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.     

Weak covering properties and selection principles

No convenient internal characterization of spaces that are productively Lindelöf is known. Perhaps the best general result known is Alster?s internal characterization, under the Continuum Hypothesis, of productively Lindelöf spaces which have a basis of cardinality at most _ℵ1${\aleph }_1$. It turns out that topological spaces having Alster?s property are also productively weakly Lindelöf. The weakly Lindelöf spaces form a much larger class of spaces than the Lindelöf spaces. In many instances spaces having Alster?s property satisfy a seemingly stronger version of Alster?s property and consequently are productively X, where X is a covering property stronger than the Lindelöf property. This paper examines the question: When is it the case that a space that is productively X is also productively Y, where X and Y are covering properties related to the Lindelöf property.     

A characterization of the duality mapping for convex bodies

We characterize the duality of convex bodies in d-dimensional Euclidean vector space, viewed as a mapping from the space of convex bodies containing the origin in the interior into the same space. The question for such a characterization was posed by Vitali Milman. The property that the duality interchanges pairwise intersections and convex hulls of unions is sufficient for a characterization, up to a trivial exception and the composition with a linear transformation. Received: March 2007, Accepted: April 2007     

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.     

Superconvergent Nyström and degenerate kernel methods for eigenvalue problems

The Nyström and degenerate kernel methods, based on projections at Gauss points onto the space of (discontinuous) piecewise polynomials of degree ?r-1, for the approximate solution of eigenvalue problems for an integral operator with a smooth kernel, exhibit order 2r. We propose new superconvergent Nyström and degenerate kernel methods that improve this convergence order to 4r for eigenvalue approximation and to 3r for spectral subspace approximation in the case where the kernel is sufficiently smooth. Moreover for a simple eigenvalue, we show that by using an iteration technique, an eigenvector approximation of order 4r can be obtained. The methods introduced here are similar to that studied by Kulkarni in [10] and exhibit the same convergence orders, so a comparison with these methods is worked out in detail. Also, the error terms are analyzed and the obtained methods are numerically tested. Finally, these methods are extended to the case of discontinuous kernel along the diagonal and superconvergence results are also obtained.     

Hölder estimates of p-harmonic extension operators

It is now a well-known fact that for 1<p<∞ the p-harmonic functions on domains in metric measure spaces equipped with a doubling measure supporting a (1,p)-Poincaré inequality are locally Hölder continuous. In this note we provide a characterization of domains in such metric spaces for which p-harmonic extensions of Hölder continuous boundary data are globally Hölder continuous. We also provide a link between this regularity property of the domain and the uniform p-fatness of the complement of the domain.     

A CHARACTERIZATION OF PONTRYAGIN-VAN KAMPEN DUALITY FOR COMPLEX LOCALLY CONVEX SPACES

ABSTRACT

Topological vector spaces (TVS) are topological Abelian groups when considered under the operation of addition. It is therefore natural to ask when they satisfy Pontryagin-van Kampen (P-vK) duality. A number of attempts have been considered for real LCS. In particular, in a previous paper we showed a characterization of (P-vK) duality for real LCS. In this article we show that this characterization also holds for complex spaces.     

Löwner chains and a subclass of biholomorphic mappings

In this paper, we give characterization of almost starlike functions of order α (respectively almost starlike mappings of order α) on the unit disc in C (respectively the unit ball in a finite-dimensional complex Banach space) in terms of Löwner chains. Furthermore, using the properties of Löwner chains, we can easily prove that two classes of generalized Roper-Suffridge extension operators preserve almost starlikeness of order α on two important classes of Reinhardt domains in Cn, respectively.     

Symmetric (41,16,6)-designs with a nontrivial automorphism of odd order

If a symmetric (41,16,6)-design has an automorphism σ of odd prime order q then q = 3 or 5. In the case q = 5 we determine all such designs and find a total of 419 nonisomorphic ones, of which 15 are self-dual. When q = 3 a combinatorial explosion occurs and the complete classification becomes impracticable. However, we give a characterization in the particular case when σ has order 3 and fixes 11 points, and find that there are 3,076 nonisomorphic designs with this property, all of them being non self-dual. The other remaining possibility is that σ, of order 3, fixes 5 points. In this case there are 960 orbit matrices (up to isomorphism and duality) and all but one of them yield designs. Here an incomplete investigation shows that in total there are at least 112,000 nonisomorphic designs. © 1993 John Wiley & Sons, Inc.     

Sustainability in bankruptcy problems

This paper focuses on a new property for bankruptcy rules, calledsustainability, which requires that the agents with small enough claims be fully reimbursed. We show that the constrained equal-awards rule is the only rule that satisfies path independence and sustainability. Exploiting duality relations, we also provide a characterization of the constrained equal-losses rule, as the only one that satisfies composition and independence of residual claims (the dual property of sustainability).     

Minimal Gröbner bases and the predictable leading monomial property

We focus on Gröbner bases for modules of univariate polynomial vectors over a ring. We identify a useful property, the “predictable leading monomial (PLM) property” that is shared by minimal Gröbner bases of modules in F[x]^q, no matter what positional term order is used. The PLM property is useful in a range of applications and can be seen as a strengthening of the wellknown predictable degree property (= row reducedness), a terminology introduced by Forney in the 70’s. Because of the presence of zero divisors, minimal Gröbner bases over a finite ring of the type Z_p^r (where p is a prime integer and r is an integer >1) do not necessarily have the PLM property. In this paper we show how to derive, from an ordered minimal Gröbner basis, a so-called “minimal Gröbner p-basis” that does have a PLM property. We demonstrate that minimal Gröbner p-bases lend themselves particularly well to derive minimal realization parametrizations over Z_p^r. Applications are in coding and sequences over Z_p^r.     

Second order mixed type duality in multiobjective programming problems

Second order mixed type dual is introduced for multiobjective programming problems. Results about weak duality, strong duality, and strict converse duality are established under generalized second order (F,ρ)-convexity assumptions. These results generalize the duality results recently given by Aghezzaf and Hachimi involving generalized first order (F,ρ)-convexity conditions.     

Strong Duality and Dual Pricing Properties in Semi-Infinite Linear Programming: A non-Fourier–Motzkin Elimination Approach

Following the idea of the conjecture for semi-infinite programming in a paper by Kortanek and Zhang, recently published in Optimization, in this paper we show that the Fourier–Motzkin elimination is not needed in the study of the strong duality and dual pricing properties for semi-infinite programming. We also prove several new results on the strong duality and dual pricing properties. Specifically, we propose a new subspace, under which the strong duality property holds. We give a necessary and sufficient condition for the dual pricing property to hold under this subspace, which is further used to examine the examples presented in the Basu–Martin–Ryan paper.     

Optimality and duality for nondifferentiable multiobjective variational problems

The concept of efficiency is used to formulate duality for nondifferentiable multiobjective variational problems. Wolfe and Mond-Weir type vector dual problems are formulated. By using the generalized Schwarz inequality and a characterization of efficient solution, we established the weak, strong, and converse duality theorems under generalized (F,ρ)-convexity assumptions.     

Duality and operator algebras: automatic weak* continuity and applications

We investigate some subtle and interesting phenomena in the duality theory of operator spaces and operator algebras, and give several applications of the surprising fact that certain maps are always weak^*-continuous on dual spaces. In particular, if X is a subspace of a C^*-algebra A, and if a∈A satisfies aX⊂X, then we show that the function x?ax on X is automatically weak^* continuous if either (a) X is a dual operator space, or (b) a^*X⊂X and X is a dual Banach space. These results hinge on a generalization to Banach modules of Tomiyama's famous theorem on contractive projections onto a C^*-subalgebra. Applications include a new characterization of the σ-weakly closed (possibly nonunital and nonselfadjoint) operator algebras, and a generalization of the theory of W^*-modules to the framework of modules over such algebras. We also give a Banach module characterization of σ-weakly closed spaces of operators which are invariant under the action of a von Neumann algebra.     

Quantum convex support

Convex support, the mean values of a set of random variables, is central in information theory and statistics. Equally central in quantum information theory are mean values of a set of observables in a finite-dimensional C^∗-algebra A, which we call (quantum) convex support. The convex support can be viewed as a projection of the state space of A and it is a projection of a spectrahedron.Spectrahedra are increasingly investigated at least since the 1990s boom in semi-definite programming. We recall the geometry of the positive semi-definite cone and of the state space. We write a convex duality for general self-dual convex cones. This restricts to projections of state spaces and connects them to results on spectrahedra.Our main result is an analysis of the face lattice of convex support by mapping this lattice to a lattice of orthogonal projections, using natural isomorphisms. The result encodes the face lattice of the convex support into a set of projections in A and enables the integration of convex geometry with matrix calculus or algebraic techniques.     

Dualities for quasi-varieties of bands

We complete the characterization of finite bands admitting a natural duality, by showing that every finite normal band admits a natural duality. In particular we show that every finite normal band is finitely related.     

Cluster-tilted algebras are Gorenstein and stably Calabi-Yau

We prove that in a 2-Calabi-Yau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its Cohen-Macaulay modules is 3-Calabi-Yau. We deduce in particular that cluster-tilted algebras are Gorenstein of dimension at most one, and hereditary if they are of finite global dimension. Our results also apply to the stable (!) endomorphism rings of maximal rigid modules of [Christof Geiß, Bernard Leclerc, Jan Schröer, Rigid modules over preprojective algebras, arXiv: math.RT/0503324, Invent. Math., in press]. In addition, we prove a general result about relative 3-Calabi-Yau duality over non-stable endomorphism rings. This strengthens and generalizes the Ext-group symmetries obtained in [Christof Geiß, Bernard Leclerc, Jan Schröer, Rigid modules over preprojective algebras, arXiv: math.RT/0503324, Invent. Math., in press] for simple modules. Finally, we generalize the results on relative Calabi-Yau duality from 2-Calabi-Yau to d-Calabi-Yau categories. We show how to produce many examples of d-cluster tilted algebras.     

Lagrangian Duality and Cone Convexlike Functions

In this paper, we consider first the most important classes of cone convexlike vector-valued functions and give a dual characterization for some of these classes. It turns out that these characterizations are strongly related to the closely convexlike and Ky Fan convex bifunctions occurring within minimax problems. Applying the Lagrangian perturbation approach, we show that some of these classes of cone convexlike vector-valued functions show up naturally in verifying strong Lagrangian duality for finite-dimensional optimization problems. This is achieved by extending classical convexity results for biconjugate functions to the class of so-called almost convex functions. In particular, for a general class of finite-dimensional optimization problems, strong Lagrangian duality holds if some vector-valued function related to this optimization problem is closely K-convexlike and satisfies some additional regularity assumptions. For K a full-dimensional convex cone, it turns out that the conditions for strong Lagrangian duality simplify. Finally, we compare the results obtained by the Lagrangian perturbation approach worked out in this paper with the results achieved by the so-called image space approach initiated by Giannessi.     

A characterization of uniquely representable interval graphs

An alternative to Hanlon's buried-subgraph characterization of interval graphs that have unique (up to duality) agreeing interval orders is proveded. The alternative is based on a relation L between ordered pairs of points that are adjacent in the interval graph. The interpretation of abLxy is that if the interval for a precedes (follows) the interval for b in a representation of the interval graph, then the interval for x must precede (follow) the interval for y.