Search Results
Search type | Search syntax |
---|---|
Tags | [tag] |
Exact | "words here" |
Author |
user:1234 user:me (yours) |
Score |
score:3 (3+) score:0 (none) |
Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
Views | views:250 |
Code | code:"if (foo != bar)" |
Sections |
title:apples body:"apples oranges" |
URL | url:"*.example.com" |
Saves | in:saves |
Status |
closed:yes duplicate:no migrated:no wiki:no |
Types |
is:question is:answer |
Exclude |
-[tag] -apples |
For more details on advanced search visit our help page |
Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.
5
votes
Why are there so many smooth functions?
Here is a somewhat different answer:
Smooth functions always admit asymptotic expansions in each point, convergent or not. One difference between analytic functions and smooth functions is, of course …
1
vote
1
answer
84
views
ODE estimate for boundary value problem
Let $X$ be a solution to the boundary value problem
$$ X^{\prime\prime}(s) = A(s)X(s), \quad X(0) = X_0, ~~X(t) = X_1,$$
where $0 < t \leq 1$, $A$ is some matrix-valued function, defined on $[0, 1]$, …
10
votes
Accepted
Relationship between Laplacian and Hessian on compact Lie groups
This has nothing to do with Lie groups, I believe. Let $M$ be a Riemannian manifold. The Bochner formula on $1$-forms states that
$$\nabla^* \nabla \omega = (d \delta + \delta d)\omega - \mathrm{Ric}\ …
1
vote
1
answer
135
views
Positive Definiteness of a certain function
Recall that a function $f: \mathbb{R}^d \longrightarrow \mathbb{C}$ is positive definite, iff for all numbers $N$ and $x_1, \dots, x_N \in \mathbb{R}^d$, the matrix $(a_{ij})$ with entries
$$a_{ij} = …
3
votes
Spectrum of Mathieu equation
The Wikipedia article (http://en.wikipedia.org/wiki/Mathieu_function) is quite extensive, I believe, and in general there is NO way to explicitly calculate either eigenvalues or eigenfunctions of the …
5
votes
1
answer
1k
views
Exponential mapping versus flow
In Hamilton's article on the Nash-Moser Theorem, he gives the map that maps a vector field $X$ to its flow $e^{tX}$ in $\mathrm{Diff}(M)$ as an example where the implicit function theorem in Frechet s …
3
votes
Approximation with continuous functions
All the answers here rely on measure-theoretic ideas to give a negative answer. I feel that this does not exactly meet the point, instead I would say: You did not quite ask the right question, but if …
5
votes
1
answer
1k
views
Existence of Geodesics in continuous metrics
I learned that if we are given a $C^0$ Riemannian metric on a smooth manifold $M$, geodesics (i.e. length minimizing curves) are absolutely continuous, and if the metrics is $C^{0,\alpha}$, then the g …
3
votes
Accepted
Existence of Geodesics in continuous metrics
Ok, thank you Misha for the comments, let me try to fill out the hints you gave myself. I try to prove the following:
Let $g_n$ be a sequence of complete smooth metrics that converge in $C^0$ agains …