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What are the Longest Ropes on the Unit Sphere?

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Which spaces look like an n-sphere through the eyes of the n-th Postnikov section functor and the n-connected cover functor? The answer is what we call the Postnikov genus of the n-sphere. We define in fact the notion of localization genus for any homotopi ...

Univ Autonoma Barcelona2017

An Approach for Imitation Learning on Riemannian Manifolds

Sylvain Calinon

In imitation learning, multivariate Gaussians are widely used to encode robot behaviors. Such approaches do not provide the ability to properly represent end-effector orientation, as the distance metric in the space of orientations is not Euclidean. In thi ...

2017

We study time and space equivariant wave maps from

M \times \mathbb{R} \rightarrow S^2

, where

M

is dieomorphic to a two dimensional sphere and admits an action of

SO(2)

by isometries. We assume that metric on

M

can be written as $dr^2+f^2(r)d\theta ...

2016

We analyze the critical connectivity of systems of penetrable d-dimensional spheres having size distributions in terms of weighed random geometrical graphs, in which vertex coordinates correspond to random positions of the sphere centers, and edges are for ...

Amer Physical Soc2015

Full blow-up range for co-rotaional wave maps to surfaces of revolution

Can Gao

We construct blow-up solutions of the energy critical wave map equation on

\mathbb{R}^{2+1}\to \mathcal N

with polynomial blow-up rate (

t^{-1-\nu}

for blow-up at

t=0

) in the case when

\mathcal N

is a surface of revolution. Here we extend the blow-u ...

2014

On the number of lattice points in a small sphere and a recursive lattice decoding algorithm

Annika Meyer

Let L be a lattice in . This paper provides two methods to obtain upper bounds on the number of points of L contained in a small sphere centered anywhere in . The first method is based on the observation that if the sphere is sufficiently small then the la ...

Springer US2013

Asymptotic linearity and Hadamard differentiability

Charles Stuart

For a mapping between Banach spaces, two weaker variants of the usual notion of asymptotic linearity are defined and explored. It is shown that, under inversion through the unit sphere, they correspond to Hadamard and weak Hadamard differentiability at the ...

2012

Blow Up Construction and Stability of Stationary Maps

Sohrab Mirshams Shahshahani

This thesis is a study of wave maps from a curved background to the standard two dimensional sphere S2. The target is always assumed to be embedded in R3 in the standard way. The do- main manifold (the "curved background") will be diffeomorphic to S2 × R, ...

EPFL2012

On Sphere-Filling Ropes

Henryk Gerlach

What is the longest rope on the unit sphere? Intuition tells us that the answer to this packing problem depends on the rope's thickness. For a countably infinite number of prescribed thickness values we construct and classify all solution curves. The simpl ...

2011

Ideal knots and other packing problems of tubes