General Topology Research Articles

Soft Weakly Quasi-Continuous Functions Between Soft Topological Spaces

As an extension of quasi-continuity in general topology, we define soft quasi-continuity. We show that this notion is equivalent to the known notion of soft semi-continuity. Next, we define soft weak quasi-continuity. With the help of examples, we prove that soft weak quasi-continuity is strictly weaker than both soft semi-continuity and soft weak continuity. We introduce many characterizations of soft weak quasi-continuity. Moreover, we study the relationship between soft quasi-continuity and weak quasi-continuity with their analogous notions in general topology. Furthermore, we show that soft regularity of the co-domain of a soft function is a sufficient condition for equivalence between soft semi-continuity and soft weakly quasi-continuity. Furthermore, we provide several results of soft composition, restrictions, preservation, and soft graph theorems in terms of soft weak quasi-continuity.

Topological laws of the Rayleigh wave scattering on a statistical inhomogeneity of isotropic solid in the Rayleigh limit

Topological laws of the Rayleigh wave scattering on a statistical inhomogeneity of isotropic solid are obtained theoretically in the Rayleigh limit. They are completely defined by the inhomogeneity structure and include the Rayleigh law of scattering as a particular case. They violate the Rayleigh law in the case of a more general inhomogeneity topology, then the Rayleigh one. It enables first to construct theoretically arbitrary spectrum of scattering up to its oscillations and a strong angular anisotropy.

Review of recent trends of advancements in multilevel inverter topologies with reduced power switches and control techniques

AbstractCurrently, multilevel inverters (MLI) are comprehensively used to integrate renewable energy sources with the grid or high‐power applications. MLI has outstanding properties such as high‐level output voltage with tremendous power quality and negligible harmonics distortion. General MLI topologies have a more complex circuit, high overall cost and installation area. Due to a large number of power switches and isolation of control circuit, fewer power electronic switches and DC source along with its suitable control logic is the need of the hour. Therefore, a lot of research scope exists in this area. Here, some of the topologies with a fewer number of power switches are reviewed and scrutinized. An extensive analysis of the topologies, control strategy and applications are presented here, suggesting suitable multilevel inverter solutions to the known industrial application or new research.

A one-time training machine learning method for general structural topology optimization

Machine learning (ML) methods have found some applications in structural topology optimization. In the existing methods, however, the ML models need to be retrained when the design domains and supporting conditions have been changed, posing a limitation to their wide applications. In this paper, we propose a one-time training ML (OTML) method for general topology optimization, where the self-attention convolutional long short-term memory (SaConvLSTM) model is introduced to update the design variables. An extension–division approach is used to enrich the training sets. By developing a splicing strategy, the training results of a small design space (i.e., a basic cell of either two- or three-dimensions) can be extended to tackling the optimization problem of a large design domain with arbitrary geometric shapes. Using the OTML method, the ML model needs to be trained for only one time, and the trained model can be used directly to solve various optimization problems with arbitrary shapes of design domains, loads, and boundary conditions. In the SaConvLSTM model, the material volume of the post-processed thresholded designs can be precisely controlled, though the control precision of the gray-scale designs might be slightly sacrificed. The effects of model parameters on the computational cost and the result quality are examined. Four examples are provided to demonstrate the high performance of this structural design method. For large-scale optimization problems, the present method can accelerate the structural form-finding process. This study holds a promise in the high-resolution structural form-finding and transdisciplinary computational morphogenesis.

THE NEUTROSOPHIZE OF NEW CONTINUITY SPECIES

In this study, after giving the necessary definitions in the preliminaries and explaining the process that make this study necessary is explained in the introduction, in the third section, some types of open set that were previously defined in general topology and various non-standard topological spaces are presented and the relationships between them are explained with the help of a diagram. Then, the concept of neutrosophic af-open set is defined and its relations with other open set types are examined and their properties are investigated in neutrosophic topology. In the following sections, the concept of af-open set is generalized and different types of continuities are introduced using these new concepts of open set, and the connections between them are illustrated with examples and diagrams.

Digital Experiences of Mathematical Cognitive Functions in Learning the Basic Concepts of General Topology

Digital Experiences of Mathematical Cognitive Functions in Learning the Basic Concepts of General Topology

Symmetry-free routing and spectrum assignment: a universal algorithm based on first-fit

First-fit (FF) is a well-known and widely deployed algorithm for spectrum assignment (SA), but until our recent study [J. Opt. Commun. Netw. 14, 165 (2022)JOCNBB1943-062010.1364/JOCN.445492], investigations of the algorithm had been experimental in nature and no formal properties of the algorithm with respect to SA were known. In this work, we make two contributions. First, we show that FF is a universal algorithm for the SA problem in the sense that, for any variant, 1) it can be used to construct solutions equivalent to, or better than, any solution obtained by any other algorithm, and 2) it can construct an optimal solution. This universality property applies to both the min-max and min-frag objectives and to variants of the SA problem with or without guard band constraints. Consequently, the spectrum symmetry-free model of our recent study [J. Opt. Commun. Netw. 14, 165 (2022)JOCNBB1943-062010.1364/JOCN.445492] extends to all known SA variants, which therefore reduce to permutation problems. Second, we extend the spectrum symmetry-free model to the routing and spectrum assignment (RSA) problem in general topologies. This model allows for the design of more efficient algorithms as it eliminates from consideration an exponential number of equivalent symmetric solutions. By sidestepping symmetry, the RSA solution space is naturally and optimally decomposed into a routing space and a connection permutation space. Building upon this property, we introduce a two-parameter, symmetry-free universal algorithm that can be used to tackle any RSA variant in a uniform manner. The algorithm is amenable to multi-threaded execution to speed up the search process, and the value of the parameters can be adjusted to strike a balance between running time and solution quality. Our evaluation provides insight into the relative benefits of path diversity (which determines the size of the routing space) and connection diversity (which determines the size of the permutation space).

Computation of the exact time delay margin for vehicle platoon under generic communication topologies

Due to the limited bandwidth and transmission congestion of the vehicle platoon's communication, it is inevitable to induce time delay, which significantly degrades the control performance of the vehicle platoon, even resulting in instability. This paper focuses on analyzing the internal stability under generic communication topologies and presents a method of computing the exact time delay margin (ETDM). The proposed method can offer a necessary and sufficient internal stability condition with no conservatism. Firstly, to reduce the analytical complexity and computational burden elegantly, we decompose the closed-loop platoon dynamics into a set of individual subsystems via similarity transformation and matrix factorization. This decomposition approach is applicable for any general communication topology. Secondly, an explicit formula is deduced to compute the ETDM by surveying the characteristic roots' distribution of all these individual subsystems. It is further demonstrated that only the positive purely imaginary roots need to be considered to compute the ETDM. Finally, simulations are conducted to demonstrate the effectiveness of the theoretical claims.

Near open generalizations of rough sets and their applications

BackgroundThe concept of near open sets is a potent tool that empowers researchers to achieve a more encompassing approximation of rough sets, thereby enhancing the accuracy of measurements. The evolution of rough set theory into various generalized forms, based on topological structures, has emerged as a significant approach in the realm of knowledge discovery within databases.ResultsThis paper’s primary contribution lies in the introduction of a novel category of generalized near open sets, termed “inverse simply open sets,” within the context of the j\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{j}$$\\end{document}-neighborhood space. The paper proposes diverse methods for extending the Pawlak’s rough approximations leading to the definition of new approximations in the j\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{j}$$\\end{document}-neighborhood space. By employing these newly introduced generalizations, we establish fresh connections between two pivotal theories, namely “general topology and rough set theory”. Through a comprehensive investigation, we conduct multiple comparisons between our methodology and classical approaches. Furthermore, we showcase practical applications of these techniques within real-life scenarios, demonstrating their utility in decision-making processes.ConclusionsWe reduced the data’s ambiguity while increasing its accuracy measure. As a result, we may conclude that the proposed approximations were more precise than earlier techniques and contributed to the elimination of data ambiguity in real-world scenarios requiring accurate decisions.

WEAKER FORMS ON SEPARATION AXIOMS

In general topology, the concepts of separation axioms play an important role. The main goal of this paper is to present the study of weaker forms of separation axioms called spgwα-T1, spgwα-T2, spgwα-regular and spgwα-normal spaces using spgwα-open sets in topological spaces. Further, the properties spgwα-compact, spgwα-connected and spgwα-Lindelof spaces have been defined and studied their basic characterizations in topological spaces.

Equisingularity in pencils of curves on germs of reduced complex surfaces

Abstract We study pencils of curves on a germ of complex reduced surface $(S,0)$ . These are families of curves parametrized by $ \mathbb{P}^1 $ having 0 as the unique common point. We prove that for $w\in \mathbb{P}^1$ , the corresponding curve of the pencil does not have the generic topology if and only if either the corresponding curve of the pulled-back pencil to the normalized surface has a non generic topology or w is a limit value for the function $ f/g $ along the singular locus of $(S,0)$ , where f and g are generators of the pencil.

Structural and functional insights into the C-terminal signal domain of the Bacteroidetes type-IX secretion system.

Gram-negative bacteria from the Bacteroidota phylum possess a type-IX secretion system (T9SS) for protein secretion, which requires cargoes to have a C-terminal domain (CTD). Structurally analysed CTDs are from Porphyromonas gingivalis proteins RgpB, HBP35, PorU and PorZ, which share a compact immunoglobulin-like antiparallel 3+4 β-sandwich (β1-β7). This architecture is essential as a P. gingivalis strain with a single-point mutant of RgpB disrupting the interaction of the CTD with its preceding domain prevented secretion of the protein. Next, we identified the C-terminus ('motif C-t.') and the loop connecting strands β3 and β4 ('motif Lβ3β4') as conserved. We generated two strains with insertion and replacement mutants of PorU, as well as three strains with ablation and point mutants of RgpB, which revealed both motifs to be relevant for T9SS function. Furthermore, we determined the crystal structure of the CTD of mirolase, a cargo of the Tannerella forsythia T9SS, which shares the same general topology as in Porphyromonas CTDs. However, motif Lβ3β4 was not conserved. Consistently, P. gingivalis could not properly secrete a chimaeric protein with the CTD of peptidylarginine deiminase replaced with this foreign CTD. Thus, the incompatibility of the CTDs between these species prevents potential interference between their T9SSs.

Spaces determined by countably many locally compact subspaces

The theory of compactly generated spaces, alternatively k-spaces, plays an important role in general and algebraic topology. In this paper, we develop a theory of compact generation for non-Hausdorff spaces using locally compact spaces. We are particularly interested in the case that countably many locally compact spaces suffice and restrict our attention to that case. The notion of ℓcω-spaces (which can be considered as a type of kω-spaces) is introduced, which are determined by countably many locally compact spaces. It is proved that the product space of two ℓcω-spaces is still an ℓcω-space. We slightly modify the notion of an ℓcω-space to apply it to posets equipped with the Scott topology. The notions of ℓcω-posets, ℓfω-posets and c-posets are introduced. We develop a corresponding theory for these three kinds of posets and investigate the conditions under which the Scott topology on the product of two posets is equal to the product of the individual Scott topologies and under which the Scott topology on a dcpo is sober. Several such conditions are given.

PerfTop: Towards performance prediction of distributed learning over general topology

Distributed learning with multiple GPUs has been widely adopted to accelerate the training process of large-scale deep neural networks. However, misconfiguration of the GPU clusters with various communication primitives and topologies could potentially diminish the gains in parallel computation and lead to significant degradation in training efficiency. Predicting the performance of distributed learning enables service providers to identify potential bottlenecks beforehand. In this work, we propose a Performance prediction framework over General Topologies, called PerfTop, for accurate estimation of per-iteration execution time. The main strategy is to integrate computation time prediction with an analytical model to map the nonlinearity in communication and fine-grained computation-communication patterns. This enables accurate prediction of a variety of neural network models over general topologies, such as tree, hierarchical, and exponential. Our extensive experiments show that PerfTop outperforms existing methods in estimating both computation and communication time, particularly for communication, surpassing the existing methods by over 45%. Meanwhile, it achieves an accuracy of above 85% in predicting the execution time over general topologies compared to simple topologies such as star and ring from the previous works.

Fixed-time consensus control for uncertain heterogeneous multi-agent systems with high-order dynamics and time-varying delay under generic topologies

This article is concerned with the fixed-time control problem of uncertain high-order heterogeneous multi-agent systems with time-varying delays for both directed and undirected topologies. Applying the free-weighting matrix approach, a sliding mode controller is designed through the Lyapunov–Krasovskii functional and the conditions are derived with the help of linear matrix inequalities to attain fixed-time consensus tracking control of such systems. Two numerical simulations are included to verify the capability of the derived control law with respect to different initializations.

A survey of generalized metrizable properties in topological groups and weakly topological groups

The theory of generalized metrizable spaces is an important topic of general topology. This paper is a survey of research methods and achievements on generalized metrizable properties in topological groups and weakly topological groups. We mainly study this kind of properties in topological groups, semitopological groups, paratopological groups, quasitopological groups and free topological groups, and focus on the influence of separation properties, conditions for weakly topological groups to become topological groups, cardinal invariants, weak first-countability, three-space properties and remainders in compactifications on topological groups and related structures. Finally, some unsolved problems in this field are listed for researchers.

Discrete Weber Inequalities and Related Maxwell Compactness for Hybrid Spaces over Polyhedral Partitions of Domains with General Topology

Discrete Weber Inequalities and Related Maxwell Compactness for Hybrid Spaces over Polyhedral Partitions of Domains with General Topology

Soft super-continuity and soft delta-closed graphs.

Introducing a strong form of soft continuity between soft topological spaces is significant because it can contribute to our growing understanding of soft topological spaces and their features, provide a basis for creating new mathematical tools and methods, and have significant applications in various fields. In this paper, we define soft super-continuity as a new form of soft mapping. We present various characterizations of this soft concept. Also, we show that soft super-continuity lies strictly between soft continuity and soft complete continuity and that soft super-continuity is a strong form of soft δ-continuity. In addition, we give some sufficient conditions for the equivalence between soft super-continuity and other related concepts. Moreover, we characterize soft semi-regularity in terms of super-continuity. Furthermore, we provide several results of soft composition, restrictions, preservation, and products by soft super-continuity. In addition to these, we study the relationship between soft super-continuity and soft δ-continuity with their analogous notions in general topology. Finally, we give several sufficient conditions on a soft mapping to have a soft δ-closed graph.

Fractal networks: Topology, dimension, and complexity.

Over the past two decades, the study of self-similarity and fractality in discrete structures, particularly complex networks, has gained momentum. This surge of interest is fueled by the theoretical developments within the theory of complex networks and the practical demands of real-world applications. Nonetheless, translating the principles of fractal geometry from the domain of general topology, dealing with continuous or infinite objects, to finite structures in a mathematically rigorous way poses a formidable challenge. In this paper, we overview such a theory that allows to identify and analyze fractal networks through the innate methodologies of graph theory and combinatorics. It establishes the direct graph-theoretical analogs of topological (Lebesgue) and fractal (Hausdorff) dimensions in a way that naturally links them to combinatorial parameters that have been studied within the realm of graph theory for decades. This allows to demonstrate that the self-similarity in networks is defined by the patterns of intersection among densely connected network communities. Moreover, the theory bridges discrete and continuous definitions by demonstrating how the combinatorial characterization of Lebesgue dimension via graph representation by its subsets (subgraphs/communities) extends to general topological spaces. Using this framework, we rigorously define fractal networks and connect their properties with established combinatorial concepts, such as graph colorings and descriptive complexity. The theoretical framework surveyed here sets a foundation for applications to real-life networks and future studies of fractal characteristics of complex networks using combinatorial methods and algorithms.

On T-continuous map

It is no secret to the observer that the new open sets play a role in general topology. and they represent research topic for many topologists in different parts of the world. In fact, there is an important topics in general topology and real analysis that deals with continuity of maps and their different forms by relying on new open set. In this work, we presented and studied a new class of continuous maps called T-continuous by using the concepts of T-open sets and T-closed sets, where important and basic properties of T-continuous map and its relationship with other maps were verified.