Topological Concepts Research Articles

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.

Topological Metamaterials.

The topological properties of an object, associated with an integer called the topological invariant, are global features that cannot change continuously but only through abrupt variations, hence granting them intrinsic robustness. Engineered metamaterials (MMs) can be tailored to support highly nontrivial topological properties of their band structure, relative to their electronic, electromagnetic, acoustic and mechanical response, representing one of the major breakthroughs in physics over the past decade. Here, we review the foundations and the latest advances of topological photonic and phononic MMs, whose nontrivial wave interactions have become of great interest to a broad range of science disciplines, such as classical and quantum chemistry. We first introduce the basic concepts, including the notion of topological charge and geometric phase. We then discuss the topology of natural electronic materials, before reviewing their photonic/phononic topological MM analogues, including 2D topological MMs with and without time-reversal symmetry, Floquet topological insulators, 3D, higher-order, non-Hermitian and nonlinear topological MMs. We also discuss the topological aspects of scattering anomalies, chemical reactions and polaritons. This work aims at connecting the recent advances of topological concepts throughout a broad range of scientific areas and it highlights opportunities offered by topological MMs for the chemistry community and beyond.

Topos of Noise

This paper focuses on the significance of the concept of noise for cognition and computation. The concept of noise was massively transformed in the twentieth century with the advent of information theory, cybernetics, and computer science, all of which provide formal accounts of information and noise centrally concerned with contingency. We show how the concept has changed from these classical formulations, through developments in mathematics (topology and topos theory), computing (interactive computing and univalent foundations), and cognitive science (predictive processing and cognitive morphodynamics). Ultimately it argues for the central importance of noise not only within a topological conception of cognition and computation, but also in the transcendental-empirical torsion of image schemata and the social interactive elaboration of freedom.

Topology optimization of a rotating electric machine by the topological derivative

AbstractWe consider the topology optimization of a rotating electric machine in magnetoquasistatic operation in two space dimensions. This amounts to a topology optimization problem subject to a parabolic PDE constraint on a moving domain which we intend to solve by means of the topological derivative concept. For that purpose, we consider a topological perturbation of the materials in the space‐time cylinder along a trajectory given by the rotation of the machine. Using a Lagrangian approach, we derive the topological derivative formula, which depends on the solution of an exterior problem that is bounded in time and unbounded in the space directions. We use this sensitivity information for the design optimization of an electric motor by means of a level set method.

Estimating uncertainties in 3-D models of complex fold-and-thrust belts: A case study of the Eastern Alps triangle zone

Geological modeling commonly results in a single prescribed geometric representation of the subsurface with no consideration of uncertainties. Accounting for uncertainties is of particular importance in the triangle zone at the leading edge of deformation of the foreland fold-thrust belt of the European Alps, the Subalpine Molasse. Here, interpretations of the complex structures are limited to 2D and are based almost exclusively on subsurface data, lacking the constraint of surface data.Implicit modeling can be used to create automated 3D model realizations considering parameter uncertainty. In this sense, multiple possible models can be assessed within the uncertainty range assigned. As implicit modeling often yields artifacts or geologically unfeasible scenarios, the concept of geological topology can be used to constrain the modeling outputs, so that only models without artifacts are considered into the final model ensemble.Two experiments are designed to test out this workflow. The methodology is first tested using a very simple synthetic model, and later applied to a portion of the Subalpine Molasse, where different 2-D subsurface geometries have been proposed. For the first time, a 3D implicit model ensemble incorporating a full assessment of uncertainties is built in this area. Results show that it is feasible to use topological information for posterior model constraint, even in models of high structural complexity. To eliminate all non-meaningful models, however, it is necessary to use more than one topological constraint. A comparison of the prior and posterior model ensembles shows a correlation between model parameters and a shift in parameter probability density curves in the synthetic model experiment, and a decrease in entropy for both experiments. A topological constraint should thus be applied routinely when building a stochastic implicit model ensemble.

Topological phases in polar oxide nanostructures

Topology provides a framework for an understanding of emergent phenomena in various fields such as the appearance of new results in the field of ferroelectrics. A historical introduction and a primer on the concepts of topology are followed by a discussion of experimental and theoretical methods that elucidate the exotic textures that appear in polar oxide nanostructures and superlattices. The complex interplay of several fundamental processes occurring on different energy scales leads to ground states that compete with one another and have large and/or novel responses with respect to external stimuli.

Entropy measures of Y-junction based nanostructures

Recent research on nanostructures proved their importance and implementation in various areas. In drug delivery systems, medicine and pharmaceutical, biological sensors, photodetectors, transistors, optical and electronic devices, in all these topics nanostructures are implemented directly or indirectly. Future advanced electronic devices are a motivation for the discovery of carbon nanotubes made Y-shaped junctions. Electronic switches, amplifiers, and three-terminal transistors are of great interest due to their interaction with Y-junctions. The concept of entropy determines the uncertainty of a system or network. Concepts of entropy are also commonly used in biology, chemistry, and applied mathematics. Entropy in the form of a graph can be classified into several types based on the requirements. Graph-based entropy was introduced in 1955. Edge-weighted-based entropy is one of the types of entropy. In this study, we look closely at the abstract form of Y-shaped junctions. For the generic view of Y-shaped junctions, we developed some edge-weight-based entropy formulas. On Y-shaped junctions, we discussed some edge-weighted and topological index-based concepts.

An Efficient Method for Predicting the Shielding Effectiveness of an Apertured Enclosure with an Interior Enclosure based on Electromagnetic Topology

A fast analytical method has been proposed for predicting the shielding effectiveness (SE) and resonances of an apertured enclosure with an interior enclosure. Under the concept of electromagnetic topology, the monitor point and the walls are treated as nodes, and the space between them is treated as tubes. The propagation relationships at tube level and reflection relationships at node level are derived as the propagation matrix. After modeling the front wall of the interior enclosure as a junction between two waveguides, an equivalent circuital model of the enclosures is derived. The front wall and the window structure in front of the adjacent space of the interior enclosure are considered as a three-port scattering matrix. Then we can use the extended BLT equations to calculate the voltage response at each node. Results from the proposed method are compared with those from the numerical method, and the results have a good agreement while it can dramatically save calculation time.

Bloch points and topological dipoles observed by X-ray vector magnetic tomography in a ferromagnetic microstructure

Bloch points are small 3D magnetic textures with unit topological charge that require the use of advanced vector imaging techniques for their direct experimental characterization. Here we show results from the reconstruction of the magnetization field {{{{{bf{m}}}}}}left({{{{{bf{r}}}}}}right) of an elongated permalloy microstructure by X-ray vector magnetic tomography. A central asymmetric Bloch domain wall is observed, decorated by Bloch points arranged in several dipoles and a triplet. The analysis of the {{{{{bf{m}}}}}}left({{{{{bf{r}}}}}}right) map in terms of topological concepts provides a quantitative description of the Bloch points as topological monopoles connected by bundles of emergent magnetic field lines carrying a fractional topological flux. It also reveals the topological constraints that determine chirality transitions of the central domain wall in the vicinity of Bloch points, independent of specific material properties. This approach could be readily extended to the study of magnetic microstructures in arbitrary remanent configurations.

Perovskite quantum dot one-dimensional topological laser

Various topological laser concepts have recently enabled the demonstration of robust light-emitting devices that are immune to structural deformations and tolerant to fabrication imperfections. Current realizations of photonic cavities with topological boundaries are often limited by outcoupling issues or poor directionality and require complex design and fabrication that hinder operation at small wavelengths. Here we propose a topological cavity design based on interface states between two one-dimensional photonic crystals with distinct Zak phases. Using a few monolayers of solution-processed all-inorganic cesium lead halide perovskite quantum dots as the ultrathin gain medium, we demonstrate a lithography-free, vertical-emitting, low-threshold, and single-mode laser emitting in the green. We show that the topological laser, akin to vertical-cavity surface-emitting lasers (VCSELs), is robust against local perturbations of the multilayer structure. We argue that the design simplicity and reduction of the gain medium thickness enabled by the topological cavity make this architecture suitable for low-cost and efficient quantum dot vertical emitting lasers operating across the visible spectral region.

New Structures of Continuous Functions

Continuous functions are novel concepts in topology. Many topologists contributed to the theory of continuous functions in topology. The present authors continued the study on continuous functions by utilizing the concept of gpα-closed sets in topology and introduced the concepts of weakly, subweakly and almost continuous functions. Further, the properties of these functions are established.

Global Topology Concept Neighborhood Graph and Its Application in Topological Similarity Measurement

A topological relationship is a key factor in the spatial data processing. Aiming at the problem of topological similarity measurement when the dimension of spatial objects changes in the process of map generalization, this paper proposes a topological similarity measurement method based on the global topological concept neighborhood graph. Firstly, all the topological relationships that can be described by the nine intersection model are abstracted, generalized, and classified, and the classified point/point, point/line, point/polygon, line/line, line/polygon, and polygon/polygon local topological concept neighborhood graphs are established. Then, according to the map generalization rules, the mapping between different types of topological concept neighborhood graphs is established, and the global topological concept neighborhood graph is obtained, Finally, the similarity of the topological relationship between different morphological objects is measured. The experimental results show that the proposed method can better meet the topological similarity measurement when the shape of spatial objects changes.

Topological phases and non-Hermitian topology in photonic artificial microstructures

Abstract In the past few decades, the discovery of topological matter states has ushered in a new era in topological physics, providing a robust framework for strategically controlling the transport of particles or waves. Topological photonics, in particular, has sparked considerable research due to its ability to construct and manipulate photonic topological states via photonic artificial microstructures. Although the concept of topology originates from condensed matter, topological photonics has given rise to new fundamental ideas and a range of potential applications that may lead to revolutionary technologies. Here, we review recent developments in topological photonics, with a focus on the realization and application of several emerging research areas in photonic artificial microstructures. We highlight the research trend, spanning from the photonic counterpart of topological insulator phases, through topological semimetal phases, to other emerging non-Hermitian topologies.

On strongly faintly M-θ-i-continuous functions in Bi-Supra Topological Space

In this paper we introduces a new definition , called i- open and via this definition we introduce class of topological concepts(µ-θ-i-open set, µ-θ-i-closed, strong faintly µ-θ-continuity, strong µ-θ-continuity )and we generalized these concepts in bi -supra topological space .At last many important theorems in strongly faintly M-θ-i-continuous functions are investigated. And study the relationships among these functions and other forms are discussed.

Various topological phases and their abnormal effects of topological acoustic metamaterials

AbstractThe last 20 years have witnessed growing impacts of the topological concept on the branches of physics, including materials, electronics, photonics, and acoustics. Topology describes objects with some global invariant property under continuous deformation, which in mathematics could date back to the 17th century and mature in the 20th century. In physics, it successfully underpinned the physics of the Quantum Hall effect in 1984. To date, topology has been extensively applied to describe topological phases in acoustic metamaterials. As artificial structures, acoustic metamaterials could be well theoretically analyzed, on‐demand designed, and easily fabricated by modern techniques, such as three‐dimensional printing. Some new theoretical topological models were first discovered in acoustic metamaterials analogous to electronic counterparts, associated with novel effects for acoustics closer to applications. In this review, we focused on the concept of topology and its realization in airborne acoustic crystals, solid elastic phononic crystals, and surface acoustic wave systems. We also introduced emerging concepts of non‐Hermitian, higher‐order, and Floquet topological insulators in acoustics. It has been shown that the topology theory has such a powerful generality that among the disciplines from electron to photon and phonon, from electronic to photonics and acoustics, from acoustic topological theory to acoustic devices, could interact and be analogous to fertilize fantastic new ideas and prototype devices, which might find applications in acoustic engineering and noise‐vibration control engineering in the near future.

A Novel Framework for Generalizations of Soft Open Sets and Its Applications via Soft Topologies

Soft topological spaces (STSs) have received a lot of attention recently, and numerous soft topological ideas have been created from differing viewpoints. Herein, we put forth a new class of generalizations of soft open sets called “weakly soft semi-open subsets” following an approach inspired by the components of a soft set. This approach opens the door to reformulating the existing soft topological concepts and examining their behaviors. First, we deliberate the main structural properties of this class and detect its relationships with the previous generalizations with the assistance of suitable counterexamples. In addition, we probe some features that are obtained under some specific stipulations and elucidate the properties of the forgoing generalizations that are missing in this class. Next, we initiate the interior and closure operators with respect to the classes of weakly soft semi-open and weakly soft semi-closed subsets and look at some of their fundamental characteristics. Ultimately, we pursue the concept of weakly soft semi-continuity and furnish some of its descriptions. By a counterexample, we elaborate that some characterizations of soft continuous functions are invalid for weakly soft semi-continuous functions.

Covering Approximations Approach to Interval Ordered Information Systems

It is been proved that the theory of rough set is very beneficial in working with conflict problems induced by the information endearment The original idea of the rough set is not accurate, however, when preference orders of characteristics domains (standard) are to be considered. Seeking the issue of covering approximation relative to a control relation in interval-ordered information systems is a new mathematical tool to set up a control-based rough set approach. These Mathematical tools apply some topological generalizations, and control relations, covering approximations, and interval information systems. These results using our approach are more accurate than using a classical approach such as the approach of Pawlak. Topologies reproduce types of information systems with one value. By defining a control relation to interval information systems, we generalized the Pawlak approximation space to a covering approximation space and then we involve this approach to work with the interval-ordered information systems. Results gained by the proposed approach to generate two different rough approximations called j-lower and j-upper approximations. We applied j-rough notions such as j-rough membership, j-rough equality, and j-rough inclusion relations using topological generalizations. Our proposed method has improved the results gained from interval information systems. In this Research, there are many improved approaches to investing generalized approximation relative to a control relation in interval information systems, and we use the covering approximation in the rough set approach. This is a generalization of the Pawlak approach applied to interval information systems. This approach opens the way for other generalizations by making new algorithms, which simplify the calculations on it, and we can add more topological concepts to support decision-making in real-life applications.

Observation of Topologically Enabled Complete Polarization Conversion

AbstractExploiting topological ideas has been a major theme in modern photonics, which provides unprecedented opportunities to design photonic devices with robustness against defects. While most previous works in topological photonics have focused on band theory, recent theories extend the topological concepts to the analysis of scattering matrices and suggest a topological route to complete polarization conversion (CPC). Here, the experimental observation of the topological CPC is reported. Using angle‐resolved reflection measurements, it is unveiled experimentally that the CPC between arbitrary two polarizations occurs at vortex singularities of reflection coefficients in momentum space, reflecting its topological nature. Besides, it is visualized directly that for a given input polarization, the output one can cover the entire Poincaré sphere over a wide frequency range by varying the incident angle, guaranteed by the topological nature of CPC, which is in sharp contrast to the conventional polarization–conversion approaches that usually suffer from the bulky volume, limited choice of eigen‐polarization states, or narrow operation bandwidths. Remarkably, that BICs lie on the critical coupling curves that define the condition for CPC is experimentally demonstrated. This work paves the way to exploring the topological properties in scattering matrices for controlling light polarization and creating robust photonics devices.

On L-valued tree automata: The topological characterization

The current study aims to investigate the L-valued tree automata theory based on t-norm/t-conorm and it further examines their algebraic and L-valued topological properties. Specifically, the concept of L-valued operators with t-norm/t-conorm is introduced and the existing relationships between them are also studied. Interestingly, we associate L-valued co-topologies/topologies for a given L-valued tree automaton, using them to characterize some algebraic concepts. Further, we introduce the concepts of Alexandroff L-graded co-topologies and Alexandroff L-graded topologies which correspond to the L-valued operators with t-norm and L-valued operators with t-conorm/implicator, respectively. In addition, we aim to specify the relationship between the L-graded co-topologies/topologies, showing that the introduced L-graded co-topologies/topologies have some interesting consequences under homomorphism.