Phase Shift Modulations for Stable,Oscillatory, Traveling,Strongly Nonlinear Waves

Oscillatory dispersive waves propagating in a slowly varying medium are analyzed for Klein-Gordon equations with perturbations. The method of multiple scales is extended to include two fast scales, the usual traveling-wave phase and time, in order to allow initial conditions not usually permitted. An exact wave-action equation is introduced if the traveling wave is stable, involving averages over the periodic wave as well as time. This is equivalent to an extended averaged Lagrangian principle. The equation for the slow modulations of the phase shift of the traveling wave is derived from the higher order terms in the exact action equation and is shown to be the same as in earlier more restrictive studies.     

Molecular addressing? Studies on light-induced reorientation in liquid-crystalline side chain polymers

Polarized light leads to an effective reorientation of the optic axis in the glassy state of liquid-crystalline side chain polymers containing azobenzene mesogenic groups, via a trans-cis and cis-trans isomerization. Using a combination of U V and IR dichroitic studies it is shown for copolymers consisting of chromophores (azobenzene) and non-chromophores (phenylbenzoate) that only the chromophores are reoriented by light as far as the glassy state is concerned. Individual chromophores are thus addressed by photoselection. Photoselection in the fluid nematic state, on the other hand, leads also to a reorientation of the non-chromophores.     

Liquid crystal side chain polysiloxanes containing various proportions of non-mesogenic units

Recent studies have established that side chain polymeric liquid crystals composed of mesogenic and non-mesogenic side groups keep their liquid-crystalline properties even for a low proportion of mesogens. We show that the detailed structures of three kinds of new diluted liquid crystal polysiloxanes depend on the nature of the co-substituent as well as on the proportion of the silicon sites occupied by the mesogenic groups. Mixtures of these systems with low molar mass liquid crystals were also investigated in terms of compatibility and/or stabilization of smectic A phases.     

On worst-case aggregation analysis for network location problems

Network location problems occur when new facilities must be located on a network, and the network distances between new and existing facilities are important. In urban, regional, or geographic contexts, there may be hundreds of thousands (or more) of existing facilities, in which case it is common to aggregate existing facilities, e.g. represent all the existing facility locations in a zip code area by a centroid. This aggregation makes the size of the problem more manageable for data collection and data processing purposes, as well as for purposes of analysis; at the same time, it introduces errors, and results in an approximating location problem being solved. There seems to be relatively little theory for doing aggregation, or evaluating the results of aggregation; most approaches are based on experimentation or computational studies. We propose a theory that has the potential to improve the means available for doing aggregation.This research was supported in part by the National Science Foundation, Grant No. DDM-9023392.     

M/G/1/N vacation model with varying E-limited service discipline

We define and analyze anM/G/1/N vacation model that uses a service discipline that we call theE-limited with limit variation discipline. According to this discipline, the server provides service until either the system is emptied (i.e. exhausted) or a randomly chosen limit ofl customers has been served. The server then goes on a vacation before returning to service the queue again. The queue length distribution and the Laplace-Stieltjes transforms of the waiting time, busy period and cycle time distributions are found. Further, an expression for the mean waiting time is developed. Several previously analyzed service disciplines, including Bernoulli scheduling, nonexhaustive service and limited service, are special cases of the general varying limit discipline that is analyzed in this paper.     

Whodunit? Exploring Proportional Reasoning Through the Footprint Problem

This paper describes a proportional reasoning problem set within a real‐life context and a complete analysis of one small group discussion of this problem over the course of a 90‐minute block. The seventh‐grade students' discourse is described to provide insights into typical mathematical interpretations of this problem, as well as some generalizations for other problems of this type. The interpretations provided reveal the gradual development of proportional reasoning in a local context from additive to multiplicative understandings.     

Synthesis and conformational studies of a new host system based on cholic acid

New synthetic hosts have been designed incorporating two molecules of cholic acid linked by a rigid diamine. Proton NMR studies indicate that the compounds exist in a rigid conformation with the steroid hydroxyl groups intramolecularly hydrogen-bonded. Heat or addition of methanol leads to conformational isomerism due to insertion of methanol into the cavity.A preliminary report of this work was presented at the 4th International Symposium on Inclusion Phenomena, July 20–25, 1986, Lancaster, U.K.