Topological Properties Research Articles

Exploring topological properties in partial metric spaces: compactness and paracompactness

Exploring topological properties in partial metric spaces: compactness and paracompactness

On Fuzzy Soft α-open Sets, α-continuity, andα-compactness: Some Novel Results

In this paper, we defined the notions of fuzzy soft α-interior (α-closure) operators via fuzzy soft topologies based on the sense of Sostak and studied some topological properties of them. Also, the notion of r-fuzzy soft α-connected sets was introduced and investigated. Thereafter, we defined and characterized the notions of fuzzy soft weakly (almost) α-continuous mappings, which are weaker forms of fuzzy soft α-continuous mappings. Moreover, we established that fuzzy soft α-continuity → fuzzy soft almost α-continuity → fuzzy soft weakly α-continuity, but the converse may not be true. In addition, we investigated some properties of fuzzy soft α-continuity. It is also we showed that the composition is fuzzy soft almost α-continuous mapping if is fuzzy soft α-continuous mapping and is fuzzy soft almost continuous mapping. Finally, some novel types of fuzzy soft compactness via r-fuzzy soft α-open sets were introduced and the relationships between them were explored with the help of some examples.

Identification of asymmetrical abnormalities in functional connectivity and brain network topology for migraine sufferers: A preliminary study based on resting-state fMRI data

Research on the neural mechanisms underlying brain asymmetry in patients with migraine patients using fMRI is insufficient. This study proposed using lateralized algorithms for functional connectivity and brain network topology and investigated changes in their lateralization in patients with migraine. In study 1, laterality indices of functional connectivity (LFunctionCorr) and brain network topological properties (LBetweennessCentrality, LDegree, and LStrength) were defined. Differences between migraineurs and normal subjects were compared at whole-brain, half-brain, and region levels. In study 2, laterality indices were used to classify migraine and were validated using independent samples and the segment method for repeatability. In study 3, abnormal brain regions related to migraine were extracted based on the classification results and differences analysis. Study 1 found no significant differences related to in for migraine at the whole-brain level; however, significant differences were identified at the half-brain level for the hemispheric lateralization of the LFunctionCorr, while 11 significantly different brain regions were also identified at the brain region level. Furthermore, the classification accuracy in study 2 was 0.9366. With repeated validation, the accuracy reached 0.8561. Furthermore, after extending the samples according to the segmentation strategy, the classification accuracies were improved to 0.9408 and 0.8585. Study 3 identified 10 crucial brain regions with asymmetrical specificity based on laterality indices distributed across the visual network, the frontoparietal control network, the default mode network, the salience/ventral attention network and the limbic system. The results revealed novel insights and avenues for research into the mechanisms of migraine asymmetry and showed that the laterality indices could be used as a potential diagnostic imaging marker for migraine.

Some New Types of Fuzzy Closed Sets, SeparationAxioms, and Compactness via Double Fuzzy Topologies

In this article, we first defined a stronger form of (r,s)-generalized fuzzy semi-closed sets (briefly, (r,s)-gfsc sets) called (r,s)-g*fsc sets and investigated some of its features. Moreover, we showed that (r,s)-fsc set → (r,s)-g*fsc set → (r,s)-gfsc set, but the converse may not be true. In addition, we explored novel types of fuzzy generalized mappings between double fuzzy topological spaces (U, τ, τ*) and (V, η, η*), and the relationships between these classes of mappings were examined with the help of some illustrative examples. Thereafter, we introduced novel types of higher separation axioms called (r,s)-GFS-regular and (r,s)-GFS-normal spaces with the help of (r,s)-gfsc sets and discussed some topological properties of them. Finally, some novel types of compactness via (r,s)-gfso sets were defined and the relationships between them were introduced.

A semi-definite optimization method for maximizing the shared band gap of topological photonic crystals

Topological photonic crystals (PCs) can support robust edge modes to transport electromagnetic energy in an efficient manner. Such edge modes are the eigenmodes of the PDE operator for a joint optical structure formed by connecting together two photonic crystals with distinct topological invariants, and the corresponding eigenfrequencies are located in the shared band gap of two individual photonic crystals. This work is concerned with maximizing the shared band gap of two photonic crystals with different topological features in order to increase the bandwidth of the edge modes. We develop a semi-definite optimization framework for the underlying optimal design problem, which enables efficient update of dielectric functions at each time step while respecting symmetry constraints and, when necessary, the constraints on topological invariants. At each iteration, we perform sensitivity analysis of the band gap function and the topological invariant constraint function to linearize the optimization problem and solve a convex semi-definite programming (SDP) problem efficiently. Numerical examples show that the proposed algorithm is superior in generating optimized optical structures with robust edge modes.

Spectroscopic Evidence of Ultrafast Topological Phase Transition by Light-Driven Strain.

Enabling reversible control over the topological invariants, transitioning them from nontrivial to trivial states, has fundamental implications for quantum information processing and spintronics. It offers a promising avenue for establishing an efficient on/off switch mechanism for robust and dissipationless spin-currents. While mechanical strain has traditionally been advantageous for such manipulation of topological invariants, it often comes with the drawback of in-plane fractures, rendering it unsuitable for high-speed, time-dependent operations. This study employs ultrafast optical and THz spectroscopy to explore topological phase transitions induced by light-driven strain in Bi2Se3. Bi2Se3 requires substantial strain for Z2 switching. Our observations provide experimental evidence of ultrafast switching behavior, demonstrating a transition from a topological insulator with spin-momentum-locked surfaces to hybridized states and normal insulating phases under ambient conditions. Notably, applying light-induced strong out-of-plane strain effectively suppresses surface-bulk coupling, facilitating the differentiation of surface and bulk conductance even at room temperature─significantly surpassing the Debye temperature. We expect various time-dependent sequences of transient hybridization and manipulation of topological invariant through photoexcitation intensity adjustments. The sudden surface and bulk transport alterations near the transition point enable coherent conductance modulation at hypersound frequencies. Our findings on the potential of light-triggered ultrafast switching of topological invariants hold promise for high-speed topological switching and its related applications.

Exploring Bonding Configurations in MnBi2Te4-Type Materials.

We perform a systematic investigation of several crystal structures, based on monolayer MnBi2Te4, of the form MnBiBiiXi2Xii2 using first-principles calculations. Our analysis shows that the most energetically favorable bonding configuration of the constituent elements in monolayer MnBiBiiXi2Xii2 is determined by the bond length between the Mn atom and its nearest X-site atoms. Tuning the bonding configuration of the material alters the magnetic, electronic, and topological properties. We also calculate the magnetic exchange parameters and magnetic anisotropy energy of the predicted structures. The calculations show that the elements at the X sites mainly determine the magnetic properties. Finally, we propose a stable phase of monolayer MnBi2S2Te2 (i.e., γ-MnBi2S2Te2) that exhibits the quantum anomalous Hall effect (QAHE). This study demonstrates that the bonding configuration of MnBi2Te4-type materials provides avenues for tuning the magnetic, electronic, and topological properties of van der Waals (vdW) materials.

Topology and Automorphism Structures in General Linear Group

The general linear group, as a significant topic in algebra and topology, has a wide-ranging research background and application value. With the continuous advancement of mathematical research, the combination of topological structures and automorphism structures has become a key entry point for exploring the properties of the general linear group. This paper investigates the topological properties of general linear group GL (n, R) , specifically focusing on compactness, connectedness and the fundamental group, and the automorphism. The first part of the study is dedicated to a detailed analysis of these properties, providing new insights into the topological structural characteristics of GL (n, R) . This paper scrutinized the homotopy type within it, giving proof to the fundamental group of GL (n, R) , which is showed to be isomorphism to a trivial group. In the second half, This paper introduce and examine the function φ (G A GL n G A G ) = { ( , )| • } ∈ =  , which is used to identify the automorphism group of a dense subgroup G of Rn . Specifically, the automorphism group of Qn is investigated. This paper show that the automorphism group of it is isomorphic to the subgroup GL (n, Q) of general linear group GL (n, R) . Through this function, this paper offer an innovative approach to understanding the automorphisms within general linear groups, revealing the deep connections between algebraic and topological aspects of these groups.

Topological Properties on Neural Networks Using Graph Properties

Topological Properties on Neural Networks Using Graph Properties

Topological organization of the brain network in thyroid-associated ophthalmopathy using graph theoretical analysis.

Mounting neuroimaging evidence indicates that patients with thyroid-associated ophthalmopathy (TAO) demonstrate altered brain function and structure. Nonetheless, the alterations in the topological properties of the functional brain connectome in TAO patients are not yet fully understood. This study aimed to investigate the topological organization of the functional brain connectome in TAO patients using graph-theoretic methods. Twenty-five TAO patients (10 males and 15 females) and 25 age-, sex-, and education-matched healthy controls (HCs) (10 males and 15 females) (the TAO and HC data are from the same dataset in previous studies) underwent resting-state MRI scans. Graph-theoretic analysis was used to study the global, nodal, and edge topological properties of the brain's functional connectome. Both the TAO and HC groups exhibited high-efficiency small-world networks in their brain functional networks. However, there were no significant differences in small-world properties (Cp, γ, λ, Lp, and σ) and network efficiency [global and local efficiencies (Eloc)] between the two groups. In addition, the TAO group demonstrated reduced betweenness centrality in the right fusiform and increased nodal Eloc in the right intraparietal sulcus (P < 0.05, Bonferroni-corrected). Furthermore, the TAO group displayed altered functional connections among the default-mode network (DMN), visual network (VN), sensorimotor network (SMN), and cingulo-opercular network (CON). Patients with TAO exhibited abnormal topological organization of the human brain connectome, including decreased betweenness centrality and increased nodal Eloc. Moreover, the TAO group displayed altered functional connections primarily within the DMN, VN, SMN, and CON. These findings provide crucial insights into the neural mechanisms underlying visual loss, abnormal emotion regulation, and cognitive deficits in TAO patients.

Double-pentagon silicon chains in a quasi-1D Si/Ag(001) surface alloy

Silicon surface alloys and silicide nanolayers are highly important as contact materials in integrated circuit devices. Here we demonstrate that the submonolayer Si/Ag(001) surface reconstruction, reported to exhibit interesting topological properties, comprises a quasi-one-dimensional Si-Ag surface alloy based on chains of planar double-pentagon Si moieties. This geometry is determined using a combination of density functional theory calculations, scanning tunnelling microscopy, and grazing incidence x-ray diffraction simulations, and yields an electronic structure in excellent agreement with photoemission measurements. This work provides further evidence of pentagonal geometries in 2D materials and heterostructures and elucidates the importance of surface alloying in stabilizing their formation.

Transcendence and Normality of Complex Numbers via Hurwitz Continued Fractions

Abstract We study the topological, dynamical, and descriptive set-theoretic properties of Hurwitz continued fractions. Hurwitz continued fractions associate an infinite sequence of Gaussian integers to every complex number that is not a Gaussian rational. The resulting space of sequences of Gaussian integers $\Omega $ is not closed. Using an iterative procedure, we show that $\Omega $ contains a natural subset whose closure $\overline{\textsf{R}}$ encodes continued fraction expansions of complex numbers that are not Gaussian rationals. We prove that $(\overline{\textsf{R}}, \sigma )$ is a subshift with a feeble specification property. As an application, we determine the rank in the Borel hierarchy of the set of Hurwitz normal numbers with respect to the complex Gauss measure. We also construct a family of complex transcendental numbers with bounded partial quotients.

On Knots with Cyclic Symmetries

This review paper investigates the relationship between symmetries of knots and links in the three-dimensional sphere, with a focus on cyclic symmetries, and their associated polynomial invariants. It examines the behavior of these polynomials in the case where the knot is set-wise fixed by the action of finite cyclic group on the 3-sphere. The study highlights how these invariants can obstruct the existence of such symmetries, providing valuable insights into the topological properties of these knots. The paper reviews established results on polynomial criteria for periodicity and their extensions to freely periodic links. It also explores generalizations to study the symmetries of virtual knots and spatial graphs.

Multivariate growth analysis on D019–phase Mn3Ga kagome–based topological antiferromagnets

The combination of antiferromagnetism and topological properties in Mn3X (X = Sn,Ge,Ga) offers a unique platform to explore novel spin-dependent phenomena and develop innovative spintronic devices. Here, we have systematically investigated the phase transition of Mn3Ga thin films on SiO2(001)/Si substrates under various growth parameters such as seeding layer structure, annealing conditions, and film thickness. The relatively thick Mn3Ga films grown with Ru seeding exhibit a variety of polycrystalline hexagonal phases, including (002), and (201). The addition of a Ta layer to the conventional Ru seeding layer promotes the formation of nearly single-crystal antiferromagnetic (AF) Mn3Ga(002) phase from the relatively thin Mn3Ga films after annealing at 773 K. The investigation of the growth mechanism of Mn3Ga polycrystalline thin films provides a reference strategy for exploring Mn-based AF spintronic devices.

Explicit construction of Hermitian Yang-Mills instantons on coset manifolds

In four dimensions, ’t Hooft symbols offer a compact and powerful framework for describing the self-dual structures fundamental to instanton physics. Extending this to six dimensions, the six-dimensional ’t Hooft symbols can be constructed using the isomorphism between the Lorentz group Spin(6) and the unitary group SU(4). We demonstrate that the six-dimensional self-dual structures governed by the Hermitian Yang-Mills equations can be elegantly organized using these generalized ’t Hooft symbols. We also present a systematic method for constructing Hermitian Yang-Mills instantons from spin connections on six-dimensional manifolds using the generalized ’t Hooft symbols. We provide a thorough analysis of the topological invariants such as instanton and Euler numbers.

A novel approach to detecting epileptic patients: complex network-based EEG classification

Abstract Detection of epileptic seizures is important for early diagnosis and treatment. It is known that the behavioral patterns of the brain in electroencephalogram (EEG) signals have huge and complex fluctuations. Diagnosing epilepsy by analyzing signals are costly process. Various methods are used to classify epileptic seizures. However, the inadequacy of these approaches in classifying signals makes it difficult to diagnose epilepsy. Complex network science produces effective solutions for analyzing interrelated structures. Using methods based on complex network analysis, it is possible to EEG signals analyze the relationship between signals and perform a classification process. In this study proposes a novel approach for classifying epileptic seizures by utilizing complex network science. In addition, unlike the studies in the literature, classification processes were carried out with lower dimensional signals by using 1-s EEG signals instead of 23.6-s full-size EEG signals. Using the topological properties of the EEG signal converted into a complex network, the classification process has been performed with the Jaccard Index method. The success of the classification process with the Jaccard Index was evaluated using Accuracy, F1 Score, Recall, and K-Fold metrics. In the results obtained, the signals of individuals with epileptic seizures were separated with an accuracy rate of 98.15%.

Two types of corner states in two dimensional photonic crystals with finite sizes

Abstract Using two-dimensional (2D) square lattice photonic crystals (PCs) with different topological properties, we design different combined structures to construct two types of topological corner states (CSs), named as Type I and Type II CSs. Then by tuning sizes of inner PCs in the combined structures, we systematically investigate size effects on the two types of CSs. Numerical results demonstrate as the structures decrease to their critical sizes, due to the interactions of opposite interfaces and the couplings of corners, size changes of inner PCs in the combined structures have significant effects on the frequencies, degeneracies and mode field distributions of the two types of CSs. Moreover, Type I CSs peform better topological stability than Type II CSs during the size changes of structures. We also monitor mode field localizations of the two types of CSs and reveal that their localizations are only related to the types of the CSs, and have no relations to sizes and overall symmetries of the combined structures. Our research enriches the study of higher order topological CSs and paves the way for design and manufacture of optical micro-nano devices with photonic topological CSs.

Interaction-Induced Crystalline Topology of Excitons.

We apply the topological theory of symmetry indicators to interaction-induced exciton band structures in centrosymmetric semiconductors. Crucially, we distinguish between the topological invariants inherited from the underlying electron and hole bands and those that are intrinsic to the exciton wave function itself. Focusing on the latter, we show that there exists a class of exciton bands for which the maximally localized exciton Wannier states are shifted with respect to the electronic Wannier states by a quantized amount; we call these excitons shift excitons. Our analysis explains how the exciton spectrum can be topologically nontrivial and sustain exciton edge states in open boundary conditions even when the underlying noninteracting bands have a trivial atomic limit. We demonstrate the presence of shift excitons as the lowest energy neutral excitations of the Su-Schrieffer-Heeger model in its trivial phase when supplemented by local two-body interactions.

A new model describing improved bound-state and statistical properties in the framework of deformed Schrödinger equation for the improved Coshine Yukawa potential in 3D-NR(NCPS) symmetries

In this paper, within the three-dimensional non-relativistic non-commutative phase–space (3D-NR(NCPS)) symmetries, we examined the 3D deformed Schrödinger equation (3D-DSE) using the improved Coshine Yukawa potential (ICYP) model composed from Coshine Yukawa potential (CYP) ([Formula: see text]) and other terms produced from the effect of phase–space deformation ([Formula: see text] and [Formula: see text]). For this study, the 3D-DSE in the 3D-NR(NCPS) symmetry is solved and discussed with standard independent time perturbation theory and the generalized Bopp’s shifts method. For the homogeneous (H2 and N2) and heterogeneous (LiH, ScH, and HCL) diatomic molecules, it is obtained that the improved non-relativistic energy equation and eigenfunction for the ICYP in the presence of deformation phase–space are dependent on the discrete atomic quantum numbers ([Formula: see text],m), the dissociation energy [Formula: see text], the equilibrium bond length the screening parameter [Formula: see text], the deformation phase parameters ([Formula: see text]) and the deformation space parameters ([Formula: see text]). Additionally, the thermal properties of the CYP and ICYP with 3D Schrödinger equation (3D-SE) and 3D-DSE are thoroughly examined in the three-dimensional non-relativistic quantum mechanics (3D-NR(QM)) known in the literature symmetry and 3D-NR(NCPS) symmetries, including the partition function, mean energy, free energy, specific heat, and entropy. Furthermore, we discussed particular cases of thermodynamic characteristics for the ICYP model. In our study, we have shown that all the physical values related to energy and thermodynamic properties within the framework of the 3D(NR)NCPS symmetry are equal to the corresponding values within the framework of 3D-NR(QM) symmetry known in the literature, in addition to minor effects resulting from their interaction with the topological properties of the deformed phase–space. This study has multiple applications in various domains, including atomic and molecular physics.

Multiferroic properties and the layer-stacking quantum anomalous Hall effect in two-dimensional RuOHX (X = F, Cl, Br)

Exploring the physics coupled with layer degrees of freedom in materials has become a hot topic in quantum layertronics. We propose a robust second-order topological insulator monolayer RuOHX (X = F, Cl, and Br), a two-dimensional ferromagnetic semiconductor with large valley polarization, capable of undergoing topological phase transition induced by strain effect. In the bilayer RuOHX, we achieve layer-polarized anomalous Hall effect through interlayer sliding, originating from layer-stacking Berry curvature. Moreover, it can be controlled and reversed by the direction of ferroelectric polarization. Under appropriate biaxial strain, the bilayer RuOHX exhibits quantum layer spin Hall effect in which the helical edge states are manifested as spin-chirality-locking, due to the degeneracy of layer-polarized quantum anomalous Hall effect. Our work explores the potential application via layer-stacking topological properties for future quantum device applications.