Invariant Quantum States of Quadratic Hamiltonians

The problem of finding covariance matrices that remain constant in time for arbitrary multi-dimensional quadratic Hamiltonians (including those with time-dependent coefficients) is considered. General solutions are obtained.     

A universal result for consecutive random subdivision of polygons

We consider consecutive random subdivision of polygons described as follows. Given an initial convex polygon with edges, we choose a point at random on each edge, such that the proportions in which these points divide edges are i.i.d. copies of some random variable ξ. These new points form a new (smaller) polygon. By repeatedly implementing this procedure we obtain a sequence of random polygons. The aim of this paper is to show that under very mild non‐degenerateness conditions on ξ, the shapes of these polygons eventually become “flat” The convergence rate to flatness is also investigated; in particular, in the case of triangles (d = 3), we show how to calculate the exact value of the rate of convergence, connected to Lyapunov exponents. Using the theory of products of random matrices our paper greatly generalizes the results of [11] which are achieved mostly by using ad hoc methods. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 341–371, 2017     

Improving the stability and robustness of incomplete symmetric indefinite factorization preconditioners

Sparse symmetric indefinite linear systems of equations arise in numerous practical applications. In many situations, an iterative method is the method of choice but a preconditioner is normally required for it to be effective. In this paper, the focus is on a class of incomplete factorization algorithms that can be used to compute preconditioners for symmetric indefinite systems. A limited memory approach is employed that incorporates a number of new ideas with the goal of improving the stability, robustness, and efficiency of the preconditioner. These include the monitoring of stability as the factorization proceeds and the incorporation of pivot modifications when potential instability is observed. Numerical experiments involving test problems arising from a range of real‐world applications demonstrate the effectiveness of our approach.     

Structure of Fukui matrices

Fukui matrices considered as the generalization of the concept of Fukui densities are decomposed into their pairing and unpairing contributions within the theory of the reduced density matrices. Their algebraic structure become clear from this decomposition providing their relationships with the spin density matrices and the irreducible part of the second‐order reduced density matrix cumulant, that is, the explicit contributions of the many‐body or correlation effects. The uncorrelated state function approximation is a simple way to emphasize the physical meaning of these matrices and represents the appropriate starting point for the treatment of a quasi‐analytical model to denote the occurrence of correlation effects.