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

Digital homotopic distance between digital functions

[EN] In this paper, we define digital homotopic distance and give its relation with LS category of a digital function and of a digital image. Moreover, we introduce some properties of digital homotopic distance such as being digitally homotopy invariance.The author would like to thank Tane Vergili and the referees for their helpful suggestions. In particular, the author would like to thank the referee who contributed Proposition 3.2 and Example 4.3.Borat, A. (2021). Digital homotopic distance between digital functions. Applied General Topology. 22(1):183-192. https://doi.org/10.4995/agt.2021.14542OJS183192221C. Berge, Graphs and Hypergraphs, 2nd edition, North-Holland, Amsterdam, 1976.A. Borat and T. Vergili, Digital Lusternik-Schnirelmann category, Turkish Journal of Mathematics 42, no 1 (2018), 1845-1852. https://doi.org/10.3906/mat-1801-94A. Borat and T. Vergili, Higher homotopic distance, Topological Methods in Nonlinear Analysis, to appear.L. Boxer, Digitally continuous functions, Pattern Recognit. Lett. 15 (1994), 883-839. https://doi.org/10.1016/0167-8655(94)90012-4L. Boxer, A classical construction for the digital fundamental group, Journal of Mathematical Imaging and Vision 10 (1999), 51-62. https://doi.org/10.1023/A:1008370600456L. Boxer, Homotopy properties of sphere-like digital images, J. Math. Imaging Vision, 24 (2006), 167-175. https://doi.org/10.1007/s10851-005-3619-xL. Boxer, Alternate product adjacencies in digital topology, Applied General Topology 19, no. 1 (2018), 21-53. https://doi.org/10.4995/agt.2018.7146O. Cornea, G. Lupton, J. Oprea and D. Tanre, Lusternik-Schnirelmann Category, Mathematical Surveys and Monographs, vol. 103, American Mathematical Society 2003. https://doi.org/10.1090/surv/103M. Farber, Topological complexity of motion planning, Discrete and Computational Geometry 29 (2003), 211-221. https://doi.org/10.1007/s00454-002-0760-9S. E. Han, Non-product property of the digital fundamental group, Information Sciences 171 (2005), 73-91. https://doi.org/10.1016/j.ins.2004.03.018M. Is and I. Karaca, The higher topological complexity in digital images, Applied General Topology 21, no. 2 (2020), 305-325. https://doi.org/10.4995/agt.2020.13553I. Karaca and M. Is, Digital topological complexity numbers, Turkish Journal of Mathematics 42, no. 6 (2018), 3173-3181. https://doi.org/10.3906/mat-1807-101E. Khalimsky, Motion, deformation, and homotopy in finite spaces, Proceedings IEEE International Conference on Systems, Man, and Cybernetics (1987), 227-234.G. Lupton, J. Oprea and N. A. Scoville, Homotopy theory in digital topology, ArXiv: 1905.07783.G. Lupton, J. Oprea and N. A. Scoville, Subdivisions of maps of digital images, ArXiv: 1906.03170.E. Macias-Virgos and D. Mosquera-Lois, Homotopic distance between maps, Math. Proc. Cambridge Philos. Soc., to appear.G. Sabidussi, Graph multiplication, Math. Z. 72 (1960), 446-457. https://doi.org/10.1007/BF01162967T. Vergili and A. Borat, Digital Lusternik-Schnirelmann category of digital functions, Hacettepe Journal of Mathematics and Statistics 49, no. 4 (2020), 1414-1422. https://doi.org/10.15672/hujms.55979

Remarks on fixed point assertions in digital topology, 2

[EN] Several recent papers in digital topology have sought to obtain fixed point results by mimicking the use of tools from classical topology, such as complete metric spaces. We show that in many cases, researchers using these tools have derived conclusions that are incorrect, trivial, or limited.Boxer, L. (2019). Remarks on fixed point assertions in digital topology, 2. Applied General Topology. 20(1):155-175. https://doi.org/10.4995/agt.2019.10667SWORD155175201L. Boxer, A classical construction for the digital fundamental group, Journal of Mathematical Imaging and Vision 10 (1999), 51-62. https://doi.org/10.1023/A:1008370600456L. Boxer, Generalized normal product adjacency in digital topology, Applied General Topology 18, no. 2 (2017), 401-427. https://doi.org/10.4995/agt.2017.7798L. Boxer, Alternate product adjacencies in digital topology, Applied General Topology 19, no. 1 (2018), 21-53. https://doi.org/10.4995/agt.2018.7146L. Boxer, O. Ege, I. Karaca, J. Lopez and J. Louwsma, Digital fixed points, approximate fixed points, and universal functions, Applied General Topology 17, no. 2 (2016), 159-172. https://doi.org/10.4995/agt.2016.4704L. Boxer and P. C. Staecker, Remarks on fixed point assertions in digital topology, Applied General Topology, Applied General Topology 20, no. 1 (2019), https://doi.org/10.4995/agt.2019.10474S. Dalal, I. A. Masmali and G. Y. Alhamzi, Common fixed point results for compatible map in digital metric space, Advances in Pure Mathematics 8 (2018), 362-371. https://doi.org/10.4236/apm.2018.83019U. P. Dolhare and V. V. Nalawade, Fixed point theorems in digital images and applications to fractal image compression, Asian Journal of Mathematics and Computer Research 25, no. 1 (2018), 18-37.O. Ege and I. Karaca, Banach fixed point theorem for digital images, Journal of Nonlinear Sciences and Applications 8 (2015), 237-245. https://doi.org/10.22436/jnsa.008.03.08G. Herman, Oriented surfaces in digital spaces, CVGIP: Graphical Models and Image Processing 55 (1993), 381-396. https://doi.org/10.1006/cgip.1993.1029S.-E. Han, Banach fixed point theorem from the viewpoint of digital topology, Journal of Nonlinear Science and Applications 9 (2016), 895-905. https://doi.org/10.22436/jnsa.009.03.19A. Hossain, R. Ferdausi, S. Mondal and H. Rashid, Banach and Edelstein fixed point theorems for digital images, Journal of Mathematical Sciences and Applications 5, no. 2 (2017), 36-39. https://doi.org/10.12691/jmsa-5-2-2D. Jain, Common fixed point theorem for intimate mappings in digital metric spaces, International Journal of Mathematics Trends and Technology 56, no. 2 (2018), 91-94. https://doi.org/10.14445/22315373/IJMTT-V56P511K. Jyoti and A. Rani, Fixed point results for expansive mappings in digital metric spaces, International Journal of Mathematical Archive 8, no. 6 (2017), 265-270.K. Jyoti and A. Rani, Digital expansions endowed with fixed point theory, Turkish Journal of Analysis and Number Theory 5, no. 5 (2017), 146-152. https://doi.org/10.12691/tjant-5-5-1K. Jyoti and A. Rani, Fixed point theorems for β−ψ−φ-expansive type mappings in digital metric spaces, Asian Journal of Mathematics and Computer Research 24, no. 2 (2018), 56-66.L. N. Mishra, K. Jyoti, A. Rani and Vandana, Fixed point theorems with digital contractions image processing, Nonlinear Science Letters A 9, no. 2 (2018), 104-115.C. Park, O. Ege, S. Kumar, D. Jain and J. R. Lee, Fixed point theorems for various contraction conditions in digital metric spaces, Journal of Computational Analysis and Applications 26, no. 8 (2019), 1451-1458.A. Rani, K. Jyoti and A. Rani, Common fixed point theorems in digital metric spaces, International Journal of Scientific & Engineering Research 7, no. 12 (2016), 1704-1715.A. Rosenfeld, 'Continuous' functions on digital images, Pattern Recognition Letters 4 (1986), 177-184. https://doi.org/10.1016/0167-8655(86)90017-6B. Samet, C. Vetro, and P. Vetro, Fixed point theorems for contractive mappings, Nonlinear Analysis: Theory, Methods & Applications 75, no. 4 (2012), 2154-2165. https://doi.org/10.1016/j.na.2011.10.014K. Sridevi, M. V. R. Kameswari and D. M. K. Kiran, Fixed point theorems for digital contractive type mappings in digital metric spaces, International Journal of Mathematics Trends and Technology 48, no. 3 (2017), 159-167. https://doi.org/10.14445/22315373/IJMTT-V48P522K. Sridevi, M. V. R. Kameswari and D. M. K. Kiran, Common fixed points for commuting and weakly compatible self-maps on digital metric spaces, International Advanced Research Journal in Science, Engineering and Technology 4, no. 9 (2017), 21-27

A "poor man's" approach for high-resolution three-dimensional topology optimization of natural convection problems

This paper treats topology optimization of natural convection problems. A simplified model is suggested to describe the flow of an incompressible fluid in steady state conditions, similar to Darcy's law for fluid flow in porous media. The equations for the fluid flow are coupled to the thermal convection-diffusion equation through the Boussinesq approximation. The coupled non-linear system of equations is discretized with stabilized finite elements and solved in a parallel framework that allows for the optimization of high resolution three-dimensional problems. A density-based topology optimization approach is used, where a two-material interpolation scheme is applied to both the permeability and conductivity of the distributed material. Due to the simplified model, the proposed methodology allows for a significant reduction of the computational effort required in the optimization. At the same time, it is significantly more accurate than even simpler models that rely on convection boundary conditions based on Newton's law of cooling. The methodology discussed herein is applied to the optimization-based design of three-dimensional heat sinks. The final designs are formally compared with results of previous work obtained from solving the full set of Navier-Stokes equations. The results are compared in terms of performance of the optimized designs and computational cost. The computational time is shown to be decreased to around 5-20% in terms of core-hours, allowing for the possibility of generating an optimized design during the workday on a small computational cluster and overnight on a high-end desktop