denote the order statistics, and let (1) Fn(x) 1 n-L l[xiSx) for-00 < x < 00 n i=l denote the empirical df; here 1.4 denotes the indicator function of the set A. Then (2) is called the empirical process.

We have already considered instances of the following type of problem: given a bounded subset Ω of Euclidean space R N , to determine #(Ω ∩ Z N), the number of integral points in Ω. It is clear however that there is no answer to the problem in this level of generality: an arbitrary Ω can have any number of lattice points whatsoever, including none at all. In [Gauss's Circle Problem], we counted...

In this paper, we consider the problem of minimizing the sum of nondifferentiable, convex functions over a closed convex set in a real Hilbert space, which is simple in the sense that the projection onto it can be easily calculated. We present a parallel subgradient method for solving it and the two convergence analyses of the method. One analysis shows that the parallel method with a small con...

In this paper some concepts of convex analysis are extended in an intrinsic way from the Euclidean space to the sphere. In particular, relations between convex sets in the sphere and pointed convex cones are presented. Several characterizations of the usual projection onto a Euclidean convex set are extended to the sphere and an extension of Moreau’s theorem for projection onto a pointed convex...

Definition 1.1. Let C be a subset of R. We say C is convex if αx+ (1− α)y ∈ C, ∀x, y ∈ C, ∀α ∈ [0, 1]. Definition 1.2. Let C be a convex subset of R. A function f : C 7→ R is called convex if f(αx+ (1− α)y) ≤ αf(x) + (1− α)f(y), ∀x, y ∈ C, ∀α ∈ [0, 1]. The function f is called concave if −f is convex. The function f is called strictly convex if the above inequality is strict for all x, y ∈ C wi...

In many scientific and engineering applications, the fuzzy set concept plays an important role. The fuzziness appears when we need to perform, on manifold, calculations with imprecision variables. The fuzzy set theory was introduced initially by Zadeh [1] in 1965. In the theory and applications of fuzzy sets, convexity is a most useful concept. In fact, in the basic and classical paper [1], Zad...

The paper is devoted to two real problems that generate a cooperative game model with a so-called k-convex characteristic function when certain conditions are fulfilled. Both a bankruptcy problem and an information trading problem are modelled as a cooperative game by constructing the corresponding bankruptcy game as well as the information market game. Firstly, it is established that the bankr...

The positive characteristic function-field Mordell-Lang conjecture for finite rank subgroups is resolved for curves as well as for subvarieties of semiabelian varieties defined over finite fields. In the latter case, the structure of the division points on such subvarieties is determined.

Suppose that A and B are real stable matrices, and that their difference A − B is rank one. Then A and B have a common quadratic Lyapunov function if and only if the product AB has no real negative eigenvalue. This result is due to Shorten and Narendra, who showed that it follows as a consequence of the Kalman-Yacubovich-Popov solution of the Lur’e problem. Here we present a new and independent...

We consider Calderón’s inverse problem on planar domains Ω with conductivities in fractional Sobolev spaces. When Ω is Lipschitz, the problem was shown to be stable in the L–sense in [18]. We remove the Lipschitz condition on the boundary. To this end, we analyse the Sobolev regularity of the characteristic function of Ω. For Ω a quasiball, we compute ‖χΩ‖W s,p(Rd) in terms of the δ–neighbourho...