The Multiplicities of a Dual-thin Q-polynomial Association Scheme

Let Y=(X,{Ri}1≤i≤D) denote a symmetric association scheme, and assume that Y is Q-polynomial with respect to an ordering E0,...,ED of the primitive idempotents. Bannai and Ito conjectured that the associated sequence of multiplicities mi (0≤i≤D) of Yis unimodal. Talking to Terwilliger, Stanton made the related conjecture that mi≤mi+1 and mi≤mD−i for i\u3cD/2. We prove that if Y is dual-thin in the sense of Terwilliger, then the Stanton conjecture is true

Kernels of Directed Graph Laplacians

Let G denote a directed graph with adjacency matrix Q and in- degree matrix D. We consider the Kirchhoff matrix L = D − Q, sometimes referred to as the directed Laplacian. A classical result of Kirchhoff asserts that when G is undirected, the multiplicity of the eigenvalue 0 equals the number of connected components of G. This fact has a meaningful generalization to directed graphs, as was observed by Chebotarev and Agaev in 2005. Since this result has many important applications in the sciences, we offer an independent and self-contained proof of their theorem, showing in this paper that the algebraic and geometric multiplicities of 0 are equal, and that a graph-theoretic property determines the dimension of this eigenspace--namely, the number of reaches of the directed graph. We also extend their results by deriving a natural basis for the corresponding eigenspace. The results are proved in the general context of stochastic matrices, and apply equally well to directed graphs with non-negative edge weights

Shift-Symmetric Configurations in Two-Dimensional Cellular Automata: Irreversibility, Insolvability, and Enumeration

The search for symmetry as an unusual yet profoundly appealing phenomenon, and the origin of regular, repeating configuration patterns have been for a long time a central focus of complexity science, and physics. Here, we introduce group-theoretic concepts to identify and enumerate the symmetric inputs, which result in irreversible system behaviors with undesired effects on many computational tasks. The concept of so-called configuration shift-symmetry is applied on two-dimensional cellular automata as an ideal model of computation. The results show the universal insolvability of “non-symmetric” tasks regardless of the transition function. By using a compact enumeration formula and bounding the number of shift-symmetric configurations for a given lattice size, we efficiently calculate how likely a configuration randomly generated from a uniform or density-uniform distribution turns shift-symmetric. Further, we devise an algorithm detecting the presence of shift-symmetry in a configuration. The enumeration and probability formulas can directly help to lower the minimal expected error for many crucial (non-symmetric) distributed problems, such as leader election, edge detection, pattern recognition, convex hull/minimum bounding rectangle, and encryption. Besides cellular automata, the shift-symmetry analysis can be used to study the non-linear behavior in various synchronous rule-based systems that include inference engines, Boolean networks, neural networks, and systolic arrays

Cycle Structures of Orthomorphisms Extending Partial Orthomorphisms of Boolean Groups

A partial orthomorphism of a group GG (with additive notation) is an injection π:S→G for some S⊆G such that π(x)−x ≠ π(y) for all distinct x,y∈S. We refer to |S| as the size of π, and if S=G, then π is an orthomorphism. Despite receiving a fair amount of attention in the research literature, many basic questions remain concerning the number of orthomorphisms of a given group, and what cycle types these permutations have. It is known that conjugation by automorphisms of G forms a group action on the set of orthomorphisms of G. In this paper, we consider the additive group of binary n-tuples, Z, where we extend this result to include conjugation by translations in Z2n, where we extend this result to include conjugation by translations in Z2n and related compositions. We apply these results to show that, for any integer n\u3e1, the distribution of cycle types of orthomorphisms of the group Z2n that extend any given partial orthomorphism of size two is independent of the particular partial orthomorphism considered. A similar result holds for size one. We also prove that the corresponding result does not hold for orthomorphisms extending partial orthomorphisms of size three, and we give a bound on the number of cycle-type distributions for the case of size three. As a consequence of these results, we find that all partial orthomorphisms of Z2n of size two can be extended to complete orthomorphisms

The Multiplicities of a Dual-thin Q-polynomial Association Scheme

. Let Y = (X; fR i g 0iD ) denote a symmetric association scheme, and assume that Y is Q-polynomial with respect to an ordering E 0 ; :::; ED of the primitive idempotents. In [1, p.205], Bannai and Ito conjectured that the associated sequence of multiplicities m i (0 i D) of Y is unimodal. Stanton [7] made the related conjecture that m i m i+1 and m i mD\Gammai for i ! D=2. We prove that if Y is dual-thin in the sense of Terwilliger, then the Stanton conjecture is true. 1 Introduction For a general introduction to association schemes, we refer to [1], [2], [5], or [8]. Our notation follows that found in [3]. Throughout this article, Y = (X; fR i g 0iD ) will denote a symmetric, D- class association scheme. Our point of departure is the following well-known result of Taylor and Levingston. 1.1 Theorem. [6] If Y is P -polynomial with respect to an ordering R 0 ; :::; RD of the associate classes, then the corresponding sequence of valencies k 0 ; k 1 ; : : : ; kD is unimodal. Fur..

Decentralized Control of Vehicle Formations

This paper investigates a method for decentralized stabilization of vehicle formations using techniques from algebraic graph theory. The vehicles exchange information according to a pre-specified communication digraph, G. A feedback control is designed using relative information between a vehicle and its in-neighbors in G. We prove that a necessary and sufficient condition for an appropriate decentralized linear stabilizing feedback to exist is that G has a rooted directed spanning tree. We show the direct relationship between the rate of convergence to formation and the eigenvalues of the (directed) Laplacian of G. Various special situations are discussed, including symmetric communication graphs and formations with leaders. Several numerical simulations are used to illustrate the results

Flocks and Formations

Given a large number (the “flock”) of moving physical objects, we investigate physically reasonable mechanisms of influencing their orbits in such a way that they move along a prescribed course and in a prescribed and fixed configuration (or “in formation”). Each agent is programmed to see the position and velocity of a certain number of others. This flow of information from one agent to another defines a fixed directed (loopless) graph in which the agents are represented by the vertices. This graph is called the communication graph. To be able to fly in formation, an agent tries to match the mean position and velocity of his neighbors (his direct antecedents on the communication graph) to his own. This operation defines a (directed) Laplacian on the communication graph. A linear feedback is used to ensure stability of the coherent flight patterns. We analyze in detail how the connectedness of the communication graph affects the coherence of the stable flight patterns and give a characterization of these stable flight patterns. We do the same if in addition the flight of the flock is guided by one or more leaders. Finally we use this theory to develop some applications. Examples of these are: flight guided by external controls, flocks of flocks, and some results about flocks whose formation is always oriented along the line of flight (such as geese)