The chromatic index of simple graphs

The object of this thesis is twofold: (i) to study the structural properties of graphs which are critical with respect to edge-colourings; (ii) to apply the results obtained to the classification problem arising from Vizing's Theorem. Chapter 1 contains a historical, non-technical introduction, general graph-theoretic definitions and notation, a discussion of Vizing's Theorem as well as a survey of the main results obtained to date in Vizing's classification problem. Chapter 2 introduces the notion of criticality in the first section; the second section contains both well-known and new constructions of critical graphs which will be used in later chapters. The third and final section contains new results concerning elementary properties of critical graphs. Chapter 3 deals with uniquely-colourable graphs and their relationship to critical graphs. Chapter 4 contains results on the connectivity of critical graphs, whereas Chapter 5 deals with bounds on the number of edges of these graphs. In particular, bounds improving those given by Vizing are presented. These results are applied to problems concerning planar graphs. In Chapter 6, critical graphs of small order are discussed. All such graphs of order at most 8 are determined, while the 'critical graph conjecture’ of Beineke & Wilson and Jakobsen is shown to be true for all graphs on at most 10 vertices. The seventh and final chapter deals with circuit length properties of critical graphs. In particular, the minimal order of certain critical graphs with given girth and maximum valency is determined. Results improving Vizing’s estimate of the circumference of critical graphs are also given. The Appendix includes a computer programme which generates critical graphs from simpler ones using a constructive algorithm given in Chapter 2

The regular cosmic string in Born-Infeld gravity

It is shown that Born-Infeld gravity --a high energy deformation of Einstein gravity-- removes the singularities of a cosmic string. The respective vacuum solution results to be free of conical singularity and closed timelike curves. The space ends at a minimal circle where the curvature invariants vanish; but this circle cannot be reached in a finite proper time.Comment: 4 pages, submitted to Proceedings of Spanish Relativity Meeting 2010 (ERE2010, Granada, Spain

On the existence of 0/1 polytopes with high semidefinite extension complexity

In Rothvo\ss{} it was shown that there exists a 0/1 polytope (a polytope whose vertices are in \{0,1\}^{n}) such that any higher-dimensional polytope projecting to it must have 2^{\Omega(n)} facets, i.e., its linear extension complexity is exponential. The question whether there exists a 0/1 polytope with high PSD extension complexity was left open. We answer this question in the affirmative by showing that there is a 0/1 polytope such that any spectrahedron projecting to it must be the intersection of a semidefinite cone of dimension~2^{\Omega(n)} and an affine space. Our proof relies on a new technique to rescale semidefinite factorizations

Stochastic resonance in a suspension of magnetic dipoles under shear flow

We show that a magnetic dipole in a shear flow under the action of an oscillating magnetic field displays stochastic resonance in the linear response regime. To this end, we compute the classical quantifiers of stochastic resonance, i.e. the signal to noise ratio, the escape time distribution, and the mean first passage time. We also discuss limitations and role of the linear response theory in its applications to the theory of stochastic resonance.Comment: 17 pages, 5 figures, approved for publication in PR

PARTITION OF THE ORGANOCHLORINE INSECTICIDE LINDANE INTO THE HUMAN SPERM SURFACE INDUCES MEMBRANE DEPOLARIZATION AND CALCIUM INFLUX

The effects of the insecticide lindane (the gamma-isomer of 1,2,3,4,5,6-hexachlorocyclohexane) on membrane potential, cytosolic free Ca2+ concentration ([Ca2+]i) and surface biophysical properties were studied in human spermatozoa. The insecticide induces rapid, transient and reproducible membrane depolarization and opening of voltage-dependent Ca2+ channels leading to an increase in [Ca2+]i. In contrast with the effect in somatic cells, lindane did not affect gamma-aminobutyric acid receptor-linked Cl- currents. Ca2+ and K+ currents were found to drive lindane-induced membrane depolarization and repolarization respectively, whereas Na+ and Cl- fluxes appear not to have a role in the phenomenon. The insecticide was still able to produce membrane depolarization both in the combined absence of extracellular Ca2+ and Na+ and in high-K+ buffer, suggesting that lindane alters the membrane dipole potential. In agreement with this, Laurodan and Prodan fluorescence spectroscopy revealed that lindane partition into the sperm plasma membrane lowers water molecular dynamics in the uppermost region of the membrane external leaflet, probably as the result of reordering of water dipoles. We propose that the first effect of lindane partitioning into the sperm plasma membrane is a change in the membrane dipole potential, which results in the activation of membrane-located Ca2+-influx pathways

The VPN problems with concave costs

Only recently Goyal, Olver and Shepherd (Proc. STOC, 2008) proved that the symmetric Virtual Private Network Design (sVPN) problem has the tree routing property, namely, that there always exists an optimal solution to the problem whose support is a tree. Combining this with previous results by Fingerhut, Suri and Turner (J. Alg., 1997) and Gupta, Kleinberg, Kumar, Rastogi and Yener (Proc. STOC, 2001), sVPN can be solved in polynomial time. In this paper we investigate an APX-hard generalization of sVPN, where the contribution of each edge to the total cost is proportional to some non-negative, concave and non-decreasing function of the capacity reservation. We show that the tree routing property extends to the new problem, and give a constant-factor approximation algorithm for it. We also show that the undirected uncapacitated single-source minimum concave-cost flow problem has the tree routing property when the cost function has some property of symmetry

New experimental limits on the alpha decays of lead isotopes

For the first time a PbWO4 crystal was grown using ancient Roman lead and it was run as a cryogenic detector. Thanks to the simultaneous and independent read-out of heat and scintillation light, the detector was able to discriminate beta/gamma interactions with respect to alpha particles down to low energies. New more stringent limits on the alpha decays of the lead isotopes are presented. In particular a limit of T_{1/2} > 1.4*10^20 y at a 90% C.L. was evaluated for the alpha decay of 204Pb to 200Hg