Heterocyclic analogs of prostaglandins: II. Synthesis of 10-oxaprostanoids with isoxazole and isoxazoline fragments in α-and ω-chain

By the use of “nitrile oxide procedure” from 3-alkyl(arylalkyl)-substituted 2,5-dihydro-2-furanones new 10-oxaprostanoids were prepared containing an isoxazole (isoxazoline) fragment in the α-or ω-prostanoid chain.     

Rigidity of measures—The high entropy case and non-commuting foliations

We consider invariant measures for partially hyperbolic, semisimple, higher rank actions on homogeneous spaces defined by products of real andp-adic Lie groups. In this paper we generalize our earlier work to establish measure rigidity in the high entropy case in that setting. We avoid any additional ergodicity-type assumptions but rely on, and extend the theory of conditional measures. To Hillel Furstenberg with friendship and admiration Manfred Einsiedler is partially supported by the NSF Grant DMS 0400587. Anatole Katok is partially supported by the NSF Grant DMS 0071339.     

Formulas for the derivative and critical points of topological entropy for Anosov and geodesic flows

This paper represents part of a program to understand the behavior of topological entropy for Anosov and geodesic flows. In this paper, we have two goals. First we obtain some regularity results forC ¹ perturbations. Second, and more importantly, we obtain explicit formulas for the derivative of topological entropy. These formulas allow us to characterize the critical points of topological entropy on the space of negatively curved metrics.Partially supported by NSF grant DMS-8514630Chaim Weizmann Research Fellow and NSF postdoctoral Research Fellow     

Controlling the ratchet effect for cold atoms

Low-order quantum resonances manifested by directed currents have been realized with cold atoms. Here we show that by increasing the strength of an experimentally achievable delta-kicking ratchet potential, quantum resonances of a very high order may naturally emerge and can induce larger ratchet currents than low-order resonances, with the underlying classical limit being fully chaotic. The results offer a means of controlling quantum transport of cold atoms.     

Yet more on the linear search problem

The linear search problem has been discussed previously by one of the present authors. In this paper, the probability distribution of the point sought in the real line is not known to the searcher. Since there is noa priori choice of distribution which recommends itself above all others, we treat the situation as a game and obtain minimax type solutions. Different minimaxima apply depending on the factors which one wishes to minimize (resp. maximize). Certain criteria are developed which help the reader judge whether the results obtained can be considered “good advice” in the solution of real problems analogous to this one. The research of this paper has been supported in part by the following agencies: National Science Foundation, Wisconsin Alumni Research Foundation, German Academic Exchange Service (DAAD), and Air Force Office of Scientific Research.     

On Sums of Products of Bernoulli Variables and Random Permutations

Let {X _k } _k1 be independent Bernoulli random variables with parameters p _k . We study the distribution of the number or runs of length 2: that is . Let S=lim _n S _n . For the particular case p _k =1/(k+B), B being given, we show that the distribution of S is a Beta mixture of Poisson distributions. When B=0 this is a Poisson(1) distribution. For the particular case p _k =p for all k we obtain the generating function of S _n and the limiting distribution of S _n for .     

Linear extensions of dynamic systems and the reductibility problem

The relation of linear extensions of smooth dynamic systems to cohomologies and to reducibility in the case of flow is investigated. A result is obtained concerning -cohomologies in the neighborhood of a constant cocycle for the case of an arbitrary closed subgroup of the group GL (k, C).Translated from Matematicheskie Zametki, Vol. 8, No. 4, pp. 451–462, October, 1970.     

The growth rate for the number of singular and periodic orbits for a polygonal billiard

For any simply connected polygon in the plane, the number of billiard orbits which begin and end at a vertex grows subexponentially with respect to the length or to the number of reflections. This implies that the numbers of isolated periodic orbits and of families of parallel periodic orbits do grow subexponentially. The main technical device is a calculation showing that the topological entropy of the Poincaré map for the billiard flow is equal to zero.Supported in part by NSF Grant #DMS-8414400