Bounding the Roots of Independence Polynomials

title: A ramsey type problem concerning vertex colourings
abstract: Let H → k v G denote the fact that for every function π: V ( H ) → {1, …, k } there is an induced subgraph G ′ of H with G ′ ≅ G and V ( G ′) ⊆ π −1 ( i ) for some i . Folkman has shown that for all graphs G and for all positive integers k such a graph H exists. We examine here f ( G , k ), the minimum order of a graph H for which H → k v G . We show that for any fixed integer k ≥ 2 there are positive constants C 1 and C 2 such that C 1 n 2 ≤ max{ f ( G , k ): | V ( G )| = n } ≤ C 2 n 2 log 2 n .

title: The existence of uniquely -G colourable graphs
abstract: Given graphs F and G and a nonnegative integer k , a function π : V ( F ) → 1, …, k is a − G k-colouring of F if no induced copy of is monochromatic; F is − G k-chromatic if F has a − G k -colouring but no − G ( k − 1)-colouring. Further, we say F is uniquely − G k -colourable if F is − G k -chromatic and, up to a permutation of colours, it has only one − G k -colouring. Such notions are extensions of the well-known corresponding definitions from chromatic theory. It was conjectured that for all graphs G of order at least two and all positive integers k there exist uniquely − G k -colourable graphs. We prove the conjecture and show that, in fact, in all cases infinitely many such graphs exist.

title: The strongly connected reliability of complete digraphs
abstract: Given a digraph D, consider the model where each vertex is always operational, but the edges are independently operational with probability p. The strongly connected reliability of D, scRel(D, p), is the probability that the spanning subgraph of D consisting of the operational edges is strongly connected. One can view strongly connected reliability as the probability that any vertex can send information to any other vertex, given that edges fail independently. ∗Partially supported by a grant from NSERC. Communicating author.

title: On chromatic roots with negative real part

title: Average independence polynomials
abstract: The independence polynomial of a graph G is the function i(G, x) = Σ k ≥ 0 ik xk, where ik is the number of independent sets of vertices in G of cardinality k. We investigate here the average independence polynomial, where the average is taken over all independence polynomials of (labeled) graphs of order n. We prove that while almost every independence polynomial has a nonreal root, the average independence polynomials always have all real, simple roots.

title: Foundations of Envisioning
abstract: This paper explores a particular kind of qualitative reasoning, called envisioning, that is capable of producing causal explanations for device behavior. It has been implemented in a computer program, ENVISION, which can analyze a wide variety of thermal, fluid, electrical, translational and rotational devices. Rather than present the technical details of the envisioning process, this paper examines the theoretical foundations upon which it is built. Many of these considerations are ones that any builder of qualitative reasoning systems must pay attention to. Two such considerations are explanation and robustness: What notion of causality is adequate for causal explanations of device behavior? How can there be any confidence in the analysis of a novel device?

title: Plant snoRNA database
abstract: The Plant snoRNA database (http://www.scri.sari.ac.uk/plant_snoRNA/) provides information on small nucleolar RNAs from Arabidopsis and eighteen other plant species. Information includes sequences, expression data, methylation and pseudouridylation target modification sites, initial gene organization (polycistronic, single gene and intronic) and the number of gene variants. The Arabidopsis information is divided into box C/D and box H/ACA snoRNAs, and within each of these groups, by target sites in rRNA, snRNA or unknown. Alignments of orthologous genes and gene variants from different plant species are available for many snoRNA genes. Plant snoRNA genes have been given a standard nomenclature, designed wherever possible, to provide a consistent identity with yeast and human orthologues.

title: A Qualitative Physics Based on Confluences
abstract: A qualitative physics predicts and explains the behavior of mechanisms in qualitative terms. The goals for the qualitative physics are (1) to be far simpler than the classical physics and yet retain all the important distinctions (e.g., state, oscillation, gain, momentum) without invoking the mathematics of continuously varying quantities and differential equations, (2) to produce causal accounts of physical mechanisms that are easy to understand, and (3) to provide the foundations for commonsense models for the next generation of expert systems. This paper presents a fairly encompassing account of qualitative physics. First, we discuss the general subject of naive physics and some of its methodological considerations. Second, we present a framework for modeling the generic behavior of individual components of a device based on the notions of qualitative differential equations (confluences) and qualitative state. This requires developing a qualitative version of the calculus. The modeling primitives induce two kinds of behavior, intrastate and interstate, which are governed by different laws. Third, we present algorithms for determining the behavior of a composite device from the generic behavior of its components. Fourth, we examine a theory of explanation for these predictions based on logical proof. Fifth, we introduce causality as an ontological commitment for explaining how devices behave.

title: Computational and neural mechanisms of task switching
abstract: Switching between tasks that overlap in perceptual and response characteristics is assumed to rely upon the maintenance of task representations in prefrontal cortex (PFC). However, task-switching studies demonstrate ''switch costs,'' even when there is sufficient time to prepare for a new task. These costs suggest that task-switching performance reflects a complex interplay between priming and the updating and maintenance of task representations. We describe a computational model in which this interaction is made explicit and linked to the dynamics of PFC. Simulation results account for a variety of empirical phenomena and predict a double dissociation in lateral PFC that was subsequently identified.

title: G-functions for the hermeneutic circle of evolution
abstract: Evolution by natural selection may include both frequency and density-dependence. Frequency-dependent selection is a kind of hermeneutic circle. As a literary term (and a school of philosophy), a hermeneutic circle describes how each word of a sentence or paragraph simultaneously draws its meaning from its context (the other words) and contributes towards the context. Frequency dependence is like this. Under frequency dependence, the fitness of an individual is not only influenced by its own heritable phenotype (in evolutionary game theory heritable phenotype = strategy) but by the frequency of strategies found among others in the population. And, the fitness accrued by individuals with a particular strategy will in time influence the frequency of strategies in the population. The challenge is to develop conceptual and modeling tools that allow us to determine the outcome of this hermeneutic circle of evolution. We need a fitness concept that defines the fitness of (or payoff to) to individuals as influenced by both their own strategy and the strategies of others. And, this fitness formulation must describe how these payoffs translate into changes in strategy frequencies. The fitness generating function, or G-function, provides this service for evolutionary game theory.