Ordered Vector Spaces Research Articles

Calculation of the condensate fraction in liquid helium-4

We adduce the results of the condensate fraction calculation for liquid helium-4. The method is derived from the first principles and involves minimum assumptions. The only experimental quantity we need in our calculations is the static structure factor which is easily measurable. We use the approximation in which expressions contain one summation in the wave vector space (or one integration it the radius vector space) in order to demonstrate the validity of our method. The computed values of the condensate fraction lie within 8.8 ÷ 14% depending on the model potential for the short-range interaction. This result agrees with recent experimental measurements and numerical estimations.

Disjointness in Partially Ordered Vector Spaces

A notion of disjointness in arbitrary partially ordered vector spaces is intro- duced by calling two elements x and y disjoint if the set of all upper bounds of x + y and −x − y equals the set of all upper bounds of x − y and −x + y. Several elementary properties are easily observed. The question whether the disjoint complement of a subset is a linear subspace appears to be more difficult. It is shown that in directed Archime- dean spaces disjoint complements are always subspaces. The proof relies on theory on order dense embedding in vector lattices. In a non-Archimedean directed space even the disjoint complement of a singleton may fail to be a subspace. According notions of dis- jointness preserving operator, band, and band preserving operator are defined and some of their basic properties are studied. Mathematics Subject Classifications 2000: 46A40, 06F20.

Risk Neutrality and Ordered Vector Spaces

The following result clarifies when preferences over time and under risk correspond to discounting and are not risk neutral. If a binary relation on a real vector space V satisfies four axioms, then there is a utility function U=fu from V to R where u from V to R is linear as a map of vector spaces and f from R to R is continuous and weakly monotone. Three axioms are familiar: weak ordering, continuity, and non-triviality. The fourth axiom is X-Y is weakly preferred to 0 (where 0 is the zero element in V) implies X is weakly preferred to Y (X and Y in V). The function f from R to R is linear if and only if the binary relation also satisfies the converse of the fourth axiom. When V is a real vector space of stochastic processes and 0 is the zero process, it is known that the four axioms imply the existence of discount factors and the linearity of an intra-period utility function. So preferences correspond to discounting and are not risk neutral only if the converse of the fourth axiom is not satisfied.

Differential equations with positive evolutions and some applications

Relations between algebraic Riccati equations and generators of positive semigroups on ordered vector spaces have proved to be useful in the analysis of control problems for linear time-invariant systems and the numerical solution of general matrix equations, compare e.g. the recent works [22, 12, 20, 11]. In particular, this approach allows a unified treatment – up to a certain extent – of symmetric and nonsymmetric Riccati equations arising in discrete or continuous time or in a stochastic framework. The notion of positivity facilitates the design of monotonically convergent solution schemes and has a clear interpretation with respect to stability properties of the system.

Full Nuclear Cones and a Relation Between Strong Optimization and Pareto Efficiency

In this paper we present some new and pertinent connections between the Strong Optimization and the Approximate Pareto type Efficiency, in particular, with the usual Vector Optimization, at first in the Ordered Vector Spaces by the natural Convex Cones and, afterwards, in the Ordered Hausdorff Locally Convex Spaces. The main result is obtained considering the notion of full nuclear cone. Our results, is related to an appropriate scalarization method

An Elementary Example of an Order Bound Dual Space that is not Directed

This paper presents an elementary example of an order bounded linear functional on a directed partially ordered vector space that is not equal to the difference of two positive linear functionals.

Shape preserving approximation in vector ordered spaces

The aim of this note is to extend some classical results on the shape preserving approximation of real functions (of real variables) to functions with values in ordered vector spaces.

On the positivity of symmetric polynomial functions. Part II: Lattice general results and positivity criteria for degrees 4 and 5

We prove that homogeneous symmetric polynomial inequalities of degree p ∈ { 4 , 5 } in n positive 1 1 Which will mean ⩾0, according to the terminology of ordered vector spaces. variables can be algorithmically tested, on a finite set depending on the given inequality (Theorem 13); the test-set can be obtained by solving a finite number of equations of degree not exceeding p − 2 . Section 3 discusses the structure of the ordered vector spaces ( H p [ n ] , ⪯ ) and ( H p [ n ] , ⋞ ) . In Section 4, positivity criteria for degrees 4 and 5 are stated and proved. The main results are Theorems 10–14. Part III of this work will be concerned with the construction of extremal homogeneous symmetric polynomials (best inequalities) of degree 4 in n positive variables.

Vector Equilibrium Problems. Existence Theorems and Convexity of Solution Set

The natural quasi-concavity of set-valued mappings in an ordered vector space is introduced. Existence theorems for vector equilibrium problems involving set-valued monotone mappings are obtained and the convexity of the solution set is shown.

A note on measures with values in a partially ordered vector space

The goal of this note is to present an alternative, and we think simpler, proof of the following generalisation of the Riesz representation theorem due to J.D.M. Wright (Proc. London Math. Soc. 25 (1972) 675): any positive linear map Φ : C(X) →V can be represented by a V-valued measure on Baire subsets of X, where X is compact Hausdorff and V is a monotone σ-complete ordered vector space, not necessarily a lattice. Our proof suggests a purely inductive approach to measure theory, in the spirit of Borel’s original definition of measure of Borel sets.

Pure matrix states on operator systems

An operator system is a complex matricially ordered vector space that is completely order isomorphic to a unital selfadjoint subspace of a unital C*-algebra. A matrix state on an operator system V is a unital completely positive linear map of V into a full matrix algebra. Pure matrix states are studied, and a new and somewhat simplified proof of a Krein–Milman-type theorem of Webster and Winkler is given. If V is 3-dimensional, then the matrix state space of V is matrix-affinely homeomorphic to the matricial range of some Hilbert space operator. With the aid of this representation, pure matrix states on 3-dimensional operator systems are examined––and in some cases completely determined.

Subspaces of Normed Riesz Spaces

It will be shown that a normed partially ordered vector space is linearly, norm, and order isomorphic to a subspace of a normed Riesz space if and only if its positive cone is closed and its norm p satisfies p(x)≤p(y) for all x and y with -y≤x≤y. A similar characterization of the subspaces of M-normed Riesz spaces is given. With aid of the first characterization, Krein's lemma on directedness of norm dual spaces can be directly derived from the result for normed Riesz spaces. Further properties of the norms ensuing from the characterization theorem are investigated. Also a generalization of the notion of Riesz norm is studied as an analogue of the r-norm from the theory of spaces of operators. Both classes of norms are used to extend results on spaces of operators between normed Riesz spaces to a setting with partially ordered vector spaces. Finally, a partial characterization of the subspaces of Riesz spaces with Riesz seminorms is given.

On the positivity of symmetric polynomial functions.: Part I: General results

We prove that a real symmetric polynomial inequality of degree d⩾2 holds on R+n if and only if it holds for elements with at most ⌊d/2⌋ distinct non-zero components, which may have multiplicities. We establish this result by solving a Cauchy problem for ordinary differential equations involving the symmetric power sums; this implies the existence of a special kind of paths in the minimizer of some restriction of the considered polynomial function. In the final section, extensions of our results to the whole space Rn are outlined. The main results are Theorems 5.1 and 5.2 with Corollaries 2.1 and 5.2, and the corresponding results for Rn from the last subsection. Part II will contain a discussion on the ordered vector space Hd[n] in general, as well as on the particular cases of degrees d=4 and d=5 (finite test sets for positivity in the homogeneous case and other sufficient criteria).

Liftings of Diagrams of Semilattices by Diagrams of Dimension Groups

We investigate categorical and amalgamation properties of the functor Idc assigning to every partially ordered abelian group G its 〈 ∨ 0〉-semilattice of compact ideals IdcG. Our main result is the following. Theorem 1. Every diagram of finite Boolean semilattices indexed by a finite dismantlable partially ordered set can be lifted, with respect to the Idc functor, by a diagram of pseudo-simplicial vector spaces. Pseudo-simplicial vector spaces are a special kind of finite-dimensional partially ordered vector spaces with interpolation over the field of rational numbers. The methods introduced also make it possible to prove the following ring-theoretical result. Theorem 2. For any countable distributive 〈 ∨, 0〉-semilattices S and T and any field K, any 〈 ∨, 0〉-homomorphism f: S → T can be lifted, with respect to the Idc functor on rings, by a homomorphism f: A → B of K-algebras, for countably dimensional locally matricial algebras A and B over K. We also state a lattice-theoretical analogue of Theorem 2 (with respect to the Conc functor), and we provide counterexamples to various related statements. In particular, we prove that the result of Theorem 1 cannot be achieved with simplicial vector spaces alone. 2000 Mathematics Subject Classification 06A12, 06C20, 06F20, 15A03, 15A24, 15A48, 16E20, 16E50, 19A49, 19K14.

Seminorms on ordered vector spaces that extend to Riesz seminorms on larger Riesz spaces

As a generalization of the notion of Riesz seminorm, a class of seminorms on directed partially ordered vector spaces is introduced, such that (1) every seminorm in the class can be extended to a Riesz seminorm on every larger Riesz space that is majorized and (2) a seminorm on an order dense linear subspace of a Riesz space is in the class if and only if it can be extended to a Riesz seminorm on the Riesz space. The latter property yields that if a directed partially ordered vector space has an appropriate Riesz completion, then a seminorm on the space is in the class if and only if it is the restriction of a Riesz seminorm on the Riesz completion. An explicit formula for the extension is given. The class of seminorms is described by means of a notion of solid unit ball in partially ordered vector spaces. Some more properties concerning restriction and extension are studied, including extension to L- and M-seminorms.

Generalized quasiconvex set-valued maps

The aim of this paper is to introduce a concept of quasiconvexity for set-valued maps in a general framework, by only considering an abstract convexity structure in the domain and an arbitrary binary relation in the codomain. It is shown that this concept can be characterized in terms of usual quasiconvexity of certain real-valued functions. In particular, we focus on cone-quasiconvex set-valued maps with values in a partially ordered vector space.

Integral Representations for Continuous Linear Functionals in Operator-Initiated Topologies

On a given cone (resp. vector space) \(\mathcal{Q}\) we consider an initial topology and order induced by a family of linear operators into a second cone \(\mathcal{P}\) which carries a locally convex topology. We prove that monotone linear functionals on \(\mathcal{Q}\) which are continuous with respect to this initial topology may be represented as certain integrals of continuous linear functionals on \(\mathcal{P}\). Based on the Riesz representation theorem from measure theory, we derive an integral version of the Jordan decomposition for linear functionals on ordered vector spaces.

The super order dual of an ordered vector space and the Riesz–Kantorovich formula

A classical theorem of F. Riesz and L. V. Kantorovich asserts that if L L is a vector lattice and f f and g g are order bounded linear functionals on L L , then their supremum (least upper bound) f ∨ g f\lor g exists in L ∼ L^\sim and for each x ∈ L + x\in L_+ it satisfies the so-called Riesz–Kantorovich formula: \[ [ f ∨ g ] ( x ) = sup { f ( y ) + g ( z ) : y , z ∈ L + a n d y + z = x } . \bigl [f\lor g\bigr ](x)=\sup \bigl \{f(y)+g(z)\colon \ y,z\in L_+\ \,\mathrm {and}\ \, y+z=x\bigr \}\,. \] Related to the Riesz–Kantorovich formula is the following long-standing problem: If the supremum of two order bounded linear functionals f f and g g on an ordered vector space exists, does it then satisfy the Riesz–Kantorovich formula? In this paper, we introduce an extension of the order dual of an ordered vector space and provide some answers to this long-standing problem. The ideas regarding the Riesz–Kantorovich formula owe their origins to the study of the fundamental theorems of welfare economics and the existence of competitive equilibrium. The techniques introduced here show that the existence of decentralizing prices for efficient allocations is closely related to the above-mentioned problem and to the properties of the Riesz–Kantorovich formula.

Differences of Convex Compact Sets in the Space of Directed Sets. Part I: The Space of Directed Sets

A normed and partially ordered vector space of so-called ‘directed sets’ is constructed, in which the convex cone of all nonempty convex compact sets in R n is embedded by a positively linear, order preserving and isometric embedding (with respect to a new metric stronger than the Hausdorff metric and equivalent to the Demyanov one). This space is a Banach and a Riesz space for all dimensions and a Banach lattice for n=1. The directed sets in R n are parametrized by normal directions and defined recursively with respect to the dimension n by the help of a ‘support’ function and directed ‘supporting faces’ of lower dimension prescribing the boundary. The operations (addition, subtraction, scalar multiplication) are defined by acting separately on the ‘support’ function and recursively on the directed ‘supporting faces’. Generalized intervals introduced by Kaucher form the basis of this recursive approach. Visualizations of directed sets will be presented in the second part of the paper.

Differences of Convex Compact Sets in the Space of Directed Sets. Part II: Visualization of Directed Sets

This paper is a continuation of the author's first paper (Set-Valued Anal.9 (2001), pp. 217–245), where the normed and partially ordered vector space of directed sets is constructed and the cone of all nonempty convex compact sets in Rn is embedded. A visualization of directed sets and of differences of convex compact sets is presented and its geometrical components and properties are studied. The three components of the visualization are compared with other known differences of convex compact sets.