Abstract. In this paper, we define the notion of C*-affine maps in the unital *-rings and we investigate the C*-extreme points of the graph and epigraph of such maps. We show that for a C*-convex map f on a unital *-ring R satisfying the positive square root axiom with an additional condition, the graph of f is a C*-face of the epigraph of f. Moreover, we prove som...

A pair (X, ) of a finite set X and a closure operator : 2X → 2X is called a closure space. The class of closure spaces includes matroids as well as antimatroids. Associated with a closure space (X, ), the extreme point operator ex: 2X → 2X is defined as ex(A) = {p|p ∈ A,p / ∈ (A − {p})}. We give characterizations of extreme point operators of closure spaces, matroids and antimatroids, respectiv...

Extreme value theory for a class of stochastic volatility models, in which the logarithm of the conditional variance follows a Gaussian linear process, is developed. A result for the asymptotic tail behavior of the transformed stochastic volatility process is established and used to prove that the suitably normalized extremes converge in distribution to the double exponential (Gumbel) distribut...

Let φ be some integer function on words of length n and oracle Oφ gives the value φ(x) for a given x. It is shown how quantum algorithm can find a point of maximum of φ with the probability of error 2/3 applying oracle Oφ 32 √ 2n times. This algorithm is optimal in within constant factor in the following sense. Any other algorithm acting in substantially shorter time o( √ 2n) (n −→ ∞) gives inc...

This paper deals with the optimization of the ratio of two linear functions subject to a set of linear constraints with the additional restriction that the optimal solution is to be an extreme point of another convex polyhedron. In this paper, an enumerative procedure for solving such type of problems is developed. For an illustration, a numerical example is also provided.

Given a set of n colored points in some d-dimensional Euclidean space, a bichromatic closest (resp. farthest) pair is a closest (resp. farthest) pair of points of different colors. We present efficient algorithms to maintain both a bichromatic closest pair and a bichromatic farthest pair when the the points are fixed but they dynamically change color. We do this by solving the more general prob...

We continue here the study of free extreme values begun in [3]. We study the convergence of the free point processes associated with free extreme values to a free Poisson random measure ([15], [2]). We relate this convergence to the free extremal laws introduced in [3] and give the limit laws for free order statistics.