35054660
The Ultimate Categorical Independence Ratio of a Graph
Let $\beta(G)$ denote the independence number of a graph $G$. We introduce $A(G) = \lim_{k \rightarrow \infty}\beta(G^{k})/|V(G)|^{k}$, where the categorical graph product is used. This limit, surprisingly, lies in the range $(0,1/2]\cup\{1\}$. We can show that this limit can take any such rational number, but is there any $G$ for which $A(G)$ is irrational? A useful technique for bounding $A(G)$ is to consider special spanning subgraphs. These bounds allow us to efficiently compute $A(G)$ for many $G$. We give a condition which if true for $G$ shows that $A(G)>\beta(G)/|V(G)|$. This brings up the question: for which $G$ does $A(G)=\beta(G)/|V(G)|$? This happens if $G$ is a Cayley graph of an Abelian group or if $G$ is a connected graph that has an automorphism which has a single orbit.
44284809
The strongly connected reliability of complete digraphs
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.
40600226
The Complexity of Generalized Graph Colorings
Given a graph property P and positive integer k, a P k-coloring of a graph G is an assignment of one of k colors to each vertex of the graph so that the subgraphs induced by each color class have property P. This notion generalizes the standard definition of graph coloring, and has been investigated for many properties. We consider here the complexity of the decision problem. In particular, for the property —G, of not containing an induced subgraph isomorphic to G, we conjecture (and provide strong evidence) that —G k-colorability is NP-complete whenever G has order at least 3 and k ⩾ 2. The techniques rely on new NP-completeness results for hypergraph colorings.
18896340
On the Roots of Chromatic Polynomials
It is proved that the chromatic polynomial of a connected graph with n vertices and m edges has a root with modulus at least (m?1)/(n?2); this bound is best possible for trees and 2-trees (only). It is also proved that the chromatic polynomial of a graph with few triangles that is not a forest has a nonreal root and that there is a graph with n vertices whose chromatic polynomial has a root with imaginary part greater thann/4.
28649866
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 .
19128559
Bounding the Roots of Independence Polynomials
645538
Plant snoRNA database
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.
7956229
How laminar frontal cortex and basal ganglia circuits interact to control planned and reactive saccades
How does the brain learn to balance between reactive and planned behaviors? The basal ganglia (BG) and frontal cortex together allow animals to learn planned behaviors that acquire rewards when prepotent reactive behaviors are insufficient. This paper proposes a new model, called TELOS, to explain how laminar circuitry of the frontal cortex, exemplified by the frontal eye fields, interacts with the BG, thalamus, superior colliculus, and inferotemporal and parietal cortices to learn and perform reactive and planned eye movements. The model is formulated as fourteen computational hypotheses. These specify how strategy priming and action planning (in cortical layers III, Va and VI) are dissociated from movement execution (in layer Vb), how the BG help to choose among and gate competing plans, and how a visual stimulus may serve either as a movement target or as a discriminative cue to move elsewhere. The direct, indirect and hyperdirect pathways through the BG are shown to enable complex gating functions, including deferred execution of selected plans, and switching among alternative sensory-motor mappings. Notably, the model can learn and gate the use of a What-to-Where transformation that enables spatially invariant object representations to selectively excite spatially coded movement plans. Model simulations show how dopaminergic reward and non-reward signals guide monkeys to learn and perform saccadic eye movements in the fixation, single saccade, overlap, gap, and delay (memory-guided) saccade tasks. Model cell activation dynamics quantitatively simulate seventeen established types of dynamics exhibited by corresponding real cells during performance of these tasks.
15658893
Theories of Causal Ordering
Abstract This paper is a response to Iwasaki and Simon [14] which criticizes de Kleer and Brown [8]. We argue that many of their criticisms, particularly concerning causality, modeling and stability, originate from the difference of concerns between engineering and economics. Our notion of causality arises from considering the interconnections of components, not equations. When no feedback is present, the ordering produced by our qualitative physics is similar to theirs. However, when feedback is present, our qualitative physics determines a causal ordering around feedback loops as well. Causal ordering is a general technique not only applicable to qualitative reasoning. Therefore we also explore the relationship between causal ordering and propagation of constraints upon which the methods of qualitative physics are based.
38979633
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.
205225115
The ribonuclease P database
Ribonuclease P is responsible for the 5'-maturation of tRNA precursors. Ribonuclease P is a ribonucleoprotein, and in bacteria (and some Archaea) the RNA subunit alone is catalytically active in vitro, i.e. it is a ribozyme. The Ribonuclease P Database is a compilation of ribonuclease P sequences, sequence alignments, secondary structures, three-dimensional models and accessory information, available via the World Wide Web at the following URL: http://www.mbio.ncsu.edu/RNaseP/home .html