A note on the Cops & Robber game on graphs embedded in non-orientable surfaces

The Cops and Robber game is played on undirected finite graphs. A number of cops and one robber are positioned on vertices and take turns in sliding along edges. The cops win if they can catch the robber. The minimum number of cops needed to win on a graph is called its cop number. It is known that the cop number of a graph embedded on a surface $X$ of genus $g$ is at most $3g/2 + 3$, if $X$ is orientable (Schroeder 2004), and at most $2g+1$, otherwise (Nowakowski & Schroeder 1997). We improve the bounds for non-orientable surfaces by reduction to the orientable case using covering spaces. As corollaries, using Schroeder's results, we obtain the following: the maximum cop number of graphs embeddable in the projective plane is 3; the cop number of graphs embeddable in the Klein Bottle is at most 4, and an upper bound is $3g/2 + 3/2$ for all other $g$.Comment: 5 pages, 1 figur

Relations between M\"obius and coboundary polynomial

It is known that, in general, the coboundary polynomial and the M\"obius polynomial of a matroid do not determine each other. Less is known about more specific cases. In this paper, we will try to answer if it is possible that the M\"obius polynomial of a matroid, together with the M\"obius polynomial of the dual matroid, define the coboundary polynomial of the matroid. In some cases, the answer is affirmative, and we will give two constructions to determine the coboundary polynomial in these cases.Comment: 12 page

Cluster density functional theory for lattice models based on the theory of Mobius functions

Rosenfeld's fundamental measure theory for lattice models is given a rigorous formulation in terms of the theory of Mobius functions of partially ordered sets. The free-energy density functional is expressed as an expansion in a finite set of lattice clusters. This set is endowed a partial order, so that the coefficients of the cluster expansion are connected to its Mobius function. Because of this, it is rigorously proven that a unique such expansion exists for any lattice model. The low-density analysis of the free-energy functional motivates a redefinition of the basic clusters (zero-dimensional cavities) which guarantees a correct zero-density limit of the pair and triplet direct correlation functions. This new definition extends Rosenfeld's theory to lattice model with any kind of short-range interaction (repulsive or attractive, hard or soft, one- or multi-component...). Finally, a proof is given that these functionals have a consistent dimensional reduction, i.e. the functional for dimension d' can be obtained from that for dimension d (d'<d) if the latter is evaluated at a density profile confined to a d'-dimensional subset.Comment: 21 pages, 2 figures, uses iopart.cls, as well as diagrams.sty (included

Quantum Hall transitions: An exact theory based on conformal restriction

We revisit the problem of the plateau transition in the integer quantum Hall effect. Here we develop an analytical approach for this transition, based on the theory of conformal restriction. This is a mathematical theory that was recently developed within the context of the Schramm-Loewner evolution which describes the stochastic geometry of fractal curves and other stochastic geometrical fractal objects in 2D space. Observables elucidating the connection with the plateau transition include the so-called point-contact conductances (PCCs) between points on the boundary of the sample, described within the language of the Chalker-Coddington network model. We show that the disorder-averaged PCCs are characterized by classical probabilities for certain geometric objects in the plane (pictures), occurring with positive statistical weights, that satisfy the crucial restriction property with respect to changes in the shape of the sample with absorbing boundaries. Upon combining this restriction property with the expected conformal invariance at the transition point, we employ the mathematical theory of conformal restriction measures to relate the disorder-averaged PCCs to correlation functions of primary operators in a conformal field theory (of central charge $c=0$). We show how this can be used to calculate these functions in a number of geometries with various boundary conditions. Since our results employ only the conformal restriction property, they are equally applicable to a number of other critical disordered electronic systems in 2D. For most of these systems, we also predict exact values of critical exponents related to the spatial behavior of various disorder-averaged PCCs.Comment: Published versio

Relative blocking in posets

Poset-theoretic generalizations of set-theoretic committee constructions are presented. The structure of the corresponding subposets is described. Sequences of irreducible fractions associated to the principal order ideals of finite bounded posets are considered and those related to the Boolean lattices are explored; it is shown that such sequences inherit all the familiar properties of the Farey sequences.Comment: 29 pages. Corrected version of original publication which is available at http://www.springerlink.com, see Corrigendu

Co-Evolution of quasispecies: B-cell mutation rates maximize viral error catastrophes

Co-evolution of two coupled quasispecies is studied, motivated by the competition between viral evolution and adapting immune response. In this co-adaptive model, besides the classical error catastrophe for high virus mutation rates, a second ``adaptation-'' catastrophe occurs, when virus mutation rates are too small to escape immune attack. Maximizing both regimes of viral error catastrophes is a possible strategy for an optimal immune response, reducing the range of allowed viral mutation rates to a minimum. From this requirement one obtains constraints on B-cell mutation rates and receptor lengths, yielding an estimate of somatic hypermutation rates in the germinal center in accordance with observation.Comment: 4 pages RevTeX including 2 figure

RNA-binding protein CPEB1 remodels host and viral RNA landscapes.

Host and virus interactions occurring at the post-transcriptional level are critical for infection but remain poorly understood. Here, we performed comprehensive transcriptome-wide analyses revealing that human cytomegalovirus (HCMV) infection results in widespread alternative splicing (AS), shortening of 3' untranslated regions (3' UTRs) and lengthening of poly(A)-tails in host gene transcripts. We found that the host RNA-binding protein CPEB1 was highly induced after infection, and ectopic expression of CPEB1 in noninfected cells recapitulated infection-related post-transcriptional changes. CPEB1 was also required for poly(A)-tail lengthening of viral RNAs important for productive infection. Strikingly, depletion of CPEB1 reversed infection-related cytopathology and post-transcriptional changes, and decreased productive HCMV titers. Host RNA processing was also altered in herpes simplex virus-2 (HSV-2)-infected cells, thereby indicating that this phenomenon might be a common occurrence during herpesvirus infections. We anticipate that our work may serve as a starting point for therapeutic targeting of host RNA-binding proteins in herpesvirus infections

Future directions for the management of pain in osteoarthritis.

Osteoarthritis (OA) is the predominant form of arthritis worldwide, resulting in a high degree of functional impairment and reduced quality of life owing to chronic pain. To date, there are no treatments that are known to modify disease progression of OA in the long term. Current treatments are largely based on the modulation of pain, including NSAIDs, opiates and, more recently, centrally acting pharmacotherapies to avert pain. This review will focus on the rationale for new avenues in pain modulation, including inhibition with anti-NGF antibodies and centrally acting analgesics. The authors also consider the potential for structure modification in cartilage/bone using growth factors and stem cell therapies. The possible mismatch between structural change and pain perception will also be discussed, introducing recent techniques that may assist in improved patient phenotyping of pain subsets in OA. Such developments could help further stratify subgroups and treatments for people with OA in future

Radiofrequency antenna concepts for human cardiac MR at 14.0 T

OBJECTIVE: To examine the feasibility of human cardiac MR (CMR) at 14.0 T using high-density radiofrequency (RF) dipole transceiver arrays in conjunction with static and dynamic parallel transmission (pTx). MATERIALS AND METHODS: RF arrays comprised of self-grounded bow-tie (SGBT) antennas, bow-tie (BT) antennas, or fractionated dipole (FD) antennas were used in this simulation study. Static and dynamic pTx were applied to enhance transmission field (B(1)(+)) uniformity and efficiency in the heart of the human voxel model. B(1)(+) distribution and maximum specific absorption rate averaged over 10 g tissue (SAR(10g)) were examined at 7.0 T and 14.0 T. RESULTS: At 14.0 T static pTx revealed a minimum B(1)(+)(ROI) efficiency of 0.91 μT/√kW (SGBT), 0.73 μT/√kW (BT), and 0.56 μT/√kW (FD) and maximum SAR(10g) of 4.24 W/kg, 1.45 W/kg, and 2.04 W/kg. Dynamic pTx with 8 kT points indicate a balance between B(1)(+)(ROI) homogeneity (coefficient of variation  1.11 µT/√kW) at 14.0 T with a maximum SAR(10g) < 5.25 W/kg. DISCUSSION: MRI of the human heart at 14.0 T is feasible from an electrodynamic and theoretical standpoint, provided that multi-channel high-density antennas are arranged accordingly. These findings provide a technical foundation for further explorations into CMR at 14.0 T