Enumerating periodic patterns of synchrony via finite bidirectional networks

2009 ◽  
Vol 466 (2115) ◽  
pp. 891-910
Author(s):  
Ana Paula S. Dias ◽  
Eliana Manuel Pinho
Keyword(s):  
Nearest Neighbour ◽  
Periodic Patterns ◽  
Finite Dimensional ◽  
Spatial Periodicity ◽  
Coupled Cell Networks ◽  
Euclidean Lattice ◽  
Lattice Network ◽  
Bidirectional Networks ◽  
The Given ◽  
Cell Networks

Periodic patterns of synchrony are lattice networks whose cells are coloured according to a local rule, or balanced colouring, and such that the overall system has spatial periodicity. These patterns depict the finite-dimensional flow-invariant subspaces for all the lattice dynamical systems, in the given lattice network, that exhibit those periods. Previous results relate the existence of periodic patterns of synchrony, in n -dimensional Euclidean lattice networks with nearest neighbour coupling architecture, with that of finite coupled cell networks that follow the same colouring rule and have all the couplings bidirectional. This paper addresses the relation between periodic patterns of synchrony and finite bidirectional coloured networks. Given an n -dimensional Euclidean lattice network with nearest neighbour coupling architecture, and a colouring rule with k colours, we enumerate all the periodic patterns of synchrony generated by a given finite network, or graph. This enumeration is constructive and based on the automorphisms group of the graph.

scholarly journals Coupled cell networks: minimality

PAMM ◽  
2007 ◽  
Vol 7 (1) ◽  
pp. 1030501-1030502
Author(s):  
Manuela A. D. Aguiar ◽  
Ana Paula S. Dias
Keyword(s):  
Coupled Cell Networks ◽  
Cell Networks

GENERALIZED HECKE ALGEBRAS AND C*-COMPLETIONS

2009 ◽  
Vol 20 (01) ◽  
pp. 45-76
Author(s):  
MAGNUS B. LANDSTAD ◽  
NADIA S. LARSEN
Keyword(s):  
Heisenberg Group ◽  
Hecke Algebra ◽  
Continuous Extension ◽  
Finite Range ◽  
Structure And Properties ◽  
Projection Formula ◽  
Dimensional Representation ◽  
Finite Dimensional Representation ◽  
Finite Dimensional ◽  
The Given

For a Hecke pair (G, H) and a finite-dimensional representation σ of H on Vσ with finite range, we consider a generalized Hecke algebra [Formula: see text], which we study by embedding the given Hecke pair in a Schlichting completion (Gσ, Hσ) that comes equipped with a continuous extension σ of Hσ. There is a (non-full) projection [Formula: see text] such that [Formula: see text] is isomorphic to [Formula: see text]. We study the structure and properties of C*-completions of the generalized Hecke algebra arising from this corner realisation, and via Morita–Fell–Rieffel equivalence, we identify, in some cases explicitly, the resulting proper ideals of [Formula: see text]. By letting σ vary, we can compare these ideals. The main focus is on the case with dim σ = 1 and applications include ax + b-groups and the Heisenberg group.

scholarly journals Groups which satisfy a weak form of Poincaré duality

1991 ◽  
Vol 34 (3) ◽  
pp. 463-486
Author(s):  
J. E. Roberts
Keyword(s):  
Finite Rank ◽  
Weak Form ◽  
Homological Dimension ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Soluble Groups ◽  
Finite Dimensional ◽  
Duality Property ◽  
The Given ◽  
Torsion Free

Our main result is that a “restricted Poincaré duality” property with respect to finite dimensional coefficient modules over a field holds for a certain class of groups which includes all soluble groups of finite Hirsch length. This relies on a generalisation to the given class of a module construction by Stammbach; an extension of his result on homological dimension to these groups is given. We also generalise the well-known result that torsion-free soluble groups of finite rank are countable.

scholarly journals On bifurcations in lifts of regular uniform coupled cell networks

2014 ◽  
Vol 470 (2169) ◽  
pp. 20140241 ◽  
Author(s):  
Célia Sofia Moreira
Keyword(s):  
Bifurcation Point ◽  
Cell System ◽  
Point Of View ◽  
Coupled Cell Networks ◽  
Regular Network ◽  
Cell Networks

A lift of a given network is a network that admits the first network as quotient. Assuming that a bifurcation occurs for a coupled cell system consistent with the structure of a regular network (in which all cells have the same type and receive the same number of inputs and all arrows have the same type), it is well known that some lifts exhibit new bifurcating branches of solutions. In this work, we approach this problem restricting attention to uniform networks, that is, networks that have no loops and no multiple arrows. We show that, from the bifurcation point of view, rings and their lifts are special networks. We also prove that generically there are lifts that just exhibit the bifurcating branches determined by the quotient network and, moreover, we identify all generic situations where lifts exist that may exhibit bifurcating branches that do not appear in the quotient itself.

scholarly journals ON A CHARACTERISTIC PROPERTY OF FINITE-DIMENSIONAL BANACH SPACES

2011 ◽  
Vol 53 (3) ◽  
pp. 443-449 ◽  
Author(s):  
ANTONÍN SLAVÍK
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Characteristic Property ◽  
Linear Operators ◽  
Infinite Dimensional ◽  
Counter Example ◽  
Finite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
Dvoretzky’S Theorem ◽  
The Given

AbstractThis paper is inspired by a counter example of J. Kurzweil published in [5], whose intention was to demonstrate that a certain property of linear operators on finite-dimensional spaces need not be preserved in infinite dimension. We obtain a stronger result, which says that no infinite-dimensional Banach space can have the given property. Along the way, we will also derive an interesting proposition related to Dvoretzky's theorem.

scholarly journals SPECTRUM OF THE LOOP TRANSFER MATRIX ON FINITE LATTICES

2001 ◽  
Vol 16 (16) ◽  
pp. 1069-1077 ◽  
Author(s):  
GEORGIOS DASKALAKIS ◽  
GEORGE K. SAVVIDY
Keyword(s):  
Transfer Matrix ◽  
Phase Structure ◽  
Extrinsic Curvature ◽  
Critical Behaviour ◽  
Equivalent Representation ◽  
Random Surfaces ◽  
Finite Dimensional ◽  
Curvature Term ◽  
Finite Lattices ◽  
Euclidean Lattice

We consider a model of random surfaces with extrinsic curvature term embedded into 3-D Euclidean lattice Z3. On a 3-D Euclidean lattice it has an equivalent representation in terms of the transfer matrix K(Qi, Qf), which describes the propagation of the loops Q. We study the spectrum of the transfer matrix K(Qi, Qf) on finite-dimensional lattices. The renormalisation group technique is used to investigate the phase structure of the model and its critical behaviour.

scholarly journals Patterns of Synchrony in Coupled Cell Networks with Multiple Arrows

2005 ◽  
Vol 4 (1) ◽  
pp. 78-100 ◽  
Author(s):  
Martin Golubitsky ◽  
Ian Stewart ◽  
Andrei Török
Keyword(s):  
Coupled Cell Networks ◽  
Cell Networks

scholarly journals NONCOMMUTATIVE IMAGES OF COMMUTATIVE SPECTRA

2008 ◽  
Vol 07 (05) ◽  
pp. 535-552
Author(s):  
EDWARD S. LETZTER
Keyword(s):  
Semisimple Lie Groups ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Affine Spaces ◽  
Weyl Algebras ◽  
Finite Dimensional ◽  
Noncommutative Algebras ◽  
Continuous Surjection ◽  
The Given ◽  
Natural Way

We initiate a unified, axiomatic study of noncommutative algebras R whose prime spectra are, in a natural way, finite unions of commutative noetherian spectra. Our results illustrate how these commutative spectra can be functorially "sewn together" to form Spec R. In particular, we construct a bimodule-determined functor Mod Z → Mod R, for a suitable commutative noetherian ring Z, from which there follows a finite-to-one, continuous surjection Spec Z → Spec R. Algebras satisfying the given axiomatic framework include PI algebras finitely generated over fields, noetherian PI algebras, enveloping algebras of complex finite dimensional solvable Lie algebras, standard generic quantum semisimple Lie groups, quantum affine spaces, quantized Weyl algebras, and standard generic quantizations of the coordinate ring of n × n matrices. In all of these examples (except for the non-finitely-generated noetherian PI algebras), Z is finitely generated over a field, and the constructed map of spectra restricts to a surjection Max Z → Prim R.

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