# 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.

2,505
questions

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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 ...

**-1**

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**0**answers

82 views

+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 + ...

**7**

votes

**1**answer

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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**0**answers

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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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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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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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 $f(x)$ ...

**5**

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**1**answer

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(\...

**3**

votes

**1**answer

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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**3**answers

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} \} \...

**3**

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**1**answer

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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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:\...

**0**

votes

**1**answer

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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**2**answers

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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**0**answers

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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**0**answers

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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**1**answer

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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**1**answer

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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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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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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**0**answers

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 \; = ...

**1**

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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-...

**2**

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**1**answer

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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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 ...

**5**

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**0**answers

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 $\varphi>0$, $\...

**2**

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**0**answers

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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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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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 ...

**2**

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**0**answers

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) \...

**0**

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**1**answer

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**

votes

**1**answer

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(...

**0**

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**1**answer

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}^{\...

**3**

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**0**answers

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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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 ...

**2**

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**0**answers

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\...

**2**

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**1**answer

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**

votes

**1**answer

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(...

**7**

votes

**2**answers

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 ...

**1**

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**0**answers

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)}}{...

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votes

**1**answer

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 $1=a_1\leq...\leq a_n$, ...

**0**

votes

**2**answers

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**

votes

**1**answer

154 views

### Does the following function series converge?

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

**0**

votes

**0**answers

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 \\...

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votes

**2**answers

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**

vote

**1**answer

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**

votes

**0**answers

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**

vote

**1**answer

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**

votes

**1**answer

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

**5**answers

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**

votes

**0**answers

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 ...