Related papers: Optimal asymptotic bounds for spherical designs
Combining the ideas of Riesz $s$-energy and $\log$-energy, we introduce the so-called $s,\log^t$-energy. In this paper, we investigate the asymptotic behaviors for $N,t$ fixed and $s$ varying of minimal $N$-point $s,\log^t$-energy constants…
We prove the existence of optimal separators for intersection graphs of balls and spheres in any dimension $d$. One of our results is that if an intersection graph of $n$ spheres in $\mathbb{R}^d$ has $m$ edges, then it contains a balanced…
Consider the integer points lying on the sphere of fixed radius projected onto the unit sphere. Duke showed that, on congruence conditions for the radius squared, these points equidistribute. To further this study of equidistribution, we…
This paper follows a previous one in which were introduced deformation invariants $\chi^d_r$, $d \in H_2 (X ; \Z)$, $r \in \N$, of closed real symplectic four-manifolds $(X, \omega, c_X)$, invariants which produced lower bounds in real…
We prove the global in time existence of spherically symmetric solutions to an initial-boundary value problem for a system of partial differential equations, which consists of the equations of linear elasticity and a nonlinear,…
We investigate invariant random fields on the sphere using a new type of spherical wavelets, called needlets. These are compactly supported in frequency and enjoy excellent localization properties in real space, with quasi-exponentially…
We prove that the covering radius of an $N$-point subset $X_N$ of the unit sphere $S^d \subset R^{d+1}$ is bounded above by a power of the worst-case error for equal weight cubature $\frac{1}{N}\sum_{\mathbf{x} \in X_N}f(\mathbf{x}) \approx…
This paper concerns spherical adjunctions of stable $\infty$-categories and their relation to monadic adjunctions. We begin with a proof of the 2/4 property of spherical adjunctions in the setting of stable $\infty$-categories. The proof is…
We study asymptotic behaviors of solutions to the Loewner-Nirenberg problem in finite cones and establish optimal asymptotic expansions in terms of the corresponding solutions in infinite cones. The spherical domains over which cones are…
We consider the Stokes system in $\mathbb R^3,$ deprived of $N$ spheres of radius $1/N,$ completed by constant boundary conditions on the spheres. This problem models the instantaneous response of a viscous fluid to an immersed cloud of…
We establish a majorization-based theory for bounding observables of waves with varied coherence. For any measurement, exact bounds are attained by the maximal and minimal elements in the set of input coherence spectra. The set's supremum…
This paper settles the existence question for a rather general class of convex optimal design problems with a volume constraint. In low dimensions, we prove the existence of an optimal configuration for general convex minimization problems…
In this paper, we mainly consider the problem of spherical distribution of 5 points, that is, how to configure 5 points on a sphere such that the mutual distance sum attains the maximum. It is conjectured that the sum of distances is…
We show that spheres of positive constant curvature with $n$ ($n\geq3$) conic points converge to a sphere of positive constant curvature with two conic points (or called an (American) football) in Gromov-Hausdorff topology when the…
We introduce a minor variant of the approximate D-optimal design of experiments with a more general information matrix that takes into account the representation of the design space S. The main motivation (and result) is that if S in R^d is…
We prove that there exists a>0 such that for any integer d>2 and any topological types S_1,...,S_n of plane curve singularities, satisfying $\mu(S_1)+...+\mu(S_n) \leq ad^2$, there exists a reduced irreducible plane curve of degree d with…
We study lattice points in d-dimensional spheres, and count their number in thin spherical segments. We found an upper bound depending only on the radius of the sphere and opening angle of the segment. To obtain this bound we slice the…
The Solomon-Tits theorem says that the poset of proper non-trivial subspaces of a finite-dimensional vector space has realisation equivalent to a wedge of spheres. In this paper we prove a variant of this result for collections of geodesic…
We prove Manin's conjecture on the asymptotic behavior of the number of rational points of bounded anticanonical height for a spherical threefold with canonical singularities and two infinite families of spherical threefolds with log…
We discuss the constant $\sigma_{2}$ problem for conic 4-spheres. Based on earlier works of Chang-Han-Yang and Han-Li-Teixeira, we are able to find a necessary condition for the existence problem. In particular, when the condition is sharp,…