Limits of parallelism in explicit ODE methods

Numerical methods for ordinary initial value problems that do not depend on special properties of the system are usually found in the class of linear multistage multivalue methods, first formulated by J.C. Butcher. Among these the explicit methods are easiest to implement. For these reasons there has been considerable research activity devoted to generating methods of this class which utilize independent function evaluations that can be performed in parallel. Each such group of concurrent function evaluations can be regarded as a stage of the method. However, it turns out that parallelism affords only limited opportunity for reducing the computing time with such methods. This is most evident for the simple linear homogeneous constant-coefficient test problem, whose solution is essentially a matter of approximating the exponential by an algebraic function. For a given number of stages and a given number of saved values, parallelism offers a somewhat enlarged set of algebraic functions from which to choose. However, there is absolutely no benefit in having the degree of parallelism (number of processors) exceed the number of saved values of the method. Thus, in particular, parallel one-step methods offer no speedup over serial one-step methods for the standard linear test problem. Although the implication of this result for general nonlinear problems is unclear, there are indications that dramatic speedups are not possible in general. Also given are some results relevant to the construction of methods.Work supported in part by National Science Foundation grants DMS 89 11410 and DMS 90 15533 and by US Department of Energy grant DOE DEFG02-87ER25026. Work of the second author was completed while at the University of Illinois.     

FT-EPR and HPLC study of the mechanism of 4-chlorophenol photolysis

The mechanism of 4-chlorophenol (4CP) photolysis was investigated with the aid of Fourier Transform Electron Paramagnetic Resonance (FT-EPR) and pulsed-laser photolysis combined with High Performance Liquid Chromatography (HPLC) detection-of stable (diamagnetic) products. With FT-EPR transient free radicals produced by pulsed-laser excitation of solutions of 4CP in alcohols could be identified. Time profiles of the FT-EPR spectra provided information on reaction kinetics and Chemically Induced Dynamic, Electron Polarization (CIDEP) effects. It was found that 4CP photolysis in alcohols leads to the simultaneous formation of the phenoxyl radical and radicals produced by hydrogen abstraction from the solvent. CIDEP patterns establish that these radicals are formed in a reaction sequence involving a triplet state precursor and radical pair intermediate. Results of earlier transient optical absorption measurements indicate that the triplet precursor must be the carbene 4-oxocyclohexa-2,5-dienylidene. This assignment is supported by the finding that photolysis of quinone diazide in a hydrogen-donating solvent gives the same free radical products as those obtained from 4CP. The formation of the phenoxyl radical intermediate accounts for the finding that photolysis of deoxygenated solutions of 4CP in alcohols gives phenol as stable diamagnetic product. By contrast, photolysis of aerated and deoxygenated aqueous solutions of 4CP produces benzoquinone and hydroquinone as primary products, respectively.     

Oxidation of olefins with alkanesulfoperacids

Alkanesulfonic peracids RSO₄H effectively oxidize olefins to the corresponding epoxides. The kinetics and mechanism of this reaction are studied. It was shown that both the monomer and dimer form of the peracid act as oxidizing agents. The dependence of the weight constant of the reaction of monomer RSO₄H with substituted acyclic olefins on the structure of the latter is described by the Taft equation. A linear correlation is established between log k₂ and the inductive constants of the substituents on the oxygen atom in the peracid.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 2, pp. 333–337, February, 1991.     

Calculated elastic constants of dilute alloys based on bcc metals

The elastic constants of dilute alloys based on bcc metals have been calculated using the Green’s function method obtaining explicit expressions for change in elastic constants in terms oft-matrix. The crystal impurity problem is discussed within an impurity model containing central and non-central force constant changes extended up to second neighbours of the impurity. The effect of volume change on elastic constants and a contribution from electron pressure term are considered. Numerical results for changes in three elastic moduli have been presented for a number of dilute alloys based on Mo, Nb, W, Ta and V.     

Necessary and sufficient conditions for existence of exponential dichotomy of impulsive equations

Necessary and sufficient conditions for the existence of an exponential dichotomy of impulsive differential equations in a Hilbert space are found.     

Theoretical temperature-electric field phase diagram for deuterated thiourea SC(ND2)2

A theoretical temperature-electric field phase diagram is constructed for thiourea SC(ND₂)₂ using a phenomenological approach based on the representation of the thermodynamic potentials for all phases. The theoretical diagram is compared with the experimental diagram. Fiz. Tverd. Tela (St. Petersburg) 39, 1282–1286 (July 1997)     

Synthesis and properties of polyethersulphone–polydimethyisiloxane block copolymers

High molecular weight alternating block copolymers of polyethesulphone (PES) and polydimethylsiloxane (PDMS) were prepared by the condensation of dimethylamino-terminated PDMS oligomers and hydroxy-terminated PES oligomers in 1,2-dichlorobenzene. Microphase separation of the block copolymers at exceptionally short block lengths was observed by differential scanning calorimetry (DSC) and transmission electron microscopy (TEM). The Si? O? C intersegment linkage in these materials appeared to display poor hydrolytic stability which is contrary to results obtained for other block copolymers.     

The vertex-cover polynomial of a graph

In this paper we define the vertex-cover polynomial Ψ(G,τ) for a graph G. The coefficient of τ^r in this polynomial is the number of vertex covers V′ of G with |V′|=r. We develop a method to calculate Ψ(G,τ). Motivated by a problem in biological systematics, we also consider the mappings f from {1, 2,…,m} into the vertex set V(G) of a graph G, subject to f⁻¹(x)f⁻¹(y)≠ for every edge xy in G. Let F(G,m) be the number of such mappings f. We show that F(G,m) can be determined from Ψ(G,τ).