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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 $n-1$.

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.