9,179 research outputs found
Dequantisation of the Dirac Monopole
Using a sheaf-theoretic extension of conventional principal bundle theory,
the Dirac monopole is formulated as a spherically symmetric model free of
singularities outside the origin such that the charge may assume arbitrary real
values. For integral charges, the construction effectively coincides with the
usual model. Spin structures and Dirac operators are also generalised by the
same technique.Comment: 22 pages. Version to appear in Proc. R. Soc. London
A note on the consensus time of mean-field majority-rule dynamics
In this work, it is pointed out that in the mean-field version of
majority-rule opinion dynamics, the dependence of the consensus time on the
population size exhibits two regimes. This is determined by the size
distribution of the groups that, at each evolution step, gather to reach
agreement. When the group size distribution has a finite mean value, the
previously known logarithmic dependence on the population size holds. On the
other hand, when the mean group size diverges, the consensus time and the
population size are related through a power law. Numerical simulations validate
this semi-quantitative analytical prediction.Comment: 4 pages, 3 figures, Commentary and Reply available in Papers in
Physic
Negative forms and path space forms
We present an account of negative differential forms within a natural
algebraic framework of differential graded algebras, and explain their
relationship with forms on path spaces.Comment: 12 pp.; the Introduction has been rewritten and mention of cohomology
dropped in Proposition 3.2; material slightly reorganize
Gauge Group and Topology Change
The purpose of this study is to examine the effect of topology change in the
initial universe. In this study, the concept of -cobordism is introduced to
argue about the topology change of the manifold on which a transformation group
acts. This -manifold has a fiber bundle structure if the group action is
free and is related to the spacetime in Kaluza-Klein theory or
Einstein-Yang-Mills system. Our results revealed that fundamental processes of
compactification in -manifolds. In these processes, the initial high
symmetry and multidimensional universe changes to present universe by the
mechanism which lowers the dimensions and symmetries.Comment: 8 page
A nonaspherical cell-like 2-dimensional simply connected continuum and related constructions
We prove the existence of a 2-dimensional nonaspherical simply connected
cell-like Peano continuum (the space itself was constructed in one of our
earlier papers). We also indicate some relations between this space and the
well-known Griffiths' space from the 1950's
On the homology of the Harmonic Archipelago
We calculate the singular homology and \v{C}ech cohomology groups of the
Harmonic archipelago. As a corollary, we prove that this space is not homotopy
equivalent to the Griffiths space. This is interesting in view of Eda's proof
that the first singular homology groups of these spaces are isomorphic
Transform of Riccati equation of constant coefficients through fractional procedure
We use a particular fractional generalization of the ordinary differential
equations that we apply to the Riccati equation of constant coefficients. By
this means the latter is transformed into a modified Riccati equation with the
free term expressed as a power of the independent variable which is of the same
order as the order of the applied fractional derivative. We provide the
solutions of the modified equation and employ the results for the case of the
cosmological Riccati equation of FRW barotropic cosmologies that has been
recently introduced by FaraoniComment: 7 pages, 2 figure
Future asymptotic expansions of Bianchi VIII vacuum metrics
Bianchi VIII vacuum solutions to Einstein's equations are causally
geodesically complete to the future, given an appropriate time orientation, and
the objective of this article is to analyze the asymptotic behaviour of
solutions in this time direction. For the Bianchi class A spacetimes, there is
a formulation of the field equations that was presented in an article by
Wainwright and Hsu, and in a previous article we analyzed the asymptotic
behaviour of solutions in these variables. One objective of this paper is to
give an asymptotic expansion for the metric. Furthermore, we relate this
expansion to the topology of the compactified spatial hypersurfaces of
homogeneity. The compactified spatial hypersurfaces have the topology of
Seifert fibred spaces and we prove that in the case of NUT Bianchi VIII
spacetimes, the length of a circle fibre converges to a positive constant but
that in the case of general Bianchi VIII solutions, the length tends to
infinity at a rate we determine.Comment: 50 pages, no figures. Erronous definition of Seifert fibred spaces
correcte
The Non-Trapping Degree of Scattering
We consider classical potential scattering. If no orbit is trapped at energy
E, the Hamiltonian dynamics defines an integer-valued topological degree. This
can be calculated explicitly and be used for symbolic dynamics of
multi-obstacle scattering.
If the potential is bounded, then in the non-trapping case the boundary of
Hill's Region is empty or homeomorphic to a sphere.
We consider classical potential scattering. If at energy E no orbit is
trapped, the Hamiltonian dynamics defines an integer-valued topological degree
deg(E) < 2. This is calculated explicitly for all potentials, and exactly the
integers < 2 are shown to occur for suitable potentials.
The non-trapping condition is restrictive in the sense that for a bounded
potential it is shown to imply that the boundary of Hill's Region in
configuration space is either empty or homeomorphic to a sphere.
However, in many situations one can decompose a potential into a sum of
non-trapping potentials with non-trivial degree and embed symbolic dynamics of
multi-obstacle scattering. This comprises a large number of earlier results,
obtained by different authors on multi-obstacle scattering.Comment: 25 pages, 1 figure Revised and enlarged version, containing more
detailed proofs and remark
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