Unsteady laminar boundary layers on an infinite plate in an optically thick radiating gas

Unsteady laminar boundary layers on an infinite plate including the effects of radiative heat transfer and injection or suction on the surface of the plate have been studied. Similar solutions for the velocity and for the temperature are analyzed. Numerical examples for velocity and temperature profiles are given. The effectiveness of transpiration cooling decreases with increasing radiation parameter R _F. The peak temperature in the boundary layer is suppressed if R _F is high enough. This work was initiated when the authors were Summer Consultants at North American Aviation, Inc., Space and Information System Division in 1965 and completed at the University of Maryland and Wayne State University and was supported in part by U.S. Air Force through AFOSR Grant No. AFOSR-1015A-67.     

Numerical calculation of cutoff frequencies of optical fibers by a variational technique

A numerical method for calculating cutoff frequencies of modes of weakly guiding optical fibers is described. Starting from a properly constructed field representation, the method follows a variational approach to obtain an eigenvalue problem for the cutoff frequencies which can be easily solved by a standard numerical routine. The field representation uses an expansion in terms of the Laguerre–Gauss functions in the fiber core and limiting form for small arguments of modified Bessel functions in the fiber cladding. Fibers with power-law refractive index profiles and fibers with a profile showing a dip on the axis are analyzed. Results obtained for modes of arbitrary order (both azimuthal and radial) show that our method is very efficient and accurate.     

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.     

First synthesis of enantiopure 1,6-difunctionalized dodecahydrobenz[f]indenes

An enantiospecific route to the previously unreported 1,6-difunctionalized dodecahydrobenz[f]indene ring system is described. Optically pure Hajos-Parrish ketone is used as the building block for preparation of a 6-methyleneinden-5-ol. This allylic alcohol is then utilized in a Claisen rearrangement under Johnson's conditions to introduce a side chain that is further modified and cyclized to produce the benz[f]indene ring system.