The rational self-assembly of mono-lacunary Keggin clusters with 4d and 4f metal salts via a conventional method has yielded two novel polyoxometalate-based 4d–4f heterometallic compounds containing lacunary Keggin anions connected to ten metal atoms: {[Ag{Ag2(H2O)4}{Ln(H2O)6}2H ⊂ {SiW11Ln(H2O)4O39}2]·nH2O (Ln = Ce and n = 7 for 1, Ln = Pr and n = 3 for 2). Their structures were determined by s...

where E c [0, 1] is any given set o f positive measure and {ak} any given sequence of real numbers. This theorem was first proved by R. Salem and A. Zygmund in case of a -0, where {flk} satisfies the so-called Hadamard's gap condition (cf. [4], (5.5), pp. 264-268). In that case they also remarked that under the hypothesis (1.2) the condition (1.3) is necessary for the validity of (1.5) (cf. [4]...

We offer some new identities for a bipartition function, which has a relation to a Hecke-type identity of Andrews. Further, we show this partition function is lacunary, and relate it to a real quadratic field.

We find asymptotics for lacunary sums of binomial coefficients. As an application we determine the exact and approximate probability that the first card has the uniquely highest rank among the top l cards of a standard card deck.

Transition-metal substituted Keggin-type polyoxometalates (POMs) are of great interest for applications in biomedicine, material science and catalysis. The synthesis transition metal-substituted polytungstates via the formation a lacunary...

We provide an analogue of Gundy’s decomposition for L1-bounded non-commutative martingales. An important difference from the classical case is that for any L1-bounded non-commutative martingale, the decomposition consists of four martingales. This is strongly related with the row/column nature of non-commutative Hardy spaces of martingales. As applications, we obtain simpler proofs of the weak ...