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.
3,707 questions
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How small can $|\frac{a}{b-c}| + |\frac{b}{a-c}| + |\frac{c}{a-b}|$ get for distinct positive $a,b,c$?
Let $\newcommand{\Rplus}{\mathbb{R}_+}\Rplus$ denote the set of positive reals. What is the value of
$$\inf\Big\{\Big|\frac{a}{b-c}\Big| + \Big|\frac{b}{a-c}\Big| + \Big|\frac{c}{a-b}\Big|:
a,b,c \in \...
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Can positive decreasing functions $f$ with $f' \geq -f$? be such that $\int_{0}^{x} f(t) \mathrm{d}t$ diverges arbitrarily slow for $x \to \infty$?
Given an increasing function $G \colon [0, \infty) \to [0, \infty)$ with $G(x) \rightarrow \infty$ for $x \to \infty$, is there a (strictly) decreasing (differentiable) function $f \colon [0, \infty) \...
- 784
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Continuous vs discrete
Suppose that $z_1,\dotsc,z_n$ are complex numbers situated on the unit circle
$C:=\{z\colon |z|=1\}$. Let $|\cdot|$ denote the uniform, continuous,
probabilistic measure on $C$, and let $\mu$ be the ...
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Bounds on integrals of $1/(1+|\arccos(a+b\cos(t))|^k)$
In my research, I have come across a particularly nasty integral. Let $a$ and $\delta$ be such that $-1 \le a+\delta\cos(\psi) \le 1$ for all $\psi \in [0,\pi]$ and $\delta>0$. I would like to have ...
- 13
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75
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Control of an ODE to attain target
Consider the following ODE system:
$$
\begin{aligned}
dR_1(t) &= -\lambda_1 R_1(t)\,dt + \lambda_1 \left( \beta_0 - \beta_1 R_1(t) + \beta_2 R_2(t) \right) C\,dt, \\
dR_2(t) &= -\lambda_2 R_2(...
- 696
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Picard-Fuchs equation, Schwarzian derivative and Bers embedding of Teichmuller Space
Let $X$ be the elliptic curve
$$y^2=x^3-g_2x+g_3$$
The $j$ invariant of $X$ is
$$j=\frac{g_2^3}{g_2^3-27g_3^2}$$
I came across the formula of Dedekind
$S(\tau)(j)=\frac{1-\frac{1}{2^2}}{(1-j)^2}+\frac{...
- 515
4
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A system of T-functions on $(0,\infty)$
Let $r>1$ be real and
\begin{align*}
f_1(x) &= 1,\\
f_2(x) &= (x+4)^r,\\
f_3(x) &=(x+4)^r(x+3)^r,\\
f_4(x) &= (x+4)^r(x+3)^r(x+2)^r,\\
f_5(x) &=(x+4)^r(x+3)^r(x+2)^r(x+1)^r.
\...
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Can I extend a function while preserving that its off-diagonal Hessian is in $L^\infty$?
Recall the Kirszbraun extension theorem for Lipschitz maps. Here is a very weak form of it:
Let $E$ be a subset of $\mathbb R^d$ and let $f$ be a Lipschitz function on $E$. Then there is an open set $...
- 1,270
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Reference Request: Second Order Tauberian Theorem
I am interested in whether there exist Tauberian theorems for deducing the second order asymptotics of a particular function.
Throughout this post I am using the notation from Kwaśnicki's answer to ...
- 101
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Staying away from points/minimizing potential energy
Let $x_1,\dotsc,x_n$ be points on $\mathbb{R}/\mathbb{Z}$. Write $\|x-y\|$ for the distance between two points $x,y$ in $\mathbb{R}/\mathbb{Z}$. Let $V$ be one of the following functions $V_j:\mathbb{...
- 22k
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Ergodicity in the Wiener-Wintner Ergodic Theorem [cross-post from MSE]
I'm studying the Wiener-Wintner (and related) ergodic theorems, and I've been running into a bit of confussion when passing the result from ergodic systems to non-ergodic ones. In most of the ...
- 61
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$L^2$-functions orthogonal to their own Fourier transform
It is well-known that, besides the standard Gaussian $e^{-|x|^2/2}$, there are many interesting functions which are eigenfunctions of the Fourier transform, for example the Hermite functions.
Mainly ...
- 2,184
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Inhomogeneous Strichartz estimates for Schrödinger's equation on torus $\mathbb{T}^2$
As we all know, the Strichartz estimates for Schrödinger's equation on $\mathbb{R}^2$:
$$ \|e^{i t (\Delta -1) } f\|_{L^4_{T} L^4_{x}} \leq C \| f\|_{L^2_x} $$
And the corresponding inhomogeneous ...
- 51
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Continuous analogue of the discrete simple continued fraction
Background
The classical Riemann integral of a function $f : [a,b] \to \mathbb{R}$ can be defined by setting $$\int_{a}^{b} f(x) \ dx := \lim_{\Delta x \to 0} \sum f(x_{i}) \ \Delta x. $$ Here, the ...
- 5,065
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Inequality for Cauchy surface area formula
The following formula is known due to Cauchy. Suppose that $E \subset \mathbb R^n$ is a convex body. Then
\begin{align*}
|\partial E| = \frac{1}{\omega_{n-1}}\int_{S^{n-1}} |\pi_u(E)| \mathrm d \...
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