Eigenvalue Separation in Some Random Matrix Models

The eigenvalue density for members of the Gaussian orthogonal and unitary ensembles follows the Wigner semi-circle law. If the Gaussian entries are all shifted by a constant amount c/Sqrt(2N), where N is the size of the matrix, in the large N limit a single eigenvalue will separate from the support of the Wigner semi-circle provided c > 1. In this study, using an asymptotic analysis of the secular equation for the eigenvalue condition, we compare this effect to analogous effects occurring in general variance Wishart matrices and matrices from the shifted mean chiral ensemble. We undertake an analogous comparative study of eigenvalue separation properties when the size of the matrices are fixed and c goes to infinity, and higher rank analogues of this setting. This is done using exact expressions for eigenvalue probability densities in terms of generalized hypergeometric functions, and using the interpretation of the latter as a Green function in the Dyson Brownian motion model. For the shifted mean Gaussian unitary ensemble and its analogues an alternative approach is to use exact expressions for the correlation functions in terms of classical orthogonal polynomials and associated multiple generalizations. By using these exact expressions to compute and plot the eigenvalue density, illustrations of the various eigenvalue separation effects are obtained.Comment: 25 pages, 9 figures include

A phase 1 clinical trial of vorinostat in combination with decitabine in patients with acute myeloid leukaemia or myelodysplastic syndrome

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/108599/1/bjh13016.pd

Liquid-Solid Transition of Hard Spheres Under Gravity

We investigate the liquid-solid transition of two dimensional hard spheres in the presence of gravity. We determine the transition temperature and the fraction of particles in the solid regime as a function of temperature via Even-Driven molecular dynamics simulations and compare them with the theoretical predictions. We then examine the configurational statistics of a vibrating bed from the view point of the liquid-solid transition by explicitly determining the transition temperature and the effective temperature, T, of the bed, and present a relation between T and the vibration strength.Comment: 14 total pages, 4 figure

Facile synthesis and proposed mechanism of α,ω‐oxetanyl-telechelic poly(3-nitratomethyl-3-methyl oxetane) by an SN2(i) nitrato displacement method in basic media

The synthesis of a novel heterocyclic–telechelic polymer, α,ω-oxetanyl-telechelic poly(3-nitratomethyl-3-methyl oxetane), is described. Infrared spectroscopy (IR), gel permeation chromatography (GPC), and nuclear magnetic resonance (NMR) spectroscopy have been used to confirm the successful synthesis, demonstrating the presence of the telechelic-oxetanyl moieties. Synthesis of the terminal functionalities has been achieved via displacement of nitrato groups, in a manner similar to that employed with other leaving groups such as azido, bromo, and nitro, initiated by nucleophiles. In the present case, displacement occurs on the ends of a nitrato-functionalized polymer driven by the formation of sodium nitrate, which is supported by the polar aprotic solvent N,N-dimethyl formamide. The formation of an alkoxide at the polymer chain ends is favored and allows internal back-biting to the nearest carbon bearing the nitrato group, intrinsically in an SN2(i) reaction, leading to α,ω-oxetanyl functionalization. The telechelic-oxetanyl moieties have the potential to be cross-linked by chemical (e.g., acidic) or radiative (e.g., ultraviolet) curing methods without the use of high temperatures, usually below 100°C. This type of material was designed for future use as a contraband simulant, whereby it would form the predominant constituent of elastomeric composites comprising rubbery polymer with small quantities of solids, typically crystals of contraband substances, such as explosives or narcotics. This method also provides an alternative approach to ring closure and synthesis of heterocycles

Quantized vortices and superflow in arbitrary dimensions: Structure, energetics and dynamics

The structure and energetics of superflow around quantized vortices, and the motion inherited by these vortices from this superflow, are explored in the general setting of the superfluidity of helium-four in arbitrary dimensions. The vortices may be idealized as objects of co-dimension two, such as one-dimensional loops and two-dimensional closed surfaces, respectively, in the cases of three- and four-dimensional superfluidity. By using the analogy between vorticial superflow and Ampere-Maxwell magnetostatics, the equilibrium superflow containing any specified collection of vortices is constructed. The energy of the superflow is found to take on a simple form for vortices that are smooth and asymptotically large, compared with the vortex core size. The motion of vortices is analyzed in general, as well as for the special cases of hyper-spherical and weakly distorted hyper-planar vortices. In all dimensions, vortex motion reflects vortex geometry. In dimension four and higher, this includes not only extrinsic but also intrinsic aspects of the vortex shape, which enter via the first and second fundamental forms of classical geometry. For hyper-spherical vortices, which generalize the vortex rings of three dimensional superfluidity, the energy-momentum relation is determined. Simple scaling arguments recover the essential features of these results, up to numerical and logarithmic factors.Comment: 35 pages, 7 figure