Investigation of the Montmorillonite clay-catalyzed [1,3] shift reaction of 3-methyl-2-butenyl phenyl ether

The [1,3] shift reaction of 3-methyl-2-butenyl phenyl ether was catalyzed more effectively by Montmorillonite K10 clay than Montmorillonite KSF clay, and proceeded with greatest ortho-selectivity in carbon tetrachloride at room temperature.     

Arm cavity resonant sideband control for laser interferometric gravitational wave detectors

We present a new optical control scheme for a laser interferometric gravitational wave detector that has a high degree of tolerance to interferometer spatial distortions and noise on the input light. The scheme involves resonating the rf sidebands in an interferometer arm cavity.     

Simple Zeros of the Riemann Zeta-Function

Assuming the Riemann Hypothesis, Montgomery showed by meansof his pair correlation method that at least two-thirds of thezeros of Riemann's zeta-function are simple. Later he and Taylorimproved this to 67.25 percent and, more recently, Cheer andGoldston increased the percentage to 67.2753. Here we proveby a new method that if the Riemann and Generalized LindelöofHypotheses hold, then at least 70.3704 percent of the zerosare simple and at least 84.5679 percent are distinct. Our methoduses mean value estimates for various functions defined by Dirichletseries sampled at the zeros of the Riemann zeta-function. 1991Mathematics Subject Classification: 11M26.     

Existence and Uniqueness of Fixed Points for Markov Operators and Markov Processes

This paper concerns a Markov operator T on a space L₁, and aMarkov process P which defines a Markov operator on a spaceM of finite signed measures. For T, the paper presents necessaryand sufficient conditions for:

a the existence of invariant probabilitydensities (IPDs)

b the existence of strictly positive IPDs,and

c the existence and uniqueness of IPDs.

Similar resultson invariant probability measures for P are presented. The basicapproach is to pose a fixed-point problem as the problem ofsolving a certain linear equation in a suitable Banach space,and then obtain necessary and sufficient conditions for thisequation to have a solution. 1991 Mathematics Subject Classification:60J05, 47B65, 47N30.