Colourings of cubic graphs inducing isomorphic monochromatic subgraphs

A $k$-bisection of a bridgeless cubic graph $G$ is a $2$-colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes (monochromatic components in what follows) have order at most $k$. Ban and Linial conjectured that every bridgeless cubic graph admits a $2$-bisection except for the Petersen graph. A similar problem for the edge set of cubic graphs has been studied: Wormald conjectured that every cubic graph $G$ with $|E(G)| \equiv 0 \pmod 2$ has a $2$-edge colouring such that the two monochromatic subgraphs are isomorphic linear forests (i.e. a forest whose components are paths). Finally, Ando conjectured that every cubic graph admits a bisection such that the two induced monochromatic subgraphs are isomorphic. In this paper, we give a detailed insight into the conjectures of Ban-Linial and Wormald and provide evidence of a strong relation of both of them with Ando's conjecture. Furthermore, we also give computational and theoretical evidence in their support. As a result, we pose some open problems stronger than the above mentioned conjectures. Moreover, we prove Ban-Linial's conjecture for cubic cycle permutation graphs. As a by-product of studying $2$-edge colourings of cubic graphs having linear forests as monochromatic components, we also give a negative answer to a problem posed by Jackson and Wormald about certain decompositions of cubic graphs into linear forests.Comment: 33 pages; submitted for publicatio

On Metric Dimension of Functigraphs

The \emph{metric dimension} of a graph $G$, denoted by $\dim(G)$, is the minimum number of vertices such that each vertex is uniquely determined by its distances to the chosen vertices. Let $G_1$ and $G_2$ be disjoint copies of a graph $G$ and let $f: V(G_1) \rightarrow V(G_2)$ be a function. Then a \emph{functigraph} $C(G, f)=(V, E)$ has the vertex set $V=V(G_1) \cup V(G_2)$ and the edge set $E=E(G_1) \cup E(G_2) \cup \{uv \mid v=f(u)\}$. We study how metric dimension behaves in passing from $G$ to $C(G,f)$ by first showing that $2 \le \dim(C(G, f)) \le 2n-3$, if $G$ is a connected graph of order $n \ge 3$ and $f$ is any function. We further investigate the metric dimension of functigraphs on complete graphs and on cycles.Comment: 10 pages, 7 figure

Speed of synchronization in complex networks of neural oscillators Analytic results based on Random Matrix Theory

We analyze the dynamics of networks of spiking neural oscillators. First, we present an exact linear stability theory of the synchronous state for networks of arbitrary connectivity. For general neuron rise functions, stability is determined by multiple operators, for which standard analysis is not suitable. We describe a general non-standard solution to the multi-operator problem. Subsequently, we derive a class of rise functions for which all stability operators become degenerate and standard eigenvalue analysis becomes a suitable tool. Interestingly, this class is found to consist of networks of leaky integrate and fire neurons. For random networks of inhibitory integrate-and-fire neurons, we then develop an analytical approach, based on the theory of random matrices, to precisely determine the eigenvalue distribution. This yields the asymptotic relaxation time for perturbations to the synchronous state which provides the characteristic time scale on which neurons can coordinate their activity in such networks. For networks with finite in-degree, i.e. finite number of presynaptic inputs per neuron, we find a speed limit to coordinating spiking activity: Even with arbitrarily strong interaction strengths neurons cannot synchronize faster than at a certain maximal speed determined by the typical in-degree.Comment: 17 pages, 12 figures, submitted to Chao

Divergence in Dialogue

Copyright: 2014 Healey et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.This work was supported by the Economic and Social Research Council (ESRC; http://www.esrc.ac.uk/) through the DynDial project (Dynamics of Conversational Dialogue, RES-062-23-0962) and the Engineering and Physical Sciences Research Council (EPSRC; http://www.epsrc.ac.uk/) through the RISER project (Robust Incremental Semantic Resources for Dialogue, EP/J010383/1). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript

Gully Formation at the Haughton Impact Structure (Arctic Canada) Through the Melting of Snow and Ground Ice, with Implications for Gully Formation on Mars

The formation of gullies on Mars has been the topic of active debate and scientific study since their first discovery by Malin and Edgett in 2000. Several mechanisms have been proposed to account for gully formation on Mars, from dry mass movement processes, release of water or brine from subsurface aquifers, and the melting of near-surface ground ice or snowpacks. In their global documentation of martian gullies, report that gullies are confined to ~2783S and ~2872N latitudes and span all longitudes. Gullies on Mars have been documented on impact crater walls and central uplifts, isolated massifs, and on canyon walls, with crater walls being the most common situation. In order to better understand gully formation on Mars, we have been conducting field studies in the Canadian High Arctic over the past several summers, most recently in summer 2018 and 2019 under the auspices of the Canadian Space Agency-funded Icy Mars Analogue Program. It is notable that the majority of previous studies in the Arctic and Antarctica, including our recent work on Devon Island, have focused on gullies formed on slopes generated by regular endogenic geological processes and in regular bedrock. How-ever, as noted above, meteorite impact craters are the most dominant setting for gullies on Mars. Impact craters provide an environment with diverse lithologies including impact-generated and impact-modified rocks and slope angle, and thus greatly variable hill slope processes could occur within a localized area. Here, we investigate the formation of gullies within the Haughton impact structure and compare them to gullies formed in unimpacted target rock in the nearby Thomas Lee Inle

On Graph-Theoretic Identifications of Adinkras, Supersymmetry Representations and Superfields

In this paper we discuss off-shell representations of N-extended supersymmetry in one dimension, ie, N-extended supersymmetric quantum mechanics, and following earlier work on the subject codify them in terms of certain graphs, called Adinkras. This framework provides a method of generating all Adinkras with the same topology, and so also all the corresponding irreducible supersymmetric multiplets. We develop some graph theoretic techniques to understand these diagrams in terms of a relatively small amount of information, namely, at what heights various vertices of the graph should be "hung". We then show how Adinkras that are the graphs of N-dimensional cubes can be obtained as the Adinkra for superfields satisfying constraints that involve superderivatives. This dramatically widens the range of supermultiplets that can be described using the superspace formalism and organizes them. Other topologies for Adinkras are possible, and we show that it is reasonable that these are also the result of constraining superfields using superderivatives. The family of Adinkras with an N-cubical topology, and so also the sequence of corresponding irreducible supersymmetric multiplets, are arranged in a cyclical sequence called the main sequence. We produce the N=1 and N=2 main sequences in detail, and indicate some aspects of the situation for higher N.Comment: LaTeX, 58 pages, 52 illustrations in color; minor typos correcte

Bend it like Beckham: embodying the motor skills of famous athletes.

Observing an action activates the same representations as does the actual performance of the action. Here we show for the first time that the action system can also be activated in the complete absence of action perception. When the participants had to identify the faces of famous athletes, the responses were influenced by their similarity to the motor skills of the athletes. Thus, the motor skills of the viewed athletes were retrieved automatically during person identification and had a direct influence on the action system of the observer. However, our results also indicated that motor behaviours that are implicit characteristics of other people are represented differently from when actions are directly observed. That is, unlike the facilitatory effects reported when actions were seen, the embodiment of the motor behaviour that is not concurrently perceived gave rise to contrast effects where responses similar to the behaviour of the athletes were inhibited