103,897 research outputs found
Computing all maps into a sphere
Given topological spaces X and Y, a fundamental problem of algebraic topology
is understanding the structure of all continuous maps X -> Y . We consider a
computational version, where X, Y are given as finite simplicial complexes, and
the goal is to compute [X,Y], i.e., all homotopy classes of such maps. We solve
this problem in the stable range, where for some d >= 2, we have dim X <= 2d -
2 and Y is (d - 1)-connected; in particular, Y can be the d-dimensional sphere
S^d. The algorithm combines classical tools and ideas from homotopy theory
(obstruction theory, Postnikov systems, and simplicial sets) with algorithmic
tools from effective algebraic topology (locally effective simplicial sets and
objects with effective homology). In contrast, [X,Y] is known to be
uncomputable for general X,Y, since for X = S^1 it includes a well known
undecidable problem: testing triviality of the fundamental group of Y. In
follow-up papers, the algorithm is shown to run in polynomial time for d fixed,
and extended to other problems, such as the extension problem, where we are
given a subspace A of X and a map A -> Y and ask whether it extends to a map X
-> Y, or computing the Z_2-index---everything in the stable range. Outside the
stable range, the extension problem is undecidable.Comment: 42 pages; a revised and substantially updated version (referring to
follow-up papers and results
Full-sky lensing shear at second order
We compute the reduced cosmic shear up to second order in the gravitational
potential without relying on the small angle or thin-lens approximation. This
is obtained by solving the Sachs equation which describes the deformation of
the infinitesimal cross-section of light bundle in the optical limit, and maps
galaxy intrinsic shapes into their angular images. The calculation is done in
the Poisson gauge without a specific matter content, including vector and
tensor perturbations generated at second order and taking account of the
inhomogeneities of a fixed redshift source plane. Our final result is expressed
in terms of spin-2 operators on the sphere and is valid on the full sky. Beside
the well known lens-lens and Born corrections that dominate on small angular
scales, we find new non-linear couplings. These are a purely general
relativistic intrinsic contribution, a coupling between the gravitational
potential at the source with the lens, couplings between the time delay with
the lens, couplings between two photon deflections, as well as non-linear
couplings due to the second-order vector and tensor components. The
inhomogeneity in the redshift of the source induces a coupling between the
photon redshift with the lens. All these corrections become important on large
angular scales and should thus be included when computing higher-order
observables such as the bispectrum, in full or partially full-sky surveys.Comment: 29 pages, discussion about the first-order convergence added, matches
published versio
\v{C}ech-Delaunay gradient flow and homology inference for self-maps
We call a continuous self-map that reveals itself through a discrete set of
point-value pairs a sampled dynamical system. Capturing the available
information with chain maps on Delaunay complexes, we use persistent homology
to quantify the evidence of recurrent behavior. We establish a sampling theorem
to recover the eigenspace of the endomorphism on homology induced by the
self-map. Using a combinatorial gradient flow arising from the discrete Morse
theory for \v{C}ech and Delaunay complexes, we construct a chain map to
transform the problem from the natural but expensive \v{C}ech complexes to the
computationally efficient Delaunay triangulations. The fast chain map algorithm
has applications beyond dynamical systems.Comment: 22 pages, 8 figure
A Metric for genus-zero surfaces
We present a new method to compare the shapes of genus-zero surfaces. We
introduce a measure of mutual stretching, the symmetric distortion energy, and
establish the existence of a conformal diffeomorphism between any two
genus-zero surfaces that minimizes this energy. We then prove that the energies
of the minimizing diffeomorphisms give a metric on the space of genus-zero
Riemannian surfaces. This metric and the corresponding optimal diffeomorphisms
are shown to have properties that are highly desirable for applications.Comment: 33 pages, 8 figure
Computing simplicial representatives of homotopy group elements
A central problem of algebraic topology is to understand the homotopy groups
of a topological space . For the computational version of the
problem, it is well known that there is no algorithm to decide whether the
fundamental group of a given finite simplicial complex is
trivial. On the other hand, there are several algorithms that, given a finite
simplicial complex that is simply connected (i.e., with
trivial), compute the higher homotopy group for any given .
%The first such algorithm was given by Brown, and more recently, \v{C}adek et
al.
However, these algorithms come with a caveat: They compute the isomorphism
type of , as an \emph{abstract} finitely generated abelian
group given by generators and relations, but they work with very implicit
representations of the elements of . Converting elements of this
abstract group into explicit geometric maps from the -dimensional sphere
to has been one of the main unsolved problems in the emerging field
of computational homotopy theory.
Here we present an algorithm that, given a~simply connected space ,
computes and represents its elements as simplicial maps from a
suitable triangulation of the -sphere to . For fixed , the
algorithm runs in time exponential in , the number of simplices of
. Moreover, we prove that this is optimal: For every fixed , we
construct a family of simply connected spaces such that for any simplicial
map representing a generator of , the size of the triangulation of
on which the map is defined, is exponential in
Coherent States Measurement Entropy
Coherent states (CS) quantum entropy can be split into two components. The
dynamical entropy is linked with the dynamical properties of a quantum system.
The measurement entropy, which tends to zero in the semiclassical limit,
describes the unpredictability induced by the process of a quantum approximate
measurement. We study the CS--measurement entropy for spin coherent states
defined on the sphere discussing different methods dealing with the time limit
. In particular we propose an effective technique of computing
the entropy by iterated function systems. The dependence of CS--measurement
entropy on the character of the partition of the phase space is analysed.Comment: revtex, 22 pages, 14 figures available upon request (e-mail:
[email protected]). Submitted to J.Phys.
Computing Teichm\"{u}ller Maps between Polygons
By the Riemann-mapping theorem, one can bijectively map the interior of an
-gon to that of another -gon conformally. However, (the boundary
extension of) this mapping need not necessarily map the vertices of to
those . In this case, one wants to find the ``best" mapping between these
polygons, i.e., one that minimizes the maximum angle distortion (the
dilatation) over \textit{all} points in . From complex analysis such maps
are known to exist and are unique. They are called extremal quasiconformal
maps, or Teichm\"{u}ller maps.
Although there are many efficient ways to compute or approximate conformal
maps, there is currently no such algorithm for extremal quasiconformal maps.
This paper studies the problem of computing extremal quasiconformal maps both
in the continuous and discrete settings.
We provide the first constructive method to obtain the extremal
quasiconformal map in the continuous setting. Our construction is via an
iterative procedure that is proven to converge quickly to the unique extremal
map. To get to within of the dilatation of the extremal map, our
method uses iterations. Every step of the iteration
involves convex optimization and solving differential equations, and guarantees
a decrease in the dilatation. Our method uses a reduction of the polygon
mapping problem to that of the punctured sphere problem, thus solving a more
general problem.
We also discretize our procedure. We provide evidence for the fact that the
discrete procedure closely follows the continuous construction and is therefore
expected to converge quickly to a good approximation of the extremal
quasiconformal map.Comment: 28 pages, 6 figure
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