Spherical Space Research Articles

Robust Spherical Laplacian Embedding.

Laplacian embedding (LE) aims to project high-dimensional input data samples, which often contain nonlinear structures, into a low-dimensional space. However, existing distance functions used in the embedding space fail to provide discriminative representations for real-world datasets, especially those related to text analysis or image processing. Cosine similarity measurements are advantageous in dealing with sparse data but are fragile to the impact of outliers and noise from samples or data features. In this work, we propose robust spherical LE (RS-LE), a powerful method that builds the LE on a novel metric that unifies the Euclidean distance and cosine similarity in a spherical space using a robust lp,p -norm. We introduce an efficient iterative algorithm, the proximal alternating linearized minimization (ALM) method, to solve the difficult optimization problem that arises from the new metric, which is neither convex nor smooth. This algorithm allows the solution to achieve both global and objective convergence. Our findings are presented in rigorous theoretical analysis and validated in experiments.

Fractional optical normalization operator of magnetic field and electroosmotic optimistic energy

In this manuscript, we introduce optical fractional normalization and fractional recursive operators for a particle. The significance of this research lies in obtaining optical fractional recursive operators and normalized operators of magnetic fields through the utilization of a fractional spherical frame within spherical space. Throughout the study, we employ the general form of fractional derivatives, as it enables us to derive more generalized results compared to classical derivatives. Additionally, we derive the microfluidic optical fractional electroosmotic magnetic optimistic fractional energy. We propose a design for optical recursion-based fractional electroosmotic magnetic optimistic fractional energy. This innovative study holds significant potential for applications across various fields, including fluid dynamics, optics, and energy conversion.

Some Aspects of Maxwell’s Equations, Klein-Gordon Equations, and Heat and Mass Transfer Equations in an n-Dimensional Maximally Symmetric Space-Time from the Classical and Quantum Mechanical Standpoints

This manuscript examines Maxwell’s equations, Klein-Gordon equations, and heat and mass transfer equations in n-dimensional maximally symmetric space-time. It investigates these equations in spherical and hyperbolic spaces embedded in higher-dimensional Euclidean and Minkowski spaces. The study focuses on the implications of these geometries and symmetries on the behaviour of the equations, highlighting how specific transformations and parametrizations impact their solutions. The findings reveal the underlying connections between geometric symmetries and physical laws, providing insights into their possible applications in theoretical physics. We touch upon both classical and quantum mechanical aspects of density and velocity evolutions with time in the universe. Quantum mechanical aspects of single and two-particle state evolution and statistical moments of the matter four-current are derived from the quantum Boltzmann equation and Feynman’s path integral method for fields applied to gravity interacting with electrons and positrons.

Outer billiards in the spaces of oriented geodesics of the three‐dimensional space forms

AbstractLet be the three‐dimensional space form of constant curvature , that is, Euclidean space , the sphere , or hyperbolic space . Let be a smooth, closed, strictly convex surface in . We define an outer billiard map on the four‐dimensional space of oriented complete geodesics of , for which the billiard table is the subset of consisting of all oriented geodesics not intersecting . We show that is a diffeomorphism when is quadratically convex. For , has a Kähler structure associated with the Killing form of . We prove that is a symplectomorphism with respect to its fundamental form and that can be obtained as an analogue to the construction of Tabachnikov of the outer billiard in defined in terms of the standard symplectic structure. We show that does not preserve the fundamental symplectic form on associated with the cross product on , for . We initiate the dynamical study of this outer billiard in the hyperbolic case by introducing and discussing a notion of holonomy for periodic points.

Spherical Cinema: Transformation of Film Language in Virtual Reality

The article analyzes the formation of a new media language in the context of spherical film space. It examines the features and the construction of spherical cinema, and raises issues of adapting classical film language to a new developing cinematic format, taking into account its artistic uniqueness and the peculiarities of the audience’s perception.

Non‐Euclidean Sliced Optimal Transport Sampling

AbstractIn machine learning and computer graphics, a fundamental task is the approximation of a probability density function through a well‐dispersed collection of samples. Providing a formal metric for measuring the distance between probability measures on general spaces, Optimal Transport (OT) emerges as a pivotal theoretical framework within this context. However, the associated computational burden is prohibitive in most real‐world scenarios. Leveraging the simple structure of OT in 1D, Sliced Optimal Transport (SOT) has appeared as an efficient alternative to generate samples in Euclidean spaces. This paper pushes the boundaries of SOT utilization in computational geometry problems by extending its application to sample densities residing on more diverse mathematical domains, including the spherical space 𝕊d, the hyperbolic plane ℍd, and the real projective plane ℙd. Moreover, it ensures the quality of these samples by achieving a blue noise characteristic, regardless of the dimensionality involved. The robustness of our approach is highlighted through its application to various geometry processing tasks, such as the intrinsic blue noise sampling of meshes, as well as the sampling of directions and rotations. These applications collectively underscore the efficacy of our methodology.

Convolutional Neural Networks Approximation in Quasi-Orlicz Spaces on Sphers

It is necessary to study the theoretical bases of an approximation deep convolutional neural networks, because of its interesting developments in vital domains. The approximation abilities of deep-convolution neural networks produced by downsampling operators in quasi- Orlicz spaces have been studied, since this space is wider and more important than other spaces. In this paper, we define quasi-Orlicz norm on spherical spaces. In addition, modulus of smoothness is also studied in terms of quasi-Orlicz norm. Finally, Function approximation theorems are studied by using convolution neural networks with

An efficient method for electro-thermo-mechanical coupling effect in electrical contact on rough surfaces

In this study, a numerical strategy based on the boundary element method (BEM) is developed to simulate the electrical contact behavior on rough surfaces. By using the Fourier transform, the solutions of the electric potential, temperature rise, and displacement fields in the Fourier space are derived with finite boundaries, in which the electro-thermo-mechanical coupling effect is considered. With the consideration of the contact condition, an iterative algorithm is proposed to seek the solutions of rough surface electrical contact under a given contact force and total electric potential drop. The contact pressure, real contact area, electrical contact resistance, electric current, and temperature rise distribution are predicted. For the validation, the electrical contact results of a spherical punch-half space contact and a rough surface-elastic substrate contact are calculated based on the finite element method. The good agreement between the results from the BEM-based method and the finite element method indicates the accuracy of the presented method. Then, the effect of the total electric potential and the thermal expansion on the electrical contact behavior, the relationship between the contact conductance and contact stiffness, and the probability density function of the contact pressure and temperature rise distributions are evaluated using the proposed method.

Position Checking-Based Sampling Approach Combined with Attraction Point Local Optimization for Safe Flight of UAVs.

Trading off the allocation of limited computational resources between front-end path generation and back-end trajectory optimization plays a key role in improving the efficiency of unmanned aerial vehicle (UAV) motion planning. In this paper, a sampling-based kinodynamic planning method that can reduce the computational cost as well as the risks of UAV flight is proposed. Firstly, an initial trajectory connecting the start and end points without considering obstacles is generated. Then, a spherical space is constructed around the topological vertices of the environment, based on the intersections of the trajectory with the obstacles. Next, some unnecessary sampling points, as well as node rewiring, are discarded by the designed position-checking strategy to minimize the computational cost and reduce the risks of UAV flight. Finally, in order to make the planning framework adaptable to complex scenarios, the strategies for selecting different attraction points according to the environment are designed, which further ensures the safe flight of the UAV while improving the success rate of the front-end trajectory. Simulations and real-world experiment comparisons are conducted on a vision-based platform to verify the performance of the proposed method.

Extension of Hayward black hole in [formula omitted] gravity coupled with a scalar field

This study looks into regular black hole solutions in a theory of gravity called f(R) gravity, which also involves a scalar field. The f(R) theory changes Einstein’s ideas by adding a new function related to something called the Ricci scalar. This lets us tweak the equations that describe how gravity works. Adding a scalar field makes the theory more interesting, giving us more ways to investigate and understand it. The main goal of this research is to create regular black holes using a combination of f(R) gravitational theory and a scalar field. Regular solutions do not have any singularities, which are points where certain physical quantities, like invariants, become really big or undefined. In this context, we find two regular black hole solutions by using a spherical space with either an equal or unequal approach. For the solutions where we use the equal approach, we figure out the shape of f(R) and how it changes, along with its first and second derivatives. We demonstrate that Hayward’s solution in this theory stays steady because all the shapes of f(R) and their first and second derivatives are positive. Next, we focus on the case where the metric is not equal and figure out the black hole solution. We also find out what f(R) and the scalar field look like in this situation. We demonstrate that the solution in this case is a broader version of the Hayward solution. When certain conditions are met, we end up back at the scenario where the metrics are equal. We also prove that this model is stable because f(R), along with its first and second derivatives, are all positive. We analyze the trajectories of these black hole solutions and determine the forms of their conserved quantities that remain same along those trajectories.

Charged AdS black holes in presence of string cloud and Cardy-Verlinde formula

In this study, charged AdS black holes with a string cloud are taken into consideration. We identify three distinct black hole solutions within a specific temperature range, which subsequently merge into a single black hole beyond this range in the case of spherical space with low chemical potential and string density. However, in the case of large chemical potential and string density, only one black hole solution exists for all temperatures. Notably, we observe that for flat and hyperbolic space, there is only one black hole solution irrespective of the chemical potential and string density. By subtracting the contribution of the extremal black hole from the on-shell Euclidean action, the thermodynamical quantities associated with these black holes are calculated. We also investigate the stability of these black holes and find that medium-sized black hole is unstable. In the dual boundary theory side we verify the existence of bound state of quark-antiquark pair. Finally, we establish the Cardy-Verlinde formula for the boundary theory and we observed an additional energy term arising because of the presence of string cloud.

Unsupervised and Semi-Supervised Robust Spherical Space Domain Adaptation.

Adversarial domain adaptation has been an effective approach for learning domain-invariant features by adversarial training. In this paper, we propose a novel adversarial domain adaptation approach defined in the spherical feature space, in which we define spherical classifier for label prediction and spherical domain discriminator for discriminating domain labels. In the spherical feature space, we develop a spherical robust pseudo-label loss to utilize pseudo-labels robustly, which weights the importance of the estimated labels of target domain data by the posterior probability of correct labeling, modeled by the Gaussian-uniform mixture model in the spherical space. Our proposed approach can be generally applied to both unsupervised and semi-supervised domain adaptation settings. In particular, to tackle the semi-supervised domain adaptation setting where a few labeled target domain data are available for training, we propose a novel reweighted adversarial training strategy for effectively reducing the intra-domain discrepancy within the target domain. We also present theoretical analysis for the proposed method based on the domain adaptation theory. Extensive experiments are conducted on multiple benchmarks for object recognition, digit recognition, and face recognition. The results show that our method either surpasses or is competitive compared with the recent methods for both unsupervised and semi-supervised domain adaptation. Ablation studies also confirm the effectiveness of the spherical classifier, spherical discriminator, spherical robust pseudo-label loss, and reweighted adversarial training strategy.

Unraveling Meso-Substituent Steric Effects on the Mechanism of Hydrogen Evolution Reaction in NiII Porphyrin Hydrides Using DFT Method.

Substituents at the meso-site of metalloporphyrins profoundly influence the hydrogen evolution reaction (HER) mechanism. This study employs density functional theory (DFT) to computationally analyze NiII-porphyrin and its hydrides derived from tetrakis(pentafluorophenyl)porphyrin molecules, presenting stereoisomers in ortho- or para-positions. The results reveal that the spatial resistance effect of meso-substituted groups at the ortho- and para-positions induces significant changes in Ni-N bond lengths, angles, and reaction dynamics. For ortho-position substituents forming complex I, a favorable 88.88 ų spherical space was created, facilitating proton coordination and the formation of H2 molecules; conversely, para-position substituents forming complex II impeded H2 formation until bimolecular complexes arose. Molecular dynamics (MD) analysis and comparison were conducted on the intermediation products of I-H2 and (II-H)2, focusing on the configuration and energy changes. In the I-H2 products, H2 molecules underwent separation after 150 fs and overcame the 2.2 eV energy barrier. Subsequently, significant alterations in the spatial structure were observed as complex I deformed. In the case of (II-H)2, it was influenced by the distinctive "sandwich" configuration; the spatial structure necessitated overcoming a 6.7 eV energy barrier for H2 detachment and a process observed after 2400 fs.

Optical wave propagation phase for mKdV spherical electric flux density in sphere space

In this article, we illustrate surface flow alfa-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$alfa-$$\\end{document} microfluidical mKdV spherical electromotive phase in sphere space. Also, we obtain antiferromagnetic axially electrical alfa-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$alfa-$$\\end{document}microfluidical mKdV electric flux path circuit. We have electrical alfa-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$alfa-$$\\end{document}microfluidical geometrical mKdV free surface flow density in sphere space. Finally, we design antiferromagnetic wave propagation for alfa-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$alfa-$$\\end{document} microfluidical geometric mKdV surface flux density.

Yolk-shell construction of Co0.7Fe0.3 modified with dual carbon for broadband microwave absorption

The rational and effective combination of multicomponent materials and the design of subtle microstructure for efficient microwave absorption are still challenging. In this study, carbon-coated CoFe with heterogeneous interfaces was space-restricted in the void space of hollow mesoporous carbon spheres through a facile approach involving electrostatic adsorption and annealing, and a high-performance microwave absorber (MAs) (denoted as Co0.7Fe0.3@C@void@C) was successfully prepared. The heterostructure, three-dimensional lightweight porous morphology, and electromagnetic synergy strategy enabled the Co0.7Fe0.3@C@void@C material with yolk-shell structure to exhibit surprising microwave absorption properties. When the annealing temperature and filler loading were 550° C and 15 wt%, respectively, the composites exhibited an effective absorption bandwidth (EAB) of 7.16 GHz at 2.48 mm and a minimum reflection loss of −24.1 dB at 2.11 mm. A maximum EAB of 7.21 GHz at 2.37 mm could be achieved for the composite prepared with an annealing temperature of 650° C. In addition, radar cross-section experiments demonstrated, the potential practical applicability of Co0.7Fe0.3@C@void@C. This work expands a new avenue to develop high-performance and lightweight MAs with ingenious microstructure.

GLPanoDepth: Global-to-Local Panoramic Depth Estimation.

Depth estimation is a fundamental task in many vision applications. With the popularity of omnidirectional cameras, it becomes a new trend to tackle this problem in the spherical space. In this paper, we propose a learning-based method for predicting dense depth values of a scene from a monocular omnidirectional image. An omnidirectional image has a full field-of-view, providing much more complete descriptions of the scene than perspective images. However, fully-convolutional networks that most current solutions rely on fail to capture rich global contexts from the panorama. To address this issue and also the distortion of equirectangular projection in the panorama, we propose Cubemap Vision Transformers (CViT), a new transformer-based architecture that can model long-range dependencies and extract distortion-free global features from the panorama. We show that cubemap vision transformers have a global receptive field at every stage and can provide globally coherent predictions for spherical signals. As a general architecture, it removes any restriction that has been imposed on the panorama in many other monocular panoramic depth estimation methods. To preserve important local features, we further design a convolution-based branch in our pipeline (dubbed GLPanoDepth) and fuse global features from cubemap vision transformers at multiple scales. This global-to-local strategy allows us to fully exploit useful global and local features in the panorama, achieving state-of-the-art performance in panoramic depth estimation.

Interactive Design and Optics-Based Visualization of Arbitrary Non-Euclidean Kaleidoscopic Orbifolds.

Orbifolds are a modern mathematical concept that arises in the research of hyperbolic geometry with applications in computer graphics and visualization. In this paper, we make use of rooms with mirrors as the visual metaphor for orbifolds. Given any arbitrary two-dimensional kaleidoscopic orbifold, we provide an algorithm to construct a Euclidean, spherical, or hyperbolic polygon to match the orbifold. This polygon is then used to create a room for which the polygon serves as the floor and the ceiling. With our system that implements Möbius transformations, the user can interactively edit the scene and see the reflections of the edited objects. To correctly visualize non-Euclidean orbifolds, we adapt the rendering algorithms to account for the geodesics in these spaces, which light rays follow. Our interactive orbifold design system allows the user to create arbitrary two-dimensional kaleidoscopic orbifolds. In addition, our mirror-based orbifold visualization approach has the potential of helping our users gain insight on the orbifold, including its orbifold notation as well as its universal cover, which can also be the spherical space and the hyperbolic space.

A deep clustering framework integrating pairwise constraints and a VMF mixture model

<abstract><p>We presented a novel deep generative clustering model called Variational Deep Embedding based on Pairwise constraints and the Von Mises-Fisher mixture model (VDEPV). VDEPV consists of fully connected neural networks capable of learning latent representations from raw data and accurately predicting cluster assignments. Under the assumption of a genuinely non-informative prior, VDEPV adopted a von Mises-Fisher mixture model to depict the hyperspherical interpretation of the data. We defined and established pairwise constraints by employing a random sample mining strategy and applying data augmentation techniques. These constraints enhanced the compactness of intra-cluster samples in the spherical embedding space while improving inter-cluster samples' separability. By minimizing Kullback-Leibler divergence, we formulated a clustering loss function based on pairwise constraints, which regularized the joint probability distribution of latent variables and cluster labels. Comparative experiments with other deep clustering methods demonstrated the excellent performance of VDEPV.</p></abstract>

Gaussian process regression models for the properties of micro-tearing modes in spherical tokamaks

Spherical tokamaks have many desirable features that make them an attractive choice for a future fusion power plant. Power-plant viability is intrinsically related to plasma heat and particle confinement, and this is often determined by the level of micro-instability-driven turbulence. Accurate calculation of the properties of turbulent microinstabilities is therefore critical for tokamak design; however, the evaluation of these properties is computationally expensive. The considerable number of geometric and thermodynamic parameters and the high resolutions required to accurately resolve these instabilities make repeated use of direct numerical simulations in integrated modeling workflows extremely computationally challenging and create the need for fast, accurate, reduced-order models. This paper outlines the development of a data-driven reduced-order model, often termed a surrogate model for the properties of micro-tearing modes (MTMs) across a spherical tokamak reactor-relevant parameter space utilizing Gaussian process regression and classification, techniques from machine learning. These two components are used in an active learning loop to maximize the efficiency of data acquisition, thus minimizing computational cost. The high-fidelity gyrokinetic code GS2 is used to calculate the linear properties of the MTMs: the mode growth rate, frequency, and normalized electron heat flux, and core components of a quasi-linear transport model. Data cross-validation and direct validation on unseen data are used to ascertain the performance of the resulting surrogate models.

Теоремы о дифференцируемых сферах для многообразий с ограниченными сверху кривизнами Риччи

In the present paper, we prove that if is an -dimensional compact Riemannian manifold and if where , and are the sectional and Ricci curvatures of respectively, then is diffeomorphic to a spherical space form where is a finite group of isometries acting freely. In particular, if is simply connected, then it is diffeo­mor­phic to the Euclidian sphere