International Journal of Topology doi: 10.3390/ijt3020009

Authors: Jaehong Oh

The advancement of autonomous robotic systems has led to significant capabilities in perception, localization, mapping, and control, yet a critical challenge remains in representing and preserving relational semantics, contextual reasoning, and cognitive transparency essential for collaboration in dynamic, human-centric environments. This paper introduces a unified architecture comprising the Ontology Neural Network (ONN) and the Ontological Real-Time Semantic Fabric (ORTSF) to address this challenge. The ONN formalizes relational semantic reasoning as a dynamic topological process by embedding Forman–Ricci curvature, persistent homology, and semantic tensor structures within a unified loss formulation, aiming to maintain relational integrity as scenes evolve. Building upon ONN, the ORTSF transforms reasoning traces into actionable control commands while compensating for system delays through predictive operators designed to preserve phase margins. Theoretical analysis and extensive simulations demonstrate that ORTSF maintains designed phase margins, offering advantages over classical delay compensation methods. Empirical studies indicate the framework’s effectiveness in unifying semantic cognition and robust control, providing a mathematically principled solution for cognitive robotics.

International Journal of Topology doi: 10.3390/ijt3020008

Authors: Bin Li

We develop a topological classification of admissible reconstruction operations in generative systems where extended structure is built through repeated local extension subject to compatibility constraints. Reconstruction is formalized as a feasibility-governed process rather than a dynamical or metric one, with admissibility determined by the accumulation of obstruction under composition. Using loop diagnostics, we identify global incompatibilities that are invisible to local extension rules but become unavoidable under closed composition. Under mild and realization-independent assumptions, including indefinite continuation and finite interface capacity, we show that persistent nontrivial obstruction is possible only when it is supported on codimension-2 subsets of the reconstructed domain. This result induces a small number of topological universality classes distinguished by the existence and stability of loop-detectable obstruction. The framework is model-agnostic and applies equally to discrete, combinatorial, and continuum reconstructions, providing a topological explanation for the ubiquity of codimension-2 defects in generative systems.

International Journal of Topology doi: 10.3390/ijt3020007

Authors: Fatemeh Fogh Sara Behnamian

We study Geraghty-type non-self mappings within the framework of best proximity point theory. By introducing auxiliary functions with subsequential convergence, we establish general conditions ensuring the existence and uniqueness of best proximity points. Our results extend and unify earlier work on proximal and Kannan-type contractions under a Geraghty setting, and provide counterexamples showing that the auxiliary assumptions are essential. As an illustration, we construct an explicit non-self alignment mapping on subsets of R2 for which all hypotheses can be verified and the unique best proximity point, as well as the convergence of the associated proximal iteration, can be computed in closed form.

International Journal of Topology doi: 10.3390/ijt3010006

Authors: Hélène Canot Philippe Durand Emmanuel Frenod

Vertical wind shear plays a crucial role in the organization and persistence of mesoscale convective systems, yet its geometrical and topological effects remain challenging to quantify. In this study, we introduce a shear-induced anisotropic metric, denoted dS, which embeds the direction and magnitude of environmental wind shear directly into the framework of persistent homology. The metric deforms the ambient geometry by weighting distances differently along and across the shear direction, enabling topological descriptors to respond dynamically to the flow environment. We establish the analytical properties of dS, and demonstrate its compatibility with Vietoris–Rips filtrations. The method is applied to the Corsican bow–echo event of 18 August 2022, where shear vectors are derived from ERA5 reanalysis data. Two complementary topological analyses are performed: a transport analysis on H0 using Wasserstein distances, and a structural analysis on H1 persistent generators under parallel and perpendicular shear metrics. The results reveal distinct topological evolutions associated with different shear orientations, highlighting the sensitivity of persistent homology to shear-induced deformation. Overall, the framework provides a mathematically consistent bridge between dynamical meteorology and topological data analysis, extending persistent homology to anisotropic metric spaces.

International Journal of Topology doi: 10.3390/ijt3010005

Authors: Christopher L. Duston

In this paper we present an implementation of a computer algorithm that automatically determines the topological structure of spacetime, using a branched covering space representation. This algorithm is applied to a few simple examples in dimension 3, and a complete set of the fundamental groups realized over several graphs is found. We also include some new visualizations of the branched covering construction, in order to aid and clarify the understanding of how these structures can be used in quantum gravity to realize the topological nature of the spacetime foam.

International Journal of Topology doi: 10.3390/ijt3010004

Authors: Yasuhiko Kamiyama

Let η:X→R be a Morse function on a connected closed manifold X. We denote by C(η) the fiber product of two copies of η. For Morse functions f:M→R and g:N→R, we define the function f∗g:M×N→R by (f∗g)(p,q):=f(p)·g(q). The purpose of this paper is twofold: Firstly, we study the sufficient condition for which χ(C(f∗g))=χ(C(f))·χ(C(g)) holds, where χ denotes the Euler characteristic. Secondly, for the case that f is the well-known Morse function on CPn, we determine χ(C(f∗f)).

International Journal of Topology doi: 10.3390/ijt3010003

Authors: Tzu-Miao Chou

This paper investigates quantum contextuality, a central nonclassical aspect of quantum mechanics, by employing the algebraic and topological structures of modular tensor categories. The analysis establishes that braid group representations constructed from modular categories, including the SU(2)k and Fibonacci anyon models, inherently produce state-dependent contextuality, as revealed by measurable violations of noncontextuality inequalities. The explicit construction of unitary representations on fusion spaces allows this paper to identify a direct structural correspondence between braiding operations and logical contextuality frameworks. The results offer a comprehensive topological framework to classify and quantify contextuality in low-dimensional quantum systems, thereby elucidating its role as a resource in topological quantum computation and advancing the interface between quantum algebra, topology, and quantum foundations.

International Journal of Topology doi: 10.3390/ijt3010002

Authors: Yoonkang Kim

We present a comprehensive investigation into the emergence of interface-bound states, particularly Majorana zero modes (MZMs), in a lateral heterostructure composed of two three-dimensional topological insulators (TIs), Bi2Se3 and Sb2Te3, under the influence of proximity-induced superconductivity from niobium (Nb) contacts. We develop an advanced two-dimensional Dirac model for the topological surface states (TSS), incorporating spatially varying chemical potentials and s-wave superconducting pairing. Using the Bogoliubov–de Gennes (BdG) formalism, we derive analytical solutions for the bound states and compute the local density of states (LDOS) at the interface, revealing zero-energy modes characteristic of MZMs. The topological nature of these states is rigorously analyzed through winding numbers and Pfaffian invariants, and their robustness is explored under various physical perturbations, including gating effects. Our findings highlight the potential of this heterostructure as a platform for topological quantum computing, with detailed predictions for experimental signatures via tunneling spectroscopy.

International Journal of Topology doi: 10.3390/ijt3010001

Authors: Michel Planat

Microtubules are cylindrical protein polymers that organize the cytoskeleton and play essential roles in intracellular transport, cell division, and possibly cognition. Their highly ordered, quasi-crystalline lattice of tubulin dimers, notably tryptophan residues, endows them with a rich topological and arithmetic structure, making them natural candidates for supporting coherent excitations at optical and terahertz frequencies. The Penrose–Hameroff Orch OR theory proposes that such coherences could couple to gravitationally induced state reduction, forming the quantum substrate of conscious events. Although controversial, recent analyses of dipolar coupling, stochastic resonance, and structured noise in biological media suggest that microtubular assemblies may indeed host transient quantum correlations that persist over biologically relevant timescales. In this work, we build upon two complementary approaches: the parametric resonance model of Nishiyama et al. and our arithmetic–geometric framework, both recently developed in Quantum Reports. We unify these perspectives by describing microtubules as rectangular lattices governed by the imaginary quadratic field Q(i), within which nonlinear dipolar oscillations undergo stochastic parametric amplification. Quantization of the resonant modes follows Gaussian norms N=p2+q2, linking the optical and geometric properties of microtubules to the arithmetic structure of Q(i). We further connect these discrete resonances to the derivative of the elliptic L-function, L′(E,1), which acts as an arithmetic free energy and defines the scaling between modular invariants and measurable biological ratios. In the appended adelic extension, this framework is shown to merge naturally with the Bost–Connes and Connes–Marcolli systems, where the norm character on the ideles couples to the Hecke character of an elliptic curve to form a unified adelic partition function. The resulting arithmetic–elliptic resonance model provides a coherent bridge between number theory, topological quantum phases, and biological structure, suggesting that consciousness, as envisioned in the Orch OR theory, may emerge from resonant processes organized by deep arithmetic symmetries of space, time, and matter.

International Journal of Topology doi: 10.3390/ijt2040021

Authors: Michael D. Rice

The field of probabilistic metric spaces has an intrinsic interest based on a blend of ideas drawn from metric space theory and probability theory. The goal of the present paper is to introduce and study new ideas in this field. In general terms, we investigate the following concepts: linearly ordered families of distances and associated continuity properties, geometric properties of distances, finite range weak probabilistic metric spaces, generalized Menger spaces, and a categorical framework for weak probabilistic metric spaces. Hopefully, the results will contribute to the foundations of the subject.

International Journal of Topology doi: 10.3390/ijt2040020

Authors: Sidney A. Morris

We prove that in every nonempty perfect Polish space, every dense Gδ subset contains strictly decreasing and strictly increasing chains of dense Gδ subsets of length c, the cardinality of the continuum. As a corollary, this holds in Rn for each n≥1. This provides an easy answer to a question of Erdős since the set of Liouville numbers admits a descending chain of cardinality c, each member of which has the Erdős property. We also present counterexamples demonstrating that the result fails if either the perfection or the Polishness assumption is omitted. Finally, we show that the set T of real Mahler T-numbers is a dense Borel set and contains a strictly descending chain of length c of proper dense Borel subsets.

International Journal of Topology doi: 10.3390/ijt2040019

Authors: Hélène Canot Philippe Durand Emmanuel Frenod

We propose a geometry topological framework to analyze storm dynamics by coupling persistent homology with Anti-de Sitter (AdS)-inspired metrics. On radar images of a bow echo event, we compare Euclidean distance with three compressive AdS metrics (α = 0.01, 0.1, 0.3) via time-resolved H1 persistence diagrams for the arc and its internal cells. The moderate curvature setting (α=0.1) offers the best trade-off: it suppresses spurious cycles, preserves salient features, and stabilizes lifetime distributions. Consistently, the arc exhibits longer, more dispersed cycles (large-scale organizer), while cells show shorter, localized patterns (confined convection). Cross-correlations of H1 lifetimes reveal a temporal asymmetry: arc activation precedes cell activation. A differential indicator Δ(t) based on Wasserstein distances quantifies this divergence and aligns with the visual onset in radar, suggesting early warning potential. Results are demonstrated on a rapid Corsica bow echo; broader validation remains future work.

International Journal of Topology doi: 10.3390/ijt2040018

Authors: Jitka Matějková

This article develops an interdisciplinary framework that applies topological and graph-theoretical methods to public procurement markets and digital platform economies. Conceptualizing legal–economic interactions as dynamic networks of nodes and edges, we show how structural properties—centrality, clustering, connectivity, and boundary formation—shape contestability, resilience, and compliance. Using EU-relevant contexts (public procurement directives and the Digital Markets Act), we formalize network representations for buyers, suppliers, platforms, and regulators; define operational indicators; and illustrate an empirical, value-weighted buyer → supplier network to reveal a sparse but highly modular architecture with a high-value backbone. We then map these structural signatures to concrete legal levers (lotting and framework design, modification scrutiny, interoperability and data-access duties) and propose dashboard-style diagnostics for proactive oversight. The findings demonstrate how topological modelling complements doctrinal analysis by making hidden architectures visible and by linking measurable structure to regulatory outcomes. We conclude with implications for evidence-informed regulatory design and a research agenda integrating graph analytics, comparative evaluation across jurisdictions, and machine-learning-assisted anomaly detection.

International Journal of Topology doi: 10.3390/ijt2040017

Authors: Innocent Abaa Umar Islambekov

We propose a nested ensemble learning framework that utilizes Topological Data Analysis (TDA) to extract and integrate topological features from graph data, with the goal of improving performance on classification and regression tasks. Our approach computes persistence diagrams (PDs) using lower-star filtrations induced by three filter functions: closeness, betweenness, and degree 2 centrality. To overcome the limitation of relying on a single filter, these PDs are integrated through a data-driven, three-level architecture. At Level-0, diverse base models are independently trained on the topological features extracted for each filter function. At Level-1, a meta-learner combines the predictions of these base models for each filter to form filter-specific ensembles. Finally, at Level-2, a meta-learner integrates the outputs of these filter-specific ensembles to produce the final prediction. We evaluate our method on both simulated and real-world graph datasets. Experimental results demonstrate that our framework consistently outperforms base models and standard stacking methods, achieving higher classification accuracy and lower regression error. It also surpasses existing state-of-the-art approaches, ranking among the top three models across all benchmarks.

International Journal of Topology doi: 10.3390/ijt2040016

Authors: Michał Jabłonowski

In this study of the Reidemeister moves within the classical knot theory, we focus on hard diagrams of knots and links, categorizing them as either rigid or shaky based on their adaptability to certain moves. We establish that every link possesses a diagram that is a rigid hard diagram, and we provide an upper limit for the number of crossings in such diagrams. Furthermore, we investigate rigid hard diagrams for specific knots or links to determine their rigid hard index. In the topic of shaky hard diagrams, we demonstrate the existence of such diagrams for the unknot and unlink of any number of components and present examples of shaky hard diagrams.

International Journal of Topology doi: 10.3390/ijt2030015

Authors: Marian Przemski

This article proposes a unified concept of topological types of convergence for nets of multifunctions between topological spaces. Any kind of convergence is representable by a (2n + 2)-tuple, n = 0, 1, …, of two special functions u and l, such that their compositions ul and lu create the Choquet supremum and infimum operations, respectively, on the filters considered in terms of the upper Vietoris topology on the range hyperspace of the considered multifunctions. Convergence operators are defined by establishing the order of composition of the functions from such (2n + 2) tuples. An allocation of places for the two distinguished functions in a convergence operator reflects the structure of the used (2n + 2)-tuple. A monoid of special three-parameter functions called products describes the set of all possible structures. The monoid of products is the domain space of the convergence operators. The family of all convergence operators forms a finite monoid whose neutral element determines the pointwise convergence and possesses the structure determined by the neutral element of the monoid of products. We demonstrate the construction process of every convergence operator and show that the notions of the presented concept can characterize many well-known classical types of convergence. Of particular importance are the types of convergence derived from the concept of continuous convergence. We establish some general theorems about the necessary and sufficient conditions for the continuity of the limit multifunctions without any assumptions about the type of continuity of the members of the nets.

International Journal of Topology doi: 10.3390/ijt2030014

Authors: Punit Boolchand James Charles Phillips Matthieu Micoulaut Aaron Welton

The origin of glass formation has been one of the greatest mysteries of science. The first clues emerged in GexSe1-x glasses, where the bond-stretching and bond angle-bending constraints are countable, and it was found that the most favorable compositions for glass formation involved matching constraints with the degrees of freedom. Modulated-Differential Scanning Calorimetric (MDSC) studies on GexSe1-x chalcogenide glasses revealed two elastic phase transitions—a stiffness transition at x = 0.20 and a stress transition at x = 0.26—leading to the observation of three topological phases: a flexible phase at x < 0.20, an intermediate phase in the 0.20 < x < 0.26 range, and a stressed–rigid phase for compositions x > 0.26. The three topological phases (TPs) have now been generically observed in more than two dozen chalcogenides and modified oxide glasses. In proteins, the transition from the unfolded (flexible) to the folded (isostatically rigid intermediate) phase represents the stiffness transition. Self-organization causes proteins to display a dynamic reversibility of the folding process. The evolutions of protein dynamics may also exhibit stiffness phase transitions similar to those seen in glasses.

International Journal of Topology doi: 10.3390/ijt2030013

Authors: Bryan Gerardo Castro Herrejón Frédéric Mynard

Though a convergence space is connected if and only if its topological modification is connected, connected subsets of a convergence space differ from those of its topological modification. We explore which subsets exhibit connectedness for the convergence or for the topological modification. In particular, we show that connectedness of a subset is equivalent for a convergence or for its reciprocal modification and that the largest set enclosing a given connected subset of a convergence space is the adherence of the connected set for the reciprocal modification of the convergence.

International Journal of Topology doi: 10.3390/ijt2030012

Authors: Dieter Leseberg Zohreh Vaziry

We introduce the notion of bornological approach nearness as a unified extension of various classical nearness structures. By redefining completeness within this framework, we establish a generalized version of the Niemytzki–Tychonoff theorem. Our results not only extend known compactness criteria in nearness spaces but also offer a new perspective that incorporates boundedness and bornological methods in the theory of approach spaces.

International Journal of Topology doi: 10.3390/ijt2030011

Authors: Takaaki Fujita

Graph theory models relationships by representing entities as vertices and their interactionsas edges. To handle directionality and multiple head–tail assignments, various extensions—directed, bidirected, and multidirected graphs—have been introduced, with the multidirected graph unifying the first two. In this work, we further enrich this landscape by proposing the Multidirected hypergraph, which merges the flexibility of hypergraphs and superhypergraphs to describe higher-order and hierarchical connections. Building on this, we introduce five uncertainty-aware Multidirected frameworks—fuzzy, neutrosophic, plithogenic, rough, and soft multidirected graphs—by embedding classical uncertainty models into the Multidirected setting. We outline their formal definitions, examine key structural properties, and illustrate each with examples, thereby laying groundwork for future advances in uncertain graph analysis and decision-making.

International Journal of Topology doi: 10.3390/ijt2030010

Authors: Takaaki Fujita

Classical algebraic structures—such as magmas, groups, and Lie groups—are characterized by increasingly strong requirements in binary operation, ranging from no additional constraints to associativity, identity, inverses, and smooth-manifold structures. The hyperstructure paradigm extends these notions by allowing the operation to return subsets of elements, giving rise to hypermagmas, hypergroups, and Lie hypergroups, along with their variants such as quotient, reduced, and fuzzy hypergroups. In this work, we introduce the concept of superhyperstructures, obtained by iterating the powerset construction, and develop the theory of superhypermagmas and Lie superhypergroups. We further define and analyze quotient superhypergroups, reduced superhypergroups, and fuzzy superhypergroups, exploring their algebraic properties and interrelationships.

International Journal of Topology doi: 10.3390/ijt2030009

Authors: Gabriel Katz

Let Y be a smooth compact n-manifold. We studied smooth embeddings and immersions β:M→R×Y of compact n-manifolds M such that β(M) avoids some priory chosen closed poset Θ of tangent patterns to the fibers of the obvious projection π:R×Y→Y. Then, for a fixed Y, we introduced an equivalence relation between such β’s; creating a crossover between pseudo-isotopies and bordisms. We called this relation quasitopy. In the presented study of quasitopies, the spaces PdcΘ of real univariate polynomials of degree d with real divisors, whose combinatorial patterns avoid a given closed poset Θ, play the classical role of Grassmanians. We computed the quasitopy classes Qdemb(Y,cΘ) of Θ-constrained embeddings β in terms of homotopy/homology theory of spaces Y and PdcΘ. We proved also that the quasitopies of embeddings stabilize, as d→∞.

International Journal of Topology doi: 10.3390/ijt2020008

Authors: José A. Rodrigues

Persistent homology is a powerful tool in topological data analysis that captures the multi-scale topological features of data. In this work, we provide a mathematical introduction to persistent homology and demonstrate its application to protein–protein interaction networks. We combine persistent homology with algebraic connectivity, a graph-theoretic measure of network robustness, to analyze the topology and stability of PPI networks. An example is provided to illustrate the methodology and its potential applications in systems biology.

International Journal of Topology doi: 10.3390/ijt2020007

Authors: Joaquín Díaz Boils

A multicomplex structure is defined from an ordered lattice of multigraphs. This structure will help us to observe the features of persistent homology in this context, its interaction with the ordering, and the repercussions of merging multigraphs in the calculation of Betti numbers. For the latter, an extended version of the incremental algorithm is provided. The ideas developed here are mainly oriented to the original example described by the author and others in the context of the formalization of the notion of embodiment in Neuroscience.

International Journal of Topology doi: 10.3390/ijt2020006

Authors: Arturo Tozzi

Raynaud’s Phenomenon (RP), characterized by episodic reductions in peripheral blood flow, leads to significant discomfort and functional impairment. Existing therapeutic strategies focus on pharmacological treatments, external heat supplementation and exercise-based rehabilitation, but fail to address biomechanical contributions to vascular dysfunction. We introduce a computational approach rooted in topological transformations of hand prehension, hypothesizing that specific hand postures can generate transient geometric structures that enhance thermal and hemodynamic properties. We examine whether a flexed hand posture—where fingers are brought together to form a closed-loop toroidal shape—may modify heat transfer patterns and blood microcirculation. Using a combination of heat diffusion equations, fluid dynamics models and topological transformations, we implement a heat transfer and blood flow simulation to examine the differential thermodynamic behavior of the open and closed hand postures. We show that the closed-hand posture may preserve significantly more heat than the open-hand posture, reducing temperature loss by an average of 1.1 ± 0.3 °C compared to 3.2 ± 0.5 °C in the open-hand condition (p < 0.01). Microvascular circulation is also enhanced, with a 53% increase in blood flow in the closed-hand configuration (p < 0.01). Therefore, our findings support the hypothesis that maintaining a closed-hand posture may help mitigate RP symptoms by preserving warmth, reducing cold-induced vasoconstriction and optimizing peripheral flow. Overall, our topologically framed approach provides quantitative evidence that postural modifications may influence peripheral vascular function through biomechanical and thermodynamic mechanisms, elucidating how shape-induced transformations may affect physiological and pathological dynamics.

International Journal of Topology doi: 10.3390/ijt2020005

Authors: Logan Nye

We establish a comprehensive framework demonstrating that physical reality can be understood as a holographic encoding of underlying computational structures. Our central thesis is that different geometric realizations of the same physical system represent equivalent holographic encodings of a unique computational structure. We formalize quantum complexity as a physical observable, establish its mathematical properties, and demonstrate its correspondence with geometric descriptions. This framework naturally generalizes holographic principles beyond AdS/CFT correspondence, with direct applications to black hole physics and quantum information theory. We derive specific, quantifiable predictions with numerical estimates for experimental verification. Our results suggest that computational structure, rather than geometry, may be the more fundamental concept in physics.

International Journal of Topology doi: 10.3390/ijt2020004

Authors: Michel Planat

We find that the whole set of known baryons of spin parity JP=12+ (the ground state) and JP=32+ (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)≅GL(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 SU(2)2 Ising anyons, we explore the view that baryonic matter has a topological origin. We are interested in the holographic gravity dual AdS3/QFT2 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.

International Journal of Topology doi: 10.3390/ijt2010003

Authors: Dongsheng Zhao

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.

International Journal of Topology doi: 10.3390/ijt2010002

Authors: Logan Nye

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.

International Journal of Topology doi: 10.3390/ijt2010001

Authors: Orchidea Maria Lecian

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

International Journal of Topology doi: 10.3390/ijt1010005

Authors: Orchidea Maria Lecian

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.

International Journal of Topology doi: 10.3390/ijt1010004

Authors: Spyridon Louvros

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.

International Journal of Topology doi: 10.3390/ijt1010003

Authors: Dan-Marian Joița Lorentz Jäntschi

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.

International Journal of Topology doi: 10.3390/ijt1010002

Authors: Michel Planat

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

International Journal of Topology doi: 10.3390/ijt1010001

Authors: Michael O’Keeffe Michael M. J. Treacy

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.