A consistent value for games with n players and r alternatives

In Bolger [1993], an efficient value was obtained for a class of games called games with n players and r alternatives. In these games, each of the n players must choose one and only one of the r alternatives. This value can be used to determine a player’s “a priori” value in such a game. In this paper, we show that the value has a consistency property similar to the “consistency” for TU games in Hart/Mas-Colell [1989] and we present a set of axioms (including consistency) which characterizes this value.  The games considered in this paper differ from the multi-choice games considered by Hsiao and Raghavan [1993]. They consider games in which the actions of the players are ordered in the sense that, if i >j, then action i carries more “weight” than action j.  These games also differ from partition function games in that the worth of a coalition depends not only on the partitioning of the players but also on the action chosen by each subset of the partition. Received: April 1994/final version: June 1999     

On steiner systems

Steiner’s “combinatorial problems” have so far been solved only fork=3 [5, 3] and fork=4 [1,2]. In this paper a complete solution of the problem is given for “closed” Steiner systems, i.e. systems havingn=2 ^k−1−1 elements. Use is made of methods developed by Zaremba [7] for abelian groups. This research was supported by the United States Air Force under Grant No. AF-EOAR-6360 and monitored by the European Office of Aerospace Research.     

KAM Stability for a three-body problem of the Solar system

A new (iso-energetic) KAM method is tested on a specific three-body problem “extracted” from the Solar system (Sun-Jupiter + asteroid 12 Victoria). Analytical results in agreement with the observed data are established. This paper is a concise presentation of [2]. Supported by the MIUR projects: “Dynamical Systems: Classical, Quantum, Stochastic” and “Variational Methods and Nonlinear Differential Equations” Received: February 3, 2004     

Enclosure theorems for generalized mean curvature flows

We define a generalized notion of mean curvature for regular hypersurfaces in . This enables us to introduce a new class of geometric curvature flows for which we prove enclosure theorems, using methods of Dierkes [D] and Hildebrandt [H]. In particular, we obtain “neck-pinching” results that generalize previous observations by Ecker [E] concerning the classical mean curvature flow. Received: 8 October 2001 / Accepted: 1 March 2002 / Published online: 23 May 2002     

An Asymmetric Shapley–Shubik Power Index

This paper extends the traditional “pivoting” and “swing” schemes in the Shapley–Shubik (S-S) power index and the Banzhaf index to the case of “blocking”. Voters are divided into two groups: those who vote for the bill and those against the bill. The uncertainty of the division is described by a probability distribution. We derive the S-S power index, based on a priori ignorance about the random bipartition.     

The projective rank of a Hermitian symmetric space: A geometric approach and consequences

The “Projective Rank” of a compact connected irreducible Hermitian symmetric space M has been defined as the maximal complex dimension of the compact totally geodesic complex submanifolds having positive holomorphic bisectional curvature with the induced K?hler metric. We present a geometric way to compute this invariant for the space M based on ideas developed in [1], [13] and [14]. As a consequence we obtain the following inequality relating the Projective Rank, the usual rank, and the 2-number (which is known to be equal to the Euler-Poincare characteristic in these spaces). Received: 6 June 2000 / Revised version: 6 August 2001 / Published online: 4 April 2002     

A propos du lemme fondamental pondéré tordu

In order to use the trace formula of Arthur–Selberg in the twisted case, we need to prove the “twisted weighted fundamental lemma”, that is a sophisticated version of the fundamental lemma. Here, we prove that this twisted weighted fundamental lemma follows from two others lemmas, where the torsion has disappeared: the weighted fundamental lemma for Lie algebras and a “non-standard weighted fundamental lemma”, concerning Lie algebras too.     

Notes on trimodal singularities

Singularities of functions of modality 1 or 2 were classified by Arnold [3]. Types of those of modality 3 were listed in [14]. Here we show that these are all Newton non-degenerate in the extended sense introduced in [17], and use this fact to obtain explicit regular bases and precise normal forms for right and for contact equivalence. This requires care since the only existing method for non-semiquasihomogeneous germs (using Arnold's “Condition A”) does not apply to all our cases. Received: 8 February 1999 / Revised version: 28 May 1999     

Modified potential energy functions for constrained molecular dynamics

In molecular dynamics the highly oscillatory vibrations in the chemical bonds are often replaced by holonomic constraints that freeze the bond length/angle to its equilibrium value. In some cases this approach can be justified if the force constants of the bond vibrations are sufficiently large. However, for moderate values of the force constant, the constrained system might lead to a dynamical behavior that is too “rigid” compared to the flexible model. To compensate for this effect, the concept of soft constraints was introduced in [7,12,13]. However, its implementation is rather expensive. In this paper, we suggest an alternative approach that modifies the force field instead of the constraint functions. This leads to a more efficient method that avoids the resonance induced instabilities of multiple-time-stepping [5] and the above described effect of standard constrained dynamics. This revised version was published online in June 2006 with corrections to the Cover Date.     

Diametral theory of algebraic surfaces and geometric theory of invariants of groups generated by reflections. I

A review of the current state of the diametral theory of algebraic hypersurfaces in the real Euclidean space is given. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 5, pp. 639–653, May, 1998. The present paper is based on talks delivered by the author at the International Conference on Geometry “as a Whole” [1] and the Second Crimean Mathematical School “Method of Lyapunov Functions and Its Applications” [2].     

A unifying approach to abstract matrix algebra preconditioning

In earlier papers Tyrtyshnikov [42] and the first author [14] considered the analysis of clustering properties of the spectra of specific Toeplitz preconditioned matrices obtained by means of the best known matrix algebras. Here we generalize this technique to a generic Banach algebra of matrices by devising general preconditioners related to “convergent” approximation processes [36]. Finally, as case study, we focus our attention on the Tau preconditioning by showing how and why the best matrix algebra preconditioners for symmetric Toeplitz systems can be constructed in this class. Received April 25, 1997 / Revised version received March 13, 1998     

On the quantitative subspace theorem

In this survey, we give an overview of recent improvements to the quantitative subspace theorem, obtained jointly with R. Ferretti, which follow from the work in [9]. Further, we give a new gap principle with which we can estimate the number of subspaces containing the “small solutions” of the systems of inequalities under consideration. As an introduction, we start with a quantitative version of Roth’s theorem. Bibliography: 28 titles.     

Operatortheoretische Behandlung und Verallgemeinerung eines Problemkreises in der komplexen Funktionentheorie

Ohne Zusammenfassung Vorliegende Arbeit gibt eine einheitliche und systematische Darstellung von Resultaten der Verfasser, über die teils schon in den Aufs?tzen [2], [7] (“Problem C”) und [8] (“Problem A”) berichtet wurde.     

Structure of radicals in bands of semigroups

It is well-known that semigroups of many important classes are decomposable into bands of their subsemigroups with more “rigid” structure. For efficient applications of radicals to semigroups of that kind, an information on the shape of radicals in bands of semigroups may be useful. Here we will investigate the Jacobson, Baer, Brown-McCoy and the least special radicals. The results were announced in [3]. The interaction of bands and radicals was also considered in [4] for the class of semigroups with zero; analogous problems for bands and radicals of associative rings were investigated in [5], [6] and [7].     

Some observations on repeated spring balance weighing designs

Summary Dey [3] has suggested a spring balance weighing design in preference to “repeated designs”, and later, Kulshreshtha and Dey [5] have suggested yet one more weighing design which, they say, would be preferred to “repeated designs” and to those suggested in [3], provided one is interested in estimating the weights of some of the objects with increased precision at the cost of precision for others. It has been shown here that, while the above findings may be true in some situations, one might, in a given problem, prefer “repeated designs” to those suggested in [3] and [5]. NSF Grant No. GP-28312 and GP-36562.     

An integral test for a critical multitype spatially homogeneous branching particle process and a related reaction-diffusion system

An integral test (Theorem 5) is established for the dichotomy concerning local extinction and survival (even persistence) at late times for critical multitype spatially homogeneous branching particle systems in continuous time. Our conditions on the branching mechanism are close to the ones known from “classical” processes without motion component. This generalizes and complements results of López-Mimbela and Wakolbinger [LMW96] and others. Our approach is based on some genealogical tree analysis combined with the study of the long-term behavior of L ¹-norms of solutions of related systems of reaction-“diffusion” equations, which is perhaps also of some independent interest. Received: 13 August 1997 / Revised version: 12 May 1998 / Published online: 14 February 2000     

Three-page approach to knot theory. Encoding and local moves

In the present paper, we suggest a new combinatorial approach to knot theory based on embeddings of knots and links into a union of three half-planes with the same boundary. The idea to embed knots into a “book” is quite natural and was considered already in [1]. Among recent papers on embeddings of knots into a book with infinitely many pages, we mention [2] and [3] (see also references therein). The restriction of the number of pages to three (or any other number ≥3) provides a convenient way toencode links by words in a finite alphabet. For those words, we give a finite set of local changes that realizes the equivalence of links by analogy with the Reidemeister moves for planar link diagrams. This work is partially supported by Russian Foundation for Basic Research grant No. 99-01-00090. Moscow State University. Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 33, No. 4, pp. 25–37, October–December, 1999. Translated by I. A. Dynnikov     

Non-embeddable simple relation algebras

This paper presents solutions or partial solutions for several problems in the theory of relation algebras. In a simple relation algebra an element x satisfying the condition (a) must be an atom of . It follows that x must also be an atom in every simple extension of . Andréka, Jónsson and Németi [1, Problem 4] (see [12, Problem P5]) asked whether the converse holds: if x is an atom in every simple extension of a simple relation algebra, must it satisfy (a)? We show that the answer is “no”.? The only known examples of simple relation algebras without simple proper extensions are the algebras of all binary relations on a finite set. Jónsson proposed finding all finite simple relation algebras without simple proper extensions [12, Problem P6]. We show how to construct many new examples of finite simple relation algebras that have no simple proper extensions, thus providing a partial answer for this second problem. These algebras are also integral and nonrepresentable.? Andréka, Jónsson, Németi [1, Problem 2] (see [12, Problem P7]) asked whether there is a countable simple relation algebra that cannot be embedded in a one-generated relation algebra. The answer is “yes”. Givant [3, Problem 9] asked whether there is some k such that every finitely generated simple relation algebra can be embedded in a k-generated simple relation algebra. The answer is “no”. Received November 27, 1996; accepted in final form July 3, 1997.