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Öğe A randomized greedy algorithm for piecewise linear motion planning(MDPI, 2021) Ortiz C.; Lara A.; González J.; Borat, AyşeWe 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.Öğe Digital homotopic distance between digital functions(Universidad Politecnica de Valencia, 2021) Borat, AyşeIn 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.Öğe Digital Lusternik-Schnirelmann category(Scientific Technical Research Council Turkey-Tubitak, 2018) Borat, Ayşe; Vergili, TaneIn 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.Öğe Digital Lusternik-Schnirelmann category of digital functions(2020) Vergili, Tane; Borat, AyşeRoughly speaking, the digital Lusternik-Schnirelmann category of digital images studies how far a digital image is away from being digitally contractible. The digital LusternikSchnirelmann category (digital LS category, for short) is defined in [A. Borat and T. Vergili, Digital Lusternik-Schnirelmann category, Turkish J. Math. 2018]. In this paper, we introduce the digital LS category of digital functions. We will give some basic properties and discuss how this new concept will behave if we change the adjacency relation in the domain and in the image of the digital function and discuss its relation with the digital LS category of a digital image.Öğe Digital Lusternik–Schnirelmann category(2018) Borat, Ayşe; Vergili, TaneIn this paper, we define the digital Lusternik–Schnirelmann category cat? , introduce some of its properties, and discuss how the adjacency relation affects the digital Lusternik–Schnirelmann categoryÖğe Directed topological complexity of spheres(Springer Nature, 2020) Borat, Ayşe; Grant, MarkWe show that the directed topological complexity [as defined by Goubault (On directed homotopy equivalences and a notion of directed topological complexity, 2017. arXiv:1709.05702)] of the directed n-sphere is 2, for all n≥ 1. © 2019, The Author(s).Öğe Higher dimensional simplicial complexity(Electronic Journals Project, 2020) Borat, AyşeIn this paper we generalize the simplicial complexity which is defined by Gonzalez in [5], to higher dimensions. We introduce some of its properties such as its relation with the topological complexity and the relation between the dimensions of the simplicial complexity. At the last section an example of a motion planner for a complex of S-1 is given.Öğe Higher homotopic distance(Juliusz Schauder Center for Nonlinear Analysis, 2021) Borat, Ayşe; Vergili T.The concept of the homotopic distance and its higher analogs are introduced in [7]. In this paper we introduce some important properties of higher homotopic distances, investigate the conditions under which cat, secat and higher dimensional topological complexity categories are equal to the higher homotopic distance, and give alternative proofs, using higher homotopic distances, to some TCn-related theorems.Öğe MOTION PLANNING ALGORITHMS FOR CONFIGURATION SPACES IN THE HIGHER DIMENSIONAL CASE(Juliusz Schauder Ctr Nonlinear Studies, 2016) Borat, AyşeThe aim of this paper is to give an explicit motion planning algorithm for configuration spaces in the higher dimensional case.Öğe Symplectically aspherical manifolds with nontrivial Flux groups(Springer, 2016) Borat, AyşeWe investigate the fundamental group of symplectically aspherical manifolds with nontrivial Flux groups and conclude that such manifolds cannot be symplectically hyperbolic.
