Non-separability Does Not Relieve the Problem of Bell’s Theorem

This paper addresses arguments that “separability” is an assumption of Bell’s theorem, and that abandoning this assumption in our interpretation of quantum mechanics (a position sometimes referred to as “holism”) will allow us to restore a satisfying locality principle. Separability here means that all events associated to the union of some set of disjoint regions are combinations of events associated to each region taken separately. In this article, it is shown that: (a) localised events can be consistently defined without implying separability; (b) the definition of Bell’s locality condition does not rely on separability in any way; (c) the proof of Bell’s theorem does not use separability as an assumption. If, inspired by considerations of non-separability, the assumptions of Bell’s theorem are weakened, what remains no longer embodies the locality principle. Teller’s argument for “relational holism” and Howard’s arguments concerning separability are criticised in the light of these results. Howard’s claim that Einstein grounded his arguments on the incompleteness of QM with a separability assumption is also challenged. Instead, Einstein is better interpreted as referring merely to the existence of localised events. Finally, it is argued that Bell rejected the idea that separability is an assumption of his theorem.     

Cationic Surfactants in Interfacial Synthesis of Linear Aromatic Polyester

Abstract

The polycondensation of terephthaloyl chloride and bisphenol A was used as a model reaction for the production of linear aromatic polyesters in stirred interfacial polymerization. The evaluation of several catalytic and surfactant additives to this system was based upon yields and intrinsic viscosities of the products obtained with both low and high concentrations of cationic surfactants of the quaternary ammonium type, of an anionic surfactant, and of a non-micelle forming quaternary ammonium salt. Although yields were similar in most cases, viscosity differences were marked. The highest intrinsic viscosities, hence highest degrees of polymerization, were found for preparations where cationic surfactant in excess of the critical micelle concentration was employed. Modes of action for such surfactants are suggested. The possibilities include, but are not limited to, solubilization of the product, solubilization of either or both monomers, emulsification of the liquid phases, catalytic phase transfer of one monomer, and micellar catalysis.     

Synthesis of 1,3-diketones through ring-opening of ketoketene dimer β-lactones

The reaction of ketoketene dimers with organolithium reagents afforded 1,3-diketones in good to excellent yields, and with good diastereoselectivity in some cases.     

Book Review

An Introduction to Difference Equations. Second Edition by Saber N. Elaydi, New York: Springer—Verlag, 1999. ISBN 0-387-98830-0. $54.95. Gone are the days when difference equations arose mainly in the context of sections of flows or as finite difference approximations to PDE's. Today difference equations have come into their own, both as objects of intrinsic mathematical interest and as dynamical models in their own right. Discrete models form an important part of dynamical systems theory independently from their continuous cousins. In Saber Elaydi's book dynamicists have the long awaited discrete counterpart to standard textbooks such as Hirsch and Smale (“Differential Equations, Dynamical Systems, and Linear Algebra”). The first edition of this book appeared in 1996. The second edition includes substantial new material including appendices on global stability and periodic solutions, a section on applications to mathematical biology, and a new chapter entitled “Applications to Continued Fractions and Orthogonal Polynomials”. Additional material on Birkhoff's theory now appears in the chapter on asymptotic behavior. The initial chapter covers first order equations, including equilibria, cobwebbing, stability, cycles, and the bifurcations of the discrete logistic equation. Chapter 2 moves on to higher order linear equations and briefly treats the difference calculus (for an in—depth treatment, see “Difference Equations: Theory and Applications. Second Edition” by Ronald E. Mickens, New York: Van Nostrand Reinhold, 1990). The subsequent chapters include systems of difference equations, stability theory, Z—transforms, control theory, oscillation theory, asymptotic behavior, and applications to continued fractions and orthogonal polynomials.

The chapters are composed of short sections, each of which ends with a nice selection of exercises. Answers to the odd—numbered problems appear in the back of the book. The core chapters include sections of applications to various fields such as population biology, economics, and physics. Several famous examples and topics are treated in the applications, including Gambler's Ruin, the Nicholson—Bailey host/parasitoid model, the heat equation, and Markov chains. Many discrete models are noninvertible, yet as many frustrated modelers know, most of the old standard treatments of linearization and the Stable Manifold Theorem., coming as they do from the context of sections of flows, require invertibility. Commendably, Elaydi avoids the needless assumption of invertibility in his stability theorems, and also in the Stable Manifold Theorem. However, invertibility is assumed in the Hartman—Grobman Theorem, where indeed it is necessary to establish conjugacy between the map and its linearization (see “An Introduction to Structured Population Dynamics”, CBMS—NSF Regional Conference Series in Applied Mathematics, Vol. 71, SIAM, Philadelphia, 1998 by J. M. Gushing, for an example of a noninvertible map for which the conjugacy fails. Readers may be interested to know that in this reference a weaker version of the Hartman—Grobman Theorem is proved that does not require invertibility but does establish the desired correspondence between types of hyperbolic equilibria in maps and their linearizations.)

This book is in Springer's Undergraduate Texts in Mathematics series and is indeed a very readable and appropriate text for advanced undergraduates or beginning graduate students. According to the author, the main prerequisites for such a course are calculus and linear algebra, with basic advanced calculus and complex analysis needed only for some topics in the later chapters. This is true; however in most situations the book would be best appreciated by students with a bit more mathematical maturity than is engendered by today's calculus and beginning linear algebra courses.Elaydi's book is a valuable reference for anyone who models discrete systems. It is so well written and well designed, and the content is so interesting to me, that I had a difficult time putting it down. I am pleased to own a copy for reference purposes, and am looking forward to using it to teach a senior topics course in difference equations.