Pseudo memory effects, majorization and entropy in quantum random walks

A quantum random walk on the integers exhibits pseudo memory effects, in that its probability distribution after N steps is determined by reshuffling the first N distributions that arise in a classical random walk with the same initial distribution. In a classical walk, entropy increase can be regarded as a consequence of the majorization ordering of successive distributions. The Lorenz curves of successive distributions for a symmetric quantum walk reveal no majorization ordering in general. Nevertheless, entropy can increase, and computer experiments show that it does so on average. Varying the stages at which the quantum coin system is traced out leads to new quantum walks, including a symmetric walk for which majorization ordering is valid but the spreading rate exceeds that of the usual symmetric quantum walk.

The role of habitat disturbance and recovery in metapopulation persistence

Classical metapopulation theory assumes a static landscape. However, empirical evidence indicates many metapopulations are driven by habitat succession and disturbance. We develop a stochastic metapopulation model, incorporating habitat disturbance and recovery, coupled with patch colonization and extinction, to investigate the effect of habitat dynamics on persistence. We discover that habitat dynamics play a fundamental role in metapopulation dynamics. The mean number of suitable habitat patches is not adequate for characterizing the dynamics of the metapopulation. For a fixed mean number of suitable patches, we discover that the details of how disturbance affects patches and how patches recover influences metapopulation dynamics in a fundamental way. Moreover, metapopulation persistence is dependent not only oil the average lifetime of a patch, but also on the variance in patch lifetime and the synchrony in patch dynamics that results from disturbance. Finally, there is an interaction between the habitat and metapopulation dynamics, for instance declining metapopulations react differently to habitat dynamics than expanding metapopulations. We close, emphasizing the importance of using performance measures appropriate to stochastic systems when evaluating their behavior, such as the probability distribution of the state of the. metapopulation, conditional on it being extant (i.e., the quasistationary distribution).

Anisotropy-based robust performance analysis of finite horizon linear discrete time varying systems

We consider a problem of robust performance analysis of linear discrete time varying systems on a bounded time interval. The system is represented in the state-space form. It is driven by a random input disturbance with imprecisely known probability distribution; this distributional uncertainty is described in terms of entropy. The worst-case performance of the system is quantified by its a-anisotropic norm. Computing the anisotropic norm is reduced to solving a set of difference Riccati and Lyapunov equations and a special form equation.

Stochastic modelling of the invasion process of nematodes in fly larvae

Stochastic models based on Markov birth processes are constructed to describe the process of invasion of a fly larva by entomopathogenic nematodes. Various forms for the birth (invasion) rates are proposed. These models are then fitted to data sets describing the observed numbers of nematodes that have invaded a fly larval after a fixed period of time. Non-linear birthrates are required to achieve good fits to these data, with their precise form leading to different patterns of invasion being identified for three populations of nematodes considered. One of these (Nemasys) showed the greatest propensity for invasion. This form of modelling may be useful more generally for analysing data that show variation which is different from that expected from a binomial distribution.

A liberation model for comminution based on probability theory

Mineral processing plants use two main processes; these are comminution and separation. The objective of the comminution process is to break complex particles consisting of numerous minerals into smaller simpler particles where individual particles consist primarily of only one mineral. The process in which the mineral composition distribution in particles changes due to breakage is called 'liberation'. The purpose of separation is to separate particles consisting of valuable mineral from those containing nonvaluable mineral. The energy required to break particles to fine sizes is expensive, and therefore the mineral processing engineer must design the circuit so that the breakage of liberated particles is reduced in favour of breaking composite particles. In order to effectively optimize a circuit through simulation it is necessary to predict how the mineral composition distributions change due to comminution. Such a model is called a 'liberation model for comminution'. It was generally considered that such a model should incorporate information about the ore, such as the texture. However, the relationship between the feed and product particles can be estimated using a probability method, with the probability being defined as the probability that a feed particle of a particular composition and size will form a particular product particle of a particular size and composition. The model is based on maximizing the entropy of the probability subject to mass constraints and composition constraint. Not only does this methodology allow a liberation model to be developed for binary particles, but also for particles consisting of many minerals. Results from applying the model to real plant ore are presented. A laboratory ball mill was used to break particles. The results from this experiment were used to estimate the kernel which represents the relationship between parent and progeny particles. A second feed, consisting primarily of heavy particles subsampled from the main ore was then ground through the same mill. The results from the first experiment were used to predict the product of the second experiment. The agreement between the predicted results and the actual results are very good. It is therefore recommended that more extensive validation is needed to fully evaluate the substance of the method. (C) 2003 Elsevier Ltd. All rights reserved.

Rank test of location optimal for hyperbolic secant distribution

There are at least two reasons for a symmetric, unimodal, diffuse tailed hyperbolic secant distribution to be interesting in real-life applications. It displays one of the common types of non normality in natural data and is closely related to the logistic and Cauchy distributions that often arise in practice. To test the difference in location between two hyperbolic secant distributions, we develop a simple linear rank test with trigonometric scores. We investigate the small-sample and asymptotic properties of the test statistic and provide tables of the exact null distribution for small sample sizes. We compare the test to the Wilcoxon two-sample test and show that, although the asymptotic powers of the tests are comparable, the present test has certain practical advantages over the Wilcoxon test.

R-estimator of location of the generalized secant hyperbolic distribution

The generalized secant hyperbolic distribution (GSHD) proposed in Vaughan (2002) includes a wide range of unimodal symmetric distributions, with the Cauchy and uniform distributions being the limiting cases, and the logistic and hyperbolic secant distributions being special cases. The current article derives an asymptotically efficient rank estimator of the location parameter of the GSHD and suggests the corresponding one- and two-sample optimal rank tests. The rank estimator derived is compared to the modified MLE of location proposed in Vaughan (2002). By combining these two estimators, a computationally attractive method for constructing an exact confidence interval of the location parameter is developed. The statistical procedures introduced in the current article are illustrated by examples.