Dimensional Sphere Research Articles

Improved spectral convergence rates for graph Laplacians on ε-graphs and k-NN graphs

In this paper we improve the spectral convergence rates for graph-based approximations of weighted Laplace-Beltrami operators constructed from random data. We utilize regularity of the continuum eigenfunctions and strong pointwise consistency results to prove that spectral convergence rates are the same as the pointwise consistency rates for graph Laplacians. In particular, for an optimal choice of the graph connectivity ε, our results show that the eigenvalues and eigenvectors of the graph Laplacian converge to those of a weighted Laplace-Beltrami operator at a rate of O(n−1/(m+4)), up to log factors, where m is the manifold dimension and n is the number of vertices in the graph. Our approach is general and allows us to analyze a large variety of graph constructions that include ε-graphs and k-NN graphs. We also present the results of numerical experiments analyzing convergence rates on the two dimensional sphere.

Detection of the Neurodegenerative Drug in a Biological Sample Using Three-Dimensional Sphere Mixed Metal Oxide Tailored with Carbon Fiber as an Electrocatalyst by Voltammetry Technique

The scientific community has shown a lot of interest in the design and engineering of electrodes as sensing platforms for sensitive and selective drug detection in recent years. The sensor based on three-dimensional spheres like NiCo2O4 tailored on carbon nanofiber CNF as a hybrid composite was prepared by hydrothermal-assisted synthesis. This combination was preferred due to its superior electro-catalytic activity. Various physical characterizations, such as morphological analysis and spectroscopic tests, were performed on the as-synthesized composite. The electrochemical performance of the modified NCO@CNF electrode is investigated for the detection of clioquinol. The structural property of NCO@CNF promoted a high charge transfer ratio with outstanding conductivity. Upon the amperometric analysis, a low detection limit (LOD) of 8 nM with a very wide linear range of 0.01–301.4 μM and good sensitivity of 0.177 μA μM−1 cm−2 was witnessed for clioquinol detection. Additionally, NCO@CNF modified electrode demonstrated superior anti-interference capability, repeatability, reproducibility, cycle stability (30 cycles) and operational stability (2000 s). The proposed NCO@CNF electrode was tested with diverse biological samples like human blood and urine and showed remarkable recovery.

Highly efficient visible/near-infrared light photocatalytic degradation of antibiotic wastewater over 3D yolk-shell ZnFe2O4 supported 0D carbon dots with up-conversion property

The development of effective visible and near-infrared photocatalysts is highly promising in the current field of photocatalysis. Herein, carbon dots/ZnFe 2 O 4 (CDs/ZFO) with coating zero dimensional (0D) CDs on the surface of three dimensional (3D) yolk-shell ZFO spheres was designed and synthesized via a self-templated solvothermal method. The as-prepared CDs/ZFO composites displayed outstanding visible and near-infrared photocatalytic degradation activity of tetracycline (TC), and the optimal 3% CDs/ZFO sample with loading 3% (mass) CDs displayed the highest photocatalytic TC degradation ability under visible light (79.5% within 120 min) and near-infrared light (41% within 120 min). The enhancement of photocatalytic activity for CDs/ZFO composite is mainly ascribed to the fact that 0D/3D yolk-shell CDs/ZFO structure not only effectively reflect the incident light to increase the utilization efficiency of solar light, but also utilize the up-conversion photoluminescence and electronic conductivity properties of CDs to broaden sunlight absorption range and promote separation and transfer of electron-hole pairs.

Computing the Conformal Barycenter

The conformal barycenter of a point cloud on the sphere at infinity of the Poincare ball model of hyperbolic space is a hyperbolic analogue of the geometric median of a point cloud in Euclidean space. It was defined by Douady and Earle as part of a construction of a conformally natural way to extend homeomorphisms of the circle to homeomorphisms of the disk, and it plays a central role in Millson and Kapovich's model of the configuration space of cyclic linkages with fixed edgelengths. In this paper we consider the problem of computing the conformal barycenter. Abikoff and Ye have given an iterative algorithm for measures on $\mathbb{S}^1$ which is guaranteed to converge. We analyze Riemannian versions of Newton's method computed in the intrinsic geometry of the Poincare ball model. We give Newton-Kantorovich (NK) conditions under which we show that Newton's method with fixed step size is guaranteed to converge quadratically to the conformal barycenter for measures on any $\mathbb{S}^d$ (including infinite-dimensional spheres). For measures given by $n$ atoms on a finite dimensional sphere which obey the NK conditions, we give an explicit linear bound on the computation time required to approximate the conformal barycenter to fixed error. We prove that our NK conditions hold for all but exponentially few $n$ atom measures. For all measures with a unique conformal barycenter we show that a regularized Newton's method with line search will always converge (eventually superlinearly) to the conformal barycenter. Though we do not have hard time bounds for this algorithm, experiments show that it is extremely efficient in practice and in particular much faster than the Abikoff-Ye iteration.

회전체의 특성이 시각적 선호에 미치는 영향

This study assumed two key parameters that describe smoothness. Previously, the number of studies that defined smoothness with the concept of curvature has been limited. The parameters were the two physical characteristics of curvature: the number of waves and the amplitude of them. The purpose of the study was to observe which of the two parameters influences the perception of smoothness and to examine the relationship between the sense of aesthetic beauty and smoothness. The visual stimuli used in the study were transformed three-dimensional spheres, based on the combination of the three levels of the two parameters of nine distinct conditions. We analyzed the three responses that measured the preference of each visual stimuli, the familiarity score, and the smoothness evaluation score, each with the linear mixed model whose fixed effects were the two parameters and random effects were the participants’ individual differences. Nearly the eighty percent of the variance of the smoothness evaluation score was explained by the linear model with the two key parameters and their interaction. The physical characteristics of a viewed object were far more significant than individual differences such as personality factors and the manner of art appreciation. In conclusion, the study examined the perception of smoothness based on the change of the physical characteristics of a shape. The study further recognized the relationship between smoothness and the aesthetic preference. No significant influence of the participants’ individual difference such as gender, the degree in fine arts, personality factors, and the manner of art appreciation was observed. The amplitude of waves, rather than the number of them, was far more significant to the perception of smoothness.

Five-dimensional Poincaré sphere system for representing azimuthally varying vector optical fields

Vector optical fields (VOFs) with space-variant polarization on the wave front, have attracted considerable attention and have been applied in many realms from focal engineering to light-matter interaction. Recently, some upgraded Poincar\'e sphere (PS) models have been presented to describe VOFs, providing new insights in studying and applying structured light. Here, we report a model of five-dimensional (5D) PS system consisting of a series of three-dimensional (3D) spheres (Bloch spheres) located on a two-dimensional (2D) plane. In fact, this model is geometrically similar to the solar system, and the 3D Bloch spheres and the 2D plane are analogous to the ``planets'' and their orbital plane, respectively. This model is the most generalized model describing the azimuthally varying VOFs we know so far. A reliable and flexible experimental scheme is exploited to generate the VOFs represented by the 5D PS system. Furthermore, the 5D PS system is a complete tool to represent the azimuthally varying VOFs with polarization changing along arbitrary circular path on any 3D PS, and can be implemented for representing and designing the spin and orbital angular momenta. The 5D PS system graphically simplifies the representation of complex structured light and provides a prominent toolkit to the design and application of VOFs as well as the corresponding optical angular momentum.

Gravothermal phase transition, black holes and space dimensionality

In the framework of gravothermal evolution of an ideal monatomic fluid, I examine the dynamical instability of the fluid sphere in ($N$+1) dimensions by exploiting Chandrasekhar's criterion to each quasistatic equilibrium along the sequence of the evolution. Once the instability is triggered, it would probably collapse into a black hole if no other interaction halts the process. From this viewpoint, the privilege of (3+1)-dimensional spacetime is manifest, as it is the marginal dimensionality in which the ideal monatomic fluid is stable but not too stable. Moreover, it is the unique dimensionality that allows stable hydrostatic equilibrium with positive cosmological constant. While all higher dimensional ($N>3$) spheres are genuinely unstable. In contrast, in (2+1)-dimensional spacetime it is too stable either in the context of Newton's theory of gravity or Einstein's general relativity. It is well known that the role of negative cosmological constant is crucial to have the Ba\~nados-Teitelboim-Zanelli (BTZ) black hole solution and the equilibrium configurations of a fluid disk. Owing to the negativeness of the cosmological constant, there is no unstable configuration for a homogeneous fluid disk to collapse into a naked singularity, which supports the cosmic censorship conjecture. However, BTZ holes of mass $\mathcal{M}_{\rm BTZ}>0$ could emerge from collapsing fluid disks. The implications of spacetime dimensionality are briefly discussed.

Slice conformality and Riemann manifolds on quaternions and octonions

In this paper we establish quaternionic and octonionic analogs of the classical Riemann surfaces. The construction of these manifolds has nice peculiarities and the scrutiny of Bernhard Riemann approach to Riemann surfaces, mainly based on conformality, leads to the definition of slice conformal or slice isothermal parameterization of quaternionic or octonionic Riemann manifolds. These new classes of manifolds include slice regular quaternionic and octonionic curves, graphs of slice regular functions, the 4 and 8 dimensional spheres, the helicoidal and catenoidal 4 and 8 dimensional manifolds. Using appropriate Riemann manifolds, we also give a unified definition of the quaternionic and octonionic logarithm and n-th root function.

Extended DMPs Framework for Position and Decoupled Quaternion Learning and Generalization

Dynamic movement primitives (DMPs) as a robust and efficient framework has been studied widely for robot learning from demonstration. Classical DMPs framework mainly focuses on the movement learning in Cartesian or joint space, and can't properly represent end-effector orientation. In this paper, we present an extended DMPs framework (EDMPs) both in Cartesian space and 2-Dimensional (2D) sphere manifold for Quaternion-based orientation learning and generalization. Gaussian mixture model and Gaussian mixture regression (GMM-GMR) are adopted as the initialization phase of EDMPs to handle multi-demonstrations and obtain their mean and covariance. Additionally, some evaluation indicators including reachability and similarity are defined to characterize the learning and generalization abilities of EDMPs. Finally, a real-world experiment was conducted with human demonstrations, the endpoint poses of human arm were recorded and successfully transferred from human to the robot. The experimental results show that the absolute errors of the Cartesian and Riemannian space skills are less than 3.5 mm and 1.0°, respectively. The Pearson’s correlation coefficients of the Cartesian and Riemannian space skills are mostly greater than 0.9. The developed EDMPs exhibits superior reachability and similarity for the multi-space skills’ learning and generalization. This research proposes a fused framework with EDMPs and GMM-GMR which has sufficient capability to handle the multi-space skills in multi-demonstrations.

Nonautonomous vector fields on $$S^3$$: Simple dynamics and wild embedding of separatrices

We construct new substantive examples of non-autonomous vector fields on 3-dimensional sphere having a simple dynamics but non-trivial topology. The construction is based on two ideas: the theory of diffeomorpisms with wild separatrix embedding (Pixton, Bonatti-Grines, etc.) and the construction of a non-autonomous suspension over a diffeomorpism (Lerman-Vainshtein). As a result, we get periodic, almost periodic or even nonrecurrent vector fields which have a finite number of special integral curves possessing exponential dichotomy on $\R$ such that among them there is one saddle integral curve (with an exponential dichotomy of the type (3,2)) having wildly embedded two-dimensional unstable separatrix and wildly embedded three-dimensional stable manifold. All other integral curves tend, as $t\to \pm \infty,$ to these special integral curves. Also we construct another vector fields having $k\ge 2$ special saddle integral curves with tamely embedded two-dimensional unstable separatrices forming mildly wild frames in the sense of Debrunner-Fox. In the case of periodic vector fields, corresponding specific integral curves are periodic with the period of the vector field, and they are almost periodic for the case of almost periodic vector fields.

Dividing a Sphere Hierarchically into a Large Number of Spherical Pentagons Using Equal Area or Equal Length Optimization

Dividing a 2-dimensional sphere uniformly into a large number of spherical polygons is a challenging mathematical problem, which has been studied across many disciplines due to its important practical applications. Most sphere subdivisions are achieved using spherical triangles, quadrangles, or a combination of hexagons and pentagons. However, spherical pentagons, which may create elegant configurations, remain under-explored. This study presents a new sphere subdivision method to generate a large number of spherical pentagons based on successively subdividing a module of an initial spherical dodecahedron. The new method can conveniently control the shapes of generated spherical pentagons through specified design parameters. Two optimization problems have been investigated: (I) dividing a sphere into spherical pentagons of equal area; (II) minimizing the number of different arc lengths used in the pentagonal subdivision. A variety of examples are presented to demonstrate the effectiveness of the new method. This study shows that treating the mathematical challenge of dividing a sphere uniformly into a large number of spherical polygons as an optimization problem can effectively obtain equal area or equal length sphere subdivisions. Furthermore, considering additional constraints on the optimization problem may achieve sphere subdivisions of specific characteristics. • A new sphere subdivision method is developed to generate a large number of spherical pentagons. • The subdivision approach is performed locally and hierarchically on a module of a spherical dodecahedron. • The mathematical challenge of dividing a sphere uniformly is treated as an optimization problem. • Optimization algorithms are used to achieve sphere subdivisions of specific characteristics.

Independence complexes of strongly orderable graphs

We prove that for any finite strongly orderable (generalized strongly chordal) graph G, the independence complex Ind(G) is either contractible or homotopy equivalent to a wedge of spheres of dimension at least bp(G)−1, where bp(G) is the biclique vertex partition number of G. In particular, we show that if G is a chordal bipartite graph, then Ind(G) is either contractible or homotopy equivalent to a sphere of dimension at least bp(G) − 1.

Spherical flow diagram with finite hyperbolic chain-recurrent set

In this paper, authors examine flows with a finite hyperbolic chain-recurrent set without heteroclinic intersections on arbitrary closed n-manifolds. For such flows, the existence of a dual attractor and a repeller is proved. These points are separated by a (n−1)-dimensional sphere, which is secant for wandering trajectories in a complement to attractor and repeller. The study of the flow dynamics makes it possible to obtain a topological invariant, called a spherical flow scheme, consisting of multi-dimensional spheres that are the intersections of a secant sphere with invariant saddle manifolds. It is worth known that for some classes of flows spherical scheme is complete invariant. Thus, it follows from G. Fleitas results that for polar flows (with a single sink and a single source) on the surface, it is the spherical scheme that is complete equivalence invariant.

Heteroatom‐Doped Three Dimensional Hierarchical Porous Carbon Sphere from Melamine and Glucose for High‐Performance Supercapacitor

AbstractHeteroatom‐doped‐carbon with high electrochemical performance is crucial for the development of advanced carbonaceous materials for energy use. However, achieving homogeneous doping while maintaining electrochemically active microstructure remains challenging. Herein, a feasible hydrothermal protocol to fabricate uniformly N‐doped carbon spheres by in‐situ treatment of glucose and melamine was shown. Melamine served as the soft template to introduce heteroatom dopants while releasing gases to enrich the porous structure. Profiting from the structural directing effect of melamine along with an effective fabrication route, the as‐prepared material presented multiple electrochemically advantageous features including a high surface area, a hierarchical porous structure at different levels, abundant heteroatom doping, etc., and thus delivered high supercapacitance. This work provided an effective strategy for fabrication of carbonaceous scaffold for advanced heterostructured energy materials.

Nonsymmetric wormholes and localized big rip singularities in Einstein-Weyl gravity

The inclusion of the Weyl squared term in the gravitational action is one of the most simple, yet non trivial modifications to General Relativity at high energies. Nevertheless the study of the spherically-symmetric vacuum solutions of this theory has received much attention only in recent times. A new type of asymptotically flat wormhole which does not match symmetrically at a finite radius with another sheet of the spacetime is presented. The outer spacetime is characterized by a newtonian potential with a Yukawa correction, and has gravitational properties that can be arbitrarily close to the ones of a Schwarzschild black hole. The internal spacetime instead possesses a singularity at $r=\infty$ with the topology of a 2-dimensional sphere. The expansion scalar of geodesics reaching this singularity diverges in a finite amount of proper time, with a striking resemblance with the future singularity of the Big Rip cosmological scenario. In terms of the external Yukawa hair and mass $M$, these new wormholes fill a large region of the two-dimensional parameter space of physical solutions with $M>0$. On the contrary black holes, both of Schwarzschild and non-Schwarzschild nature, are confined on a line. We argue that this type of wormholes are ideal black hole mimickers.

Ru-doped 3D porous Ni3N sphere as efficient Bi-functional electrocatalysts toward urea assisted water-splitting

Developing efficient, stable and ideal urea oxide (UOR) electrocatalyst is key to produce green hydrogen in an economical way. Herein, Ru doped three dimensional (3D) porous Ni3N spheres, with tannic acid (TA) and urea as the carbon and nitrogen resources, is synthesized via hydrothermal and low-temperature treated process (Ru–Ni3N@NC). The porous nanostructure of Ni3N and the nickel foam provide abundant active sites and channel during catalytic process. Moreover, Ru doping and rich defects favor to boost the reaction kinetics by optimizing the adsorption/desorption or dissociation of intermediates and reactants. The above advantages enable Ru–Ni3N@NC to have good bifunctional catalytic performance in alkaline media. Only 43 and 270 mV overpotentials are required for hydrogen evolution (HER) and oxygen evolution (OER) reactions to drive a current of 10 mA cm−2. Moreover, it also showed good electrocatalytic performance in neutral and alkaline seawater electrolytes for HER with 134 mV to drive 10 mA cm−2 and 83 mV to drive 100 mA cm−2, respectively. Remarkably, the as-designed Ru–Ni3N@NC also owns extraordinary catalytic activity and stability toward UOR. Moreover, using the synthesized Ru–Ni3N@NC nanomaterial as the anode and cathode of urea assisted water decomposition, a small potential of 1.41 V was required to reach 10 mA cm−2. It can also be powered by sustainable energy sources such as wind, solar and thermal energies. In order to make better use of the earth's abundant resources, this work provides a new way to develop multi-functional green electrocatalysts.

A Topological sphere theorem for contact CR-warped product submanifolds of an odd-dimensional unit sphere

AbstractIn this paper, we investigate topological sphere theorems for compact minimal contact CR-submanifolds of odd dimensional unit sphere. We show that if an inequality involving the warping function and the scalar curvature of the fibers is satisfied, a compact minimal contact CR-warped product submanifold of the odd dimensional unit sphere is homeomorphic to the sphere. In particular case, for 5-dimensional unit sphere, we show that a 4-dimensional compact minimal contact CR-warped product submanifold is homeomorphic to a sphere if ∣∇lnf∣2 < 1 is satisfied. By using Bonnet-Myers’s theorem we give a result about fundamental group and by Leung’s theorem we obtain a result about homology groups of a contact CR-warped submanifold.

Ultrathin graphene@NiCo2S4@Ni-Mo layered double hydroxide with a 3D hierarchical flowers structure as a high performance positive electrode for hybrid supercapacitor

This study presents three-dimensional flower-like spheres composed of NiCo2S4@Ni-Mo layered double hydroxide (LDH) nanocomposites, which are grown in situ on reduced graphene oxide (RGO). The RGO@NiCo2S4@NiMo-LDH nanocomposites exhibit remarkable electrochemical properties and have a specific capacitance of 1346 F g−1 at the current density of 1 A g−1. In addition, using RGO@NiCo2S4 @NiMo-LDH hybrid nanosheets and activated carbon (AC) as the positive and negative electrodes, the asymmetric supercapacitor device was successfully assembled, and it showed a significant energy density of 59.38 Wh kg−1 at a power density of 808.19 W kg−1, with good cycle stability. The initial capacity retention rate is 80 % after 10,000 cycles. These results demonstrate the great potential of the synthesized RGO@NiCo2S4@NiMo-LDH composite for the development of high-performance electrode materials for supercapacitors.

Trivializable and Quaternionic Subriemannian Structures on {mathbb {S}}^7 and Subelliptic Heat Kernel

On the seven dimensional Euclidean sphere {mathbb {S}}^7 we compare two subriemannian structures with regards to various geometric and analytical properties. The first structure is called trivializable and the underlying distribution {mathcal {H}}_T is induced by a Clifford module structure of {mathbb {R}}^8. More precisely, {mathcal {H}}_T is rank 4, bracket generating of step two and generated by globally defined vector fields. The distribution {mathcal {H}}_{Q} of the second structure is of rank 4 and step two as well and obtained as the horizontal distribution in the quaternionic Hopf fibration {mathbb {S}}^3hookrightarrow {mathbb {S}}^7rightarrow {mathbb {S}}^4. Answering a question in: Markina and Godoy Molina (Rev Mat Iberoam 27(3), 997–1022, 2011) we first show that {mathcal {H}}_{Q} does not admit a global nowhere vanishing smooth section. In both cases we determine the Popp measures [20], the intrinsic sublaplacians Delta _{text {sub}}^T and Delta _{text {sub}}^{Q} and the nilpotent approximations. We conclude that both subriemannian structures are not locally isometric and we discuss properties of the isometry group. By determining the first heat invariant of the sublaplacians it is shown that both structures are also not isospectral in the subriemannian sense.

Numerical analysis of debris flows along the Sichuan-Tibet railway based on an improved 3D sphere DDA model and UAV-based photogrammetry

This paper develops an improved three-dimensional sphere discontinuous deformation analysis (SDDA) model and uses UAV-based photogrammetry to enable SDDA to numerically analyze potential long-runout debris flows in Zheduotang area along the Sichuan-Tibet railway. First, the original SDDA is improved by proposing a sphere-sphere bonding algorithm to bond spheres with tetrahedral layout to form four-sphere clusters to approximate non-sphere particles in debris flow. Second, the capacity of the improved SDDA is verified by comparing modelling results with four groups of laboratory tests of granular flow. Third, UAV-based photogrammetry is used to obtain the real morphology of the extremely complex terrain surface of Zheduotang area which is finally transferred into a large number of triangular faces as the fixed boundary faces of SDDA. Then, three potential debris flow scenarios on this terrain surface are numerically studied, and some suggestions for engineering disaster prevention are drawn. Model performance shows that, the improved SDDA is a feasible tool to simulate debris flows. With the help of UAV-based photogrammetry, it shows significantly helpful for geohazard prevention for infrastructures in mountainous areas like the Sichuan-Tibet railway.