Topological Point Of View Research Articles

Some aspects of the five-dimensional Lovelock black hole spacetime: Strong homotopy retract, perihelion precession and quasistationary levels

In this work we explore some mathematical physics aspects of the spherically symmetric Lovelock black hole in high dimensions. Intended for this aim, we thoroughly consider the metric corresponding to the five-dimensional Lovelock black hole spacetime. We construct the strong retractions by the geodesic equations on the background under consideration. As a result, from the topological point of view, we construct the theory of strong homotopy retract, which will allow us, in principle, to better understand some of its suitable applications on astrophysics and cosmology, in particular, in the analysis of the spacetime singularities. We find the solutions of the equation of motion for both radial and angular coordinates, and then we describe the outer (“exterior”) and lower (“interior”) apparent horizons. Indeed, the outer apparent horizon is the last surface from which the light waves could still escape from the black hole. Thus, it is meaningful to analyze some physical phenomena related to quantum particles propagating outside the exterior apparent horizon, in particular, we discuss the quasistationary levels of scalar fields and their radial wave functions, which are given in terms of the general Heun functions. We also calculate the perihelion precession in this background.

A Graphic Method for Detecting Multiple Roots Based on Self-Maps of the Hopf Fibration and Nullity Tolerances

The aim of this paper is to study, from a topological and geometrical point of view, the iteration map obtained by the application of iterative methods (Newton or relaxed Newton’s method) to a polynomial equation. In fact, we present a collection of algorithms that avoid the problem of overflows caused by denominators close to zero and the problem of indetermination which appears when simultaneously the numerator and denominator are equal to zero. This is solved by working with homogeneous coordinates and the iteration of self-maps of the Hopf fibration. As an application, our algorithms can be used to check the existence of multiple roots for polynomial equations as well as to give a graphical representation of the union of the basins of attraction of simple roots and the union of the basins of multiple roots. Finally, we would like to highlight that all the algorithms developed in this work have been implemented in Julia, a programming language with increasing use in the mathematical community.

Synthesis and crystal structure of a new three-dimensional Pb(II)/Ni(II) coordination polymer based on 2,6-dimethyl-pyridine-3,5-dicarboxylic acid

A new Ni(II) and Pb(II) polymer [NiPb(dpdc)]n 1 (H2dpdc = 2,6-dimethyl-pyridine-3,5-dicarboxylic acid) was synthesized under hydrothermal conditions. The compound crystallizes in the triclinic system, space group P-1 with the final R = 0.0243 and wR = 0.0579 for 3590 observed reflections with I > 2σ(I). The compound presents a three-dimensional (3 D) network extended by dpdc2- anions employing a chelating coordination mode. From the topology point of view, the structure could be described as a (3,4)-connected 3 D network with (4.8.10)(82.10)2(4.8.103.12) topology net. Infrared spectroscopy, elemental analyses, thermogravimetric analysis and the single-crystal X-ray diffraction properties of 1 are also investigated.

Poisson Dixmier-Moeglin equivalence from a topological point of view

A complex affine Poisson algebra A is said to satisfy the Poisson Dixmier-Moeglin equivalence if the Poisson cores of maximal ideals of A are precisely those Poisson prime ideals that are locally closed in the Poisson prime spectrum P.spec A and if, moreover, these Poisson prime ideals are precisely those whose extended Poisson centers are exactly the complex numbers. In this paper, we provide some topological criteria for the Poisson Dixmier-Moeglin equivalence for A in terms of the poset (P.spec A, ⊆) and the symplectic leaf or core stratification on its maximal spectrum. In particular, we prove that the Zariski topology of the Poisson prime spectrum and of each symplectic leaf or core can detect the Poisson Dixmier-Moeglin equivalence for any complex affine Poisson algebra. Moreover, we generalize the weaker version of the Poisson Dixmier-Moeglin equivalence for a complex affine Poisson algebra proved in [J. Bell, S. Launois, O. L. Sanchez and B. Moosa, Poisson algebras via model theory and differential-algebraic geometry, J. Eur. Math. Soc. (JEMS) 19 (2017), 2019–2049] to the general context of a commutative differential algebra.

Coordination Properties of Hydroxyisophthalic Acids: Topological Correlations, Synthesis, Structural Analysis, and Properties of New Complexes.

Hydroxyisophthalic acids are valuable polytopic ligands for the design of functional materials based on coordination polymers due to the variety of charges and coordination modes they possess. Herein, we describe the synthesis, thermal stability, nonlinear optical (NLO) and spectroscopic properties of five novel coordination compounds, [K2 L(H2 O)2 ], [MgL(H2 O)2 ] ⋅ 3H2 O, [CaL(H2 O)3 ], [SrL(H2 O)3 ] ⋅ H2 O, [BaL(H2 O)(H2 O)5 ], and one salt, (NH4 )2 L ⋅ 2H2 O, with 4,5,6-trihydroxyisophthalic acid (H2 L), which has not been tested in assembling crystalline coordination networks before. The peculiarities of the structural organization of the compounds were analyzed and compared with those for other hydroxyisophthalates. The coordination properties of hydroxyisophthalic acids were studied from the topological point of view, and a comparative topological analysis of coordination and H-bonded networks was performed. Structural correlations revealed in this study could be useful for the design of hydroxyisophthalate-based coordination networks, including porous metal-organic frameworks, proton conductors, and NLO materials.

Spatial Configurations and Walkability Potentials. Measuring Urban Compactness with Space Syntax

This contribution demonstrates how space syntax methods on various scale levels can be used to identify and describe the spatial features of a compact city. Firstly, the term urban compactness is discussed. A short discussion of some writings on the compact city are elaborated. As it transpired, urban compactness can best be approached from a spatial topological point of view, since compactness is a topological property. Secondly, urban compactness will be reconsidered in spatial configurative terms through the use of space syntax and urban micro scale tools. Examples from car-, pedestrian-, and public transport-based centres in Oslo and Bergen will be used throughout this contribution. Discussions of the examples in this contribution are discussed with references to other space syntax research results. As the case studies show, enhancing compact neighbourhoods with good walkability potential from a spatial perspective relies on spatial interaccessibility on all scale levels. Accessibility depends on spatial configurative compactness. Seemingly, it depends on the following complex set of sufficient conditions: a spatially integrated street network on all scale levels, short urban blocks and streets with building entrances with windows and doors on the ground floor level.

Gene Set Enrichment Analysis of Interaction Networks Weighted by Node Centrality.

Gene set enrichment analysis (GSEA) is a powerful tool to associate a disease phenotype to a group of genes/proteins. GSEA attributes a specific weight to each gene/protein in the input list that depends on a metric of choice, which is usually represented by quantitative expression data. However, expression data are not always available. Here, GSEA based on betweenness centrality of a protein–protein interaction (PPI) network is described and applied to two cases, where an expression metric is missing. First, personalized PPI networks were generated from genes displaying alterations (assessed by array comparative genomic hybridization and whole exome sequencing) in four probands bearing a 16p13.11 microdeletion in common and several other point variants. Patients showed disease phenotypes linked to neurodevelopment. All networks were assembled around a cluster of first interactors of altered genes with high betweenness centrality. All four clusters included genes known to be involved in neurodevelopmental disorders with different centrality. Moreover, the GSEA results pointed out to the evidence of “cell cycle” among enriched pathways. Second, a large interaction network obtained by merging proteomics studies on three neurodegenerative disorders was analyzed from the topological point of view. We observed that most central proteins are often linked to Parkinson’s disease. The selection of these proteins improved the specificity of GSEA, with “Metabolism of amino acids and derivatives” and “Cellular response to stress or external stimuli” as top-ranked enriched pathways. In conclusion, betweenness centrality revealed to be a suitable metric for GSEA. Thus, centrality-based GSEA represents an opportunity for precision medicine and network medicine.

Topological Quantum Computing and 3-Manifolds

In this paper, we will present some ideas to use 3D topology for quantum computing. Topological quantum computing in the usual sense works with an encoding of information as knotted quantum states of topological phases of matter, thus being locked into topology to prevent decay. Today, the basic structure is a 2D system to realize anyons with braiding operations. From the topological point of view, we have to deal with surface topology. However, usual materials are 3D objects. Possible topologies for these objects can be more complex than surfaces. From the topological point of view, Thurston’s geometrization theorem gives the main description of 3-dimensional manifolds. Here, complements of knots do play a prominent role and are in principle the main parts to understand 3-manifold topology. For that purpose, we will construct a quantum system on the complements of a knot in the 3-sphere. The whole system depends strongly on the topology of this complement, which is determined by non-contractible, closed curves. Every curve gives a contribution to the quantum states by a phase (Berry phase). Therefore, the quantum states can be manipulated by using the knot group (fundamental group of the knot complement). The universality of these operations was already showed by M. Planat et al.

Geometrical and topological study of the Kosterlitz–Thouless phase transition in the XY model in two dimensions

Phase transitions do not necessarily correspond to a symmetry-breaking phenomenon. This is the case of the Kosterlitz–Thouless (KT) phase transition in a two-dimensional classical XY model, a typical example of a transition stemming from a deeper phenomenon than a symmetry-breaking. Actually, the KT transition is a paradigmatic example of the successful application of topological concepts to the study of phase transition phenomena in the absence of an order parameter. Topology conceptually enters through the meaning of defects in real space. In the present work, the same kind of KT phase transition in a two-dimensional classical XY model is tackled by resorting again to a topological viewpoint, however focussed on the energy level sets in phase space rather than on topological defects in real space. Also from this point of view, the origin of the KT transition can be attributed to a topological phenomenon. In fact, the transition is detected through peculiar geometrical changes of the energy level sets which, after a theorem in differential topology, are direct probes of topological changes of these level sets.

Soft topological hyperstructure

Molodtsov proposed the theory of soft sets which can be considered as a recent mathematical tool to deal with uncertainties. The main purpose of this paper is to give the definition of soft topological hypergrupoid by examining the concept of hypergrupoid which is one of the hyperystructures with soft set theory from the topological point of view. Also, the relation between soft topological hypergroupoids and soft hypergroupoids is examined and some theoretical results are obtained. By introducing the concept of soft good topological homomorphism, the category of soft topological hypergrupoids is constructed. At last, the definition of soft topological subhypergrupoid is presented and some related properties are studied.

A neutrosophic folding and retraction on a single-valued neutrosophic graph

The main aim of this article is to present neutrosophic folding and neutrosophic retractions on a single-valued neutrosophic graph Ğ from the viewpoint of geometry and topology. For this reason, we use a sequence of neutrosophic transformations on Ğ to obtain a new single-valued neutrosophic graph G ˇ 1 which contains different parameters under new conditions. We deduce the isometric neutrosophic folding on neutrosophic spheres and neutrosophic torii. Also, we determine the relationship between the limit neutrosophic folding and the limit of neutrosophic retraction on Ğ. Theorems regulating these relations are attained.

Geometry and integrability of quadratic systems with invariant hyperbolas

Let QSH be the family of non-degenerate planar quadratic differential systems possessing an invariant hyperbola. We study this class from the viewpoint of integrability. This is a rich family with a variety of integrable systems with either polynomial, rational, Darboux or more general Liouvillian first integrals as well as nonintegrable systems. We are interested in studying the integrable systems in this family from the topological, dynamical and algebraic geometric viewpoints. In this work we perform this study for three of the normal forms of QSH, construct their topological bifurcation diagrams as well as the bifurcation diagrams of their configurations of invariant hyperbolas and lines and point out the relationship between them. We show that all systems in one of the three families have a rational first integral. For another one of the three families, we give a global answer to the problem of Poincare by producing a geometric necessary and sufficient condition for a system in this family to have a rational first integral. Our analysis led us to raise some questions in the last section, relating the geometry of the invariant algebraic curves (lines and hyperbolas) in the systems and the expression of the corresponding integrating factors.

Synthesis and crystal structure of a new three-dimensional cd(II) coordination polymer based on 5-(3′,4′-dicarboxylphenoxy)-isophthalic acid

Under hydrothermal conditions a new Cd(II) polymer [Cd(dic)]n 1 (H4dic = 5-(3′,4′-Dicarboxylphenoxy)-isophthalic acid) was synthesized. The compound crystallizes in the space group C2 and the Monoclinic system, with b = 8.0043(9), a = 18.523(2), c = 16.5238(19) Å, β = 118.742(2)°,α = 90°, γ = 90°, Z = 1, V = 2148.1(4) Å3, Mr = 2728.38, Dc = 2.109 Mg/m3, F(000) = 1332, μ = 2.059 mm−1, the final wR = 0.0801 and R = 0.0317 for 3598 with I > 2σ(I) observed reflections. A three-dimensional (3D) network of the compound presents extended by both btc4- anions for a chelating coordination mode in which the dic4- anions employ. From the topology point of view, the structure with (4.62)(4.65.84) topology net could be described as a (3,5)-connected 3D network. Elemental analyses, Infrared spectroscopy, powder X-ray diffraction properties of 1 are also investigated and thermogravimetric analysis.

Synthesis and crystal structure of a new three-dimensional Cd(II) coordination polymer based on 4,4′-bipyridine and 3,3′,4,5′-biphenyl tetracarboxylic acid

A new Cd(II) polymer [Cd(btc)(4,4′-bipyridine)] n 1 (H4btc = 3,3′,4,5′-biphenyl tetracarboxylic acid) was synthesized under hydrothermal conditions. The compound crystallizes with b = 20.223(3), c = 13.478(2) Å, a = 11.9134(18), β = 110.486(2)°, α = 90°, γ = 90°, V = 3041.8(8) Å3, Z = 4, in the monoclinic system, space group P2(1)/c, μ = 1.468 mm−1, M r = 690.33, F(000) = 1600, D c = 1.771 mg/m3, the final R = 0.0546 and wR = 0.1480 for 5647 observed reflections with I > 2σ(I). The compound presents a three-dimensional (3D) network extended by both 4,4′-bipyridine ligands and btc4- anions in which the btc4- anions employ a chelating coordination mode. The structure could be described as a (3,4,5)-connected 3D network with (42.63.74.8)(42.63.7)(6.72) topology net from the topology point of view. Elemental analyses, infrared spectroscopy, thermogravimetric analysis, and powder X-ray diffraction properties of 1 are also investigated.

Implications of Metastable Nicks and Nicked Holliday Junctions in Processing Joint Molecules in Mitosis and Meiosis.

Joint molecules (JMs) are intermediates of homologous recombination (HR). JMs rejoin sister or homolog chromosomes and must be removed timely to allow segregation in anaphase. Current models pinpoint Holliday junctions (HJs) as a central JM. The canonical HJ (cHJ) is a four-way DNA that needs specialized nucleases, a.k.a. resolvases, to resolve into two DNA molecules. Alternatively, a helicase–topoisomerase complex can deal with pairs of cHJs in the dissolution pathway. Aside from cHJs, HJs with a nick at the junction (nicked HJ; nHJ) can be found in vivo and are extremely good substrates for resolvases in vitro. Despite these findings, nHJs have been neglected as intermediates in HR models. Here, I present a conceptual study on the implications of nicks and nHJs in the final steps of HR. I address this from a biophysical, biochemical, topological, and genetic point of view. My conclusion is that they ease the elimination of JMs while giving genetic directionality to the final products. Additionally, I present an alternative view of the dissolution pathway since the nHJ that results from the second end capture predicts a cross-join isomerization. Finally, I propose that this isomerization nicely explains the strict crossover preference observed in synaptonemal-stabilized JMs in meiosis.

Topological Strain-Induced Regioselective Linker Elimination in a Chiral Zr(IV)-Based Metal-Organic Framework

Summary Zr-carboxylate metal-organic frameworks (MOFs) are structurally robust materials, in part due to their strong coordination bonds. The regioselective Zr–O bond cleavage and formation between 3D architectures are thus challenging and are heretofore unexplored. In this work, by introducing highly flexible 18-crown-6-ether functionalities into a homochiral Zr-MOF, we report an unprecedented topology transition in which a 4,10-connected framework undergoes a rapid solid-state transition into a thermodynamically more stable 4,8-connected analog by a regioselective-linker-elimination under ambient conditions. The transition process was unambiguously unraveled by single-crystal and powder X-ray diffraction studies, and we proposed a possible transition mechanism based on various control experiments and theoretical calculations. The excellent chemical stability and substantially expanded porosity and pore apertures endowed the transformed chiral MOF with an exceptional capacity for the enantioadsorptive and solid-phase extractive separation of the racemic drug molecule of lansoprazole with 98% ee and 93% ee, respectively.

Anthropology of a city: organization, communication, information

The article is concerned with the anthropology of a city. As an instrument of conceptualization, we used a three-dimensional scheme: organization — communication — information, which allows giving the urban interpre-tation for each of the coordinates, and then, using the two-dimensional projections (organization-information, or-ganization-communication, information-communication), to develop a concept of the three-dimensional image of city. It has been proposed to distinguish three types of organizational structures when considering the city as an organization: hierarchical, algorithmic, and relational. The necessity of curbing the growth of the city and solidat-ing it in the context of communication has produced a number of projections. One of them, "A compact city or city of short distances" has been considered. The concept of a compact city is based on the cost-effective public transport system, and it encourages pedestrian traffic and cycling. When considering the city as an organization, the main focus lies on its structure. From a topological point of view, it has been proposed to distinguish between three types of such structures: hierarchical, algorithmic, and relational. A hierarchical structure represents a tree of power hierarchy, but its content is not necessarily associated with consistent delegation of authority power (from top to bottom) or gradual accumulation of information (from the bottom up). Thus, a library catalogue, as well as other classification means, is arranged on the principle of hierarchy. An algorithmic structure is a scheme of production process which consists of a set of sequential operations. Its mathematical model appears as a net-work diagram. Typical examples of such structures include an algorithm of construction of a building, from founda-tion to roof, a conveyed assembly of complex technical devices, etc. In relation to a city, this structure can be filled with different content. Thus, carrying out of repair and maintenance in one of the city networks often requires not only formal coordination, but also a network schedule of joint work with other community services (water, electric-ity grid, heating, communication lines, etc.). A relational structure is a group of objects of arbitrary nature, usually of the same type, singled out from the total quantity of objects on the basis of any common feature. A complex of educational or medical institutions, trade companies, domestic services, etc. could be an example of such group within the organizational structure of a city.

INFINITARY COMMUTATIVITY AND ABELIANIZATION IN FUNDAMENTAL GROUPS

AbstractInfinite product operations are at the forefront of the study of homotopy groups of Peano continua and other locally path-connected spaces. In this paper, we define what it means for a space X to have infinitely commutative $\pi _1$ -operations at a point $x\in X$ . Using a characterization in terms of the Specker group, we identify several natural situations in which this property arises. Maintaining a topological viewpoint, we define the transfinite abelianization of a fundamental group at any set of points $A\subseteq X$ in a way that refines and extends previous work on the subject.

Some global, analytical, and topological properties of regular black holes

In this work we study regular black holes from a global perspective looking for evading some of the well-known singularity theorems by using their "reverses". Then, model geometries for the slices of typical spherically symmetric, (locally) static four dimensional regular black hole solutions are described from both an analytical and a topological point of view. While the finiteness of both the scalar and Kretchmann curvature of the slices around the regular center determines the geometry of the core, the positive answer to the Poincar\'e conjecture assures that, under two assumptions, its topology is that of a three-sphere. However, in general, the cores are shown to be $S^3$, $H^3$, $\mathbb{R}\times S^1$ or $S^1\times S^2$, depending whether a de Sitter, anti de Sitter, Nariai or Bertotti-Robinson geometries are employed to describe the slices at the regular center. Then, a description of the aforementioned slices in terms of Seifert fibre spaces is given in order to show that the Euler characteristic of the bundle can be used to track the transition between the core of the regular black hole and the rest of the slices in most of the cases considered in the literature. After Geroch and Tipler's theorems are employed to study the consequences of topology change on regular black hole spacetimes, we show that Borde and Vilenkin's singularity theorem is used to restrict their possible types. We end by noting that Nariai cores can be safely used to construct regular black holes without topology changes.

Random covering sets in metric space with exponentially mixing property

Let {B(ξn,rn)}n≥1 be a sequence of random balls whose centers {ξn}n≥1 is a stationary process, and {rn}n≥1 is a sequence of positive numbers decreasing to 0. Our object is the random covering set E=lim supn→∞B(ξn,rn), that is, the points covered by B(ξn,rn) infinitely often. The sizes of E are investigated from the viewpoint of measure, dimension and topology.