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.
2,440
questions
1
vote
0answers
59 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)...
0
votes
0answers
123 views
Boundedness of total current in electrical network (Banded graph)
Consider the following symmetric matrix (adjacency matrix):
$$A=(a_{ij})_{1\leq i,j\leq n}$$
such that $a_{ij}=a_{ji}, a_{ii}=0$ and $a_{ij}=0$ for $|i-j|\geq k$ where $k\geq3$. We also have $1\leq a_{...
-4
votes
1answer
42 views
How do you calculate residual risk. Is it possible to aggregate control scores? [closed]
I'm sure what I need is possible as I did it years ago in excel but can't remember!
I work in cyber security and I am trying to calculate residual risk, example..
Lets take phishing for example and ...
1
vote
1answer
31 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 $|x|^p$?
3
votes
0answers
107 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
31 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
votes
0answers
49 views
Analytical solution for boundary problem of convection diffusion equation [closed]
I am trying to solve the boundary problem for the convection diffusion equation. The conditions are following:
$$
\begin{cases}
\frac{\partial c}{\partial t}=\frac{\partial^2c}{\partial z^2}=+\frac{\...
1
vote
2answers
229 views
Linear independence of exponential functions: a reference
Is there a publication containing this obvious fact: For any real $T>0$, any natural $n$, any complex $c_1,\dots,c_n$, and any distinct complex $z_1,\dots,z_n$ such that $\sum_1^n c_k e^{tz_k}=0$ ...
0
votes
1answer
75 views
Analyze a complicated double summation
Let $f(x)$ be a real-valued twice continuously differentiable function, and considered the below double sum $$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
45 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
$$
L[Y](r)\simeq-Y''(r)-\frac{1}{r}Y'(r)...
-1
votes
0answers
21 views
Unable to do question 3 in 7.3 from Folland's Fourier Analysis and its Application [migrated]
I'm unable to answer this question, where we were given $f(x)$:
$$f(x)=\begin{cases}
1, & \text{if }-1<x<1 \\
0, & \text{otherwise}\end{cases}$$
The questions asks me to compute $f*f$ ...
2
votes
0answers
48 views
Separating a Riemann-Hilbert problem
Consider a RHP on the real line a jump is piece-wise H\"older continuous(or $L^2$), say for example the jump is
$$g(x)=g_1(x)\chi_1+g_2(x)\chi_2,$$
where $g_j(x)$ are Holder continuous functions and $...
0
votes
1answer
119 views
How to solve the integral equation $f(x)=\frac{\lambda\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_1}{\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_2}$?
Recently, I have asked a question about variational analysis (First moments of uniform distribution on a curve from (0,0) to (1,1) in two-space). Such the question can be addressed in some cases by ...
1
vote
0answers
159 views
Problem with completeness of an orthogonal system
For $\nu\in (-1, \infty)$ denote by $\{\lambda_{k,\nu}\}_{k\in\mathbb{N}}$ the sequence of positive zeros of the Bessel function $J_{\nu}$. The Fourier-Bessel "Laplacean" is given by
\begin{equation}...
4
votes
0answers
33 views
Results on: (path/initial condition)-dependent variant of exponential map generates compactly supported diffeomorphisms
Let $M$ be a connected and simply connected Riemannian manifold, non-compact, and suppose that $\{V_p\}_{p \in M}$ is a family of vector fields on $M$ indexed by $M$. Suppose moreover that the map
$$
...
2
votes
1answer
190 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 $$\bigg(\frac{1}{h}∫_{-h}^...
3
votes
1answer
93 views
Example of a bounded function whose mean-zero mollification diverges at a point
For a Schwartz function $\psi(x)=xe^{-x^2}$ define $\varphi(x):=\psi'(x)$ and consider a family of $L^1$-dilations of $\varphi$ given by:
$$
\varphi_t(x)=\frac{1}{t}\varphi(x/t), \qquad t>0.
$$
$\...
-2
votes
0answers
51 views
the subdifferential at points of differentiability in infinite dimensional space
Let $ f:X \longrightarrow (-\infty,\infty] $ that $X$ is infinite dimensional space and $f$ be a proper convex function and $ x\in int(dom(f))$.
Is it the case that: if $f$ is differentiable at $x$, ...
2
votes
2answers
472 views
Monotonic and bounded sequences throughout mathematics [closed]
When I refer to the Monotone Convergence Theorem below, I refer to the very simple claim that if a non-decreasing sequence has an upper bound then it converges. I don't refer to the claim from Measure ...
5
votes
0answers
130 views
Minimizing total variation
Let $f:\mathbb{R}\to \mathbb{R}$ be a function of bounded variation. Define $\overline{f}(x)$ by
$$\overline{f}(x) = \limsup_{\mu(I)\to 0} \frac{1}{\mu(I)} \int_I f(x) dx,$$ where $I$ ranges over ...
1
vote
1answer
73 views
Gradient flows: convex potential vs. contractive flow?
Take a $\mathcal C^2$ potential $V:\mathbb R^d\to \mathbb R$, and assume that it is bounded from below (say $\min V=0$ for simplicity, so that $V\geq 0$).
Consider the autonomous gradient-flow
$$
\dot ...
3
votes
0answers
31 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
463 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 ...
2
votes
1answer
197 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 ...
39
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 ...
1
vote
0answers
40 views
Alternate characterization of floquet multipliers: Floquet theory
Given an autonomous ode $\dot{x}=f(x)$ in $\mathbb{R}^n$ possessing a period-p time-periodic solution $\bar x(t)$, one can use the so-called variational equation about $\bar x$ to study its stability. ...
3
votes
1answer
98 views
Euler–Maclaurin formula in $\mathbb{Z}^d$
I was wondering whether there is a Euler–Maclaurin formula of sorts for expressions such as
$$
\sum_{x \in [a,b]^d\cap \mathbb{Z}^d} f(x) - \int_{[a,b]^d}f(x)
$$
where $d\ge 2$ is an integer, $a,b \...
3
votes
1answer
98 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=...
2
votes
0answers
84 views
Inequality about exponential integrals
I am reading about Dirichlet polynomials in the book Analytic Number Theory by Iwaniec-Kowalski.
During the proof of Theorem 9.1 for any positive real numbers $T, N$ they define a piecewise linear and ...
1
vote
1answer
159 views
How to compute integral of a gaussian over a noncentered ball?
Let $\mathcal{B}(x,r)$ the ball of center $x \in \mathbb{R}^n$ and radius $r>0$ (so $\mathcal{B}(x,r) = \{y \in \mathbb{R}^n : \|y-x\| \leq r\}$, where all norms are $\ell^2$-norms).
I would like ...
2
votes
1answer
77 views
A density question
Suppose $\Omega= (0,1)\times(0,1)\subset \mathbb R^2$. Assume that $f, g \in C^{\infty}(\Omega)$ and that
$$ \int_\Omega \left(f(x_1,x_2)- \frac{m}{(n+1)}g(x_1,x_2)\right) x_1^n \,x_2^m \,dx_1\,dx_2 = ...
6
votes
2answers
301 views
Why is $-\int_{-\infty}^\infty \log\left[1+2f'(x)(1-\cos\phi)\right]\,dx$ equal to $\phi^2$?
I came across this integral involving the derivative $f'(x)$ of the Fermi function $f(x)=(1+e^x)^{-1}$:
$$I(\phi)=-\int_{-\infty}^\infty \log\left[1+2f'(x)(1-\cos\phi)\right]\,dx.$$
I'm pretty certain ...
5
votes
1answer
84 views
When is the log-permanent concave?
Let $\operatorname{PSD}_n$ be the cone of $n\times n$ semidefinite positive matrices. For any $X\in \operatorname{PSD}_n$, define $$f(X)=\log(\det(X)).$$ Then $f$ is a concave function on $\...
4
votes
1answer
170 views
Idea behind Carleson's theorem modern proof “intitial reductions”
I'm having troubles to understand the philosophy behind the modern proof of Carleson's theorem. For convenience, let me state precisely what I am asking for.
For any $f \in L^2(\mathbb{R})$, let $\...
4
votes
1answer
97 views
Continuous dependence on initial parameters of an ODE for non-Lipschitz functions?
For ODEs, the standard theorem of continuous dependence of initial parameters deals only with functions that are Lipschitz. Do there exist more general results holding for non-Lipschitz functions? If ...
71
votes
4answers
6k views
Note rejected from arXiv: what to do next?
Short version: A note of mine was rejected by the arXiv moderation (something I didn't even know was possible) on account of being “unrefereeable”. The moderation process provides absolutely no ...
7
votes
1answer
273 views
Compactly supported probability measure in high dimensions with fast Fourier decay?
For any sufficiently large $d\in\mathbb{N}$, does there exist a probability measure $\Psi$ supported on the Euclidean ball in $\mathbb{R}^d$ for which $|\widehat{\Psi}[\omega]|\le C\cdot \exp(-\|\...
5
votes
0answers
81 views
Points where singular sum is small
We consider $x_1,..,x_N$ points in the plane $\mathbb{R}^2.$
We define the sum
$$F(x):=\frac{1}{N^2}\sum_{i=1}^N \sum_{j \neq i} \vert x_i-x_j \vert^{-2}.$$
I am looking for a statement of the ...
4
votes
1answer
88 views
Singular Sturm-Liouville problems: criterion for discrete spectrum for zero potential ($q=0$) and Hermite Polynomials
There are some known criteria for the Sturm-Liouville Problem
\begin{equation} \tag{1}
\frac {\mathrm {d} }{\mathrm {d} x}\left[p(x){\frac {\mathrm {d} y}{\mathrm {d} x}}\right]+q(x)y=-\lambda w(x)y
\...
24
votes
1answer
763 views
Solving a delay-differential equation related to epidemiology
For some inexplicable reason, I have recently been interested in epidemiology. One of the classical and simplistic models in epidemiology is the SIR model given by the following system of first-order ...
2
votes
2answers
215 views
An ODE comparison problem
Recently I met an ODE problem but after thinking for quite a while I still could not find an answer. Here is the question, which looks very simple:
Let $y=y(t)$ be a smooth function defined on $[0,\...
3
votes
0answers
15 views
A (discontinuous) Initial value problem with exactly two solutions
It is an old result of Kneser that if $f$ is a continuous function, and there are two solutions to the IVP $$y'=f(x,y), \quad y(x_0)=y_0,$$
then there are uncountably many solutions.
I am interested ...
0
votes
0answers
27 views
Consistency results of optimal control solutions using direct transcription
I have read a number of books on discrete time solutions to continuous time optimal control problems.
What is not clear to me is what consistency results exist with respect to showing the discretized ...
0
votes
0answers
65 views
ODE operator splitting with second order time discretization not possible?
I am trying to solve an ordinary differential equation (ODE) using an operator splitting approach:
$\frac{\partial f}{\partial t} = A(f) + B(f)$
Let's assume that $A$ and $B$ are very simple:
$\...
6
votes
2answers
213 views
Movement of repelled particles in a ball
EDIT:
Given a system of $N\geq 3$ charged point particles in $\mathbb{R}^3$ of the same charge which interact according to Coulomb law (thus they repell one from each other). Is it possible that ...
7
votes
3answers
408 views
A generalization of discrete Hibert's transform (Montgomery's inequality)
In the paper "Hilbert's inequality", Montgomery and Vaughan proved that a generalization of the discrete Hilbert transform is bounded in $\ell^2$. The inequality reads as follows
$$ \Big| \sum_{k\neq ...
1
vote
1answer
145 views
Square-integrable unbounded function
In R.D. Richtmyer, Principles of Advanced Mathematical Physics, p.85 an example is given of a continuous and square-integrable on $\bf{R}$ function, which is not bounded at infinity:
$$f(x)=x^2\exp{(−...
1
vote
1answer
104 views
Can a real (1,1) form $\phi$ be represented by $u\sqrt{-1}\partial\bar{\partial}u$ on Kahler manifold?
Let M be a 2-dimension (complex dimension) K\"{a}ler maifold and $\phi$ be a real $(1,1)$ form. Is it possible that there exists a function $u$ such that $\phi=u\sqrt{-1}\partial\bar{\partial}u$
