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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.

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      4
      votes
      1answer
      46 views

      Riemann $P$-symbol for ODEs

      Good afternoon, everyone. Does anyone know what the Riemann $P$-symbols mean when they contain more than three columns (i.e., to what ordinary differential equations they correspond)? Examples: $P\...
      -4
      votes
      1answer
      80 views

      When $0<\alpha<\pi$, what is $\iint_{0}^{\infty}e^{-(x^2+y^2+2xy\cos\alpha)}dxdy$ [on hold]

      For $0<\alpha<\pi$, what is $$\iint_{0}^{\infty}e^{-(x^2+y^2+2xy\cos\alpha)}dxdy\, =?$$
      3
      votes
      1answer
      72 views

      Decay of Laplace (or Mellin) transform beyond region of convergence?

      Let $f:[0,\infty)\to \mathbb{R}$ be a piecewise differentiable function with $f(0)=0$ and $f'(t)$ of bounded variation. Its Laplace transform $\mathcal{L}f$ converges for $\Re s > 0$. Assume it can ...
      10
      votes
      1answer
      329 views

      turn $\pi/n$, move $1/n$ forward

      start at the origin, first step number is 1. turn $\pi/n$ move $1/n$ units forward Angles are cumulative, so this procedure is equivalent (finitely) to $$ u(k):=\sum_{n=1}^{k} \frac{\exp(\pi i H_{n}...
      8
      votes
      1answer
      1k views

      An equality about sin function?

      Empirical evidence suggests that, for each positive integer $n$, the following equality holds: \begin{equation*} \prod_{s=1}^{2n}\sum_{k=1}^{2n}(-i)^k\sin\frac{sk\pi}{2n+1}=(-1)^n\frac{2n+1}{2^n}, \...
      13
      votes
      1answer
      343 views

      Computing spectra without solving eigenvalue problems

      There is a rather remarkable conjecture formulated in this paper, "Computing spectra without solving eigenvalue problems," https://arxiv.org/pdf/1711.04888.pdf and in this talk by Svitlana Mayboroda ...
      4
      votes
      1answer
      128 views

      Functions orthogonal to powers of $1/{\left(1+x^2\right)}$

      Let $f,g:\mathbb{R}\to\mathbb{R}$ be continuous functions with the following properties: $f(x)$ and ${g(x)}/x$ are bounded; ${g(x)}/{\left(1+x^2\right)}\in L^1\left(\mathbb{R}\right)$; $\lim_{x\to0}f(...
      0
      votes
      0answers
      28 views

      Asymptotic dispersion of a periodic Sturm–Liouville problem

      From a physical problem of waves in periodic waveguides, I obtain the following Sturm–Liouville equation: $$ \left(\frac{d^2}{dx^2}-k^2+\omega^2\ V(x)\right)\psi(x)=0 $$ where $V(x+L)=V(x)>0$ is a ...
      2
      votes
      0answers
      167 views

      Packing number of a subset of Stiefel manifold

      Let $\lambda_1\geq...\geq\lambda_r>0$ be a sequence of real numbers such that $\sum_{i=1}^r\lambda_i^2=1$. Define a subset of the Stiefel manifold $O(n,r)$ as $$S_t=\{U\in O(n,r): 1-\sum_{i=1}^{r}\...
      20
      votes
      4answers
      2k views

      Prove that $\left(\frac{x^n+1}{x^{n-1}+1}\right)^n+\left(\frac{x+1}{2}\right)^n\geq x^n+1$

      Let $x>0$ and $n$ be a natural number. Prove that: $$\left(\frac{x^n+1}{x^{n-1}+1}\right)^n+\left(\frac{x+1}{2}\right)^n\geq x^n+1.$$ This question is very similar to many contests problems, but ...
      1
      vote
      0answers
      57 views

      How to tanslate the problem of computing a Kakutani fixed point to the problem of computing a Brouwer fixed point?

      I read from some paper that the problem of computing a Kakutani fixed point can be reduced to the problem of computing a Brouwer fixed point? How to make this translation? Can this translation be ...
      1
      vote
      0answers
      123 views

      Lax Milgram for non coercive problem?

      I obtained the variational form of my problem. and the bilinear form is below. Bilinear Form Let $\Omega\subset\mathbb{R}$ be an open set. For $u,v\in H^1_0(\Omega)$. I have $$a(u,v)=\int_\Omega u(t)...
      -1
      votes
      0answers
      42 views

      Prove or find correct 4-order power series expansion of a multivariate function [on hold]

      $a(x)$ is an $n$-order positive definite matrix, $$F(x)=(1/2)?(?_{x}^{T}F(x))?a(x)?(?_{x}F(x)),$$ $F(x)=0$ if and only if $x=x_?$, $x$ is an $n$-dimensional vector, $F(x)$ is a function of vector $x$,...
      3
      votes
      0answers
      126 views

      A combinatorial / geometric interpretation of compositional inversion via matrix inversion

      There are several ways of finding the power or Taylor series for the compositional inverse of a function $f(x)$ with $f(0)=0\;$ given its series expansion, e.g., by using the classic Lagrange ...
      11
      votes
      2answers
      543 views

      Function orthogonal to powers of $1/\left(1+x^2\right)$

      Does there exist any continuous function $f:\mathbb{R}\to\mathbb{R}$, $f(x)/(1+x^2)\in L^1(\mathbb R)$, such that $f(0)=1$ and $$\int_{-\infty}^{\infty}\frac{f(x)}{\left(1+x^2\right)^p}dx=0$$ for ...

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