# 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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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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**2**answers

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.

**5**

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**1**answer

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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**0**answers

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-...

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**1**answer

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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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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**0**answers

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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**0**answers

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 ...

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**1**answer

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**

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**1**answer

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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**1**answer

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 ...

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**1**answer

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**

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**0**answers

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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**0**answers

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}{...

**-1**

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**0**answers

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 $N$...