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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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        1answer
        27 views

        On the convergence of a sqeuence of functions with decomposition

        Considering a sequence of function $f_n(x,y)$ converges to $F(x,y)$ as $n\to+\infty$, we also know that the function $f_n(x,y)$ can be decoupled as the summation of two functions with $x$, $y$ as ...
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        vote
        2answers
        146 views

        A Fredholm equation with a particular kernel

        How to solve this kind of Fredholm’s equation? $$ x(t)+\lambda \int\limits_{0}^{1}\! \big[ts - \min\{t,s\}\big]x(s)ds=t $$ Thanks for any help.
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        votes
        1answer
        226 views

        Sub-Gaussian decay of convolution of $L^1$ function with Gaussian kernel

        I think it might be helpful to put the new statement at the beginning and put the original post at the end. This new statement is more mathematically elegant. Let $f\geq0$ be in $L^1(\mathbb{R}^d)$ ...
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        votes
        0answers
        66 views

        How can I solve this issue? [closed]

        I have a problem related to this theme system analysis and action research. If you can please solve this, in another case please take me more resource for understanding this. The system consists of N-...
        1
        vote
        1answer
        72 views

        Relation between the measures of two sets defined via Lebesgue integration

        I posted this question on StackExchange, people have upvoted it but I have not received any response. I read up here that it is okay to post unanswered StackExchange questions on Mathoverflow. So, ...
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        votes
        0answers
        89 views

        Stability analysis of a differential equation

        My question is about stability analysis and whether the equilibrium point is well-defined in example 2?. If they are not defined, how can one approach the stability analysis for those cases. Example ...
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        0answers
        77 views

        Estimate on first derivatives given $L^2$-norm of Laplacian

        Let $B$ be the unit ball in the Euclidean space $\mathbb{R}^n$. Consider the set of functions $$X=\{u\in C^2(\bar B) \mid u|_{\partial B}=0 \text{ and } \|\Delta u\|_{L^2(B)}\leq 1\},$$ where $\Delta$ ...
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        0answers
        31 views

        Under what conditions is this family normal?

        Let $\mathcal{S} = \{s \in \mathbb{C}\,\mid\,|\Im(s)| < 1\}$ be a strip of the complex plane. Let $q(s,z)$ be a holomorphic function on $\mathcal{S} \times \mathbb{C}$. Letting $\mathcal{K}$ be a ...
        3
        votes
        1answer
        159 views

        Simple proof of Prékopa's Theorem: log-concavity is preserved by marginalization

        The following result is well-known: Suppose that $H(x,y)$ is a log-concave distribution for $(x,y) \in \mathbb R^{m \times n}$ so that by definition we have $$H \left( (1 - \lambda)(x_1,y_1) + \...
        1
        vote
        1answer
        60 views

        Area formula for parametric surfaces

        Assume for $\xi\in S^{n-1}$ the parametrization of a closed hypersurface is given by $x(\xi)=R(\xi)\xi\in \mathbb R^n$. Here $R: S^{n-1}\to \mathbb R$ is a positive function. Is there a reference for ...
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        votes
        1answer
        106 views

        On exponential polynomials

        Suppose we have the following function $f:\mathbb{R}^{+}\mapsto \mathbb{R}$ $$f(t)=\sum_{i=1}^k P_i(t)\exp(\alpha_i t),$$ where $\alpha_i$s are all algebraic numbers and $P_i(t)$ are all polynomials ...
        1
        vote
        1answer
        100 views

        Moment generating function of random unit vector

        Let $X$ be uniformly distributed on the unit sphere $S^{n-1}$. Is there any result concerning the calculation or bound (particularly lower bound) of $$\mathbb{E}[\exp(X^Tv)]$$ for any $v$?
        0
        votes
        0answers
        55 views

        Ratio of exponentially weighted Selberg integrals

        I'm interested in bounding the following ratio of integral: $$\frac{\int_{0<x_k<...<x_1<1}\prod_{i=1}^kx_i^{m-\frac{k+1}{2}}\prod_{i<j}(x_i-x_j)\exp(-\sum_{i=1}^kw_ix_i)}{\int_{0<x_k&...
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        0answers
        65 views

        Help me prove this is identically zero! [duplicate]

        I am having trouble with the definition of functional derivatives to see that the following expression is zero. The expression is: $$\int{\nabla \left(\frac{\delta F}{\delta B}\times \frac{\delta G}{...
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        0answers
        34 views

        Derivatives of descriptors with respect to cartesian coordinates

        I am trying to take the derivatives of some complicated descriptors in order to derive forces for usage in molecular dynamics. We start with a "lab frame" of cartesian coordinates, with a system of $N$...

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