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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Polya-Szego inequality w.r.t. partial direction

Let $f\in H^1(\mathbb{R}^d)$ and let $f^*$ be its symmetric deceasing rearragement. Then the Polya-Szego inequality tells us that $f^*$ is also in $H^1(\mathbb{R}^d)$ and $$\|\nabla f^*\|_2\leq \|\...
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Bounding integral expression with Sobolev norm of integrand

Consider the following integral expression: $$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$ for $\epsilon>0$, $f \in L^\...
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Hamiltonian, energy, and conservation laws of nonlinear PDEs

In many PDEs, I see the papers mention the energy of the PDE. And some papers and books mention Hamiltonians. I know that integrable systems have infinitely many conservation laws and these laws are ...
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Integral representation of $\frac{355}{113}-\pi$?

It is well known that $$ \frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 \frac{dx}{1+x^2}, $$ Given that $\frac{355}{113}$ is an excellent approximation of $\pi$, is there any known integral representation of ...
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Determine $\alpha \in (0,1)$ such that $J_{\alpha}(\phi):=\int \psi/\phi^{\alpha}$ exists?

Fix $\alpha \in (0,1)$ and $\psi\in C^{\infty}_{c}(\mathbb{R}\to \mathbb{R})$. For a smooth function $\phi\geq 0$ define the integral $$J_{\alpha}(\phi):=\int \frac{\psi}{\phi^{\alpha}}.$$ If $|\phi^{...
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Bounding integral expression with BV norm of integrand

Consider the following integral expression: $$\mathcal I :=\iint_{\epsilon \leq|x-y| \leq 1/2} f(x) f(y) \frac{\langle g(x)-g(y), x-y\rangle}{|x-y|^{n+2}} d x d y $$ for $\epsilon>0$, $f \in L^\...
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If $\lambda$ is a closed orbit of the field $X=(P,Q)$ and $D$ is the disc limitaded by $\lambda$ then $D$ has a unique singularity of $X$

Let $X=(P,Q)$ a vector field on $\mathbb{R}^2$ where $P$ and $Q$ are polynomials of degree $2$ (i.e is of the form $ax^2+bxy+y^2+dx...$). Let also $\lambda$ be a closed orbit of $X$ and $D\subset\...
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On the number of limit cycles for a differential equation

Let us consider the following differential equation: $$x′=f(x,t)$$ where $f$ is a continuous function and $t,x∈ℝ$ When one search for the number of limit cycles of this equation, the following ...
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The energy of a semilinear ODE

I'm currently reading Caffarelli, Gidas, Spruck's paper "Asymptotic Symmetry and Local Behavior of Semilinear Elliptic Equations with Critical Sobolev Growth". For some background, we ...
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A sharp estimate for an oscillatory integral with a simple phase

Let $a>1$ not necessarily an integer, and let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a smooth function with compact support. Consider the oscillatory integral $$I(\lambda):=\int_{0}^{\infty}\int_{...
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Eigenvalues come in pairs

Consider the two matrices with some parameter $s \in \mathbb R$ $$A_1= \begin{pmatrix} s& -1 &0& 0 \\1&0 &0&0 \\ 0&0&1&0 \\0&0&0&1 \end{pmatrix}$$ and $$...
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2 votes
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measure corresponding to certain orthogonal polynomials

Given a scalar $\lambda\in (0,1)$, consider a sequence of monic polynomials $\{p_n(x)\}_{n\geq 0}$ over real variables satisfying the following recurrence relations: $xp_n(x)=p_{n+1}(x)+\lambda p_{n}(...
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the limit of a scalar sequence in a sequence of scalars

Let $(a_{n})_{n}$ be a bounded real-valued sequence. Suppose that $(b_{n})_{n}$ is a sequence (not necessarily a subsequence) in the set $A:=(a_{n})_{n}$. Assume that the limit $\lim\limits_{n}b_{n}$ ...
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Limit cycles or stable solutions for k-dimensional piece-wise linear ODEs

As a branch of reinforcement learning, restless multi-armed bandits have been shown PSPACE-HARD but Whittle has offered an implementable solution called the Whittle Index Policy. Weber and Weiss ...
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$BMO^{-1}$ functions and $L^p$ space

It is well known that $BMO(\mathbb R^n)$ functions are in $L^p_{loc}(\mathbb R^n)$ if $1 \le p < \infty$, but need not be locally bounded. How about a function $f \in BMO^{-1}(\mathbb R^n)$? Does ...
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Is integration by parts allowed for the $J^s$ derivative, where $s \in \mathbb R$

I am having the following integral: $$I = \int u\, J^s(\partial_x \overline{u})- \overline{u}\, J^s(\partial_x u))dxdy$$ where $J^S= (I-\Delta)^\frac{2}{2}$, $\mathbb{R} \ni s \geq 1$ and $u=u(x,y)$, $...
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1 answer
193 views

Fundamental theorem of calculus for Lebesgue–Stieltjes integrals?

Note: Throughout, we denote by $\mathcal L$ the Lebesgue measure on $\mathbb R$. Let $g: [0, 1] \to \mathbb R$ be a continuous function of bounded variation. Denote by $\mu_g$ its associated Lebesgue–...
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The meromorphic continuation of Selberg-like integrals in the symmetric case

Introduction. In connection with the question (1) (link below), I've been trying to understand the meromorphic continuation in $\alpha,\beta,\gamma$ of the Selberg-like integral $$ S_N(\alpha,\beta,\...
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1 answer
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Equidifferentiable functions

Let $f_n: [0, 1] \to \mathbb R$ be a sequence of continuously differentiable functions. We say that the sequence $f_n$ is equidifferentiable if for every $x \in [0, 1]$ and every $\varepsilon > 0$, ...
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Is every sequence of functions with uniformly bounded variation almost equicontinuous?

Let $f_n: [0, 1] \to \mathbb R$ be a sequence of functions. Given a measurable subset $E$ of $[0, 1]$, we say that the sequence $f_n$ is equicontinuous on $E$ if for every $x \in E$, and $\varepsilon &...
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Bounding $2$-Wasserstein distance and the $L^1$ distance

My questions come from the paper Logarithmic Sobolev inequalities for some nonlinear PDE’s written by F. Malrieu (May 2001) where author omitted a good amount of details to be filled. Suppose that $W$ ...
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Zeros of entire functions

Let $f_w:\mathbb C \to \mathbb C$ be an entire function such that $(0,1) \ni w \mapsto f_w$ is real-analytic. Assuming that there is a dense subset $D \subset (0,1)$ such that for $w \in D$ the ...
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3 votes
2 answers
236 views

Growth of $L^p$ norms as $p \to \infty$

Let $f$ be a non-negative function defined on the unit interval. It is well known that $N(p) := \left(\int_0^1 f^p(t) dt\right)^{\frac{1}{p}} $ converges to $\operatorname{esssup}_{[0,1]} f$ when $p \...
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Zeros of entire functions with parameter

Let $f_w:\mathbb C \to \mathbb C$ be an entire function with $f_w(0)=1$ and at least one root for any choice of $w \in (0,1)$. Assume further that for a dense set of $w$ the function $f_w$ has ...
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  • 192
2 votes
1 answer
258 views

Measurability of a net

Let $(f_\epsilon)_{\epsilon>0}$ be a family of positive measurable functions on $L_p(\mathbb R)$ where $1<p<\infty.$ Assume that the pointwise supremum $f^*(x)=\sup_{\epsilon>0}|f_\epsilon(...
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7 votes
2 answers
358 views

Kneser theorem about the Klein bottle

I know that in $1923$ H. Kneser showed that a continuous flow in a Klein bottle without singular points has a periodic trajectory. The original article is this, but does anyone know another old or new ...
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Asymptotic expansion of a *specific* ODE solution

Suppose I have a second-order ODE of the form $$y'' + p(x) y' + q(x) y = 0.$$ There are plenty of techniques for expressing the asymptotic behavior of the solutions of such an equation as $x \...
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1 answer
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Mikusiński's approach to Bochner integrals; replace absolute by unconditional?

In the book The Bochner Integral, Mikusiński described an approach to Lebesgue and Bochner integrals via absolutely convergent series corresponding to step functions: Defn. Let $X$ be a Banach space. ...
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3 votes
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98 views

Tauberian theorem with flatness condition

Suppose $f(z)=\sum_{n=0}^{\infty}a_nz^n$ is a series with $a_n\in \mathbb{R}$ and radius of convergence $1$ and such that $f$ restricted to $[0,1[$ admits a smooth extension to $[0,1]$ with $f^{(n)}(1)...
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Convolution definition in an old educational article

I was reading an old article in IEEE Education magazine by Robbins and Fawcett titled "A Classroom Demonstration of Correlation, Convolution and the Superposition Integral" DOI: 10.1109/TE....
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Construction of holomorphic function

I was trying to construct a holomorphic function $f$ on $\mathbb{C}$ such that $|f|^2(z)=e^{(|z|^2-\frac{1}{2})^2}$. I will be happy if someone can give me an idea how to do that. I would like also ...
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1 vote
1 answer
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Is this long closed form for pi trivial?

With the help of wolfram alpha we got very long closed form for $\pi$ in terms of algebraic numbers, logarithms of algebraic numbers and $cot^{-1},coth^{-1}$ which can be expressed as logarithms. From ...
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1 vote
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Explanation of a step in a preprinted work

I have been studying this preprinted paper and the references therein. I believe that there are some typos; However, since I am not an expert yet, I would like to make sure I am correct. I do not ...
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7 votes
3 answers
452 views

Why mpmath computes $\sum_{n=2}^\infty (-1)^n\log(n)=\log\left(\frac1 2 \sqrt{2} \sqrt{\pi}\right)$

Working with precision 500 decimal digits, mpmath in sage computes: $$\sum_{n=2}^\infty (-1)^n\log(n)=\log\left(\frac1 2 \sqrt{2} \sqrt{\pi}\right).\tag{1}\label{1}$$ We believe the LHS of \eqref{1} ...
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5 votes
1 answer
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Duality argument

$\newcommand\norm[1]{\lVert#1\rVert}\newcommand\abs[1]{\lvert#1\rvert}$Throughout my studying for some papers, in particular, the proof of localized Strichartz estimates, I encountered a use of the ...
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3 votes
2 answers
137 views

Continuity of Radon transform w.r.t the angle

Let $f \in L^1(\mathbb R^n)$ (or in case it helps, actually a probability density on $\mathbb R^n$). Define the Radon transform $R[f]:S_{n-1} \times \mathbb R \to \mathbb R$ of $f$ by $$ R[f](w,b) := ...
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$L^p$ inequality for "positively correlated" random variables

Suppose that we have $m$ complex-valued random variables $\xi_1,\ldots,\xi_m$ and assume the following "positive correlation" property: for all non-negative integers $\alpha_1,\ldots,\...
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3 votes
1 answer
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Explanation of a step in a work by C. E. Kenig and A.D. Ionescu

I am studying the work Ionescu, A. D.; Kenig, C. E., Local and global wellposedness of periodic KP-I equations, Bourgain, Jean (ed.) et al., Mathematical aspects of nonlinear dispersive equations. ...
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Value of divergent alternating series

I was curious if there is any method to be able to sum series like this $$\sum _{n=1}^{\infty } (-1)^n n^n$$ or similar $$\sum _{n=1}^{\infty } n^n (-z)^n$$ for any value os z , I see solutions using ...
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Generalization of identity for terminating hypergeometric function

Let ${}_2F_1(a,b;c;z)$ be the ordinary hypergeometric function for $z \in \mathbb{C}$ \begin{equation} {}_2F_1(a,b;c;z) = \sum_{k=0}^{\infty} \frac{z^k}{k!} \frac{(a)_{k} (b)_k}{(c)_k}\,, \end{...
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5 votes
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Helmholtz decomposition of compactly supported fields

Let $\mathbf F$ be a smooth vector field in $\mathbb R^3$, which is null outside a finite compact domain $V$. By the Helmholtz decomposition theorem, there exist a scalar field $\Phi$ and a vector ...
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7 votes
2 answers
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A counterexample to: $\frac{1-f(x)^2}{1-x^2}\le f'(x)$ — revisited

Can we find a counterexample to the following assertion? Assume that $f:[-1,1]\to [-1,1]$ an odd function of class $C^3$, and assume thaht $f$ is a concave increasing diffeomorphism of $[0,1]$ onto ...
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3 votes
1 answer
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Estimate for an oscillatory integral of the first kind

I am confused in finding the right bound for the following oscillatory integral $$I = \int_\mathbb{R} (\psi(2^{-k} \xi))^2 e^{i (y \xi - 3 \eta \xi^2 t)} d\xi.$$ Where $\psi(2^{-k} \xi)$ is a smooth ...
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Identity principle of solutions of SL-problems with matching values on open set

Situation (cut short): Corresponding solutions (by eigenvalue) of two given regular Sturm-Liouville problems with homogeneous Neumann BC, same spectrum but possibly distinct coefficient functions, &...
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6 votes
2 answers
939 views

Exercise 8.13 - Brezis

Let $1 \leq p < \infty$ and $u \in W^{1,p}(\mathbb{R}$). Set $$ D_{h}u(x) = \frac{1}{h}(u(x+h) - u(x)), \ \ x \in \mathbb{R}, h> 0 $$ Show that $D_{h}u \to u'$ in $L^{p}(\mathbb{R}$) as $h \to ...
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2 votes
1 answer
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For estimation on the integral $g(t)=\int_{-t}^{t}\left\vert\sum_{k=1}^Ne^{ikx}\right\rvert^2dx$ for small $t>0$

For real numbers $t>0$ and $x$, let $f(x)=\sum_{k=1}^Ne^{ikx}$ and $g(t)=\int_{-t}^{t}\lvert f(x)\rvert^2dx$. Then $g(\pi)=\int_{-\pi}^{\pi}\lvert f(x)\rvert^2dx=2\pi N$. I want to know is there ...
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  • 376
2 votes
1 answer
135 views

Coefficients of certain Taylor series

For $t\in(-1,1)$, let $$f(t):=\left(\frac{1+t}{1-t}\right)^{(1-t)/2}+\left(\frac{1-t}{1+t}\right)^{(1+t)/2}$$ and $$g(t):=\frac1{f(t)}.$$ Note that the functions $f$ and $g$ are even. Question 1: Is ...
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Nonlinear wave equation : ODE blow-up construction

Consider the focusing non-linear wave equation (NLW) \begin{align*}\partial_{tt} u(t,x) - \Delta u(t,x) &= -|u|^{p-1}u, \quad (t,x) \in [0,+\infty) \times \mathbb R^d \\ u(0,x) &= u_0(x) \in \...
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11 votes
1 answer
266 views

Does every smooth map of rank at most d factor through a d-manifold?

Suppose $d≥0$, $m≥0$, $n≥0$, and $\def\R{{\bf R}} f\colon \R^m→\R^n$ is a smooth map whose rank at any point of $\R^m$ is at most $d$. Here and below, smooth means infinitely differentiable. Can we ...
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0 answers
69 views

About convex functions

Is there a non negative, convex, and decreasing function $g$ on $[0,\infty)$, with $g(0)=1$, such that $g(s+t)< g(t)g(s)$ for $s,t \in (0,\infty)$?
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