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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.

        1
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        0answers
        40 views

        Algebraic Riccati and WKB

        It's a one-liner to show that the algebraic Riccati equation (ARE) and the lowest order form of WKB for a linear ode are the same. But I've looked all over the web and there does not seem to be a ...
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        votes
        1answer
        101 views

        Brascamp-Lieb inequalities on the sphere

        In the paper [CLL], Carlen, Lieb, and Loss demonstrate a version of the Young inequality on the sphere $S^{N-1}$ in $\mathbb{R}^N$. For positive functions $f_j$ on $[-1,1]$, the following bound holds:...
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        55 views

        Does there exist $\alpha>0, \beta\in (0,1)$ such that $\dfrac{\sum_{k=1}^n a_k}{n}\le \alpha (a_1\cdots a_n)^{1/n} + \beta \max_i(a_i)$ holds?

        Let $a_1\ge a_2\ge \cdots\ge a_n\ge 0$ be given non-negative numbers. My question is the following: Is there any $\beta \in(0,1),\ \alpha>0$, such that $$\dfrac{a_1+\cdots+a_n}{n}\le \alpha (a_1\...
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        0answers
        29 views

        Unique solution for a difference ODE?

        Any idea how to find general solution $$a'_{n}(t)= (n+\alpha )a_{n}(t) + \beta a_{n+1}(t) + \gamma a_{n+2}(t)$$ for some coefficients $\alpha, \beta, \gamma$?, Where $a'_{n}(t)=\frac{d}{dt} a_{n}(t)...
        2
        votes
        1answer
        73 views

        Failure of Falconer distance problem in one dimension

        I am told that the Falconer distance conjecture fails trivially in one dimension, but I really cannot find any reference for that. Precisely, I am asking the following question: For a compact set $E\...
        8
        votes
        1answer
        101 views

        Log-concavity of repeated convolution

        Let $A = (a_0,a_1,\ldots,a_k)$ be a sequence of strictly positive numbers, and let $A^{\ast k}$ denote the $k$-fold repeated convolution (defined by $A^{\ast 1} = A$ and $A^{\ast k+1} = A^{\ast k} \...
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        votes
        1answer
        183 views

        Dependence of a solution of a linear ODE on parameter

        Is the following theorem known, or can be easily derived from known results? Consider the differential equation $$w''-kz^{-1}w'=(\lambda+\phi(z))w,$$ where $k>0$ is fixed, $\lambda$ is a large (...
        2
        votes
        1answer
        77 views

        Discrete dynamical system and bound on norm

        Let $z \in \mathbb R\backslash \left\{2 \right\}$ then I would like to understand the following: Consider the dynamical system with $x_i \in \mathbb C^2:$ $$ x_{i} = \left(\begin{matrix} z &&...
        2
        votes
        0answers
        60 views

        Off-diagonal estimates for Poisson kernels on manifolds

        Let $(M,g)$ be a complete Riemannian manifold, $\Delta$ its Laplace-Beltrami operator and $T_t = (e^{t \Delta})_{t \geq 0}$ the associated heat semigroup. We can define the subordinated Poisson ...
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        0answers
        22 views

        Independence of eigenfunction evaluations at random locations

        My question is based on this one. Let the Gaussian kernel be $k(x,y) = \exp(-(x-y)^2/2)$ for $x, y \in R$. Then the Hilbert-Schmidt integral operator is defined as $T_k: (T_k f)(x) = \int_y k(x,y) f(...
        8
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        1answer
        262 views
        +50

        Constants of motion for Droop equation

        There is an important ODE system in biochemistry, Droop's equations: $$s'=1-s-\frac{sx}{a_1+s}$$ $$x'=a_2\big(1-\frac{1}{q}\big)x-x$$ $$q'=\frac{a_3s}{a_1+s}-a_2(q-1)$$ Relatively easy one finds a ...
        2
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        0answers
        46 views

        Traceless sobolev forms on compact manifolds with boundary

        Let $(M,g)$ be a smooth, compact, oriented Riemannian manifold with smooth, oriented boundary $\partial M$. Further, let $\Omega^p(M)$ and $\Omega^p(\partial M)$ be the spaces of smooth differential $...
        2
        votes
        1answer
        105 views

        Eigenvalues Sturm-Liouville Operator

        Is the eigenvalue decomposition of the Sturm-Liouville operator $$ Lf(x)=-f''(x)+h\sin(x)f'(x),\quad h>0, $$ with Neumann boundary conditions $f'(-\pi)=f'(\pi)=0$ on the Hilbert space $L^2([-\pi,\...
        2
        votes
        1answer
        85 views

        How to get asymptotic expansion of the sum of modified Bessel function $\sum_{n=1}^\infty K_0(s\, n)$ as $s\to 0^+$?

        I guess for the modified Bessel funcion $K_0(z)$, $$\sum_{n=1}^\infty K_0(s\, n) \sim \frac{-2\log 2 - \log \pi + \gamma}{2} + \frac{\log s}{2} + \frac{\pi}{2\, s}, \quad s\to 0^+,$$ if taking $$\...
        6
        votes
        1answer
        150 views

        Eigenvalue estimates for operator perturbations

        I edited the question to a general mathematical question, since I found the answer in Carlo Beenakker's reference and think that my initial question was mathematically misleading. What was behind ...

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