Predator–prey model of Beddington–DeAngelis type with maturation and gestation delays

A predator–prey model of Beddington–DeAngelis type with maturation and gestation delays is formulated and analyzed. This two-delay model is similar to the stage-structured model by Liu and Beretta [S. Liu, E. Beretta, Stage-structured predator–prey Model with the Beddington–DeAngelis functional response, SIAM J. Appl. Math. 66 (2006) 1101–1129] but contains an extra gestation delay term. Criteria for permanence and for predator extinction as well as the global attractiveness of the interior equilibrium are derived. The combined effects of the two delays and the degree of predator interference on the dynamical behaviors of the coexistence equilibrium are also studied both analytically and numerically. It is shown that complicated behaviors including chaotic and multi-periodic solutions may occur with the introduction of gestation delay, and that the predator interference can stabilize the system by simplifying the dynamical behaviors and enlarging the stability parameter fields.     

Free Diffusions and Matrix Models with Strictly Convex Interaction

We study solutions to the free stochastic differential equation , where V is a locally convex polynomial potential in m non-commuting variables and S an m-dimensional free Brownian motion. We prove that such free processes have a unique stationary distribution μV. When the potential V is self-adjoint, we show that the law μV is the limit law of a random matrix model, in which an m-tuple of self-adjoint matrices are chosen according to the law exp. If V = V_β depends on complex parameters , we prove that the moments of the law μV are analytic in β at least for those β for which V_β is locally convex. In particular, this gives information on the region of convergence of the generating function for the enumeration of related planar maps. We prove that the solution X_t has nice convergence properties with respect to the operator norm as t goes to infinity. This allows us to show that the C* and W* algebras generated by an m-tuple with law μV share many properties with those generated by a semi-circular system. Among them is the lack of projections, exactness, the Haagerup property, and embeddability into the ultrapower of the hyperfinite II₁ factor. We show that the microstates free entropy χ(μV ) is finite when V is self-adjoint. A corollary of these results is the fact that the support of the law of any self-adjoint polynomial in under the law μV is connected, vastly generalizing the case of a single random matrix. We also deduce from this dynamical approach that the convergence of the operator norms of independent matrices from the GUE proved by Haagerup and Thorbjornsen [HT] extends to the context of matrices interacting via a convex potential. Received: February 2007, Revision: July 2007, Accepted: July 2007     

Computer-Assisted Statistics Education at Delft University of Technology

Abstract

At Delft University of Technology many students experience difficulties in mastering basic concepts of probability and statistics. In the past few years the lectures have undergone a radical change—the lecture notes now contain modern data analysis techniques, like kernel density estimation, simulation, and bootstrapping. In the TWI-Stat project, a computer-aided instruction course was developed to help students become more familiar with modern statistical analysis. The course presents itself as a dynamic, interactive, personal book. Highly interactive analysis tools are available. The software will be available for MS-Windows.     

Unprecedented Route to Ordered Polyaniline: Direct Synthesis of Highly Crystalline Fibrillar Films with Strong π‐π Stacking Alignment

Films of polyaniline (PANI) featuring about 80% crystallinity and characterised with strong π‐π stacking alignment parallel to the film surface have been obtained directly after the original synthesis upon simple drying of the aqueous PANI suspension. A strong anisotropy in the growth of the nano‐sized crystals produced during the synthesis results in the formation of micrometer‐length fibrils perpendicular to the film surface in the course of water evaporation. The regular intercalation of water molecules between the PANI chains seems to be crucial for their ordering throughout the synthesis and film formation.