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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Generalization of the triple tangent identity

It is well known that if $x + y + z = \pi$ then $$\tan x \times \tan y \times \tan z = \tan x+ \tan +\tan z.$$ I came across the following generalization of this equality: $$\sqrt{1-k^2} {\rm sc}(...
Kilian Raschel's user avatar
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Can we implicitly fit a system of linear ODEs by reduced information?

Assume we are given several initial vectors $x^{(1)},\ldots,x^{(r)} \in \mathbb{R}^n$, where the dimension $n$ is in the range of 50 to 100, and the number of initial vectors $r$ is in the range of ...
tobias's user avatar
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Polynomials orthogonal w.r.t. the logarithmic weight

Recently, I have encountered the family of orthogonal polynomials $p_{n}(x)$ which is orthogonal w.r.t. the function $-\ln(x)$ on $(0,1)$. This means we have $$\int_{0}^{1}p_{n}(x)p_{m}(x)\ln(1/x)dx=\...
Twi's user avatar
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The asymptotic distribution of a subset of Bessel function zeroes

For a research problem I am working on in PDE, I need to obtain asymptotics for the counting function of $$\{0<\alpha <\lambda: \exists n\in \mathbb{N} \textrm{ such that }J_n(\alpha)=0 \textrm{ ...
Goonfiend's user avatar
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Lyapunov stability, nonlinear system

Please, is there any reference for proposition below or does it perhaps follow from a standard fact? I've got it for some other problem but I actually do not know how to comment it in my article. ...
freddy's user avatar
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Lyapunov stability of linear system

Consider a linear ODE system $$\dot x_k=\sum_{j=1}^ma_{kj}(t)x_j,\qquad k=1,\ldots, m,\quad a_{kj}(t)\in C[0,\infty).\quad (1)$$ Proposition. Suppose that $$\sup_{t\ge 0}\Big\{\int_0^t\Big(a_{kk}(...
Clive's user avatar
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How to study analytically this ODE?

I have tried to find some known ODEs before posting on this forum, but I did not find anything about this kind of ODE: $y'(x)^2 + a(x)*y(x)^2 = 1$ with $a\in C^∞(\mathbb{R},\mathbb{R})$ and $y\in C^∞...
Linkis's user avatar
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How naturally can functions defined by parametric integrals be interpolated from $\mathbb N$ to $\mathbb R^+$?

It is well known that several definite integrals $I_n$ containing a parameter $n\in\mathbb N$ can be expressed recursively (e.g. doing integration by parts) in terms of $I_{n-1} $ or $I_{n-2} $, and ...
Wolfgang's user avatar
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Can a smooth function on a cross be extended to the whole plane?

Consider a real function on the union of two lines R×0 and 0×R in R² whose restrictions to R×0 and 0×R are smooth functions R→R. Is it possible to extend this function to a smooth function on R²? ...
Dmitri Pavlov's user avatar
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Hardy-type inequality for point boundary

Let $f$ be in $W^{2,p}(\mathbb{R}^n)$ for $n\geq 3$ and $p>n/2$, with $f=0$ at the origin. I want to show that the integral $$\int_{B(0,r)} (f |x|^{-2})^p dV <\infty$$ for some small $r>0$. A ...
James Dilts's user avatar
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A question about the existence of a solution for an ODE

Let $\gamma_+$, $\gamma_-:\mathbb{R}_+\to\mathbb{R}$ be two given functions. Assume that $\gamma_+$ ($\gamma_-$) is smooth, strictly increasing (decreasing) and $\gamma_{+}(+\infty)=+\infty$ ($\gamma_{...
CodeGolf's user avatar
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path integral and index theorem

I actually have an integral which is used to prove Atiyah-Singer index theorem for spin complex in a path integral fashion. The integral I need to evaluate is following (in simplified form) $\int \...
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Zeroes of Sturm-Liouville solutions as a function of the (complex) eigenvalue

Given the Sturm-Liouville type (time independent Schroedinger) equation \begin{equation} \frac{d^2 y}{d x^2} - \left(\mu + V(x)\right) y = \lambda \, y,\quad x \in \mathbb{R} \end{equation} where $V(...
Frits Veerman's user avatar
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Generalizing "variation of parameters"

I'm stuck on generalizing an ODE formula and could use your help! One way to think about "variation of parameters" is that it bakes the solution $z(t)=e^{At}z_0$ of $z'=Az$ (here $z(t)\in\mathbb{R}^n$...
Justin's user avatar
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A linear operator equation (PDE) with non-monotone term

I'm interested in the existence and/or uniqueness to the following problem. Let $V$ and $H$ be Hilbert spaces and $V \subset H \subset V^*$ form a Gelfand triple. There is a linear operator $L:{D}(L) ...
AACA's user avatar
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2 answers
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Prove that the Dirichlet eta function is monotonic

Let us consider $\eta(p):= \sum\limits_{n=1}^\infty \frac{(-1)^{n+1}}{n^p}$ for $p>0$. Has anyone come along with an elementary proof that $\eta(x)$ is monotonically increasing on this set? By ...
Alexander Kuleshov's user avatar
2 votes
0 answers
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Operator theory of initial-value ODE problems

The theory of elliptic boundary value problems is usually treated from the perspective of functional analysis, and the theory of operators between Hilbert spaces. In contrast to that, the theory of ...
shuhalo's user avatar
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the existence of a real polynomial satisfying the following property

It is easy to verify that $$ \frac{t}{2}\leq \frac{1}{4}+\frac{1}{4}t^2\leq \frac{t}{1-(1-t^2)^2}-\frac{t}{2} \quad \quad 0<t\leq1$$ I want to ask if there exist a real polynomial $h(t)$ such ...
user173856's user avatar
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Does the following measurable Halmilton-Jacobian equation admit a Lipschitz solution?

I have the following question: Let $F:\Omega\times \mathbb{R}^n\to [0,\infty)$ be a convex Finsler norm, which means that $F(x,\cdot)$ is convex with respect to the second variable. $F(\cdot,v)$ is ...
Changyu Guo's user avatar
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4 votes
2 answers
317 views

Uniqueness of solutions to an ODE system

For each $i$ (up to infinity), let $u_i \in C^1(0,T)$ satisfy $$\frac{d}{dt}u_i(t) + \sum_{j=1}^\infty b(t;w_j,w_i)u_j(t) = 0$$ $$u_i(0) = u_i(T)$$ where $b(t;\cdot,\cdot)$ is an inner product on some ...
assa888's user avatar
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18 votes
3 answers
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Poincare lemma for non-smooth differentiable forms

The Poincare lemma is almost always formulated for differential forms with smooth coefficients (or sometimes for currents that have distributional coefficients). I would like to have it for $C^k$-...
Jochen Wengenroth's user avatar
14 votes
4 answers
1k views

$L^p$ norm means

Consider the unit sphere $S_p^{n-1}$ of an $L^p$ normin $\mathbb{R}^n.$ The question is: what is the expected value of the $L^q$ norm on $S_p^{n-1}?$ Since (I assume) this is intractable in closed ...
Igor Rivin's user avatar
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2 answers
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A book about almost periodic functions [closed]

Can anyone give me suggestions for new books about Besicovitch's almost periodic functions? Thanks a lot.
Madarb's user avatar
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Solving polynomials of arbitrary degree

Is there any analogue of the method for expressing roots of polynomials of degree 5 with elliptic and η-functions that generalizes to polynomials of degrees n>5? More specifically: does there exist ...
user57888's user avatar
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Extending point-wise bound to uniform bound

Suppose $f(t,x)\in \mathcal{C}^0([0,1]\times \mathbb{R}^n)$. Further suppose that for each $t$ $$ C(t):= \sup_{x\in\mathbb{R}^n} |f(t,x)|<\infty \, .$$ Does it follow that $f$ is bounded? Note ...
Benji's user avatar
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2 answers
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How much can one say about this differential equation?

Consider the ODE $y^{\prime \prime}(x) = \cos(x) y(x)$ with boundary value conditions $y(0)=1$, $y(1)=2$. Solving it results in a linear combination of Mathieu functions, but what I find more ...
Igor Rivin's user avatar
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7 votes
1 answer
391 views

A conjecture about the measure estimates of a trigonometric polynomial

Formulation of the Conjecture Let $\Omega =(0,\pi)\times (0,2\pi)\subset\mathbb R^2$ and let $\psi:\Omega\to \mathbb{R}$ defined by $$\psi(x,t)=\sum_{k\in S \,j\in S'} \sin(kx)\left( a_{kj}\sin(jt)+...
Arturo Sanjuán's user avatar
6 votes
1 answer
405 views

Are there superexponential Pfaffian functions?

This question is motivated by model theory, but it's really an analysis question (which means it may have an easy analysis answer that I just don't have the background for). Here's the main question, ...
Richard Rast's user avatar
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0 answers
101 views

The trivility of Besov space for large parameter

For all $s>0$, $1\le p<+\infty$ and $u\in L^p(\mathrm{d}x)$, we define $$D_{s,p}(u)=\sup_{r>0}\frac{1}{r^{sp+n}}\int_{\mathbb{R}^n}\int_{B(x,r)}|u(y)-u(x)|^p\mathrm{d}y\mathrm{d}x$$ and $$W^{...
yangmengqh's user avatar
5 votes
0 answers
894 views

Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?

Preliminary Definitions Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...
juan rojo's user avatar
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12 votes
1 answer
715 views

If the generating function summation and zeta regularized sum of a divergent series exist, do they always coincide?

One could assign a value to divergent series by means of several summation methods. One summation method we could consider is the generating function method. Let's sum, for example, the fibonacci ...
Max Muller's user avatar
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1 vote
0 answers
178 views

Laplacian mapping on various function spaces

I have a question related to a certain elliptic operator on $R^N$ but I think i can clarify my confusion if I just consider the Laplacian $\Delta$ on the unit ball in $R^N$. If $ 1 <p< \infty$...
Craig's user avatar
  • 539
2 votes
1 answer
129 views

Controling mixed derivatives

This is a cross-post from Math.SE since the question got nothing (but upvotes) even after offering a decent bounty. If it is too trivial or in other ways not suited for this site, please let me know ...
AlexR's user avatar
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2 votes
2 answers
206 views

Caratheodory equations

Ok, I am reading Fillipov book on discontinuous right hand side differential equations (the red book). He states the next lemma: " Let the function $f(t,x)$ satisfy the Caratheodory conditions and ...
Alan's user avatar
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3 votes
2 answers
261 views

Some Questions from Reading on Wave Front Set from Hormander's Linear PDE Vol. 1

In Hormander's Linear PDE Vol. 1 (pg 252-253, before the definition of wave front set is introduced), Lemma $8.1.1$ says that if $\phi \in C_{0}^{\infty}$ and $v \in \mathcal{E}^{\prime}$, then $\...
user57055's user avatar
1 vote
0 answers
94 views

Estimating convolutions of powers

I would like an asymptotic estimate of $$ \sum_{y \in \mathbb{Z}^d} \frac{1}{|y-a_1|^{d-1} \ldots |y-a_n|^{d-1}} $$ that does not involve any infinite summation. In order to lighten the notation, I ...
Elwood's user avatar
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1 vote
1 answer
470 views

Is there an asymptotic bound for this oscillatory integral?

I have an oscillatory integral: $$ \int u(x,y) e^{i\lambda f(x,y)} dx $$ with $f(x,y)\in \mathbb{C}^{\infty}$ a complex-valued function in a neighborhood of $(0,0)$ satisfying: $$ \text{Im} f \geq ...
teagut's user avatar
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1 vote
1 answer
156 views

Solution of General Parametric Oscillator

I am wondering if there is a general solution for this ODE $\ddot X +2\gamma \alpha \dot X + (\alpha+S(t)) X = \beta $ the dot represents time derivative, and $\gamma>1$, so it is in the over-...
Pablo Riera's user avatar
5 votes
1 answer
477 views

Spectrum of this ODE

I noticed something interesting studying this Sturm-Liouville Problem: $$ \frac{d}{dx}\left(\sqrt{(1-x^{2})}\frac{df}{dx} \right)+\frac{\left(n \alpha x+\alpha^2 x^{2} + \lambda\right)f}{\sqrt{(1-x^{...
user avatar
7 votes
1 answer
489 views

Prove that ..., f(x-2), f(x-1), f(x), f(x+1), f(x+2),... is algebraically linearly independent without the Fourier transform

Suppose that $f$ is in $L^2(\mathbb{R})$ and consider the set of integer translates of this function, $V=\{f(x-k):k\in\mathbb{Z}\}$. This set is linearly independent: taking the Fourier transform of ...
Peter Luthy's user avatar
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5 votes
1 answer
266 views

Is this graph of reciprocal power means always convex?

Let $$ p = (p_1, \ldots, p_n) $$ be a finite probability distribution, which for convenience I'll assume to have no zeroes: thus, $p_i > 0$ for all $i$ and $\sum_i p_i = 1$. Is the function $$ q \...
Tom Leinster's user avatar
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-4 votes
1 answer
556 views

Derivatives of infinite order [closed]

Is there any sense of taking an infinite number of derivatives? Is it discussed in the literature? For example, can one make sense of $$\frac{\partial^{\infty}f(x_1,x_2,\cdots)}{\partial x_1 \...
Alan's user avatar
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3 votes
0 answers
146 views

Variational Principle for a System of Differential Equations

I am studying a differential operator of the form $$ L\left(\begin{array}{c} u \\ v \end{array}\right) = -\Delta \left(\begin{array}{c} u \\ v \end{array}\right) + V(x)\left(\begin{array}{c} u \\ v \...
k3thomps's user avatar
  • 516
1 vote
1 answer
135 views

Positive Definiteness of a certain function

Recall that a function $f: \mathbb{R}^d \longrightarrow \mathbb{C}$ is positive definite, iff for all numbers $N$ and $x_1, \dots, x_N \in \mathbb{R}^d$, the matrix $(a_{ij})$ with entries $$a_{ij} = ...
Matthias Ludewig's user avatar
4 votes
1 answer
197 views

M-Wright function asymptotics

Let $M(z;\nu):= \frac{1}{\pi}\sum_{n=1}^{\infty} \frac{(-z)^{n-1}}{(n-1)!}\Gamma(\nu n)\sin(\nu n\pi)=\frac{1}{2\pi i}\int_{\text{H}_a}\exp(\sigma -z\sigma^{\nu})/\sigma^{1-\nu} d\sigma$, $0<\nu<...
Xorwell's user avatar
  • 434
1 vote
1 answer
84 views

Bäcklund transformation related to two nonlinear differential equations

I'm looking for a Bäcklund transformation linking the following two nonlinear differential equations for real $t$: $$\dfrac{d^2}{dt^2}f(t)=\cos\left[f(t)\right]$$ $$\dfrac{d^2}{dt^2}g(t)=\sinh\left[g(...
Riccardo.Alestra's user avatar
1 vote
0 answers
345 views

Integral of Bessel function of 1st kind with complex exponential

Does someone know the solution (simple closed form) of one of theses integrals: $$\int_0^t J_l(s) e^{-iA(t-s)}ds$$ $$\int_0^t \frac{J_l(s)}{s} e^{-iA(t-s)}ds$$ with $l>0$, $t>0$, $\Re(A)>0$, ...
user2460530's user avatar
11 votes
2 answers
776 views

Functions that Calculate their $L_p$ Norm

are there any examples of functions $f:x\in\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ and intervals $(a,b), 0\le a \lt b \le \infty$ , for which $$\Big(\int_a^b{|f(x)|^p dx}\Big)^\frac{1}{p} = f(p)$$ $$\...
Manfred Weis's user avatar
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9 votes
1 answer
378 views

Differential operator simplification

Does anyone know the explicit formulation for the $q_k$'s in, $$(x+D)^n=\sum_{k=0}^n q_k(x)D^k\ \ \ \ ?$$ I know that $e^{-x^2/2+x}$ is a fixed point of $(x+D)$. I also, know that $$(x+D)H_n(x)e^{-x^...
Bobby Ocean's user avatar
2 votes
1 answer
150 views

A question on existence of solutions of a linear ODE system

I am working on a problem of harmonic functions on surfaces, and in one step I got the following system of ODEs with prescribed asymptotes. I was wondering what methods could give us the existence or ...
littlelittlelittle's user avatar

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