Jean van Heijenoort’s Contributions to Proof Theory and Its History

Jean van Heijenoort was best known for his editorial work in the history of mathematical logic. I survey his contributions to model-theoretic proof theory, and in particular to the falsifiability tree method. This work of van Heijenoort??s is not widely known, and much of it remains unpublished. A complete list of van Heijenoort??s unpublished writings on tableaux methods and related work in proof theory is appended.     

An efficient algorithm for the “stable roommates” problem

The stable marriage problem is that of matching n men and n women, each of whom has ranked the members of the opposite sex in order of preference, so that no unmatched couple both prefer each other to their partners under the matching. At least one stable matching exists for every stable marriage instance, and efficient algorithms for finding such a matching are well known. The stable roommates problem involves a single set of even cardinality n, each member of which ranks all the others in order of preference. A stable matching is now a partition of this single set into n/2 pairs so that no two unmatched members both prefer each other to their partners under the matching. In this case, there are problem instances for which no stable matching exists. However, the present paper describes an O(n²) algorithm that will determine, for any instance of the problem, whether a stable matching exists, and if so, will find such a matching.     

Voltage security and reactive power management

The focus of this paper is on voltage security as a functionof ancillary services that can be described in the context ofreactive power markets and associated pricing mechanisms. Firstly,an introduction to voltage security and reactive power managementis presented and the consequences of recent electricity industryrestructuring in the UK are highlighted. Finally, several recentlydeveloped algorithmic techniques are presented in detail. Thetechniques will introduce the concepts of transition-optimizationand congestion management in the context of a complete frameworkfor the monitoring, assessment and control of voltage security.     

Kinetics and mechanism of the reaction between thiosulfate and chlorite ions at 90°C

The kinetics of the oxidation of S₂O₃^2? by ClO₂^? have been studied in aqueous alkaline solution at 90⁰C using classical titrimetric methods to follow the course of the reaction. The reaction takes place according to the stoichiometry S₂O₃^2? + 2ClO₂^? + 2OH^? = 2SO₄^2? + 2Cl^? + H₂O even in large S₂O₃^2? excess. There is some indication of a complex reaction pattern, but 70% of the ClO₂^? disappearance can be best described by the autocatalytic rate equation -d[ClO₂^?]/dt = k[S₂O₃^2?] [ClO₂^?] [H⁺] with k = (1.3 ± 0.2) ×10⁸ M^?2 sec^?1. The mechanism is explained by postulating nucleophilic attack of S₂O₃^2? on HClO₂ to form a chlorine-containing intermediate.     

Geometric phases in dissipative systems

It is shown that a phenomenon analogous to the geometric phase shifts of Berry and Hannay occurs for dissipative oscillatory systems and can be detected in numerical simulations of chemical oscillators. The approach herein to the theory of geometric phases begins with a study of simple first-order differential equations on the circle (circle dynamics). It is shown how more complicated systems exhibit geometric phases through reduction to a circle dynamics. In this way, the various manifestations of the phenomenon are seen from a single unified perspective. The results are illustrated in numerical experiments on several model systems ranging from analytically solvable, but contrived, to realistic models of chemical oscillators.     

The initiation process in the polymerization of tetrahydrofuran with carbonium ion salts

The initiation process for the polymerization of tetrahydrofuran with (C₆H₅)₃C⁺SbCl₆^? has been studied. Two mechanisms have been considered: a cation-addition process, and a process in which tetrahydrofuran donates a hydride ion to the cation of the initiator to form triphenylmethane. The biscarbonium salt [(C₆H₅)₂C⁺C₆H₄CH₂]₂(SbCl₆^?)₂ has been synthesized and used to initiate the polymerization of tetrahydrofuran. The results are consistent with the hydride-ion mechanism but may be inconclusive because of chain transfer. NMR experiments with 0.05–0.2M solutions of initiator in tetrahydrofuran show that triphenylmethane is rapidly produced in an amount equal to the molar amount of initiator originally present. Some NMR evidence for the presence of an acetal end group in the polymer has been obtained. It is concluded that the initiation process in this system definitely involves the formation of triphenylmethane, although a detailed, unique mechanism cannot be selected at this time.