On some guiding principles in mathematical modelling with special emphasis on determinism

Various principles guiding the construction of mathematical models are isolated-selection of features to be considered in the model, character of time flow, character of space representation, determinism, causality, etc. It is shown how different choices of each of these items give rise to different classes of models—e.g., static vs dynamic, discrete time vs continuous time, lumped space vs distributed space, input–output systems vs stochastic systems, etc. Both causality and determinism are formalized in set theoretical terms. The validation problem of the models arrived at on deterministic-causal assumptions is dealt with along empiricist lines, the need for considering stochastic models emerging out of these considerations. A critical examination of determinism is given, out of which results the contention that determinism in a model merely expresses an attitude of the modeller towards the real world rather than an intrinsic property of a real system. Finally the conventional character of determinism is seen to be shared by the various other principles guiding the activity of the modeller, principles whose adoption is justified or invalidated only in terms of the comparison of the resulting model's predictions with actual observations.     

Napier revisited or A new look at the computation of his logarithms

After 400 years it seems improbable that something new can be said about the way Napier computed his logarithms. Time has passed over these logarithms and made them as well as the ways of their computation obsolete almost immediately after their publication. In particular we had to wait exactly 270 years until a translation of the Latin original of the Constructio into English came into being, and this shows clearly that the interest in Napier’s work is confined to the realm of history. But even there we find nothing but the facts which are well known for almost two centuries. In contrast, we propose to follow the path opened by one of us and to have a new and fresh look at the rules and methods which had to be formulated from scratch by John Napier, as well as their possible applications.     

A geometric look at nuisance parameter effect of local powers in testing hypothesis

This paper is concerned with the theory of testing hypothesis with composite null hypothesis or with nuisance parameters. The asymptotic behaviour of the likelihood ratio and the associated test statistics are investigated. Under a class of local alternatives with local orthogonality relative to the nuisance parameter vector, a unique decomposition of local power is presented. The decomposition consists of two parts; one is the influence of nuisance parameters and the other is the power corresponding to the simple case where the nuisance parameters are known. The decomposition formula is applied to some examples, including the gamma, Weibull and location-scale family.     

Modified weighted squared error estimation procedures with special emphasis on the stable laws

Two families of parameter estimation procedures for the stablelaws based on a variant of the characteristic function are provided. The methodology which produces viable computational procedures for the stable laws is generally applicable to other families of distributions across avariety of settings. Both families of procedures may be described as a modified weighted chi-squared minimization procedure, and both explicitlytake account of constraints on the parameter space. Influence functions for and efficiencies of the estimators a r e given. If _l, x₂, ..., x_n is a random sample from an unknown distribution F, a method for determining the stable law to which F is attracted is developed. Procedures for regression and autoregression with stable error structurear e provided. A number of examples are given.     

A prospective look at the OR system

The paper examines the long term problems of operational research through an analysis of OR as a system. The general environment will change as a result of the transformations occurring within the developed societies and of the mutations in the North-South relations; the interaction between OR, mass media and public opinion will be influenced by the emerging ideologies; the naive conceptions of the division of labour between policy makers and OR specialists will have to be revised as OR studies will be more and more developed for governments through national administrations, international organizations or public opinions; the formulation of goals will no longer be separated from the search for institutions liable to elaborate suitable goals; the tendency towards a systems analysis inspired by general systems theory should be reinforced; and finally the OR people will have to rethink the relations between their applied studies, the central core of OR itself and the various social sciences. Time seems to have come for an epistemology and a prospective study of OR.     

A new look at the brachistochrone problem

Summary The classic brachistochrone problem is solved as an optimal control problem in conjunction with the use of Chebyshev series.

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Résumé Le problème classique du brachistochrone est résolu comme un problème de contrôle optimale à l'aide des séries de Chebyshev.

     

Analysis of variable-stepsize linear multistep methods with special emphasis on symmetric ones

In this paper we deal with several issues concerning variable-stepsize linear multistep methods. First, we prove their stability when their fixed-stepsize counterparts are stable and under mild conditions on the stepsizes and the variable coefficients. Then we prove asymptotic expansions on the considered tolerance for the global error committed. Using them, we study the growth of error with time when integrating periodic orbits. We consider strongly and weakly stable linear multistep methods for the integration of first-order differential systems as well as those designed to integrate special second-order ones. We place special emphasis on the latter which are also symmetric because of their suitability when integrating moderately eccentric orbits of reversible systems. For these types of methods, we give a characterization for symmetry of the coefficients, which allows their construction, and provide some numerical results for them.

     

Estimates for the probability of ruin with special emphasis on the possibility of large claims

The present paper investigates, for the general Andersen model, the asymptotic behaviour of the probability of ruin function when the initial risk reserve tends to infinity. Whereas the exponential (Cramér) case is well understood, in the past, less attention has been paid to a systematic study of a model taking big claim sizes into account. We give a thorough treatment of the latter and also review previously known but mostly scattered results to show how they all follow from essentially one mathematical model.     

A new look at the optimal assignment problem

Let E=[e_ij] be a matrix with integral elements. We study matrices of the form X=[e^e_ij], where x is an indeterminate defined over the rational field Q. There is a fascinating interplay between the combinatorial structures of the matrices E and X. For example, in the case of square matrices the number of optimal assignments in the matrix E is precisely the coefficient of the term of highest degree in the polynomial per(X). We allow the multiplication of rows and columns of X by arbitrary integral powers of x, and we study topics such as the diagonal products of X. We apply certain familiar results to this “exponential” setting, and we also introduce some new concepts, such as the class of matrices S(X) generated by a matrix X and the notion of an invariant 1 within such a class.     

A new look at the homogeneous risk model

The present paper aims to revisit the homogeneous risk model investigated by [De Vylder and Goovaerts, 1999] and [De Vylder and Goovaerts, 2000]. First, a claim arrival process is defined on a fixed time interval by assuming that the arrival times satisfy an order statistic property. Then, the variability and the covariance of an aggregate claim amount process is discussed. The distribution of the aggregate discounted claims is also examined. Finally, a closed-form expression for the non-ruin probability is derived in terms of a family of Appell polynomials. This formula holds for all claim distributions, even dependent. It generalizes several results obtained so far.     

A new look at the homogeneous risk model

The present paper aims to revisit the homogeneous risk model investigated by De Vylder and Goovaerts, 1999, De Vylder and Goovaerts, 2000. First, a claim arrival process is defined on a fixed time interval by assuming that the arrival times satisfy an order statistic property. Then, the variability and the covariance of an aggregate claim amount process is discussed. The distribution of the aggregate discounted claims is also examined. Finally, a closed-form expression for the non-ruin probability is derived in terms of a family of Appell polynomials. This formula holds for all claim distributions, even dependent. It generalizes several results obtained so far.     

Reciprocity principles associated with the general linear gravity wave problem

Summary Reziprocity principles are presented which hold for the general problem of small amplitude wave motion on the surface of an incompressible liquid. The effects of viscosity, surface tension, irrotationality, free surface sources and solid boundary geometry at liquid-solid intersections are discussed.

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Zusammenfassung Reziprozitätsprinzipien, die für das allgemeine Problem der Wellenbewegung mit kleinen Amplituden auf der Oberfläche einer inkompressiblem Flüssigkeit gelten, werden angegeben. Die Einflüsse von Viskosität, Oberflächenspannung, Wirbelfreiheit, Quellen an der freien Oberfläche un die Geometrie fester Ränder bei flüssig-festen Übergängen werden diskutiert.

     

A new look at the chemical master equation

This paper discusses the numerical simulation of stochastic chemical kinetics from a point of view that is opposite to the one prevalent in the literature. Specifically, we re-derive existing methods by first discretizing in time the chemical master equation, then sampling the resulting numerical solution (which is a probability density). This analysis reveals the hidden approximations made by the stochastic simulation algorithm and by the tau-leaping method, and opens the way for constructing new families of solvers.     

A Hamiltonian look at the Rikitake two-disk dynamo system

In this paper we will describe some relevant dynamical properties of the Rikitake dynamo system from the Poisson geometry point of view.