Let H(S) be the Hardy space on the unit sphere S in C, n ≥ 2. Consider the Hankel operator Hf = (1 − P )Mf |H(S), where the symbol function f is allowed to be arbitrary in L(S, dσ). We show that for p > 2n, Hf is in the Schatten class Cp if and only if f − Pf belongs to the Besov space Bp. To be more precise, the “if” part of this statement is easy. The main result of the paper is the “only if ...

Let X1, . . . ,Xn be a random sample from a p-dimensional population distribution. Assume that c1n α ≤ p≤ c2n for some positive constants c1, c2 and α. In this paper we introduce a new statistic for testing independence of the p-variates of the population and prove that the limiting distribution is the extreme distribution of type I with a rate of convergence O((logn)/ √ n). This is much faster...

Cooling of primordial gas plays a crucial role in the birth of the first structures in our Universe. Due to the low fractional abundance of molecular species at high redshifts, spontaneous emission rather than collisions represents the most efficient way to cool the pristine plasma. In the present work, radiative cooling functions are evaluated for the diatomic species HD, HD+, HeH+, LiH and Li...

An N-tangle can be defined for the finite dimensional Hilbert space H = C2N , with N = 3 or N even. We give an orthonormal basis which is fully entangled with respect to this measure. We provide a spin Hamilton operator which has this entangled basis as normalized eigenvectors if N is even. From these normalized entangled states a Bell matrix is constructed and the cosine–sine decomposition is ...

Let t(n, d) be the minimum number t such that there are t of the n lattice points {(x1, . . . , xd) : 1 ≤ xi ≤ n} so that the ( t 2 ) lines that they determine cover all the above n lattice points. We prove that for every integer d ≥ 2 there are two positive constants c1 = c1(d) and c2 = c2(d) such that for every n c1n d(d−1)/(2d−1) ≤ t(n, d) ≤ c2n log n. The special case d = 2 settles a proble...