On Enumeration of Conjugacy Classes of Coxeter Elements

In this paper we study the equivalence relation on the set of acyclic orientations of a graph Y that arises through source-to-sink conversions. This source-to-sink conversion encodes, e.g. conjugation of Coxeter elements of a Coxeter group. We give a direct proof of a recursion for the number of equivalence classes of this relation for an arbitrary graph Y using edge deletion and edge contraction of non-bridge edges. We conclude by showing how this result may also be obtained through an evaluation of the Tutte polynomial as T(Y,1,0), and we provide bijections to two other classes of acyclic orientations that are known to be counted in the same way. A transversal of the set of equivalence classes is given.Comment: Added a few results about connections to the Tutte polynomia

Equivalences on Acyclic Orientations

The cyclic and dihedral groups can be made to act on the set Acyc(Y) of acyclic orientations of an undirected graph Y, and this gives rise to the equivalence relations ~kappa and ~delta, respectively. These two actions and their corresponding equivalence classes are closely related to combinatorial problems arising in the context of Coxeter groups, sequential dynamical systems, the chip-firing game, and representations of quivers. In this paper we construct the graphs C(Y) and D(Y) with vertex sets Acyc(Y) and whose connected components encode the equivalence classes. The number of connected components of these graphs are denoted kappa(Y) and delta(Y), respectively. We characterize the structure of C(Y) and D(Y), show how delta(Y) can be derived from kappa(Y), and give enumeration results for kappa(Y). Moreover, we show how to associate a poset structure to each kappa-equivalence class, and we characterize these posets. This allows us to create a bijection from Acyc(Y)/~kappa to the union of Acyc(Y')/~kappa and Acyc(Y'')/~kappa, Y' and Y'' denote edge deletion and edge contraction for a cycle-edge in Y, respectively, which in turn shows that kappa(Y) may be obtained by an evaluation of the Tutte polynomial at (1,0).Comment: The original paper was extended, reorganized, and split into two papers (see also arXiv:0802.4412

Cycle Equivalence of Graph Dynamical Systems

Graph dynamical systems (GDSs) can be used to describe a wide range of distributed, nonlinear phenomena. In this paper we characterize cycle equivalence of a class of finite GDSs called sequential dynamical systems SDSs. In general, two finite GDSs are cycle equivalent if their periodic orbits are isomorphic as directed graphs. Sequential dynamical systems may be thought of as generalized cellular automata, and use an update order to construct the dynamical system map. The main result of this paper is a characterization of cycle equivalence in terms of shifts and reflections of the SDS update order. We construct two graphs C(Y) and D(Y) whose components describe update orders that give rise to cycle equivalent SDSs. The number of components in C(Y) and D(Y) is an upper bound for the number of cycle equivalence classes one can obtain, and we enumerate these quantities through a recursion relation for several graph classes. The components of these graphs encode dynamical neutrality, the component sizes represent periodic orbit structural stability, and the number of components can be viewed as a system complexity measure

Coxeter Groups and Asynchronous Cellular Automata

The dynamics group of an asynchronous cellular automaton (ACA) relates properties of its long term dynamics to the structure of Coxeter groups. The key mathematical feature connecting these diverse fields is involutions. Group-theoretic results in the latter domain may lead to insight about the dynamics in the former, and vice-versa. In this article, we highlight some central themes and common structures, and discuss novel approaches to some open and open-ended problems. We introduce the state automaton of an ACA, and show how the root automaton of a Coxeter group is essentially part of the state automaton of a related ACA.Comment: 10 pages, 4 figure

Order Independence in Asynchronous Cellular Automata

A sequential dynamical system, or SDS, consists of an undirected graph Y, a vertex-indexed list of local functions F_Y, and a permutation pi of the vertex set (or more generally, a word w over the vertex set) that describes the order in which these local functions are to be applied. In this article we investigate the special case where Y is a circular graph with n vertices and all of the local functions are identical. The 256 possible local functions are known as Wolfram rules and the resulting sequential dynamical systems are called finite asynchronous elementary cellular automata, or ACAs, since they resemble classical elementary cellular automata, but with the important distinction that the vertex functions are applied sequentially rather than in parallel. An ACA is said to be pi-independent if the set of periodic states does not depend on the choice of pi, and our main result is that for all n>3 exactly 104 of the 256 Wolfram rules give rise to a pi-independent ACA. In 2005 Hansson, Mortveit and Reidys classified the 11 symmetric Wolfram rules with this property. In addition to reproving and extending this earlier result, our proofs of pi-independence also provide significant insight into the dynamics of these systems.Comment: 18 pages. New version distinguishes between functions that are pi-independent but not w-independen

Posets from Admissible Coxeter Sequences

We study the equivalence relation on the set of acyclic orientations of an undirected graph G generated by source-to-sink conversions. These conversions arise in the contexts of admissible sequences in Coxeter theory, quiver representations, and asynchronous graph dynamical systems. To each equivalence class we associate a poset, characterize combinatorial properties of these posets, and in turn, the admissible sequences. This allows us to construct an explicit bijection from the equivalence classes over G to those over G' and G", the graphs obtained from G by edge deletion and edge contraction of a fixed cycle-edge, respectively. This bijection yields quick and elegant proofs of two non-trivial results: (i) A complete combinatorial invariant of the equivalence classes, and (ii) a solution to the conjugacy problem of Coxeter elements for simply-laced Coxeter groups. The latter was recently proven by H. Eriksson and K. Eriksson using a much different approach.Comment: 16 pages, 4 figures. Several examples have been adde

Dynamics Groups of Asynchronous Cellular Automata

We say that a finite asynchronous cellular automaton (or more generally, any sequential dynamical system) is pi-independent if its set of periodic points are independent of the order that the local functions are applied. In this case, the local functions permute the periodic points, and these permutations generate the dynamics group. We have previously shown that exactly 104 of the possible 256 cellular automaton rules are pi-independent. In this article, we classify the periodic states of these systems and describe their dynamics groups, which are quotients of Coxeter groups. The dynamics groups provide information about permissible dynamics as a function of update sequence and, as such, connect discrete dynamical systems, group theory, and algebraic combinatorics in a new and interesting way. We conclude with a discussion of numerous open problems and directions for future research.Comment: Revised per referee's comment

The role of attenuated astrocyte activation in infantile neuronal Ceroid Lipofuscinosis

Infantile neuronal ceroid lipofuscinosis (INCL) is an inherited neurodegenerative disorder affecting the CNS during infancy. INCL is caused by mutations in the CLN1 gene that leads to a deficiency in the lysosomal hydrolase, palmitoyl protein thioesterase (PPT1). A murine model of INCL, the PPT1(−/−) mouse, is an accurate phenocopy of the human disease. The first pathological change observed in the PPT1(−/−) brain is regional areas of GFAP upregulation, which predicts future areas of neurodegeneration. We hypothesized that preventing GFAP upregulation in reactive astrocytes will alter the CNS disease. To test this hypothesis, we generated mice simultaneously carrying null mutations in the GFAP, Vimentin, and PPT1 genes (GFAP(−/−)Vimentin(−/−)PPT1(−/−)). Although the clinical and pathological features of the GFAP(−/−)Vim(−/−)PPT1(−/−) mice are similar to INCL, the disease appears earlier and progresses more rapidly. One mechanism underlying this accelerated phenotype is a profound neuroinflammatory response within the CNS. Thus, our data identify a protective role for intermediate filament upregulation during astrocyte activation in INCL, a model of chronic neurodegeneration

Editorial: Addressing roles for glycans in immunology using chemical biology

Glycoconjugates, macromolecules containing carbohydrates (glycans) conjugated to proteins or lipids, are a diverse class of biopolymers capable of regulating cell-cell interactions. They are present in a high, natural heterogeneity, which originates from the complex mechanisms involved in their biosynthesis. Genetic and environmental factors determine the ensemble of glycans on any particular cell type, in a non-template encoded manner. As a consequence, the cell surface glycan profile provides a tightly-regulated temporal and spatial signature containing crucial biological information. This information is translated into biological functions by glycan binding proteins (GBPs), also called lectins. Importantly, our immune system is modulated by three major GBP families: C-type lectins, galectins, and Siglecs. The abilities of these GBPs to modulate immune cell function is intimately connected to their ability to differentiate ?self? or ?non-self? glycans from our own cells or pathogens, respectively. Hence, GBP?glycan interactions are critical mediators in immune cell homeostasis. Genetic manipulation of glycan processing enzymes has shed light on the roles of glycans in pathologies such as autoimmune diseases and cancer. However, genetic tools such as genomic manipulation and transgenic animal models have shown to be insufficient to fully untangle the roles of GBP-glycan interactions. Accordingly, recent advances in our understanding of GBPs and how they control immune cell function via glycan recognition has been driven by the development of chemical tools.In this Research Topic, we explore recent work illuminating the various roles of glycans and/or GBPs in controlling immune cell function with special emphasis placed on chemical biology approaches that have been instrumental in such efforts. Potential subjects covered may include:? Immunological roles of Glycan-binding proteins? Glycans as immunomodulators? Development of ligands to probe glycan-binding proteins? Chemical biology approaches to modulate glycan-binding proteins and their glycan ligands? Glycans and synthetic derivatives as novel adjuvants? Glycan-based targeted delivery? Intracellular glycosylation in immune cells? Tissue homing of immune cells mediated by glycans? Glycolipid presentation to immune cells? Glycan-based vaccines? Analytical methods for functional characterization of lectin-glycan interactionsFil: Macauley, Matthew S.. University of Alberta; CanadáFil: Rademacher, Christoph. Max Planck Institute of Colloids and Interfaces; AlemaniaFil: Mariño, Karina Valeria. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto de Biología y Medicina Experimental. Fundación de Instituto de Biología y Medicina Experimental. Instituto de Biología y Medicina Experimental; Argentin