Homotopy in digital spaces

Large scale and very large scale systems are considered in computer sciences, economics, ecology, biology, engineering, cybernetics and many other areas. Interconnection of their elements, sels or subsystems and complexity of their topological structure is not less important than their quantative characteristics or parameters. In this paper we give brief review of topological methods of pattern analysis of n-dimensional digital images in finite spaces, which was presented on two previous SMS conferences and in some other publications, New properties of digital fundamental group, which are important for their calculation, are investigated. Digital fundamental group of product spaces is calculated through fundamental group of multipliers. Some other higher dimensional invariants of finite digital spaces are considered and their application to pattern analysis is investigated. These methods of algebraic topological shape analysis uniquely characterize digital patterns with precision to homotopy equivalence. It shows that pattern analysis has to be conducted on interdisciplinary basis.

Intersection number and topology preservation within digital surfaces

The fundamental group of a directed graph admits a natural sequence of quotient groups called ‐fundamental groups, and the ‐fundamental groups can capture properties of a directed graph that the fundamental group cannot capture. The fundamental group of a directed graph is related to path homology through the Hurewicz theorem. The magnitude‐path spectral sequence connects magnitude homology and path homology of a directed graph, and it may be thought of as a sequence of homology of a directed graph, including path homology. In this paper, we study relations of the ‐fundamental groups and the magnitude‐path spectral sequence through the Hurewicz theorem and the Seifert–van Kampen theorem.

Ultra regular covering space and its automorphism groupIn order to classify digital spaces in terms of digital-homotopic theoretical tools, a recent paper by Han (2006b) (see also the works of Boxer and Karaca (2008) as well as Han (2007b)) established the notion of regular covering space from the viewpoint of digital covering theory and studied an automorphism group (or Deck's discrete transformation group) of a digital covering. By using these tools, we can calculate digital fundamental groups of some digital spaces and classify digital covering spaces satisfying a radius 2 local isomorphism (Boxer and Karaca, 2008; Han, 2006b; 2008b; 2008d; 2009b). However, for a digital covering which does not satisfy a radius 2 local isomorphism, the study of a digital fundamental group of a digital space and its automorphism group remains open. In order to examine this problem, the present paper establishes the notion of an ultra regular covering space, studies its various properties and calculates an automorphism group of the ultra regular covering space. In particular, the paper develops the notion of compatible adjacency of a digital wedge. By comparing an ultra regular covering space with a regular covering space, we can propose strong merits of the former.

Several recent papers have adapted notions of geometric topology to the emerging field of `digital topology.' An important notion is that of digital homotopy. In this paper, we study a variety of digitally-continuous functions that preserve homotopy types or homotopy-related properties such as the digital fundamental group.

The paper [9] introduces the important tool of the digital covering space for studying the digital fundamental group. From a classical construction of algebraic topology [16,17,19], we show the existence of digital universal covering spaces and their significance for the study of the digital fundamental group.

The big fundamental group, big Hawaiian earrings, and the big free groups

The topological fundamental group and free topological groups

Every finitely presented group is the fundamental group of the total space of a Lefschetz fibration. This follows from results of Gompf and Donaldson, and was also proved by Amoros-Bogomolov-Katzarkov-Pantev. We give another proof by providing the monodromy explicitly. We then define the genus of a finitely presented group Γ to be the minimal genus of a Lefschetz fibration with fundamental group Γ. We also give some estimates of the genus of certain groups.

We consider limits of inverse sequences of closed manifolds, whose consecutive terms are obtained by connect summing with closed manifolds, which are in turn trivialized by the bonding maps. Such spaces, which we refer to as trees of manifolds, need not be semilocally simply connected at any point and can have complicated fundamental groups. Trees of manifolds occur naturally as visual boundaries of standard nonpositively curved geodesic spaces, which are acted upon by right-angled Coxeter groups whose nerves are closed PL-manifolds. This includes, for example, those Coxeter groups that act on Davis' exotic open contractible manifolds. Also, all of the homogeneous cohomology manifolds constructed by Jakobsche are trees of manifolds. In fact, trees of manifolds of this type, when constructed from PL-homology spheres of common dimension at least 4, are boundaries of negatively curved geodesic spaces. We prove that if Z is a tree of manifolds, the natural homomorphism φ: π 1 (Z) → π 1 (Z) from its fundamental group to its first shape homotopy group is injective. If Z = bdy X is the visual boundary of a nonpositively curved geodesic space X, or more generally, if Z is a Z-set boundary of any ANR X, then the first shape homotopy group of Z coincides with the fundamental group at infinity of X: π 1 (Z) = π 1 ∞ (X). We therefore obtain an injective homomorphism ψ: π 1 (Z) → π 1 ∞ (X), which allows us to study the relationship between these groups. In particular, if Z = bdy r is the boundary of one of the Coxeter groups Γ mentioned above, we get an injective homomorphism ψ: π 1 (bdy F) → π 1 ∞ (Γ).

Polyhedra for which every homotopy domination over itself is a homotopy equivalence

We show that every locally compact Polish group is isomorphic to the isometry group of a proper separable metric space. This answers a question of Gao and Kechris. We also analyze the natural action of the isometry group of a separable ultrametric space on the space. This leads us to a structure theorem representing an arbitrary separable ultrametric space as a bundle with an ultrametric base and with ultrahomogeneous fibers which are invariant under the action of the isometry group.

The paper examines the general theoretical and civilistic construction of subjective law. Analyzing the general options for possible behavior of legal entities, such as the right to commit their own actions, the right to demand the fulfillment of a legal obligation, the right to apply to a competent state body for protection, the author concludes that the structure of legitimate behavior of participants in civil transactions in relation to digital objects is identical. In order to clarify the foundations of the functioning of the digital environment and the specifics of the protection of subjective rights in the digital environment, the extrapolation method is used and the possibility of extending the general civil law principles to relations arising in the digital space is considered. The paper substantiates the expediency of applying general civil law principles to disputes arising in connection with the violation of subjective rights in the digital environment. At the same time, the author argues that such principles do not take into account all the specifics of the interaction of subjects, and therefore formulates special principles: the principle of total accounting for actions, the principle of using agreed technologies, and the principle of access control. The paper also provides a scientific search for an answer to the question of whether the grounds for the emergence of subjective rights coincide with the grounds for their protection. The author substantiates the affirmative answer in connection with the correlation of new objects of civil transactions with the general provisions of civil law. The author formulates additional conditions for the implementation of the mechanism for the protection of the relevant digital right.

The article highlights the essence of the components of digital personal intelligence, which largely determines the effectiveness of human activity in the modern world.It is noted that digital intelligence in humans today is a necessity, as modern learning models will become effective only if there is a sufficient level of mastery of digital technologies by all participants in the educational process, the ability to build effective relationships in the digital space. The essence of digital intelligence (DQ), its components (digital memory, digital empathy and digital footprint) and their place in the modern educational environment have been represented.The concept of digital empathy and its significance in the organization of the educational process in the context of online learning have been revealed. It is indicated that the manifestation of empathy in the online environment is more complex than in face-to-face communication.Some strategies that help the speaker to evoke emotions are described: feeling and understanding of the target audience, instructing students to master the material, colours, emotions, words, maintaining regular online communication, chatting and active listening.It is emphasized that the concept of digital emotional memory is associated with digital empathy, as the mastery of information and digital technologies is accompanied by a significant emotional colour. The concept of "digital footprint" is revealed as information that remains as a result of the user’s work in the network, as well as "digital memories", which are based on emotional contact with digital objects and enhance learning motivation by creating a situation of success.It is concluded that in the context of global digitalization of the modern world, the presence of advanced41digital intelligence and its components is a necessity for each individual, and the general formula of a modern successful personality is a harmonious combination of cognitive, emotional and digital intelligences.

Two basic tasks in graph analysis are: (1) computing the distance between two graphs and (2) embedding of the graph elements (i.e., nodes or links) into a lower-dimensional space. The former task has numerous applications from k-nearest neighbor search, to clustering a collection of graphs, to transfer learning. Unfortunately, there exists no canonical way of computing the distance between two finite graphs because the dissimilarity problem is ill-defined. Despite this fact, the relevant literature contains many methods for measuring graph distance with different heuristics, computational efficiency, interpretability, and theoretical soundness [1]. We introduce a graph distance, called the Non-Backtracking Spectral Distance (NBD), which is theoretically sound and interpretable. NBD is based on a mathematical construct from algebraic topology (in particular, homotopy) called the length spectrum. The length spectrum of a graph is a function defined on the graph's first homotopy group (a.k.a. the fundamental group); and it uniquely characterizes a graph's 2-core up to isometry [2].1 Thus, if two graphs have the same length spectra, then their 2-cores are isometric. We show the relationship between a graph's length spectrum and its non-backtracking cycles; and present a method based on computing the eigenvalues of a graph's non-backtracking matrix. For the second task (i.e., graph embedding), most existing methods are stochastic and depend on black-box models such as deep networks. Both of these characteristics make their output difficult to analyze. We propose the Non-Backtracking Embedding Dimensions (NBED) for finding a graph embedding in low-dimensional space by computing the eigenvectors of the non-backtracking matrix. Both NBD and NBED are interpretable in terms of features of complex networks such as hubs, triangles, edge centrality, and communities. We showcase the usefulness of both NBD and NBED in experiments on synthetic and real-world networks.