p-topological and p-regular: dual notions in convergence theory

The natural duality between "topological" and "regular," both considered as convergence space properties, extends naturally to p-regular convergence spaces, resulting in the new concept of a p-topological convergence space. Taking advantage of this duality, the behavior of p-topological and p-regular convergence spaces is explored, with particular emphasis on the former, since they have not been previously studied. Their study leads to the new notion of a neighborhood operator for filters, which in turn leads to an especially simple characterization of a topology in terms of convergence criteria. Applications include the topological and regularity series of a convergence space.Comment: 12 pages in Acrobat 3.0 PDF forma

On the continuity of separately continuous bihomomorphisms

Separately continuous bihomomorphisms on a product of convergence or topological groups occur with great frequency. Of course, in general, these need not be jointly continuous. In this paper, we exhibit some results of Banach-Steinhaus type and use these to derive joint continuity from separate continuity. The setting of convergence groups offers two advantages. First, the continuous convergence structure is a powerful tool in many duality arguments. Second, local compactness and first countability, the usual requirements for joint continuity, are available in much greater abundance for convergence groups

Block-Coordinate Frank-Wolfe Optimization for Structural SVMs

We propose a randomized block-coordinate variant of the classic Frank-Wolfe algorithm for convex optimization with block-separable constraints. Despite its lower iteration cost, we show that it achieves a similar convergence rate in duality gap as the full Frank-Wolfe algorithm. We also show that, when applied to the dual structural support vector machine (SVM) objective, this yields an online algorithm that has the same low iteration complexity as primal stochastic subgradient methods. However, unlike stochastic subgradient methods, the block-coordinate Frank-Wolfe algorithm allows us to compute the optimal step-size and yields a computable duality gap guarantee. Our experiments indicate that this simple algorithm outperforms competing structural SVM solvers.Comment: Appears in Proceedings of the 30th International Conference on Machine Learning (ICML 2013). 9 pages main text + 22 pages appendix. Changes from v3 to v4: 1) Re-organized appendix; improved & clarified duality gap proofs; re-drew all plots; 2) Changed convention for Cf definition; 3) Added weighted averaging experiments + convergence results; 4) Clarified main text and relationship with appendi

Convergence rates in homogenization of higher order parabolic systems

This paper is concerned with the optimal convergence rate in homogenization of higher order parabolic systems with bounded measurable, rapidly oscillating periodic coefficients. The sharp O(\va) convergence rate in the space L^2(0,T; H^{m-1}(\Om)) is obtained for both the initial-Dirichlet problem and the initial-Neumann problem. The duality argument inspired by \cite{suslinaD2013} is used here.Comment: 28 page

Duality of locally quasi-convex convergence groups

[EN] In the realm of the convergence spaces, the generalisation of topological groups is the convergence groups, and the corresponding extension of the Pontryagin duality is the continuous duality. We prove that local quasi-convexity is a necessary condition for a convergence group to be c-reflexive. Further, we prove that every character group of a convergence group is locally quasi-convex.We thank Prof. H.-P. Butzmann and the anonymous reviewers for their many insightful comments and suggestions.Sharma, P. (2021). Duality of locally quasi-convex convergence groups. Applied General Topology. 22(1):193-198. https://doi.org/10.4995/agt.2021.14585OJS193198221L. Außenhofer, Contributions to the Duality Theory of Abelian Topological Groups and to the Theory of Nuclear Groups, Dissertationes mathematicae. Institute of Mathematics, Polish Academy of Sciences, 1999.W. Banaszczyk, Additive Subgroups of Topological Vector Spaces, Lecture Notes in Matheatics, Springer Berlin Heidelberg, 1991. https://doi.org/10.1007/BFb0089147R. Beattie and H.-P. Butzmann, Convergence Structures and Applications to Functional Analysis, Bücher, Springer Netherlands, 2013.M. Bruguera, Topological groups and convergence groups: Study of the Pontryagin duality, Thesis, 1999.H.-P. Butzmann, Über diec-Reflexivität von Cc (X), Comment. Math. Helv. 47, no. 1 (1972), 92-101. https://doi.org/10.1007/BF02566791H.-P. Butzmann, Duality theory for convergence groups, Topology Appl. 111, no. 1 (2000), 95-104. https://doi.org/10.1016/S0166-8641(99)00188-1M. J. Chasco and E. Martín-Peinador, Binz-Butzmann duality versus Pontryagin duality, Arch. Math. (Basel) 63, no. 3 (1994), 264-270. https://doi.org/10.1007/BF01189829M. J. Chasco, D. Dikranjan and E. Martín-Peinador, A survey on reflexivity of abelian topological groups, Topology Appl. 159, no. 9 (2012), 2290-2309. https://doi.org/10.1016/j.topol.2012.04.012S. Dolecki and F. Mynard, Convergence Foundations of Topology, World Scientific Publishing Company, 2016. https://doi.org/10.1142/9012E. Martín-Peinador, A reflexive admissible topological group must be locally compact, Proc. Amer. Math. Soc. 123, no. 11 (1995), 3563-3566. https://doi.org/10.2307/2161108E. Martín-Peinador and V. Tarieladze, A property of Dunford-Pettis type in topological groups, Proc. Amer. Math. Soc. 132, no. 6 (2004), 1827-1837. https://doi.org/10.1090/S0002-9939-03-07249-6P. Sharma, Locally quasi-convex convergence groups, Topology Appl. 285 (2020), 107384. https://doi.org/10.1016/j.topol.2020.107384P. Sharma and S. Mishra, Duality in topological and convergence groups, Top. Proc., to appear