Direct product and decomposition of certain physically important algebras

I consider the direct product algebra formed from two isomorphic Clifford algebras. More specifically, for an element x in each of the two component algebras I consider elements in the direct product space with the form x x. I show how this construction can be used to model the algebraic structure of particular vector spaces with metric, to describe the relationship between wavefunction and observable in examples from quantum mechanics, and to express the relationship between the electromagnetic field tensor and the stress-energy tensor in electromagnetism. To enable this analysis I introduce a particular decomposition of the direct product algebra.     

Hydraulic conductivity of rock fractures

The flow of a single-phase fluid through a rough-walled rock fracture is discussed within the context of fluid mechanics. The derivation of the cubic law is given as the solution to the Navier-Stokes equations for flow between smooth, parallel plates - the only fracture geometry that is amenable to exact treatment. The various geometric and kinematic conditions that are necessary in order for the Navier-Stokes equations to be replaced by the more tractable lubrication or Hele-Shaw equations are studied and quantified. In general, this requires a sufficiently low flow rate, and some restrictions on the spatial rate of change of the aperture profile. Various analytical and numerical results are reviewed pertaining to the problem of relating the effective hydraulic aperture to the statistics of the aperture distribution. These studies all lead to the conclusion that the effective hydraulic aperture is less than the mean aperture, by a factor that depends on the ratio of the mean value of the aperture to its standard deviation. The tortuosity effect caused by regions where the rock walls are in contact with each other is studied using the Hele-Shaw equations, leading to a simple correction factor that depends on the area fraction occupied by the contact regions. Finally, the predicted hydraulic apertures are compared to measured values for eight data sets from the literature for which aperture and conductivity data were available on the same fracture. It is found that reasonably accurate predictions of hydraulic conductivity can be made based solely on the first two moments of the aperture distribution function, and the proportion of contact area.     

Variable step size destabilizes the Störmer/leapfrog/verlet method

For fixed step-sizeh the Störmer method is stable for the standard test equationÿ= ² y,>0, if and only ifh<2. We show that for variable step sizeh _n there does not exist a (positive) limit onh which ensures stability. Nor can we guarantee stability if, in addition, we limit the step size ratioh _n/h _n–1.This work was supported in part by National Science Foundation Grant DMS 90 15533.     

Substituent effects in the fragmentation reactions of acetanilides and phenyl acetates

The mass spectra of a group of m- and p-substituted acetanilides were measured to determine the effect of the substituents, if any, on the rate of C₂H₂O expulsion (rearrangement) vs. the rate of [CH₃CO]⁺ formation (simple cleavage). Our results agree with observations of others in that substituents which raise the ionization potential of the aromatic ring appear to localize, on the average, less charge density on this locus, and conversely. The atomic composition of the substituent, however, irrespective of its position, seems to determine the relative rates of the competing reaction. Although in several cases Z_m/Z_p values were close to unity, rearrangement of isomeric molecular ions to a common species is shown not to occur.     

Study of cross-conjugated iso-polytriacetylenes and related oligoenynes

Two series of monodisperse cross-conjugated oligomers based on enyne repeat units have been realized. The first class of molecules, iso-polytriacetylenes (iso-PTAs, 2), was divergently synthesized using an iterative sequence of palladium-catalyzed cross-coupling reactions of vinyl triflate 5 with terminal alkynes. The second series of oligoenynes (17-20) are based on an octatetrayne backbone, and result from homocoupling of the differentially protected iso-PTA oligomers 8-11. The longest member of this series, 20, spans ca. 5.6 nm from Si atom to Si atom and is composed of a contiguous sequence of 44 sp and sp(2) carbons. The lowest energy electronic absorption band for iso-PTA dimers in the progression 13 --> 9 --> 16 is consistently red-shifted as a result of extending the cross-conjugated structure. A similar comparison within each series (i.e., 16, 6-7, or 17-20), however, suggests little effect on the electronic characteristics of these molecules as oligomer length is increased. The solid-state properties of one derivative, 17, are also described.     

Excited state solvation dynamics of 2-anilinonaphthalene

Nanosecond fluorescence spectroscopy has been used to study the interaction of 2-anilinonaphthalene with polar solvent molecules which is shown to result in stoichiometric complex formation at low polar solvent concentrations. This is followed by reorientation of the solvent cage when the concentration of polar solvent is high.     

A theoretical study of configurational inversion of 1,1′-binaphthyl by molecular mechanics

The molecular mechanics method has been applied to the study of various pathways and transition states for the configurational inversion of 1,1′-binaphthyl. The preferred pathway is found to be one on the anti (“trans”) side involving one point of H?H nonbonded contact in each of two identical transition states, separated by a very shallow minimum. Very satisfactory agreement with available experimental values for the enthalpy of activation is achieved.     

Salting-in of alcohols in aqueous solutions by tetraalkylammonium bromides at the freezing temperature

The freezing-point depression of the ternary systems tetraalkylammonium bromides-t-butanol-water for the first five homologs of R₄NBr was measured. In the case of Bu₄NBr, the effect of size of the alcohol (methahol ton-butanol) was also investigated. From the corresponding freezing-point data for the binary systems the apparent salting constants were calculated. The true salting constantsk _s were obtained by extrapolation to infinite dilution. These are all very close to zero at the freezing temperature. From the corresponding thermochemical data the temperature dependence ofk _s was calculated, and above 5°C all the R₄NBr salts int-butanol; the salting-in increases with temperature and with the size of the hydrophobic cations. The scaled-particle theory is at present the only one which can account semiquantitatively for the temperature dependence of the salting-in effect. On leave of absence from Chemistry Department, The University, Sheffield S3 7HF, England To whom correspondence should be addressed.