Smart Geodesic Tracing in GRworkbench

We have developed a new tool for numerical work in General Relativity: GRworkbench. We discuss how GRworkbench's implementation of a numerically-amenable analogue to Differential Geometry facilitates the development of robust and chart-independent numerical algorithms. We consider, as an example, geodesic tracing on two charts covering the exterior Schwarzschild space-time.     

Primal cutting plane algorithms revisited

Dual fractional cutting plane algorithms, in which cutting planes are used to iteratively tighten a linear relaxation of an integer program, are well-known and form the basis of the highly successful branch-and-cut method. It is rather less well-known that various primal cutting plane algorithms were developed in the 1960s, for example by Young. In a primal algorithm, the main role of the cutting planes is to enable a feasible solution to the original problem to be improved. Research on these algorithms has been almost non-existent.  In this paper we argue for a re-examination of these primal methods. We describe a new primal algorithm for pure 0-1 problems based on strong valid inequalities and give some encouraging computational results. Possible extensions to the case of general mixed-integer programs are also discussed.     

To-and-fro motion of W5Si3 on a beta-Si3N4 substrate at elevated temperatures.

Behavior of fine crystalline particles of W5Si3 on a beta-Si3N4 substrate at high temperatures was observed by an in situ heating experiment in a transmission electron microscope. Some of the fine particles of W5Si3 moved in a to-and-fro manner.     

On the Reality of the Eigenvalues for a Class of -Symmetric Oscillators

 We study the eigenvalue problem with the boundary conditions that decays to zero as z tends to infinity along the rays , where is a real polynomial and . We prove that if for some we have for all , then the eigenvalues are all positive real. We then sharpen this to a larger class of polynomial potentials. In particular, this implies that the eigenvalues are all positive real for the potentials when with , and with the boundary conditions that decays to zero as z tends to infinity along the positive and negative real axes. This verifies a conjecture of Bessis and Zinn-Justin. Received: 16 January 2002 / Accepted: 1 May 2002 Published online: 6 August 2002     

Eigenvalue distributions of random unitary matrices

 Let U be an n × n random matrix chosen from Haar measure on the unitary group. For a fixed arc of the unit circle, let X be the number of eigenvalues of M which lie in the specified arc. We study this random variable as the dimension n grows, using the connection between Toeplitz matrices and random unitary matrices, and show that (X -E [X])/(\Var (X))^1/2 is asymptotically normally distributed. In addition, we show that for several fixed arcs I ₁ , ..., I _m , the corresponding random variables are jointly normal in the large n limit. Received: 15 November 2000 / Revised version: 27 September 2001 / Published online: 17 May 2002     

A primer on perfect simulation for spatial point processes

This primer provides a self-contained exposition of the case where spatial birth-and-death processes are used for perfect simulation of locally stable point processes. Particularly, a simple dominating coupling from the past (CFTP) algorithm and the CFTP algorithms introduced in [13], [14], and [5] are studied. Some empirical results for the algorithms are discussed. Received: 30 June 2002     

Structural evolution in amorphous silicon and germanium thin films.

The structural evolution in amorphous silicon and germanium thin films has been investigated by high-resolution transmission electron microscopy (HRTEM) in conjunction with autocorrelation function (ACF) analysis. The results established that the structure of as-deposited semiconductor films is of a high density of nanocrystallites embedded in the amorphous matrix. In addition, from ACF analysis, the structure of a-Ge is more ordered than that of a-Si. The density of embedded nanocrystallites in amorphous films was found to diminish with annealing temperature first, then to increase. The conclusions also corroborate well with the results of diminished medium-range order in annealed amorphous films determined previously by a variable coherence microscopy method.     

Rank-perfect and" weakly rank-perfect graphs

An edge e of a perfect graph G is critical if G−e is imperfect. We would like to decide whether G−e is still “almost perfect” or already “very imperfect”. Via relaxations of the stable set polytope of a graph, we define two superclasses of perfect graphs: rank-perfect and weakly rank-perfect graphs. Membership in those two classes indicates how far an imperfect graph is away from being perfect. We study the cases, when a critical edge is removed from the line graph of a bipartite graph or from the complement of such a graph.     

Possible Loss and Recovery of Gibbsianness¶During the Stochastic Evolution of Gibbs Measures

We consider Ising-spin systems starting from an initial Gibbs measure ν and evolving under a spin-flip dynamics towards a reversible Gibbs measure μ≠ν. Both ν and μ are assumed to have a translation-invariant finite-range interaction. We study the Gibbsian character of the measure νS(t) at time t and show the following: (1) For all ν and μ, νS(t) is Gibbs for small t. (2) If both ν and μ have a high or infinite temperature, then νS(t) is Gibbs for all t > 0. (3) If ν has a low non-zero temperature and a zero magnetic field and μ has a high or infinite temperature, then νS(t) is Gibbs for small t and non-Gibbs for large t. (4) If ν has a low non-zero temperature and a non-zero magnetic field and μ has a high or infinite temperature, then νS(t) is Gibbs for small t, non-Gibbs for intermediate t, and Gibbs for large t. The regime where μ has a low or zero temperature and t is not small remains open. This regime presumably allows for many different scenarios. Received: 26 April 2001 / Accepted: 10 October 2001     

Every diassociative A-loop is Moufang

An A-loop is a loop in which every inner mapping is an automorphism. A problem which had been open since 1956 is settled by showing that every diassociative A-loop is Moufang.