9,017 research outputs found
Commensurators of cusped hyperbolic manifolds
This paper describes a general algorithm for finding the commensurator of a
non-arithmetic cusped hyperbolic manifold, and for deciding when two such
manifolds are commensurable. The method is based on some elementary
observations regarding horosphere packings and canonical cell decompositions.
For example, we use this to find the commensurators of all non-arithmetic
hyperbolic once-punctured torus bundles over the circle.
For hyperbolic 3-manifolds, the algorithm has been implemented using
Goodman's computer program Snap. We use this to determine the commensurability
classes of all cusped hyperbolic 3-manifolds triangulated using at most 7 ideal
tetrahedra, and for the complements of hyperbolic knots and links with up to 12
crossings.Comment: 32 pages, 46 figures; to appear in "Experimental Mathematics
Numerical analysis of semilinear elliptic equations with finite spectral interaction
We present an algorithm to solve - \lap u - f(x,u) = g with Dirichlet
boundary conditions in a bounded domain . The nonlinearities are
non-resonant and have finite spectral interaction: no eigenvalue of -\lap_D
is an endpoint of \bar{\partial_2f(\Omega,\RR)}, which in turn only contains
a finite number of eigenvalues. The algorithm is based in ideas used by Berger
and Podolak to provide a geometric proof of the Ambrosetti-Prodi theorem and
advances work by Smiley and Chun for the same problem.Comment: 20 pages, 15 figures (34 .eps files
Canonical Decompositions of Affine Permutations, Affine Codes, and Split -Schur Functions
We study the unique maximal decomposition of an arbitrary affine permutation
into a product of cyclically decreasing elements, providing a new perspective
on work of Thomas Lam. This decomposition is closely related to the affine
code, which generalizes the -bounded partition associated to Grassmannian
elements. We also show that the affine code readily encodes a number of basic
combinatorial properties of an affine permutation. As an application, we prove
a new special case of the Littlewood-Richardson Rule for -Schur functions,
using the canonical decomposition to control for which permutations appear in
the expansion of the -Schur function in noncommuting variables over the
affine nil-Coxeter algebra.Comment: 51 pages, 15 figure
Lattice Surfaces and smallest triangles
We calculate the area of the smallest triangle and the area of the smallest
virtual triangle for many known lattice surfaces. We show that our list of the
lattice surfaces for which the area of the smallest virtual triangle greater
than .05 is complete. In particular, this means that there are no new lattice
surfaces for which the area of the smallest virtual triangle is greater than
.05. Our method follows an algorithm described by Smillie and Weiss and
improves on it in certain respects.Comment: Minor revisions thanks to the suggestions of Anja Randecker and Alex
Wrigh
Immersed cycles and the JSJ decomposition
We present an algorithm to construct the JSJ decomposition of one-ended
hyperbolic groups which are fundamental groups of graphs of free groups with
cyclic edge groups. Our algorithm runs in double exponential time, and is the
first algorithm on JSJ decompositions to have an explicit time bound. Our
methods are combinatorial/geometric and rely on analysing properties of
immersed cycles in certain CAT(0) square complexes.Comment: 35 pages, 9 figures. v2: Incorporated referee comments. Results have
been strengthened. To appear in Algebraic and Geometric Topolog
Beating the integrality ratio for s-t-tours in graphs
Among various variants of the traveling salesman problem, the s-t-path graph
TSP has the special feature that we know the exact integrality ratio, 3/2, and
an approximation algorithm matching this ratio. In this paper, we go below this
threshold: we devise a polynomial-time algorithm for the s-t-path graph TSP
with approximation ratio 1.497. Our algorithm can be viewed as a refinement of
the 3/2-approximation algorithm by Seb\H{o} and Vygen [2014], but we introduce
several completely new techniques. These include a new type of
ear-decomposition, an enhanced ear induction that reveals a novel connection to
matroid union, a stronger lower bound, and a reduction of general instances to
instances in which s and t have small distance (which works for general
metrics)
Tractable Optimization Problems through Hypergraph-Based Structural Restrictions
Several variants of the Constraint Satisfaction Problem have been proposed
and investigated in the literature for modelling those scenarios where
solutions are associated with some given costs. Within these frameworks
computing an optimal solution is an NP-hard problem in general; yet, when
restricted over classes of instances whose constraint interactions can be
modelled via (nearly-)acyclic graphs, this problem is known to be solvable in
polynomial time. In this paper, larger classes of tractable instances are
singled out, by discussing solution approaches based on exploiting hypergraph
acyclicity and, more generally, structural decomposition methods, such as
(hyper)tree decompositions
Sylvester-t' Hooft generators of sl(n) and sl(n|n), and relations between them
Among the simple finite dimensional Lie algebras, only sl(n) possesses two
automorphisms of finite order which have no common nonzero eigenvector with
eigenvalue one. It turns out that these automorphisms are inner and form a pair
of generators that allow one to generate all of sl(n) under bracketing. It
seems that Sylvester was the first to mention these generators, but he used
them as generators of the associative algebra of all n times n matrices Mat(n).
These generators appear in the description of elliptic solutions of the
classical Yang-Baxter equation, orthogonal decompositions of Lie algebras, 't
Hooft's work on confinement operators in QCD, and various other instances. Here
I give an algorithm which both generates sl(n) and explicitly describes a set
of defining relations. For simple (up to center) Lie superalgebras, analogs of
Sylvester generators exist only for sl(n|n). The relations for this case are
also computed.Comment: 14 pages, 6 figure
Lossless Image and Intra-frame Compression with Integer-to-Integer DST
Video coding standards are primarily designed for efficient lossy
compression, but it is also desirable to support efficient lossless compression
within video coding standards using small modifications to the lossy coding
architecture. A simple approach is to skip transform and quantization, and
simply entropy code the prediction residual. However, this approach is
inefficient at compression. A more efficient and popular approach is to skip
transform and quantization but also process the residual block with DPCM, along
the horizontal or vertical direction, prior to entropy coding. This paper
explores an alternative approach based on processing the residual block with
integer-to-integer (i2i) transforms. I2i transforms can map integer pixels to
integer transform coefficients without increasing the dynamic range and can be
used for lossless compression. We focus on lossless intra coding and develop
novel i2i approximations of the odd type-3 DST (ODST-3). Experimental results
with the HEVC reference software show that the developed i2i approximations of
the ODST-3 improve lossless intra-frame compression efficiency with respect to
HEVC version 2, which uses the popular DPCM method, by an average 2.7% without
a significant effect on computational complexity.Comment: Draft consisting of 16 page
Extreme-Point Symmetric Mode Decomposition Method for Data Analysis
An extreme-point symmetric mode decomposition (ESMD) method is proposed to
improve the Hilbert-Huang Transform (HHT) through the following prospects: (1)
The sifting process is implemented by the aid of 1, 2, 3 or more inner
interpolating curves, which classifies the methods into ESMD_I, ESMD_II,
ESMD_III, and so on; (2) The last residual is defined as an optimal curve
possessing a certain number of extreme points, instead of general trend with at
most one extreme point, which allows the optimal sifting times and
decompositions; (3) The extreme-point symmetry is applied instead of the
envelop symmetry; (4) The data-based direct interpolating approach is developed
to compute the instantaneous frequency and amplitude. One advantage of the ESMD
method is to determine an optimal global mean curve in an adaptive way which is
better than the common least-square method and running-mean approach; another
one is to determine the instantaneous frequency and amplitude in a direct way
which is better than the Hilbert-spectrum method. These will improve the
adaptive analysis of the data from atmospheric and oceanic sciences,
informatics, economics, ecology, medicine, seismology, and so on.Comment: 40 pages, 28 figure
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