All Questions
Tagged with approximation-theory dg.differential-geometry
13 questions
17
votes
2
answers
750
views
Approximation of smooth diffeomorphisms by polynomial diffeomorphisms?
Is it possible to (locally) approximate an arbitrary smooth diffeomorphism by a polynomial diffeomorphism?
More precisely: Let $f:\mathbb{R}^d\rightarrow\mathbb{R}^d$ be a smooth diffeomorphism for $d&...
- 173
8
votes
1
answer
600
views
Can we perturb a map $\mathbb{R}^n \to \mathbb{R}^n$ to have distinct singular values?
$\newcommand{\SO}[1]{\text{SO}(#1)}$
$\newcommand{\dist}{\operatorname{dist}}$
Let $\mathbb{D}^n$ be the closed $n$-dimensional unit ball, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be smooth.
Set
$$...
- 6,741
6
votes
1
answer
182
views
Can we approximate a vector field on the plane with non-vanishing vector fields in $W^{1,2}$?
Let $V$ be a compactly-supported vector field on $\mathbb{R}^2$, whose zeros inside some open neighbourhood of the closed unit disk $\mathbb{D}^2$ are isolated.
Does there exist a sequence of ...
- 6,741
6
votes
0
answers
104
views
Uniform approximation of a continuous flow by a $\mathcal{C}^1$ flow
Setup: Consider a (smooth) compact Riemannian manifold $M$, whose distance is denoted by $d$. Let $\Phi$ be a continuous flow, namely a continuous application from $\mathbb{R} \times M $ to $M$ ...
3
votes
1
answer
955
views
Least-squares regression and differential geometry
For $k, n \in \mathbb{N}$, let $\mathcal{C}_n \mathbb{R}^k$ denote the configuration space of $n$ distinct points in $\mathbb{R}^k$.
(1) Is there a description of the tangent space $T_C \mathcal{C}...
- 952
3
votes
1
answer
81
views
Estimating the Size of an Approximating Polyline
let $\gamma(s) = \left(x(s),y(s)\right), s\in[0,1]; \gamma'(s) = 1$ be a length-parameterized curve in the plane, with finite and strictly positive curvature.
Questions:
is it possible to estimate ...
- 13.2k
3
votes
0
answers
269
views
covariant derivative of manifold-valued function and logarithm map
Let $M$ be a Riemannian manifold and $f\colon \Omega\subset \mathbb{R}^d\rightarrow M$ a smooth, i.e. $C^\infty$, function. For any $p\in M$ let $T_pM$ be the tangent space at $p$ and $\log_p\colon U\...
- 2,286
2
votes
1
answer
262
views
Bounding the determinant of the Jacobian between a set and its polyhedral approximation
My question is, essentially, suppose I have two simply connected subset of $\mathbb{R}^n$, if I know that the boundaries of both are very close, how can I bound the determinant of the Jacobian between ...
- 21
2
votes
0
answers
42
views
Find weak approximation by smooth unit vector fields for Sobolev fields on manifold
I am considering the Sobolev space of unit tangent vector fields on a compact manifold:
$Γ_{W^{1,2}}(M, UTM)$.
I would like to approximate those weakly with smooth vector fields ($Γ_{C^∞}(M, UTM)$).
...
- 121
2
votes
0
answers
87
views
Approximating the lateral surface area of a generalized cylinder with arc lengths of cross section curves
Given a generalized cylinder1 whose axis is defined by $r(s) = (x(s), y(s), z(s))$, we want to derive a lower bound of the lateral surface area between the cross sections of $r(a)$ and $r(b)$, where $...
- 69
1
vote
1
answer
1k
views
Approximation of a continuous function by a smooth one on an open set
I'm interested in the following kind of theorems :
Let $M$ be a real analytic manifold and $U$ an open set of $M$. Let $f : U \to \mathbb{R}$ a continuous function. Then, there is a $C_{\infty}$ ...
- 193
1
vote
1
answer
248
views
Approximating a compact $C^1$ hypersurface without boundary
Can we approximate (arbitrarily closely) a compact $C^1$ hypersurface in Euclidean space without boundary with a polygonal hypersurface, such as a simplicial complex? To clarify, I want to have the $\...
- 323
0
votes
0
answers
105
views
Wavelet decomposition of $C^{k}$-functions on smooth manifolds
Background (compactly supported wavelet decomposition of $\mathbb{R}^n$):
Fix compactly supported “mother and father wavelets” $\phi,\psi^{\epsilon}:\mathbb{R}^n\rightarrow \mathbb{R}$ where $\epsilon$...
- 5,405