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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16 views

Basic Math Question for Data Analysis?

This is probably a really basic question for all you math wizards. I am not even sure how to phrase this question per se. I feel like this should be easy and yet I am questioning my thinking. Here's ...
2
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1answer
66 views

Two questions about an integral involving double product of Bessel functions

Let us define the following integral : $$W_n(r)=r\int_0^{+\infty} J_1(rt)[J_0(t)]^n dt,$$ with $r>0$ a real number and $n\in\mathbb{N}$ and where $J_0(x)$ and $J_1(x)$ are Bessel functions of the ...
3
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1answer
97 views

The square root of natural number expressed by an infinite series

Can you prove or disprove the following claim: Let $U(n,P,Q)$ be the nth generalized Lucas number of the first kind and let $m$ be a natural number. Then, $$\sqrt{m}=1+\displaystyle\sum_{n=1}^{\infty}...
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1answer
314 views

The constant $\pi$ expressed by an infinite series

I am looking for the proof of the following claim: First, define the function $\operatorname{sgn_1}(n)$ as follows: $$\operatorname{sgn_1}(n)=\begin{cases} -1 \quad \text{if } n \neq 3 \text{ and } n \...
6
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2answers
276 views

Matrix-valued ordinary differential equation with symmetry

I am considering the following equation $$\begin{pmatrix} -\frac{d}{dx} + \lambda \sin(2\pi x) & \lambda - \lambda \cos(2\pi x) \\ -\lambda-\lambda \cos(2\pi x) & -\frac{d}{dx} - \lambda \sin(...
4
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1answer
215 views

The constant $e$ represented by an infinite series

In this Wikipedia article the constant $\pi$ is represented by the following infinite series: $$\pi=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}-\...
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2answers
153 views

Given a total variation distance from uniform, how well can we bound the probabilities of sub-intervals?

I made the following claim, which I now see that I don't know how to prove. Can anyone prove it? Claim. Let $f$ be a concave and non-negative function on $[0,1]$ with $$\int_0^1 f = 1,$$ $$\int_0^1 |f-...
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1answer
354 views
+100

Real rootedness of a polynomial with binomial coefficients

It is possible to show using diverse techniques that the following polynomial: $$P_n(x)=1 + \binom{n}{2} x + \binom{n}{4} x^2 + \binom{n}{6} x^3 + \binom{n}{8} x^4 +\ldots + \binom{n}{2\lfloor\tfrac{n}...
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1answer
54 views

A bound on an oscillatory solution of an ODE

This question was restated as follows: Let $V\colon[a,b]\to\mathbb{R}$ be smooth, strictly decreasing and $V(b) = 0$. Suppose that $f\colon[a,b]\to\mathbb{R}$ is smooth and satisfies $f''(x)+V(x) f(x)...
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0answers
55 views

Laguerre polynomials variations

The generalized Laguerre polynomial is written by the following form: $$L_{n}^{(\alpha)}(x)=\sum_{k=0}^{n} \begin{pmatrix} n+ \alpha \\ n-k \end{pmatrix} (-1)^{k} \frac{x^{k}}{k!}$$ I have found the ...
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1answer
78 views

A bound on a solution of an ODE, given some bounds on endpoints

Let $V : [a,b] \to \mathbb{R}$ be smooth, strictly increasing and $V(a) = 0$. Suppose that $f : [a,b] \to \mathbb{R}$ is smooth and satisfies $f^{\prime \prime} (x) + V(x) f(x) = 0$ on $[a,b]$. Can we ...
2
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1answer
83 views

An inequality for an integral transform of a function

Let $$J_{f;y}(u):=3 u^3 \int_u^1\frac{dt}{t^4} \,e^{-i y t}f(t)- e^{-i u y}f(u),$$ where $y\in(0,\infty)$, $u\in(0,1)$, and $$f(t):=t+\pi (1-t) t \cot (\pi t).$$ Here are the graphs of $f$ (black), ...
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205 views

How is this expression for the regularization of integrals of monomials, given in a paper, justified? How strong is argument in favor?

In this answer by Carlo Beenakker he cites the following regularization formula: $$\int_0^\infty x^p\,dx\mathrel{"="}\frac{(-1)^{p+1}}{(p+1)(p+2)},\;\;p=0,1,2,\dotsc,$$ citing Tafazoli - Calculation ...
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0answers
35 views

Uniform bound on a certain family of hypergeometric functions

We have the following problem, which we can't solve. Let $a \in \mathbb{C}$ be fixed, with real part $1/2$ and imaginary part $\neq 0$. We consider parameters $n \in \mathbb{Z}$ and $k \in \mathbb{Z}_{...
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2answers
77 views

Lipschitz continuity of eigenvalues and eigenvectors of Hermitian matrices

It is well-known that the eigenvalues (in decreasing order) of a Hermitian matrix $A$ are Lipschitz continuous functions of $A$. Do there exist orthonormal eigenvectors that vary in a Lipschitz ...
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0answers
43 views

What is known about discrete versions of the spatially homogenous Boltzmann equation with finitely many (but arbitrarily many) velocities?

Consider a discrete version of spatially homogenous Boltzmann equation with finitely many (but arbitrarily many) velocities $v_i \in \mathbb R^n$ with $i \in I$. Equivalently, consider a system of ...
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0answers
102 views

Is TREE(3) really that big? [duplicate]

Grahams number is pretty big and cannot be written. TREE(3) is the length of the largest set of trees with 3 seeds such that the first tree can have 1 seed, second tree, 2 seeds, third tree, 3 seeds ...
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1answer
231 views

Closed form of $\prod_{k=1}^{n}\left(\cos(kx)-1\right)$

Is there any closed form of $$\prod_{k=1}^{n}\left(\cos(kx)-1\right)?$$ I failed to find references on this problem in the internet.
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4answers
264 views

Joint boundedness of solutions of a family of Sturm-Liouville ODE

Let us fix $0 \neq \lambda \in \mathbb{R}$. Let us consider the following ODE, on $[0,\infty)$: $$ y^{\prime \prime} (x) + \frac{r e^{-x}}{(1+e^{-x})^2} y(x) = -\lambda^2 y(x).$$ Here $r \ge 1$ is a ...
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0answers
141 views

$\int_{\mathbb{R}^{N}\setminus\Omega}\vert x-z\vert^{-N-\alpha} dz = c \ \forall x\in\partial U$ implies $dist(x,\partial\Omega)=c, x \in \partial U$?

Let $\alpha \in \mathbb R_+$, $\Omega \subset \mathbb R^N$ and $U \subset \Omega$. Is it true that if $$\int_{\mathbb R^N \setminus \Omega} |x - z|^{-N-\alpha} dz = \text{constant} \quad \text{for all ...
3
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2answers
250 views

Second order differential equation with oscillating behavior

I consider a differential equation $y^{\prime \prime} (x) + V(x) y(x) = 0$ in the interval $[0,\infty)$, where $C_1 \leq V(x) \leq C_2$ for all $x \in [0,\infty)$ for some constants $C_2 > C_1 >...
10
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1answer
705 views

Imaginary eigenvalues

Consider the matrix $$A(\mu) = \begin{pmatrix} 0 & 1& 0 & 0 \\ -1 & -i\mu & 0 & i \\ 0 & 0 & 0 & 1 \\ 0 &i & -1 & i\mu \end{pmatrix}.$$ This matrix is ...
7
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4answers
322 views

Improper integral $\int_0^\infty\frac{x^{2n+1}}{(1+x^2)^2(2+x^2)^N}dx,\ \ \ n\le N$

How can I evaluate this integral? $$\int_0^\infty\frac{x^{2n+1}}{(1+x^2)^2(2+x^2)^N}dx,\ \ \ n\le N$$ Maybe there is a recurrence relation for the integral?
1
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1answer
90 views

Asymptotic behavior of an ODE

Consider the following ODE eigenproblem of $y(x)$ \begin{equation} y'' + [\varepsilon + b^2 x - (a + \frac{b^2}{2}x^2)^2 ] y=0 \end{equation} with eigenvalue $\varepsilon$, real constants $a,b$. ...
2
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1answer
155 views

Similarity of two matrices

Consider the matrix, for some $\lambda \in \mathbb R$ . $$A=\begin{pmatrix} i \lambda & -1 & i & 0 \\ 1 & 0 & 0& 0 \\ i & 0 & - i \lambda & -1 \\ 0 & 0 & 1 ...
13
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3answers
1k views

Eigenvalue pattern

We consider a matrix $$M_{\mu} = \begin{pmatrix} 1 & \mu & 1 & 0 \\ -\mu & 1 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 &-1 & 0 & 0 \end{pmatrix}$$ One easily ...
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0answers
117 views

Prove an inequality or find a counterexample

Suppose $\mathcal{M}_1$ represents the space of smooth probability density functions with unit mean, whose support is contained in $[0,\infty)$ (or $\mathbb{R}_+$). Define the following functional $$\...
1
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0answers
43 views

Initial-boundary value problem for transport equation with $W^{1,p}$ velocity

Let us consider $v:\mathbb R_+ \times \mathbb R \to \mathbb R_+$ such that $v \in L^1(0,\infty, W^{1,p}(\mathbb R))$ and the transport equation $$ \begin{cases} u_t + v(t,x) u_x = 0 \qquad & (...
2
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2answers
45 views

Asymptotic behavior of a Bessel function on a sequence on zeros with a shifted parameter of type

Let $J_\nu$ be a Bessel function of the first kind and let $\{\lambda_{n, \nu}\}_{n\ge 1}$ be a sequence of its zeroes. I claim that $$ \inf_{n\ge 1}\bigg|\sqrt{\lambda_{n,\nu}} J_{\nu+1}(\lambda_{n,\...
6
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1answer
135 views

Littlewood-Paley theory and dense property of Sobolev spaces [duplicate]

I'm learning the Littlewood-Paley theory by myself and I encounter the following claim: Pick a smooth function $\chi$ such that: $$\chi(\xi) = \begin{cases} 1 &|\xi| \leq \frac{1}{2}\\ 0 &|\...
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0answers
31 views

On a question relating integral equation:

I don't know if the following question qualifies as research level. If it isn't, sorry. Set the following terminology: $ \alpha_1 =\alpha_1(t,x)=t(\tan^{-1}(x)+c)$ $\alpha_2=\alpha_2(s,x)=s(\tan^{-1}(...
1
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0answers
28 views

Pontryagin's principle with Lebesgue-integrable control

Does there exist a (weak) version of Pontryagin's minimum principle in which the control is allowed to be just Lebesgue integrable? I am mostly familiar with the 1975 text of Fleming & Rishel, ...
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0answers
174 views

Inverse of block matrix II

This is a follow-up question on a previous question of mine that had a negative answer. I tried some examples and believe the following has a chance to be true. Let $V$ be a finite-dimensional vector ...
1
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0answers
51 views

domain of flow of an inner vector field is a manifold with corners

Crosspost. Let $X$ be a manifold with corners. Let $\vec v$ be an inner vector field on $X$. The existence and uniqueness theorem for ODE says there's a domain of flow $\mathfrak D(\vec v)\subset \...
3
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2answers
199 views

An integral transform and the Stone-Weierstrass theorem

For a bounded function $\operatorname{F}: \mathbb{R}_{\,\ge\ 0} \to \mathbb{R}$ (not necessarily non-negative), if $$ \int_{0}^{\infty}\frac{x^{k}\,s}{(s^{2} + x^{2})^{\left(k + 3\right)/2}\,\,}\, \...
1
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3answers
166 views

Can we calculate the probability that $f(x)$ is positive for a randomly chosen value of $x\in(0,m)$ as $m\to\infty$? (uniform distribution)

Following my previous question here, I have this function $$f(x)=10+3 \cos (ax-bx)+13 \cos (ax+bx)+2 \cos (\frac32 a x)+17 \cos (b x),$$ with $\frac ab \notin \mathbb{Q}$. What is the limit $$ \lim_{m\...
10
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1answer
514 views

Have any proposals been advanced for the analytic continuation of the divisor function?

While I was working on the evaluation of a certain series, the following limit came up: \begin{align} \lim_{n \to 1} \frac{d(n)-1}{n(n-1)} &= \lim_{n \to 1} \frac{d'(n)}{2n-1} \\ &= d'(1) .\...
3
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1answer
97 views

Small power series “approximating” a Dirac

Does there exist a (sequence of) power series $\sum_{i\geq 0} a_{n,i} x^i$ that is $1$ at $0$ and $0$ at integers from $1$ to $n$, and such that $\sum_{i\geq 0} \vert a_{n,i}\vert n^i=O(n^p)$ for some ...
2
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0answers
51 views

Sequence tending to modulus function

I am looking for a sequence of smooth functions $f_n: \mathbb R^2 \rightarrow \mathbb R$such that the 1.) the nodal set of $f_n$, i.e. $N_n:=f_n^{-1}(0)$ converges to the graph of the absolute value ...
4
votes
1answer
131 views

Scaling of double convolution

I am interested in the scaling of $$F(x_1,x_4)=\int_{\mathbb R^2} e^{-\vert x_1 -x_2 \vert -\varepsilon \vert x_2 -x_3 \vert- \vert x_3 -x_4 \vert } \ dx_2 dx_3 $$ In particular, I suspect that $$F(...
5
votes
0answers
124 views

Lavrentiev phenomenon between $C^1$ and $C^2$

Does there exist a (onedimensional) functional that exhibits the Lavrentiev phenomenon between $C^1$ and $C^2$ that is $ F(y)=\int_a^b f(t,y(t),y'(t))\,dt \hspace{1cm}$ or possibly $ \hspace{1cm} F(y)=...
1
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1answer
33 views

Green function of symmetric stable process in dimension 1 and 2

Are the results in this paper on the Green function of a symmetric stable process available also in space dimension $d =1$ and $d=2$? The main theorems here are stated only for $d \ge 3$.
9
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1answer
574 views

Simple-looking problem with integrals

Let $f: [0,\infty) \rightarrow \mathbb{R}$ be a continuous function such that $f(0) = 0$. Is it true that if the integral $$ \int_0^{\pi/2} \sin(\theta) f(\lambda \sin(\theta)) \, d\theta $$ is zero ...
5
votes
1answer
288 views

Existence of periodic solution to ODE

We shall consider the matrix-valued differential operator $$(L u)(x) :=u'(x) - \begin{pmatrix} 0 & \sin(2\pi x-\frac{\pi}{6})\\ - 2\sin(2\pi x+\frac{\pi}{6}) & 0 \end{pmatrix} u(x).$$ This is ...
3
votes
1answer
80 views

On integral representation of Whittaker $W$ functions

According to NIST, the integral representation of Whittaker $W$ functions $$ W_{\kappa,\mu}\left(z\right)=\frac{z^{\mu+\frac{1}{2}}2^{-2\mu}}{\Gamma\left(% \frac{1}{2}+\mu-\kappa\right)}\int_{1}^{\...
7
votes
2answers
303 views

Energy levels of double well potential

Consider the (quantum) Hamiltonian on the real line $$H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x).$$ Let us assume that the potential $V$ is an even smooth functions with exactly two non-degenerate ...
4
votes
1answer
179 views

Prove $\int_{\mathbb R^N \setminus \Omega} |x - z|^{-N-s} dz \approx dist(x,\partial \Omega)^{-s}$, $s \in (0,2)$

Let $\Omega \subset \mathbb R^N$ and $s \in (0,2)$. Under what assumptions on $\partial \Omega$ do we have $$\int_{\mathbb R^N \setminus \Omega} |x - z|^{-N-s} dz \approx \mathrm{dist}(x,\partial \...
7
votes
0answers
206 views

Is this function concave?

Let $$h(u):=u^3 \left|\int_u^\infty \frac{e^{-i t}}{t^3} \, dt\right|$$ for $u>0$. Is the function $h$ concave on $(0,\infty)$? (For context, see Proposition 4.4.4 and formula (4.4.21) in this ...
0
votes
0answers
34 views

Almost-differential functional equations

The ODE $y'(x)+P(x)y(x)=Q(x)$ has solution $$I(x)y(x)=\int I(x)Q(x)\,dx$$ where $I(x)=\exp\int P(x)\,dx$. Equivalently, $$Y(x)+P(x)\int_0^xY(t)\,dt=Q(x)\tag1$$ has solution $$Y(x)=\frac d{dx}\frac{\...
4
votes
1answer
213 views

Rouché's Theorem in complex analysis on the relation of the number of zeros and poles of meromorphic functions in a region [closed]

This question is from my son referenced in my earlier question, Need advice or assistance for son who is in prison. His interest is scattering theory . He asked me to post this question: Hello and ...

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