Stability of a Model of Relativistic Quantum Electrodynamics

The relativistic “no pair” model of quantum electrodynamics uses the Dirac operator, D(A) for the electron dynamics together with the usual self-energy of the quantized ultraviolet cutoff electromagnetic field A– in the Coulomb gauge. There are no positrons because the electron wave functions are constrained to lie in the positive spectral subspace of some Dirac operator, D, but the model is defined for any number, N, of electrons, and hence describes a true many-body system. In addition to the electrons there are a number, K, of fixed nuclei with charges ≤Z. If the fields are not quantized but are classical, it was shown earlier that such a model is always unstable (the ground state energy E=−∞) if one uses the customary D(0) to define the electron space, but is stable (E > − const.(N+K)) if one uses D(A) itself (provided the fine structure constant α and Z are not too large). This result is extended to quantized fields here, and stability is proved for α= 1/137 and Z≤ 42. This formulation of QED is somewhat unusual because it means that the electron Hilbert space is inextricably linked to the photon Fock space. But such a linkage appears to better describe the real world of photons and electrons. Received: 8 September 2001 / Accepted: 18 March 2002     

Segregation in the Falicov--Kimball Model

The Falicov–Kimball model is a simple quantum lattice model that describes light and heavy electrons interacting with an on-site repulsion; alternatively, it is a model of itinerant electrons and fixed nuclei. It can be seen as a simplification of the Hubbard model; by neglecting the kinetic (hopping) energy of the spin up particles, one gets the Falicov–Kimball model. We show that away from half-filling, i.e. if the sum of the densities of both kinds of particles differs from 1, the particles segregate at zero temperature and for large enough repulsion. In the language of the Hubbard model, this means creating two regions with a positive and a negative magnetization. Our key mathematical results are lower and upper bounds for the sum of the lowest eigenvalues of the discrete Laplace operator in an arbitrary domain, with Dirichlet boundary conditions. The lower bound consists of a bulk term, independent of the shape of the domain, and of a term proportional to the boundary. Therefore, one lowers the kinetic energy of the itinerant particles by choosing a domain with a small boundary. For the Falicov- Kimball model, this corresponds to having a single “compact” domain that has no heavy particles. Received: 21 June 2001 / Accepted: 4 January 2002     

A representation of acoustic waves in unbounded domains

Compact, time‐harmonic, acoustic sources produce waves that decay too slowly to be square‐integrable on a line away from the sources. We introduce an inner product, arising directly from Green's second theorem, to form a Hilbert space of these waves and present examples of its computation.¹ © 2005 Wiley Periodicals, Inc.     

Resolution of Some Open Problems Concerning Multiple Zeta Evaluations of Arbitrary Depth

We prove some new evaluations for multiple polylogarithms of arbitrary depth. The simplest of our results is a multiple zeta evaluation one order of complexity beyond the well-known Broadhurst–Zagier formula. Other results we provide settle three of the remaining outstanding conjectures of Borwein, Bradley, and Broadhurst. A complete treatment of a certain arbitrary depth class of periodic alternating unit Euler sums is also given.     

Experimental design and the optimization of a polymer supported palladium complex for use in the Heck reaction

Optimization of the Heck reaction of 4-bromoacetophenone with styrene by a polymer supported, sulfur-containing palladacycle, varying 6 factors at a total of 28 different levels, corresponding to 5760 different possibilities was undertaken. Conversion improved from 34%, with large observable leaching to 88% with no leaching. This was accomplished using a Design of Experiments approach facilitated by the Statistical Design Package, MODDE 7.0^TM.     

Measure change techniques in optimal control

Fully and partially observed stochastic control of systems with nonlinear dynamics and terminal and running costs are considered. Measure changes are introduced which allow both state and observation dynamics to be thought of as linear. In the case when the terms of the cost have a special form the measure change transformation “cancels out” the nonlinearities and changes the original nonlinear problem into a classical LQG one and standard results can be applied. We also consider unnormalized conditional densities of the whole path as state variables and obtain dynamic programming and verification results. R. J. Elliott wishes to acknowledge support of the Natural Sciences and Engineering Research Council of Canada, Grant A7964.     

Characterisation of the photocatalyst Pilkington Activ™: a reference film photocatalyst?

Pilkington Glass Activ™ represents a possible suitable successor to P25 TiO₂, especially as a benchmark photocatalyst film for comparing other photocatalyst or PSH self-cleaning films. Activ™ is a glass product with a clear, colourless, effectively invisible, photocatalytic coating of titania that also exhibits PSH. Although not as active as a film of P25 TiO₂, Activ™ vastly superior mechanical stability, very reproducible activity and widespread commercial availability makes it highly attractive as a reference photocatalytic film. The photocatalytic and photo-induced superhydrophilitic (PSH) properties of Activ™ are studied in some detail and the results reported. Thus, the kinetics of stearic acid destruction (a 10⁴ electron process) are zero order over the stearic acid range 4–129 monolayers and exhibit formal quantum efficiencies (FQE) of 0.7×10⁻⁵ and 10.2×10⁻⁵ molecules per photon when irradiated with light of 365±20 and 254 nm, respectively; the latter appears also to be the quantum yield for Activ™ at 254 nm. The kinetics of stearic acid destruction exhibit Langmuir–Hinshelwood-like saturation type kinetics as a function of oxygen partial pressure, with no destruction occurring in the absence of oxygen and the rate of destruction appearing the same in air and oxygen atmospheres. Further kinetic work revealed a Langmuir adsorption type constant for oxygen of 0.45±0.16 kPa⁻¹ and an activation energy of 19±1 kJ mol⁻¹. A study of the PSH properties of Activ™ reveals a high water contact angle (67°) before ultra-bandgap irradiation reduced to 0° after prolonged irradiation. The kinetics of PSH are similar to those reported by others for sol–gel films using a low level of UV light. The kinetics of contact angle recovery in the dark appear monophasic and different to the biphasic kinetics reported recently by others for sol–gel films [J. Phys. Chem. B 107 (2003) 1028]. Overall, Activ™ appears a very suitable reference material for semiconductor film photocatalysis.     

Algorithm for Optimal Triangulations in Scattered Data Representation and Implementation

Scattered data collected at sample points may be used to determine simple functions to best fit the data. An ideal choice for these simple functions is bivariate splines. Triangulation of the sample points creates partitions over which the bivariate splines may be defined. But the optimality of the approximation is dependent on the choice of triangulation. An algorithm, referred to as an Edge Swapping Algorithm, has been developed to transform an arbitrary triangulation of the sample points into an optimal triangulation for representation of the scattered data. A Matlab package has been completed that implements this algorithm for any triangulation on a given set of sample points.     

Image Formation in Low Vacuum SEM

The role of positive gaseous ions in the formation of secondary electron images in low vacuum scanning electron microscopes is discussed. This paper describes the charging processes and related effects that occur during high vacuum imaging of insulators and then discusses the influence of ions on those processes. The ions are responsible for a number of phenomena, including distortion of the electric field above and below the specimen surface due to space charge, removal of excess negative charge from the specimen, alteration of the specimen surface barrier, and scavenging/filtering of the secondary electron emission. The resulting electron-specimen-ion interactions can give rise to interesting contrast effects that are unique to this class of instruments.