Topological Notions Research Articles

On neutrosophic sets and topology

Recently, F.Smarandache generalized the Atanassov's intuitionistic fuzzy sets and other kinds of sets to neutrosophic sets. Also, this author defined the notion of neutrosophic topology on the non-standard interval. One can expect some relation between the intuitionistic fuzzy topology on an IFS and the neutrosophic topology. We show in this work that this is false.

Empirical evidence on the topological properties of structural paths and some notes on its theoretical explanation

The mathematical literature has developed a large pool of topological concepts and theorems for dynamic systems analysis. The aim of our paper is to make a first step towards the application of these concepts and theorems in the analysis of (long-run) structural change (in the three-sector framework). Our approach focuses on two of the most basic topological notions, namely intersection and self-intersection of trajectories on a two-dimensional domain. We discuss the mathematical foundations of the application of these concepts in structural change analysis, use them for analyzing empirical data, and elaborate new stylized facts stating that different countries’ structural change trajectories are (non-self-)intersecting. Finally, we discuss briefly the theoretical explanations of (non-self-)intersection and a wide range of new research topics relating to (a) the topological classification and comparison of models and evidence and (b) the application of (further) topological concepts in standard branches of growth and development theory.

Representability in supergeometry

In this paper we use the notion of Grothendieck topology to present a unified way to approach representability in supergeometry, which applies to both the differential and algebraic settings.

Stacked spaces: Mapping digital infrastructures

This article turns towards the spatial life of ‘digital infrastructures’, i.e. code, protocols, standards, and data formats that are hidden from view in everyday applications of computational technologies. It does so by drawing on the version control system Git as a case study, and telling the story of its initial development in order to reconstruct the circumstances and technical considerations surrounding its conception. This account engages with computational infrastructures on their own terms by adopting the figure of the ‘stack’ to frame a technically informed analysis, and exploring its implications for a different kind of geographic inquiry. Drawing on topology as employed by Law and Mol, attention is given to the multiplicity of spatialities and temporalities enrolled in digital infrastructures in general, and Git specifically. Along the lines of the case study and by reading it against other literatures, this notion of topology is expanded to include the material performation of fundamentally arbitrary, more-than-human topologies, as well as their nested articulation, translation and negotiation within digital infrastructures.

A Diagram of Galois Connections of Functorial Topologies

A Galois connection between functorial topologies on abelian groups and subclasses of abelian groups is constructed by means of the notion of indiscrete topology. It is shown that the composition of this Galois connection with a previously introduced one coincides with the classical Galois connection induced by the notion of constant morphism. Examples are provided.

Smoothing discrete Morse theory

t. After surveying classical notions of PL topology of the Seventies, we clarify the relation between Morse theory and its discretization by Forman. We show that PL handlestheory and discrete Morse theory are equivalent, in the sense that every discrete Morse vector on some PL triangulation is also a PL handle vector, and conversely, every PL handle vector is also a discrete Morse vector on some PL triangulation. It follows that, in dimension up to 7, every discrete Morse vector on some PL triangulation is also a smooth Morse vector; the viceversa is true in all dimensions. This revises and improves a result by Gallais. Some further consequences of our work are: (1) For d 6= 4, every simply connected smooth d-manifold admits locally constructible triangulations. In contrast, the Mazur 4-manifold has no locally constructible triangulation. (This solves a question by Zˇivaljevic` and completes a work by the author and Ziegler.) (2) The Heegaard genus of 3-manifolds can be characterized as the smallest g for which some triangulation of the manifold has discrete Morse vector (1, g, g, 1). (This allows for heuristics to bound the Heegaard genus of any 3-manifold.) (3) Some non-PL 5-spheres admit discrete Morse functions with only 2 critical faces. (This result, joint with Adiprasito, completes the Sphere Theorem by Forman.)

Network topology and mean infection times

A fundamental concept of social network analysis is centrality. Many analyses represent the network topology in terms of concept transmission/variation, e.g., influence, social structure, community or other aggregations. Even when the temporal nature of the network is considered, analysis is conducted at discrete points along a continuous temporal scale. Unfortunately, well-studied metrics of centrality do not take varying probabilities into account. The assumption that social and other networks that may be physically stationary, e.g., hard wired, are conceptually static in terms of information diffusion or conceptual aggregation (communities, etc.) can lead to incorrect conclusions. Our findings illustrate, both mathematically and experimentally, that if the notion of network topology is not stationary or fixed in terms of the concept, e.g., groups, belonging, community or other aggregations, centrality should be viewed probabilistically. We show through some surprising examples that study of transmission behavior based solely on a graph’s topological and degree properties is lacking when it comes to modeling network propagation or conceptual (vs. physical) structure.

Weak Finite Intersection Characterizations of Existence in Optimization

We propose a pure topological notion of a finite-intersection map (FI-map), which captures the idea of a typical generalized KKM-map, with no use of any generalized convexity structure and prove its full characterizations in terms of recent generalized KKM and connectedness structures. We establish an intersection point theorem in terms of FI-maps. Then, a half of the paper is devoted to apply this result to obtain characterizations of the existence of solutions in optimization. First, we discuss a general model of variational relations. Then, we move on to more particular, but typical and important cases: equilibrium problems, saddle points, constrained minimization, and Nash equilibria. We include discussions and examples to illustrate that our results improve or extend many recent contributions in the literature.

On topological filtrations of groups

The (weak) geometric simple connectivity and the quasi-simple filtration are topological notions of manifolds, which may be defined for discrete groups too. It turns out that they are equivalent for finitely presented groups, but the main problem is the absence of examples of groups which do not satisfy them. In this note we study some algebraic classes of groups with respect to these properties.

Some network topological notions of the Mycielskian of a graph

Efficiency and reliability are two important criteria in the designing of a good interconnection network. Network topological notions such as wide diameter, fault diameter, diameter vulnerability and (l,k)-domination can be used to study the efficiency and reliability of a network. In this paper we study these notions in the Mycielskian of a graph and its iterates.

Soft topological space and topology on the Cartesian product

The paper deals with a soft topological space which is dened over an initial universe set U with a xed set of parameters E . The main goal is to point out that any soft topological space is homeomorphic to a topological space $( E \times U, \tau )$ where $\tau$ is an arbitrary topology on the product $E \times U$ , consequently many soft topological notions and results can be derived from general topology. Furthermore, in many papers some notions are introduced by dierent ways and it would be good to give a unied approach for a transfer of topological notions to a soft set theory and to create a bridge between general topology and soft set theory.

A Variational Definition for Limit and Derivative

Using the topological notion of compacity, we present a variational definition for the concepts of limit and derivative of a function. The main result of these new definition is that they produce implementable tests to check whether a value is the limit or the derivative of a differenciable function.

The holographic Weyl semi-metal

We present a holographic model of a Weyl semi-metal. We show the evidences that upon varying a mass parameter the model undergoes a sharp crossover at small temperature from a topologically non-trivial state to a trivial one. The order parameter is the anomalous Hall effect (AHE) and we find that it is very strongly suppressed above a critical value of the mass parameter. This can be taken as a hint for an underlying topological quantum phase transition. We give an interpretation of the results in terms of a holographic RG flow and compare to a weakly coupled field theoretical model. Since there are no fermionic quasiparticle excitations in the strongly coupled holographic model the presence of an anomalous Hall effect cannot be bound to notions of topology in momentum spaces.

On a generalization of density topologies on the real line

Density topologies appear naturally in many considerations in real analysis. The most fundamental one is the classical density topology in the family of Lebesgue measurable sets. Namely, A⊂R is open iff A⊂Φ(A), whereΦ(A)={x∈R:limh→0+⁡λ((A−x)∩[−h,h])2h=1} that is, when all points of A are Lebesgue density points of A.One of deeply studied problems in the last years (by, for example, M. Filipczak, J. Hejduk and R. Wiertelak) was the problem of weakening the definition of the Lebesgue density points (and, in turn, the definition of the operator Φ). Our paper deals with this problem – we try to define the most general (from a certain point of view) notion of density topology on the real line.

Linear structures, causal sets and topology

Causal set theory and the theory of linear structures (which has recently been developed by Tim Maudlin as an alternative to standard topology) share some of their main motivations. In view of that, I raise and answer the question how these two theories are related to each other and to standard topology. I show that causal set theory can be embedded into Maudlin׳s more general framework and I characterise what Maudlin׳s topological concepts boil down to when applied to discrete linear structures that correspond to causal sets. Moreover, I show that all topological aspects of causal sets that can be described in Maudlin׳s theory can also be described in the framework of standard topology. Finally, I discuss why these results are relevant for evaluating Maudlin׳s theory. The value of this theory depends crucially on whether it is true that (a) its conceptual framework is as expressive as that of standard topology when it comes to describing well-known continuous as well as discrete models of spacetime and (b) it is even more expressive or fruitful when it comes to analysing topological aspects of discrete structures that are intended as models of spacetime. On one hand, my theorems support (a). The theory is rich enough to incorporate causal set theory and its definitions of topological notions yield a plausible outcome in the case of causal sets. On the other hand, the results undermine (b). Standard topology, too, has the conceptual resources to capture those topological aspects of causal sets that are analysable within Maudlin׳s framework. This fact poses a challenge for the proponents of Maudlin׳s theory to prove it fruitful.

The Wijsman topology of a fuzzy metric space

We introduce and study the notions of lower Wijsman topology, upper Wijsman topology and Wijsman topology of a fuzzy metric space in the sense of Kramosil and Michalek. In particular, quasi-uniformizability, uniformizability, quasi-metrizability and metrizability of these topologies are discussed. Their relations with other hypertopologies are also analyzed. Corresponding results to the Wijsman topology of a metric space are deduced from our approach with the help of the standard fuzzy metric.

Lusternik–Schnirelmann category of simplicial complexes and finite spaces

In this paper we establish a natural definition of Lusternik–Schnirelmann category for simplicial complexes via the well known notion of contiguity. This simplicial category has the property of being invariant under strong equivalences, and it only depends on the simplicial structure rather than its geometric realization.In a similar way to the classical case, we also develop a notion of simplicial geometric category. We prove that the maximum value over the simplicial homotopy class of a given complex is attained in the core of the complex.Finally, by means of well known relations between simplicial complexes and posets, specific new results for the topological notion of LS-category are obtained in the setting of finite topological spaces.

Robustness-Based Simplification of 2D Steady and Unsteady Vector Fields.

Vector field simplification aims to reduce the complexity of the flow by removing features in order of their relevance and importance, to reveal prominent behavior and obtain a compact representation for interpretation. Most existing simplification techniques based on the topological skeleton successively remove pairs of critical points connected by separatrices, using distance or area-based relevance measures. These methods rely on the stable extraction of the topological skeleton, which can be difficult due to instability in numerical integration, especially when processing highly rotational flows. In this paper, we propose a novel simplification scheme derived from the recently introduced topological notion of robustness which enables the pruning of sets of critical points according to a quantitative measure of their stability, that is, the minimum amount of vector field perturbation required to remove them. This leads to a hierarchical simplification scheme that encodes flow magnitude in its perturbation metric. Our novel simplification algorithm is based on degree theory and has minimal boundary restrictions. Finally, we provide an implementation under the piecewise-linear setting and apply it to both synthetic and real-world datasets. We show local and complete hierarchical simplifications for steady as well as unsteady vector fields.

A Topological Approach to Cheeger‐Gromov Universal Bounds for von Neumann ρ‐Invariants

Using deep analytic methods, Cheeger and Gromov showed that for any smooth (4k‐1)‐manifold there is a universal bound for the von Neumann L2 ρ‐invariants associated to arbitrary regular covers. We present a proof of the existence of a universal bound for topological (4k‐1)‐manifolds, using L2‐signatures of bounding 4k‐manifolds. We give explicit linear universal bounds for 3‐manifolds in terms of triangulations, Heegaard splittings, and surgery descriptions. We show that our explicit bounds are asymptotically optimal. As an application, we give new lower bounds of the complexity of 3‐manifolds that can be arbitrarily larger than previously known lower bounds. As ingredients of the proofs that seem interesting on their own, we develop a geometric construction of efficient 4‐dimensional bordisms of 3‐manifolds over a group and develop an algebraic topological notion of uniformly controlled chain homotopies.© 2016 Wiley Periodicals, Inc.

Formal Characteristics of the Architectural Type of contemporary Mosques within the concept of Topology

Mosques could be considered as one of the most powerful architectural types throughout historical ages. With their highly symbolic formal legacy, Mosques play an essential role in providing the Islamic city with its special identity. Nevertheless, the advent of digital technology and its ubiquity at different levels of architectural design marked the emergence of new tendencies in the Architecture of Mosques, represented by various models added to the storage of this architectural type. Consequently a review of these tendencies would be needed, aiming at pointing out the formal transformations and new suggested characteristics.
 The paper investigates the surviving and the disappearing formal components of the Mosque Type, and reviews the different ways they are contemporarily employed within the notion of Topology in Architecture. This is achieved through the study and analysis of several examples of contemporary mosques , with comparison to the development of the formal components undergone by the Mosque type.