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For a collection of $N$ unit vectors $mathbf{X}={x_i}_{i=1}^N$, define the $p$-frame energy of $mathbf{X}$ as the quantity $sum_{i eq j} |langle x_i,x_j rangle|^p$. In this paper, we connect the problem of minimizing this value to another optimization problem, so giving new lower bounds for such energies. In particular, for $p<2$, we prove that this energy is at least $2(N-d) p^{-frac p 2} (2-p)^{frac {p-2} 2}$ which is sharp for $dleq Nleq 2d$ and $p=1$. We prove that for $1leq m<d$, a repeated orthonormal basis construction of $N=d+m$ vectors minimizes the energy over an interval, $pin[1,p_m]$, and demonstrate an analogous result for all $N$ in the case $d=2$. Finally, in connection, we give conjectures on these and other energies.
We provide new answers about the placement of mass on spheres so as to minimize energies of pairwise interactions. We find optimal measures for the $p$-frame energies, i.e. energies with the kernel given by the absolute value of the inner product rai
The space of constructible functions form a dense subspace of the space of generalized valuations. In this note we prove a somewhat stronger property that the sequential closure, taken sufficiently many (in fact, infinitely many) times, of the former
We consider the general physical situation of a quantum system $H_0$ interacting with a chain of exterior systems $bigotimes_N H$, one after the other, during a small interval of time $h$ and following some Hamiltonian $H$ on $H_0 otimes H$. We discu
Let $mathcal F_0={f_i}_{iinmathbb{I}_{n_0}}$ be a finite sequence of vectors in $mathbb C^d$ and let $mathbf{a}=(a_i)_{iinmathbb{I}_k}$ be a finite sequence of positive numbers. We consider the completions of $cal F_0$ of the form $cal F=(cal F_0,cal
We consider a version of the fractional Sobolev inequality in domains and study whether the best constant in this inequality is attained. For the half-space and a large class of bounded domains we show that a minimizer exists, which is in contrast to the classical Sobolev inequalities in domains.