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Int. J. Topol., Volume 1, Issue 1 (December 2024) – 5 articles

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17 pages, 319 KiB  
Article
Sheaf Cohomology of Rectangular-Matrix Chains to Develop Deep-Machine-Learning Multiple Sequencing
by Orchidea Maria Lecian
Int. J. Topol. 2024, 1(1), 55-71; https://doi.org/10.3390/ijt1010005 - 16 Dec 2024
Viewed by 715
Abstract
The sheaf cohomology techniques are newly used to include Morse simplicial complexes in a rectangular-matrix chain, whose singular values are compatible with those of a square matrix, which can be used for multiple sequencing. The equivalence with the simplices of the corresponding graph [...] Read more.
The sheaf cohomology techniques are newly used to include Morse simplicial complexes in a rectangular-matrix chain, whose singular values are compatible with those of a square matrix, which can be used for multiple sequencing. The equivalence with the simplices of the corresponding graph is proven, as well as that the filtration of the corresponding probability space. The new protocol eliminates the problem of stochastic stability of deep Markov models. The paradigm can be implemented to develop deep-machine-learning multiple sequencing. The construction of the deep Markov models for sequencing, starting from a profile Markov model, is analytically written. Applications can be found as an amino-acid sequencing model. As a result, the nucleotide-dependence of the positions on the alignments are fully modelized. The metrics of the manifolds are discussed. The instance of the application of the new paradigm to the Jukes–Cantor model is successfully controlled on nucleotide-substitution models. Full article
28 pages, 415 KiB  
Review
On Linear Operators in Hilbert Spaces and Their Applications in OFDM Wireless Networks
by Spyridon Louvros
Int. J. Topol. 2024, 1(1), 27-54; https://doi.org/10.3390/ijt1010004 - 29 Nov 2024
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Abstract
This paper explores the application of Hilbert topological spaces and linear operator algebra in the modelling and analysis of OFDM signals and wireless channels, where the channel is considered as a linear time-invariant (LTI) system. The wireless channel, when subjected to an input [...] Read more.
This paper explores the application of Hilbert topological spaces and linear operator algebra in the modelling and analysis of OFDM signals and wireless channels, where the channel is considered as a linear time-invariant (LTI) system. The wireless channel, when subjected to an input OFDM signal, can be described as a mapping from an input Hilbert space to an output Hilbert space, with the system response governed by linear operator theory. By employing the mathematical framework of Hilbert spaces, we formalise the representation of OFDM signals, which are interpreted as elements of an infinite-dimensional vector space endowed with an inner product. The LTI wireless channel is characterised by using bounded linear operators on these spaces, allowing for the decomposition of complex channel behaviour into a series of linear transformations. The channel’s impulse response is treated as a kernel operator, facilitating a functional analysis approach to understanding the signal transmission process. This representation enables a more profound understanding of channel effects, such as fading and interference, through the eigenfunction expansion of the operator, leading to a spectral characterization of the channel. The algebraic properties of linear operators are leveraged to develop optimal solutions for mitigating channel distortion effects. Full article
14 pages, 1697 KiB  
Perspective
Counting Polynomials in Chemistry II
by Dan-Marian Joița and Lorentz Jäntschi
Int. J. Topol. 2024, 1(1), 13-26; https://doi.org/10.3390/ijt1010003 - 23 Oct 2024
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Abstract
Some polynomials find their way into chemical graph theory less often than others. They could provide new ways of understanding the origins of regularities in the chemistry of specific classes of compounds. This study’s objective is to depict the place of polynomials in [...] Read more.
Some polynomials find their way into chemical graph theory less often than others. They could provide new ways of understanding the origins of regularities in the chemistry of specific classes of compounds. This study’s objective is to depict the place of polynomials in chemical graph theory. Different approaches and notations are explained and levelled. The mathematical aspects of a series of such polynomials are put into the context of recent research. The directions in which this project was intended to proceed and where it stands right now are presented. Full article
(This article belongs to the Special Issue Feature Papers in Topology and Its Applications)
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2 pages, 243 KiB  
Editorial
International Journal of Topology
by Michel Planat
Int. J. Topol. 2024, 1(1), 11-12; https://doi.org/10.3390/ijt1010002 - 1 Jul 2024
Viewed by 1399
Abstract
Welcome to the new open access journal: the International Journal of Topology (IJT), published by MDPI [...] Full article
10 pages, 7158 KiB  
Article
Embeddings of Graphs: Tessellate and Decussate Structures
by Michael O’Keeffe and Michael M. J. Treacy
Int. J. Topol. 2024, 1(1), 1-10; https://doi.org/10.3390/ijt1010001 - 29 Mar 2024
Viewed by 1097
Abstract
We address the problem of finding a unique graph embedding that best describes a graph’s “topology” i.e., a canonical embedding (spatial graph). This question is of particular interest in the chemistry of materials. Graphs that admit a tiling in 3-dimensional Euclidean space are [...] Read more.
We address the problem of finding a unique graph embedding that best describes a graph’s “topology” i.e., a canonical embedding (spatial graph). This question is of particular interest in the chemistry of materials. Graphs that admit a tiling in 3-dimensional Euclidean space are termed tessellate, those that do not decussate. We give examples of decussate and tessellate graphs that are finite and 3-periodic. We conjecture that a graph has at most one tessellate embedding. We give reasons for considering this the default “topology” of periodic graphs. Full article
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