On the domain of formal balls of the Sorgenfrey quasi-metric space

[EN] We show that the poset of formal balls of the Sorgenfrey quasi-metric space is an omega-continuous domain, and deduce that it is also a computational model, in the sense of R.C. Flagg and R. Kopperman, for the Sorgenfrey line. Furthermore, we study its structure of quantitative domain in the sense of P. Waszkiewicz. (C) 2016 Elsevier B.V. All rights reserved.Supported by the Ministry of Economy and Competitiveness of Spain, under grant MTM2012-37894-C02-01.Romaguera Bonilla, S.; Schellekens, M.; Tirado Peláez, P.; Valero Sierra, Ó. (2016). On the domain of formal balls of the Sorgenfrey quasi-metric space. Topology and its Applications. 203:177-187. https://doi.org/10.1016/j.topol.2015.12.086S17718720

Complete partial metric spaces have partially metrizable computational models

We show that the domain of formal balls of a complete partial metric space (X, p) can be endowed with a complete partial metric that extends p and induces the Scott topology. This result, that generalizes well-known constructions of Edalat and Heckmann [A computational model for metric spaces, Theoret. Comput. Sci. 193 (1998), pp. 53-73] and Heckmann [Approximation of metric spaces by partial metric spaces, Appl. Cat. Struct. 7 (1999), pp. 71-83] for metric spaces and improves a recent result of Romaguera and Valero [A quantitative computational model for complete partial metric spaces via formal balls, Math. Struct. Comput. Sci. 19 (2009), pp. 541-563], motivates a notion of a partially metrizable computational model which allows us to characterize those topological spaces that admit a compatible complete partial metric via this model.The authors acknowledge the support of the Spanish Ministry of Science and Innovation, under grant MTM2009-12872-C02-01.Romaguera Bonilla, S.; Tirado Peláez, P.; Valero Sierra, Ó. (2012). Complete partial metric spaces have partially metrizable computational models. International Journal of Computer Mathematics. 89(3):284-290. https://doi.org/10.1080/00207160.2011.559229S284290893ALI-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/s0960129509007439Edalat, A., & Heckmann, R. (1998). A computational model for metric spaces. Theoretical Computer Science, 193(1-2), 53-73. doi:10.1016/s0304-3975(96)00243-5Edalat, A., & Sünderhauf, P. (1999). Computable Banach spaces via domain theory. Theoretical Computer Science, 219(1-2), 169-184. doi:10.1016/s0304-3975(98)00288-6Flagg, B., & Kopperman, R. (1997). Computational Models for Ultrametric Spaces. Electronic Notes in Theoretical Computer Science, 6, 151-159. doi:10.1016/s1571-0661(05)80164-1Heckmann, R. (1999). Applied Categorical Structures, 7(1/2), 71-83. doi:10.1023/a:1008684018933Kopperman, R., Künzi, H.-P. A., & Waszkiewicz, P. (2004). Bounded complete models of topological spaces. Topology and its Applications, 139(1-3), 285-297. doi:10.1016/j.topol.2003.12.001Krötzsch, M. (2006). Generalized ultrametric spaces in quantitative domain theory. Theoretical Computer Science, 368(1-2), 30-49. doi:10.1016/j.tcs.2006.05.037Künzi, H.-P. A. (2001). Nonsymmetric Distances and Their Associated Topologies: About the Origins of Basic Ideas in the Area of Asymmetric Topology. History of Topology, 853-968. doi:10.1007/978-94-017-0470-0_3LAWSON, J. (1997). Spaces of maximal points. Mathematical Structures in Computer Science, 7(5), 543-555. doi:10.1017/s0960129597002363Martin, K. (1998). Domain theoretic models of topological spaces. Electronic Notes in Theoretical Computer Science, 13, 173-181. doi:10.1016/s1571-0661(05)80221-xMatthews, S. G.Partial metric topology. Procedings of the 8th Summer Conference on General Topology and Applications, Ann. New York Acad. Sci. 728 (1994), pp. 183–197Rodríguez-López, J., Romaguera, S., & Valero, O. (2008). Denotational semantics for programming languages, balanced quasi-metrics and fixed points. International Journal of Computer Mathematics, 85(3-4), 623-630. doi:10.1080/00207160701210653Romaguera, S., & Valero, O. (2009). A quasi-metric computational model from modular functions on monoids. International Journal of Computer Mathematics, 86(10-11), 1668-1677. doi:10.1080/00207160802691652ROMAGUERA, 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/s0960129510000010Rutten, J. J. M. M. (1998). Weighted colimits and formal balls in generalized metric spaces. Topology and its Applications, 89(1-2), 179-202. doi:10.1016/s0166-8641(97)00224-1Schellekens, 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)00705-3Smyth, M. B. (2006). The constructive maximal point space and partial metrizability. Annals of Pure and Applied Logic, 137(1-3), 360-379. doi:10.1016/j.apal.2005.05.032Waszkiewicz, 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/s096012950600519

Almost every path structure is not variational

Given a smooth family of unparameterized curves such that through every point in every direction there passes exactly one curve, does there exist a Lagrangian with extremals being precisely this family? It is known that in dimension 2 the answer is positive. In dimension 3, it follows from the work of Douglas that the answer is, in general, negative. We generalise this result to all higher dimensions and show that the answer is actually negative for almost every such a family of curves, also known as path structure or path geometry. On the other hand, we consider path geometries possessing infinitesimal symmetries and show that path and projective structures with submaximal symmetry dimensions are variational. Note that the projective structure with the submaximal symmetry algebra, the so-called Egorov structure, is not pseudo-Riemannian metrizable; we show that it is metrizable in the class of Kropina pseudo-metrics and explicitly construct the corresponding Kropina Lagrangian

Computable Stone spaces

We investigate computable metrizability of Polish spaces up to homeomorphism. In this paper we focus on Stone spaces. We use Stone duality to construct the first known example of a computable topological Polish space not homeomorphic to any computably metrized space. In fact, in our proof we construct a right-c.e. metrized Stone space which is not homeomorphic to any computably metrized space. Then we introduce a new notion of effective categoricity for effectively compact spaces and prove that effectively categorical Stone spaces are exactly the duals of computably categorical Boolean algebras. Finally, we prove that, for a Stone space $X$, the Banach space $C(X;\mathbb{R})$ has a computable presentation if, and only if, $X$ is homeomorphic to a computably metrized space. This gives an unexpected positive partial answer to a question recently posed by McNicholl.Comment: 16 page

Computational Models of Certain Hyperspaces of Quasi-metric Spaces

In this paper, for a given sequentially Yoneda-complete T_1 quasi-metric space (X,d), the domain theoretic models of the hyperspace K_0(X) of nonempty compact subsets of (X,d) are studied. To this end, the $\omega$-Plotkin domain of the space of formal balls BX, denoted by CBX is considered. This domain is given as the chain completion of the set of all finite subsets of BX with respect to the Egli-Milner relation. Further, a map $\phi:K_0(X)\rightarrow CBX$ is established and proved that it is an embedding whenever K_0(X) is equipped with the Vietoris topology and respectively CBX with the Scott topology. Moreover, if any compact subset of (X,d) is d^{-1}-precompact, \phi is an embedding with respect to the topology of Hausdorff quasi-metric H_d on K_0(X). Therefore, it is concluded that (CBX,\sqsubseteq,\phi) is an $\omega$-computational model for the hyperspace K_0(X) endowed with the Vietoris and respectively the Hausdorff topology. Next, an algebraic sequentially Yoneda-complete quasi-metric D on CBX$is introduced in such a way that the specialization order$\sqsubseteq_D$is equivalent to the usual partial order of CBX and, furthermore,$\phi:({\cal K}_0(X),H_d)\rightarrow({\bf C}{\bf B}X,D)$is an isometry. This shows that (CBX,\sqsubseteq,\phi,D) is a quantitative$\omega$-computational model for (K_(X),H_d).Comment: 25 page

On generally covariant mathematical formulation of Feynman integral in Lorentz signature

It is widely accepted that the Feynman integral is one of the most promising methodologies for defining a generally covariant formulation of nonperturbative interacting quantum field theories (QFTs) without a fixed prearranged causal background. Recent literature suggests that if the spacetime metric is not fixed, e.g. because it is to be quantized along with the other fields, one may not be able to avoid considering the Feynman integral in the original Lorentz signature, without Wick rotation. Several mathematical phenomena are known, however, which are at some point showstoppers to a mathematically sound definition of Feynman integral in Lorentz signature. The Feynman integral formulation, however, is known to have a differential reformulation, called to be the master Dyson--Schwinger (MDS) equation for the field correlators. In this paper it is shown that a particular presentation of the MDS equation can be cast into a mathematically rigorously defined form: the involved function spaces and operators can be strictly defined and their properties can be established. Therefore, MDS equation can serve as a substitute for the Feynman integral, in a mathematically sound formulation of constructive QFT, in arbitrary signature, without a fixed background causal structure. It is also shown that even in such a generally covariant setting, there is a canonical way to define the Wilsonian regularization of the MDS equation. The main result of the paper is a necessary and sufficient condition for the regularized MDS solution space to be nonempty, for conformally invariant Lagrangians. This theorem also provides an iterative approximation algorithm for obtaining regularized MDS solutions, and is guaranteed to be convergent whenever the solution space is nonempty. The algorithm could eventually serve as a method for putting Lorentz signature QFTs onto lattice, in the original metric signature.Comment: 28 pages + appendix and supplementary material about topological vector spaces; final published versio