^2( \mathbb{Z}^2)$ and suppose that I have a unitary band matrix. I.e. $ \langle e_j , U {e_k} \rangle = 0 $ for say $\vert \vert j-k \vert \vert > 2 $ (in say taxi-cap ...
1
vote
1answer
64 views
Variational formulation of abstract Cauchy problem, possible?
Recently I have come across a method known as "variational method" in which we try to establish weak solutions of various boundary value problems involving ordinary derivatives, partial ...
2
votes
1answer
93 views
Functions with a Jacobian whose columns are orthogonal
I am interested in vector fields whose Jacobian has orthogonal columns; i.e. if $\mathbf{f}(\cdot):\mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a function where $\mathbf{f}(\mathbf{x})=[f_1(\mathbf{x}...
1
vote
0answers
45 views
Optimal control of nonlinear harmonic oscillator
Consider the ODE
$$
\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$$
for $...
12
votes
1answer
379 views
Possible limit involving the gamma function
Does $$\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)) $$
decays to zero as $t \to \infty$. ...
-1
votes
0answers
50 views
transform $ \phi '' + ( 1 +c^2/4 -|\phi |^2)\phi = 0 $ into $ \varphi '' + ( 1 - |\varphi |^2)\varphi = 0$ [closed]
Assume that $\psi: \mathbb{R}\to\mathbb{C}$ is a solution of $\psi '' + i c\psi ' + (1-\vert\psi\vert^2)\psi = 0$, where $i^2 = -1$ and $c\in (0,\sqrt{2})$.
Applying the transformation $\Phi (\psi)=e^{...
2
votes
1answer
76 views
Fourier transform of a function of bounded variation
I know if $f\in L^2(\mathbb R)$ is two times continuously differentiable, then we must have that the Fourier transform is integrable. Is there any more relaxed condition than this? For example if $f$ ...
1
vote
0answers
38 views
Integrability of Fourier transform of truncated fractional power
Is the Fourier transform of the function $f$ which agrees with $1_{[-1.1]}|x|^\alpha$ on $[-1,1]$ and then decays very fast to zero to become a compactly supported continuous function, is in $L^1(\...
1
vote
1answer
144 views
Existence of entire function that yields periodicity
I have the following question:
Does there exist an entire function $f(z)$ where $z=x+iy$ such that
$$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(...
5
votes
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 ...
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 $b$...
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 ...
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 ...
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
$$...
5
votes
1answer
439 views
What fraction of fractions does Cantor's famous sequence enumerate?
Cantor's famous sequence
$\frac{1}{1},\frac{1}{2},\frac{2}{1},\frac{1}{3},\frac{3}{1},\frac{1}{4}, \frac{2}{3},\frac{3}{2},\frac{4}{1}, \frac{1}{5},\frac{5}{1},\frac{1}{6}, ...$
provides a ...
0
votes
0answers
39 views
What are the weakest conditions for this inequality to be true?
Let $h:\mathbb{R}_+\to \mathbb{R}_+$ such that, for every $t\geqslant 0$,
$$ h(t) \geqslant h(0) + \int_0^t \Lambda(h(s))\ ds $$
with $\Lambda(x) = a-bx - c x^2$ ($a,b,c\in\mathbb{R}-\{0\}$).
a) ...
-2
votes
1answer
67 views
Shortest Path finding in vector fields (2D and 3D) [closed]
Hoping someone may be able to point me in the right direction so I can research this topic further.
Scenario: You have a vector field (either 2D or 3D) and you wish to find the shortest path between ...
0
votes
0answers
69 views
Intuition behind topological equivalence in dynamical systems
I have two dynamical systems defined on real line $X =\mathbb{R}$ and continuous time. They are defined by:
$x' = \alpha − x^2$
$x' = \alpha − 2x^2 - 3$
When I plot the bifurcation diagrams of ...
5
votes
0answers
61 views
Reference request: sufficiently smooth functions on the plane belong to the projective tensor square of $L^2$ of the line
Let $\newcommand{\ptp}{\widehat{\otimes}}\ptp$ denote the projective tensor product of Banach spaces. Some back of the envelope calcuations, using the Fourier transform and Plancherel/Parseval, ...
3
votes
2answers
164 views
Morse theory for vector-valued functions
Let $f:\mathbb{R}^{m+k}\mapsto\mathbb{R}^k$ be a smooth function. I have seen quite a few books for Morse theory for $f$ when $k=1$. Is there a generalization to $k\geq2$? When $k=1$, we can define a ...
2
votes
1answer
93 views
Dynamical system described by coupled nonlinear differential equations
Suppose a dynamical system is described by two variables, $x$ and $y$, and they change over time according to the following two coupled nonlinear differential equations:
\begin{equation}
\begin{split}
...
11
votes
2answers
272 views
Examples of Stokes data
I'm trying to learn Stokes data but can't find an example to get my teeth into it.
Background. It's well-known that on a complex manifold $X$, there is the Riemann Hilbert equivalence
$$\text{regular ...
3
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
$$
\dot x(t) = A(t) x(t) \...
1
vote
1answer
78 views
Cyclic vector of holomorphic vector bundle with flat connection over compact Riemann surface
I originally posted the question on math.stackexhange, but there doesn't seem to be an answer. I apalogize in advance for cross posting.
Let $E\rightarrow X$ be a holomorphic vector bundle over a ...
1
vote
1answer
79 views
Alternate proof of uniqueness of integral curves to vector fields
Let $V$ be a continuous vector field on an open set $U \subset \mathbb{R}^n$ and let $p_0 \in U$. There are many ways to construct local integral curves of $V$ through $p_0$, i.e. differentiable maps ...
2
votes
0answers
88 views
Solutions of the differential equation $f'=(f^{-1})^{[n]}$
For clarity, we use the notations $f^{[n]}=f\circ \dots\circ f$ for nesting and $f^{(n)}=\frac d{dx}\left(\frac{d}{dx}\dots\right)f$ for differentiation.
After reading these two posts (here and here)...
1
vote
1answer
40 views
Computing the fractional Laplacian of power function
Is it possible to compute explicitly the fractional Laplacian (in $\mathbb R^n$) of a power function $|x|^p$?
15
votes
1answer
440 views
Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$
Can one show that Fourier transform of
$$ f_a(x) = a^{-2} \exp(-|x|^a), \qquad a \in (0,2)$$
is decreasing in $a$?
I have a solution for $a \in (0,1]$ which cannot be used for $a\in (1,2)$.
2
votes
1answer
33 views
Does a scalar LTV system with odd-periodic coefficients and even-periodic inputs have no periodic solutions?
Problem Setup
Suppose we have the following scalar, linear time-varying (LTV) system with parameter $\mu \in [0,\pi[$:
\begin{cases}
\dot{x_1}(t,\mu) = a(t,\mu)x_1(t,\mu) + b(t,\mu) \\
x_1(0,\mu) = ...
1
vote
2answers
245 views
Linear independence of exponential functions: a reference
Is there a publication containing this obvious fact: For any real $T>0$, any natural $n$, any complex $c_1,\dots,c_n$, and any distinct complex $z_1,\dots,z_n$ such that $\sum_1^n c_k e^{tz_k}=0$ ...
0
votes
1answer
81 views
Analyze a complicated double summation
Let $f(x)$ be a real-valued twice continuously differentiable function, and considered the below double sum $$F(t,f(x)):=\dfrac{1}{t}\Big(\sum_{k=0}^{\infty}\sum_{m=0}^{\infty}f(x+(k-m)/\sqrt{n})\...
3
votes
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
$$
L[Y](r)\simeq-Y''(r)-\frac{1}{r}Y'(r)...
2
votes
0answers
48 views
Separating a Riemann-Hilbert problem
Consider a RHP on the real line a jump is piece-wise H\"older continuous(or $L^2$), say for example the jump is
$$g(x)=g_1(x)\chi_1+g_2(x)\chi_2,$$
where $g_j(x)$ are Holder continuous functions and $...
0
votes
1answer
143 views
How to solve the integral equation $f(x)=\frac{\lambda\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_1}{\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_2}$?
Recently, I have asked a question about variational analysis (First moments of uniform distribution on a curve from (0,0) to (1,1) in two-space). Such the question can be addressed in some cases by ...
1
vote
0answers
219 views
Problem with completeness of an orthogonal system
For $\nu\in (-1, \infty)$ denote by $\{\lambda_{k,\nu}\}_{k\in\mathbb{N}}$ the sequence of positive zeros of the Bessel function $J_{\nu}$. The Fourier-Bessel "Laplacean" is given by
\begin{equation}...
4
votes
0answers
33 views
Results on: (path/initial condition)-dependent variant of exponential map generates compactly supported diffeomorphisms
Let $M$ be a connected and simply connected Riemannian manifold, non-compact, and suppose that $\{V_p\}_{p \in M}$ is a family of vector fields on $M$ indexed by $M$. Suppose moreover that the map
$$
...
2
votes
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 $$\bigg(\frac{1}{h}∫_{-h}^...
3
votes
1answer
108 views
Example of a bounded function whose mean-zero mollification diverges at a point
For a Schwartz function $\psi(x)=xe^{-x^2}$ define $\varphi(x):=\psi'(x)$ and consider a family of $L^1$-dilations of $\varphi$ given by:
$$
\varphi_t(x)=\frac{1}{t}\varphi(x/t), \qquad t>0.
$$
$\...
2
votes
2answers
497 views
Monotonic and bounded sequences throughout mathematics [closed]
When I refer to the Monotone Convergence Theorem below, I refer to the very simple claim that if a non-decreasing sequence has an upper bound then it converges. I don't refer to the claim from Measure ...
5
votes
0answers
145 views
Minimizing total variation
Let $f:\mathbb{R}\to \mathbb{R}$ be a function of bounded variation. Define $\overline{f}(x)$ by
$$\overline{f}(x) = \limsup_{\mu(I)\to 0} \frac{1}{\mu(I)} \int_I f(x) dx,$$ where $I$ ranges over ...
1
vote
2answers
147 views
Gradient flows: convex potential vs. contractive flow?
Take a $\mathcal C^2$ potential $V:\mathbb R^d\to \mathbb R$, and assume that it is bounded from below (say $\min V=0$ for simplicity, so that $V\geq 0$).
Consider the autonomous gradient-flow
$$
\dot ...
3
votes
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
votes
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
votes
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
votes
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 ...
