493 research outputs found
Effective action in spherical domains
The effective action on an orbifolded sphere is computed for minimally
coupled scalar fields. The results are presented in terms of derivatives of
Barnes zeta-functions and it is shown how these may be evaluated. Numerical
values are shown. An analytical, heat-kernel derivation of the Ces\`aro-Fedorov
formula for the number of symmetry planes of a regular solid is also presented.Comment: 18 pages, Plain TeX (Mailer oddities possibly corrected.
Geometrical Frustration and Static Correlations in Hard-Sphere Glass Formers
We analytically and numerically characterize the structure of hard-sphere
fluids in order to review various geometrical frustration scenarios of the
glass transition. We find generalized polytetrahedral order to be correlated
with increasing fluid packing fraction, but to become increasingly irrelevant
with increasing dimension. We also find the growth in structural correlations
to be modest in the dynamical regime accessible to computer simulations.Comment: 21 pages; part of the "Special Topic Issue on the Glass Transition
Some integrals ocurring in a topology change problem
In a paper presented a few years ago, De Lorenci et al. showed, in the
context of canonical quantum cosmology, a model which allowed space topology
changes (Phys. Rev. D 56, 3329 (1997)). The purpose of this present work is to
go a step further in that model, by performing some calculations only estimated
there for several compact manifolds of constant negative curvature, such as the
Weeks and Thurston spaces and the icosahedral hyperbolic space (Best space).Comment: RevTeX article, 4 pages, 1 figur
Functional Maps Representation on Product Manifolds
We consider the tasks of representing, analyzing and manipulating maps
between shapes. We model maps as densities over the product manifold of the
input shapes; these densities can be treated as scalar functions and therefore
are manipulable using the language of signal processing on manifolds. Being a
manifold itself, the product space endows the set of maps with a geometry of
its own, which we exploit to define map operations in the spectral domain; we
also derive relationships with other existing representations (soft maps and
functional maps). To apply these ideas in practice, we discretize product
manifolds and their Laplace--Beltrami operators, and we introduce localized
spectral analysis of the product manifold as a novel tool for map processing.
Our framework applies to maps defined between and across 2D and 3D shapes
without requiring special adjustment, and it can be implemented efficiently
with simple operations on sparse matrices.Comment: Accepted to Computer Graphics Foru
Regular Incidence Complexes, Polytopes, and C-Groups
Regular incidence complexes are combinatorial incidence structures
generalizing regular convex polytopes, regular complex polytopes, various types
of incidence geometries, and many other highly symmetric objects. The special
case of abstract regular polytopes has been well-studied. The paper describes
the combinatorial structure of a regular incidence complex in terms of a system
of distinguished generating subgroups of its automorphism group or a
flag-transitive subgroup. Then the groups admitting a flag-transitive action on
an incidence complex are characterized as generalized string C-groups. Further,
extensions of regular incidence complexes are studied, and certain incidence
complexes particularly close to abstract polytopes, called abstract polytope
complexes, are investigated.Comment: 24 pages; to appear in "Discrete Geometry and Symmetry", M. Conder,
A. Deza, and A. Ivic Weiss (eds), Springe
Hard Discs on the Hyperbolic Plane
We examine a simple hard disc fluid with no long range interactions on the
two dimensional space of constant negative Gaussian curvature, the hyperbolic
plane. This geometry provides a natural mechanism by which global crystalline
order is frustrated, allowing us to construct a tractable model of disordered
monodisperse hard discs. We extend free area theory and the virial expansion to
this regime, deriving the equation of state for the system, and compare its
predictions with simulation near an isostatic packing in the curved space.Comment: 4 pages, 3 figures, included, final versio
Graphs and Reflection Groups
It is shown that graphs that generalize the ADE Dynkin diagrams and have
appeared in various contexts of two-dimensional field theory may be regarded in
a natural way as encoding the geometry of a root system. After recalling what
are the conditions satisfied by these graphs, we define a bilinear form on a
root system in terms of the adjacency matrices of these graphs and undertake
the study of the group generated by the reflections in the hyperplanes
orthogonal to these roots. Some ``non integrally laced " graphs are shown to be
associated with subgroups of these reflection groups. The empirical relevance
of these graphs in the classification of conformal field theories or in the
construction of integrable lattice models is recalled, and the connections with
recent developments in the context of supersymmetric theories and
topological field theories are discussed.Comment: 42 pages TEX file, harvmac and epsf macros, AMS fonts optional,
uuencoded, 8 figures include
Gauge Theoretic Invariants of, Dehn Surgeries on Knots
New methods for computing a variety of gauge theoretic invariants for
homology 3-spheres are developed. These invariants include the Chern-Simons
invariants, the spectral flow of the odd signature operator, and the rho
invariants of irreducible SU(2) representations. These quantities are
calculated for flat SU(2) connections on homology 3-spheres obtained by 1/k
Dehn surgery on (2,q) torus knots. The methods are then applied to compute the
SU(3) gauge theoretic Casson invariant (introduced in [H U Boden and C M
Herald, The SU(3) Casson invariant for integral homology 3--spheres, J. Diff.
Geom. 50 (1998) 147-206]) for Dehn surgeries on (2,q) torus knots for q=3,5,7
and 9.Comment: Version 3: minor corrections from version 2. Published by Geometry
and Topology at http://www.maths.warwick.ac.uk/gt/GTVol5/paper6.abs.htm
Spherical Orbifolds for Cosmic Topology
Harmonic analysis is a tool to infer cosmic topology from the measured
astrophysical cosmic microwave background CMB radiation. For overall positive
curvature, Platonic spherical manifolds are candidates for this analysis. We
combine the specific point symmetry of the Platonic manifolds with their deck
transformations. This analysis in topology leads from manifolds to orbifolds.
We discuss the deck transformations of the orbifolds and give eigenmodes for
the harmonic analysis as linear combinations of Wigner polynomials on the
3-sphere. These provide new tools for detecting cosmic topology from the CMB
radiation.Comment: 17 pages, 9 figures. arXiv admin note: substantial text overlap with
arXiv:1011.427
Ultrametric spaces of branches on arborescent singularities
Let be a normal complex analytic surface singularity. We say that is
arborescent if the dual graph of any resolution of it is a tree. Whenever
are distinct branches on , we denote by their intersection
number in the sense of Mumford. If is a fixed branch, we define when and
otherwise. We generalize a theorem of P{\l}oski concerning smooth germs of
surfaces, by proving that whenever is arborescent, then is an
ultrametric on the set of branches of different from . We compute the
maximum of , which gives an analog of a theorem of Teissier. We show that
encodes topological information about the structure of the embedded
resolutions of any finite set of branches. This generalizes a theorem of Favre
and Jonsson concerning the case when both and are smooth. We generalize
also from smooth germs to arbitrary arborescent ones their valuative
interpretation of the dual trees of the resolutions of . Our proofs are
based in an essential way on a determinantal identity of Eisenbud and Neumann.Comment: 37 pages, 16 figures. Compared to the first version on Arxiv, il has
a new section 4.3, accompanied by 2 new figures. Several passages were
clarified and the typos discovered in the meantime were correcte
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