Optimal strategies for a game on amenable semigroups

The semigroup game is a two-person zero-sum game defined on a semigroup S as follows: Players 1 and 2 choose elements x and y in S, respectively, and player 1 receives a payoff f(xy) defined by a function f from S to [-1,1]. If the semigroup is amenable in the sense of Day and von Neumann, one can extend the set of classical strategies, namely countably additive probability measures on S, to include some finitely additive measures in a natural way. This extended game has a value and the players have optimal strategies. This theorem extends previous results for the multiplication game on a compact group or on the positive integers with a specific payoff. We also prove that the procedure of extending the set of allowed strategies preserves classical solutions: if a semigroup game has a classical solution, this solution solves also the extended game.Comment: 17 pages. To appear in International Journal of Game Theor

Quantum entanglement between a nonlinear nanomechanical resonator and a microwave field

We consider a theoretical model for a nonlinear nanomechanical resonator coupled to a superconducting microwave resonator. The nanomechanical resonator is driven parametrically at twice its resonance frequency, while the superconducting microwave resonator is driven with two tones that differ in frequency by an amount equal to the parametric driving frequency. We show that the semi-classical approximation of this system has an interesting fixed point bifurcation structure. In the semi-classical dynamics a transition from stable fixed points to limit cycles is observed as one moves from positive to negative detuning. We show that signatures of this bifurcation structure are also present in the full dissipative quantum system and further show that it leads to mixed state entanglement between the nanomechanical resonator and the microwave cavity in the dissipative quantum system that is a maximum close to the semi-classical bifurcation. Quantum signatures of the semi-classical limit-cycles are presented.Comment: 36 pages, 18 figure

Emergence of Symmetry in Complex Networks

Many real networks have been found to have a rich degree of symmetry, which is a very important structural property of complex network, yet has been rarely studied so far. And where does symmetry comes from has not been explained. To explore the mechanism underlying symmetry of the networks, we studied statistics of certain local symmetric motifs, such as symmetric bicliques and generalized symmetric bicliques, which contribute to local symmetry of networks. We found that symmetry of complex networks is a consequence of similar linkage pattern, which means that nodes with similar degree tend to share similar linkage targets. A improved version of BA model integrating similar linkage pattern successfully reproduces the symmetry of real networks, indicating that similar linkage pattern is the underlying ingredient that responsible for the emergence of the symmetry in complex networks.Comment: 7 pages, 7 figure

The Complexity of the Empire Colouring Problem

We investigate the computational complexity of the empire colouring problem (as defined by Percy Heawood in 1890) for maps containing empires formed by exactly $r > 1$ countries each. We prove that the problem can be solved in polynomial time using $s$ colours on maps whose underlying adjacency graph has no induced subgraph of average degree larger than $s/r$. However, if $s \geq 3$, the problem is NP-hard even if the graph is a forest of paths of arbitrary lengths (for any $r \geq 2$, provided $s < 2r - \sqrt(2r + 1/4+ 3/2)$. Furthermore we obtain a complete characterization of the problem's complexity for the case when the input graph is a tree, whereas our result for arbitrary planar graphs fall just short of a similar dichotomy. Specifically, we prove that the empire colouring problem is NP-hard for trees, for any $r \geq 2$, if $3 \leq s \leq 2r-1$ (and polynomial time solvable otherwise). For arbitrary planar graphs we prove NP-hardness if $s<7$ for $r=2$, and $s < 6r-3$, for $r \geq 3$. The result for planar graphs also proves the NP-hardness of colouring with less than 7 colours graphs of thickness two and less than $6r-3$ colours graphs of thickness $r \geq 3$.Comment: 23 pages, 12 figure

Crystal constructions in Number Theory

Weyl group multiple Dirichlet series and metaplectic Whittaker functions can be described in terms of crystal graphs. We present crystals as parameterized by Littelmann patterns and we give a survey of purely combinatorial constructions of prime power coefficients of Weyl group multiple Dirichlet series and metaplectic Whittaker functions using the language of crystal graphs. We explore how the branching structure of crystals manifests in these constructions, and how it allows access to some intricate objects in number theory and related open questions using tools of algebraic combinatorics

Vertex labeling and routing in expanded Apollonian networks

We present a family of networks, expanded deterministic Apollonian networks, which are a generalization of the Apollonian networks and are simultaneously scale-free, small-world, and highly clustered. We introduce a labeling of their vertices that allows to determine a shortest path routing between any two vertices of the network based only on the labels.Comment: 16 pages, 2 figure

On the problem of reconstructing a tournament from subtournaments

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41629/1/605_2005_Article_BF01299955.pd

Hamiltonian submanifolds of regular polytopes

We investigate polyhedral $2k$-manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex {\it $k$-Hamiltonian} if it contains the full $k$-skeleton of the polytope. Since the case of the cube is well known and since the case of a simplex was also previously studied (these are so-called {\it super-neighborly triangulations}) we focus on the case of the cross polytope and the sporadic regular 4-polytopes. By our results the existence of 1-Hamiltonian surfaces is now decided for all regular polytopes. Furthermore we investigate 2-Hamiltonian 4-manifolds in the $d$-dimensional cross polytope. These are the "regular cases" satisfying equality in Sparla's inequality. In particular, we present a new example with 16 vertices which is highly symmetric with an automorphism group of order 128. Topologically it is homeomorphic to a connected sum of 7 copies of $S^2 \times S^2$. By this example all regular cases of $n$ vertices with $n < 20$ or, equivalently, all cases of regular $d$-polytopes with $d\leq 9$ are now decided.Comment: 26 pages, 4 figure

Sometimes Sperm Whales (Physeter macrocephalus) Cannot Find Their Way Back to the High Seas: A Multidisciplinary Study on a Mass Stranding

BACKGROUND: Mass strandings of sperm whales (Physeter macrocephalus) remain peculiar and rather unexplained events, which rarely occur in the Mediterranean Sea. Solar cycles and related changes in the geomagnetic field, variations in water temperature and weather conditions, coast geographical features and human activities have been proposed as possible causes. In December 2009, a pod of seven male sperm whales stranded along the Adriatic coast of Southern Italy. This is the sixth instance from 1555 in this basin. METHODOLOGY/PRINCIPAL FINDINGS: Complete necropsies were performed on three whales whose bodies were in good condition, carrying out on sampled tissues histopathology, virology, bacteriology, parasitology, and screening of veins looking for gas emboli. Furthermore, samples for age determination, genetic studies, gastric content evaluation, stable isotopes and toxicology were taken from all the seven specimens. The animals were part of the same group and determined by genetic and photo-identification to be part of the Mediterranean population. Causes of death did not include biological agents, or the "gas and fat embolic syndrome", associated with direct sonar exposure. Environmental pollutant tissue concentrations were relatively high, in particular organochlorinated xenobiotics. Gastric content and morphologic tissue examinations showed a prolonged starvation, which likely caused, at its turn, the mobilization of lipophilic contaminants from the adipose tissue. Chemical compounds subsequently entered the blood circulation and may have impaired immune and nervous functions. CONCLUSIONS/SIGNIFICANCE: A multi-factorial cause underlying this sperm whales' mass stranding is proposed herein based upon the results of postmortem investigations as well as of the detailed analyses of the geographical and historical background. The seven sperm whales took the same "wrong way" into the Adriatic Sea, a potentially dangerous trap for Mediterranean sperm whales. Seismic surveys should be also regarded as potential co-factors, even if no evidence of direct impact has been detected

Morbillivirus Glycoprotein Expression Induces ER Stress, Alters Ca2+ Homeostasis and Results in the Release of Vasostatin

Although the pathology of Morbillivirus in the central nervous system (CNS) is well described, the molecular basis of neurodegenerative events still remains poorly understood. As a model to explore Morbillivirus-mediated CNS dysfunctions, we used canine distemper virus (CDV) that we inoculated into two different cell systems: a monkey cell line (Vero) and rat primary hippocampal neurons. Importantly, the recombinant CDV used in these studies not only efficiently infects both cell types but recapitulates the uncommon, non-cytolytic cell-to-cell spread mediated by virulent CDVs in brain of dogs. Here, we demonstrated that both CDV surface glycoproteins (F and H) markedly accumulated in the endoplasmic reticulum (ER). This accumulation triggered an ER stress, characterized by increased expression of the ER resident chaperon calnexin and the proapoptotic transcription factor CHOP/GADD 153. The expression of calreticulin (CRT), another ER resident chaperon critically involved in the response to misfolded proteins and in Ca2+ homeostasis, was also upregulated. Transient expression of recombinant CDV F and H surface glycoproteins in Vero cells and primary hippocampal neurons further confirmed a correlation between their accumulation in the ER, CRT upregulation, ER stress and disruption of ER Ca2+ homeostasis. Furthermore, CDV infection induced CRT fragmentation with re-localisation of a CRT amino-terminal fragment, also known as vasostatin, on the surface of infected and neighbouring non-infected cells. Altogether, these results suggest that ER stress, CRT fragmentation and re-localization on the cell surface may contribute to cytotoxic effects and ensuing cell dysfunctions triggered by Morbillivirus, a mechanism that might potentially be relevant for other neurotropic viruses