Non-degenerate colorings in the Brook's Theorem

Let $c\geq 2$ and $p\geq c$ be two integers. We will call a proper coloring of the graph $G$ a \textit{$(c,p)$-nondegenerate}, if for any vertex of $G$ with degree at least $p$ there are at least $c$ vertices of different colors adjacent to it. In our work we prove the following result, which generalizes Brook's Theorem. Let $D\geq 3$ and $G$ be a graph without cliques on $D+1$ vertices and the degree of any vertex in this graph is not greater than $D$. Then for every integer $c\geq 2$ there is a proper $(c,p)$-nondegenerate vertex $D$-coloring of $G$, where $p=(c^3+8c^2+19c+6)(c+1).$ During the primary proof, some interesting corollaries are derived.Comment: 18 pages, 10 figure

Photo-response of the conductivity in functionalized pentacene compounds

We report the first investigation of the photo-response of the conductivity of a new class of organic semiconductors based on functionalized pentacene. These materials form high quality single crystals that exhibit a thermally activated resistivity. Unlike pure pentacene, the functionalized derivatives are readily soluble in acetone, and can be evaporated or spin-cast as thin films for potential device applications. The electrical conductivity of the single crystal materials is noticeably sensitive to ambient light changes. The purpose, therefore, of the present study, is to determine the nature of the photo-response in terms of carrier activation vs. heating effects, and also to measure the dependence of the photo-response on photon energy. We describe a new method, involving the temperature dependent photo-response, which allows an unambiguous identification of the signature of heating effects in materials with a thermally activated conductivity. We find strong evidence that the photo-response in the materials investigated is predominantly a highly localized heating mechanism. Wavelength dependent studies of the photo-response reveal resonant features and cut-offs that indicate the photon energy absorption is related to the electronic structure of the material.Comment: Preprint: 18 pages total,7 figure

Solid H\u3csub\u3e2\u3c/sub\u3e and D\u3csub\u3e2\u3c/sub\u3e: Remarkable Differences in Some NMR Properties

The differences in the observed properties of solid H2 and D2 are reviewed, and in particular those encountered in NMR experiments. The failure to detect a sharp NMR (I=1) impurity ‘‘isolated pair’’ spectrum in p-D2 is discussed in terms of a larger crystalline field than in H2, where an intense and sharp pair spectrum has been observed. Furthermore, we discuss the dramatic (I=1) solid echo signal loss with decreasing temperature which is observed in solid D2, but not for solid H2. A theory of the solid echo damping through orientational fluctuations is developed. This theory accounts for the observed solid echo decay in D2 as a function of the pulse spacing time τ and leads to an estimation of the order parameter fluctuation amplitude and the correlation time τc. However, the theory cannot account for the loss of spin (as determined from Curie’s law), which must occur for very small values of τ that are not covered by the theory

77^{77}Se and 63^{63}Cu NMR studies of the electronic correlations in Cux_xTiSe2_2 (x=0.05,0.07x=0.05, 0.07)

We report $^{77}$Se and $^{63}$Cu nuclear magnetic resonance (NMR) investigation on the charge-density-wave (CDW) superconductor Cu$_x$TiSe$_2$ ($x=0.05$ and 0.07). At high magnetic fields where superconductivity is suppressed, the temperature dependence of $^{77}$Se and $^{63}$Cu spin-lattice relaxation rates 1/T_{1}$follow a linear relation. The slope of$^{77}1/T_{1}$ vs \emph{T} increases with the Cu doping. This can be described by a modified Korringa relation which suggests the significance of electronic correlations and the Se 4\emph{p}- and Ti 3\emph{d}-band contribution to the density of states at the Fermi level in the studied compounds.Comment: Revised manuscript. Submitted to Journal of Physics: Condensed Matte

Redox kinetics of the amyloid-β-Cu complex and its biological implications

The ability of the amyloid-β peptide to bind to redox active metals and act as a source of radical damage in Alzheimer’s disease has been largely accepted as contributing to the disease’s pathogenesis. However, a kinetic understanding of the molecular mechanism, which underpins this radical generation, has yet to be reported. Here we use a sensitive fluorescence approach, which reports on the oxidation state of the metal bound to the amyloid-β peptide and can therefore shed light on the redox kinetics. We confirm that the redox goes via a low populated, reactive intermediate and that the reaction proceeds via the Component I coordination environment rather than Component II. We also show that while the reduction step readily occurs (on the 10 ms time scale) it is the oxidation step that is rate-limiting for redox cycling

Return to driving after traumatic brain injury : a British perspective

Primary Objective: to identify current legal situation, and professional practice in assisting persons with traumatic brain injury (TBI) to return to safe driving after injury. Methods and Procedures A brief review of relevant literature, a description of the current statutory and quasi-statutory authorities regulating return to driving after TBI in the UK, and a description of the nature and resolution of clinical and practical dilemmas facing professionals helping return to safe driving after TBI. Each of the 15 UK mobility centres was contacted and literature requested; in addition a representative of each centre responded to a structured telephone survey. Main Outcome and Results: The current situation in Great Britain is described, with a brief analysis of the strengths and weaknesses both of the current statutory situation, and also the practical situation (driving centres), with suggestions for improvements in practice. Conclusion Although brain injury may cause serious limitations in driving ability, previous drivers are not routinely assessed or advised regarding return to driving after TBI

Tur\'an Graphs, Stability Number, and Fibonacci Index

The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the stability number and the order of general graphs and connected graphs. Tur\'an graphs frequently appear in extremal graph theory. We show that Tur\'an graphs and a connected variant of them are also extremal for these particular problems.Comment: 11 pages, 3 figure

In-situ characterization of the thermal state of resonant optical interferometers via tracking of their higher-order mode resonances

Thermal lensing in resonant optical interferometers such as those used for gravitational wave detection is a concern due to the negative impact on control signals and instrument sensitivity. In this paper we describe a method for monitoring the thermal state of such interferometers by probing the higher-order spatial mode resonances of the cavities within them. We demonstrate the use of this technique to measure changes in the Advanced LIGO input mode cleaner cavity geometry as a function of input power, and subsequently infer the optical absorption at the mirror surfaces at the level of 1 ppm per mirror. We also demonstrate the generation of a useful error signal for thermal state of the Advanced LIGO power recycling cavity by continuously tracking the first order spatial mode resonance frequency. Such an error signal could be used as an input to thermal compensation systems to maintain the interferometer cavity geometries in the presence of transients in circulating light power levels, thereby maintaining optimal sensitivity and maximizing the duty-cycle of the detectors

Multi-color Cavity Metrology

Long baseline laser interferometers used for gravitational wave detection have proven to be very complicated to control. In order to have sufficient sensitivity to astrophysical gravitational waves, a set of multiple coupled optical cavities comprising the interferometer must be brought into resonance with the laser field. A set of multi-input, multi-output servos then lock these cavities into place via feedback control. This procedure, known as lock acquisition, has proven to be a vexing problem and has reduced greatly the reliability and duty factor of the past generation of laser interferometers. In this article, we describe a technique for bringing the interferometer from an uncontrolled state into resonance by using harmonically related external fields to provide a deterministic hierarchical control. This technique reduces the effect of the external seismic disturbances by four orders of magnitude and promises to greatly enhance the stability and reliability of the current generation of gravitational wave detector. The possibility for using multi-color techniques to overcome current quantum and thermal noise limits is also discussed

Super Weyl invariance: BPS equations from heterotic worldsheets

It is well-known that the beta functions on a string worldsheet correspond to the target space equations of motion, e.g. the Einstein equations. We show that the BPS equations, i.e. the conditions of vanishing supersymmetry variations of the space-time fermions, can be directly derived from the worldsheet. To this end we consider the RNS-formulation of the heterotic string with (2,0) supersymmetry, which describes a complex torsion target space that supports a holomorphic vector bundle. After a detailed account of its quantization and renormalization, we establish that the cancellation of the Weyl anomaly combined with (2,0) finiteness implies the heterotic BPS conditions: At the one loop level the geometry is required to be conformally balanced and the gauge background has to satisfy the Hermitean Yang-Mills equations.Comment: 1+31 pages LaTeX, 5 figures; final version, discussion relation Weyl invariance and (2,0) finiteness extended, typos correcte