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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.

201 questions from the last 365 days
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Linear and non-linear intersection to solve ODE

Consider a linear operator $$L(u(t)) = \dfrac{d}{dt}u(t)+p(t)u(t)$$ for known function $p(t)$. It is well known the homogeneous equation $$L(u) = 0 ~~\text{or}~~\dfrac{d}{dt}u(t)+p(t)u(t)= 0$$ has ...
John Wayne's user avatar
1 vote
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Explicit rate of decay of the positive standing wave of the subcritical nonlinear Schrödinger equation

Consider the following semilinear problem: $$ \begin{cases} - \Delta u + u = u |u|^{p - 2} &\text{in} ~ \mathbb{R}^N; \\ u (x) \to 0 &\text{as} ~ |x| \to \infty, \end{cases} $$ where $N \geq 2$...
gpr1's user avatar
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Stability of flow map

$\DeclareMathOperator\Diff{Diff}$Setting: Let $(M,g)$ be a compact and connected $C^{\infty}$-Riemannian manifold. Let $d_g$ denote the induced shorted path metric and equip $C^{\infty}(M)$ with the ...
ABIM's user avatar
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If $u \in H^2(\mathbb{R}^3)$, does $r^{-1} u \in H^{\alpha}(\mathbb{R}^3)$ for some $\alpha > 0$?

Let $u$ belong to the Sobolev space $H^1(\mathbb{R}^3)$. We have the classical Hardy inequality \begin{equation*} \int_{\mathbb{R}^3} \frac{|u|^2}{|x|^2} dx \le 4\int_{\mathbb{R}^3} |\nabla u(x)|^2 dx,...
JZS's user avatar
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Existence and uniqueness of heteroclinic solution of Allen–Cahn on $\mathbb R$ with driving-damping term

The Allen–Cahn equations on $\mathbb R$ are $u'' = u^3 - u$. It is well-known that all the solutions of this equation which satisfy the asymptotic boundary conditions $\lim_{x \to \pm \infty} u\left(x\...
Ervin's user avatar
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+50

When is a first-order delay differential equation equivalent to a higher-order ordinary differential equation?

The proportional delay differential equation $$ xf'(x)+2xf'(x/2)+C+4f(x/2)-5f(x)=0 $$ with initial condition $f(0)=C$ expresses that Simpson's rule exactly integrates $f$ over any interval $[0,x]$ and ...
gmvh's user avatar
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Fourier decay implies what kind of regularity

We consider a function $f:\mathbb R^2 \to \mathbb C$ that is compactly supported and bounded. In addition, we know that $$\lim_{\vert x\vert \to \infty} \vert x \vert^2 \vert \hat{f}(x)\vert =0,$$ ...
Yizheng Yuan's user avatar
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1 answer
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Singularities at the circle of convergence: generalization of Cauchy-Hadamard theorem

Consider a series $\sum a_n z^n$ with finite radius of convergence $R$. Cauchy-Hadamard theorem gives $1/R = lim\ sup |a_n|^{1/n}$. Q: Suppose for some reason (e.g. numerical) we know that there is ...
0x11111's user avatar
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Fourier transform of exponential over torus

I found the following formula for the Fourier transform on a flat 2-torus, but I don't quite know how to derive it. We have a variable $q=(q_x,q_y) \in [0,2\pi)^2$ and by considering it in polar ...
António Borges Santos's user avatar
7 votes
1 answer
160 views

When is a non-linear first-order ODE equivalent to a linear second-order ODE?

The Riccati equation $y'(x)+y(x)^2=f(x)$ is non-linear, but can be transformed into the linear equation $-u''(x)+f(x)u(x)=0$ via $y(x)=\frac{u'(x)}{u(x)}$. Is there a general statement known about ...
gmvh's user avatar
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1 answer
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When is Laplace transform of a function power-law and relation to the behavior of the function near zero?

I want to see when the Laplace transform of a non-negative function $f$ defined on $[0, +\infty)$ is a power function in the loose sense, i.e., $$g(s) = \mathcal L\{f\}(x) = \int_0^\infty f(x) e^{-sx} ...
Yfiua's user avatar
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1 answer
436 views

Function $\phi$ such that $f(\phi(x,y)) = f(x) + f(y)$

I have a continuous function $f:\mathbb{R}^n\to\mathbb{R}$, and I am looking for a continuous (or at least measurable) function $\phi:\mathbb{R}^{2n}\to\mathbb{R}^n$ such that $f(\phi(x,y))=f(x)+f(y)$....
gmvh's user avatar
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Prove or disprove that $|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$

$|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$. I had this conjecture for a long time. I tried various methods and techniques but they all failed. It might also be wrong ...
Haidara's user avatar
  • 178
7 votes
1 answer
553 views

Example of continuous function which is not differentiable everywhere in a strong sense

Is there a continuous function $$u\colon (0,1)\to \mathbb{R}$$ such that at every point $x\in (0,1)$ one has $$\lim\sup_{y\to x+0}\frac{u(y)-u(x)}{y-x}=+\infty?$$ In particular $u$ is not ...
asv's user avatar
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Derivative bounds for self convolution of the spherical measure in $R^d$

While reading this article on near $L^1$ estimates for the spherical lacunary maximal function, I came across the estimate $$ |\partial^{\gamma} (\widetilde{\sigma} \ast \sigma)(x)| \lesssim |x|^{-(1 +...
Zygmund's user avatar
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1 answer
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Upper bound on higher order derivatives of $\frac{1}{v(t)}$

Suppose that $ v(t) >l>0$ and $$ \vert v^{(k)}(t) \vert \leq c \frac{k!}{r^k}. $$ Can we give an upper bound for $$ (\frac{1}{v(t)})^{(k)} $$ ? Attempt: We first compute the first fourth order ...
Yidong Luo's user avatar
5 votes
0 answers
204 views

A proof for an $L^p$-$L^p$ inequality

This is a transfer of the question https://math.stackexchange.com/questions/4996853/an-lp-lp-inequality Let $a\in (0,1)$ and $1<p<\infty$ and use $L^{p}$ to denote the space $L^{p}([0,\infty))$ ...
Medo's user avatar
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3 votes
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Deeper reason for why classical orthogonal polynomials have simple generating functions?

Is there a known reason why all classical families of orthogonal polynomials have simple generating functions? I was wondering whether one could get an explanation using the connection with Sturm-...
Plemath's user avatar
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2 answers
148 views

Asymptotic behavior of the integral of Hermite functions/polynomials on half-lines

I would like to understand the asymptotic behaviour of the following integrals with fixed $x_0>0$: $$J_m=\int^{+\infty}_{x_0}|H_m(x)|^2 e^{-x^2}dx,$$ where $H_m(x)$ is the $m-$th Hermite polynomial....
Darius's user avatar
  • 21
4 votes
4 answers
473 views

A certain inequality involving square roots of polynomials

I want to prove the inequality $$\begin{aligned} &\sqrt{(x - 1)^2 + y^2}\Big[y^2(9x - 6) - 9x^2 + 9x^3\Big]+ y^2(16x^2 - 16x + 7)\\ &- \sqrt{x^2 + y^2}\Big[9x + y^2(9x - 3) + \sqrt{(x - 1)^2 + ...
Benjamin L. Warren's user avatar
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1 answer
53 views

Exponentially weighted norms are not equivalent

Let $\|u\|^2_{L^2_\eta}$ be the exponentially weighted norm of the space of functions $u(x)$ for which $u(x)\mathrm{e}^{\eta\cdot x}$ with $\eta\in \mathbb{R}$ is in $L^2(\mathbb{R})$. How can I show ...
Lars Siemer's user avatar
1 vote
0 answers
59 views

Asymptotic behavior of positive solution to nonlinear scalar field equation

It is well-known that the radial positive solution $u=u(r)$ to nonlinear scalar field equation $$-\Delta u+u=u^p\text{ in } ~\mathbb{R}^d, 1<p<\frac{d+2}{d-2}$$ has the following asymptotic ...
sorrymaker's user avatar
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1 answer
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The asymptotic at infinity of ODE

It may be a simple problem in ODE. Let $u$ be the positiv solution to $$u''(t)-f(u,t)u(t)=0, u(0)=1, \lim_{t\to+\infty}=0,$$ with $f>0$ and $\lim_{t\to+\infty}f(u(t),t)=1$. Can we prove that there ...
sorrymaker's user avatar
2 votes
0 answers
43 views

A distribution defined via an ODE for its Laplace trnsform

Fix a parameter $0 < c < \infty$. As the solution to a certain problem, there is a probability density function $f_c(t)$ on $0 < t < \infty$ with mean $1$ and whose Laplace transform $L(\...
David Aldous's user avatar
2 votes
0 answers
85 views

Can an SDE be made to follow the flow lines of a vector field?

Let $V: \mathbb R^n \to \mathbb R^n$ be a Lipschitz vector field. Consider a one dimensional Brownian motion $W$ and the SDE $$dX_t = V(X_t) \, dW_t,$$ where we identify $V(X_t) \in \mathbb R^n$ with ...
Nate River's user avatar
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1 vote
1 answer
189 views

How to evaluate the following integral?

How to (analytically) calculate the following integral, $$I = \int_{S_{2n-1}} \left( \langle z, \zeta \rangle \right)^a e^{ b \langle z, \zeta \rangle} \, d\sigma(\zeta),$$ where $\langle z, \zeta \...
zoran  Vicovic's user avatar
1 vote
1 answer
188 views

Can one show $h(x)=|2(\zeta'(x))^2-\zeta''(x)\zeta(x)|$ is a decreasing function for $x\in\mathbb{R}\cap [1,\infty)$?

This question is related to This question. When I tried to approach it I couldn't even proof that the LHS is a decreasing function on the given domain using regular methods. I have tried to write the ...
Haidara's user avatar
  • 178
4 votes
1 answer
195 views

Asymptotic spectrum of a complex Sturm-Liouville differential operator

Let $\varepsilon > 0$ and consider the (complex) Sturm–Liouville differential operator on $[0,1]$ given by $$ \mathcal{L}_\varepsilon f(x) = \varepsilon^2 f''(x) + i V(x) f(x), $$ with Neumann ...
Matheus Manzatto's user avatar
0 votes
2 answers
364 views

Can one show $\left|\frac{2(\zeta'(x))^2-\zeta''(x)\zeta(x)}{\zeta^3(x)}\right|\leq \frac{2}{(x-\frac{1}{2})^2}$ for $x\in\mathbb{R}\cap [1,\infty)$?

I have found that $\left|\frac{2(\zeta'(x))^2-\zeta''(x)\zeta(x)}{\zeta^3(x)}\right|\leq \frac{2}{(x-\frac{1}{2})^2}$ for all real $x$ such that $x>1$ seems to be true. I have plotted the ...
Haidara's user avatar
  • 178
2 votes
1 answer
93 views

Reference needed: estimate of the second order derivatives

In $\mathbb{R}^d$ there is estimate (see 1.3, Chapter III of E.M.Stein' book Singular Integrals and Differentiability Properties of Functions) $$\left\|\frac{\partial^2 f}{\partial x_i \partial x_j} \...
Michael Perelmuter's user avatar
3 votes
1 answer
80 views

Solution of $d Y_t/dt = A(t) Y_t, Y_0 = I_d$ is positive definite?

Let $\{A(t)\}_{t \in [0,1]}$ be time-varying symmetric matrices in $\mathbb{R}^{d\times d}$. We consider the following ODE for $Y_t \in \mathbb{R}^{d \times d}$ $$ \tag{1} \frac{d Y_t}{dt} = A(t) Y_t, ...
De vinci's user avatar
  • 399
6 votes
1 answer
568 views

Can one show that $|\zeta'(x) / \zeta^2(x)| \leq 1/(x-.5)$ for $x\in\mathbb{R}\cap [1,\infty)$?

I have found that $\left|\frac{\zeta'(x)}{\zeta^2(x)}\right|\leq \frac{1}{x-\frac{1}{2}}$ for all real $x$ such that $x>1$ seems to be true. I have plotted the inequality and got this inequality ...
Haidara's user avatar
  • 178
2 votes
1 answer
104 views

Looking for review of delay differential equations involving $f(x)$ and $f(x/k)$

A research problem unexpectedly leads me to a delay differential equation of the form $$ f(x)=\alpha(f(x),f(x/2))\,f'(x)+\beta(f(x),f(x/2))\,f'(x/2)+\gamma(f(x),f(x/2)) $$ For special cases of $\alpha,...
gmvh's user avatar
  • 3,065
2 votes
1 answer
208 views

Proving an exponential sum inequality for symmetric Hamming distance sequences in binary vectors

Background: Let $X = \{0,1\}^k$ represent the set of all binary vectors of length $k$. For two binary vectors $x, y \in X$, the Hamming distance $d_H(x, y)$ is defined as the number of positions where ...
tom jerry's user avatar
  • 349
0 votes
0 answers
33 views

Non-positive definite solution for differential Riccati equation

Consider the matrix-valued differential Riccati equation (DRE): $$ \dot P_t +PA+A^\top P+Q-(B^\top P+S)^\top (B^\top P+S)=0, \quad t\in [0,T];\quad P(T)=G, $$ where all coefficients are continuous. ...
John's user avatar
  • 503
0 votes
0 answers
57 views

Double-periodic functions with (possible) poles

Consider the set of double-periodic function $f:\mathbb C/(\mathbb Z+i \mathbb Z) \setminus \{z_0\} \to \mathbb C$, where $z_0$ is a fixed point inside $\mathbb C/(\mathbb Z+i \mathbb Z),$ that have a ...
António Borges Santos's user avatar
16 votes
0 answers
351 views

The convergence domain of the function $\sum \{n!x\}$

This is a problem from a mathematics competition: Does there exist an irrational number $x$ such that the series $$\sum_{n=1}^{\infty}\{n!x\}<+\infty$$ where $\{ \}$ means the fractional part of a ...
Fate Lie's user avatar
  • 505
2 votes
2 answers
175 views

Great literature on discrete dynamical systems and/or qualitative theory of difference equations

I am asking for the great literature on topics of discrete dynamical systems and/or qualitative theory of difference equations especially aimed on pure mathematicians. Could you please give me some ...
1 vote
0 answers
52 views

Stability of Euler discretization

I am looking at the discretization of an ODE: $$x_{n+1} = x_n + \alpha f(x_n),$$ where $x_n\in R^d$ and $f$ is continuously differentiable and such that $f(0)=0$ and $f'(0)$ is Hurwitz (i.e., the real ...
N. Gast's user avatar
  • 562
0 votes
0 answers
78 views

Is there an asymptotic expansion for the reciprocal of the classical Euler beta function?

The classical Euler beta function can be defined by $$ B(p,q)=\int_0^1t^{p-1}(1-t)^{q-1}\operatorname{d\!}t $$ for $\Re(p),\Re(q)>0$. The beta function and the classical Euler gamma function $\...
qifeng618's user avatar
  • 1,091
3 votes
1 answer
309 views

Extremizing sequence consists of two elements

Let $\mathcal A_{s}$ be the set of sequences $X=(x_m)_{m \in I}$ where $I=\{1,2,...,n\}$ with $n \ge 2$ and possibly $n =\infty$ is an index set with $x_1=0$, $x_2=s>0$ and $x_m>x_{m-1}$ for $m,...
António Borges Santos's user avatar
2 votes
0 answers
179 views

Analytic continuation of $\int_V (f(x_1,\cdots,x_n))^s dx_i$

Let $V$ be an $n$-dimensional simplex, let $f(\boldsymbol{x}) = f(x_1,\cdots,x_n)\in \mathbb{C}[x_1,\cdots,x_n]$ be a product of linear polynomials that is non-zero in interior of $V$. Also let $E(\...
pisco's user avatar
  • 528
9 votes
1 answer
429 views

A curious norm related to the L¹ norm

If $f \in C^0([0,1])$, one can define $$\Vert f \Vert_? = \sup_{J \subset [0,1]} \left\lvert \int_J f \right\rvert,$$ where $J$ runs among all subintervals of $[0,1]$. This is a norm on $C^0([0,1])$ (...
PseudoNeo's user avatar
  • 575
0 votes
1 answer
119 views

Detecting singular points from a parametrization

Suppose $r(t)$ parametrizes some, say algebraic, curve in the plane. It can certainly be that $r$ is smooth but the curve is not, since $r$ resolves double points by passing through them at different ...
tex.support's user avatar
0 votes
0 answers
61 views

Bounding the coefficients of a polynomial written as the sum of powers of linear forms

Crosspost: Just a heads up, I've posted this question on math.stackexchange as well. I have made new attempts at solving it though, and figured I'd ask here since it's more of a research-level ...
AlkaKadri's user avatar
  • 121
1 vote
2 answers
120 views

Expansion of the associated Legendre polynomials $P^m_l(\cos\vartheta)$ for $\vartheta \rightarrow 0$

A simple test with Mathematica indicates that for the associated Legendre polynomials $P^m_l$ the following relation should hold: $$ \lim_{\vartheta\rightarrow 0} P^m_l(\cos\vartheta) =a_{lm} \...
p6majo's user avatar
  • 369
3 votes
1 answer
111 views

Sobolev inequalities and Wiener algebra

It follows from the Gagliardo-Nirenberg inequality that for a locally integrable function $f$ defined on $\mathbb R^d$ (we assume $d\ge 3$) such that $\nabla f$ belongs to $L^2(\mathbb R^d)$ and $$ \...
Bazin's user avatar
  • 16.2k
2 votes
1 answer
160 views

Is the function $(z-1)(2^{z}-1)\zeta(z)$ logarithmically concave and convex in $z\in(0,\infty)$?

For proving that the sequence \begin{equation}\label{first-proof-decreas-seq} \frac{1}{(2k-1)(k+1)} \frac{2^{2k+2}-1}{2^{2k}-1} \biggl|\frac{B_{2k+2}}{B_{2k}}\biggr| \end{equation} is decreasing in $k\...
qifeng618's user avatar
  • 1,091
2 votes
1 answer
315 views

Where to find or how to establish a general formula for the improper integral $\int_{0}^{\infty}\frac{\sin t}{t}(\ln t)^k\operatorname{d\!} t$?

When I tried to establish the Maclaurin power series expansion of the reciprocal $\frac{1}{\Gamma(z)}$ of the Euler gamma function $\Gamma(z)$, I came across the improper integral $$ I_k=\int_{0}^{\...
qifeng618's user avatar
  • 1,091
3 votes
1 answer
240 views

Solutions and asymptotics of the ODE $ f''=f^{-\alpha} $

Consider the ODE $ f''=f^{-\alpha} $, where $ \alpha>1 $ and $ f>0 $ in $ \mathbb{R} $. Assume that for $ [f]_{\frac{2}{\alpha+1}}\leq A $, where $ A>0 $ is a constant and $$ [f]_{\frac{2}{\...
Luis Yanka Annalisc's user avatar

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