Non-classical law of the iterated logarithm behaviour for trimmed sums

Summary We study the law of the iterated logarithm for the partial sum of i.i.d. random variables when the r _n largest summands are excluded, where r _n=o(log logn). This complements earlier work in which the case log log_n=O(r_n) was considered. A law of the iterated logarithm is again seen to prevail for a wide class of distributions, but for reasons quite different from previously.Research supported in part by NSF Grant DMS-8501732     

Double deoxygenation of PPN(NO2) using metal carbonyl clusters of cobalt,rhodium, and iridium

The reaction of PP(NO₂) with M₄(CO)₁₂ (M = Co, Rh) gives the nitrido clusters [M₆N(CO)₁₅]^? in 13 and 21% yields, respectively. A high yield synthesis (77%) of [Rh₆N(CO)₁₅)]^? directly from Rh₆(CO)₁₆ and PPN(NO₂) is also presented. PPN(NO₂) reacts with Ir₄(CO)₁₂ to give the new isocyanato cluster, [Ir₄(NCO)(CO)₁₁]^? in 34% yield, while the direct synthesis of this isocyanate product occurs in 77% yield from PPN(N₃) and Ir₄(CO)₁₂. Modifications of published procedures for the preparation of [N(C₂H₅)₄]₂ [Ir₆(CO)₁₅] and Ir₆(CO)₁₆ are reported that allow shorter reaction times and give higher yields. The reaction of Ir₆(CO)₁₆ with one equivalent of PPN(NO₂) generates a new cluster, PPN[Ir₆(CO)₁₅(NO)], in 57% yield which is proposed to contain a bent nitrosyl ligand. An additional equivalent of PPN(NO₂) gives (PPN)₂[Ir₆(CO)₁₅] in 84% yield with the evolution of N₂O as well as CO₂.     

On distinct representatives and mapping theorems

We present an elementary proof of a transfinite symmetric form of Philip Hall's theorem on systems of distinct representatives and derive some wellknown mapping theorems from it to illustrate the ease with which this form of the theorem may be used.