6,995 research outputs found
Topological complexity of motion planning and Massey products
We employ Massey products to find sharper lower bounds for the Schwarz genus
of a fibration than those previously known. In particular we give examples of
non-formal spaces for which the topological complexity \TC(X) (defined to
be the genus of the free path fibration on ) is greater than the
zero-divisors cup-length plus one.Comment: 11 pages; minor revisions and 1 added reference; to appear in the
Proceedings of the M. M. Postnikov Memorial Conferenc
Sphalerons with Two Higgs Doublets
We report on work studying the properties of the sphaleron in models of the
electroweak interactions with two Higgs doublets in as model-independent a way
as possible: by exploring the physical parameter space described by the masses
and mixing angles of the Higgs particles. If one of the Higgs particles is
heavy, there can be several sphaleron solutions, distinguished by their
properties under parity and the behaviour of the Higgs field at the origin. In
general, these solutions are not spherically symmetric, although the departure
from spherical symmetry is small.Comment: Talk given at Strong and Electroweak Matter, Marseille, 14-17 June
2000. Uses World Scientific proceedings class ws-p8-50x6-00.cl
Homologies are infinitely complex
We show that for any k>1, stratified sets of finite complexity are
insufficient to realize all homology classes of codimension k in all smooth
manifolds. We also prove a similar result concerning smooth generic maps whose
double-point sets are co-oriented.Comment: 5 pages; v2: Very minor corrections, to appear in Topol. Methods
Nonlinear Ana
Sequential motion planning of non-colliding particles in Euclidean spaces
In terms of Rudyak's generalization of Farber's topological complexity of the
path motion planning problem in robotics, we give a complete description of the
topological instabilities in any sequential motion planning algorithm for a
system consisting of non-colliding autonomous entities performing tasks in
space whilst avoiding collisions with several moving obstacles. The Isotopy
Extension Theorem from manifold topology implies, somewhat surprisingly, that
the complexity of this problem coincides with the complexity of the
corresponding problem in which the obstacles are stationary.Comment: 10 pages; Final version, to appear in Proc. Amer. Math. So
On realizing homology classes by maps of restricted complexity
We show that in every codimension greater than one there exists a mod 2
homology class in some closed manifold (of sufficiently high dimension) which
cannot be realized by an immersion of closed manifolds. The proof gives
explicit obstructions (in terms of cohomology operations) for realizability of
mod 2 homology classes by immersions. We also prove the corresponding result in
which the word `immersion' is replaced by `map with some restricted set of
multi-singularities'.Comment: 13 pages; Final version, to appear in Bull. Lond. Math. So
Do Two Temperature Debris Disks Have Multiple Belts?
We present a study of debris disks whose spectra are well modelled by dust
emission at two different temperatures. These disks are typically assumed to be
a sign of multiple belts, which in only a few cases have been confirmed via
high resolution observations. We first compile a sample of two-temperature
disks to derive their properties, summarised by the ratios of the warm and cool
component temperatures and fractional luminosities. The ratio of warm to cool
temperatures is constant in the range 2-4, and the temperatures of both warm
and cool components increases with stellar mass. We then explore whether this
emission can arise from dust in a single narrow belt, with the range of
temperatures arising from the size variation of grain temperatures. This model
can produce two-temperature spectra for Sun-like stars, but is not supported
where it can be tested by observed disk sizes and far-IR/mm spectral slopes.
Therefore, while some two-temperature disks arise from single belts, it is
probable that most have multiple spatial components. These disks are plausibly
similar to the outer Solar System's configuration of Asteroid and
Edgeworth-Kuiper belts separated by giant planets. Alternatively, the inner
component could arise from inward scattering of material from the outer belt,
again due to intervening planets. In either case, we suggest that the ratio of
warm/cool component temperatures is indicative of the scale of outer planetary
systems, which typically span a factor of about ten in radius.Comment: accepted to MNRA
New lower bounds for the topological complexity of aspherical spaces
Date of Acceptance: 5/04/2015 15 pages, 4 figuresPeer reviewedPostprin
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