Topological Concepts Research Articles

Typed topology and its application to data sets

The concept of typedtopology is introduced. In a typed topological space, some open sets are assigned “types”, and topological concepts such as closure, connectedness can be defined using types. A finite data set in R2 is a typically typed topological space. Clusters calculated by the DBSCAN algorithm for data clustering can be well represented in a finite typed topological space. Other concepts such as tracks, port (starting points), type-p-connectedness, p-closure-connectedness, indexing, branches are also introduced for finite typed topological spaces. Finally, left−r and up−left−r type open sets are introduced for finite data sets in R2, so that tracks, port, branches can be calculated.

Some concepts of topology and questions in topological algebra

Some concepts of topology and questions in topological algebra

Two-dimensional topological insulators: past, present and future

Two-dimensional (2D) topological insulators (TIs) are special quantum conductors that possess an insulating bulk but metallic edge with quantized charge and spin conductance protected by electron band topology. The concept of topology and TI has not only renewed our fundamental understanding of electronic properties of solid materials, but also opened an exciting avenue towards potential applications of topological quantum devices with minimized heat dissipation and robustness against disorder. In this video article, I will first introduce and review the concept of TIs within the context of transport properties of solid-state materials. I will then use two examples, the organic 2D TIs and the surface-based 2D topological states, to recap the rapid theoretical and experimental developments made in this emerging field. The existence of quantized edge conductance and topological edge states of 2D TIs has been so far confirmed experimentally in several systems, such as semiconductor quantum wells, 2D transition metal dichalcogenides, metallic overlayer of bismuth on a semiconductor surface. However, discovery of high-temperature 2D TIs and construction of functional TI-based quantum devices remain largely elusive. At the end of this video article, I will offer briefly my personal perspective and possible future directions in low-dimensional topological materials.

Room-temperature continuous-wave topological Dirac-vortex microcavity lasers on silicon

Robust laser sources are a fundamental building block for contemporary information technologies. Originating from condensed-matter physics, the concept of topology has recently entered the realm of optics, offering fundamentally new design principles for lasers with enhanced robustness. In analogy to the well-known Majorana fermions in topological superconductors, Dirac-vortex states have recently been investigated in passive photonic systems and are now considered as a promising candidate for robust lasers. Here, we experimentally realize the topological Dirac-vortex microcavity lasers in InAs/InGaAs quantum-dot materials monolithically grown on a silicon substrate. We observe room-temperature continuous-wave linearly polarized vertical laser emission at a telecom wavelength. We confirm that the wavelength of the Dirac-vortex laser is topologically robust against variations in the cavity size, and its free spectral range defies the universal inverse scaling law with the cavity size. These lasers will play an important role in CMOS-compatible photonic and optoelectronic systems on a chip.

On the relationship between solid particle attractors and thermal inhomogeneities in vibrationally driven fluid-particle systems

The present analysis extends earlier authors' work [Crewdson et al., “Two-dimensional vibrationally-driven solid particle structures in non-uniformly heated fluid containers,” Chaos 32, 103119 (2022); M. Lappa, “Characterization of two-way coupled thermovibrationally driven particle attractee,” Phys. Fluids 34(5), 053109 (2022); M. Lappa and T. Burel, “Symmetry breaking phenomena in thermovibrationally driven particle accumulation structures,” ibid.32(5), 053314 (2020); and M. Lappa, “The patterning behavior and accumulation of spherical particles in a vibrated non-isothermal liquid,” ibid.26(9), 093301 (2014)] on the existence of solid particle attractee in thermovibrational flow in order to identify new physical principles and enable increased control over the ability of particles to target desired locations into the host fluid. The causality between the thermal boundary conditions and the multiplicity and morphology of emerging particle structures is discussed, and new fundamental topological concepts are harnessed through the combination of two-dimensional and three-dimensional simulations. It is shown that the threefold relationship among the inclination of vibrations, the multi-directional nature of the imposed temperature gradient, and the dimensionality of the system itself can open up new pathways for additional classes of attractors. These can manifest themselves as compact particle structures or completely disjoint sets, apparently behaving as they were driven by different clustering mechanisms (coexisting in the physical space, but differing in terms of characteristic size, shape, and position). A variety of new solutions are presented for a geometry as simple as a cubic enclosure in the presence of localized spots of temperature on otherwise uniformly heated or cooled walls. In order to filter out possible asymmetries due to fluid-dynamic instabilities induced by the back influence of the solid mass on the fluid flow, the analysis is conducted under the constraint of one-way coupled phases.

Philosophy of Body: Movement and Thought

Human thinking is modeled on the body in much the same way as hominid thinking was (Sheets-Johnstone, 1990, 4, 25). The hominid experience of moving was the cornerstone of the experience of the precinct of I can. The experience of I can subsequently formed the basis for corporeal and topological concepts. These, in turn, generated analogical insights. One sees, accordingly, not only an intermingling of the bodily experience and discernment, but also everywhere indications of analogical thinking at the root of hominid tool-building. Self-movement is the mainspring of one’s experience of oneself as an agent endowed with free-will-related activities. It is the creative fount of one’s ideas of space and time. The phenomenon of moving oneself structures how one knows the world. Moving oneself is a manner of coming to knowledge. The Lives of species-specific bodies are built on thinking in movement. Studies provide ample evidence to show a fundamental correlation between motor and cognitive development. This developmental correlation is also found in both the prefrontal cortex and the cerebellum. Both cognitive and motor developments reveal a prolonged developmental timetable. Both the prefrontal cortex and the cerebellum attain maturity much later. Studies by Adele Diamond and others show that the cerebellum is not only involved in motor operations but also in cognitive functions. Conversely, the prefrontal cortex, with (the help of) its connection with the cortical and subcortical areas that are critical for movement control, is involved in both motor and cognitive functions. Thus both motor and cognitive behaviours seem to belong together; and the body appears crucially involved in the activities of the mind (Diamond, 2000, 44, 49; Bushnell & Boudreau, 1993, 1008, 1015-1017; Burns et al., 2004, 19-29)

Weakly Generalized β- Continuous Mapping in Neutrosophic Bitopological Spaces

This paper explores the notion of β-continuity in neutrosophic bitopological spaces, a specialized area of mathematics that extends classical topological concepts to handle indeterminate or uncertain information. The study begins with the introduction of τ₁τ₂ semi-closed sets and τ₁τ₂-weakly continuous functions, which are fundamental building blocks. Key results include Proposition 2.1.3, which characterizes τ₁τ₂-weakly β-continuous mappings in terms of pre-images and β-interiors of open sets in the codomain space. Propositions 2.1.4 and 2.1.5 establish equivalent conditions for τ₁τ₂-weakly β-continuous functions involving pre-images, closures, and regular closed sets. Propositions 2.1.6 and 2.1.7 provide alternative characterizations of τ₁τ₂-weakly β-continuous functions, revealing connections with β-interiors and pre-image relationships. These findings contribute to the understanding of topological properties in neutrosophic bitopological spaces, offering valuable insights for further research in this intricate field.

THE USE OF POLYGONS IN VARIOUS FILE FORMATS IN THE DEVELOPING DESIGN OF 3D MODELS

The article describes the basic concepts of topology, which are fundamental for specialists in the field of three-dimensional design. All polygons can be divided into triangles, squares and n-gons, and of these polygons are the most undesirable. To consider this statement, a number of models of varying complex- ity have been developed and transferred to the UNITY game engine in two forms obj and fbx. Both formats are the most commonly used in game design, character design, locations, or object modeling, but they con- vert the polygonal grid of an object in different ways, which is why the shape of the model itself will change in the presence of n-gons.

The local properties in topological groups and related concepts and questions

The next general question is considered below: when certain open subspaces of topological groups are homeomorphic to some topological group? A version of this question has been posed in [4] (open problem 1.4.1). In it, the complements to the elements in any zero-dimensional topological group were considered. A result below in this direction is Theorem 3.1 saying that if G is a σ-compact topological group and F is a nonempty compact subspace of G such that the complement G∖F is homeomorphic to some topological group, then F is a Gδ-subset of G (and therefore, by a remarkable theorem of M.M. Choban [6], F is a dyadic compactum). Another conclusion in this theorem says that if, moreover, the tightness of F is countable (in particular, if F is first countable), or if F is topologically linearly orderable, then G has a countable network. Lemma 4.1 says that if X is a Lindelöf homogeneous space with countable Souslin number and X is puncturable at some point a, then a is a Gδ-point in X, the subspace Y=X∖{a} is Lindelöf, and X is submetrizable. It follows from this that every Lindelöf topological field with countable Souslin number is submetrizable (its topology contains a metrizable topology). A natural Abelian topological group with few open subspaces that are homeomorphic to topological groups is identified in Example 3.2.

Direct Observation of Topological Phonons in Graphene.

Phonons, as the most fundamental emergent bosons in condensed matter systems, play an essential role in the thermal, mechanical, and electronic properties of crystalline materials. Recently, the concept of topology has been introduced to phonon systems, and the nontrivial topological states also exist in phonons due to the constraint by the crystal symmetry of the space group. Although the classification of various topological phonons has been enriched theoretically, experimental studies were limited to several three-dimensional (3D) single crystals with inelastic x-ray or neutron scatterings. The experimental evidence of topological phonons in two-dimensional (2D) materials is absent. Here, using high-resolution electron energy loss spectroscopy following our theoretical predictions, we directly map out the phonon spectra of the atomically thin graphene in the entire 2D Brillouin zone, and observe two nodal-ring phonons and four Dirac phonons. The closed loops of nodal-ring phonons and the conical structure of Dirac phonons in 2D momentum space are clearly revealed by our measurements, in nice agreement with our theoretical calculations. The ability of 3D mapping (2D momentum space and energy space) of phonon spectra opens up a new avenue to the systematic identification of the topological phononic states. Our work lays a solid foundation for potential applications of topological phonons in superconductivity, dynamic instability, and phonon diode.

Topologies of belonging in the digital university

ABSTRACT Belonging is a complex relational concept. It has been shown to be processual, emergent, and dynamic. And yet this relationality, and complexity, sits in tension with increasingly voluble calls to measure, manage and maintain students’ sense of belonging to an ostensibly fixed space of higher education. This article reports on research that invited students to not only define how they experience belonging, but also to surface belonging’s relationality: how it is entangled with material spaces and artefacts and enacted through evolving behaviours and communities. Our data depict modulations of belonging which disrupt dominant discourses of simplicity, stability and uniformity. Engaging the generative concept of social topologies, we offer a rethinking of both space and belonging as material and relational. These findings enable us to consider more nuanced perspectives about how belonging is both understood and also enacted, surfacing the complex tapestries of belonging and non-belonging experiences within education, as well as the increasing departure from a coherent delineated conception of ‘the university’. Given the diversity of both learners and the spaces in which they learn, interrogating the nature of belonging is urgently needed if we are to understand students’ diverse experiences of education in more meaningful ways.

Nano βθ-Open Sets in Nano Topological Spaces

This paper's primary objective is to present and study a different class of nano sets known as nano βθ-open sets in nano topological spaces. Additionally, introduced some nano topological concepts by using nano βθ-open set.

Interval-Valued Topology on Soft Sets

In this paper, we study the concept of interval-valued fuzzy set on the family SSX,E of all soft sets over X with the set of parameters E and examine its basic properties. Later, we define the concept of interval-valued fuzzy topology (cotopology) τ on SSX,E. We obtain that each interval-valued fuzzy topology is a descending family of soft topologies. In addition, we study some topological structures such as interval-valued fuzzy neighborhood system of a soft point, base and subbase of τ and investigate some relationships among them. Finally, we give some concepts such as direct sum, open mapping and continuous mapping and consider connections between them. A few examples support the presented results.

Numerical simulations and complex valued fractional order neural networks via ([formula omitted] – μ)-uniformly contractive mappings

In this article, we present some fixed point results for (ɛ−μ)-uniformly contractive mappings and Ćirić–Riech–Rus type (ɛ−μ)-uniformly locally contractive mappings. As an application, the fixed point approach in suprametric spaces is used to examine the uniform stability problem of fractional-order complex-valued neural networks with time delay and equilibrium point results for impulsive fractional-order complex-valued neural networks with time delay. In addition, we used topological concepts such as compactness, closed convex sets, etc. instead of the traditional method that would have been used in the literature for many years. A few numerical results are provided to show the feasibility and correctness of the results presented.

Modeling of COVID-19 in View of Rough Topology

Rough-based topology has become an important technique for decision making in numerous real-life problems. The main purpose of this research paper is to use some topological notions in the approximation space of a rough set. Firstly, some concepts of topological near open sets and rough concepts are presented. A new approximation structure based on the topological near open sets is introduced. We debate the properties of the new approximation structure. We included an algorithm for detecting COVID-19 infection by its side effects. We believe that our approach will be helpful for any future detection.

Connectomes and properties of quantum entanglement

Topological quantum field theories (TQFT) encode properties of quantum states in the topological features of abstract manifolds. One can use the topological avatars of quantum states to develop intuition about different concepts and phenomena of quantum mechanics. In this paper we focus on the class of simplest topologies provided by a specific TQFT and investigate what the corresponding states teach us about entanglement. These “planar connectome” states are defined by graphs of simplest topology for a given adjacency matrix. In the case of bipartite systems the connectomes classify different types of entanglement matching the classification of stochastic local operations and classical communication (SLOCC). The topological realization makes explicit the nature of entanglement as a resource and makes apparent a number of its properties, including monogamy and characteristic inequalities for the entanglement entropy. It also provides tools and hints to engineer new measures of entanglement and other applications. Here the approach is used to construct purely topological versions of the dense coding and quantum teleportation protocols, giving diagrammatic interpretation of the role of entanglement in quantum computation and communication. Finally, the topological concepts of entanglement and quantum teleportation are employed in a simple model of information retrieval from a causally disconnected region, similar to the interior of an evaporating black hole.

Application of Intuitionistic Fuzzy Topological Operators in Spatial Objects Modeling

The concept of topology is widely used in mathematical modeling of spatial objects and in GIS. One such application of fuzzy topological operators, detecting areas affected by Mikania micrantha, was studied by Shi and Liu. The precision of the results was evaluated by applying the intuitionistic fuzzy pre-interior and pre-clouser operators defined by the author to the data included in this study.
 With this application example, it has been shown that the newly defined operators give results closer to the real bounds.

Berry curvature contributions of kagome-lattice fragments in amorphous Fe–Sn thin films

Amorphous semiconductors are widely applied to electronic and energy-conversion devices owing to their high performance and simple fabrication processes. The topological concept of the Berry curvature is generally ill-defined in amorphous solids, due to the absence of long-range crystalline order. Here, we demonstrate that the Berry curvature in the short-range crystalline order of kagome-lattice fragments effectively contributes to the anomalous electrical and magneto-thermoelectric properties in Fe–Sn amorphous films. The Fe–Sn films on glass substrates exhibit large anomalous Hall and Nernst effects comparable to those of the single crystals of topological semimetals Fe3Sn2 and Fe3Sn. With modelling, we reveal that the Berry curvature contribution in the amorphous state likely originates from randomly distributed kagome-lattice fragments. This microscopic interpretation sheds light on the topology of amorphous materials, which may lead to the realization of functional topological amorphous electronic devices.

Electrically-pumped compact topological bulk lasers driven by band-inverted bound states in the continuum

One of the most exciting breakthroughs in physics is the concept of topology that was recently introduced to photonics, achieving robust functionalities, as manifested in the recently demonstrated topological lasers. However, so far almost all attention was focused on lasing from topological edge states. Bulk bands that reflect the topological bulk-edge correspondence have been largely missed. Here, we demonstrate an electrically pumped topological bulk quantum cascade laser (QCL) operating in the terahertz (THz) frequency range. In addition to the band-inversion induced in-plane reflection due to topological nontrivial cavity surrounded by a trivial domain, we further illustrate the band edges of such topological bulk lasers are recognized as the bound states in the continuum (BICs) due to their nonradiative characteristics and robust topological polarization charges in the momentum space. Therefore, the lasing modes show both in-plane and out-of-plane tight confinements in a compact laser cavity (lateral size ~3λlaser). Experimentally, we realize a miniaturized THz QCL that shows single-mode lasing with a side-mode suppression ratio (SMSR) around 20 dB. We also observe a cylindrical vector beam for the far-field emission, which is evidence for topological bulk BIC lasers. Our demonstration on miniaturization of single-mode beam-engineered THz lasers is promising for many applications including imaging, sensing, and communications.

Directional Connectivity-based Segmentation of Medical Images.

Anatomical consistency in biomarker segmentation is crucial for many medical image analysis tasks. A promising paradigm for achieving anatomically consistent segmentation via deep networks is incorporating pixel connectivity, a basic concept in digital topology, to model inter-pixel relationships. However, previous works on connectivity modeling have ignored the rich channel-wise directional information in the latent space. In this work, we demonstrate that effective disentanglement of directional sub-space from the shared latent space can significantly enhance the feature representation in the connectivity-based network. To this end, we propose a directional connectivity modeling scheme for segmentation that decouples, tracks, and utilizes the directional information across the network. Experiments on various public medical image segmentation benchmarks show the effectiveness of our model as compared to the state-of-the-art methods. Code is available at https://github.com/Zyun-Y/DconnNet.