On self duality of pathwidth in polyhedral graph embeddings

Let G be a 3‐connected planar graph and G^* be its dual. We show that the pathwidth of G^* is at most 6 times the pathwidth of G. We prove this result by relating the pathwidth of a graph with the cut‐width of its medial graph and we extend it to bounded genus embeddings. We also show that there exist 3‐connected planar graphs such that the pathwidth of such a graph is at least 1.5 times the pathwidth of its dual. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 42–54, 2007     

Gale duality and Koszul duality

Given a hyperplane arrangement in an affine space equipped with a linear functional, we define two finite-dimensional, noncommutative algebras, both of which are motivated by the geometry of hypertoric varieties. We show that these algebras are Koszul dual to each other, and that the roles of the two algebras are reversed by Gale duality. We also study the centers and representation categories of our algebras, which are in many ways analogous to integral blocks of category O.     

Wolfe duality and Mond-Weir duality via perturbations

Considering a general optimization problem, we attach to it by means of perturbation theory two dual problems having in the constraints a subdifferential inclusion relation. When the primal problem and the perturbation function are particularized different new dual problems are obtained. In the special case of a constrained optimization problem, the classical Wolfe and Mond-Weir duals, respectively, follow as particularizations of the general duals by using the Lagrange perturbation. Examples to show the differences between the new duals are given and a gate towards other generalized convexities is opened.     

Binz-Butzmann duality versus Pontryagin duality

Supported by Ministerio de Education y Ciencia, grant #BE91-031     

Stone duality and Gleason covers through de Vries duality

We introduce zero-dimensional de Vries algebras and show that the category of zero-dimensional de Vries algebras is dually equivalent to the category of Stone spaces. This shows that Stone duality can be obtained as a particular case of de Vries duality. We also introduce extremally disconnected de Vries algebras and show that the category of extremally disconnected de Vries algebras is dually equivalent to the category of extremally disconnected compact Hausdorff spaces. As a result, we give a simple construction of the Gleason cover of a compact Hausdorff space by means of de Vries duality. We also discuss the insight that Stone duality provides in better understanding of de Vries duality.     

Sheaves and duality

It has long been known in universal algebra that any distributive sublattice of congruences of an algebra which consists entirely of commuting congruences yields a sheaf representation of the algebra. In this paper we provide a generalization of this fact and prove a converse of the generalization. To be precise, we exhibit a one-to-one correspondence (up to isomorphism) between soft sheaf representations of universal algebras over stably compact spaces and frame homomorphisms from the dual frames of such spaces into subframes of pairwise commuting congruences of the congruence lattices of the universal algebras. For distributive-lattice-ordered algebras this allows us to dualize such sheaf representations.     

Sheaf coalgebras and duality

In this paper we construct an adjoint pair of functors between the category of sheaves on a smooth manifold M and the category of coalgebras over the ring of smooth functions with compact support on M. We show that the sheaf coalgebra associated to a sheaf E on M determines the sheaf E uniquely up to an isomorphism, and that the adjoint functors restrict to an equivalence between sheaves on M and sheaf coalgebras over .     

Weak minimization and duality

Sufficient and necessary optimality conditions are given for weakly minimized optimization problems in terms of a vector valued Lagrangian. Lagrangian and Wolfe type duals are constructed and duality established using an ordering that accords with the definition of a weak minimum. The results for differentiable problems continue to hold under weakened convexity assumptions and for problems which quasiminimize rather than minimize.     

B-frame duality

This paper introduces the category of b-frames as a new tool in the study of complete lattices. B-frames can be seen as a generalization of posets, which play an important role in the representation theory of Heyting algebras, but also in the study of complete Boolean algebras in forcing. This paper combines ideas from the two traditions in order to generalize some techniques and results to the wider context of complete lattices. In particular, we lift a representation theorem of Allwein and MacCaull to a duality between complete lattices and b-frames, and we derive alternative characterizations of several classes of complete lattices from this duality. This framework is then used to obtain new results in the theory of complete Heyting algebras and the semantics of intuitionistic propositional logic.     

Random duality

The purpose of this paper is to provide a random duality theory for the further development of the theory of random conjugate spaces for random normed modules. First, the complicated stratification structure of a module over the algebra L(μ, K) frequently makes our investigations into random duality theory considerably different from the corresponding ones into classical duality theory, thus in this paper we have to first begin in overcoming several substantial obstacles to the study of stratification structure on random locally convex modules. Then, we give the representation theorem of weakly continuous canonical module homomorphisms, the theorem of existence of random Mackey structure, and the random bipolar theorem with respect to a regular random duality pair together with some important random compatible invariants.     

Invex-convexlike functions and duality

We define a class of invex-convexlike functions, which contains all convex, pseudoconvex, invex, and convexlike functions, and prove that the Kuhn-Tucker sufficient optimality condition and the Wolfe duality hold for problems involving such functions. Applications in control theory are given.The author is grateful to Professor W. Stadler and the referees for many valuable remarks and suggestions, which have enabled him to improve considerably the paper.     

Equivariant homology and duality

This note is concerned with stable G-equivariant homology and cohomology theories (G a compact Lie group). In important cases, when H-equivariant theories are defined naturally for all closed subgroups H of G, we show that the G-(co)homology groups of G x_H X are isomorphic with H-(co)homology groups of X. We introduce the concept of orientability of G-vector bundles and manifolds with respect to an equivariant cohomology theory and prove a duality theorem which implies an equivariant analogue of Poincaré-Lefschetz duality.

     

Grassmannians and Koszul duality

Let \(X\) be a partial flag variety, stratified by orbits of the Borel. We give a criterion for the category of modular perverse sheaves to be equivalent to modules over a Koszul ring. This implies that modular category \(\mathcal O\) is governed by a Koszul-algebra in small examples.     

Invex multifunctions and duality

For optimization problems with multifunction objective and constraints, duality theorems are proved for analogs of the Wolfe and Mond–Weir dual problems, assuming that the multifunctions satisfy a generalization of the invex property for functions. Several characterizations of generalized invexity are obtained.     

Quasi-free divisors and duality

We prove a duality theorem for some logarithmic $\text{D}$-modules associated with a class of divisors. We also give some results for the locally quasi-homogeneous case. To cite this article: F.J. Castro-Jiménez, J.M. Ucha-Enr??quez, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Critical duality

We look for a general framework in which the Ekeland duality can be formulated. We propose a scheme in which the parameter sets are provided with a coupling function which induces a conjugacy. The decision spaces are not supposed to have any special structure. We examine several examples. In particular, we consider some special classes of generalized convex functions.     

Second order duality and symmetric duality for non-linear programs in complex space

A second order dual to a non-linear program with linear constraints in complex space over arbitrary polyhedral convex cones is presented and appropriate duality theorems are established. Also, a pair of second order non-linear symmetric dual programs is formulated and appropriate duality results are proved. It is shown that a second order dual gives a better bound for the objective function than does the first order dual. Some special cases are discussed.     

Constrained duality via unconstrained duality in generalized geometric programming

A specialization of unconstrained duality (involving problems without explicit constraints) to constrained duality (involving problems with explicit constraints) provides an efficient mechanism for extending to the latter many important theorems that were previously established for the former.This research was sponsored by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant No. AFOSR-73-2516.