Computational modelling of nonlinear calcium waves

The calcium transport in biological systems is modelled as a reaction–diffusion process. Nonlinear calcium waves are then simulated using a stochastic cellular automaton whose rules are derived from the corresponding coupled partial differential equations. Numerical simulations show self-organized criticality in the complex calcium waves and patterns. Both the stochastic cellular automaton approach and the equation-based simulations can predict the characteristics of calcium waves and complex pattern formation. The implication of locality of calcium distribution with positional information in biological systems is also discussed.     

Stochastic modelling and optimization of water resources systems

This work involves the simultaneous optimization of the initial design and operating policy over the life of multipurpose multireservoir water resources systems receiving stochastic inflows. The approach is based on the division of the reservoir into two imaginary water storage pools, namely, the conservation and flood pools. Based on this treatment, the optimization problem is stated using the concepts of Lagrange multipliers and parameter optimization. Two nonlinear programming techniques, namely, the generalized reduced gradient technique and the gradient projection technique, combined independently with Markovian decision are proposed to solve such a problem. To illustrate the use of the proposed techniques, the Walnut River Basin in southeastern Kansas, is employed in this work.     

Qualitative theory of nonlinear steady water waves

In this paper, the nonlinear boundary problem describing two-dimensional steady waves on the surface of water with finite depth is discussed. The problem is formulated in the conventional statement (the gravity is taken into account, but the surface tension is neglected). The latter one allows discussing the whole class of bounded waves that includes periodic waves, solitary waves, and other types of waves (for instance, almost-periodic waves, although their existence has not been established yet). This fact suggests that the results obtained fall within the domain of the qualitative theory of differential equations (investigation of the properties of solutions without finding them). In this paper, two approaches to the qualitative theory are discussed. The first approach consists in averaging the solution along the vertical sections of the region, and the second approach is based on the authors’ modification of Byatt-Smith’s integro-differential equation. Thus, the paper contains an overview of the results obtained for the problem of nonlinear stationary waves on water with finite depth. Two approaches to this problem form a basis of the qualitative theory of such waves, because there are no constraints imposed on the waves except for the boundedness of their profiles and steepness restrictions.     

Stability properties of steady water waves with vorticity

We present two stability analyses for exact periodic traveling water waves with vorticity. The first approach leads in particular to linear stability properties of water waves for which the vorticity decreases with depth. The second approach leads to a formal stability property for long water waves that have small vorticity and amplitude although we do not use a small‐amplitude or long‐wave approximation. © 2006 Wiley Periodicals, Inc.     

Stochastic modelling of reservoir operations

The nature of hydrologic parameters in reservoir management models is uncertain. In mathematical programming models the uncertainties are dealt with either indirectly (sensitivity analysis of a deterministic model) or directly by applying a chance-constrained type of formulation or some of the stochastic programming techniques (LP and DP based models). Various approaches are reviewed in the paper. Moran's theory of storage is an alternative stochastic modelling approach to mathematical programming techniques. The basis of the approach and its application is presented. Reliability programming is a stochastic technique based on the chance-constrained approach, where the reliabilities of the chance constraints are considered as extra decision variables in the model. The problem of random event treatment in the reservoir management model formulation using reliability programming is addressed in this paper.     

Linear and nonlinear properties of electron-acoustic solitary waves with non-thermal electrons

Propagation of linear and nonlinear electron-acoustic waves (EAWs) in an unmagnetized collisionless plasma consisting of a cold electron fluid, non-thermal hot electrons and stationary ions are investigated. The standard normal-mode analysis is used to study the stability condition of linear (EAWs) waves. For nonlinear (EAWs) waves, a reductive perturbation method was employed to obtain a Korteweg–de Vries (KdV) equation for the first-order potential. The effects of a non-thermal distribution of hot electrons on the amplitude, width and energy of electrostatic solitary structures are also discussed.     

Regularity and symmetry properties of rotational solitary water waves

We prove that a solitary water wave driven by gravity has real-analytic streamlines for arbitrary vorticity functions if the flow contains no stagnation points. Based on this property, we show that if all the streamlines attain their global maximum (resp. minimum) on the same vertical line, then the solitary wave has to be symmetric and strictly monotone away from the crest (resp. trough). Our results are true for sub- and supercritical solitary waves as well.     

Stochastic nonlinear beam equations

An extensible beam equation with a stochastic force of a white noise type is studied, Lyapunov functions techniques being used to prove existence of global mild solutions and asymptotic stability of the zero solution.This research was supported in part by the GA R Grants no. 201/98/1454, 201/01/1197 and by a Royal Society grant     

Stochastic modelling of long-term investment risks

There are many risks that individuals, firms, and societieshave to face, and among them are the uncertainties of futureinvestment variables, which include inflation (both of pricesand earnings), interest rates,exchange rates, and returns onordinary shares (including both dividend income and changesin capital values).These investment risks affect inviduals intheir own financial planning; affect companies in planning investmentprojects and in arrangements for raising capital;affect governmentsand government institutions that have to borrow in the capitalmarkets; and especially affect investment institutions and intermediarieswho take on borrowings, deposits, insurance contracts, or pensionfund liabilities on the one hand and invest assets in loans,ordinary shares, property, or other investments on the other. A great deal of work done by financial economists in recentdecades has established reasonable models for describing movementsof many investment variables in the short run. Typically thesemodels are based on a 'random walk' or Gauss-Wiener continuousdiffusion process. This sort of model has been particularlyvaluable to market-makers and other investment participantswhose time horizon is short. But these short-term models oftendo not provide a satisfactory structure for the long term. Thispresentation will describe some of the author's work in thestatistical analysis of long-term investment series, both inthe United Kingdom and in other countries, based on statisticaltime-series analysis of historical data. Although many of the series could be valued using multivariatemethods, such as vector autoregressive (VAR) models, preliminaryinvestigation showed that many of the series could be investigatedin a 'cascade' fashion, with price inflation being put as theinitial 'driver'. A very long historic series shows long periodswhen changes in prices in successive years could be taken asrandom, with zero drift, and other periods (including most ofthis century) when inflation rates in successive years werecorrelated. A similar pattern has applied in recent years inmany other countries. It is postulated that the prices of ordinary shares in aggregateare closely related to the dividends paid on them, so that theratio between dividend and price, i.e. the dividend yield, isstationary—fluctuating around a constant mean. The dividend-yieldseries can be described by means of a first-order autoregressivetime-series model,while the dividend series can be describedby a model that depends on inflation in the current and precedingyears, with an appropriate time lag. Interest rates, both long-term and short-term, are first decomposedinto an allowance for prospective future inflation and a 'real'rate of interest, comparable to the yield on index-linked stocks.The real rate of interest can also be modelled as a mean-revertingautoregressive model. The allowance for future inflationcanbe derived as a moving average of past inflation rates.In orderto link models for different countries, it is necessary to havea model for currency exchange rates. This can be done by postulatinga hypothetical 'purchasingpower parity' exchange rate, whichexactly allows for changes in inflation, and then by modellingthe deviation of the actual rate from the hypothetical rateby means of yet another autoregressive model. It is necessaryalso to keep an appropriate structure for cross rates betweenany pair of currencies. This series of stochastic models is particularly useful forestimating future scenarios of all the variables in a consistentmanner, and for estimating their likely variability. In some cases this can be done analytically, but in generalit requires ’Monte Carlo‘ simulations. Various possibleapplications of the model in different fields will be described.     

Stochastic modelling of combat with reinforcement

The homogeneous combat when reinforcements are made at prespecified force levels has been modelled as a continuous-time discrete-state space Markov process. The effect of reinforcement on various combat characteristics, viz., the mean combat duration, the winning probabilities, the expected survivors, etc., is studied using the uniformization method. Two new concepts, i.e., Reinforcement Effectiveness Index and Reinforcement Parity Curve, have also been defined and are illustrated numerically.     

Stochastic modelling of tropical cyclone tracks

A stochastic model for the tracks of tropical cyclones that allows for the computerised generation of a large number of synthetic cyclone tracks is introduced. This will provide a larger dataset than previously available for the assessment of risks in areas affected by tropical cyclones. To improve homogeneity, the historical tracks are first split into six classes. The points of cyclone genesis are modelled as a spatial Poisson point process, the intensity of which is estimated using a generalised version of a kernel estimator. For these points, initial values of direction, translation speed, and wind speed are drawn from histograms of the historical values of these variables observed in the neighbourhood of the respective points, thereby generating a first 6-h segment of a track. The subsequent segments are then generated by drawing changes in theses variables from histograms of the historical data available near the cyclone’s current location. A termination probability for the track is determined after each segment as a function of wind speed and location. In the present paper, the model is applied to historical cyclone data from the western North Pacific, but it is general enough to be transferred to other ocean basins with only minor adjustments. A version for the North Atlantic is currently under preparation.     

Numerical modelling of surface water waves generated by moving underwater landslide on irregular bottom

The numerical modelling of surface water waves generated by a moving underwater landslide on irregular bottom is considered. The non-linear shallow water model is used with taking into account bottom mobility. The equations are obtained for an underwater landslide movement under the action of gravity force, buoyancy force, friction force and water resistance force. The predictor-corrector scheme [5], preserving the monotonicity of the numerical solution profiles in a linear case, is used on adaptive grids. The scheme is validated on the problem with a known analytical solution. The analysis is done for the dependencies of wave regimes on bottom slope, initial landslide depth, its length and width. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stochastic Lipschitz dynamic programming

Mathematical Programming - We propose a new algorithm for solving multistage stochastic mixed integer linear programming (MILP) problems with complete continuous recourse. In a similar way to...     

Multi-solitonic solutions for the variable-coefficient variant Boussinesq model of the nonlinear water waves

For the nonlinear and dispersive long gravity waves traveling in two horizontal directions with varying depth of the water, we consider a variable-coefficient variant Boussinesq (vcvB) model with symbolic computation. We construct the connection between the vcvB model and a variable-coefficient Ablowitz-Kaup-Newell-Segur (vcAKNS) system under certain constraints. Using the N-fold Darboux transformation of the vcAKNS system, we present two sets of multi-solitonic solutions for the vcvB model, which are expressed in terms of the Vandermonde-like and double Wronskian determinants, respectively. Dynamics of those solutions are analyzed and graphically discussed, such as the parallel solitonic waves, shape-changing collision, head-on collision, fusion-fission behavior and elastic-fusion coupled interaction.     

Enriched workflow modelling and Stochastic Branch-and-Bound

Workflow systems provide means and techniques for modelling, designing, performing and controlling repetitive (business) processes. The quality of commercial workflow systems is usually determined to a large extent by their versatility and multi-purpose application. One of the current trends in improving workflow systems lies in enriching modelling methods and techniques in order to enlarge design alternatives.The need for such advanced methods is particularly apparent in those fields in which the process duration can be determined only vaguely, but whose completion schedules are at the same time strictly enforced by a highly competitive market by means of fines and penalties. The risk of an overrun has to be weighed against the expected costs and benefits of certain measures reducing turn-around time and their combinations. Because they can help to avoid such penalties—or, at least, keep any potential losses low by identifying critical subprocesses and evaluate appropriate measures—modelling and evaluation techniques are becoming essential features of workflow systems.Methodologically, we use Stochastic Branch-and-Bound as a technique for finding “optimal” bundles of measures. A numerical study shows the benefits of this meta-approach by means of five stepwise-developed decision scenarios requiring rich modelling. Petri nets as a modelling tool and Stochastic Branch-and-Bound as an optimization technique determine for multi-mode resource constrained workflows of varying complexity an optimal workforce strategy with respect to the number of workers and their qualification.     

Local defect modelling and nonlinear dynamic analysis for the inter-shaft bearing in a dual-rotor system

This paper presents a force model for the inter-shaft bearing with a local defect on the surface of the outer race or the inner race, and the nonlinear dynamic characteristics of a dual-rotor system affected by the local defect are investigated. A simplified dual-rotor system is presented with the consideration of the inter-shaft bearing's nonlinearities such as the Hertzian contact force and the radial clearance. The local defect is considered as a regular dent with a constant depth, thus the radial clearance will increase when rolling elements go through the range of the local defect. The motion equations of the system with eight degrees of freedom are formulated by using the Lagrange's equation. The nonlinear vibration responses of the dual-rotor system affected by the local defect are obtained using numerical method. The results show that there exist four abnormal resonances on the amplitude frequency curves of the system due to the effect of the local defect, apart from the couple of primary resonances excited by the unbalance excitations of the two rotors. With the aid of the characteristic defect frequency analysis, it is revealed that one pair of the abnormal resonances are excited by the characteristic defect frequency, and the other pair of the abnormal resonances are excited by the combination frequency. Furthermore, a comprehensive parametric analysis is carried out to give an insight into the nonlinear resonant response characteristics affected by parameters such as the depth and the span of the defect, the rotation speed ratio, the unbalances of two rotors, the stiffness and the damping of the linear elastic spring, and the radial clearance, the stiffness and the roller number of the inter-shaft bearing. The results show that the vibration amplitudes for the abnormal resonances are mainly determined by the depth and the span of the defect, while the resonant frequencies for the abnormal resonances are mainly influenced by the rotation speed ratio and the roller number of the inter-shaft bearing. However, the rotors’ unbalances mainly affect the corresponding primary resonance rather than the abnormal resonances. The obtained results will contribute to a better understanding of the nonlinear resonant response characteristics of dual-rotor systems with a local defect on the inter-shaft bearing, which are helpful for the fault diagnostics of the inter-shaft bearing in a dual-rotor system.     

On a high-dimensional objective genetic algorithm and its nonlinear dynamic properties

The revival of multi-objective optimization is mainly resulted from the recent development of multi-objective evolutionary optimization that allows the generation of the overall Pareto front. This paper presents an algorithm called HOGA (High-dimensional Objective Genetic Algorithm) for high-dimensional objective optimization on the basis of evolutionary computing. It adopts the principle of Shannon entropy to calculate the weight for each object since the well-known multi-objective evolutionary algorithms work poorly on the high-dimensional optimization problem. To further discuss the nonlinear dynamic property of HOGA, a martingale analysis approach is then employed; some mathematical derivations of the convergent theorems are obtained. The obtained results indicate that this new algorithm is indeed capable of achieving convergence and the suggested martingale analysis approach provides a new methodology for nonlinear dynamic analysis of evolutionary algorithms.