Topological Entities Research Articles

Computing the cut locus, Voronoi diagram, and signed distance function of polygons

This paper presents a new method for the computation of the generalized Voronoi diagram of planar polygons. First, we show that the vertices of the cut locus can be computed efficiently. This is achieved by enumerating the tripoints of the polygon, a superset of the cut locus vertices. This is the set of all points that are of equal distance to three distinct topological entities. Then our algorithm identifies and connects the appropriate tripoints to form the cut locus vertex connectivity graph, where edges define linear or parabolic boundary segments between the Voronoi regions, resulting in the generalized Voronoi diagram. Our proposed method is validated on complex polygon soups. We apply the algorithm to represent the exact signed distance function of the polygon by augmenting the Voronoi regions with linear and radial functions, calculating the cut locus both inside and outside.

A parametric and feature-based CAD dataset to support human-computer interaction for advanced 3D shape learning

3D shape learning is an important research topic in computer vision, in which the datasets play a critical role. However, most of the existing 3D datasets use voxels, point clouds, mesh, and B-rep, which are not parametric and feature-based. Thus they can not support the generation of real-world engineering computer-aided design (CAD) models with complicated shape features. Furthermore, they are based on 3D geometry results without human-computer interaction (HCI) history. This work is the first to provide a full parametric and feature-based CAD dataset with a selection mechanism to support HCI in 3D learning. First, unlike existing datasets, mainly composed of simple features (typical sketch and extrude), we devise complicated engineering features, such as fillet, chamfer, mirror, pocket, groove, and revolve. Second, different from the monotonous combination of features, we invent a select mechanism to mimic how human focuses on and selects a particular topological entity. The proposed mechanism establishes the relationships among complicated engineering features, which fully express the design intention and design knowledge of human CAD engineers. Therefore, it can process advanced 3D features for real-world engineering shapes. The experiments show that the proposed dataset outperforms existing CAD datasets in both reconstruction and generation tasks. In quantitative experiment, the proposed dataset demonstrates better prediction accuracy than other parametric datasets. Furthermore, CAD models generated from the proposed dataset comply with semantics of the human CAD engineers and can be edited and redesigned via mainstream industrial CAD software.

Emergence of polar skyrmions in 2D Janus CrInX3 (X=Se, Te) magnets

In the realm of multiferroicity in 2D magnets, whether magnetic and polar skyrmions can coexist within a single topological entity has emerged as an important question. Here, we study Janus 2D magnets CrInX3 (X=Se, Te) for a comprehensive investigation of the magnetic ground state, magnetic excited state, and corresponding ferroelectric polarization by first-principles electronic structure calculations and Monte Carlo simulations. Specifically, we have thoroughly elucidated the magnetic exchange mechanisms, and have fully exemplified the magnetic field dependence of the magnon spectrum. More importantly, our study reveals a previously unrecognized, remarkably large spin-spiral-induced ferroelectric polarization (up to 194.9 μC/m2) in both compounds. We propose an approach to identify polar skyrmions within magnetic skyrmions, based on the observed direct correlation between spin texture and polarization density. Elucidating this correlation not only deepens our understanding of magnetic skyrmions but also paves the way for innovative research in the realm of multiferroic skyrmions.

Electric-field-induced multiferroic topological solitons.

Topologically protected spin whirls in ferromagnets are foreseen as the cart-horse of solitonic information technologies. Nevertheless, the future of skyrmionics may rely on antiferromagnets due to their immunity to dipolar fields, straight motion along the driving force and ultrafast dynamics. While complex topological objects were recently discovered in intrinsic antiferromagnets, mastering their nucleation, stabilization and manipulation with energy-efficient means remains an outstanding challenge. Designing topological polar states in magnetoelectric antiferromagnetic multiferroics would allow one to electrically write, detect and erase topological antiferromagnetic entities. Here we stabilize ferroelectric centre states using a radial electric field in multiferroic BiFeO3 thin films. We show that such polar textures contain flux closures of antiferromagnetic spin cycloids, with distinct antiferromagnetic entities at their cores depending on the electric field polarity. By tuning the epitaxial strain, quadrants of canted antiferromagnetic domains can also be electrically designed. These results open the path to reconfigurable topological states in multiferroic antiferromagnets.

Modification and completion of geological structure knowledge graph based on pattern matching

As a knowledge representation method, knowledge graph is widely used in intelligent question answering systems and recommendation systems. At present, the research on knowledge graph mainly focuses on information query and retrieval based on knowledge graph. In some domain knowledge graphs, specific subgraph structures (patterns) have specific physical meanings. Aiming at this problem, this paper proposes a method and framework of knowledge graph pattern mining based on gat. Firstly, the patterns with specific physical meaning were transformed into subgraph structures containing topological structures and entity attributes. Secondly, the subgraph structure of the pattern is regarded as the query graph, and the knowledge graph is regarded as the data graph, so that the problem is transformed into an approximate subgraph matching problem. Then, the improved relational graph attention network is used to fuse the adaptive edge deletion mechanism to realize the approximate subgraph matching of subgraph structure and attribute, so as to obtain the best matching subgraph. The proposed method is trained in an end-to-end manner. The approximate subgraph matching is realized on the existing data set, and the research work of key pattern mining of complex geological structure knowledge graph is carried out.

Water-Wave Vortices and Skyrmions.

Topological wave structures-phase vortices, skyrmions, merons, etc.-are attracting enormous attention in a variety of quantum and classical wave fields. Surprisingly, these structures have never been properly explored in the most obvious example of classical waves: water-surface (gravity-capillary) waves. Here, we fill this gap and describe (i)water-wave vortices of different orders carrying quantized angular momentum with orbital and spin contributions, (ii)skyrmion lattices formed by the instantaneous displacements of the water-surface particles in wave interference, and (iii)meron (half-skyrmion) lattices formed by the spin-density vectors, as well as (iv)spatiotemporal water-wave vortices and skyrmions. We show that all these topological entities can be readily generated in linear water-wave interference experiments. Our findings can find applications in microfluidics and show that water waves can be employed as an attainable playground for emulating universal topological wave phenomena.

Formal topological description of three-dimensional topological entities based on k-dimensional pseudo-manifold

This paper clarifies the necessity of studying three-dimensional spatial data models, analyzes the research progress and existing problems of topological spatial relationship description methods, and proposes a complete and formal description framework of three-dimensional topological spatial relations based on point set topological theory and dimension expansion method. On this basis, the topological spatial relations existing in three-dimensional spatial objects are classified, and five fundamental topological spatial relations are defined. The mutual exclusivity and completeness of the minimal set of 3D topological space relations are proved. Based on point set topology and k-dimensional pseudo-manifold, this paper proposes the formal description of three-dimensional topological properties and spatial relations between spatial entities. The simplicial homology is applied to manifolds, the topological properties of three-dimensional space entities are revealed, and the formal description method of three-dimensional space entities is given through the geometric structural relationship between simplex and simplicial k-complex.

Topological Flatband Loop States in Fractal‐Like Photonic Lattices

AbstractNoncontractible loop states (NLSs) are a recently realized topological entity in flatband lattices, arising typically from the band touching at a point where a flat band intersects one or more dispersive bands. There exists also band touching across a plane, where one flat band overlaps another all over the Brillouin zone without crossing a dispersive band. Such isolated plane‐touching flat bands remain largely unexplored. For example, what are the topological features associated with such flatband degeneracy? Here, nontrivial NLSs and robust boundary modes in a system with such degeneracy are demonstrated. Based on a tailored photonic lattice constructed from the well‐known fractal Sierpinski gasket, the wavefunction singularities and the conditions for the existence of the NLSs are theoretically analyzed. It is shown that the NLSs can exist in both singular and nonsingular flat bands, as a direct reflection of the real‐space topology. Experimentally, directly such flatband NLSs in a laser‐written Corbino‐shaped fractal‐like lattice are observed. This work not only leads to a deep understanding of the mechanism behind the nontrivial flatband states, but also opens up new avenues to explore fundamental phenomena arising from the interplay of flatband degeneracy, fractal structures, and band topology.

Hotness prediction of scientific topics based on a bibliographic knowledge graph

As a part of innovation in forecasting, scientific topic hotness prediction plays an essential role in dynamic scientific topic assessment and domain knowledge transformation modeling. To improve the topic hotness prediction performance, we propose an innovative model to estimate the co-evolution of scientific topic and bibliographic entities, which leverages a novel dynamic Bibliographic Knowledge Graph (BKG). Then, one can predict the topic hotness by using various kinds of topological entity information, i.e., TopicRank, PaperRank, AuthorRank, and VenueRank, along with pre-trained node embedding, i.e., node2vec embedding, and different pooling techniques. To validate the proposed method, we constructed a new BKG by using 4.5 million PubMed Central publications plus MeSH (Medical Subject Heading) thesaurus and witnessed the essential prediction improvement with extensive experiment outcomes over 10 years observations.

String diagram rewrite theory II: Rewriting with symmetric monoidal structure

AbstractSymmetric monoidal theories (SMTs) generalise algebraic theories in a way that make them suitable to express resource-sensitive systems, in which variables cannot be copied or discarded at will. In SMTs, traditional tree-like terms are replaced by string diagrams, topological entities that can be intuitively thought of as diagrams of wires and boxes. Recently, string diagrams have become increasingly popular as a graphical syntax to reason about computational models across diverse fields, including programming language semantics, circuit theory, quantum mechanics, linguistics, and control theory. In applications, it is often convenient to implement the equations appearing in SMTs as rewriting rules. This poses the challenge of extending the traditional theory of term rewriting, which has been developed for algebraic theories, to string diagrams. In this paper, we develop a mathematical theory of string diagram rewriting for SMTs. Our approach exploits the correspondence between string diagram rewriting and double pushout (DPO) rewriting of certain graphs, introduced in the first paper of this series. Such a correspondence is only sound when the SMT includes a Frobenius algebra structure. In the present work, we show how an analogous correspondence may be established for arbitrary SMTs, once an appropriate notion of DPO rewriting (which we call convex) is identified. As proof of concept, we use our approach to show termination of two SMTs of interest: Frobenius semi-algebras and bialgebras.

Liquid Crystalline Half‐Skyrmions and Their Optical Properties

AbstractLiquid crystals have intrigued physicists as a platform that facilitates direct visualization of topological concepts. Here, the theoretical and experimental studies demonstrating that a thin film of a chiral liquid crystal forms topological entities known as half‐Skyrmions, swirl‐like orientational order that spans the hemisphere of the order parameter space , are reviewed. They appear in the form of a hexagonal lattice or in an isolated manner, accompanied by topological line defects to satisfy topological constraints. Under an optical microscope, half‐Skyrmions appear as dark spots, and their hexagonal lattice yields a Kossel diagram comprising a hexagonal arrangement of circular arcs. These experimental observations are corroborated by numerical calculations of the orientational order based on the Landau‐de Gennes theory with an orientational order parameter of a second‐rank tensor, and construction of its optical images and Kossel diagrams by directly solving Maxwell equations for light wave. A thin film of a chiral liquid crystal thus provides an interesting system for the investigation of topological concepts, and spontaneous formation of a half‐Skyrmion lattice whose periodicity is of the order of the wavelength of visible light has a potential for photonics applications as well as poses a challenging problem on optics.

CONSTRUCTION OF CATMULL-CLARK SUBDIVISION SCHEME

Subdivision surface is a versatile tool for representing a smooth surface with any topology. This research explains how a smooth polyhedron surface is made using the Catmull-Clark subdivision method. The approach is based on the consideration of the regular topological entities of polyhedron on a cube. Construction is seen as a generalization of an arbitrary control point mesh recurrent subdivision algorithm. For faces, edges and arbitrary net points, the process uses the same expression that is formed in the cube. The process of the scheme will produce a smooth surface as the result. The important criteria for the construction also presented.

CONSTRUCTION OF CATMULL-CLARK SUBDIVISION SCHEME

Subdivision surface is a versatile tool for representing a smooth surface with any topology. This research explains how a smooth polyhedron surface is made using the Catmull-Clark subdivision method. The approach is based on the consideration of the regular topological entities of polyhedron on a cube. Construction is seen as a generalization of an arbitrary control point mesh recurrent subdivision algorithm. For faces, edges and arbitrary net points, the process uses the same expression that is formed in the cube. The process of the scheme will produce a smooth surface as the result. The important criteria for the construction also presented.

Shaping the topology of light with a moving Rabi-oscillating vortex.

Quantum vortices are the analogue of classical vortices in optics, Bose-Einstein condensates, superfluids and superconductors, where they provide the elementary mode of rotation and orbital angular momentum. While they mediate important pair interactions and phase transitions in nonlinear fluids, their linear dynamics is useful for the shaping of complex light, as well as for topological entities in multi-component systems, such as full Bloch beams. Here, setting a quantum vortex into directional motion in an open-dissipative fluid of microcavity polaritons, we observe the self-splitting of the packet, leading to the trembling movement of its center of mass, whereas the vortex core undergoes ultrafast spiraling along diverging and converging circles, in a sub-picosecond precessing fashion. This singular dynamics is accompanied by vortex-antivortex pair creation and annihilation and a periodically changing topological charge. The spiraling and branching mechanics represent a direct manifestation of the underlying Bloch pseudospin space, whose mapping is shown to be rotating and splitting itself. Its reshaping is due to three simultaneous drives along the distinct directions of momentum and complex frequency, by means of the differential group velocities, Rabi frequency and dissipation rates, which are natural assets in coupled fields such as polaritons. This state, displaying linear momentum dressed with oscillating angular momentum, confirms the richness of multi-component and open quantum fluids and their innate potentiality to implement sophisticated and dynamical topological textures of light.

Implementation of Structural Assemblies by Simultaneous Projection and Density-Based Topology Optimization

This article proposed a methodology that combines two well-known projections and modified density-based optimized techniques in one formulation methodology. This methodology contains an effective explicit geometric entity identified by shape variables that provide easy control in desired particular regions; implicit density-based topological optimization entities utilized topological variables that offer critical design elsewhere. Our main attractive key point of this combined formulation approach is structural assemblies. Structure always manufactures in many patches and joins them by utilizing the structural assemblies such as welding and riveting. It is not easy to execute the structural patches at the required region without acknowledging their dimensions. This proposed approach demonstrates the competence to impose the restrictions related to shape and topological variables of interfaces among the specific patches. Numerous standard numerical examples make sure the validation of the introduced methodology. It remarked that the optimal design could minimize compliance and the minimum number of iterations through numerically performed, concerning computational cost minimized without any kind loss of accuracy of the final structure.

PCPATCH

Effective relaxation methods are necessary for good multigrid convergence. For many equations, standard Jacobi and Gauß–Seidel are inadequate, and more sophisticated space decompositions are required; examples include problems with semidefinite terms or saddle point structure. In this article, we present a unifying software abstraction, PCPATCH, for the topological construction of space decompositions for multigrid relaxation methods. Space decompositions are specified by collecting topological entities in a mesh (such as all vertices or faces) and applying a construction rule (such as taking all degrees of freedom in the cells around each entity). The software is implemented in PETSc and facilitates the elegant expression of a wide range of schemes merely by varying solver options at runtime. In turn, this allows for the very rapid development of fast solvers for difficult problems.

Aberration corrected STEM techniques to investigate polarization in ferroelectric domain walls and vortices

Ferroelectric domain wall (DW) based nano-electronics is an emerging new field of research. It is only recently with advancements in electron and atomic force microscopy instrumentation that the complex nature of these 2D entities can be probed. In this Research Update, the advances in aberration corrected scanning transmission electron microscopy applied to ferroelectric topological defects are summarized. We discuss sub-atomic imaging and diffraction techniques used to observe changes in polarization, chemical composition, charge density, and strain at DWs and vortices. We further highlight the current achievements in mapping the 3D nature of ferroelectric polar skyrmions and in situ biasing. This Review will focus on both the fundamental physics of DW and polar vortex formation and their dynamics. Finally, we discuss how electron spectroscopy can be used to relate the quantified structural distortions of polar topological entities to changes in their oxidation state and band structure.

Controlled Nucleation and Stabilization of Ferroelectric Domain Wall Patterns in Epitaxial (110) Bismuth Ferrite Heterostructures

AbstractFerroelectric domain walls, topological entities separating domains of uniform polarization, are promising candidates as active elements for nanoscale memories. In such applications, controlled nucleation and stabilization of domain walls are critical. Here, using in situ transmission electron microscopy and phase‐field simulations, a controlled nucleation of vertically oriented 109° domain walls in (110)‐oriented BiFeO3 (BFO) thin films is reported. In the switching experiment, reversed domains that are nucleated preferentially at the nanoscale edges of the “crest and sag” pattern‐like electrode under external bias subsequently grow into a stable stripe configuration. In addition, when triangular pockets (with an in‐plane polarization component) are present, these domain walls are pinned to form stable flux‐closure domains. Phase field simulations show that i) field enhancement at the edges of the electrode causes site‐specific domain nucleation, and ii) the local electrostatics at the domain walls drives the formation of flux closure domains, thus stabilizing the striped pattern, irrespective of the initial configuration. The results demonstrate how flux closure pinning can be exploited in conjunction with electrode patterning and substrate orientation to achieve a desired topological defect configuration. These insights constitute critical advancements in exploiting domain walls in next generation ferroelectronic devices.

Flatband Line States in Photonic Super‐Honeycomb Lattices

AbstractFor the first time, a photonic super‐honeycomb lattice (sHCL) is established experimentally by use of a continuous‐wave laser writing technique, and thereby two distinct flatband line states that manifest as noncontractible loop states in an infinite flatband lattice are demonstrated. These localized states (“straight” and “zigzag” lines) observed in the sHCL with tailored boundaries cannot be obtained by the superposition of conventional compact localized states because they arise from real‐space topological property of certain flatband systems. In fact, the zigzag‐line states, unique to the sHCL, are in contradistinction with those previously observed in the Kagome and Lieb lattices. Their momentum‐space spectrum emerges in the high‐order Brillouin zone where the flatband touches the dispersive bands, revealing the characteristic of topologically protected band‐touching. The experimental results are corroborated by numerical simulations. This work may provide insight into Dirac‐like 2D materials beyond graphene.

From skyrmions to Z2 vortices in distorted chiral antiferromagnets

Swirling topological spin configurations, known as magnetic skyrmions, are known to be stabilised in chiral ferromagnets with Dzyaloshinskii-Moriya interaction (DMI). In particular, for appropriate values of the external magnetic field they appear in a topological crystalline phase, termed Skyrmion crystal phase (SkX). A similar phenomenon is present in the antiferromagnetic case, for the Heisenberg triangular antiferromagnet (HTAF) with DMI. Here, the most striking feature is that the emergent topological phase consists of three SkX interpenetrated sublattices. On the other hand, the pure HTAF, being described by an SO(3) order parameter, can host Z2 vortices. This rises the fundamental question on whether both non trivial structures are related. In this paper we unravel a hidden connection between both topological entities by studying the HTAF with anisotropic DMI. To this end, we combine an effective field theory description, the Luttinger-Tisza approximation and Monte Carlo simulations. We show that even a slight anisotropy in the DMI proves to be the key ingredient to deform the interpenetrated SkX structure and reveal a Z2-vortex crystal.