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Mathematical and Molecular Topology
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Mathematical and Molecular Topology

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Algebra, Geometry and Topology".

Deadline for manuscript submissions: closed (31 January 2023) | Viewed by 13190

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editors

Prof. Dr. Lorentz Jäntschi

E-Mail Website
Guest Editor
Department of Physics and Chemistry, Technical University of Cluj-Napoca, 400641 Cluj-Napoca, Romania
Interests: applied mathematics; applied informatics
Special Issues, Collections and Topics in MDPI journals
Dr. Mihaela Tomescu

E-Mail Website
Guest Editor
Department of Mathematics - Informatics, University of Petroșani, Petroșani, Romania
Interests: banach spaces

Special Issue Information

Dear Colleagues,

This Special Issue, "Mathematical and Molecular Topology", is open for submissions and welcomes papers from a broad interdisciplinary area, since topology is concerned with the properties of objects that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending. One of the oldest problems in topology is The Seven Bridges of Königsberg.

Topology naturally finds application in all fields of engineering, physical sciences, life sciences, social sciences, medicine, business and even arts. The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together.

In about 1750, Euler stated the polyhedron formula, V − E + F = 2 (where V, E, and F respectively indicate the number of vertices, edges, and faces of the polyhedron), which may be regarded as the first theorem, signaling the birth of topology.

Subjects included in topology are graph theory and algebraic topology. MSC 2010 subjects related to topology include the following: Number theory (with Density, gaps, topology and Relations with algebraic geometry and topology); Algebraic geometry (with Topology of surfaces—Donaldson polynomials, Seiberg–Witten invariants and Topology of real algebraic varieties); Category theory & homological algebra (for categories in topology); K-theory (for obstructions from topology); Group theory and generalizations (for other groups related to topology); Topological groups & Lie groups; Real functions (for elementary topology of the line); Several complex variables and analytic spaces (for Topology of analytic spaces); Dynamical systems and ergodic theory (for Topology in relation with holomorphic dynamical systems and Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology); Functional analysis (for Embeddings of discrete metric spaces into Banach spaces; applications in topology, Hilbert and pre-Hilbert spaces: geometry and topology, Noncommutative topology, and Methods of algebraic topology in functional analysis); Quantum theory (for Feynman integrals and graphs; applications of algebraic topology); Game theory, economics, finance, and other social and behavioral sciences (for Games involving topology); and finally Biology and other natural sciences (for Molecular structure—graph-theoretic methods, methods of differential topology and alike). Finally, three sections of MSC are almost entirely devoted to or build upon topology: General topology (54-XX), Algebraic topology (55-XX) and Manifolds and cell complexes (57-XX).

Prof. Dr. Lorentz Jäntschi
Dr. Mihaela Tomescu
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • algebraic topology
  • clopen sets
  • cohomology
  • compactness
  • connectedness
  • continuity
  • differential topology
  • euclidean spaces
  • geometric topology
  • graph theory
  • grothendieck topologies
  • homeomorphisms
  • homology
  • homotopy
  • manifolds
  • metric spaces
  • molecular topology
  • point-set topology
  • surgery theory
  • topological invariants

Published Papers (5 papers)

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Research

13 pages, 1305 KiB  
Article
Entropies Via Various Molecular Descriptors of Layer Structure of H3BO3
by Muhammad Usman Ghani, Muhammad Kashif Maqbool, Reny George, Austine Efut Ofem and Murat Cancan
Mathematics 2022, 10(24), 4831; https://doi.org/10.3390/math10244831 - 19 Dec 2022
Cited by 11 | Viewed by 1382
Abstract
Entropy is essential. Entropy is a measure of a system’s molecular disorder or unpredictability, since work is produced by organized molecular motion. Entropy theory offers a profound understanding of the direction of spontaneous change for many commonplace events. A formal definition of a [...] Read more.
Entropy is essential. Entropy is a measure of a system’s molecular disorder or unpredictability, since work is produced by organized molecular motion. Entropy theory offers a profound understanding of the direction of spontaneous change for many commonplace events. A formal definition of a random graph exists. It deals with relational data’s probabilistic and structural properties. The lower-order distribution of an ensemble of attributed graphs may be used to describe the ensemble by considering it to be the results of a random graph. Shannon’s entropy metric is applied to represent a random graph’s variability. A structural or physicochemical characteristic of a molecule or component of a molecule is known as a molecular descriptor. A mathematical correlation between a chemical’s quantitative molecular descriptors and its toxicological endpoint is known as a QSAR model for predictive toxicology. Numerous physicochemical, toxicological, and pharmacological characteristics of chemical substances help to foretell their type and mode of action. Topological indices were developed some 150 years ago as an alternative to the Herculean, and arduous testing is needed to examine these features. This article uses various computational and mathematical techniques to calculate atom–bond connectivity entropy, atom–bond sum connectivity entropy, the newly defined Albertson entropy using the Albertson index, and the IRM entropy using the IRM index. We use the subdivision and line graph of the H3BO3 layer structure, which contains one boron atom and three oxygen atoms to form the chemical boric acid. Full article
(This article belongs to the Special Issue Mathematical and Molecular Topology)
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14 pages, 2288 KiB  
Article
Exact Maximum Clique Algorithm for Different Graph Types Using Machine Learning
by Kristjan Reba, Matej Guid, Kati Rozman, Dušanka Janežič and Janez Konc
Mathematics 2022, 10(1), 97; https://doi.org/10.3390/math10010097 - 28 Dec 2021
Cited by 5 | Viewed by 2206
Abstract
Finding a maximum clique is important in research areas such as computational chemistry, social network analysis, and bioinformatics. It is possible to compare the maximum clique size between protein graphs to determine their similarity and function. In this paper, improvements based on machine [...] Read more.
Finding a maximum clique is important in research areas such as computational chemistry, social network analysis, and bioinformatics. It is possible to compare the maximum clique size between protein graphs to determine their similarity and function. In this paper, improvements based on machine learning (ML) are added to a dynamic algorithm for finding the maximum clique in a protein graph, Maximum Clique Dynamic (MaxCliqueDyn; short: MCQD). This algorithm was published in 2007 and has been widely used in bioinformatics since then. It uses an empirically determined parameter, Tlimit, that determines the algorithm’s flow. We have extended the MCQD algorithm with an initial phase of a machine learning-based prediction of the Tlimit parameter that is best suited for each input graph. Such adaptability to graph types based on state-of-the-art machine learning is a novel approach that has not been used in most graph-theoretic algorithms. We show empirically that the resulting new algorithm MCQD-ML improves search speed on certain types of graphs, in particular molecular docking graphs used in drug design where they determine energetically favorable conformations of small molecules in a protein binding site. In such cases, the speed-up is twofold. Full article
(This article belongs to the Special Issue Mathematical and Molecular Topology)
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12 pages, 3905 KiB  
Article
Comparison of Molecular Geometry Optimization Methods Based on Molecular Descriptors
by Donatella Bálint and Lorentz Jäntschi
Mathematics 2021, 9(22), 2855; https://doi.org/10.3390/math9222855 - 10 Nov 2021
Cited by 17 | Viewed by 4937
Abstract
Various methods (Hartree–Fock methods, semi-empirical methods, Density Functional Theory, Molecular Mechanics) used to optimize a molecule structure feature the same basic approach but differ in the mathematical approximations used. The geometry optimization procedure calculates the energy at an initial geometry of a molecule [...] Read more.
Various methods (Hartree–Fock methods, semi-empirical methods, Density Functional Theory, Molecular Mechanics) used to optimize a molecule structure feature the same basic approach but differ in the mathematical approximations used. The geometry optimization procedure calculates the energy at an initial geometry of a molecule and then proceeds to search a new geometry with a lower energy. Using the 3D structures collected from the PubChem database, 20 amino acid geometry optimization calculations were performed with several methods. The purpose of the study was to analyze these methods (39) to find the relationship between them and to determine which to use under different circumstances. Cluster analysis and principal component analysis were performed to evaluate the similarities between the different methods. The results after the analysis can classified into three main groups and can be selected accordingly to solve different types of problems. Full article
(This article belongs to the Special Issue Mathematical and Molecular Topology)
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16 pages, 517 KiB  
Article
Convergence Analysis and Dynamical Nature of an Efficient Iterative Method in Banach Spaces
by Deepak Kumar, Sunil Kumar, Janak Raj Sharma and Lorentz Jantschi
Mathematics 2021, 9(19), 2510; https://doi.org/10.3390/math9192510 - 07 Oct 2021
Cited by 3 | Viewed by 1367
Abstract
We study the local convergence analysis of a fifth order method and its multi-step version in Banach spaces. The hypotheses used are based on the first Fréchet-derivative only. The new approach provides a computable radius of convergence, error bounds on the distances involved, [...] Read more.
We study the local convergence analysis of a fifth order method and its multi-step version in Banach spaces. The hypotheses used are based on the first Fréchet-derivative only. The new approach provides a computable radius of convergence, error bounds on the distances involved, and estimates on the uniqueness of the solution. Such estimates are not provided in the approaches using Taylor expansions of higher order derivatives, which may not exist or may be very expensive or impossible to compute. Numerical examples are provided to validate the theoretical results. Convergence domains of the methods are also checked through complex geometry shown by drawing basins of attraction. The boundaries of the basins show fractal-like shapes through which the basins are symmetric. Full article
(This article belongs to the Special Issue Mathematical and Molecular Topology)
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9 pages, 220 KiB  
Article
Some Variants of Normal Čech Closure Spaces via Canonically Closed Sets
by Ria Gupta and Ananga Kumar Das
Mathematics 2021, 9(11), 1225; https://doi.org/10.3390/math9111225 - 27 May 2021
Cited by 3 | Viewed by 1898
Abstract
New generalizations of normality in Čech closure space such as π-normal, weakly π-normal and κ-normal are introduced and studied using canonically closed sets. It is observed that the class of κ-normal spaces contains both the classes of weakly π [...] Read more.
New generalizations of normality in Čech closure space such as π-normal, weakly π-normal and κ-normal are introduced and studied using canonically closed sets. It is observed that the class of κ-normal spaces contains both the classes of weakly π-normal and almost normal Čech closure spaces. Full article
(This article belongs to the Special Issue Mathematical and Molecular Topology)
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