The uniform modulus of continuity of iterated Brownian motion

LetX be a Brownian motion defined on the line (withX(0)=0) and letY be an independent Brownian motion defined on the nonnegative real numbers. For allt0, we define theiterated Brownian motion (IBM),Z, by setting . In this paper we determine the exact uniform modulus of continuity of the process Z.Research supported by NSF grant DMS-9122242.     

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.     

Sequence distribution and intramolecular excimer formation in styrene-acrylonitrile copolymers

Styrene-acrylonitrile copolymers, like many other copolymers containing styrene, exhibit both normal and excimer fluorescence. We have shown that the ratio of the excimer to monomer fluorescence intensities in random styrene-acrylonitrile copolymers is linearly dependent upon the concentration of styrene-styrene bonds in the copolymer. This observation is consistent with a photophysical model which allows the energy absorbed by styrene units to migrate freely along the copolymer chain. Some of the energy is emitted in the form of normal fluorescence; some of the energy, trapped by neighbouring styrene-styrene pairs suitably oriented to allow excimer formation, is emitted as excimer fluorescence. The fluorescence characteristics of acrylonitrile-styrene copolymers are contrasted with those of methyl methacrylate-styrene copolymers, in which the methylmethacrylate sequences are believed to present partial barriers to energy migration along the copolymer chains.