A characterization of non-archimedeanly quasimetrizable spaces

In this paper we introduce a new structure on topological spaces which allows us to give a characterization of non-archimedeanly quasipseudometrizable spaces

Metrization of free groups on ultrametric spaces

AbstractWe consider ultrametrizations of free topological groups of ultrametric spaces. A construction is defined that determines a functor in the category UMET1 of ultrametric spaces of diameter ⩽1 and nonexpanding maps. This functor is the functorial part of a monad in UMET1 and we provide a characterization of the category of its algebras

Q-functions on quasimetric spaces and fixed points for multivalued maps

[EN] We discuss several properties of Q-functions in the sense of Al-Homidan et al.. In particular, we prove that the partial metric induced by any T0 weighted quasipseudometric space is a Q-function and show that both the Sorgenfrey line and the Kofner plane provide signi¿cant examples of quasimetric spaces for which the associated supremum metric is a Q-function. In this context we also obtain some ¿xed point results for multivalued maps by using Bianchini-Grandol¿ gauge functions.The authors thank one of the reviewers for suggesting the inclusion of a concrete example to which Theorem 3.3 applies. They acknowledge the support of the Spanish Ministry of Science and Innovation, Grant no. MTM2009-12872-C02-01.Marín Molina, J.; Romaguera Bonilla, S.; Tirado Peláez, P. (2011). Q-functions on quasimetric spaces and fixed points for multivalued maps. Fixed Point Theory and Applications. 2011:1-10. https://doi.org/10.1155/2011/603861S1102011Ekeland, I. (1979). Nonconvex minimization problems. Bulletin of the American Mathematical Society, 1(3), 443-475. doi:10.1090/s0273-0979-1979-14595-6Al-Homidan, S., Ansari, Q. H., & Yao, J.-C. (2008). Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory. Nonlinear Analysis: Theory, Methods & Applications, 69(1), 126-139. doi:10.1016/j.na.2007.05.004Alegre, C. (2008). Continuous operators on asymmetric normed spaces. Acta Mathematica Hungarica, 122(4), 357-372. doi:10.1007/s10474-008-8039-0ALI-AKBARI, M., HONARI, B., POURMAHDIAN, M., & REZAII, M. M. (2009). The space of formal balls and models of quasi-metric spaces. Mathematical Structures in Computer Science, 19(2), 337-355. doi:10.1017/s0960129509007439Cobzaş, S. (2009). Compact and precompact sets in asymmetric locally convex spaces. Topology and its Applications, 156(9), 1620-1629. doi:10.1016/j.topol.2009.01.004García-Raffi, L. M., Romaguera, S., & Sánchez-Pérez, E. A. (2009). The Goldstine Theorem for asymmetric normed linear spaces. Topology and its Applications, 156(13), 2284-2291. doi:10.1016/j.topol.2009.06.001García-Raffi, L. M., Romaguera, S., & Schellekens, M. P. (2008). Applications of the complexity space to the General Probabilistic Divide and Conquer Algorithms. Journal of Mathematical Analysis and Applications, 348(1), 346-355. doi:10.1016/j.jmaa.2008.07.026Heckmann, R. (1999). Applied Categorical Structures, 7(1/2), 71-83. doi:10.1023/a:1008684018933Romaguera, S., & Schellekens, M. (2005). Partial metric monoids and semivaluation spaces. Topology and its Applications, 153(5-6), 948-962. doi:10.1016/j.topol.2005.01.023Romaguera, S., & Tirado, P. (2011). The complexity probabilistic quasi-metric space. Journal of Mathematical Analysis and Applications, 376(2), 732-740. doi:10.1016/j.jmaa.2010.11.056ROMAGUERA, S., & VALERO, O. (2009). A quantitative computational model for complete partial metric spaces via formal balls. Mathematical Structures in Computer Science, 19(3), 541-563. doi:10.1017/s0960129509007671ROMAGUERA, S., & VALERO, O. (2010). Domain theoretic characterisations of quasi-metric completeness in terms of formal balls. Mathematical Structures in Computer Science, 20(3), 453-472. doi:10.1017/s0960129510000010Schellekens, M. P. (2003). A characterization of partial metrizability: domains are quantifiable. Theoretical Computer Science, 305(1-3), 409-432. doi:10.1016/s0304-3975(02)00705-3Waszkiewicz, P. (2003). Applied Categorical Structures, 11(1), 41-67. doi:10.1023/a:1023012924892WASZKIEWICZ, P. (2006). Partial metrisability of continuous posets. Mathematical Structures in Computer Science, 16(02), 359. doi:10.1017/s0960129506005196Proinov, P. D. (2007). A generalization of the Banach contraction principle with high order of convergence of successive approximations. Nonlinear Analysis: Theory, Methods & Applications, 67(8), 2361-2369. doi:10.1016/j.na.2006.09.008Reilly, I. L., Subrahmanyam, P. V., & Vamanamurthy, M. K. (1982). Cauchy sequences in quasi-pseudo-metric spaces. Monatshefte f�r Mathematik, 93(2), 127-140. doi:10.1007/bf01301400Rakotch, E. (1962). A note on contractive mappings. Proceedings of the American Mathematical Society, 13(3), 459-459. doi:10.1090/s0002-9939-1962-0148046-1Proinov, P. D. (2010). New general convergence theory for iterative processes and its applications to Newton–Kantorovich type theorems. Journal of Complexity, 26(1), 3-42. doi:10.1016/j.jco.2009.05.00

Metrics for generalized persistence modules

We consider the question of defining interleaving metrics on generalized persistence modules over arbitrary preordered sets. Our constructions are functorial, which implies a form of stability for these metrics. We describe a large class of examples, inverse-image persistence modules, which occur whenever a topological space is mapped to a metric space. Several standard theories of persistence and their stability can be described in this framework. This includes the classical case of sublevelset persistent homology. We introduce a distinction between `soft' and `hard' stability theorems. While our treatment is direct and elementary, the approach can be explained abstractly in terms of monoidal functors.Comment: Final version; no changes from previous version. Published online Oct 2014 in Foundations of Computational Mathematics. Print version to appea

Magnitude and Topological Entropy of Digraphs

Magnitude and (co)weightings are quite general constructions in enriched categories, yet they have been developed almost exclusively in the context of Lawvere metric spaces. We construct a meaningful notion of magnitude for flow graphs based on the observation that topological entropy provides a suitable map into the max-plus semiring, and we outline its utility. Subsequently, we identify a separate point of contact between magnitude and topological entropy in digraphs that yields an analogue of volume entropy for geodesic flows. Finally, we sketch the utility of this construction for feature engineering in downstream applications with generic digraphs.Comment: In Proceedings ACT 2022, arXiv:2307.1551

Function Spaces, Hyperspaces, and Asymmetric and Fuzzy Structures

Romaguera Bonilla, S.; Beer, G.; Sanchis, M. (2013). Function Spaces, Hyperspaces, and Asymmetric and Fuzzy Structures. Journal of Function Spaces and Applications. doi:10.1155/2013/619707

Dynamic Processes, Fixed Points, Endpoints, Asymmetric Structures, and Investigations Related to Caristi, Nadler, and Banach in Uniform Spaces

Research ArticleIn uniform spaces (...) with symmetric structures determined by the D-families of pseudometrics which define uniformity in these spaces, the new symmetric and asymmetric structures determined by the J-families of generalized pseudodistances on (...) are constructed; using these structures the set-valued contractions of two kinds of Nadler type are defined and the new and general theorems concerning the existence of fixed points and endpoints for such contractions are proved. Moreover, using these new structures, the single-valued contractions of two kinds of Banach type are defined and the new and general versions of the Banach uniqueness and iterate approximation of fixed point theorem for uniform spaces are established. Contractions defined and studied here are not necessarily continuous. One of the main key ideas in this paper is the application of our fixed point and endpoint version of Caristi type theorem for dissipative set-valued dynamic systems without lower semicontinuous entropies in uniform spaces with structures determined by J-families. Results are new also in locally convex and metric spaces. Examples are provided