Some results on linear delay integral equations

Recently Borwein has proposed a definition for extending Geoffrion's concept of proper efficiency to the vector maximization problem in which the domination cone S is any nontrivial, closed convex cone. However, when S is the non-negative orthant, solutions may exist which are proper according to Borwein's definition but improper by Geoffrion's definition. As a result, when S is the non-negative orthant, certain properties of proper efficiency as defined by Geoffrion do not hold under Borwein's definition. To rectify this situation, we propose a definition of proper efficiency for the case when S is a nontrivial, closed convex cone which coincides with Geoffrion's definition when S is the non-negative orthant. The proposed definition seems preferable to Borwein's for developing a theory of proper efficiency for the case when S is a nontrivial, closed convex cone.     

Concepts of proper efficiency

The centrol concept of proper efficiency has been largely that of Geoffrion. There are, however, other concepts, and this paper considers two of them, viz. those of Klinger and Kuhn and Tucker, in relationship to each other and to Geoffrion. This is done in terms of various properties which characterise the efficient sets. Geoffrion's concept is a global one, whereas the other two concepts are local ones, and their significance is somewhat different. This is examined specifically in the context of optimal solutions, where, for example, it is shown that Geoffrion and Kuhn and Tucker proper efficiency fails to meet an optimality condition, which is satisfied by Klinger proper efficiency.     

Convergence in Hilbert's metric and convergence in direction

Hilbert's metric on a cone K is a measure of distance between the rays of K. Hilbert's metric has many applications, but they all depend on the equivalence between closeness of two rays in the Hilbert metric and closeness of the two unit vectors along these rays (in the usual sense). A necessary and sufficient condition on K for this equivalence to hold is given.     

Obtuse cones in Hilbert spaces and applications to partial differential equations

The positive cone K in a partially ordered Hilbert space is said to be obtuse with respect to the inner product if the dual cone $\text{K}{}^1\text{? K}$. Obtuseness of cones with respect to non-symmetric bilinear forms is also defined and characterized. These results are applied to the generalized Sobolev space associated with an elliptic boundary value problem, in particular to the question of determining the non-negativity of the Green's function. A notion of strict obtuseness is defined, characterized and applied to the question of strict positivity of the Green's function. Applications to positivity preserving semi-groups are also given.     

Extension theorems for some classes of continua

It is proved that every mapping from a proper subcontinuum of a hereditarily unicoherent continuum X onto the Knaster's indecomposable continuum (onto a cone over a zerodimensional compact metric set) can be extended to a mapping defined on X.Similarly, every mapping from a proper subcontinuum of a hereditarily indecomposable continuum onto a pseudoarc can be extended to a mapping defined on the whole space.Both of the above results are generalizations of the author's earlier results to the nonmetric case. As a consequence it is obtained that a pseudoarc is continuously n-homogeneous.     

On the semiring of cone preserving maps

If K is a proper cone in Rⁿ, then the cone of all linear operators that preserve K, denoted by π(K), forms a semiring under usual operator addition and multiplication. Recently J.G. Horne examined the ideals of this semiring. He proved that if K₁, K₂ are polyhedral cones such that π(K₁) and π(K₂) are isomorphic as semirings, then K₁ and K₂ are linearly isomorphic. The study of this semiring is continued in this paper. In Sec. 3 ideals of π(K) which are also faces are characterized. In Sec. 4 it is shown that π(K) has a unique minimal two-sided ideal, namely, the dual cone of π(K¹), where K¹ is the dual cone of K. Extending Horne's result, it is also proved that the cone K is characterized by this unique minimal two-sided ideal of π(K). The set of all faces of π(K) inherits a quotient semiring structure from π(K). Properties of this face-semiring are given in Sec. 5. In particular, it is proved that this face-semiring admits no nontrivial congruence relation iff the duality operator of π(K) is injective. In Sec. 6 the maximal one-sided and two-sided ideals of π(K) are identified. In Sec. 8 it is shown that π(K) never satisfies the ascending-chain condition on principal one-sided ideals. Some partial results on the question of topological closedness of principal one-sided ideals of π(K) are also given.     

Green's and Dirichlet spaces associated with fine Markov processes

This is the second paper in a series devoted to Green's and Dirichlet spaces. In the first paper, we have investigated Green's space $\text{K}$ and the Dirichlet space $\text{H}$ associated with a symmetric Markov transition function p_t(x, B). Now we assume that p is a transition function of a fine Markov process X and we prove that: (a) the space $\text{H}$ can be built from functions which are right continuous along almost all paths; (b) the positive cone $\text{K}$⁺ in $\text{K}$ can be identified with a cone M of measures on the state space; (c) the positive cone $\text{H}$⁺ in $\text{H}$ can be interpreted as the cone of Green's potentials of measures μ?M. To every measurable set B in the state space E there correspond a subspace $\text{K}$(B) of $\text{K}$ and a subspace $\text{H}$(B) of $\text{H}$. The orthogonal projections of $\text{K}$ onto $\text{K}$ and of $\text{H}$ onto $\text{H}$(B) can be expressed in terms of the hitting probabilities of B by the Markov process X. As the main tool, we use additive functionals of X corresponding to measures μ?M.     

Lattice valued relations and automata

A lattice-valued relation, lvr for short, from a set X to a set Y is a function from the Cartesian product of X and Y to a lattice. This concept is a generalization of other structures, notably tolerance spaces, nets and automata, separately investigated by the authors elsewhere. It is adequate to admit a natural definition of homogeneity and a classification of homogeneous lvr's by their isomorphism groups. The main result of the present paper is a proof of this classification. The application of this to automata, also interpretable as lvr's, is described, and an example given. We conclude with a brief discussion of the lvr theory of fuzzy and stochastic automata.     

Conjugate cone characterization of positive definite and semidefinite matrices

Positive definite and semidefinite matrices are characterized in terms of positive definiteness and semidefiniteness on arbitrary closed convex cones in Rⁿ. These results are obtained by generalizing Moreau's polar decomposition to a conjugate decomposition. Some typical results are: The matrix A is positive definite if and only if for some closed convex cone K, A is positive definite on K and (A+A^T)^?1 exists and is semidefinite on the polar cone K°. The matrix A is positive semidefinite if and only if for some closed convex cone K such that either K is polyhedral or (A+A^T)(K) is closed, A is positive semidefinite on both K and the conjugate cone K^A={s∣x^T(A+ A^T)s?0?x∈K}, and (A+A^T)x=0 for all x in K such that x^TAx=0.     

Wagner's theorem for Torus graphs

Wagner's theorem (any two maximal plane graphs having p vertices are equivalent under diagonal transformations) is extended to maximal torus graphs, graphs embedded in the torus with a maximal set of edges present. Thus any maximal torus graph having p vertices may be diagonally transformed into any other maximal torus graph having p vertices. As with Wagner's theorem, a normal form representing an intermediate stage in the above transformation is displayed. This result, along with Wagner's theorem, may make possible constructive characterizations of planar and toroidal graphs, through a wholly combinatorial definition of diagonal transformation.     

Connectedness of efficient solutions in multiple criteria combinatorial optimization

In multiple criteria optimization an important research topic is the topological structure of the set X_e of efficient solutions. Of major interest is the connectedness of X_e, since it would allow the determination of Xe without considering non-efficient solutions in the process. We review general results on the subject, including the connectedness result for efficient solutions in multiple criteria linear programming. This result can be used to derive a definition of connectedness for discrete optimization problems. We present a counterexample to a previously stated result in this area, namely that the set of efficient solutions of the shortest path problem is connected. We will also show that connectedness does not hold for another important problem in discrete multiple criteria optimization: the spanning tree problem.     

A proof of the existence of Batanin's initial operad

In this paper we prove the existence of the n-globular operad used in Batanin's definition of weak n-category. This operad is initial in the category of n-globular operads equipped with two extra pieces of structure: a system of compositions and a contraction. Our approach closely follows a proof by Leinster of the existence of a similar n-globular operad used in his definition of weak n-category (itself a variant of Batanin's definition) – we show that there is a functor giving the free operad equipped with a contraction and system of compositions on an n-globular collection, and applying this functor to the initial collection gives the desired initial operad. Since there is no interaction between the contraction and operad structures we are able to treat their free constructions separately. This is not true of the system of compositions structure, which cannot exist separately from the operad structure, so we use an interleaving-style construction to describe the free operad with system of compositions.     

A classification of 4-connected graphs

Having observed Tutte's classification of 3-connected graphs as those attainable from wheels by line addition and point splitting and Hedetniemi's classification of 2-connected graphs as those obtainable from K₂ by line addition, subdivision and point addition, one hopes to find operations which classify n-connected graphs as those obtainable from, for example, K_n+1. In this paper I give several generalizations of the above operations and use Halin's theorem to obtain two variations of Tutte's theorem as well as a classification of 4-connected graphs.     

Optimality conditions for a class of composite multiobjective nonsmooth optimization problems

In this paper, a class of composite multiobjective nonsmooth optimization problems with cone constraints is considered. Necessary optimality conditions for weak minimum are established in terms of Semi-infinite Gordan type theorem. η-generalized null space condition, which is a proper generalization of generalized null space condition, is proposed. Sufficient optimality conditions are obtained for weak minimum, Pareto minimum, Benson’s proper minimum under K-generalized invexity and η-generalized null space condition. Some examples are given to illustrate our main results.     

On indecomposable subcones

Let K₁,K₂ be cones. We say that K₁ is a subcone of K₂ if ExtK₁?ExtK₂. Furthermore, if K₁≠K₂, K₁ is called a proper subcone; if dimK₁=dimK₂, K₁ is called a non-degenerate subcone. We first prove that every n-dimensional indecomposable cone, n?3, contains a non-degenerate indecomposable subcone which has no more than 2n-2 extremals. Then we construct for each n?3 an n-dimensional indecomposable cone with exactly 2n-2 extremals such that each of its proper non-degenerate subcones is decomposable.     

Searching for critical angles in a convex cone

The concept of antipodality relative to a closed convex cone has been explored in detail in a recent work of ours. The antipodality problem consists of finding a pair of unit vectors in K achieving the maximal angle of the cone. Our attention now is focused not just in the maximal angle, but in the angular spectrum of the cone. By definition, the angular spectrum of a cone is the set of angles satisfying the stationarity (or criticality) condition associated to the maximization problem involved in the determination of the maximal angle. In the case of a polyhedral cone, the angular spectrum turns out to be a finite set. Among other results, we obtain an upper bound for the cardinality of this set. We also discuss the link between the critical angles of a cone K and the critical angles of its dual cone. Dedicated to Boris Polyak on his 70th Birthday.     

Positive semi-definite matrices as sums of squares

If K is a field and char K ≠ 2, then an element α?K is a sum of squares in K if and only if α ? 0 for every ordering of K. This is the famous theorem of Artin and Landau. It has been generalized to symmetric matrices over K by D. Gondard and P. Ribenboim. They have also shown that Artin's theorem on positive definite rational functions has a suitable extension to positive definite matrix functions. In this paper we attain two goals. First, we show that similar theorems are valid for Hermitian matrices instead of symmetric ones. Second, we simplify D. Gondard and P. Ribenboim's proof of their second theorem and strengthen it.     

Jensen's inequality for a convex vector-valued function on an infinite-dimensional space

Jensen's inequality f(EX) ≤ Ef(X) for the expectation of a convex function of a random variable is extended to a generalized class of convex functions f whose domain and range are subsets of (possibly) infinite-dimensional linear topological spaces. Convexity of f is defined with respect to closed cone partial orderings, or more general binary relations, on the range of f. Two different methods of proof are given, one based on geometric properties of convex sets and the other based on the Strong Law of Large Numbers. Various conditions under which Jensen's inequality becomes strict are studied. The relation between Jensen's inequality and Fatou's Lemma is examined.     

Theory of cones

This survey deals with the aspects of archimedian partially ordered finite-dimensional real vector spaces and order preserving linear maps which do not involve spectral theory. The first section sketches some of the background of entrywise nonnegative matrices and of systems of inequalities which motivate much of the current investigations. The study of inequalities resulted in the definition of a polyhedral cone K and its face lattice F(K). In Section II.A the face lattice of a not necessarily polyhedral cone K in a vector space V is investigated. In particular the interplay between the lattice properties of $\text{F}$(K) and geometric properties of K is emphasized. Section II.B turns to the cones Π(K) in the space of linear maps on V. Recall that Π(K) is the cone of all order preserving linear maps. Of particular interest are the algebraic structure of Π(K) as a semiring and the nature of the group Aut(K) of nonsingular elements A?Π(K) for which A^-1?Π(K) as well. In a short final section the cone P_n of n×n positive semidefinite matrices is discussed. A characterization of the set of completely positive linear maps is stated. The proofs will appear in a forthcoming paper.