Path integrals for continuous-time Markov chains

This note presents a method of evaluating the distribution of a path integral for Markov chains on a countable state space.

Set-valued Markov chains and negative semitrajectories of discretized dynamical systems

Computer simulation of dynamical systems involves a phase space which is the finite set of machine arithmetic. Rounding state values of the continuous system to this grid yields a spatially discrete dynamical system, often with different dynamical behaviour. Discretization of an invertible smooth system gives a system with set-valued negative semitrajectories. As the grid is refined, asymptotic behaviour of the semitrajectories follows probabilistic laws which correspond to a set-valued Markov chain, whose transition probabilities can be explicitly calculated. The results are illustrated for two-dimensional dynamical systems obtained by discretization of fractional linear transformations of the unit disc in the complex plane.

Integrals for continuous-time Markov chains

This paper presents a method of evaluating the expected value of a path integral for a general Markov chain on a countable state space. We illustrate the method with reference to several models, including birth-death processes and the birth, death and catastrophe process. (C) 2002 Elsevier Science Inc. All rights reserved.

Lexicographic decomposition of chains and the concept of a planar chain

In an earlier paper [Journal of Mathematical Economics, 37 (2002) 17-38], we proved that if a preference relation on a commodity space is non-representable by a real-valued function then that chain is necessarily a long chain, a planar chain, an Aronszajn-like chain or a Souslin chain. In this paper, we study the class of planar chains, the simplest example of which is the Debreu chain (R-2, <(l)). (C) 2002 Elsevier Science B.V. All rights reserved.

Leaf area development of ABA-deficient and wild-type peas at two levels of nitrogen supply

The ABA-deficient wilty pea (Pisum sativum L.) and its wild-type (WT) were grown at two levels of nitrogen supply (0.5 and 5.0 mM) for 5-6 weeks from sowing, to determine whether leaf ABA status altered the leaf growth response to N deprivation. Plants were grown at high relative humidity to prevent wilting of the wilty peas. Irrespective of N supply, expanding wilty leaflets had ca 50% less ABA than WT leaflets but similar ethylene evolution rates. Fully expanded wilty leaflets had lower relative water contents (RWC) and were 10-60% smaller in area (according to the node of measurement) than WT leaflets. However, there were no genotypic differences in plant relative leaf expansion rate (RLER). Growth of both genotypes at 0.5 mM N increased the RWC of fully expanded leaflets, but did not alter ethylene evolution or ABA concentration of expanding leaflets. Plants grown at 0.5 mM N showed a 20-30% reduction in RLER, which was similar in magnitude in both wilty and WT peas. Thus, leaf ABA status did not alter the leaf growth response to N deprivation.