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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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)...
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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_{...
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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 ...
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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 $|x|^p$?
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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 $a\in (1,2)$.
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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) = ...
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0answers
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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{\...
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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 $\sum_1^n c_k e^{tz_k}=0$ ...
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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})\...
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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)...
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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 $f*f$ ...
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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
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 ...
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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}...
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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 $$ ...
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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}^...
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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: $$ \varphi_t(x)=\frac{1}{t}\varphi(x/t), \qquad t>0. $$ $\...
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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 $x$, ...
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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 ...
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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 ...
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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 ...
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0answers
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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'...
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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 ...
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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
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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 ...
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6answers
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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 ...
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3answers
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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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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. ...
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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
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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
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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 ...
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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
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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 = ...
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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 ...
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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 $\...
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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
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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 ...
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4answers
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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 ...
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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(-\|\...
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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 ...
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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 \...
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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
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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,\...
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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 ...
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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 ...
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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
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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
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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
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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{(−...
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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 $\phi=u\sqrt{-1}\partial\bar{\partial}u$

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