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16 pages, 890 KiB

Open AccessArticle

Baryonic Matter, Ising Anyons and Strong Quantum Gravity

by Michel Planat

Int. J. Topol. 2025, 2(2), 4; https://doi.org/10.3390/ijt2020004 - 4 Apr 2025

Viewed by 45

We find that the whole set of known baryons of spin parity

J^{P} = {\frac{1}{2}}^{+}

(the ground state) and

J^{P} = {\frac{3}{2}}^{+}

(the first excited state) is organized in multiplets which may efficiently be encoded by the [...] Read more.

We find that the whole set of known baryons of spin parity

J^{P} = {\frac{1}{2}}^{+}

(the ground state) and

J^{P} = {\frac{3}{2}}^{+}

(the first excited state) is organized in multiplets which may efficiently be encoded by the multiplets of conjugacy classes in the small finite group

G = (192, 187)

. A subset of the theory is the small group

(48, 29) ≅ G L (2, 3)

whose conjugacy classes are in correspondence with the baryon families of Gell-Mann’s octet and decuplet. G has many of its irreducible characters that are minimal and informationally complete quantum measurements that we assign to the baryon families. Since G is isomorphic to the group of braiding matrices of

S U {(2)}_{2}

Ising anyons, we explore the view that baryonic matter has a topological origin. We are interested in the holographic gravity dual

A d S_{3} / Q F T_{2}

of the Ising model. This dual corresponds to a strongly coupled pure Einstein gravity with central charge

c = 1 / 2

and AdS radius of the order of the Planck scale. Some physical issues related to our approach are discussed. Full article

(This article belongs to the Special Issue Feature Papers in Topology and Its Applications)

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9 pages, 249 KiB

Open AccessArticle

Johnstone’e Non-Sober Dcpo and Extensions

by Dongsheng Zhao

Int. J. Topol. 2025, 2(1), 3; https://doi.org/10.3390/ijt2010003 - 3 Mar 2025

Viewed by 328

Abstract

One classic result in domain theory is that the Scott space of every domain (continuous directed complete poset) is sober. Johnstone constructed the first directed complete poset (dcpo for short) whose Scott space is not sober. This non-sober dcpo has been used in [...] Read more.

One classic result in domain theory is that the Scott space of every domain (continuous directed complete poset) is sober. Johnstone constructed the first directed complete poset (dcpo for short) whose Scott space is not sober. This non-sober dcpo has been used in many other parts of domain theory and more properties of it have been uncovered. In this survey paper, we first collect and prove the major properties (some of which are new as far as we know) of Johnstone’s dcpo. We then propose a general method of constructing a dcpo from given posets and prove some properties. Some problems are posed for further investigation. This paper can serve as a relatively complete resource on Johnstone’s dcpo. Full article

(This article belongs to the Special Issue Feature Papers in Topology and Its Applications)

62 pages, 523 KiB

Open AccessArticle

Existence and Mass Gap in Quantum Yang–Mills Theory

by Logan Nye

Int. J. Topol. 2025, 2(1), 2; https://doi.org/10.3390/ijt2010002 - 25 Feb 2025

Viewed by 621

Abstract

This paper presents a novel approach to solving the Yang–Mills existence and mass gap problem using quantum information theory. We develop a rigorous mathematical framework that reformulates the Yang–Mills theory in terms of quantum circuits and entanglement structures. Our method provides a concrete [...] Read more.

This paper presents a novel approach to solving the Yang–Mills existence and mass gap problem using quantum information theory. We develop a rigorous mathematical framework that reformulates the Yang–Mills theory in terms of quantum circuits and entanglement structures. Our method provides a concrete realization of the Yang–Mills theory that is manifestly gauge-invariant and satisfies the Wightman axioms. We demonstrate the existence of a mass gap by analyzing the entanglement spectrum of the vacuum state, establishing a direct connection between the mass gap and the minimum non-zero eigenvalue of the entanglement Hamiltonian. Our approach also offers new insights into non-perturbative phenomena such as confinement and asymptotic freedom. We introduce new mathematical tools, including entanglement renormalization for gauge theories and quantum circuit complexity measures for quantum fields. The implications of our work extend beyond the Yang–Mills theory, suggesting new approaches to quantum gravity, strongly coupled systems, and cosmological problems. This quantum information perspective on gauge theories opens up exciting new directions for research at the intersection of quantum field theory, quantum gravity, and quantum computation. Full article

13 pages, 276 KiB

Open AccessArticle

The Bogomolny–Carioli Twisted Transfer Operators and the Bogomolny–Gauss Mapping Class Group

by Orchidea Maria Lecian

Int. J. Topol. 2025, 2(1), 1; https://doi.org/10.3390/ijt2010001 - 12 Jan 2025

Viewed by 641

Abstract

The twisted reflection operators are defined on the hyperbolic plane. They are then specialized in hyperbolic reflections, according to which the desymmetrized

P S L (2, Z)

group is rewritten. The Bogomolny–Carioli transfer operators are newly analytically expressed in [...] Read more.

The twisted reflection operators are defined on the hyperbolic plane. They are then specialized in hyperbolic reflections, according to which the desymmetrized

P S L (2, Z)

group is rewritten. The Bogomolny–Carioli transfer operators are newly analytically expressed in terms of the Dehn twists. The Bogomolny–Gauss mapping class group of the desymmetrized

P S L (2, Z)

domain is newly proven. The paradigm to apply the Hecke theory on the CAT spaces on which the Dehn twists act is newly established. The Bogomolny–Gauss map is proven to be one of infinite topological entropy. Full article

17 pages, 319 KiB

Open AccessArticle

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 939

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

Open AccessReview

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

Viewed by 1091

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

Open AccessPerspective

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

Viewed by 1177

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

Open AccessEditorial

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 1638

Abstract

Welcome to the new open access journal: the International Journal of Topology (IJT), published by MDPI [...] Full article

10 pages, 7158 KiB

Open AccessArticle

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

Cited by 1 | Viewed by 1323

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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