On a question of H. Kraljević

In this paper, assuming a certain set-theoretic hypothesis, a positive answer is given to a question of H. Kraljevi, namely it is shown that there exists a Lebesgue measurable subsetA of the real line such that the set {c R: A + cA contains an interval} is nonmeasurable. Here the setA + cA = {a + ca: a, a A}. Two other results about sets of the formA + cA are presented.     

Values for two-stage games: Another view of the Shapley axioms

This short study reports an application of the Shapley value axioms to a new concept of two-stage games. In these games, the formation of a coalition in the first stage entitles its members to play a prespecified cooperative game at the second stage. The original Shapley axioms have natural equivalents in the new framework, and we show the existence of (non-unique) values and semivalues for two stage games, analogous to those defined by the corresponding axioms for the conventional (one-stage) games. However, we also prove that all semivalues (hence, perforce, all values) must give patently unacceptable solutions for some two-stage majority games (where the members of a majority coalition play a conventional majority game). Our reservations about these prescribed values are related to Roth's (1980) criticism of Shapley's -transfer value for non-transferable utility (NTU) games. But our analysis has wider scope than Roth's example, and the argument that it offers appears to be more conclusive. The study also indicates how the values and semivalues for two-stage games can be naturally generalized to apply for multi-stage games.Earlier versions of this study were presented at the International Conference on Game Theory and its Applications, organized by Ohio State University in 1987, and at the Workshop on Mathematical Economics and Game Theory at Tel Aviv Unversity. We gratefully acknowledge the valuable comments received on both occasions, especially those of Robert J. Aumann, Roy Gardner, Sergiu Hart, Ehud Kalai, Michael Maschler, Alvin E. Roth, and Lloyd S. Shapley, and also those ofIJGT's anonymous referees. Of course, all responsibility lies with us.     

An extended continuous Newton method

This paper describes a numerical realization of an extended continuous Newton method defined by Diener. It traces a connected set of locally one-dimensional trajectories which contains all critical points of a smooth functionf: ⁿ . The results show that the method is effectively applicable.The authors would like to thank L. C. W. Dixon for pointing out some errors in the original version of this paper and for several suggestions of improvements.     

Mark Grigor'evich Krein

Obituary

Mark Grigor'evich Krein     

Classes of functions defined on the real line and their approximation by entire functions. I

We introduce classes of functions defined on the real line which in the periodic case become the classes defined by multipliers (·) and translations with respect to the argument equal to . We study the simplest properties of these classes needed for the study of their approximation characteristics which will be discussed in the second part of the paper.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 1, pp. 102–112, January, 1990.     

Impulse systems with fixed moments of shocks of general position: The structure of the set of moments of the shocks

Ordinary differential equations with impulse action are considered at fixed moments of time when finite limits points are admitted to the set of moments of the shocks. The topological structure of this set is clarified: it must be resolvable, in particular nowhere dense in R.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 3, pp. 378–383, March, 1990.     

Deformations of positive linear operators of type B

The concept of the deformation of positive linear operators of type B is introduced, in particular, that of a linear deformation of operators. These concepts are applied to the solving of operator equations of type B. One solves completely the problems of the determination of all positive linear operators of type B for which the covariance of the operators is 1) a polynomial of degrees 3; 2) any power function; 3) an exponential function. In addition, one finds; two families of positive linear operators of type B for which the covariance is a polynomial of degree four.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 888–900, July, 1990.     

A combinatorial property of points and ellipsoids

For eachd1 there is a constantc _d>0 such that any finite setXR ^d contains a subsetYX, |Y|[1/4d(d+3)]+1 having the following property: ifEY is an ellipsoid, then |E X|c _d |X|.On leave from the Mathematical Institute of the Hungarian Academy of Sciences, 1364 Budapest, P.O. Box 127, Hungary. Supported by a research fellowship from the Science and Engineering Research Council, U.K., and by Hungarian National Foundation for Scientific Research Grant No. 1812.     

On the connection between the one-dimensionalS=1/2 Heisenberg chain and Haldane-Shastry model

Extra integrals of motion and the Lax representation are found for interacting spin systems with the HamiltonianH = (J/2) _{j, k=1,j k} ^N P(j – k) _{j k} , where one of the periods of the WeierstrassP function is equal toN. The Heisenberg and Haldane-Shastry chains appear as limiting cases of these systems at some values of the second period. The simplest eigenvectors and eigenvalues ofH corresponding to the scattering of two spin waves are presented explicitly for these finite-dimensional systems and for their infinite-dimensional version.     

Behavior of effective charges in e6 grand unification theory in curved spacetime

The behavior of effective charges _i(t) corresponding to the parameters of nonminimal coupling of scalar and gravitational fields in E₆ Grand Unification Theory (GUT) in curve spacetimes is studied. It is shown that in a strong gravitational field ¦_i(t)¦ grows indefinitely.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 74–11, September, 1990.