Some best possible bounds concerning the traces of finite sets II

LetX be ann-element set and be a family of its subsets. Consider the family _x = {F – {x} : F } for a givenx X. We write(m, n) (m – k, n – 1), when for all with || m, there exists an elementx ofX such that| _x| m – k. We show that (m, n) (m – 10,n – 1) for allm 5n and (m, n) (m – 13,n – 1) for allm 29n/5.     

Density results for uniform families

A familyF of subsets is calledk-dense if there exists ak-element setA such that all 2 ^k of its subsets can be obtained in the formAF for someFF. Best possible bounds are obtained for the maximum number of sets in notk-densek-partite families. This is a consequence of a new rank formula for inclusion matrices.     

Observation of magnetic order of Cu-moment aroundx=0.12 in La2−x Ba x CuO4 and La2−x sr x CuO4 by μ+SR

Zero field positive muon spin rotation method ( ⁺SR) is applied on La_2–x Sr _x CuO₄ around forx=0.12 at which the high-T _c superconductivity (SC) is suppressed. The magnetically ordered state of Cu-moments, which is not a spin glass state but an antiferromagnetic like state, appears below 15 K for 0.105x0.120. The magnetic phase boundary is very similar to the one of the La_2–x Ba _x CuO₄ in which the structural transition from the low temperature orthorhombic (LTO) phase to the low temperature tetragonal (LTT) phase is observed aroundx=0.12. The present study suggests that there is no big difference of the electronic state of the CuO₂ plane between the La-Ba system and the La-Sr system and that the magnetic ordering of Cu-moments plays an important role for the suppression of the high-T _c SC aroundx=0.12 in both of the systems, although the LTO-LTT structural transition has not been observed yet in the La-Sr system.     

Nitrosation of Phenolic Substrates under Mildly Basic Conditions: Selective Preparation of p-Quinone Monooximes and Their Antiviral Activities

Nitrosation of 3-methoxyphenol and 1-naphthol were examined under both acidic (NaNO(2)-EtCO(2)H-H(2)O) and basic (i-AmNO(2)-K(2)CO(3)-DMF) conditions. Acidic nitrosations afforded ortho-directed products, whereas para-directed nitrosations were observed under basic conditions to yield p-quinone monooximes. The basic para-directed nitrosation was further examined using 15 phenols, two naphthols, and four phenolic heterocyclics. A one-pot operation of the basic nitrosation followed by methylation with dimethyl sulfate gave the corresponding methyl ethers in high yield. Two p-quinone monooximes derived from 3-methoxyphenol and 8-hydroxyquinoline showed a moderate activity against HSV-1, and the latter oxime was also effective against HSV-2. On the other hand, p-quinone monooximes derived from methyl salicylate, 1-naphthol, 7-hydroxy-2-methylbenzo[b]furan, and 8-hydroxycoumarin showed the comparable activity to that of DDI against HIV-1.