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Register a new userMinimal $N$-Point Diameters and $f$-Best-Packing Constants in $R^d$
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In terms of the minimal $N$-point diameter $D_d(N)$ for $R^d,$ we determine, for a class of continuous real-valued functions $f$ on $[0,+infty],$ the $N$-point $f$-best-packing constant $min{f(|x-y|), :, x,yin R^d}$, where the minimum is taken over point sets of cardinality $N.$ We also show that $$ N^{1/d}Delta_d^{-1/d}-2le D_d(N)le N^{1/d}Delta_d^{-1/d}, quad Nge 2,$$ where $Delta_d$ is the maximal sphere packing density in $R^d$. Further, we provide asymptotic estimates for the $f$-best-packing constants as $Ntoinfty$.
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For $N$-point best-packing configurations $omega_N$ on a compact metric space $(A,rho)$, we obtain estimates for the mesh-separation ratio $gamma(omega_N,A)$, which is the quotient of the covering radius of $omega_N$ relative to $A$ and the minimum pairwise distance between points in $omega_N$. For best-packing configurations $omega_N$ that arise as limits of minimal Riesz $s$-energy configurations as $sto infty$, we prove that $gamma(omega_N,A)le 1$ and this bound can be attained even for the sphere. In the particular case when N=5 on $S^2$ with $rho$ the Euclidean metric, we prove our main result that among the infinitely many 5-point best-packing configurations there is a unique configuration, namely a square-base pyramid $omega_5^*$, that is the limit (as $sto infty$) of 5-point $s$-energy minimizing configurations. Moreover, $gamma(omega_5^*,S^2)=1$.
We investigate the asymptotic behavior, as $N$ grows, of the largest minimal pairwise distance of $N$ points restricted to an arbitrary compact rectifiable set embedded in Euclidean space, and we find the limit distribution of such optimal configurations. For this purpose, we compare best-packing configurations with minimal Riesz $s$-energy configurations and determine the $s$-th root asymptotic behavior (as $sto infty)$ of the minimal energy constants. We show that the upper and the lower dimension of a set defined through the Riesz energy or best-packing coincides with the upper and lower Minkowski dimension, respectively. For certain sets in ${rm {bf R}}^d$ of integer Hausdorff dimension, we show that the limiting behavior of the best-packing distance as well as the minimal $s$-energy for large $s$ is different for different subsequences of the cardinalities of the configurations.
We prove that the local eigenvalue statistics at energy $E$ in the localization regime for Schrodinger operators with random point interactions on $mathbb{R}^d$, for $d=1,2,3$, is a Poisson point process with the intensity measure given by the density of states at $E$ times the Lebesgue measure. This is one of the first examples of Poisson eigenvalue statistics for the localization regime of multi-dimensional random Schrodinger operators in the continuum. The special structure of resolvent of Schrodinger operators with point interactions facilitates the proof of the Minami estimate for these models.
We provide the analytic expressions of the totally symmetric and anti-symmetric structure constants in the $mathfrak{su}(N)$ Lie algebra. The derivation is based on a relation linking the index of a generator to the indexes of its non-null elements. The closed formulas obtained to compute the values of the structure constants are simple expressions involving those indexes and can be analytically evaluated without any need of the expression of the generators. We hope that these expressions can be widely used for analytical and computational interest in Physics.
For a compact set A in Euclidean space we consider the asymptotic behavior of optimal (and near optimal) N-point configurations that minimize the Riesz s-energy (corresponding to the potential 1/t^s) over all N-point subsets of A, where s>0. For a large class of manifolds A having finite, positive d-dimensional Hausdorff measure, we show that such minimizing configurations have asymptotic limit distribution (as N tends to infinity with s fixed) equal to d-dimensional Hausdorff measure whenever s>d or s=d. In the latter case we obtain an explicit formula for the dominant term in the minimum energy. Our results are new even for the case of the d-dimensional sphere.
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