# 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

**5**answers

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

**1**answer

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

**1**answer

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

**2**answers

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**

vote

**1**answer

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

**0**answers

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

**1**answer

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

**2**answers

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

**5**answers

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

**3**answers

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

**1**answer

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**

vote

**0**answers

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

**6**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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**

votes

**0**answers

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

**1**answer

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

**4**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**4**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**8**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**3**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**6**answers

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

**1**answer

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

**3**answers

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 ...