Abstract. This is an overview of merging the techniques of Riesz space theory and convex geometry. Alexandr Danilovich Alexandrov became the first and foremost Russian geometer of the twentieth century. He contributed to mathematics under the slogan: “Retreat to Euclid, ” remarking that “the pathos of contemporary mathematics is the return to Ancient Greece. ” Hermann Minkowski revolutionized the theory of numbers with the aid of the synthetic geometry of convex surfaces. The ideas and techniques of the geometry of numbers comprised the fundamentals of functional analysis which was created by Banach. The pioneering studies of Alexandrov continued the efforts of Minkowski and enriched geometry with the methods of measure theory and functional analysis. Alexandrov accomplished the turnround to the ancient synthetic geometry in a much deeper and subtler sense than it is generally acknowledged today. Geometry in the large reduces in no way to overcoming the local restrictions of differential geometry which bases upon the infinitesimal methods and ideas of Newton, Leibniz, and Gauss.

Abstract. This is a talk delivered on September 20, 2007 at the conference “Mathematics in the Modern World ” on the occasion of the fiftieth anniversary

There is a number of completely integrable gravity theories in two dimensions. We study the metric-affine approach on a 2-dimensional spacetime and display a new integrable model. Its properties are described and compared with the known results of Poincaré gauge gravity. PACS: 04.50.+h, 04.20.Fy, 04.20.Jb, 04.60.Kz, 02.30.Ik I.

The OWA operators gained interest among researchers as they provide a continuum of aggregation operators able to cover the whole range of compensation between the minimum and the maximum. In some circumstances, it is useful to consider a wider range of values, extending below the minimum and over the maximum. ST-OWA are able to surpass the boundaries of variation of ordinary compensatory operators. Their application requires identification of the weighting vector, the t-norm, and the t-conorm. This task can be accomplished by considering both the desired analytical properties and empirical data.

Abstract. The aim of this paper is to investigate real partially ordered linear topological spaces in which directed sets admit a supremum in their closure. In particular, we point out that this property is intimately related to the normality of the ordering cone and also to the Scott continuity of functionals belonging to the nonnegative polar of the ordering cone.

Robust positively invariant (RPI) sets for linear difference inclusions are considered here under the assumption that the linear difference inclusion is absolutely asymptotically stable in the absence of additive state disturbances, which is the case for parametrically uncertain or switching linear discrete-time systems controlled by a stabilizing linear state feedback controller. The existence and uniqueness of the minimal RPI set and the minimal convex RPI set are studied. A new method for the computation of outer RPI approximations of the minimal RPI set for linear difference is presented; these approximations include a family of star–shaped RPI sets and two families of convex RPI sets. The use of a family of star–shaped RPI sets, and the characterization of the family, is reported for the first time.

convex sets and functions

convexity to the upper semilattices of gauges in finite dimensions.

This talk is devoted to some origins of abstract convexity and a few vexing limitations on the range of abstraction in convexity. Convexity is a relatively recent subject. Although the noble objects of Euclidean geometry are mostly convex, the abstract notion of a convex set appears only after the Cantor paradise was founded. The idea of convexity feeds generation, separation, calculus, and approximation. Generation appears as duality; separation, as optimality; calculus, as representation; and approximation, as stability. 1. Generation. Let E be a complete lattice E with the adjoint top ⊤: = + ∞ and bottom ⊥: = −∞. Unless otherwise stated, Y is usually a Kantorovich space which is a Dedekind complete vector lattice in another terminology. Assume further that H is some subset of E which is by implication a (convex) cone in E, and so the bottom of E lies beyond H. A subset U of H is convex relative to H or H-convex, in symbols U ∈ V(H, E), provided that U is the H-support set UH p: = {h ∈ H: h ≤ p} of some element p of E. Alongside the H-convex sets we consider the so-called H-convex elements. An element p ∈ E is H-convex provided that p = sup UH p; i.e., p represents the supremum of the H-support set of p. The H-convex elements comprise the cone which is denoted by C (H, E). We may omit the references to H when H is clear from the context. It is worth noting that convex elements and sets are “glued together ” by the Minkowski diality ϕ: p ↦ → UH p. This duality enables us to study convex elements and sets simultaneously. Since the classical results by Fenchel [1] and Hörmander [2, 3] it has been well known that the most convenient and conventional classes of convex functions and sets are C (A(X), X) and V(X ′, X). Here X is a locally convex space, X ′ is the dual of X, and A(X) is the space of affine functions on X (isomorphic with X ′ ×).