Teraoka-type two-person nonzero-sum silent duel

The paper discusses a silent nonzero-sum duel between two players each of whom has a single bullet. The duel is terminated at a random time in [0, 1] given by a cumulative distribution function. It is shown that the game has a unique Nash equilibrium under a wide range of possible payoff values for simultaneous firing. This contrasts with a very similar game considered by Teraoka for which there are many Nash equilibria.This work was carried out while the second author was visiting the University of Southampton on a Postdoctoral Fellowship of The Royal Society of London.     

On the noisy-silent versus silent-noisy duel with equal accuracy functions

This paper deals with the noisy-silent versus silent-noisy duel with equal accuracy functions. Each of player I and player II has a gun with two bullets and he can fire his bullets at any time in [0, 1] aiming at his opponent. The first bullet of player I and the second bullet of player II are noisy, and the second bullet of player I and the first bullet of player II are silent. It is assumed that both players have equal accuracy functions. If player I hits player II, not being hit himself before, the payoff of the duel is +1; if player I is hit by player II, not hitting player II before, the payoff is ?1. The value of the game and the optimal strategies are obtained. We find that the firing time of the silent bullet by player II's optimal strategy depends directly on the firing time of player I's noisy bullet.     

On a noisy-silent-versus-silent duel with equal accuracy functions

This paper deals with the noisy-silent-versus-silent duel with equal accuracy functions. Player I has a gun with two bullets and player II has a gun with one bullet. The first bullet of player I is noisy, the second bullet of player I is silent, and the bullet of player II is silent. Each player can fire their bullets at any time in [0, 1] aiming at his opponent. The accuracy function ist for both players. If player I hits player II, not being hit himself before, the payoff of the duel is +1; if player I is hit by player II, not hitting player II before, the payoff is –1. The optimal strategies and the value of the game are obtained. Although optimal strategies in past works concerning games of timing does not depend on the firing moments of the players, the optimal strategy obtained for player II depends explicitly on the firing moment of player I's noisy bullet.     

On a duel with time lag and equal accuracy functions

This paper deals with a duel with time lag that has the following structure: Each of two players I and II has a gun with one bullet and he can fire his bullet at any time in [0, 1], aiming at this opponent. The gun of player I is silent and the gun of player II is noisy with time lagt (i.e., if player II fires at timex, then player I knows it at timex+t). They both have equal accuracy functions. Furthermore, if player I hits player II without being hit himself before, the payoff is +1; if player I is hit by player II without hitting player II before, the payoff is –1; if they hit each other at the same time or both survive, the payoff is 0.This paper gives the optimal strategy for each player, the game value, and some examples.     

L ∞ — Limit theorems for Markov processes

A necessary and sufficient condition for convergence of Markov processesL _∞ is given. As a consequence we get a theorem concerning the convergence of Harris processes. This paper is a part of the author’s Ph.D. thesis to be submitted to the Hebrew University of Jerusalem. The author wishes to express his thanks to Professor S. R. Foguel for much valuable advice and encouragement.     

Two-noisy-versus-one-silent duel with equal accuracy functions

This paper deals with the two-noisy-versus-one-silent duel which is still open, as pointed out by Styszyński (Ref. 1). Player I has a noisy gun with two bullets, and player II has a silent gun with one bullet. Each player fires his bullets aiming at his opponent at any time in [0, 1]. The accuracy function (the probability that one player hits his opponent if he fires at timet) isp(t)=t for each player. If player I hits player II, without being hit himself before, the payoff of the duel is +1; if player I is hit by player II, without hitting player II before, the payoff is taken to be ?1. In this paper, we determine the optimal strategies and the value of the game. The strategy for player II depends explicitly on the firing moment of player I's first shot.     

On multiplicity theory for Boolean algebras of projections

The aim of this paper is to present a study of the connections between the commutants of a Boolean algebra of projections of finite multiplicity and the uniformly closed algebra generated by these projections. This paper is a part of the author’s Ph.D. thesis to be submitted to the Hebrew University. The author wishes to express his thanks to Professor S. R. Foguel for much valuable advice and encouragement.     

The problem of optimal stopping in a partially observable Markov chain

The optimal-stopping problem in a partially observable Markov chain is considered, and this is formulated as a Markov decision process. We treat a multiple stopping problem in this paper. Unlike the classical stopping problem, the current state of the chain is not known directly. Information about the current state is always available from an information process. Several properties about the value and the optimal policy are given. For example, if we add another stop action to thek-stop problem, the increment of the value is decreasing ink.The author wishes to thank Professor M. Sakaguchi of Osaka University for his encouragement and guidance. He also thanks the referees for their careful readings and helpful comments.     

Sets of degrees of computable fields

Given a Σ₂ (resp. Σ₁) degree of recursive unsolvability a, a computable field (resp. a computable field with a splitting algorithm)F is constructed in any given characteristic, such that the set of dimensions of all finite extensions ofF has degree a. This is part of an M.Sc. Thesis presented at The Hebrew University. The author wishes to express his indebtedness to his Supervisor Professor H. Gaifman for his constant guidance and encouragement. Thanks are also due to Professor G. Sabbagh (University of Paris VII) for several useful suggestions.     

On variable-metric algorithms

In this paper, a generalized variable-metric algorithm is presented. It is shown that this algorithm attains the minimum of a positive-definite, quadratic function in a finite number of steps and that thevariable-metric matrix tends to the inverse of the Hessian matrix. Most known variable-metric algorithms can be derived from this generalized algorithm, and some new algorithms are also obtained.The author expresses his gratitude to Professor H. Tokumaru for guidance and encouragement.     

On Minimizing Some Merit Functions for Nonlinear Complementarity Problems under <Emphasis Type="Italic">H</Emphasis>-Differentiability

In this paper, we describe the H-differentials of some well known NCP functions and their merit functions. We show how, under appropriate conditions on an H-differential of f, minimizing a merit function corresponding to f leads to a solution of the nonlinear complementarity problem. Our results give a unified treatment of such results for C ¹-functions, semismooth-functions, and locally Lipschitzian functions. Illustrations are given to show the usefulness of our results. We present also a result on the global convergence of a derivative-free descent algorithm for solving the nonlinear complementarity problem. The first author is deeply indebted to Professor M. Seetharama Gowda for his numerous helpful suggestions and encouragement. Special thanks to Professor J.-P. Crouzeix and an anonymous referees for their constructive suggestions which led to numerous improvements in the paper. The research of the first author was supported in part by the Natural Sciences and Engineering Research Council of Canada and Scholar Activity Grant of Thompson Rivers University. The research of the second author was supported by the Natural Sciences and Engineering Research Council of Canada.     

Some limit theorems for Markov processes

The aim of this paper is to prove some limit theorems for Markov processes using only functional analytic methods. Some of our results were proved in [7], [8] and [5] by probabilistic methods. We prove in the Appendix a theorem on Markov processes that have no finite invariant measure. This paper is a part of the author’s Ph.D. thesis to be submitted to the Hebrew University of Jerusalem. The author wishes to express his thanks to Professor S. R. Foguel for much valuable advice and encouragement.     

Uniformly separating families of functions

A concept concerning separation of points by functions is defined and studied. This concept has close relations with superpositions of functions and dimension theory, and these relations are investigated. A theorem concerning the dimension of projections of Cantor manifolds inR ^m is proved. This is a part of the authors Ph.D thesis prepared at the Hebrew University of Jerusalem under the supervision of Professor J. Lindenstrauss. The author wishes to thank Professor Lindenstrauss for his advice and encouragement during the preparation of this work.     

An exponential differential game which admits a simple Nash solution

This paper deals with a class ofN-person nonzero-sum differential games where the control variables enter into the state equations as well as the payoff functionals in an exponential way. Due to the structure of the game, Nash-optimal controls are easily determined. The equilibrium in open-loop controls is also a closed-loop equilibrium. An example of optimal exploitation of an exhaustible resource is presented.The helpful comments of Professor Y. C. Ho and Dipl. Ing. E. Dockner are gratefully acknowledged.     

On the Selection of One Feedback Nash Equilibrium in Discounted Linear-Quadratic Games

We study a selection method for a Nash feedback equilibrium of a one-dimensional linear-quadratic nonzero-sum game over an infinite horizon. By introducing a change in the time variable, one obtains an associated game over a finite horizon T > 0 and with free terminal state. This associated game admits a unique solution which converges to a particular Nash feedback equilibrium of the original problem as the horizon T goes to infinity.     

Farkas' theorem of nonconvex type and its application to a min-max problem

This note is concerned with the generalization of Farkas' theorem and its application to derive optimality conditions for a mix-max problem. Farkas' theorem is generalized to a system of inequalities described by sup-min type positively homogeneous functions. This generalization allows us to deal with optimization problems consisting of objective and constraint functions whose directional derivatives are not necessarily convex with respect to the directions. As an example of such problems, we formulate a min-max problem and derive its optimality conditions.The author would like to express his sincere thanks to Professors S. Suzuki and T. Asano of Sophia University and Professor K. Shimizu of Keio University for encouragement and suggestions.     

Ultraproducts and ultra-limits of near-rings

The construction of ultraproducts, ultrapowers and ultralimits have proven useful, both for logic and for special (algebraic) structures. It seems that these constructions might be especially useful in studying near-rings; they might provide solutions to long-standing problems. Among other results, we show that ultraproducts of primitive near rings are again primitive.Part of this work was done while the second author was a Visiting Professor at the University of Southwestern Louisiana (Lafayette, La). This author expresses his gratitude for the received hospitality and for most valuable discussions at this Department. Special thanks are due to Professors D. Blumberg, H. Heatherly and A. Iskander.     

Onσ-finite invariant measures for Markov processes

Sufficient conditions are given for the existence ofσ-finite invariant measure for conservative and ergodic Markov processes. This paper is a part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem uuder the supervision of Professor S. R. Foguel. The author wishes to thank him for his helpful advice and kind encouragement.     

Existence of identities inA⊗B

The following theorem is proved: ifA andB are twoPI-algebras over a fieldF, thenA⊗ _F B is aPI-algebra. This paper was written while the author was doing his Ph.D. thesis at the Hebrew University of Jerusalem under the supervision of Professor A. S. Amitsur, to whom the author wishes to express his warm thanks.     

Variational quasi-Newton methods for unconstrained optimization

In this paper, we propose new members of the Broyden family of quasi-Newton methods. We develop, on the basis of well-known least-change results for the BFGS and DFP updates, a measure for the Broyden family which seeks to take into account the change in both the Hessian approximation and its inverse. The proposal is then to choose the formula which gives the least value of this measure in terms of the two parameters available, and hence to produce an update which is optimal in the sense of the given measure. Several approaches to the problem of minimizing the measure are considered, from which new updates are obtained. In particular, one approach yields a new variational result for the Davidon optimally conditioned method and another yields a reasonable modification to this method. The paper is also concerned with the possibility of estimating, in a certain sense, the size of the eigenvalues of the Hessian approximation on the basis of two available scalars. This allows one to derive further modifications to the above-mentioned methods. Comparisons with the BFGS and Davidson methods are made on a set of standard test problems that show promising results for certain new methods.Part of this work was done during the author's visits at International Centre for Theoretical Physics, Trieste, Italy, at Systems Department, University of Calabria, Cosenza, Italy, and at Ajman University College of Science and Technology, Ajman, United Arab Emirates.The author expresses his gratitude to Professor L. Grandinetti for his encouragement and thanks the anonymous referees for their careful reading of an earlier draft of the paper and valuable comments, which led to a substantial improvement of the original paper.