On Series of Ordinals and Combinatorics

This paper deals mainly with generalizations of results in finitary combinatorics to infinite ordinals. It is well-known that for finite ordinals ∑_bT<αβ is the number of 2-element subsets of an α-element set. It is shown here that for any well-ordered set of arbitrary infinite order type α, ∑_bT<αβ is the ordinal of the set M of 2-element subsets, where M is ordered in some natural way. The result is then extended to evaluating the ordinal of the set of all n-element subsets for each natural number n ≥ 2. Moreover, series ∑_β<αf(β) are investigated and evaluated, where α is a limit ordinal and the function f belongs to a certain class of functions containing polynomials with natural number coefficients. The tools developed for this result can be extended to cover all infinite α, but the case of finite α appears to be quite problematic.     

Synthesis of β -Cyperone via Fungal Hydroxylation of thujone-derived tricyclic cyclopropanes

Synthesis of optically active sesquiterpenes with a eudesmane C-skeleton from the chiral starting material thujone involves transformation of a tricyclic intermediate (1R,2R,4S)-1,7-dimethyl-4-(1-methylethyl)tricyclo[4.4.0.0^2,4]dec-6-en-8-one ( 2 ) into the bicyclic compound β -cyperone ( 5 ). Hydroxylation of 2 at C(5) or C(11) permits subsequent opening of the cyclopropane ring and rearrangement to β -cyperone. In this publication, studies involving hydroxylation of 2 by fungal cultures are presented. The resultant products are useful intermediates in efficient synthesis of eudesmane sesquiterpenes. Of five fungi tested, Rhizopus oryzae ATCC 11145 proved most versatile. It hydroxylates at the exocyclic C(11) position in high yield (70%) and, to a lesser extent, at C(5) (5%). Enzymatic activity appears at the end of growth phase and at least 2.2 g of 2 per liter can be metabolized without significant loss of product yield. A second fungus, Cunninghamella echinulata ATCC 9244, proved most useful for hydroxylation of derivatives of 2 for the preparation of derivatives of β -cyperone, although product yields were low (2–20%), some derivatives were nonreactive, and hydroxylation at C(9) occurred. The relationship between precursor structure and enzyme affinity is discussed.     

A finite model property for RMImin

It is proved that the variety of relevant disjunction lattices has the finite embeddability property. It follows that Avron's relevance logic RMI _min has a strong form of the finite model property, so it has a solvable deducibility problem. This strengthens Avron's result that RMI _min is decidable. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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     

Elektrische und thermische Eigenschaften von Salzlösungen

Ohne Zusammenfassung     

Hypernormal form theory: foundations and algorithms

Hypernormal forms (unique normal forms, simplest normal forms) are investigated both from the standpoint of foundational theory and algorithms suitable for use with computer algebra. The Baider theory of the Campbell-Hausdorff group is refined, by a study of its subgroups, to determine the smallest substages into which the hypernormalization process can be divided. This leads to a linear algebra algorithm to compute the generators needed for each substage with the least amount of work. A concrete interpretation of Jan Sanders’ spectral sequence for hypernormal forms is presented. Examples are given, and a proof is given for a little-known theorem of Belitskii expressing the hypernormal form space (in the inner product style) as the kernel of a higher-order differential operator.     

Experimental analysis, modeling, and theoretical design of McMaster pore-filled nanofiltration membranes

A side-by-side comparison of the performance of McMaster pore-filled (MacPF) and commercial nanofiltration (NF) membranes is presented here. The single-salt and multi-component performance of these membranes is studied using experimental data and using a mathematical model. The pseudo two-dimensional model is based on the extended Nernst–Planck equation, a modified Poisson–Boltzmann equation, and hydrodynamic calculations. The model includes four structural properties of the membrane: pore radius, pure water permeability, surface charge density and the ratio of effective membrane thickness to water content. The analysis demonstrates that the rejection and transport mechanisms are the same in the commercial and MacPF membranes with different contributions from each type of mechanism (convection, diffusion and electromigration). Solute rejection in NF membranes is determined primarily by a combination of steric and electrostatic effects. The selectivity of MacPF membranes is primarily determined by electrostatic effects with a significantly smaller contribution of steric effects compared to commercial membranes. Hence, these membranes have the ability to reject ions while remaining highly permeable to low molecular weight organics. Additionally, a new theoretical membrane design approach is presented. This design procedure potentially offers the optimization of NF membrane performance by tailoring the membrane structure and operating variables to the specific process, simultaneously. The procedure is validated at the laboratory scale.     

Transport of Neutrally Buoyant and Dense Variably Sized Colloids in a Two-Dimensional Fracture with Anisotropic Aperture

The transport of monodisperse as well as polydisperse colloid suspensions in a two-dimensional, water saturated fracture with spatially variable and anisotropic aperture is investigated with a particle tracking model. Both neutrally buoyant and dense colloid suspensions are considered. Although flow and transport in fractured subsurface formations have been studied extensively by numerous investigators, the transport of dense, polydisperse colloid suspensions in a fracture with spatially variable and anisotropic aperture has not been previously explored. Simulated snapshots and breakthrough curves of ensemble averages of several realizations of a log-normally distributed aperture field show that polydisperse colloids exhibit greater spreading than monodisperse colloids, and dense colloids show greater retardation than neutrally buoyant colloids. Moreover, it is demonstrated that aperture anisotropy oriented along the flow direction substantially increases colloid spreading; whereas, aperture anisotropy oriented transverse to the flow direction retards colloid movement.