0–1 laws for maps

A class of finite structures has a 0–1 law with respect to a logic if every property expressible in the logic has a probability approaching a limit of 0 or 1 as the structure size grows. To formulate 0–1 laws for maps (i.e., embeddings of graphs in a surface), it is necessary to represent maps as logical structures. Three such representations are given, the most general being the full cross representation based on Tutte's theory of combinatorial maps. The main result says that if a class of maps has two properties, richness and large representativity, then the corresponding class of full cross representations has a 0–1 law with respect to first‐order logic. As a corollary the following classes of maps on a surface of fixed type have a first‐order 0–1 law: all maps, smooth maps, 2‐connected maps, 3‐connected maps, triangular maps, 2‐connected triangular maps, and 3‐connected triangular maps. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 14, 215–237, 1999     

Poly(aryl ether)s containing o‐terphenyl subunits. II. Random poly(ether sulfone)s

The synthesis of a new A₂X‐type difluoride monomer, N‐2‐pyridyl‐4′,4″‐bis‐(4‐fluorobenzenesulfonyl)‐o‐terphenyl‐3,6‐dimethyl‐4,5‐dicarboxylic imide ( 3 ), is described. The monomer 3 was incorporated into a series of copoly(aryl ether sulfone)s by polymerization of 4,4′‐isopropylidenediphenol and 4,4′‐difluorophenylsulfone. The incorporation of monomer 3 had an observable effect on both the glass‐transition temperature of poly(aryl ether sulfone)s and the tendency for macrocyclic oligomers to form during polymerization. Replacement of the pyridyl imide group via a transimidization reaction with propargyl amine proceeded quantitatively and without polymer degradation. The acetylene containing copoly(aryl ether sulfone) could be crosslinked by simple thermal treatment, resulting in an increase in the glass‐transition temperature and solvent resistance. © 2000 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 38: 9–17, 2000     

Exact operator solutions to Euler hamiltonians

It is nearly impossible in general to solve exactly the quantum-mechanical Heisenberg equations of motion when these operator equations are non-linear. However, in this paper we show how to obtain exact closed-form solutions for the operators p(t) and q(t) for any Euler hamiltonian (one that be expressed a function of pq).     

Optimization of enzymatic production of biodiesel from castor oil in organic solvent medium

We studied the production of fatty acid ethyl esters from castor oil using n-hexane as solvent and two commercial lipases, Novozym 435 and Lipozyme IM, as catalysts. For this purpose, a Taguchi experimental design was adopted considering the following variables: temperature (35–65°C), water (0–10 wt/wt%), and enzyme (5–20 wt/wt%) concentrations and oil-to-ethanol molar ratio (1∶3 to 1∶10). An empirical model was then built so as to assess the main and cross-variable effects on the reaction conversion and also to maximize biodiesel production for each enzyme. For the system containing Novozym 435 as tatalyst the maximum conversion obtained was 81.4% at 65°C, enzyme concentration of 20 wt/wt%, water concentration of 0 wt/wt%, and oil-to-ethanol molar ratio of 1∶10. When the catalyst was Lipozyme IM, a conversion as high as 98% was obtained at 65°C, enzyme concentration of 20 wt/wt%, water concentration of 0 wt/wt%, and oil-to-ethanol molar ratio of 1∶3.     

Sensitivity enhancement in pulse EPR distance measurements

Established pulse EPR approaches to the measurement of small dipole-dipole couplings between electron spins rely on constant-time echo experiments to separate relaxational contributions from dipolar time evolution. This requires a compromise between sensitivity and resolution to be made prior to the measurement, so that optimum data are only obtained if the magnitude of the dipole-dipole coupling is known beforehand to a good approximation. Moreover, the whole dipolar evolution function is measured with relatively low sensitivity. These problems are overcome by a variable-time experiment that achieves suppression of the relaxation contribution by reference deconvolution. Theoretical and experimental results show that this approach leads to significant sensitivity improvements for typical systems and experimental conditions. Further sensitivity improvements or, equivalently, an extension of the accessible distance range can be obtained by matrix deuteration or digital long-pass filtering of the time-domain data. Advantages and limitations of the new variable-time experiment are discussed by comparing it to the established analogous constant-time experiment for measurements of end-to-end distances of 5 and 7.5 nm on rod-like shape-persistent biradicals and for the measurement of a broadly distributed transmembrane distance in a doubly spin-labeled mutant of plant light harvesting complex II.     

Ghost busting: Making sense of non-Hermitian Hamiltonians

The Lee model is an elementary quantum field theory in which mass, wave-function, and charge renormalization can be performed exactly. In early studies of this model in the 1950's it was found that there is a critical value of g ², the square of the renormalized coupling constant, above which g ₀ ² , the square of the unrenormalized coupling constant, is negative. For g ² larger than this critical value, the Hamiltonian of the Lee model becomes non-Hermitian. In this non-Hermitian regime a new state appears whose norm is negative. This state is called a ghost. It has always been thought that in this ghost regime the Lee model is an unacceptable quantum theory because unitarity appears to be violated. However, in this regime while the Hamiltonian is not Hermitian, it does possess symmetry. It has recently been discovered that a non-Hermitian Hamiltonian having symmetry may define a quantum theory that is unitary. The proof of unitarity requires the construction of a time-independent operator called C. In terms of C one can define a new inner product with respect to which the norms of the states in the Hilbert space are positive. Furthermore, it has been shown that time evolution in such a theory is unitary. In this talk the C operator for the Lee model in the ghost regime is constructed in the V/Nθ sector. It is then shown that the ghost state has a positive norm and that the Lee model is an acceptable unitary quantum field theory for all values of g ². Presented at the 3rd International Workshop “Pseudo-Hermitian Hamiltonians in Quantum Physics”, Istanbul, Turkey, June 20–22, 2005.     

Central and local limit theorems applied to asymptotic enumeration. III. Matrix recursions

Let a multivariate sequence a_n(k) be obtained by a matrix recursion. It is shown that it is usually easy to establish central and local limit theorems for a_n(k). The proof requires a lemma on multisection of multivariate series which appears to be new. The applications of the limit theorems include covering by polyominoes, enumeration of plane animals, occupancy problems, 0–1 matrices, and nonexistence of critical phenomena.     

Central and local limit theorems applied to asymptotic enumeration II: Multivariate generating functions

Let a multivariate sequence a_n(k) ? 0 be given. Multivariate central and local limit theorems are proved for a_n(k) as n → ∞ that are based on examining the generating function. Applications are made to permutations with rises and falls, ordered partitions of sets, Tutte polynomials of recursive families, and dissections of polygons.