All Questions
Tagged with gt.geometric-topology ca.classical-analysis-and-odes
14 questions
-1
votes
0
answers
114
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Stability of flow map
$\DeclareMathOperator\Diff{Diff}$Setting:
Let $(M,g)$ be a compact and connected $C^{\infty}$-Riemannian manifold. Let $d_g$ denote the induced shorted path metric and equip $C^{\infty}(M)$ with the ...
- 5,405
4
votes
0
answers
83
views
Conditional convergence of sums over infinite sets
As undergraduates, we learn that conditional convergence of infinite series is highly sensitive to the order structure on $\mathbb N$: if $\sum_{n=0}^\infty x_n$ conditionally converges and $x \in \...
- 708
2
votes
0
answers
81
views
Extension of a tangent vector field
Let $\Omega$ be an open subset of $S^2$ with $\overline{\Omega} \neq S^2$. Suppose a continuous tangent vector field $G$ is defined on $\partial \Omega$ such that $|G(y)| = 1$ for all $y \in \partial \...
- 434
1
vote
1
answer
84
views
Simple convergence of convex compact set implies Hausdorff convergence
I am wondering about the following :
In $\mathbb{R}^n$, suppose you are given compact convex bodies $\left\{ C_k : k \geq 1 \right\}$ and $C$, such that for every $x \in \mathbb{R}^n$ $$ \mathbb{1}_{...
- 125
2
votes
1
answer
672
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What are the best definitions for smoothness of a 2D curve (real-valued function)?
Sounds like a trivial question, but could not find any answer other than the fact that there are many ways to define it. My problem is this: I look at different elevation maps,
some with sharp ...
- 3,098
10
votes
0
answers
263
views
Bi-Lipschitz mappings
Assume that we have a bi-Lipschitz mapping $f:\bar{\mathbb{B}}^n(0,1)\to\mathbb{R}^n$. The mapping need not be smooth anywhere and it may happen that it cannot be extended to a homeomorphism of a ...
- 28k
3
votes
0
answers
200
views
The best applications of the Poincaré-Bendixson theorem [closed]
I'm reading about the Poincaré-Bendixson theorem in the plane, I really liked the theorem. I have seen common applications in Sotomayor and Perko's book. But I would like to know what other ...
- 143
0
votes
0
answers
57
views
Can iterative application of ham sandwich cuts form streamlines of an ODE?
It has been known that given two probability distributions $\mu_1$ and $\mu_2$ (let us say, they are smooth for simplicity), there is a hyperplane that divides the domain into two regions (denoted as $...
- 23
10
votes
2
answers
698
views
Bi-Lipschitz extension
Given a bi-Lipschitz homeomorphism
$\Phi:\mathbb{B}^n(0,1)\to\mathbb{R}^n$, (that is a bi-Lipschitz map onto the image), can one find a bi-Lipschitz homeomorphism $\Psi:\mathbb{R}^n\to\mathbb{R}^n$ ...
- 28k
11
votes
2
answers
1k
views
Thurston-Cannon $S^2$-filling curves
I have been looking into equivariant space-filling curves $S^1\to S^2$ as discussed by Cannon & Thurston in these two papers:
Three-Dimensional Manifolds, Kleinian Groups and Hyperbolic Geometry
...
- 22.8k
8
votes
6
answers
2k
views
Uncountable preimage of every point
Let $f:[0,1]\to [0,1]$ be a continuous function. Must it have a point $x$ that $f^{-1}(x)$ is at most countable?
Added: Must it have a point $x$ that $dim_H(f^{-1}(x))=0$ ? ($dim_H$ means the ...
- 5,053
5
votes
1
answer
2k
views
Examples and importance of Embedding (and Non-Embedding) Theorems
An embedding is an injective map into a universal, simpler model object. Many embedding theorems are without obstruction, in the sense that every object which you wish to embed can be embedded. ...
3
votes
2
answers
594
views
A question about the Kakeya problem
Besicovich proved a long time ago that a straight line segment of fixed length could be rotated 360
degrees within a subset S of the Euclidean plane such that $M(S)$ is arbitrarily small-where M is ...
- 6,215
9
votes
5
answers
3k
views
Geometric group theory and analysis
Geometric group theory is mainly concerned with topological and geometric properties of groups, spaces on which they act etc., so the ideas employed in GGT are mainly algebraic/geometric/topological. ...
- 2,449