Common supports as fixed points

A family S of sets in R ^d is sundered if for each way of choosing a point from rd+1 members of S, the chosen points form the vertex-set of an (r–1)-simplex. Bisztriczky proved that for each sundered family S of d convex bodies in R ^d , and for each partition (S , S ), of S, there are exactly two hyperplanes each of which supports all the members of S and separates the members of S from the members of S . This note provides an alternate proof by obtaining each of the desired supports as (in effect) a fixed point of a continuous self-mapping of the cartesian product of the bodies.     

A law of the iterated logarithm for random walk in random scenery with deterministic normalizers

LetX, X _i ,i≥1, be a sequence of independent and identically distributed ? ^d -valued random vectors. LetS _o=0 and \(S_n = \sum\nolimits_{i = 1}^n {X_i } \) forn≤1. Furthermore letY, Y(α), α∈? ^d , be independent and identically distributed ?-valued random variables, which are independent of theX _i . Let \(Z_n = \sum\nolimits_{i = 0}^n {Y(S_i )} \) . We will call (Z _n ) arandom walk in random scenery. In this paper, we consider the law of the iterated logarithm for random walk in random scenery where deterministic normalizers are utilized. For example, we show that if (S _n ) is simple, symmetric random walk in the plane,E[Y]=0 andE[Y ²]=1, then $$\mathop {\overline {\lim } }\limits_{n \to \infty } \frac{{Z_n }}{{\sqrt {2n\log (n)\log (\log (n))} }} = \sqrt {\frac{2}{\pi }} a.s.$$      

Total Synthesis of Indole and Dihydroindole Alkaloids. XVIII. Isomers and analogues of vinblastine

The synthesis and conformational analysis of (3′R)-3-hydroxyleurosidine ( 5 ), (3′S)-3-hydroxyleurosidine ( 10 ), (3′S)-3-acetoxy-4′-deoxyleurosidine ( 15 ), (3′R)-3-acetoxy-4′-deoxyleurosidine ( 23 ), (3′R)-3-acetoxy-4′-deoxyvinblastine ( 16 ), (3′S)-3-acetoxy-4′-deoxyleurosidine ( 28 ) is discussed.     

Salting-in of alcohols in aqueous solutions by tetraalkylammonium bromides at the freezing temperature

The freezing-point depression of the ternary systems tetraalkylammonium bromides-t-butanol-water for the first five homologs of R₄NBr was measured. In the case of Bu₄NBr, the effect of size of the alcohol (methahol ton-butanol) was also investigated. From the corresponding freezing-point data for the binary systems the apparent salting constants were calculated. The true salting constantsk _s were obtained by extrapolation to infinite dilution. These are all very close to zero at the freezing temperature. From the corresponding thermochemical data the temperature dependence ofk _s was calculated, and above 5°C all the R₄NBr salts int-butanol; the salting-in increases with temperature and with the size of the hydrophobic cations. The scaled-particle theory is at present the only one which can account semiquantitatively for the temperature dependence of the salting-in effect. On leave of absence from Chemistry Department, The University, Sheffield S3 7HF, England To whom correspondence should be addressed.