İsmet Karaca

Some properties of proximal homotopy theory

Filomat

Nearness theory comes into play in homotopy theory because the notion of closeness between points... more Nearness theory comes into play in homotopy theory because the notion of closeness between points is essential in determining whether two spaces are homotopy equivalent. While nearness theory and homotopy theory have different focuses and tools, they are intimately connected through the concept of a metric space and the notion of proximity between points, which plays a central role in both areas of mathematics. This manuscript investigates some concepts of homotopy theory in proximity spaces. Moreover, these concepts are taken into account in descriptive proximity spaces.

Discrete Topological Complexities of Simplical Maps

arXiv (Cornell University), Mar 9, 2024

In this study, we delve into the discrete TC of surjective simplicial fibrations, aiming to unrav... more In this study, we delve into the discrete TC of surjective simplicial fibrations, aiming to unravel the interplay between topological complexity, discrete geometric structures, and computational efficiency. Moreover, we examine the properties of the discrete TC number in higher dimensions and its relationship with scat. We also touch on the basic properties of the notion of higher contiguity distance, and show that it is possible to consider discrete TC computations in a simpler sense.

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Homotopy extension property for multi-valued functions

Mathematica Moravica, 2023

In this study, we introduce some well-known definitions and properties for multi-valued functions... more In this study, we introduce some well-known definitions and properties for multi-valued functions. We present new definitions such as, m-retraction, m-section, m-homeomorphism, m-HEP and reducible function. We give a new result on the relation between multi-homotopy groups and m-homeomorphism. We also deal with some properties of m-HEP.

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Computing Higher Dimensional Digital Homotopy Groups

Applied mathematics & information sciences, Sep 1, 2014

In this paper, we study out a method for computing digital homotopy groups in higher dimensions. ... more In this paper, we study out a method for computing digital homotopy groups in higher dimensions. We investigate the relation between a digital image and its n th homotopy group when n is greater than 1 and show that a digital covering map which is a radius 2 local isomorphism induces an isomorphism between digital homotopy groups in higher dimensions.

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Monomial Bases in the Mod-p Steenrod Algebra

Czechoslovak Mathematical Journal, Sep 1, 2005

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more

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Nilpotence relations in the mod-p Steenrod algebra

Journal of Pure and Applied Algebra, Jun 1, 2002

In our paper we generalize some results proved by Monks (Bol. Mex. Mat. Soc. 37 (1992) 401-416) a... more

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Some new results on the connected sum of certain digital surfaces

Malaya journal of matematik, Apr 1, 2019

In this paper, we construct some new digital surfaces from the topological sum of two digital sur... more In this paper, we construct some new digital surfaces from the topological sum of two digital surfaces. Also, we compute the digital simplicial homology groups of these digital surfaces. We calculate the Euler characteristics of certain digital connected surfaces. Moreover, we obtain some results of Euler characteristics of certain minimal simple closed surfaces.

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Conjectures on the anti-automorphism of Z-basis of the Steenrod algebra

Journal of Linear and Topological Algebra, Mar 1, 2021

Some formulas for nilpotence of Steenrod operations

Invariant elements in the mod-p Steenrod algebra

Digital Borsuk-Ulam theorem

Bulletin of The Iranian Mathematical Society, 2017

The aim of this paper is to compute a simplicial cohomology group of some specific digital images... more

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The excision theorem for digital Khalimsky spaces

New trends in mathematical sciences, Jun 27, 2018

Unlike the digital images, Khalimsky digital spaces have topological structures in addition to th... more Unlike the digital images, Khalimsky digital spaces have topological structures in addition to their adjacency relations. The construction of their digital singular homology groups is given in [23]. The functorial properties of digital singular homology theory allow us to characterize and classify the digital spaces. In algebraic topology, additivity and excision axioms are one of Eilenberg-Steenrod axioms in homology theory and in this paper, we check their validity for digital singular homology theory.

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Some Properties of Steenrod Squares on Digital Images

Mathematical Sciences Letters, May 1, 2016

Some Properties Of Proximal Homotopy Theory

arXiv (Cornell University), Jun 13, 2023

Nearness theory comes into play in homotopy theory because the notion of closeness between points... more Nearness theory comes into play in homotopy theory because the notion of closeness between points is essential in determining whether two spaces are homotopy equivalent. While nearness theory and homotopy theory have different focuses and tools, they are intimately connected through the concept of a metric space and the notion of proximity between points, which plays a central role in both areas of mathematics. This manuscript investigates some concepts of homotopy theory in proximity spaces. Moreover, these concepts are taken into account in descriptive proximity spaces.

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Proximal Motion Planning Algorithms

arXiv (Cornell University), May 23, 2023

In this paper, we transfer the problem of measuring navigational complexity in topological spaces... more In this paper, we transfer the problem of measuring navigational complexity in topological spaces to the nearness theory. We investigate the most important component of this problem, the topological complexity number (denoted by TC), with its different versions including relative and higher TC, on the proximal Schwarz genus as well as the proximal (higher) homotopic distance. We outline the fundamental properties of some concepts related to the proximal (or descriptive proximal) TC numbers. In addition, we provide some instances of (descriptive) proximity spaces, specifically on basic robot vacuum cleaners, to illustrate the results given on proximal and descriptive proximal TC.

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An Algorithm for Computing Digital Cohomology Groups

Applied mathematics & information sciences, May 1, 2016

Several recent papers have discussed the digital version of the cohomology group for digital imag... more Several recent papers have discussed the digital version of the cohomology group for digital images, and some researchers calculate digital cohomology groups of some special two or three-dimensional digital images. In this paper, we determine the simplicial cohomology groups of some minimal simple closed curves and the digital surface MSS 6. Also we give a general algorithm for computing digital cohomology groups of finite dimensional digital images.

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Simplicial Relative Cohomology Rings of Digital Images

Applied Mathematics & Information Sciences, 2014

The first goal of this paper is to show that the relative cohomology groups of digital images are... more The first goal of this paper is to show that the relative cohomology groups of digital images are determined algebraically by the relative homology groups of digital images. Then we state simplicial cup product for digital images and use it to establish ring structure of digital cohomology. Furthermore we give a method for computing the cohomology ring of digital images and give some examples concerning cohomology ring.

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Fixed point theorems and an Application in Parametric Metric Spaces

Azerbaijan Journal of Mathematics, Jun 20, 2016

In this paper, we give concepts of couple fixed and coupled coincidence point in parametric metri... more

Some Results on Simplical Homology Groups of 2D Digital Images

International journal of information and computer science, Nov 1, 2012

In this paper we study some results related to the simplicial homology groups of 2D digital image... more In this paper we study some results related to the simplicial homology groups of 2D digital images. We show that if a bounded digital image Z X ⊂ is nonempty and κ -connected, then its homology groups at the first dimension are a trivial group. In general, we prove that the homology groups of the operands of a wedge of digital images need not

The digital smash product

Electronic research archive, 2020

In this paper, we construct the smash product from the digital viewpoint and prove some its prope... more In this paper, we construct the smash product from the digital viewpoint and prove some its properties such as associativity, distributivity, and commutativity. Moreover, we present the digital suspension and the digital cone for an arbitrary digital image and give some examples of these new concepts.

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