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In this paper, we investigate the deterministic learning problem associated with spherical scattered data for the first time. We design the quadrature-weighted kernel regularized regression learning schemes associated with deterministic scattered data on the unit sphere and the spherical cap. By employing the minimal norm interpolation technique and leveraging results from the numerical integration of spherical radial basis functions over these surfaces, we derive the corresponding learning rates. Notably, our algorithm design and error analysis methods diverge from those typically employed in randomized learning algorithms. Our findings suggest that the learning rates are influenced by both the mesh norm of the scattered data and the smoothness of the radial basis function. This implies that when the radial basis function exhibits sufficient smoothness, the learning rate achieved with deterministic samples outperforms that obtained with random samples. Furthermore, our results provide theoretical support for the feasibility of deterministic spherical learning, which may bring potential applications in tectonic plate geology and Earth sciences.

In the process of finding Einstein metrics in dimension $n\ge 3$, we can search metrics critical for the scalar curvature among fixed-volume metrics of constant scalar curvature on a closed oriented manifold. This leads to a system of PDEs (which we call the Fischer–Marsden Equation, after a conjecture concerning this system) for scalar functions, involving the linearization of the scalar curvature. The Fischer–Marsden conjecture said that, if the equation admits a solution, the underlying Riemannian manifold is Einstein. Counter-examples are known by Kobayashi and Lafontaine, and by our first paper. Multiple solutions to this system yield Killing vector fields. We showed in our first paper that the dimension of the solution space $W$ can be at most $n+1$, with equality implying that $(M,g)$ is a sphere with constant sectional curvatures. Moreover, we also showed there that the identity component of the isometry group has a factor $\mathrm{\text{SO}}(dimW)$. In this second paper, we apply our results in the first paper to show that either $(M,g)$ is a standard sphere or the dimension of the space of Fischer–Marsden solutions can be at most $n-1$.

In this paper we compute the leading order of the Casimir energy for a free massless scalar field confined in a sphere in three spatial dimensions, with the Dirichlet boundary condition. When one tabulates all of the reported values of the Casimir energies for two closed geometries, cubical and spherical, in different space–time dimensions and with different boundary conditions, one observes a complicated pattern of signs. This pattern shows that the Casimir energy depends crucially on the details of the geometry, the number of the spatial dimensions, and the boundary conditions. The dependence of the sign of the Casimir energy on the details of the geometry, for a fixed spatial dimensions and boundary conditions has been a surprise to us and this is our main motivation for doing the calculations presented in this paper. Moreover, all of the calculations for spherical geometries include the use of numerical methods combined with intricate analytic continuations to handle many different sorts of divergences which naturally appear in this category of problems. The presence of divergences is always a source of concern about the accuracy of the numerical results. Our approach also includes numerical methods, and is based on Boyer's method for calculating the electromagnetic Casimir energy in a perfectly conducting sphere. This method, however, requires the least amount of analytic continuations. The value that we obtain confirms the previously established result.

This study investigates the interaction between fluid dynamics and electromagnetic fields, a complex problem that has not been extensively studied. The Lorentz force, which arises due to the interaction between magnetic fields and currents in a fluid is considered in this study. This research investigates the effects of a magnetic field, couple stress, and slip velocity on the behavior of a squeeze film (SF) formed between a porous flat and spherical plate. The Stokes equation for couple stress fluids is used to produce a generalized version of the Reynolds equation, which is then used to determine the film pressure. Also, this study considers the impact of a constant magnetic field orthogonal to the plate. The fluid in the porous region is governed by modified Darcy law. The effect of a uniform magnetic field perpendicular to the plate is considered. The bearing characteristics pressure, squeeze film time, and load-carrying capacity are graphically presented. The results revealed that the load-carrying capacity, pressure, and squeeze film time are reduced with a rise in slip and porousness parameters. The slip parameter decreases the values of film pressure, squeeze time, and load-carrying capacity as related to the no-slip case.

Point-, line- or boundary-sampled intercepts may be measured inside a particle as a measure of particle size. Each intercept is primarily characterized by three geometric properties: length, location and orientation. Circle and sphere models are used in the present study to analyze these properties. The probability distribution function, probability density function, expectation and coefficient of variation for each of the properties were presented based on geometric probability and mathematical statistics. Such presentation would be helpful for potential users of stereology to better understand the concept of intercepts and implement stereological intercept measurement for estimation of particle sizes in practice.

We generalize the classical two-body problem from flat space to spherical space and realize much of the complexity of the classical three-body problem with only two bodies. We show analytically, by perturbation theory, that small, nearly circular orbits of identical particles in a spherical universe precess at rates proportional to the square root of their initial separations and inversely proportional to the square of the universe's radius. We show computationally, by graphically displaying the outcomes of large open sets of initial conditions, that large orbits can exhibit extreme sensitivity to initial conditions, the signature of chaos. Although the spherical curvature causes nearby geodesics to converge, the compact space enables infinitely many close encounters, which is the mechanism of the chaos.

Spherical splines are used to define approximate solutions to strongly elliptic pseudodifferential equations on the unit sphere. These equations arise from geodesy. The approximate solutions are found by using Galerkin method. We prove optimal convergence (in Sobolev norms) of the approximate solution by spherical splines to the exact solution. Our numerical results underlie the theoretical result.

In this paper, we study propagation phenomena on the sphere using the bistable reaction–diffusion formulation. This study is motivated by the propagation of waves of calcium concentrations observed on the surface of oocytes, and the propagation of waves of kinase concentrations on the B-cell membrane in the immune system. To this end, we first study the existence and uniqueness of mild solutions for a parabolic initial-boundary value problem on the sphere with discontinuous bistable nonlinearities. Due to the discontinuous nature of reaction kinetics, the standard theories cannot be applied to the underlying equation to obtain the existence of solutions. To overcome this difficulty, we give uniform estimates on the Legendre coefficients of the composition function of the reaction kinetics function and the solution, and a priori estimates on the solution, and then, through the iteration scheme, we can deduce the existence and related properties of solutions. In particular, we prove that the constructed solutions are of $C^{2,1}$ class everywhere away from the discontinuity point of the reaction term. Next, we apply this existence result to study the propagation phenomenon on the sphere. Specifically, we use stationary solutions and their variants to construct a pair of time-dependent super/sub-solutions with different moving speeds. When applied to the case of sufficiently small diffusivity, this allows us to infer that if the initial concentration of the species is above the inhomogeneous steady state, then the species will exhibit the propagating behavior.

The Green function of the Klein–Gordon equation in black-hole coordinates is calculated. This function is a sum on the harmonic modes of the sphere. The first term is a double integration on the spectrum of energy, and the momentum of the particle. Far from the horizon, the double integration is approximated by an integration on a line defined by the relation of energy and momentum of a free particle. Assumptions of time-independence and radial symmetry are made.

In this paper, we are interested in studying the initial value problem for parabolic problem associated with the Caputo–Fabrizio derivative. We deal the problem in two cases: linear inhomogeneous case and nonlinearity source term. For the linear case, we derive the convergence result of the mild solution when the fractional order $\alpha \to 1^{-}$ under some various assumptions on the initial datum. For the nonlinear problem, we show the existence and uniqueness of the mild solution using Banach fixed point theory. We also prove the convergence result of the mild solution when the fractional order $\alpha \to 1^{-}$.

My purpose here is to give some background to a series of sculptures I have been creating in which nonorientable surfaces, especially crosscaps, play a mathematical content role. The fundamental circle of theorems involved is on the classification of surfaces in terms of handles and crosscaps. The most basic examples are spheres, tori, möbius bands, projective planes, and klein bottles, cf., [CF1, HF1] for discussion of my earlier sculpture involving these concepts. For an exposition and further references to the mathematical background for the circle of theorems I refer to, see [FG1].

In this paper, we prove the long-time existence of the CR Yamabe flow on the compact strictly pseudoconvex CR manifold with positive CR invariant. We also prove the convergence of the CR Yamabe flow on the sphere by proving that: the contact form which is pointwise conformal to the standard contact form on the sphere converges exponentially to a contact form of constant pseudo-Hermitian sectional curvature. We also show that the eigenvalues of some geometric operators are non-decreasing under the unnormalized CR Yamabe flow provided that the pseudo-Hermitian scalar curvature satisfies certain conditions.

Here, we prove an isoperimetric inequality for the harmonic mean of the first $N-1$ non-trivial Neumann eigenvalues of the Laplace–Beltrami operator for domains contained in a hemisphere of ${𝕊}^N$.

We present a unified approach for constructing Slepian functions — also known as prolate spheroidal wave functions — on the sphere for arbitrary tensor ranks including scalar, vectorial, and rank 2 tensorial Slepian functions, using spin-weighted spherical harmonics. For the special case of spherical cap regions, we derived commuting operators, allowing for a numerically stable and computationally efficient construction of the spin-weighted spherical-harmonic-based Slepian functions. Linear relationships between the spin-weighted and the classical scalar, vectorial, tensorial, and higher-rank spherical harmonics allow the construction of classical spherical-harmonic-based Slepian functions from their spin-weighted counterparts, effectively rendering the construction of spherical-cap Slepian functions for any tensorial rank a computationally fast and numerically stable task.

Optimal estimates of Kolmogorov's n-widths, linear n-widths and Gelfand's n-widths of the weighted Sobolev classes on the unit sphere 𝕊^d are established. Similar results are also established on the unit ball B^d and on the simplex T^d.

The aim of this paper is to investigate the application of radial basis function-generated finite difference (RBF-FD) methods for convection–diffusion partial differential equations (PDEs) on a sphere. In the application of RBF-FD method, choosing a reasonable value of shape parameter is important to the computation of PDEs. The work is devoted to the numerical study of the range of near optimal shape parameters for the convection–diffusion equations. Because the RBF-FD Direct method often leads to ill-conditioned problems for small shape parameters, the RBF-QR method is applied locally to overcome the ill-conditioning in the context of RBF-FD mode. Additionally, for convection-dominated problems, it can be found that the results of using central-type stencil present spurious oscillations. Therefore, we propose an upwind RBF-FD (URBF-FD) scheme to overcome the problems, which is well adapted to the problems on the sphere and easy to be implemented. Further numerical results show that the proposed URBF-FD method is stable and effective for convection-dominated PDEs on the sphere.

There are several well-known characterizations of the sphere as a regular surface in the Euclidean space. On the contrary, there are not so many characterizations of the hyperbolic space, the spacelike sphere in the Minkowski space. By means of a purely synthetic technique, we get a rigidity result for the sphere in 𝔼ⁿ⁺¹ without any curvature conditions, nor completeness or compactness, as well as a dual result for the n-dimensional hyperbolic space in 𝕃ⁿ⁺¹.

There are two smooth functions $\sigma$ and $\rho$ associated to a nontrivial concircular vector field $v$ on a connected Riemannian manifold $(M,g)$, called potential function and connecting function. In this paper, we show that presence of a timelike nontrivial concircular vector field influences the geometry of generalized Robertson–Walker space-times. We use a timelike concircular vector field $v$ on an $n$ -dimensional connected conformally flat Lorentzian manifold, $n>2$, to find a characterization of generalized Robertson–Walker space-time with fibers Einstein manifolds. It is interesting to note that for $n=4$ the concircular vector field annihilates energy-momentum tensor and also that in this case the potential function $\sigma$ is harmonic. In the second part of this paper, we show that presence of a nontrivial concircular vector field $v$ with connecting function $\rho$ on a complete and connected $n$ -dimensional conformally flat Riemannian manifold $(M,g)$, $n>2$, with Ricci curvature $\mathrm{\text{Ric}}(v,v)$ non-negative, satisfying $n(n-1)\rho +\tau =0$, is necessary and sufficient for $(M,g)$ to be isometric to either a sphere $S^n(c)$ or to the Euclidean space $E^n$, where $\tau$ is the scalar curvature.

In this paper, the authors compute the explicit formulas for the joint distributions of the hitting time and place for a sphere or concentric spherical shell by Brownian motion, when the process starts either outside the sphere or the region bounded by concentric spheres.

This paper addresses the issue of obtaining the optimal rotation to match two functions on the sphere by minimizing the squared error norm and the Kullback–Leibler information criteria. In addition, the accuracy in terms of the band-limited approximations in both cases are also discussed. Algorithms for fast and accurate rotational matching play a significant role in many fields ranging from computational biology to spacecraft attitude estimation. In electron microscopy, peaks in the so-called "rotation function" determine correlations in orientation between density maps of macromolecular structures when the correspondence between the coordinates of the structures is not known. In X-ray crystallography, the rotational matching of Patterson functions in Fourier space is an important step in the determination of protein structures. In spacecraft attitude estimation, a star tracker compares observed patterns of stars with rotated versions of a template that is stored in its memory. Many algorithms for computing and sampling the rotation function have been proposed over the years. These methods usually expand the rotation function in a bandlimited Fourier series on the rotation group. In some contexts the highest peak of this function is interpreted as the optimal rotation of one structure into the other, and in other contexts multiple peaks describe symmetries in the functions being compared. Prior works on rotational matching seek to maximize the correlation between two functions on the sphere. We also consider the use of the Kullback–Leibler information criteria. A gradient descent algorithm is proposed for obtaining the optimal rotation, and a measure is defined to compare the convergence of this procedure applied to the maximal correlation and Kullback–Leibler information criteria.