arXiv subjects

Alexander V. Evako

Publications and source records attributed to Alexander V. Evako.

17 recordsLinked to original sources

arXiv2023

Introduction to the theory of digital spaces

This book provides an introduction to the theory of digital (molecular) spaces (TDS). Digital spaces are combinatorial models of continuous spaces. TDS is one of alternative branches of digital topology that studies constructing and modifying 2, 3 and n-dimensional digital image arrays in a computer and its memory. Demands for mathematical theory of multidimensional spaces built of finite number of points appeared in connection with the use of many dimensional images in computer programs. This work presents, as examples, digital models of n-dimensional Euclidean spaces, n-dimensional spheres, a torus, a projective plane, etc. Methods of TMS can work successfully in such areas as biology, chemistry, industry, medicine, etc. The book is organized as follows. Digital models of continuous spaces are the intersection graphs of special locally centered lamp covers of continuous spaces. Digital models are represented by graphs, algebraic matrices and a set of unit cubes in n-dimensional Euclidean space. The main mathematical characteristics of digital models of continuous spaces, such as the dimension, the Euler characteristic, the homology groups and others, are the same as those of their continuous counterparts. Contractible transformations of digital spaces, that change the number of elements, do not change the mathematical characteristics and properties of digital spaces. The main goal is to construct digital models of continuous spaces and to explain how to use the new methods to work with mathematical objects. The emphasis is on introducing the readers to the basics and conceptual understanding of what TDS is.

math.GN↗

arXiv2017

Properties of Digital n-Dimensional Spheres and Manifolds. Separation of Digital Manifolds

In the present paper, we study basic properties of digital n-dimensional manifolds and digital simply connected spaces. An important property of a digital n-manifold is that M is a digital n-sphere if and only if for any point v of M, M-v is a digital n-disk. It is proved that a digital (n-1)-sphere S contained a digital n-sphere M is a separating space of M. We show that a digital n-manifold can be converted to the compressed form by sequential contractions of simple pairs of adjacent points. We study structural features of digital simply connected spaces. In particular, a digital (n-1)-sphere S in a digital simply connected n-manifold M is a separating space of M.

cs.DM↗

arXiv2017

Classification of graphs based on homotopy equivalence. Homotopy equivalent graphs. Basic graphs and complexity of homotopy equivalence classes of graphs

Graph classification plays an important role is data mining, and various methods have been developed recently for classifying graphs. In this paper, we propose a novel method for graph classification that is based on homotopy equivalence of graphs. Graphs are called homotopy equivalent if one of them can be converted to the other one by contractible transformations. A basic graph and the complexity of a homotopy equivalence class are defined and investigated. It is shown all graphs belonging to a given homotopy equivalence class have similar topological properties and are represented by a basic graph with the minimal number of points and edges. Diagrams are given of basic graphs with the complexity N<7. The advantage of this classification is that it relies on computer experiments demonstrating a close connection between homotopy equivalent topological spaces and homotopy equivalent graphs.

cs.DM↗

arXiv2017

Graph Theoretical Models of Closed n-Dimensional Manifolds: Digital Models of a Moebius Strip, a Torus, a Projective Plane a Klein Bottle and n-Dimensional Spheres

In this paper, we show how to construct graph theoretical models of n-dimensional continuous objects and manifolds. These models retain topological properties of their continuous counterparts. An LCL collection of n-cells in Euclidean space is introduced and investigated. If an LCL collection of n-cells is a cover of a continuous n-dimensional manifold then the intersection graph of this cover is a digital closed n-dimensional manifold with the same topology as its continuous counterpart. As an example, we prove that the digital model of a continuous n-dimensional sphere is a digital n-sphere with at least 2n+2 points, the digital model of a continuous projective plane is a digital projective plane with at least eleven points, the digital model of a continuous Klein bottle is the digital Klein bottle with at least sixteen points, the digital model of a continuous torus is the digital torus with at least sixteen points and the digital model of a continuous Moebius band is the digital Moebius band with at least twelve points.

math.GT↗

arXiv2017

Solution of the Hyperbolic Partial Differential Equation on Graphs and Digital Spaces: a Klein Bottle a Projective Plane and a 4D Sphere

In many cases, analytic solutions of partial differential equations may not be possible. For practical problems, it is more reasonable to carry out computational solutions. However, the standard grid in the finite difference approximation is not a correct model of the continuous domain in terms of digital topology. In order to avoid serious problems in computational solutions it is necessary to use topologically correct digital domains. This paper studies the structure of the hyperbolic partial differential equation on graphs and digital n-dimensional manifolds, which are digital models of continuous n-manifolds. Conditions for the existence of solutions are determined and investigated. Numerical solutions of the equation on graphs and digital n-manifolds are presented.

math.NA↗

arXiv2017

Properties of Periodic Fibonacci-like Sequences

Generalized Fibonacci-like sequences appear in finite difference approximations of the Partial Differential Equations based upon replacing partial differential equations by finite difference equations. This paper studies properties of the generalized Fibonacci-like sequence F_(n+2)=A+BF_(n+1)+CF_n. It is shown that this sequence is periodic with the period T>2 if C=-1,|B|<2.

cs.DM↗

arXiv2017

Structure of a Parabolic Partial Differential Equation on Graphs and Digital spaces. Solution of PDE on Digital Spaces: a Klein Bottle, a Projective Plane, a 4D Sphere and a Moebius Band

This paper studies the structure of a parabolic partial differential equation on graphs and digital n-dimensional manifolds, which are digital models of continuous n-manifolds. Conditions for the existence of solutions of equations are determined and investigated. Numerical solutions of the equation on a Klein bottle, a projective plane, a 4D sphere and a Moebius strip are presented.

cs.DM↗

arXiv2015

Parabolic equations on digital spaces. Solutions on the digital Moebius strip and the digital projective plane

In this work, we define a parabolic equation on digital spaces and study its properties. The equation can be used in investigation of mechanical, aerodynamic, structural and technological properties of a Moebius strip, which is used as a basic element of a new configuration of an airplane wing. Condition for existence of exact solutions by a matrix method and a method of separation of variables are studied and determined. As examples, numerical solutions on Moebius strip and projective plane are presented.

cs.DM↗

arXiv2015

Topology-preserving digitization of n-dimensional objects by constructing cubical models

This paper proposes a new cubical space model for the representation of continuous objects and surfaces in the n-dimensional Euclidean space by discrete sets of points. The cubical space model concerns the process of converting a continuous object in its digital counterpart, which is a graph, enabling us to apply notions and operations used in digital imaging to cubical spaces. We formulate a definition of a simple n-cube and prove that deleting or attaching a simple n-cube does not change the homotopy type of a cubical space. Relying on these results, we design a procedure, which preserves basic topological properties of an n-dimensional object, for constructing compressed cubical and digital models.

cs.DM↗

arXiv2015

Properties of simple sets in digital spaces. Contractions of simple sets preserving the homotopy type of a digital space

A point of a digital space is called simple if it can be deleted from the space without altering topology. This paper introduces the notion simple set of points of a digital space. The definition is based on contractible spaces and contractible transformations. A set of points in a digital space is called simple if it can be contracted to a point without changing topology of the space. It is shown that contracting a simple set of points does not change the homotopy type of a digital space, and the number of points in a digital space without simple points can be reduces by contracting simple sets. Using the process of contracting, we can substantially compress a digital space while preserving the topology. The paper proposes a method for thinning a digital space which shows that this approach can contribute to computer science such as medical imaging, computer graphics and pattern analysis.

cs.CV↗

arXiv2014

Simple pairs of points in digital spaces. Topology-preserving transformations of digital spaces by contracting simple pairs of points

Transformations of digital spaces preserving local and global topology play an important role in thinning, skeletonization and simplification of digital images. In the present paper, we introduce and study contractions of simple pair of points based on the notions of a digital contractible space and contractible transformations of digital spaces. We show that the contraction of a simple pair of points preserves local and global topology of a digital space. Relying on the obtained results, we study properties if digital manifolds. In particular, we show that a digital n-manifold can be transformed to its compressed form with the minimal number of points by sequential contractions of simple pairs. Key Words: Graph, digital space, contraction, splitting, simple pair, homotopy, thinning

cs.DM↗

arXiv2014

Classification of digital n-manifolds

This paper presents the classification of digital n-manifolds based on the notion of complexity and homotopy equivalence. We introduce compressed n-manifolds and study their properties. We show that any n-manifold with p points is homotopy equivalent to a compressed n-manifold with m points, m<p. We design an algorithm for the classification of digital n-manifolds of any dimension n.

cs.DM↗

arXiv2014

Topology preserving representations of compact 2D manifolds by digital 2-surfaces. Compressed digital models and digital weights of compact 2D manifolds. Classification of closed surfaces by digital tools

Using digital topology approach, we construct digital models of closed surfaces as the intersection graphs of LCL covers of the surfaces. It is proved that digital models of closed surfaces are digital 2-dimensional surfaces preserving the geometry and topology of their continuous counterparts. In the framework of the proposed models, we show that for any closed surface there exists a compressed model of this surface with the minimal number of points. Key words: Closed Surface; Digital space; Cover; Graph; Digital model; Medical imaging;

cs.CG↗

arXiv2013

The Jordan-Brouwer theorem for the digital normal n-space Zn

In this paper we investigate properties of digital spaces which are represented by graphs. We find conditions for digital spaces to be digital n-manifolds and n-spheres. We study properties of partitions of digital spaces and prove a digital analog of the Jordan-Brouwer theorem for the normal digital n-space Zn.

cs.DM↗

arXiv2006

The Poincare conjecture for digital spaces. Properties of digital n-dimensional disks and spheres

Motivated by the Poincare conjecture, we study properties of digital n-dimensional spheres and disks, which are digital models of their continuous counterparts. We introduce homeomorphic transformations of digital manifolds, which retain the connectedness, the dimension, the Euler characteristics and the homology groups of manifolds. We find conditions where an n-dimensional digital manifold is the n-dimensional digital sphere and discuss the link between continuous closed n-manifolds and their digital models.

cs.DM↗

arXiv2005

The consistency principle for a digitization procedure. An algorithm for building normal digital spaces of continuous n-dimensional objects

This paper considers conditions, which allow to preserve important topological and geometric properties in the process of digitization. For this purpose, we introduce a triplet {C,M,D} consisting of a continuous object C, an intermediate model M, which is a collection of subregions whose union is C, a digital model D, which is the intersection graph of M, and apply the consistency principle and criteria of similarity to M in order to make its mathematical structure consistent with the natural structure of D. Specifically, this paper introduces a locally centered lump collection of subregions and shows that for any locally centered lump cover of an n-dimensional continuous manifold, the digital model of the manifold is a digital normal n-dimensional space. In addition, we give examples of locally centered lump tilings of two-manifolds. We propose an algorithm for constructing normal digital models of continuous objects.

cs.CV↗

arXiv1994

Dimension on Discrete Spaces

In this paper we develop some combinatorial models for continuous spaces. In this spirit we study the approximations of continuous spaces by graphs, molecular spaces and coordinate matrices. We define the dimension on a discrete space by means of axioms, and the axioms are based on an obvious geometrical background. This work presents some discrete models of n-dimensional Euclidean spaces, n-dimensional spheres, a torus and a projective plane. It explains how to construct new discrete spaces and describes in this connection several three-dimensional closed surfaces with some topological singularities It also analyzes the topology of (3+1)-spacetime. We are also discussing the question by R. Sorkin [19] about how to derive the system of simplicial complexes from a system of open covering of a topological space S.

gr-qc↗