Characterization of regular singular linear systems of difference equations

This paper deals with linear systems of difference equations whose coefficients admit generalized factorial series representations atz=. We are concerned with the behavior of solutions near the pointz= (the only fixed singularity for difference equations). It is important to know whether a system of linear difference equations has a regular singularity or an irregular singularity. To a given system () we can assign a number , called the Moser's invariant of (), so that the system is regular singular if and only if 1. We shall develop an algorithm, implementable in a computer algebra system, which reduces in a finite number of steps the system of difference equations to an irreducible form. The computation ot the number can be done explicitly from this irreducible form.     

A Niche Width Model of Optimal Specialization

Niche width theory, a part of organizational ecology, predicts whether specialist or generalist forms of organizations have higher fitness, in a continually changing environment. To this end, niche width theory uses a mathematical model borrowed from biology. In this paper, we first loosen the specialist-generalist dichotomy, so that we can predict the optimal degree of specialization. Second, we generalize the model to a larger class of environmental conditions, on the basis of the model's underlying assumptions. Third, we criticize the way the biological model is treated in sociological theory. Two of the model's dimensions seem to be confused, i.e., that of trait and environment; the predicted optimal specialization is a property of individual organizations, not of populations; and, the distinction between fine and coarse grained environments is superfluous.     

Compact elements in the lattice of radicals

Summary A radical R, in the sense of Kurosh--Amitsur, is said to be compact if, given any collection of radicals X such that R ≤;VX, we have R ≤;VX' for some finite subcollection X' of X. A ring A is said to be radical compact if the lower radical on the singleton {A} is compact. This paper explores the relationship between radical compact rings and rings satisfying certain finiteness conditions. Closure properties of the class of all radical compact rings are also investigated.     

Klein polyhedra for three extremal cubic forms

Davenport and Swinnerton-Dyer found the first 19 extremal ternary cubic forms g _i, which have the same meaning as the well-known Markov forms in the binary quadratic case. Bryuno and Parusnikov recently computed the Klein polyhedra for the forms g ₁ – g ₄. They also computed the convergents for various matrix generalizations of the continued fractions algorithm for multiple root vectors and studied their position with respect to the Klein polyhedra. In the present paper, we compute the Klein polyhedra for the forms g ₅, – g ₇ and the adjoint form g ₇ ^* . Their periods and fundamental domains are found and the expansions of the multiple root vectors of these forms by means of the matrix algorithms due to Euler, Jacobi, Poincaré, Brun, Parusnikov, and Bryuno, are computed. The position of the convergents of the continued fractions with respect to the Klein polyhedron is used as a measure of quality of the algorithms. Eulers and Poincarés algorithms proved to be the worst ones from this point of view, and the Bryuno one is the best. However, none of the algorithms generalizes all the properties of continued fractions.Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 566–583.Original Russian Text Copyright © 2005 by V. I. Parusnikov.This revised version was published online in April 2005 with a corrected issue number.     

“Quasiphoton” in an active medium. II. An example: The one-dimensional wave equation with a small deviating argument

In the present work we use the example of the one-dimensional wave equation with a small deviating argument to illustrate the constructions of the author's previous paper Quasiphoton in an active medium. The significance of the concept of propagation velocity introduced there, as well as the relation between the change in the propagation frequency along a ray and the parameters characterizing the concentrated solution, are discussed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 156, pp. 136–142, 1986.In conclusion, the author wants to express his sincrere gratitude to I. A. Molotkov for suggesting this problem, and to the participants of the diffraction seminar of LGU-LOMI for stimulating discussions.     

A Measure of Simultaneous Approximation for Quasi-Modular Functions

We improve on a recent result of Saradha giving a transcendence measure for the quotient of a period of an elliptic curve defined over by its associated quasi-period. In an (almost successful) attempt to include in a single measure both this result and that obtained by Reyssat in 1980, we blend into the modular method ideas related to modular but also hypergeometric functions, as appearing e.g. in André's work, as well as some Galois considerations.     

Some Issues in the Theory of Optimal Algorithms and Informational Complexity

The article presents the basic concepts and reviews the main results of the theory of optimal algorithms and informational complexity. Informational complexity bounds are provided for Lipschitzian multi-criterion problems that construct the approximate Pareto-optimal strategy set under different interpretations of approximation—approximation by the functional and approximation by the argument. The informational complexity is compared for the scalar global optimization problem and the problem of finding the roots of nonlinear equations by global search methods.     

Remarks on dynamical systems with weak forces

We study the existence of T-periodic solutions for the forced system of differential equations –ÿ=V(y)+h(t), where h is T-periodic and V is defined in an open, bounded and convex subset of R^N and V– as y tends to the boundary of the set. We prove existence of at least one solution only requiring convexity of V near the boundary.Supported by Ministero Pubblica Istruzione, Gruppo Nazionale (40%) Calcolo delle Variazioni     

Bounds of mixed semi-invariants of random processes

The paper deals with questions of bounds on mixed semi-invariants of a random process x_t which satisfies a mixing condition either according to Rosenblatt or according to Ibragimov.Translated from Matematicheskie Zametki, Vol. 13, No. 4, pp. 581–586, April, 1973.     

A characteristic property of elliptic Plücker transformations

We discuss elliptic Plücker transformations of three-dimensional elliptic spaces. These are permutations on the set of lines such that any two related (orthogonally intersecting or identical) lines go over to related lines in both directions. It will be shown that for classical elliptic 3-spaces a bijection of its lines is already a Plücker transformation, if related lines go over to related lines. Moreover, if the ground field admits only surjective monomorphisms, then bijection can be replaced by injection.Dedicated to Walter Benz on the occasion of his 65th birthday     

A result from diophantine approximation which has applications in economics

We say that a real number allows poor approximations if we can find 0<=()<1 and a sequence of integers n₁2<... such that for all rationals p/q with qn. we have |–.p/q| > Kn _j ^–l– where K is a constant depending only on .In this note we prove that the set of numbers which allow poor approximations are precisely the very well-approximable numbers.The existence of numbers with poor approximations has been used by Cheng [1] to show the existence of a dense set of economies whose cone converges to the Walras equilibrium as slowly as 0(n^–1/2–) after n replications.     

Parameters of the periodic response of a system with friction-affected constraints and harmonic excitation

Summary The periodic response of a single-freedom multi-body system with friction-affected constraints acted upon by a harmonic excitation is determined by numerical simulation. The contribution of the constraints to the generalized friction force, the influence of gravity and zero-gravity environment, and the consequences, when several constraints are considered without friction, are investigated. The effects of a small artificial damping on the response, and the possibilities of an equivalent damping for the friction-affected constraints are also examined.

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Zusammenfassung Die periodische Bewegung eines Mehrkörpersystems mit einem Freiheitsgrad und reibungsbehafteten Bindungen bei harmonischer Erregung wird durch numerische Simulation bestimmt. Untersucht werden der Beitrag der Bindungen zur verallgemeinerten Reibungskraft, der Einfluß des Schwerefeldes und der Schwerelosigkeit und die Folgen, wenn einige Bindungen als reibungsfrei angenommen werden. Die Auswirkungen einer kleinen künstlichen Dämpfung auf die Bewegung, und die Möglichkeiten einer äquivalenten Dämpfung an Stelle der reibungsbehafteten Bindungen wird auch untersucht.

     

Theory of the classical gravitational field and Mach's principle

Equations for a massive gravitational field are found in the framework of the special theory of relativity on the geometrization principle. The existence of a graviton mass has fundamental significance for the construction of the theory. In accordance with this theory of gravitation, a homogeneous and isotropic universe evolves cyclically from a high density to a minimum density, etc., and can only be flat. The theory predicts the existence of an appreciable amount of dark matter in the universe. The existence of black holes in the universe is completely ruled out. The theory explains all known observational facts in the solar system.Institute of High Energy Physics. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 1, pp. 3–27, October, 1994.     

Жизнь и деятельность яноша Бояи

: . BOLYAI.     

Simplifying assumptions for the order of partitioned multivalue methods

In solving a system of ordinary differential equations by multivalue methods, the concept of partitioning such a method into two parts corresponding to internal and external computations has recently been introduced by Burrage and Butcher [3]. In this paper the order conditions for such methods will be given and simplifying assumptions for them derived. Various examples are presented and, in particular, a new class of methods is introduced and its order properties are studied.     

Discrete Newton methods and iterated defect corrections

Summary In this paper we present a general theory for discrete Newton methods, iterated defect corrections via neighbouring problems and deferred corrections based on asymptotic expansions of the discretization error.Dedicated to Professor Dr. J. Weisinger on the occasion of his sixty-fifth birthday     

An Introduction to Divisible Codes

This paper presents some basic facts about divisible codes, culminating in a divisible version of the Gleason-Pierce theorem on self-dual codes.     

О предельном распределении для сумм независимых слууайных велиуин на бикомпактных коммутативных топологиуеских группах

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“Cautious” utility maximization and iterated weak dominance

This paper examines the implications of assuming that rational players in a two-player noncooperative game maximize expected utility using cautious beliefs; i.e., beliefs that give positive probability to all rational strategies of the other players. The result is a solution concept that captures some of the flavor of but is not equivalent to the iterated elimination of weakly dominated strategies. The solution concept yields attractive results for some games, but suffers from nonexistence problems in other games. We interpret the nonexistence problems as showing that it is logically inconsistent to assume that rational players analyze all games on the basis of cautious expected utility maximization.We would like to thank Pierpaolo Battigalli, two anonymous referees, and an editor for helpful comments. Tilman Börgers gratefully acknowledges financial assistance from the Schweizerischer Nationalfonds.