Equilibrium points of logarithmic potentials on convex domains

Let be a convex domain in . Let be summable constants and let . If the converge sufficiently rapidly to from within an appropriate Stolz angle, then the function has infinitely many zeros in . An example shows that the hypotheses on the are not redundant and that two recently advanced conjectures are false.

     

Equilibrium points in specialn-person games

The set of Nash equilibria is computed for some generalized games. It is also studied for a subclass of standardn-person games.The authors acknowledge the support of CONICET (Consejo de Investigaciones Cientificas y Tecnicas de la Republica Argentina). The first author acknowledges the support from TWAS (Third World Academy of Sciences), Grant No. 86-33.     

Equilibrium points of Infinite Sequential Truels

Infinite Sequential Truels are studied in the case when the players' duel values are preassigned parameters, rather than functions of the marksmanships. Such truels divide naturally into two classes, one of which includes all Infinite Sequential Truels investigated previously. All equilibrium points of all truels are determined. It is discovered that some truels possess one or several of four equilibrium points never before observed, and that others may possess no equilibrium points at all (in stationary strategies).     

Equilibrium points of logarithmic potentials induced by positive charge distributions. I. Generalized de Bruijn-Springer relations

A notion of weighted multivariate majorization is defined as a preorder on sequences of vectors in Euclidean space induced by the Choquet ordering for atomic probability measures. We characterize this preorder both in terms of stochastic matrices and convex functions and use it to describe the distribution of equilibrium points of logarithmic potentials generated by discrete planar charge configurations. In the case of positive charges we prove that the equilibrium points satisfy weighted majorization relations and are uniquely determined by such relations. It is further shown that the Hausdorff geometry of the equilibrium points and the charged particles is controlled by the weighted standard deviation of the latter. By using finite-rank perturbations of compact normal Hilbert space operators we establish similar relations for infinite charge distributions. We also discuss a hierarchy of weighted de Bruijn-Springer relations and inertia laws, the existence of zeros of Borel series with positive -coefficients, and an operator version of the Clunie-Eremenko-Rossi conjecture.

     

Equilibrium points of a class of many-person games

This paper deals with the competitive equilibrium points of a class of (n+1)-person games, the players beingn sellers and a single buyer. Each seller offers a price for his product; depending upon the prices offered, the buyer then determines the amount he wishes to buy from each seller. Two types of equilibrium points in this class of games are discussed.The authors are indebted to Professor G. Leitmann for his valuable comments on an earlier version of this paper.     

On some types of boundary points of harmonic functions

The paper deals with the Lindelöf and Fatou points of arbitrary harmonic functions defined in the unit disk. We present necessary and sufficient conditions for existence of such points on the unit circle.     

Representation of harmonic functions as potentials and the Cauchy problem

In this paper, we propose an explicit formula for the reconstruction of a harmonic function in the domain from its known values and from the values of its normal derivative on part of the boundary, i.e., we give an explicit continuation and a regularization formula of the solution of the Cauchy problem for the Laplace equation.     

Equilibrium points of nonatomic games over a nonreflexive Banach space

Schmeidler's results on the equilibrium points of nonatomic games with strategy sets in Euclidean n-space are generalized to nonatomic games with stategy sets in a separable Banach space whose dual possesses the Radon-Nikodým property.     

Extreme points of subordination and weak subordination families of harmonic mappings

The aim of the paper is to discuss the extreme points of subordination and weak subordination families of harmonic mappings. Several necessary conditions and sufficient conditions for harmonic mappings to be extreme points of the corresponding families are established.     

Multiple critical points for nondifferentiable functionals involving Hardy potentials

In this paper we study general functionals of the calculus of variations with the presence of a Hardy potential. We will improve several results obtained in the semilinear framework. We will first prove a general weak lower semicontinuity result, which will imply the existence of a minimum point whenever the functional is coercive. Then we will demonstrate existence and multiplicity results of critical points, even if our functional is not differentiable. We will apply a nonsmooth critical point theory developed in Corvellec et al. (Nonlinear Anal. 1 (1993) 151) and Degiovanni and Marzocchi (Ann. Mat. Pura Appl. 167 (1994) 73).     

Layer potentials C*-algebras of domains with conical points

To a domain with conical points Ω, we associate a natural C*-algebra that is motivated by the study of boundary value problems on Ω, especially using the method of layer potentials. In two dimensions, we allow Ω to be a domain with ramified cracks. We construct an explicit groupoid associated to ?Ω and use the theory of pseudodifferential operators on groupoids and its representations to obtain our layer potentials C*-algebra. We study its structure, compute the associated K-groups, and prove Fredholm conditions for the natural pseudodifferential operators affiliated to this C*-algebra.     

Periodic harmonic functions on lattices and points count in positive characteristic

This survey deals with pluri-periodic harmonic functions on lattices with values in a field of positive characteristic. We mention, as a motivation, the game “Lights Out” following the work of Sutner [20], Goldwasser- Klostermeyer-Ware [5], Barua-Ramakrishnan-Sarkar [2, 19], Hunzikel-Machiavello-Park [12] e.a.; see also [22, 23] for a more detailed account. Our approach uses harmonic analysis and algebraic geometry over a field of positive characteristic.      

On meromorphically harmonic starlike functions with respect to symmetric conjugate points

In a previous paper M.P. Chen, Z.-R. Wu and Z.-Z. Zou [M.P. Chen, Z.-R. Wu, Z.-Z. Zou, On functions α-starlike with respect to symmetric conjugate points, J. Math. Anal. Appl. 201 (1996) 25-34] developed a method, using some operators, to deal with functions analytic and starlike with respect to symmetric conjugate points in the unit disc. Then, the same method is employed to functions meromorphic by Z.Z. Zou and Z.-R. Wu [Zhong Zhu Zou, Zhuo-Ren Wu, On meromorphically starlike functions and functions meromorphically starlike with respect to symmetric conjugate points, J. Math. Anal. Appl. 261 (2001) 17-27]. Now, the method can be employed to functions meromorphic harmonic in the punctured disc 0<|z|<1. Especially, a sharp coefficient estimate and a structural representation of such functions are obtained.     

Refinement of the Plessner theorem and Plessner points for arbitrary harmonic functions

The well-known theorem of Plessner obtained for arbitrary harmonic functions by Japanese mathematician S. Yamashita is refined. In addition, one result obtained by Prof. V. I. Gavrilov for meromorphic functions is extended to harmonic functions.