Topological Point Of View Research Articles

Most Iterations of Projections Converge

Consider three closed linear subspaces C1,C2,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_1, C_2,$$\\end{document} and C3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_3$$\\end{document} of a Hilbert space H and the orthogonal projections P1,P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_1, P_2$$\\end{document} and P3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_3$$\\end{document} onto them. Halperin showed that a point in C1∩C2∩C3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_1\\cap C_2 \\cap C_3$$\\end{document} can be found by iteratively projecting any point x0∈H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_0 \\in H$$\\end{document} onto all the sets in a periodic fashion. The limit point is then the projection of x0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_0$$\\end{document} onto C1∩C2∩C3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_1\\cap C_2 \\cap C_3$$\\end{document}. Nevertheless, a non-periodic projection order may lead to a non-convergent projection series, as shown by Kopecká, Müller, and Paszkiewicz. This raises the question how many projection orders in {1,2,3}N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{1,2,3\\}^{\\mathbb {N}}$$\\end{document} are “well behaved” in the sense that they lead to a convergent projection series. Melo, da Cruz Neto, and de Brito provided a necessary and sufficient condition under which the projection series converges and showed that the “well behaved” projection orders form a large subset in the sense of having full product measure. We show that also from a topological viewpoint the set of “well behaved” projection orders is a large subset: it contains a dense Gδ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_\\delta $$\\end{document} subset with respect to the product topology. Furthermore, we analyze why the proof of the measure theoretic case cannot be directly adapted to the topological setting.

The simplified energy landscape of the φ 4 model and the phase transition

The on-lattice φ 4 model is a paradigmatic example of a continuous real-variable model undergoing a continuous symmetry-breaking phase transition (SBPT). Here, we study the Z2 -symmetric mean-field case without the quadratic term in the local potential. We show that the Z2 -SBPT is not affected by the quadratic term and that the potential energy landscape is greatly simplified from a geometric–topological viewpoint. In particular, only three critical points exist to confront, with a number growing as eN (N is the number of degrees of freedom) of the model with a negative quadratic term. We focus on the properties of the equipotential surfaces with the aim to deepen the link between SBPTs and the essential properties of a potential that is capable of entailing them. The results are interpreted in view of of some recent achievements regarding rigorous necessary and sufficient conditions for a Z2 -SBPT.

PyMulSim: a method for computing node similarities between multilayer networks via graph isomorphism networks

BackgroundIn bioinformatics, interactions are modelled as networks, based on graph models. Generally, these support a single-layer structure which incorporates a specific entity (i.e., node) and only one type of link (i.e., edge). However, real-world biological systems consisting of biological objects belonging to heterogeneous entities, and these operate and influence each other in multiple contexts, simultaneously. Usually, node similarities are investigated to assess the relatedness between biological objects in a network of interest, and node embeddings are widely used for studying novel interaction from a topological point of view. About that, the state-of-the-art presents several methods for evaluating the node similarity inside a given network, but methodologies able to evaluate similarities between pairs of nodes belonging to different networks are missing. The latter are crucial for studies that relate different biological networks, e.g., for Network Alignment or to evaluate the possible evolution of the interactions of a little-known network on the basis of a well-known one. Existing methods are ineffective in evaluating nodes outside their structure, even more so in the context of multilayer networks, in which the topic still exploits approaches adapted from static networks. In this paper, we presented pyMulSim, a novel method for computing the pairwise similarities between nodes belonging to different multilayer networks. It uses a Graph Isomorphism Network (GIN) for the representative learning of node features, that uses for processing the embeddings and computing the similarities between the pairs of nodes of different multilayer networks.ResultsOur experimentation investigated the performance of our method. Results show that our method effectively evaluates the similarities between the biological objects of a source multilayer network to a target one, based on the analysis of the node embeddings. Results have been also assessed for different noise levels, also through statistical significance analyses properly performed for this purpose.ConclusionsPyMulSim is a novel method for computing the pairwise similarities between nodes belonging to different multilayer networks, by using a GIN for learning node embeddings. It has been evaluated both in terms of performance and validity, reporting a high degree of reliability.

Identifying critical weak points of power-gas integrated energy system based on complex network theory

Prompt and precise identification of critical weak points is essential in integrated energy systems (IES) to mitigate system reliability issues and effectively prevent cascading failures. This study proposes methods to identify critical weak points in the power-gas integrated system (PGIS) based on complex network theory for this task. Firstly, we improve the traditional betweenness indices in complex network theory by integrating power output of energy supply equipment as weighted factors to capture diverse types of energy flows, and incorporating the efficiency coefficient to account for energy losses via the PGIS coupling equipment. By integrating these refinements, our approach could further identify some points that are not weak from the viewpoint of topology but are crucial for energy transmission. Secondly, we propose an approximate algorithm for shortest-paths calculations (AA-SPC), which preserves information of total node and some specific topological, and simplifies the calculation of target shortest path, aiming at increasing the efficiency of obtaining path information. Additionally, vulnerability assessment indicators, including system fragmentation and network betweenness loss, measure PGIS vulnerability under different attack strategies. The aforementioned methods are applied to a self-built virtual European PGIS based on the open-source database SciGRID, successfully identifying critical weak points overlooked by traditional betweenness indices.

Enhanced Catalytic Activity of a Copper(II) Metal-Organic Framework Constructed via Semireversible Single-Crystal-to-Single-Crystal Dehydration.

Herein, we present a copper(II) metal-organic framework, [Cu2(btec)(OH2)4]·2H2O (1) [(btec)4- = 1,2,4,5-benzenetetracarboxylate], that undergoes single-crystal-to-single-crystal transformations into two anhydrous phases 2' and 2″ with the chemical formula [Cu2(btec)], triggered by two-step dehydration at 403 and 433 K, respectively. After immersion in water for 3 days at room temperature, 2' transformed into [Cu2(btec)(OH2)] (3), while both 2' and 2″ took 1 week to revert to 1. Dynamic vapor sorption studies validated water-induced reversible structural transformations at 70% relative humidity (RH). According to single-crystal X-ray diffraction (SC-XRD), the local coordination geometry of the Cu2+ ion in 2' changed from a saturated octahedron to a coordinatively unsaturated square-based pyramid in 3, manifested by changes in color and dimensionality. From a topological point of view, all of the scaffolds show a binodal (3,6)-connected kgd topology with the point symbol {43}2{46}. In addition, the materials were thoroughly characterized using routine spectroscopic data and various analytical techniques. The catalytic activity of the microporous materials in the liquid-phase oxidation of styrene in acetonitrile, using 30% (wt) H2O2 as the oxidant, was investigated. The excellent performance of the monohydrous phase 3 was shown to be superior to the pristine framework and the anhydrous counterparts, as evidenced by a good turnover number (TON) and turnover frequency (TOF) = 82.6 and 21.0 h-1, respectively. Within 4 h, the substrates were catalytically oxidized to the desired products with up to 67% conversion and 100% benzaldehyde selectivity. It is worth noting that the accessible active metal sites and higher surface area enhanced the catalytic properties of 3. Furthermore, the maintenance of catalytic efficiency over five cycles and reusability are illustrated and discussed in terms of the structural differences of the microporous frameworks. Thus, a preliminary reaction mechanism for the selective oxidation of styrene is proposed. This study not only provides a fascinating example of MOF chromism achieved by thermal activation and rehydration but also sheds some light on the relationship between pore-surface- or metal-engineered sites in MOFs and their heterogeneous catalytic performances.

Nonisolated High Step-Up DC–DC Converters: Comparative Review and Metrics Applicability

Due to the extensive role of non-isolated high step-up DC-DC converters (NHSDC)s in industrial applications and academic research, many of these pulse width modulation converters have been presented in recent years. For each of these NHSDCs, some claims are introduced to verify its capabilities and features, which has been investigated in some review papers with different frameworks. Dissimilar to previous review papers, which have focused on the classification and derivation of voltage boosting techniques, this paper aims to evaluate the converters from various topological and operational points of view, and determine the superiority of each technique and converter according to applications. Some of these metrics are voltage gain, stresses, ripple, cost, power density, weight, size, control complexity and components count which lead to a comprehensive comparative study. Then, as the main purpose of this paper, effectiveness of these metrics is assessed to show how well they can lead us to the fair comparison results. Moreover, some new figures of merit are proposed in this paper to provide a helpful guideline in power electronic converters comparison studies. Finally, the feasibility discussion of single- and multi-objective figures of merit is followed by general practical conclusion and outlook about the NHSDC structures.

The equivalence between CPCP and strong regularity under Krein-Milman property

We obtain a result in the spirit of the well-known W. Schachermayer and H. P. Rosenthal research about the equivalence between Radon-Nikodym and Krein-Milman properties, by showing that, for closed, bounded and convex subsets C of a separable Banach space, under Krein-Milman property for C, one has the equivalence between convex point of continuity property and strong regularity both defined for every locally convex topology on C, containing the weak topology on C. Then, under Krein-Milman property, not only the classical convex point of continuity property and strong regularity are equivalent, but also when they are defined for an arbitrary locally convex topology containing the weak topology. We also show that while the unit ball B of c0 fails convex point of continuity property and strong regularity (both defined for the weak topology), there is a locally convex topology τ on B, containing the weak topology on B, such that B still fails convex point of continuity property for τ, but B surprisingly enjoys strong regularity for τ-open sets. Moreover, B satisfies the diameter two property for the topology τ, that is, every nonempty τ-open subset of B has diameter two even though every τ-open subset of B contains convex combinations of relative τ-open subsets with arbitrarily small diameter, that is, B fails the strong diameter two property for the topology τ. This stresses the known extreme differences up to now between those diameter two properties from a topological point of view.

Enhancement of short/medium-range order and thermal conductivity in ultrahard sp3 amorphous carbon by C70 precursor

As an advanced amorphous material, sp3 amorphous carbon exhibits exceptional mechanical, thermal and optical properties, but it cannot be synthesized by using traditional processes such as fast cooling liquid carbon and an efficient strategy to tune its structure and properties is thus lacking. Here we show that the structures and physical properties of sp3 amorphous carbon can be modified by changing the concentration of carbon pentagons and hexagons in the fullerene precursor from the topological transition point of view. A highly transparent, nearly pure sp3−hybridized bulk amorphous carbon, which inherits more hexagonal-diamond structural feature, was synthesized from C70 at high pressure and high temperature. This amorphous carbon shows more hexagonal-diamond-like clusters, stronger short/medium-range structural order, and significantly enhanced thermal conductivity (36.3 ± 2.2 W m−1 K−1) and higher hardness (109.8 ± 5.6 GPa) compared to that synthesized from C60. Our work thus provides a valid strategy to modify the microstructure of amorphous solids for desirable properties.

Construction of A sqc6 Topology Ni‐Gd Framework Employing 2,2’‐Phosphinico‐dibenzoate and Oxalate: Structure and Magnetic Properties

AbstractA 3d–4f heterometallic compound, namely [Ni6GdL4(ox)(H2O)12]⋅[NO3]⋅14.5H2O (1) (L=2,2’‐phosphinico‐dibenzoate and ox=oxalate), was obtained by self‐assembly of nickel nitrate, gadolinium nitrate, oxalic acid, and 2,2’‐phosphinico‐dibenzoic acid (H3L). Such compound has been structurally characterized via single‐crystal X‐ray diffraction analysis and infrared spectroscopy. This compound features a basic unit of {Ni4Gd} clusters and 2,2’‐phosphinico‐dibenzoate ligands. Each unit is connected by another Ni2+ through the axial coordination of benzoate oxygen to form a layer. Meanwhile, each oxalate ligand bridges the expanded Ni4Gd‐Ni4 net, constructing a 3‐D framework. From the viewpoint of network topology, this compound can be simplified as a sqc6 topology. Magnetic analysis shows ferromagnetic interactions between the Ni(II) and Gd(III) centers in Ni4Gd nodes and antiferromagnetic coupling with oxalate bridges in this compound.

Classical Notions and Problems in Thurston Geometries

Of the Thurston geometries, those with constant curvature geometries (Euclidean $ E^3$, hyperbolic $ H^3$, spherical $ S^3$) have been extensively studied, but the other five geometries, $ H^2\times R$, $ S^2\times R$, $Nil$, $\widetilde{SL_2 R}$, $Sol$ have been thoroughly studied only from a differential geometry and topological point of view. However, classical concepts highlighting the beauty and underlying structure of these geometries -- such as geodesic curves and spheres, the lattices, the geodesic triangles and their surfaces, their interior sum of angles and similar statements to those known in constant curvature geometries -- can be formulated. These have not been the focus of attention. In this survey, we summarize our results on this topic and pose additional open questions.

Analysis of Connectome Graphs Based on Boundary Scale.

The purpose of this work is to advance in the computational study of connectome graphs from a topological point of view. Specifically, starting from a sequence of hypergraphs associated to a brain graph (obtained using the Boundary Scale model, BS2), we analyze the resulting scale-space representation using classical topological features, such as Betti numbers and average node and edge degrees. In this way, the topological information that can be extracted from the original graph is substantially enriched, thus providing an insightful description of the graph from a clinical perspective. To assess the qualitative and quantitative topological information gain of the BS2 model, we carried out an empirical analysis of neuroimaging data using a dataset that contains the connectomes of 96 healthy subjects, 52 women and 44 men, generated from MRI scans in the Human Connectome Project. The results obtained shed light on the differences between these two classes of subjects in terms of neural connectivity.

Identifying s-wave pairing symmetry in single-layer FeSe from topologically trivial edge states

Determining the pairing symmetry of single-layer FeSe on SrTiO3 is the key to understanding the enhanced pairing mechanism. It also guides the search for superconductors with high transition temperatures. Despite considerable efforts, it remains controversial whether the symmetry is the sign-preserving s- or the sign-changing s±-wave. Here, we investigate the pairing symmetry of single-layer FeSe from a topological point of view. Using low-temperature scanning tunneling microscopy/spectroscopy, we systematically characterize the superconducting states at edges and corners of single-layer FeSe. The tunneling spectra collected at edges and corners show a full energy gap and a substantial dip, respectively, suggesting the absence of topologically non-trivial edge and corner modes. According to our theoretical calculations, these spectroscopic features can be considered as strong evidence for the sign-preserving s-wave pairing in single-layer FeSe.

Multiplicative lattices: Maximal implies prime and related questions

The goal of this paper is to deepen the study of multiplicative lattices in the sense of Facchini, Finocchiaro and Janelidze. We provide a sort of Prime Ideal Principle that guarantees that maximal implies prime in a variety of cases (among them the case of commutative rings with identity). This result is used to study the lattice theoretic counterpart of multiplicative closed sets, that of [Formula: see text]-systems. The notion of [Formula: see text]-system is also studied from the topological point of view.

Magnon Hall effect in antiferromagnetic lattices

Topology applied to condensed matter is an important area of research and technology, and topological magnetic excitations have recently become an active field of study. This paper presents a general discussion of magnon Hall transport in two-dimensional antiferromagnets. Although the Chern number is zero for a collinear antiferromagnet, we offer a general discussion that can be used in the more general case. First, we study the Union Jack lattice, where an effective time-reversal symmetry is broken, making the system display the magnon Hall effect. Then, we investigate the brick-wall lattice where such symmetry is present. Consequently, we have a phenomenon similar to the quantum spin Hall effect in electronic systems. Both lattices have not yet been studied from the topological point of view. The coexistence of opposite spin polarization in an antiferromagnet resembles the electron spin in various transport phenomena. We study magnon transport in the lattices mentioned above with Dzyaloshinskii-Moriya interaction and easy-axis single-ion anisotropy. We calculate the Berry curvature from the eigenvalues of the Hamiltonian. From that, we plot the spin Hall and thermal Hall conductivities, as well as the spin Nernst coefficient, as functions of the temperature. In the Union Jack lattice, we treat the effect of anharmonic interactions using a mean-field spin wave theory where the Hamiltonian becomes implicitly temperature-dependent. We determine self-consistently the renormalized dispersion and the staggered magnetization as a function of temperature. Our calculations can be applied to other antiferromagnetic lattices.

The language of knowledge spaces in pre-topology1

The theory of knowledge spaces (KST) which is regarded as a mathematical framework for the assessment of knowledge and advices for further learning. Now the theory of knowledge spaces has many applications in education. From the topological point of view, we discuss the language of the theory of knowledge spaces by the axioms of separation and the accumulation points of pre-topology respectively, which establishes some relations between topological spaces and knowledge spaces; in particular, we show that the language of the regularity of pre-topology in knowledge spaces and give a characterization for knowledge spaces by inner fringe of knowledge states. Moreover, we study the relations of Alexandroff spaces and quasi ordinal spaces; then we give an application of the density of pre-topological spaces in primary items for knowledge spaces, which shows that one person in order to master an item, she or he must master some necessary items. In particular, we give a characterization of a skill multimap such that the delineated knowledge structure is a knowledge space, which gives an answer to a problem in [14] or [18] whenever each item with finitely many competencies; further, we give an algorithm to find the set of atom primary items for any finite knowledge space.

Bulk-boundary thermodynamic equivalence: a topology viewpoint

Setting the cosmological constant to be dynamical, we study the bulk and boundary thermodynamics of charged Anti-de Sitter black holes. We develop mass/energy formulas in terms of thermodynamic state functions for the extended thermodynamics, mixed thermodynamics, and boundary conformal field theory thermodynamics. We employ the residue method to study the topological properties of the phase transitions. Our analysis reveals that the bulk and boundary thermodynamics are topologically equivalent for both criticalities and first-order phase transitions in the canonical ensembles, as well as for the Hawking-Page(-like) phase transitions in the grand canonical ensembles. Additionally, those three kinds of phase transitions are shown to be distinguished by their unique topological charges. Our results exemplify the gravity-gauge duality in terms of topology.

The Bloch–Nevanlinna Problem Under Lineability

The original version of the celebrated Bloch–Nevanlinna problem asked whether there exists or not a holomorphic function in the unit disc of bounded characteristic whose derivative is not of bounded characteristic. This problem has been solved in the affirmative by a number of mathematicians. Starting from a result on topological genericity of this class of functions due to Hahn, our work intends—under an algebraic as well as a topological point of view—to contribute to this topic. Specifically, the family of solutions of the Bloch–Nevanlinna problem is proved to be residual in appropriate topological spaces, and it contains, except for the zero function, dense maximal dimensional vector subspaces, large linear algebras and infinite-dimensional Banach spaces.

Almost all sets of nonnegative integers and their small perturbations are not sumsets

Fix α ∈ ( 0 , 1 / 3 ) \alpha \in (0,1/3) . We show that, from a topological point of view, almost all sets A ⊆ N A\subseteq \mathbb {N} have the property that, if A ′ = A A^\prime =A for all but o ( n α ) o(n^{\alpha }) elements, then A ′ A^\prime is not a nontrivial sumset B + C B+C . In particular, almost all A A are totally irreducible. In addition, we prove that the measure analogue holds with α = 1 \alpha =1 .

Heidegger’s Question of Being: the Unity of Topos and Logos

In this article, I elucidate the significance of Heidegger’s ‘question of being’ from a topological point of view by explaining the relationship between his thought of place and language. After exploring various hermeneutic strategies of reading Heidegger’s oeuvre, I turn to Richard Capobianco’s interpretation of Heidegger and critically engage with his idea of the experience of being itself as the ‘luminous self-showing of logos’. In doing so, I explain the later turn from ‘truth’ to ‘place’ and articulate why logos needs to be conceived as the gathering site of the presencing of being. In arguing against the primacy of passive receptivity and active projection, I put forward the primacy of language as the topos and logos of being. In returning to the Capobianco-Sheehan debate, I conclude by explaining why Heidegger’s place-related notions cannot be thought metaphorically.

Topological surface state: Universal catalytic descriptor in topological catalysis

Catalytic descriptors of electrocatalysts such as d-band center are the important indicators that connect the physiochemical interaction and catalytic performance and have been used for rational design of the catalysts and performance optimizations. However, these descriptors developed in traditional electrocatalysts are invalid in recent emerged topological catalysis. Here, we identify topological surface state (TSS) density at the Fermi level can serve as the universal catalytic descriptor in the family of metal diborides, the materials hosting fantasying topological nodal-net states. The catalytic activity for hydrogen evolution reaction (HER) is linearly correlated with the projected TSSs on corresponding surfaces, which are independent on the material types. The findings are further proven by the weakened HER performance under topological phase transition where the TSS density at Fermi level is reduced, and the revival of d-band center once all the TSSs are removed. Our work will open a new route for developing high-performance catalysts from the quantum topological point of view.