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.     

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.     

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     

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 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 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.     

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.     

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.     

Stochastic dynamic nursing service budgeting

We address the nursing service budgeting problem from the department manager’s point of view. The model allocates the budget dynamically to three types of nursing care capacities: 1) permanent nurses, 2) temporary nurses, and 3) overtime. The quarterly tactical decisions are the aggregate weekly shift pattern of permanent nurses and the policy for hiring temporary nurses and using overtime. The decisions are optimized with respect to nursing care shortage and a soft-constraint on the annual budget. For the aggregate weekly shift pattern, permanent nurses require a notification lead-time of one quarter to prepare the personal rosters. Our model offers a solution to the nursing service budgeting problem that extends the existing literature by using a Markovian demand model, resolving the anticipation of the operational decisions, and applying general budget as well as shortage penalty functions.     

Stochastic modelling and optimization for environmental management

In modelling and managing complex environmental systems, inherent uncertainties of all relevant natural processes are to be taken into consideration. In the present paper diverse stochastic modelling and optimization approaches for handling such problems (primarily in the field of water quality analysis and control) are highlighted, drawing on the findings of case studies and real-world applications.     

Instability of nonlinear dispersive solitary waves

We consider linear instability of solitary waves of several classes of dispersive long wave models. They include generalizations of KDV, BBM, regularized Boussinesq equations, with general dispersive operators and nonlinear terms. We obtain criteria for the existence of exponentially growing solutions to the linearized problem. The novelty is that we dealt with models with nonlocal dispersive terms, for which the spectra problem is out of reach by the Evans function technique. For the proof, we reduce the linearized problem to study a family of nonlocal operators, which are closely related to properties of solitary waves. A continuation argument with a moving kernel formula is used to find the instability criteria. These techniques have also been extended to study instability of periodic waves and of the full water wave problem.