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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68 views

Non-uniqueness of loop-erasure for continuous-time curves

Question. Is there a continuous curve in the plane that has a non-unique loop-erasure? Here is the definition of a loop-erasure. A continuous curve $Y:[c,d]\to\mathbb R^2$ is a loop-erasure of a curve ...
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0answers
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+100

A matrix Riccati differential equation with constant coefficients? Is there a solution for this in closed form?

The following is a matrix Riccati differential equation with constant coefficient matrices. $$D\frac{\partial{C(t)}}{\partial{t}}S_i + \frac{1}{n}C(t)Q_iDC(t)S_i - C(t)Q_iE = 0$$ or $$D\dot{C}(t)S_i + ...
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1answer
484 views

Resonance arising when harmonic oscillator is excited using sawtooth

Solutions to the differential equation $my'' + ky = F \sin \omega t$ show resonance when the driving frequency $\omega$ equals the natural frequency $\sqrt{k/m}$. That is, solutions are unbounded when ...
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0answers
58 views
+50

Prove convergence of whole sequence $f_n$ of solutions to a differential problem to a limit $f$ (without uniqueness assumptions)

Let $\{f_n\}_n \subset C^\infty \cap L^2(\mathbb R^N)$ be a sequence of functions that solves a linear differential equation $F_n(f_n, \nabla f_n) = 0$. Suppose that there exists a subsequence $n_k$ ...
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0answers
38 views

What exactly are the benefits of keeping a Hamiltonian system of equations Hamiltonian during solving or transformation?

When faced with a system of differential equations that happens to be Hamiltonian in form, or a perturbation of a Hamiltonian system, we often see in classical work a clear attempt to pursue solutions ...
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0answers
102 views

Notation for the set of maximal / minimal elements

Let $(X, \preceq)$ be a poset. Commonly the notation $\max X$ refers to the maximum of the set $X$, if it exists; similarly $\min X$ refers commonly to the minimum of the set $X$, if it exists. Are ...
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0answers
229 views

Transforming a continuous function into a differentiable function

Given a continuous function $f(x)$ when does there exist a non-constant continuous function $g(x)$ such that $f(g(x))$ is differentiable what about $g(f(x))$? Does there exist any examples of $f(x)$ ...
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1answer
146 views

Showing integrability of a locally integrable function on a bounded domain under some additional assumptions

Suppose $\Omega\subset \mathbb{R}^3$ is a smooth and bounded domain, and $f:\Omega\to[0,\infty]$ is a given function which is finite almost everywhere and satisfies Assumption A: For all $g\in C_0^1(\...
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1answer
118 views

Convergent of improper integral [closed]

Let $f \in C^1[0,\infty)$ be an increasing function with $f(0)>0$, suppose $\int_0^\infty \frac{1}{f(x)+f'(x)} < \infty$, prove that $\int_0^\infty \frac{1}{f(x)} < \infty$. I find it weird ...
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3answers
1k views

Value of an integral

I need to verify the value of the following integral $$ 4n(n-1)\int_0^1 \frac{1}{8t^3}\left[\frac{(2t-t^2)^{n+1}}{(n+1)}-\frac{t^{2n+2}}{n+1}-t^4\{\frac{(2t-t^2)^{n-1}}{n-1}-\frac{t^{2n-2}}{n-1} \} \...
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1answer
199 views

Simple closed forms for sums such as $\sum_{k=1}^\infty \frac{(-1)^{k+1}}{qk - p}$ and related integrals

My goal here is to get a simple expression for $\zeta(3)$. This is a follow up to my previous question posted here. Any Taylor-like expansion from everything I tried won't make it. So this is my last ...
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0answers
46 views

Integral inequality with Fractional Laplacian

Is the following inequality true $$ \int_{B_1(0)} f(x) (-\Delta)^\alpha u(x) dx - \frac{1}{|B_1(0)|}\int_{B_1(0)}(-\Delta)^\alpha u(x) dx \cdot \int_{B_1(0)}f(x) dx \ge 0 $$ for a strictly convex $f:\...
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1answer
74 views

Prove that the following running average is monotonically decreasing

Let $S_n$ be defined as $\frac{1}{n}\sum_{t=1}^{t=n} [x^2+(p-q)x]$ where $x = 1-(1-p-q)^t$. We want to find the conditions on $p$ and $q$ such that $S_n$ is monotonically decreasing for all $n$. $0 &...
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2answers
564 views

Erroneous Wolfram result for $\sum_{k=1}^\infty (k^3 + a^3)^{-1}$, looking for correct formula

I was trying to get some interesting result for $\zeta(3)$, exploring the following function: $$W(a) = \sum_{k=1}^\infty \frac{1}{k^3 + a^3}, \mbox{ with } \lim_{a\rightarrow 0} W(a) = \zeta(3).$$ Let ...
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0answers
53 views

Uniqueness for measure valued ode

Morning! Basically I'm working on a mean field scaling for some measure valued process (valued on $M_F(N)$). The limit turns up as a (deterministic) solution to a measure valued ODE. Let's say : $$ d\...
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0answers
32 views

nonholonomic and linear control systems

I'm struggling with the definitions and differences between a non-holonomic and a linear control systems. I'm working with a phase space $P\subset R^{m}$ and a control space $U\subset R^{l}$ and my ...
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1answer
85 views

Non-zero, bounded, continuous, differentiable at the origin, compactly supported functions with everywhere non-negative Fourier transforms

Do there exist functions $F(x) \! : \, \mathbb R \to \mathbb R$ which are non-zero and bounded: $$ \mathrm {Range} (F) = [l, u] \, , \quad \mathrm {where} \quad l, u \in \mathbb R \land u > l \, ; \...
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1answer
144 views

Directional gradient on sphere

We consider the following function $$f:(\mathbb S^n)^N \rightarrow \mathbb R^{n+1} \text{ such that } f(x_1,...,x_N)= \sum_{i=1}^N x_i.$$ This function can be written in Cartesian coordinates as $f(x)=...
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2answers
471 views

Estimate for an Airy integral

Let me define for $x\in\mathbb R$, $ F(x)=\int_{\mathbb R} e^{-π t^2}\cos(x t^3) dt. $ I claim that $F(x)>0$ for all $x\in\mathbb R$. Well, it is obvious for $x=0$ since $F(0)=1$ and also for $x$ ...
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0answers
34 views

Conditions for liftings of smooth functions to converge to a smooth function

Assume we are given a uniformly convergent series of analytic functions $(f_n)_n$ in $C^\infty(\mathbb{R}^3, \mathbb{R})$ and these converge to a smooth function $f$. Further assume that we have ...
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0answers
87 views

Sturm-Liouville Problem: When does $w y^2$ vanish at a singular boundary point?

It is well known (e.g. Courant, Hilbert - Methods of Mathematical Physics) that solutions of the Sturm-Liouville problem on an interval $J=(a,b)$ \begin{equation} \tag{1} \left(p y' \right)' - qy \; = ...
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0answers
32 views

Recommendation for books on boundary-value problems that include perturbed boundaries and many solved problems

I am looking for a book or resource that contains applied math analytical methods and a lot of solved problems in Boundary-Value Problems for second-order PDEs, and if it could be related to wave-...
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1answer
161 views

Flow induced by differentiable velocity field is differentiable

Let $E$ be a $\mathbb R$-Banach space, $\tau>0$ and $v:[0,\tau]\times E\to E$ such that$^1$ $$x\mapsto t\mapsto v(t,x)\tag1$$ belongs to $C^{0,\:1}(E,C^0([0,\tau],E))$. This is enough to ensure ...
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1answer
498 views

Example of a function with a curious property

Denote by $L^1(0,1)$ the space of Lebesgue integrable functions on the interval $(0,1)$. $\textbf{Question:}$ Does there exist a function $F:(0,1)\rightarrow\mathbb{R}$ such that: $\frac{F(x)}{x}\in ...
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0answers
72 views

Analysis of solutions to a system of nonlinear ODEs arising from differential geometry

Consider the system of ODEs: \begin{equation} \varphi''\varphi'^{q-1}\psi'^{p-2}=\varphi^{p-1}\psi^{q-1}, \end{equation} \begin{equation} \varphi'^2+\psi'^2=1, \end{equation} where $\varphi>0$, $\...
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0answers
76 views

What is the weak limit of $f_n \ \mathrm{sign}(f_n - 1)$ if $f_n \to f$ weakly in $L^p([0,1])$?

Let $f_n: [0,1] \to \mathbb R$ be a uniformly bounded sequence in $L^p$. Then there exists a subsequence such that $f_{n_k} \to f$ weakly in $L^p([0,1])$. What is the weak limit of the sequence of ...
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0answers
39 views

Analysis of coefficients for quickly vanishing analytic vector field

Let $u = (u_1, u_2, u_3): \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a divergence-free analytic vector field for $n =3$ or $n =4$, i.e., $u_i : \mathbb{R}^n \rightarrow \mathbb{R}$ are analytic ...
5
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2answers
149 views

Analytic approximations of smooth vector fields

Let $M$ be the set of smooth divergence-free vector fields $u$ on $\mathbb{R}^3$ with $$|\partial_x^{\alpha} u(x)| \leq C_{\alpha K}(1+|x|)^{-K}$$ on $\mathbb{R}^3$ for any $\alpha,K$. Further, we ...
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0answers
77 views

inequality for two integral expressions

Given bounded positive functions $f, g, h \in L^1$, with $g, h$ finitely supported, I would like to compare the following two expressions: $$\begin{aligned} a_1 &= \int_{0}^{\infty} \! dx \, f(x) \...
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1answer
44 views

the subdifferential at points of differentiability in infinite dimensional space

Let $ f: X\to (-\infty,+\infty]$ that $ X$ is an infinite dimensional space. What are the conditions for $f$ and space $X$ to have the following equality correct? $$\partial f(x)=\{\nabla f(x)\}$$ for ...
2
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1answer
172 views

$L^p$ compactness for a sequence of functions from compactness of product with cut-off

Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions $$\{f_{n}\psi_m(...
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1answer
88 views

Construct a function with certain growth property

I have the following question: Does there exist a non-negative function $g$ on $(0,1)$ such that $$1\leq F(x):=\dfrac{\displaystyle\sum_{k=0}^{\infty}a_{k}\,(k+1)^{2}\,x^{k}}{\displaystyle\sum_{k=0}^{\...
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0answers
30 views

Approximation of vector field by vector fields with all trajectories closed

Let $\vec j$ be a smooth compactly supported vector field in $\mathbb{R}^3$ such that $div(\vec j)=0$. Are there known either necessary or sufficient conditions such that there exists a sequence of ...
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0answers
44 views

Example of BV vector field $c$ without bounded divergence such that $u$ is bounded where $u_t + div(cu) = 0$

What is an example of vector field $c: \mathbb R_+ \times \mathbb R^N \to \mathbb R^N$ with $c \in L^1(\mathbb{R}_+, BV(\mathbb R^N))$ without bounded divergence $div_x c$ but such that there exists a ...
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0answers
55 views

Uniqueness of solution to Cauchy problem with quadratic nonlinearity

Consider the non-linear differential operator $$\mathfrak{L}: \ C^2((0,T)\times\mathbb{R}^2)\ni\varphi\equiv\varphi(t,x,y) \, \mapsto \, \partial_x^2\varphi + (\partial_x\varphi)^2.$$ For $U\subset\...
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1answer
236 views

How to mathematically characterize a feedback loop in ODEs?

I have a biological system that exhibits a feedback type of behavior. The diagram is a schematic of the system of ODEs. In this system, the total amount of $x_1, x_2, x_3$ is conserved; however, there ...
2
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1answer
285 views

$L^p$ compactness for a sequence of functions from compactness of cut-off

Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions $$\{f_{n}\psi_m(...
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2answers
407 views

Finding solutions of the differential equation $x\frac{d}{d x}\left(x\frac{d y}{d x}\right)=4(y-x^{2}y+y^{3})$

In my research I have come across the following non-linear differential equation: $$x\frac{d}{d x}\left(x\frac{d y}{d x}\right)=4(y-x^{2}y+y^{3})$$ I want to find the general solution of this equation ...
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0answers
103 views

Log-concavity inequality

Let $x,y,$ and $t$ be fixed real numbers, $1<x<y$, $0<t<1$. Does the following inequality hold for some $c$ $$\frac{\log{(tx+(1-t)y)}}{\log^t{x}\log^{(1-t)}{y}}>\frac{\log{(sw+(1-s)z)}}{...
4
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1answer
169 views

How to estimate the order of this integral with parameter

Some introduction: Given a homogeneous structure called "dilation" in $R^n$: For $t\geq 0$ $$D_t: R^n\rightarrow R^n$$ $$D_t(x)=(t^{a_1}x_1,...,t^{a_n}x_n)$$ where $1=a_1\leq...\leq a_n$, ...
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2answers
119 views

Uniform boundedness of integral?

I have perhaps a very simple question where I lack some inutition at the moment: Is the expression $$\sup_{\alpha < 0, \lambda \in \mathbb N}\int_{-\infty}^{\alpha} e^{-\lambda t^4} \ dt \int_{\...
2
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1answer
154 views

Does the following function series converge?

Let $$ f_n(x)=\frac{\frac{1}{(n-1)!}\sum_{k=0}^{\lfloor \alpha n-x\rfloor}C_{n-1}^{k}~(-1)^k(\alpha n-x-k)^{n-1}}{\frac{1}{n!}\sum_{k=0}^{\lfloor \alpha n\rfloor}C_{n}^{k}(-1)^k(\alpha n-k)^{n}}, $$ ...
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0answers
35 views

Prove compactness of approximate (finite differences) solutions of heat equation

Let $\{u^h\}$ be the family of solutions of the discretized heat equations on the interval $[0,1]$ (uniform grid of size $h>0$). $$\begin{cases} u^h_t - \Delta_hu^h = 0\\ u^h(t,0) = u^h(t,1) = 0 \\...
3
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2answers
200 views

A Covering Lemma for Arbitrary Measures

In the book "Harmonic Measure" by Garnett and Marshall, we have the following result: Lemma I.2.3 Let $\mu$ be a positive Borel measure on $\partial{\mathbb{D}}$ and let $\{I_{j}\}$ be a ...
1
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1answer
205 views

Does the following sum converge?

Does the sum $$ \lim_{n\to\infty}\sum_{k=0}^{\lfloor\alpha n \rfloor}C_n^k(-1)^k\left(1-\frac{k}{\alpha n}\right) $$ converge, where $C_n^k$ is the binomial coefficient and $0 <\alpha <1$? ...
5
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0answers
106 views

Uniqueness of solutions of Young differential equations

Consider the following one dimensional Young differential equation: $$ Y_t=\int_0^t Y_s dX_s,\quad t\in[0,1]. $$ Here the driving process $X$ is a bounded functions $[0,1]\to\mathbb{R}$, which is $\...
1
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1answer
80 views

Regular singular point of non-linear ODE: $\dot{x}(t) + t^{-1}Ax(t) = Q(x(t))$

Consider a system of ordinary differential equations of the form $$ \dot{x}(t) + \frac{1}{t}Ax(t) = Q(x(t)) $$ where $x(t) \in \mathbb{C}^n$, $A \in \mathrm{Mat}_{n\times n}(\mathbb{C})$ is a constant ...
3
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1answer
113 views

Closed form for the integral of a squared Legendre function

Is there a closed form for the integral $$\int_0^{\pi/2}(P_\nu^\mu(\cos\theta))^2\,\mathrm d\theta,\quad\mu>\nu\gt-\frac12$$ where $P_\nu^\mu(x)$ is the associated Legendre function of the first ...
4
votes
5answers
578 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
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0answers
29 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 ...

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