Sobrification and bicompletion of totally bounded quasi-uniform spaces

We observe that if is a compatible totally bounded quasi-uniformity on a T0-space (X,), then the bicompletion of (X, ) is a strongly sober, locally quasicompact space. It follows that the b-closure S of (X, ) in is homeomorphic to the sobrification of the space (X, ). We prove that S is equal to if and only if (X, ) is a core-compact space in which every ultrafilter has an irreducible convergence set and is the coarsest quasi-uniformity compatible with . If is the Pervin quasi-uniformity on X, then S is equal to if and only if X is hereditarily quasicompact, or equivalently, is the Pervin quasi-uniformity o

New results on the mathematical foundations of asymptotic complexity analysis of algorithms via complexity spaces

Schellekens [The Smyth completion: A common foundation for denotational semantics and complexity analysis, Electron. Notes Theor. Comput. Sci. 1 (1995), pp. 211-232.] introduced the theory of complexity (quasi-metric) spaces as a part of the development of a topological foundation for the asymptotic complexity analysis of programs and algorithms in 1995. The applicability of this theory to the asymptotic complexity analysis of divide and conquer algorithms was also illustrated by Schellekens in the same paper. In particular, he gave a new formal proof, based on the use of the Banach fixed-point theorem, of the well-known fact that the asymptotic upper bound of the average running time of computing of Mergesort belongs to the asymptotic complexity class of n log(2) n. Recently, Schellekens' method has been shown to be useful in yielding asymptotic upper bounds for a class of algorithms whose running time of computing leads to recurrence equations different from the divide and conquer ones reported in Cerda-Uguet et al. [The Baire partial quasi-metric space: A mathematical tool for the asymptotic complexity analysis in Computer Science, Theory Comput. Syst. 50 (2012), pp. 387-399.]. However, the variety of algorithms whose complexity can be analysed with this approach is not much larger than that of algorithms that can be analysed with the original Schellekens method. In this paper, on the one hand, we extend Schellekens' method in order to yield asymptotic upper bounds for a certain class of recursive algorithms whose running time of computing cannot be discussed following the techniques given by Cerda-Uguet et al. and, on the other hand, we improve the original Schellekens method by introducing a new fixed-point technique for providing, contrary to the case of the method introduced by Cerda-Uguet et al., lower asymptotic bounds of the running time of computing of the aforementioned algorithms and those studied by Cerda-Uguet et al. We illustrate and validate the developed method by applying our results to provide the asymptotic complexity class (asymptotic upper and lower bounds) of the celebrated algorithms Quicksort, Largetwo and Hanoi.The authors are thankful for the support from the Spanish Ministry of Science and Innovation, grant MTM2009-12872-C02-01.Romaguera Bonilla, S.; Tirado Peláez, P.; Valero Sierra, Ó. (2012). New results on the mathematical foundations of asymptotic complexity analysis of algorithms via complexity spaces. International Journal of Computer Mathematics. 89(13-14):1728-1741. https://doi.org/10.1080/00207160.2012.659246S172817418913-14Cerdà-Uguet, M. A., Schellekens, M. P., & Valero, O. (2011). The Baire Partial Quasi-Metric Space: A Mathematical Tool for Asymptotic Complexity Analysis in Computer Science. Theory of Computing Systems, 50(2), 387-399. doi:10.1007/s00224-010-9310-7Cull, P., & Ecklund, E. F. (1985). Towers of Hanoi and Analysis of Algorithms. The American Mathematical Monthly, 92(6), 407. doi:10.2307/2322448García-Raffi, L. M., Romaguera, S., & Sánchez-Pérez, E. A. (2002). Sequence spaces and asymmetric norms in the theory of computational complexity. Mathematical and Computer Modelling, 36(1-2), 1-11. doi:10.1016/s0895-7177(02)00100-0García-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.2008.07.026Künzi, 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_3Rodríguez-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:10.1080/00207160701210653Rodríguez-López, J., Schellekens, M. P., & Valero, O. (2009). An extension of the dual complexity space and an application to Computer Science. Topology and its Applications, 156(18), 3052-3061. doi:10.1016/j.topol.2009.02.009Romaguera, S., & Schellekens, M. (1999). Quasi-metric properties of complexity spaces. Topology and its Applications, 98(1-3), 311-322. doi:10.1016/s0166-8641(98)00102-3Romaguera, S., & Valero, O. (2008). On the structure of the space of complexity partial functions. International Journal of Computer Mathematics, 85(3-4), 631-640. doi:10.1080/00207160701210117Romaguera, S., Schellekens, M. P., & Valero, O. (2011). The complexity space of partial functions: a connection between complexity analysis and denotational semantics. International Journal of Computer Mathematics, 88(9), 1819-1829. doi:10.1080/00207161003631885Schellekens, M. (1995). The Smyth Completion. Electronic Notes in Theoretical Computer Science, 1, 535-556. doi:10.1016/s1571-0661(04)00029-5Scott, D. S. 1970. Outline of a mathematical theory of computation. Proceedings of the 4th Annual Princeton Conference on Information Sciences and Systems. March26–271970, Princeton, NJ. pp.169–176

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

Patients' motives for choosing a physician: comparison between conventional and complementary medicine in Swiss primary care

<p>Abstract</p> <p>Background</p> <p>The study is part of a nationwide evaluation of complementary and alternative medicine (CAM) in primary care in Switzerland. The Objective was to identify patients' expectations and reasons governing the choice of complementary medicine compared with conventional primary care (CONV).</p> <p>Methods</p> <p>The data were derived from the PEK study (Programm Evaluation Komplementärmedizin), which was conducted in 2002–2003 with 7879 adult patients and parents of 1291 underage patients, seeking either complementary (CAM) or conventional (CONV) primary care. The study was performed as a cross-sectional survey. The respondents were asked to document their (or their children's) self-perceived health status, reasons governing their choice, and treatment expectations. Physicians were practicing conventional medicine and/or complementary methods (homeopathy, anthroposophic medicine, neural therapy, and traditional Chinese medicine). Reasons governing the choice of physician were evaluated on the basis of a three-part classification (physician-related, procedure-related, and pragmatic/other reasons)</p> <p>Results and Discussion</p> <p>Patients seeing CAM physicians tend to be younger and more often female. CAM patients referred to procedure-related reasons more frequently, whereas pragmatic reasons dominated among CONV patients. CAM respondents expected fewer adverse side effects compared to conventional care patients.</p> <p>Conclusion</p> <p>The majority of alternative medicine users appear to have chosen CAM mainly because they wish to undergo a certain procedure; additional reasons include desire for more comprehensive treatment, and expectation of fewer side-effects.</p

Normally preordered spaces and utilities

In applications it is useful to know whether a topological preordered space is normally preordered. It is proved that every $k_\omega$-space equipped with a closed preorder is a normally preordered space. Furthermore, it is proved that second countable regularly preordered spaces are perfectly normally preordered and admit a countable utility representation.Comment: 17 pages, 1 figure. v2 contains a second proof to the main theorem with respect to the published version. The last section of v1 is not present in v2. It will be included in a different wor

Processing second-order stochastic dominance models using cutting-plane representations

This is the post-print version of the Article. The official published version can be accessed from the links below. Copyright @ 2011 Springer-VerlagSecond-order stochastic dominance (SSD) is widely recognised as an important decision criterion in portfolio selection. Unfortunately, stochastic dominance models are known to be very demanding from a computational point of view. In this paper we consider two classes of models which use SSD as a choice criterion. The first, proposed by Dentcheva and Ruszczyński (J Bank Finance 30:433–451, 2006), uses a SSD constraint, which can be expressed as integrated chance constraints (ICCs). The second, proposed by Roman et al. (Math Program, Ser B 108:541–569, 2006) uses SSD through a multi-objective formulation with CVaR objectives. Cutting plane representations and algorithms were proposed by Klein Haneveld and Van der Vlerk (Comput Manage Sci 3:245–269, 2006) for ICCs, and by Künzi-Bay and Mayer (Comput Manage Sci 3:3–27, 2006) for CVaR minimization. These concepts are taken into consideration to propose representations and solution methods for the above class of SSD based models. We describe a cutting plane based solution algorithm and outline implementation details. A computational study is presented, which demonstrates the effectiveness and the scale-up properties of the solution algorithm, as applied to the SSD model of Roman et al. (Math Program, Ser B 108:541–569, 2006).This study was funded by OTKA, Hungarian National Fund for Scientific Research, project 47340; by Mobile Innovation Centre, Budapest University of Technology, project 2.2; Optirisk Systems, Uxbridge, UK and by BRIEF (Brunel University Research Innovation and Enterprise Fund)

Dual Space of a Lattice as the Completion of a Pervin Space

16th International Conference, RAMiCS 2017, Lyon, France, May 15-18, 2017, ProceedingsInternational audienceThis survey paper presents well-known results from a new angle. A Pervin space is a set X equipped with a set of subsets,called the blocks of the Pervin space. Blocks are closed under finite intersections and finite unions and hence form a lattice of subsets of X. Pervin spaces are thus easier to define than topological spaces or (quasi)-uniform spaces. As a consequence, most of the standard topological notions, like convergence and cluster points, specialisation order, filtersand Cauchy filters, complete spaces and completion are much easier to define for Pervin spaces. In particular, the completion of a Pervin space turns out to be the dual space (in the sense of Stone) of the original lattice.We show that any lattice of subsets can be described by a set of inequations of the form u ≤ v, where u and v are elements of its dual space. Applications to formal languages and complexity classes are given.Cet article de synthèse présente des résultats bien connus sous un nouvel angle. Un espace de Pervin est unensemble X équipé d'un ensemble de parties, appelé les blocs de l'espace de Pervin. Les blocs sont fermés par intersection finie et union finie et forment ainsi un treillis de parties de X. Les espaces de Pervin sont doncplus faciles à définir que les espaces topologiques ou les espaces (quasi-)uniformes. Par conséquent, la plupart des notions topologiques, comme la convergence et les points d'adhérence, l'ordre de spécialisation, les filtres de Cauchy, les espaces complets et la complétion sont beaucoup plus faciles à définir pour les espaces Pervin. En particulier, la complétion d'un espace Pervin s'avère être l'espace dual (au sens de Stone) du treillis de départ.Nous montrons que tout treillis de parties peut être décrit par un ensemble d'inéquations de la forme u ≤ v, où u et v sont des éléments de son espace dual. On donne des applications aux langages formels et aux classes de complexité

A characterization of Smyth complete quasi-metric spaces via Caristi's fixed point theorem

We obtain a quasi-metric generalization of Caristi's fixed point theorem for a kind of complete quasi-metric spaces. With the help of a suitable modification of its proof, we deduce a characterization of Smyth complete quasi-metric spaces which provides a quasi-metric generalization of the well-known characterization of metric completeness due to Kirk. Some illustrative examples are also given. As an application, we deduce a procedure which allows to easily show the existence of solution for the recurrence equation of certain algorithms.The authors are grateful to the reviewers for several suggestions which have allowed to improve the first version of the paper. This research is supported by the Ministry of Economy and Competitiveness of Spain, Grant MTM2012-37894-C02-01.Romaguera Bonilla, S.; Tirado Peláez, P. (2015). A characterization of Smyth complete quasi-metric spaces via Caristi's fixed point theorem. Fixed Point Theory and Applications. 2015:183. https://doi.org/10.1186/s13663-015-0431-1S2015:183Cobzaş, S: Functional Analysis in Asymmetric Normed Spaces. Springer, Basel (2013)Künzi, HPA: Nonsymmetric distances and their associated topologies: about the origins of basic ideas in the area of asymmetric topology. In: Aull, CE, Lowen, R (eds.) Handbook of the History of General Topology, vol. 3, pp. 853-968. Kluwer Academic, Dordrecht (2001)Reilly, IL, Subrhamanyam, PV, Vamanamurthy, MK: Cauchy sequences in quasi-pseudo-metric spaces. Monatshefte Math. 93, 127-140 (1982)Künzi, HPA, Schellekens, MP: On the Yoneda completion of a quasi-metric spaces. Theor. Comput. Sci. 278, 159-194 (2002)Romaguera, S, Valero, O: Domain theoretic characterisations of quasi-metric completeness in terms of formal balls. Math. Struct. Comput. Sci. 20, 453-472 (2010)Künzi, HPA: Nonsymmetric topology. In: Proc. Szekszárd Conf. Bolyai Society of Math. 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Fixed Point Theory 14, 3-10 (2013)Aydi, H, Karapınar, E, Kumam, P: A note on ‘Modified proof of Caristi’s fixed point theorem on partial metric spaces, Journal of Inequalities and Applications 2013, 2013:210’. J. Inequal. Appl. 2013, 355 (2013)Cobzaş, S: Completeness in quasi-metric spaces and Ekeland variational principle. Topol. Appl. 158, 1073-1084 (2011)Hadžić, O, Pap, E: Fixed Point Theory in Probabilistic Metric Spaces. Kluwer Academic, Dordrecht (2001)Karapınar, E: Generalizations of Caristi Kirk’s theorem on partial metric spaces. Fixed Point Theory Appl. 2011, 4 (2011)Romaguera, S: A Kirk type characterization of completeness for partial metric spaces. Fixed Point Theory Appl. 2010, Article ID 493298 (2010)Park, S: On generalizations of the Ekeland-type variational principles. Nonlinear Anal. TMA 39, 881-889 (2000)Du, W-S, Karapınar, E: A note on Caristi type cyclic maps: related results and applications. 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Patient satisfaction and side effects in primary care: An observational study comparing homeopathy and conventional medicine

<p>Abstract</p> <p>Background</p> <p>This study is part of a nationwide evaluation of complementary medicine in Switzerland (Programme Evaluation of Complementary Medicine PEK) and was funded by the Swiss Federal Office of Public Health. The main objective of this study is to investigate patient satisfaction and perception of side effects in homeopathy compared with conventional care in a primary care setting.</p> <p>Methods</p> <p>We examined data from two cross-sectional studies conducted in 2002–2003. The first study was a physician questionnaire assessing structural characteristics of practices. The second study was conducted on four given days during a 12-month period in 2002/2003 using a physician and patient questionnaire at consultation and a patient questionnaire mailed to the patient one month later (including Europep questionnaire).</p> <p>The participating physicians were all trained and licensed in conventional medicine. An additional qualification was required for medical doctors providing homeopathy (membership in the Swiss association of homeopathic physicians SVHA).</p> <p>Results</p> <p>A total of 6778 adult patients received the questionnaire and 3126 responded (46.1%). Statistically significant differences were found with respect to health status (higher percentage of chronic and severe conditions in the homeopathic group), perception of side effects (higher percentage of reported side effects in the conventional group) and patient satisfaction (higher percentage of satisfied patients in the homeopathic group).</p> <p>Conclusion</p> <p>Overall patient satisfaction was significantly higher in homeopathic than in conventional care. Homeopathic treatments were perceived as a low-risk therapy with two to three times fewer side effects than conventional care</p