On the rooted Tutte polynomial

The Tutte polynomial is a generalization of the chromatic polynomial of graph colorings. Here we present an extension called the rooted Tutte polynomial, which is defined on a graph where one or more vertices are colored with prescribed colors. We establish a number of results pertaining to the rooted Tutte polynomial, including a duality relation in the case that all roots reside around a single face of a planar graph. The connection with the Potts model is also reviewed.Comment: plain latex, 14 pages, 2 figs., to appear in Annales de l'Institut Fourier (1999

Leptogenesis after Chaotic Sneutrino Inflation and the Supersymmetry Breaking Scale

We discuss resonant leptogenesis arising from the decays of two nearly-degenerate right-handed neutrinos, identified as the inflaton and stabiliser superfields in a model of chaotic sneutrino inflation. We compare an analytical estimate of the baryon asymmetry $\eta_B$ in the Boltzmann approximation to a numerical solution of the full density matrix equations, and find that the analytical result fails to capture the correct physics in certain regions of parameter space. The observed baryon asymmetry can be realised for a breaking of the mass degeneracy as small as $\mathcal{O}(10^{-8})$. The origin of such a small mass splitting is explained by considering supersymmetry (SUSY) breaking in supergravity, which requires a constant in the superpotential of the order of the gravitino mass $m_{3/2}$ to cancel the cosmological constant. This yields additional terms in the (s)neutrino mass matrices, lifting the degeneracy and linking $\eta_B$ to the SUSY breaking scale. We find that achieving the correct baryon asymmetry requires a gravitino mass $m_{3/2} \geq \mathcal{O}(100)$ TeV.Comment: v2: 25 pages, 4 figures; version published in NPB, minor corrections. v1: 24 pages, 4 figure

Distinguishing low frequency mutations from RT-PCR and sequence errors in viral deep sequencing data

There is a high prevalence of coronary artery disease (CAD) in patients with left bundle branch block (LBBB); however there are many other causes for this electrocardiographic abnormality. Non-invasive assessment of these patients remains difficult, and all commonly used modalities exhibit several drawbacks. This often leads to these patients undergoing invasive coronary angiography which may not have been necessary. In this review, we examine the uses and limitations of commonly performed non-invasive tests for diagnosis of CAD in patients with LBBB

String-Inspired Triplet See-Saw from Diagonal Embedding of SU(2)_L in SU(2)_A x SU(2)_B

Motivated by string constructions, we consider a variant on the Type II see-saw mechanism involving the exchange of triplet representations of SU(2)_L in which this group arises from a diagonal embedding into SU(2)_A x SU(2)_B. A natural assignment of Standard Model lepton doublets to the two underlying gauge groups results in a bimaximal pattern of neutrino mixings and an inverted hierarchy in masses. Simple perturbations around this leading-order structure can accommodate the observed pattern of neutrino masses and mixings.Comment: 8 pages; uses RevTe

LFV and Dipole Moments in Models with A4 Flavour Symmetry

It is presented an analysis on lepton flavour violating transitions, leptonic magnetic dipole moments and electric dipole moments in a class of models characterized by the flavour symmetry A4 x Z3 x U(1)_FN, whose choice is motivated by the approximate Tri-Bimaximal mixing observed in neutrino oscillations. A low-energy effective Lagrangian is constructed, where these effects are dominated by dimension six operators, suppressed by the scale M of new physics. All the flavour breaking effects are universally described by the vacuum expectation values of a set of spurions. Two separate cases, a supersymmetric and a general one, are described. An upper limit on the reactor angle of a few percent is concluded.Comment: 10 pages, 1 figure. Adapted from a talk given at "DISCRETE'08: Symposium on Prospects in the Physics of Discrete Symmetries", December 11-16 2008, Valencia, Spai

Coexpression of rat P2X2 and P2X6 subunits in Xenopus oocytes.

Transcripts for P2X(2) and P2X(6) subunits are present in rat CNS and frequently colocalize in the same brainstem nuclei. When rat P2X(2) (rP2X(2)) and rat P2X(6) (rP2X(6)) receptors were expressed individually in Xenopus oocytes and studied under voltage-clamp conditions, only homomeric rP2X(2) receptors were fully functional and gave rise to large inward currents (2-3 microA) to extracellular ATP. Coexpression of rP2X(2) and rP2X(6) subunits in Xenopus oocytes resulted in a heteromeric rP2X(2/6) receptor, which showed a significantly different phenotype from the wild-type rP2X(2) receptor. Differences included reduction in agonist potencies and, in some cases (e.g., Ap(4)A), significant loss of agonist activity. ATP-evoked inward currents were biphasic at the heteromeric rP2X(2/6) receptor, particularly when Zn(2+) ions were present or extracellular pH was lowered. The pH range was narrower for H(+) enhancement of ATP responses at the heteromeric rP2X(2/6) receptor. Also, H(+) ions inhibited ATP responses at low pH levels (<pH 6.3). The pH-dependent blocking activity of suramin was changed at this heteromeric receptor, although the potentiating effect of Zn(2+) on ATP responses was unchanged. Thus, the rP2X(2/6) receptor is a functionally modified P2X(2)-like receptor with a distinct pattern of pH modulation of ATP activation and suramin blockade. Although homomeric P2X(6) receptors function poorly, the P2X(6) subunit can contribute to functional heteromeric P2X channels and may influence the phenotype of native P2X receptors in those cells in which it is expressed

Convex Hull of Planar H-Polyhedra

Suppose are planar (convex) H-polyhedra, that is, $A_i \in \mathbb{R}^{n_i \times 2}$ and $\vec{c}_i \in \mathbb{R}^{n_i}$. Let $P_i = \{\vec{x} \in \mathbb{R}^2 \mid A_i\vec{x} \leq \vec{c}_i \}$ and $n = n_1 + n_2$. We present an $O(n \log n)$ algorithm for calculating an H-polyhedron with the smallest $P = \{\vec{x} \in \mathbb{R}^2 \mid A\vec{x} \leq \vec{c} \}$ such that $P_1 \cup P_2 \subseteq P$

Trimaximal neutrino mixing from vacuum alignment in A4 and S4 models

Recent T2K results indicate a sizeable reactor angle theta_13 which would rule out exact tri-bimaximal lepton mixing. We study the vacuum alignment of the Altarelli-Feruglio A4 family symmetry model including additional flavons in the 1' and 1" representations and show that it leads to trimaximal mixing in which the second column of the lepton mixing matrix consists of the column vector (1,1,1)^T/sqrt{3}, with a potentially large reactor angle. In order to limit the reactor angle and control the higher order corrections, we propose a renormalisable S4 model in which the 1' and 1" flavons of A4 are unified into a doublet of S4 which is spontaneously broken to A4 by a flavon which enters the neutrino sector at higher order. We study the vacuum alignment in the S4 model and show that it predicts accurate trimaximal mixing with approximate tri-bimaximal mixing, leading to a new mixing sum rule testable in future neutrino experiments. Both A4 and S4 models preserve form dominance and hence predict zero leptogenesis, up to renormalisation group corrections.Comment: 24 pages, 2 figures, version to be published in JHE