3,598 research outputs found
Zoology of Atlas-groups: dessins d'enfants, finite geometries and quantum commutation
Every finite simple group P can be generated by two of its elements. Pairs of
generators for P are available in the Atlas of finite group representations as
(not neccessarily minimal) permutation representations P. It is unusual but
significant to recognize that a P is a Grothendieck's dessin d'enfant D and
that most standard graphs and finite geometries G-such as near polygons and
their generalizations-are stabilized by a D. In our paper, tripods P -- D -- G
of rank larger than two, corresponding to simple groups, are organized into
classes, e.g. symplectic, unitary, sporadic, etc (as in the Atlas). An
exhaustive search and characterization of non-trivial point-line configurations
defined from small index representations of simple groups is performed, with
the goal to recognize their quantum physical significance. All the defined
geometries G' s have a contextuality parameter close to its maximal value 1.Comment: 19 page
On the order of a non-abelian representation group of a slim dense near hexagon
We show that, if the representation group of a slim dense near hexagon
is non-abelian, then is of exponent 4 and ,
, where is the near polygon
embedding dimension of and is the dimension of the universal
representation module of . Further, if , then
is an extraspecial 2-group (Theorem 1.6)
Compressible magnetoconvection in three dimensions: pattern formation in a strongly stratified layer
The interaction between magnetic fields and convection is interesting both because of its astrophysical importance and because the nonlinear Lorentz force leads to an especially rich variety of behaviour. We present several sets of computational results for magnetoconvection in a square box, with periodic lateral boundary conditions, that show transitions from steady convection with an ordered planform through a regime with intermittent bursts to complicated spatiotemporal behaviour. The constraints imposed by the square lattice are relaxed as the aspect ratio is increased. In wide boxes we find a new regime, in which regions with strong fields are separated from regions with vigorous convection. We show also how considerations of symmetry and associated group theory can be used to explain the nature of these transitions and the sequence in which the relevant bifurcations occur
Two-frequency forced Faraday waves: Weakly damped modes and pattern selection
Recent experiments (Kudrolli, Pier and Gollub, 1998) on two-frequency
parametrically excited surface waves exhibit an intriguing "superlattice" wave
pattern near a codimension-two bifurcation point where both subharmonic and
harmonic waves onset simultaneously, but with different spatial wavenumbers.
The superlattice pattern is synchronous with the forcing, spatially periodic on
a large hexagonal lattice, and exhibits small-scale triangular structure.
Similar patterns have been shown to exist as primary solution branches of a
generic 12-dimensional -equivariant bifurcation problem, and may
be stable if the nonlinear coefficients of the bifurcation problem satisfy
certain inequalities (Silber and Proctor, 1998). Here we use the spatial and
temporal symmetries of the problem to argue that weakly damped harmonic waves
may be critical to understanding the stabilization of this pattern in the
Faraday system. We illustrate this mechanism by considering the equations
developed by Zhang and Vinals (1997, J. Fluid Mech. 336) for small amplitude,
weakly damped surface waves on a semi-infinite fluid layer. We compute the
relevant nonlinear coefficients in the bifurcation equations describing the
onset of patterns for excitation frequency ratios of 2/3 and 6/7. For the 2/3
case, we show that there is a fundamental difference in the pattern selection
problems for subharmonic and harmonic instabilities near the codimension-two
point. Also, we find that the 6/7 case is significantly different from the 2/3
case due to the presence of additional weakly damped harmonic modes. These
additional harmonic modes can result in a stabilization of the superpatterns.Comment: 26 pages, 8 figures; minor text revisions, corrected figure 8; this
version to appear in a special issue of Physica D in memory of John David
Crawfor
Secondary instabilities of hexagons: a bifurcation analysis of experimentally observed Faraday wave patterns
We examine three experimental observations of Faraday waves generated by
two-frequency forcing, in which a primary hexagonal pattern becomes unstable to
three different superlattice patterns. We use the symmetry-based approach
developed by Tse et al. to analyse the bifurcations involved in creating the
three new patterns. Each of the three examples reveals a different situation
that can arise in the theoretical analysis.Comment: 14 pages LaTeX, Birkhauser style, 5 figures, submitted to the
proceedings of the conference on Bifurcations, Symmetry and Patterns, held in
Porto, June 200
Coordinated optimization of visual cortical maps : 1. Symmetry-based analysis
In the primary visual cortex of primates and carnivores, functional architecture can be characterized by maps of various stimulus features such as orientation preference (OP), ocular dominance (OD), and spatial frequency. It is a long-standing question in theoretical neuroscience whether the observed maps should be interpreted as optima of a specific energy functional that summarizes the design principles of cortical functional architecture. A rigorous evaluation of this optimization hypothesis is particularly demanded by recent evidence that the functional architecture of orientation columns precisely follows species invariant quantitative laws. Because it would be desirable to infer the form of such an optimization principle from the biological data, the optimization approach to explain cortical functional architecture raises the following questions: i) What are the genuine ground states of candidate energy functionals and how can they be calculated with precision and rigor? ii) How do differences in candidate optimization principles impact on the predicted map structure and conversely what can be learned about a hypothetical underlying optimization principle from observations on map structure? iii) Is there a way to analyze the coordinated organization of cortical maps predicted by optimization principles in general? To answer these questions we developed a general dynamical systems approach to the combined optimization of visual cortical maps of OP and another scalar feature such as OD or spatial frequency preference. From basic symmetry assumptions we obtain a comprehensive phenomenological classification of possible inter-map coupling energies and examine representative examples. We show that each individual coupling energy leads to a different class of OP solutions with different correlations among the maps such that inferences about the optimization principle from map layout appear viable. We systematically assess whether quantitative laws resembling experimental observations can result from the coordinated optimization of orientation columns with other feature maps
Coordinated optimization of visual cortical maps (I) Symmetry-based analysis
In the primary visual cortex of primates and carnivores, functional
architecture can be characterized by maps of various stimulus features such as
orientation preference (OP), ocular dominance (OD), and spatial frequency. It
is a long-standing question in theoretical neuroscience whether the observed
maps should be interpreted as optima of a specific energy functional that
summarizes the design principles of cortical functional architecture. A
rigorous evaluation of this optimization hypothesis is particularly demanded by
recent evidence that the functional architecture of OP columns precisely
follows species invariant quantitative laws. Because it would be desirable to
infer the form of such an optimization principle from the biological data, the
optimization approach to explain cortical functional architecture raises the
following questions: i) What are the genuine ground states of candidate energy
functionals and how can they be calculated with precision and rigor? ii) How do
differences in candidate optimization principles impact on the predicted map
structure and conversely what can be learned about an hypothetical underlying
optimization principle from observations on map structure? iii) Is there a way
to analyze the coordinated organization of cortical maps predicted by
optimization principles in general? To answer these questions we developed a
general dynamical systems approach to the combined optimization of visual
cortical maps of OP and another scalar feature such as OD or spatial frequency
preference.Comment: 90 pages, 16 figure
A new invariant on hyperbolic Dehn surgery space
In this paper we define a new invariant of the incomplete hyperbolic
structures on a 1-cusped finite volume hyperbolic 3-manifold M, called the
ortholength invariant. We show that away from a (possibly empty) subvariety of
excluded values this invariant both locally parameterises equivalence classes
of hyperbolic structures and is a complete invariant of the Dehn fillings of M
which admit a hyperbolic structure. We also give an explicit formula for the
ortholength invariant in terms of the traces of the holonomies of certain loops
in M. Conjecturally this new invariant is intimately related to the boundary of
the hyperbolic Dehn surgery space of M.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-23.abs.htm
Clar's Theory, STM Images, and Geometry of Graphene Nanoribbons
We show that Clar's theory of the aromatic sextet is a simple and powerful
tool to predict the stability, the \pi-electron distribution, the geometry, the
electronic/magnetic structure of graphene nanoribbons with different hydrogen
edge terminations. We use density functional theory to obtain the equilibrium
atomic positions, simulated scanning tunneling microscopy (STM) images, edge
energies, band gaps, and edge-induced strains of graphene ribbons that we
analyze in terms of Clar formulas. Based on their Clar representation, we
propose a classification scheme for graphene ribbons that groups configurations
with similar bond length alternations, STM patterns, and Raman spectra. Our
simulations show how STM images and Raman spectra can be used to identify the
type of edge termination
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