Structure of weak Descartes systems

A weak Descartes system is a basis of functions such that every ordered subset is a weak Tchebycheff system, the canonical example being the usual spline basis involving truncated power functions. By examining the intervals of degeneracy for a WD-system, we show that it is possible to produce a new basis that has a simple and convenient structure similar to the spline basis.     

Software for the Gale transform of fewnomial systems and a Descartes rule for fewnomials

We give a Descartes’-like bound on the number of positive solutions to a system of fewnomials that holds when its exponent vectors are not in convex position and a sign condition is satisfied. This was discovered while developing algorithms and software for computing the Gale transform of a fewnomial system, which is our main goal. This software is a component of a package we are developing for Khovanskii-Rolle continuation, which is a numerical algorithm to compute the real solutions to a system of fewnomials.     

Multiplicities of focal points for discrete symplectic systems: revisited

In this note, we define a notion of multiplicity of focal points for conjoined bases of discrete symplectic systems. We show that this definition is equivalent to the one given by Kratz in [Discrete oscillation, J. Difference Equ. Appl., 9(1), 135–147 (2003)] and, furthermore, it has a natural connection to the newly developed continuous time theory on linear Hamiltonian differential systems. Many results obtained recently by Bohner, Do?lý, and Kratz regarding the non-negativity of the corresponding discrete quadratic functionals, Sturmian separation and comparison theorems, and oscillation theorems relating the number of focal points of a certain special conjoined basis with the number of eigenvalues of the associated discrete symplectic eigenvalue problem, are now formulated in terms of this alternative definition of multiplicities.     

The nucleolus of a standard tree game revisited: a study of its monotonicity and computational properties

This paper introduces yet another algorithm to compute the nucleolus of a standard tree game. One advantage of this algorithm is that it provides a very intuitive interpretation of the nucleolus, under which the players participate in a joint enterprize in which each group sends a member to help the community. Another advantage is that it demonstrates monotonicity properties of the nucleolus within this class of games. As a consequence the nucleolus of a tree game can be extended to a population monotonic allocation scheme.     

The epistemology of complex systems

The human knowledge structure is acquired by a very complex process of learning. ‘Folk knowledge’, acquired in ordinary daily life, may be distinguished from ‘Scholarly knowledge’. Folk knowledge has images which range from about 10⁵ to 10⁻⁵ of the human size; scholarly knowledge produces images which range from about 10²⁰ to 10⁻²⁰ of the human size. Knowledge results from the interaction of internal and external messages. Internal messages produce perceptions of identities—propositions which cannot be untrue, and also perceptions of ‘near-identities’—propositions which have a high probability of being true. Beyond these are the empirical propositions, derived from ordered observation of the records of the past and the present, and from experiment. These are always subjet both to error, and the reduction of error. Each field of knowledge has to find its appropriate method, depending on the nature of the system about which knowledge is sought. These systems may be classified as predictable (with stable parameters) and partially predictable (systems involving information, or subject to parametric change.) It is a mistake to transfer methods appropriate in one field to another, where they may be inappropriate.     

The relevance of system dynamics to engineering systems design

With the recent advent of new technology using micro computers for the centralised monitoring of information, the scope for totally automatic, real time control of large engineering systems has been advanced. We are entering a period which, retrospectively, may well be seen as the era of control; where for the first time it is both feasible and necessary to design system in terms of the way in which they will be operated, as well as in the traditional terms of their capacity requirements. However, before such comprehensive design can be undertaken compatible advances are required in information usage technology.The purpose of this paper is to demonstrate how the subject of System Dynamics can be used to assist with this problem, though its capacity to model information feedback and hence to model and test alternative system control rules based on the information retrieved. This type of application in the hard system area of engineering provides a strong contrast to the more usual applications of System Dynamics in softer socio-economic systems.The problem described is taken from the coal mining industry and related to the design of large scale underground conveyor belt systems used for the clearance of coal from mines. The demonstration presented, therefore, concerns research in this field, but has considerable scope for application in the general area of large scale bulk handling system.Of particular importance in the results is an indication of the potential savings in physical capacity which can stem from the careful design and implementation of control in such systems.     

The history and ideas of Marxism: the relevance for OR

Journal of the Operational Research Society - The paper examines the origins of Marxism in Europe in the second half of the 19th century in the context of the industrial and political revolutions...     

Descartes and the birth of analytic geometry

The traditional thesis that analytic geometry evolved from the concepts of axes of reference, co-ordinates, and loci, is rejected. The origins of this science are re-defined in terms of Egyptian, Greek, Babylonian, and Arabic influences merging in Vieta's Isagoge in artem analyticam (1591) and culminating in a work of his pupil Ghetaldi published posthumously in 1630. Descartes' Vera mathesis, conceived over a decade earlier, served to revive and strengthen the important link with logic and thereby to extend the field of application of this analytic method to the corporeal and moral worlds.     

The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems

In this paper, the Conley conjecture, which was recently proved by Franks and Handel [J. Franks, M. Handel, Periodic points of Hamiltonian surface diffeomorphism, Geom. Topol. 7 (2003) 713-756] (for surfaces of positive genus), Hingston [N. Hingston, Subharmonic solutions of Hamiltonian equations on tori, Ann. Math., in press] (for tori) and Ginzburg [V.L. Ginzburg, The Conley conjecture, arXiv: math.SG/0610956v1] (for closed symplectically aspherical manifolds), is proved for C¹-Hamiltonian systems on the cotangent bundle of a C³-smooth compact manifold M without boundary, of a time 1-periodic C²-smooth Hamiltonian H:R×T^*M→R which is strongly convex and has quadratic growth on the fibers. Namely, we show that such a Hamiltonian system has an infinite sequence of contractible integral periodic solutions such that any one of them cannot be obtained from others by iterations. If H also satisfies H(−t,q,−p)=H(t,q,p) for any (t,q,p)∈R×T^*M, it is shown that the time-1-map of the Hamiltonian system (if exists) has infinitely many periodic points siting in the zero section of T^*M. If M is C⁵-smooth and dimM>1, H is of C⁴ class and independent of time t, then for any τ>0 the corresponding system has an infinite sequence of contractible periodic solutions of periods of integral multiple of τ such that any one of them cannot be obtained from others by iterations or rotations. These results are obtained by proving similar results for the Lagrangian system of the Fenchel transform of H, L:R×TM→R, which is proved to be strongly convex and to have quadratic growth in the velocities yet.     

The workshop on p-adic methods for modeling of complex systems

We present a brief information on “The Workshop on p-Adic Methods for Modeling of Complex Systems”, which was held in the Center for Interdisciplinary Research (Zentrum für interdisziplinäre Forshung — ZiF), Bielefeld University, Bielefeld, Germany, April 15–19, 2013.     

Stochastic equations for complex systems

Book Reviews

Stochastic equations for complex systemsA. V. Skorohod: D. Reidel, Dordrecht, 1988     

The coninvolutory decomposition and its computation for a complex matrix

A complex, square matrix E is called coninvolutory if EE = I, where E denotes complex conjugate of the matrix E and I is an identity matrix. In this paper we introduce the coninvolutory decomposition of a complex matrix and investigate a Newton iteration for computing the coninvolutory factor. A simple numerical example illustrates our results.     

The theory of self-organization and modelling complex urban systems

In this article we focus on the problem the social scientist is confronted with when trying to model the complex and dynamic world he lives and works in, and on the different scientific paradigms of the ‘hard sciences’ which are at his disposition. We further concentrate our attention on the EMUS (=Evolutionary Models of Urban Systems) models which we developed using the concepts of the theory of Self-Organization and which we believe to be capable of unravelling some of the complexity of the real world. We will not dwell on the mathematical foundations of the model, but concentrate on the philosophical basis, some practical considerations about modelling and forecasting and on results we obtained on different spatial scales and in different socio-economic contexts.