Axiomatic theory of convexity

The axiomatic construction of the theory of convexity proceeds from an arbitrary set M and a mapping l: M² 2^M, i.e., from a pair (M, l). It is shown that such a space of a certain type is domain finite. A condition is given which, for such spaces, implies join-hull commutativity. A connection is established between the Carathéodory number and join-hull commutativity. Conditions are given which imply a separation property of the space (M, l). Convexity spaces which are domain finite are characterized.Translated from Matematicheskie Zametki, Vol. 20, No. 5, pp. 761–770, November, 1976.     

Representation theory and convexity

IfS=G Exp (iW) is a complex open Ol'shanskiî semigroup, whereW is an open elliptic cone, then we considerG-biinvariant domainsD=G Exp (iD_g)S. First we show that the representation ofG×G on eachG-biinvariant irreducible reproducing kernel Hilbert space in Hol(D) is a highest weight representation whose kernel is the character of a highest weight representation ofG. In the second part of the paper we explain how to construct biinvariant Kähler structures on biinvariant Stein domains and show by a certain Legendre transform that the so obtained symplectic manifolds are isomorphic to domains in the cotangent bundleT^*(G).     

A general framework for convexity analysis in deterministic global optimization

In previous work, we, and also Epperly and Pistikopoulos, proposed an analysis of general nonlinear programs that identified certain variables as convex, not ever needing subdivision, and non-convex, or possibly needing subdivision in branch and bound algorithms. We proposed a specific algorithm, based on a generated computational graph of the problem, for identifying such variables. In our previous work, we identified only independent variables in the computational graph. Here, we examine alternative sets of non-convex variables consisting not just of independent variables, but of a possibly smaller number of intermediate variables. We do so with examples and theorems. We also apply variants of our proposed analysis to the well-known COCONUT Lib-1 test set. If the number of such non-convex variables is sufficiently small, it may be possible to fully subdivide them before analysis of ranges of objective and constraints, thus dispensing totally with the branch and bound process. Advantages to such a non-adaptive process include higher predictability and easier parallizability. We present an algorithm and exploratory results here, with a more complete empirical study in a subsequent paper.     

A general theory of self-similarity

A little-known and highly economical characterization of the real interval [0,1], essentially due to Freyd, states that the interval is homeomorphic to two copies of itself glued end to end, and, in a precise sense, is universal as such. Other familiar spaces have similar universal properties; for example, the topological simplices Δn may be defined as the universal family of spaces admitting barycentric subdivision. We develop a general theory of such universal characterizations.This can also be regarded as a categorification of the theory of simultaneous linear equations. We study systems of equations in which the variables represent spaces and each space is equated to a gluing-together of the others. One seeks the universal family of spaces satisfying the equations. We answer all the basic questions about such systems, giving an explicit condition equivalent to the existence of a universal solution, and an explicit construction of it whenever it does exist.     

A general theory of almost factoriality

Let R be a commutative integral domain with 1. The non-zero elements a,b of R may be calledv-coprime if aRbR=abR. A Krull domain is calledalmost factorial if for all f,g in R there is nN such that fⁿRgⁿR is principal. From this it is easy to establish that if R is almost factorial then for all x in R there is nN such that xⁿ=p₁p₂...p_r where p_i are mutually v-coprime primary elements and that this expression is unique. In this article we drop the requirement that R be Krull and replace the primary elements by elements calledprime blocks and develope a theory of almost factoriality, a special case of which is the theory of almost factorial Krull domains.     

A general theory of hypersurface potentials

Summary A general theory of hypersurface potentials in n-dimensional space is proposed. Not only smooth densities but also potentials generated either by L ¹ functions or by measures are considered.     

A general theory of fuzzy plausibility measures

The purpose of this paper is to present a fuzzification of probability theory, or more precisely to give a fuzzification of plausibility measures first introduced by Shafer in 1976. Although plausibility measures include probability measures as well as possibility measures, it is a typical result of this theory that only a fuzzification of possibility measures is attainable, while a fuzzification of probability measures seems to be impossible. Moreover with regard to fuzzy plausibility measures we specify a concept of mean values and entropies, which can be considered as a direct generalization of the classical notions of mean value and entropy based upon probability measures.     

Ordered Incidence geometry and the geometric foundations of convexity theory

An Ordered Incidence Geometry, that is a geometry with certain axioms of incidence and order, is proposed as a minimal setting for the fundamental convexity theorems, which usually appear in the context of a linear vector space, but require only incidence, order (and for separation, completeness), and none of the linear structure of a vector space.Research supported by the National Science Foundation.     

A general theory of some positive dependence notions

A general theory of concepts of positive dependence, which are weaker than association but stronger than orthant dependence, is developed. A random vector X is associated if and only if $\text{P(}\text{X}\text{∈ A ? B) ≥ P(}\text{X}\text{∈ A) P(}\text{X}\text{∈ B)}$ for all open upper sets A and B. By requiring the above inequality to hold only for some open upper sets A and B various notions of positive dependence which are weaker than association are obtained. First a general theory is given and then the results are specialized to some concepts of a particular interest. Various properties and interrelationships are derived and some applications are discussed.     

A general theory of measures for nonnegative matrices

By a measure μ on the set N of m × n nonnegative matrices we mean that μ is a function from N to the nonnegative reals such that (i) μ(λA)=λμ(A) for all nonnegative λ and all A ∈ N, and (ii) μ(A + B) ? μ(A) for all A,B ? N. This paper develops a theory of such measures and shows how this theory can be applied to particular problems.     

A general cone decomposition theory based on efficiency

LetK ₁ andK ₂ be two convex cones in some common vector space. This paper is concerned with the question of finding a good decomposition, with respect toK ₁ andK ₂, of a given element of the Minkowski sumK ₁ +K ₂. We propose the criterion of efficiency as a measure for the quality of a decomposition. This criterion allows us to set up a framework from which a general cone decomposition theory is then derived.Research supported by Dirección General de Investigación Científica y Técnica (DGICYT), under project PS89-0058.     

A general Galois theory for cofunctions and corelations

Cofunctions and corelations are introduced and a corresponding Galois connection is studied in analogy to operations and relations in universal algebra. The Galois closed sets are characterized by local closures of clones of cofunctions and corelations, respectively. As an application concrete characterization problems are considered e.g. for strong bisimulations.     

A note on NTU convexity

For cooperative games with transferable utility, convexity has turned out to be an important and widely applicable concept. Convexity can be defined in a number of ways, each having its own specific attractions. Basically, these definitions fall into two categories, namely those based on a supermodular interpretation and those based on a marginalistic interpretation. For games with nontransferable utility, however, the literature mainly focuses on two kinds of convexity, ordinal and cardinal convexity, which both extend the supermodular interpretation. In this paper, we analyse three types of convexity for NTU games that generalise the marginalistic interpretation of convexity. Received: December 2000