Tagged Questions

Special functions, orthogonal polynomials, harmonic analysis, ODE's, differential relations, calculus of variations, approximations, expansions, asymptotics.

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Dropping three bodies

Consider the usual three-body problem with Newtonian $1/r^2$ force between masses. Let the three masses start off at rest, and not collinear. Then they will become collinear a finite time ...
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Positivity of polynomial sequences via generating series

In this question I address the problem of proving the nonnegativity of a numerical sequence $a_0,a_1,a_2,\dots$ via generating series technique. In the notation $A(x)=\sum_{n=0}^\infty a_nx^n\ge0$ ...
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Spectral theory for Dirac Laplacian on a funnel

I would like to study the spectral theory of the Dirac Laplacian on a non-compact quotient of the hyperbolic plane by a discrete group (I am particularly interested in the simple case where the ...
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Birth-Death Process associated with Orthogonal Polynomials

I have read in various places the following objects are related: orthogonal polynomials birth-death processes Lattice paths continued fractions After a lot of searching online, I found sketches ...
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How to derive explicit bound for the solution of following equation?

Let's have equation $$y''(t) + \left(\frac{3}{16t^{2}} + \frac{a}{t} -\frac{b}{t^{\frac{5}{4}}}cos(2t)\right)y(t) = 0, \quad t \in (1, \infty), \quad a, b > 0$$ How to derive explicit upper bound ...
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Rate of convergence of Riemann sum of quasi-regular functions

The following result is well-known (I consider the 3-dimensional case only): Theorem: if $f \in H^s(\mathbb{R}^3)$ with $s > 3/2$ is compactly supported, then  \left| \int_{\mathbb{R}^3} f - \...
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Consider $\Omega$ a smooth bounded domain in $R^N$ and suppose $\phi_1(x)>0$ is the first eigenfunction of $-\Delta$ in $H_0^1(\Omega)$ normalized however one chooses. My interest is in how $\... 0answers 117 views Hyperbolic toral automorphisms, and maximizing over orbits the minimum along an orbit of a function Setup: Let$\phi\colon T^2 \to T^2$be a hyperbolic toral automorphism. Let$f\colon T^2 \to \mathbb{R}$be a continuous function. For$x \in T^2$, let$\underline{f}(x) = \inf_{n \in \mathbb{Z}} f(\...
Consider a surface $S$ and a vector field on the surface which has a closed orbit. The vector field on both sides of the closed orbit spirals towards it, which gives us that the linearized Poincare ...
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 ...