Some categorical properties of the functors O τ and O R of weakly additive functionals

In the paper, the spaces of weakly additive τ-smooth and Radon functionals are investigated. It is proved that the functors of weakly additive τ-smooth and Radon functionals weakly preserve the density of Tychonoff spaces, and the functor of weakly additive τ-smooth functionals forms a monad in the category of Tychonoff spaces and their continuous mappings. Examples and remarks are given showing that these functors fail to satisfy certain Shchepin normality conditions. Problems having positive solutions for normal functors are presented.     

Group duality with the topology of precompact convergence

The Pontryagin-van Kampen (P-vK) duality, defined for topological Abelian groups, is given in terms of the compact-open topology. Polar reflexive spaces, introduced by Köthe, are those locally convex spaces satisfying duality when the dual space is equipped with the precompact-open topology. It is known that the additive groups of polar reflexive spaces satisfy P-vK duality. In this note we consider the duality of topological Abelian groups when the topology of the dual is the precompact-open topology. We characterize the precompact reflexive groups, i.e., topological groups satisfying the group duality defined in terms of the precompact-open topology. As a consequence, we obtain a new characterization of polar reflexive spaces. We also present an example of a space which satisfies P-vK duality and is not polar reflexive. Some of our results respond to questions appearing in the literature.     

Fourier transformation of Sato's hyperfunctions

A new generalized function space in which all Gelfand-Shilov classes (α>1) of analytic functionals are embedded is introduced. This space of ultrafunctionals does not possess a natural nontrivial topology and cannot be obtained via duality from any test function space. A canonical isomorphism between the spaces of hyperfunctions and ultrafunctionals on R^k is constructed that extends the Fourier transformation of Roumieu-type ultradistributions and is naturally interpreted as the Fourier transformation of hyperfunctions. The notion of carrier cone that replaces the notion of support of a generalized function for ultrafunctionals is proposed. A Paley-Wiener-Schwartz-type theorem describing the Laplace transformation of ultrafunctionals carried by proper convex closed cones is obtained and the connection between the Laplace and Fourier transformations is established.     

Algebraic duality theorems for infinite LP problems

In this paper, we consider a primal-dual infinite linear programming problem-pair, i.e. LPs on infinite dimensional spaces with infinitely many constraints. We present two duality theorems for the problem-pair: a weak and a strong duality theorem. We do not assume any topology on the vector spaces, therefore our results are algebraic duality theorems. As an application, we consider transferable utility cooperative games with arbitrarily many players.     

Tent spaces associated with semigroups of operators

We study tent spaces on general measure spaces (Ω,μ). We assume that there exists a semigroup of positive operators on Lp(Ω,μ) satisfying a monotone property but do not assume any geometric/metric structure on Ω. The semigroup plays the same role as integrals on cones and cubes in Euclidean spaces. We then study BMO spaces on general measure spaces and get an analogue of Fefferman's H¹-BMO duality theory. We also get a H¹-BMO duality inequality without assuming the monotone property. All the results are proved in a more general setting, namely for noncommutative Lp spaces.     

凸函数的支撑凸集及其集映射

本文研究了线性空间中凸函数的支撑泛函存在性以及支撑泛函的数值域,利用子空间中支撑泛函延拓的方法,构造出在线性空间任意点的支撑泛函,确定在同一支撑点上支撑泛函的数值域,从而得到支撑泛函具有唯一性的充分必要条件,最后对支撑凸集到支撑泛函集的集映射进行讨论.     

Spectral-like duality for distributive Hilbert algebras with infimum

Distributive Hilbert algebras with infimum, or DH^-algebras for short, are algebras with implication and conjunction, in which the implication and the conjunction do not necessarily satisfy the residuation law. These algebras do not fall under the scope of the usual duality theory for lattice expansions, precisely because they lack residuation. We propose a new approach, that consists of regarding the conjunction as the additional operation on the underlying implicative structure. In this paper, we introduce a class of spaces, based on compactly-based sober topological spaces. We prove that the category of these spaces and certain relations is dually equivalent to the category of DH^-algebras and \({\wedge}\)-semi-homomorphisms. We show that the restriction of this duality to a wide subcategory of spaces gives us a duality for the category of DH^-algebras and algebraic homomorphisms. This last duality generalizes the one given by the author in 2003 for implicative semilattices. Moreover, we use the duality to give a dual characterization of the main classes of filters for DH^-algebras, namely, (irreducible) meet filters, (irreducible) implicative filters and absorbent filters.     

Best approximation in vector spaces

In this paper, we obtain some results on best f-approximation in quotient spaces of vector spaces and determine under what conditions f-proximinality can be transmitted to and from quotient spaces. Also, in conclusion, we consider the relationship between f-approximation subsets and linear functionals on X.     

On certain properties of the spaces of order-preserving functionals

In the present paper it is proved that the functor Oτ of τ-smooth order preserving functionals and the functor OR of Radon order preserving functionals, do not change the weight of infinite Tychonoff spaces. It is shown that the density and the weak density of infinite Tychonoff spaces do not increase under these functors. Moreover, if X is a metric space with the second axiom of countability then the spaces Oτ(X) and OR(X) are also metrizable.     

Convex conjugates of integral functionals

We consider a convex integral functional on a functional space V andcompute its greatest extension to the algebraic bidual space V^**, among all convex functions which are lower semicontinuous with respect tothe *-weak topology o(V^** ; V^*).Such computations are usually performed to extend these functionals to sometopological closures. In the present paper, no a priori topological restrictionsare imposed on the extended domain. As a consequence, this extended functionalis a valuable first step for the computation of the exact shape of the minimizersof the conjugate convex integral functional subject to a convex constraint,in full generality: without constraint qualification. These convex integralfunctionals are sometimes called entropies, divergences or energies. Our proofsmainly rely on basic convex duality and duality in Orlicz spaces.     

Analysis on Besov spaces II: Embedding and duality theorems

This paper gives several results on Besov spaces of holomorphic functions on a very large class of domains D in Cn. They include duality theorem, embedding theorem, best growth estimate, and boundedness of multiplication operators on Besov spaces.     

Duality theorems for a new class of multitime multiobjective variational problems

In this work, we consider a new class of multitime multiobjective variational problems of minimizing a vector of functionals of curvilinear integral type. Based on the normal efficiency conditions for multitime multiobjective variational problems, we study duals of Mond-Weir type, generalized Mond-Weir-Zalmai type and under some assumptions of (??, b)-quasiinvexity, duality theorems are stated. We give weak duality theorems, proving that the value of the objective function of the primal cannot exceed the value of the dual. Moreover, we study the connection between values of the objective functions of the primal and dual programs, in direct and converse duality theorems. While the results in §1 and §2 are introductory in nature, to the best of our knowledge, the results in §3 are new and they have not been reported in literature.     

Minimization of some non-differentiable functionals by the Augmented Lagrangian Method of Hestenes and Powell

The Augmented Lagrangian Method of Hestenes and Powell is presented here in a more general case, including Fenchel's duality, using some recent results of R. T. Rockafellar which are here extended. It is shown that for some functionals which contain a non-differentiable term which is the support function of a convex set, the Augmented Lagrangian Method provides a natural way to marry standard duality techniques and regularisation techniques. An application to visco-plastic flows is presented and numerical results are given. A convergence proof is given for the algorithm used. Another application to elasto-plastic torsion is also signaled.     

Evolution equations with Gel’fond-Leont’ev generalized differentiation operators: II

We establish the solvability of the Cauchy problem for evolution equations with Gel’fond-Leont’ev generalized differentiation operators in spaces of the type W as well as in spaces of generalized functions (analytic functionals) of the type W′.     

Characterizations of ɛ-duality gap statements for constrained optimization problems

In this paper we present different regularity conditions that equivalently characterize various ?-duality gap statements (with ? ≥ 0) for constrained optimization problems and their Lagrange and Fenchel-Lagrange duals in separated locally convex spaces, respectively. These regularity conditions are formulated by using epigraphs and ?-subdifferentials. When ? = 0 we rediscover recent results on stable strong and total duality and zero duality gap from the literature.     

A self-dual variational approach to stochastic partial differential equations

Unlike many of their deterministic counterparts, stochastic partial differential equations are not amenable to the methods of calculus of variations à la Euler–Lagrange. In this paper, we show how self-dual variational calculus leads to variational solutions of various stochastic partial differential equations driven by monotone vector fields. We construct solutions as minima of suitable non-negative and self-dual energy functionals on Itô spaces of stochastic processes. We show how a stochastic version of Bolza's duality leads to solutions for equations with additive noise. We then use a Hamiltonian formulation to construct solutions for non-linear equations with non-additive noise such as the stochastic Navier–Stokes equations in dimension two.     

Dual nonlinear programming problems in partially ordered Banach spaces

Summary The concept of duality plays an important role in mathematical programming and has been studied extensively in a finite dimensional Eucledian space, (see e.g. [13, 4, 6, 8]). More recently various dual problems with functionals as objective functions have been studied in infinite dimensional vector spaces [5, 7, 1, 10, 12].In this note we consider a nonlinear minimization problem in a partially ordered Banach space. It is assumed that the objective function of this problem is given by a (nonlinear) operator and that its feasible domain is defined by a system of (nonlinear) operator inequalities. In analogy to the finite dimensional case we associate with this minimization problem a dual maximization problem which is defined in the Cartesian product of certain Banach spaces. It is shown that under suitable assumptions the main results of the finite dimensional duality theory can be extended to this general case. This extension is based on optimality conditions obtained in [11].