Control problems with intermediate constraints

Necessary conditions are obtained for the general control problem with the minimizing are required to pass through intermediate-constraint manifolds. It is shown that this includes cases where the integrand or the region of admissibility changes at the constraint points.This research was partly sponsored by the Office of Naval Research under contract NONR-233(76).     

Permutations,periodicity, and chaos

Let f: I → I be a continuous function on a closed interval I. If there exists x?I which has period 3 with respect to f, then Li and Yorke [1] proved that f is chaotic in the sense that there are not only points x?I of arbitrarily large period, but also uncountably many points x?I which are not even asymptotically periodic with respect to f. By using only elementary combinatorial facts about permutations, it is shown that if there is a point x?I of period p with respect to f, where p is divisible by 3, 5, or 7, then f is chaotic. The proof is followed by a study of some related combinatorial problems in symmetric groups.     

Surfactant control of gas uptake: effect of butanol films on HCl and HBr entry into supercooled sulfuric acid

The entry of HCl into 60-68 wt % D(2)SO(4) and HBr into 68 wt % acid containing 0-0.18 M 1-butanol was monitored by measuring the fractions of impinging HCl and HBr molecules that desorb as DCl and DBr after undergoing H --> D exchange within the deuterated acid. The addition of 0.18 M butanol to the acid creates butyl films that reach approximately 80% surface coverage at 213 K. Surprisingly, this butyl film does not impede exchange but instead enhances it: the HCl --> DCl exchange fractions increase from 0.52 to 0.74 for 60 wt % D(2)SO(4) and from 0.14 to 0.27 for 68 wt % D(2)SO(4). HBr --> DBr exchange increases even more sharply, rising from 0.22 to 0.65 for 68 wt % D(2)SO(4). We demonstrate that this enhanced exchange corresponds to enhanced uptake into the butyl-coated acid for HBr and infer this equivalence for HCl. In contrast, the entry probability of the basic molecule CF(3)CH(2)OH exceeds 0.85 at all acid concentrations and is only slightly diminished by the butyl film. The OD groups of surface butanol molecules may assist entry by providing extra interfacial protonation sites for HCl and HBr dissociation. The experiments suggest that short-chain surfactants in sulfuric acid aerosols do not hinder heterogeneous reactions of HCl or HBr with other solute species.     

光解碎片平动谱中二级过程的运动学及化学动力学

本文发展了一套分析处理分子束光解反应实验中二级分解产物飞行谱的方法, 它改进了Kroger和Riley的最初讨论。本文表明许多重要的信息都可以从高度平均的实验数据中得出。这包括二级分解产物的平均平动能分布、空间各向异性参数、平行竞争通道间的反应比。模拟的结果可以表现二级分解反应的一些主要特征。     

Connected components of arithmetic graphs

Let . LetG _m (R) be the graph whose vertices are the numbers 1, 2, ...,m and whose edges are all pairs {a, b} such thata+br (modm) for somerR. LetC _m (R) be the number of connected components ofG _m (R). Letd be the greatest common divisor ofm and the differencesr _j –r _j or allr _i ,r _j R. ThenC _m (R) is equal to (i) (d+1)/2 ifd is odd, (ii)d/2 ifd is even andr is odd for allrR, or (iii) (d/2)+1 ifd is even andr is even for allrR.This research was supported in part by the National Science Foundation under grant No. MCS78-07908.     

N-graphs,Modular Sidon and Sum-Free Sets,and Partition Identities

Using a new graphical representation for partitions, the author obtains a family of partition identities associated with partitions into distinct parts of an arithmetic progression, or, more generally, with partitions into distinct parts of a set that is a finite union of arithmetic progressions associated with a modular sum-free Sidon set. Partition identities are also constructed for sets associated with modular sum-free sets.