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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 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.

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.

We extend the resolvent estimate on the sphere to exponents off the line $\frac{1}{r}-\frac{1}{s}=\frac{2}{n}$. Since the condition $\frac{1}{r}-\frac{1}{s}=\frac{2}{n}$ on the exponents is necessary for a uniform bound, one cannot expect estimates off this line to be uniform still. The essential ingredient in our proof is an $(L^r,L^s)$ norm estimate on the operator $H_k$ that projects onto the space of spherical harmonics of degree $k$. In showing this estimate, we apply an interpolation technique first introduced by Bourgain [J. Bourgain, Estimations de certaines fonctions maximales, C. R. Acad. Sci. Paris Sér. I Math.301(10) (1985) 499–502.]. The rest of our proof parallels that in Huang–Sogge [S. Huang and C. D. Sogge, Concerning $l^p$ resolvent estimates for simply connected manifolds of constant curvature, J. Funct. Anal.267(12) (2014) 4635–4666].

Uniform probability distributions on ${\ell }_p$ balls and spheres have been studied extensively and are known to behave like product measures in high dimensions. In this note we consider the uniform distribution on the intersection of a simplex and a sphere. Certain new and interesting features, such as phase transitions and localization phenomena emerge.