Nonlocal spectral properties of disordered alloys

A general method is proposed for calculating a fully k-dependent, continuous, and causal spectral function A(k,E) within the recently introduced nonlocal version of the coherent-potential approximation (NLCPA). The method involves the combination of both periodic and anti-periodic solutions to the associated cluster problem and also leads to correct bulk quantities for small cluster sizes. We illustrate the method by investigating the Fermi surface of a two-dimensional alloy. Dramatically, we find a smeared electronic topological transition not predicted by the conventional CPA.Comment: 17 pages, 5 figures, Submitted to: J. Phys.: Condens. Matter Editorial receipt 25 May 200

Global existence of classical solutions to the Vlasov-Poisson system in a three dimensional, cosmological setting

The initial value problem for the Vlasov-Poisson system is by now well understood in the case of an isolated system where, by definition, the distribution function of the particles as well as the gravitational potential vanish at spatial infinity. Here we start with homogeneous solutions, which have a spatially constant, non-zero mass density and which describe the mass distribution in a Newtonian model of the universe. These homogeneous states can be constructed explicitly, and we consider deviations from such homogeneous states, which then satisfy a modified version of the Vlasov-Poisson system. We prove global existence and uniqueness of classical solutions to the corresponding initial value problem for initial data which represent spatially periodic deviations from homogeneous states.Comment: 23 pages, Latex, report #

Existence of axially symmetric static solutions of the Einstein-Vlasov system

We prove the existence of static, asymptotically flat non-vacuum spacetimes with axial symmetry where the matter is modeled as a collisionless gas. The axially symmetric solutions of the resulting Einstein-Vlasov system are obtained via the implicit function theorem by perturbing off a suitable spherically symmetric steady state of the Vlasov-Poisson system.Comment: 32 page

On static shells and the Buchdahl inequality for the spherically symmetric Einstein-Vlasov system

In a previous work \cite{An1} matter models such that the energy density $\rho\geq 0,$ and the radial- and tangential pressures $p\geq 0$ and $q,$ satisfy $p+q\leq\Omega\rho, \Omega\geq 1,$ were considered in the context of Buchdahl's inequality. It was proved that static shell solutions of the spherically symmetric Einstein equations obey a Buchdahl type inequality whenever the support of the shell, $[R_0,R_1], R_0>0,$ satisfies $R_1/R_0<1/4.$ Moreover, given a sequence of solutions such that $R_1/R_0\to 1,$ then the limit supremum of $2M/R_1$ was shown to be bounded by $((2\Omega+1)^2-1)/(2\Omega+1)^2.$ In this paper we show that the hypothesis that $R_1/R_0\to 1,$ can be realized for Vlasov matter, by constructing a sequence of static shells of the spherically symmetric Einstein-Vlasov system with this property. We also prove that for this sequence not only the limit supremum of $2M/R_1$ is bounded, but that the limit is $((2\Omega+1)^2-1)/(2\Omega+1)^2=8/9,$ since $\Omega=1$ for Vlasov matter. Thus, static shells of Vlasov matter can have $2M/R_1$ arbitrary close to $8/9,$ which is interesting in view of \cite{AR2}, where numerical evidence is presented that 8/9 is an upper bound of $2M/R_1$ of any static solution of the spherically symmetric Einstein-Vlasov system.Comment: 20 pages, Late

Regularity results for the spherically symmetric Einstein-Vlasov system

The spherically symmetric Einstein-Vlasov system is considered in Schwarzschild coordinates and in maximal-isotropic coordinates. An open problem is the issue of global existence for initial data without size restrictions. The main purpose of the present work is to propose a method of approach for general initial data, which improves the regularity of the terms that need to be estimated compared to previous methods. We prove that global existence holds outside the centre in both these coordinate systems. In the Schwarzschild case we improve the bound on the momentum support obtained in \cite{RRS} for compact initial data. The improvement implies that we can admit non-compact data with both ingoing and outgoing matter. This extends one of the results in \cite{AR1}. In particular our method avoids the difficult task of treating the pointwise matter terms. Furthermore, we show that singularities never form in Schwarzschild time for ingoing matter as long as $3m\leq r.$ This removes an additional assumption made in \cite{A1}. Our result in maximal-isotropic coordinates is analogous to the result in \cite{R1}, but our method is different and it improves the regularity of the terms that need to be estimated for proving global existence in general.Comment: 25 pages. To appear in Ann. Henri Poincar\'

Investigation of the nonlocal coherent-potential approximation

Recently the nonlocal coherent-potential approximation (NLCPA) has been introduced by Jarrell and Krishnamurthy for describing the electronic structure of substitutionally disordered systems. The NLCPA provides systematic corrections to the widely used coherent-potential approximation (CPA) whilst preserving the full symmetry of the underlying lattice. Here an analytical and systematic numerical study of the NLCPA is presented for a one-dimensional tight-binding model Hamiltonian, and comparisons with the embedded cluster method (ECM) and molecular coherent potential approximation (MCPA) are made.Comment: 18 pages, 5 figure

The Newtonian Limit for Asymptotically Flat Solutions of the Vlasov-Einstein System

It is shown that there exist families of asymptotically flat solutions of the Einstein equations coupled to the Vlasov equation describing a collisionless gas which have a Newtonian limit. These are sufficiently general to confirm that for this matter model as many families of this type exist as would be expected on the basis of physical intuition. A central role in the proof is played by energy estimates in unweighted Sobolev spaces for a wave equation satisfied by the second fundamental form of a maximal foliation.Comment: 24 pages, plain TE

Uniqueness of the compactly supported weak solutions of the relativistic Vlasov-Darwin system

We use optimal transportation techniques to show uniqueness of the compactly supported weak solutions of the relativistic Vlasov-Darwin system. Our proof extends the method used by Loeper in J. Math. Pures Appl. 86, 68-79 (2006) to obtain uniqueness results for the Vlasov-Poisson system.Comment: AMS-LaTeX, 21 page

Central contracts in Test cricket: a model of best practice?

Across the last two decades, management of international cricket players has changed substantially, with the main Test playing nations using central contracts to guide their team selection. Increased management control over player workload has been a key focus of this. This paper aims to analyse selection in relation to performance for eight Test playing nations in 1,135 matches over thirty years (1985-2015), particularly in relation to the introduction of central contracts. The results demonstrated a relationship between selection stability (i.e. changes made) and performance (overall results and win ratio). The improvement was more pronounced immediately following the introduction of a contract system, as the competitive advantage appears to be at its highest in the two years following their introduction. The data presented argues that the implementation of central contracts as a best practice model has been a beneficial addition to nations' performance in Test matches. Despite this, team managers, coaches and selectors should focus their work on developing an organisational culture where the elite environment has long term stability as its focus. This is particularly pertinent as selection uncertainty can be a de-stabilising factor, as suggested in this paper and in previous research

Reachability in Biochemical Dynamical Systems by Quantitative Discrete Approximation (extended abstract)

In this paper, a novel computational technique for finite discrete approximation of continuous dynamical systems suitable for a significant class of biochemical dynamical systems is introduced. The method is parameterized in order to affect the imposed level of approximation provided that with increasing parameter value the approximation converges to the original continuous system. By employing this approximation technique, we present algorithms solving the reachability problem for biochemical dynamical systems. The presented method and algorithms are evaluated on several exemplary biological models and on a real case study.Comment: In Proceedings CompMod 2011, arXiv:1109.104