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

Directed topological complexity of spheres

Open access via Springer compact agreement. Acknowledgements The first author wishes to thank the University of Aberdeen for their hospitality during her stay at the Institute of Mathematics, where this work was carried out. Both authors wish to thank Eric Goubault for useful discussions and for sharing with them preliminary versions of his results, and the anonymous referees for valuable comments.Peer reviewedPreprintPublisher PD

Digital Lusternik-Schnirelmann category

WOS: 000439579600023In this paper, we define the digital Lusternik-Schnirelmann category cat(kappa), introduce some of its properties, and discuss how the adjacency relation affects the digital Lusternik-Schnirelmann category

A Randomized Greedy Algorithm for Piecewise Linear Motion Planning

We describe and implement a randomized algorithm that inputs a polyhedron, thought of as the space of states of some automated guided vehicle R, and outputs an explicit system of piecewise linear motion planners for R. The algorithm is designed in such a way that the cardinality of the output is probabilistically close (with parameters chosen by the user) to the minimal possible cardinality.This yields the first automated solution for robust-to-noise robot motion planning in terms of simplicial complexity (SC) techniques, a discretization of Farber’s topological complexity TC. Besides its relevance toward technological applications, our work reveals that, unlike other discrete approaches to TC, the SC model can recast Farber’s invariant without having to introduce costly subdivisions. We develop and implement our algorithm by actually discretizing Macías-Virgós and Mosquera-Lois’ notion of homotopic distance, thus encompassing computer estimations of other sectional category invariants as well, such as the Lusternik-Schnirelmann category of polyhedra