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.

Field quantization, photons and non-Hermitean modes

Field quantization in unstable optical systems is treated by expanding the vector potential in terms of non-Hermitean (Fox-Li) modes. We define non-Hermitean modes and their adjoints in both the cavity and external regions and make use of the important bi-orthogonality relationships that exist within each mode set. We employ a standard canonical quantization procedure involving the introduction of generalized coordinates and momenta for the electromagnetic (EM) field. Three-dimensional systems are treated, making use of the paraxial and monochromaticity approximations for the cavity non-Hermitean modes. We show that the quantum EM field is equivalent to a set of quantum harmonic oscillators (QHOs), associated with either the cavity or the external region non-Hermitean modes, and thus confirming the validity of the photon model in unstable optical systems. Unlike in the conventional (Hermitean mode) case, the annihilation and creation operators we define for each QHO are not Hermitean adjoints. It is shown that the quantum Hamiltonian for the EM field is the sum of non-commuting cavity and external region contributions, each of which can be expressed as a sum of independent QHO Hamiltonians for each non-Hermitean mode, except that the external field Hamiltonian also includes a coupling term responsible for external non-Hermitean mode photon exchange processes. The non-commutativity of certain cavity and external region annihilation and creation operators is associated with cavity energy gain and loss processes, and may be described in terms of surface integrals involving cavity and external region non-Hermitean mode functions on the cavity-external region boundary. Using the essential states approach and the rotating wave approximation, our results are applied to the spontaneous decay of a two-level atom inside an unstable cavity. We find that atomic transitions leading to cavity non-Hermitean mode photon absorption are associated with a different coupling constant to that for transitions leading to photon emission, a feature consequent on the use of non-Hermitean mode functions. We show that under certain conditions the spontaneous decay rate is enhanced by the Petermann factor.

High resolution 3-D imaging via multi-pass SAR

Spaceborne/airborne synthetic aperture radar (SAR) systems provide high resolution two-dimensional terrain imagery. The paper proposes a technique for combining multiple SAR images, acquired on flight paths slightly separated in the elevation direction, to generate high resolution three-dimensional imagery. The technique could be viewed as an extension to interferometric SAR (InSAR) in that it generates topographic imagery with an additional dimension of resolution. The 3-D multi-pass SAR imaging system is typically characterised by a relatively short ambiguity length in the elevation direction. To minimise the associated ambiguities we exploit the relative phase information within the set of images to track the terrain landscape. The SAR images are then coherently combined, via a nonuniform DFT, over a narrow (in elevation) volume centred on the 'dominant' terrain ground plane. The paper includes a detailed description of the technique, background theory, including achievable resolution, and the results of an experimental study.

Simulation of insect movement with respect to plant architecture and morphogenesis

Developments in computer and three dimensional (3D) digitiser technologies have made it possible to keep track of the broad range of data required to simulate an insect moving around or over the highly heterogeneous habitat of a plant's surface. Properties of plant parts vary within a complex canopy architecture, and insect damage can induce further changes that affect an animal's movements, development and likelihood of survival. Models of plant architectural development based on Lindenmayer systems (L-systems) serve as dynamic platforms for simulation of insect movement, providing ail explicit model of the developing 3D structure of a plant as well as allowing physiological processes associated with plant growth and responses to damage to be described and Simulated. Simple examples of the use of the L-system formalism to model insect movement, operating Lit different spatial scales-from insects foraging on an individual plant to insects flying around plants in a field-are presented. Such models can be used to explore questions about the consequences of changes in environmental architecture and configuration on host finding, exploitation and its population consequences. In effect this model is a 'virtual ecosystem' laboratory to address local as well as landscape-level questions pertinent to plant-insect interactions, taking plant architecture into account. (C) 2002 Elsevier Science B.V. All rights reserved.

Numerical simulations of wind and temperature structure within an Alpine lake basin, Lake Tekapo, New Zealand

High-resolution numerical model simulations have been used to study the local and mesoscale thermal circulations in an Alpine lake basin. The lake (87 km(2)) is situated in the Southern Alps, New Zealand and is located in a glacially excavated rock basin surrounded by mountain ranges that reach 3000 m in height. The mesoscale model used (RAMS) is a three-dimensional non-hydrostatic model with a level 2.5 turbulence closure scheme. The model demonstrates that thermal forcing at local (within the basin) and regional (coast-to-basin inflow) scales drive the observed boundary-layer airflow in the lake basin during clear anticyclonic summertime conditions. The results show that the lake can modify (perturb) both the local and regional wind systems. Following sunrise, local thermal circulations dominate, including a lake breeze component that becomes embedded within the background valley wind system. This results in a more divergent flow in the basin extending across the lake shoreline. However, a closed lake breeze circulation is neither observed nor modelled. Modelling results indicate that in the latter part of the day when the mesoscale (coast-to-basin) inflow occurs, the relatively cold pool of lake air in the basin can cause the intrusion to decouple from the surface. Measured data provide qualitative and quantitative support for the model results.

Effects of glucose-insulin-potassium infusion on chronic ischaemic left ventricular dysfunction

Background: Glucose-insulin-potassium (GIK) infusion improves cardiac function and outcome during acute ischaemia. Objective: To determine whether GIK infusion benefits patients with chronic ischaemic left ventricular dysfunction, and if so whether this is related to the presence and nature of viable myocardium. Methods: 30 patients with chronic ischaemic left ventricular dysfunction had dobutamine echocardiography and were given a four hour infusion of GIK. Segmental responses were quantified by improvement in wall motion score index (WMSI) and peak systolic velocity using tissue Doppler. Global responses were assessed by left ventricular volume and ejection fraction, measured using a three dimensional reconstruction. Myocardial perfusion was determined in 15 patients using contrast echocardiography. Results: WMSI (mean (SD)) improved with dobutamine (from 1.8 (0.4) to 1.6 (0.4), p < 0.001) and with GIK (from 1.8 (0.4) to 1.7 (0.4) p < 0.001); there was a similar increment for both. Improvement in wall motion score with GIK was observed in 55% of the 62 segments classed as viable by dobutamine echocardiography, and in 5% of 162 classed as non-viable. There was an increment in peak systolic velocity after both doputamine echocardiography (from 2.5 (1.8) to 3.2 (2.2) cm/s, p < 0.01) and GIK (from 3.0 (1.6) to 3.5 (17) cm/s, p < 0.001). The GlK effects were not mediated by changes in pulse, mean arterial pressure, lactate, or catecholamines, nor did they correlate with myocardial perfusion. End systolic volume improved after GlK (p = 0.03), but only in 25 patients who had viable myocardium on dobutom ne echocardiography. Conclusions: In patients with viable myocardium and chronic left ventricular dysfunction, GlK improves wall motion score, myocardial velocity, and end systolic volume, independent of effects on haemodynamics or catecholamines. The response to GlK is observed in areas of normal and abnormal perfusion assessed by contrast echocardiography.