On the Connectivity of Cobordisms and Half-Projective TQFT's

We consider a generalization of the axioms of a TQFT, the so-called half-projective TQFT's, where we inserted an anomaly, , in the composition law. Here μ₀ is a coboundary (in a group cohomological sense) on the cobordism categories with non-negative, integer values. The element of the ring over which the TQFT is defined does not have to be invertible. In particular, it may be zero. This modification makes it possible to extend quantum-invariants, which vanish on S ¹×S ², to non-trivial TQFT's. Note, that a TQFT in the ordinary sense of Atiyah with this property has to be trivial all together. We organize our discussions such that the notion of a half-projective TQFT is extracted as the only possible generalization under a few very natural assumptions. Based on separate work with Lyubashenko on connected TQFT's, we construct a large class of half-projective TQFT's with . Their invariants all vanish on S ¹×S ², and they coincide with the Hennings invariant for non-semisimple Hopf algebras and, more generally, with the Lyubashenko invariant for non-semisimple categories. We also develop a few topological tools that allow us to determine the cocycle μ₀ and find numbers, ϱ(M), such that the linear map associated to a cobordism, M, is of the form . They are concerned with connectivity properties of cobordisms, as for example maximal non-separating surfaces. We introduce in particular the notions of “interior” homotopy and homology groups, and of coordinate graphs, which are functions on cobordisms with values in the morphisms of a graph category. For applications we will prove that half-projective TQFT's with vanish on cobordisms with infinite interior homology, and we argue that the order of divergence of the TQFT on a cobordism, M, in the “classical limit” can be estimated by the rank of its maximal free interior group, which coincides with ϱ(M). Received: 20 October 1997 / Accepted: 18 March 1998     

Large-scale computer simulation of fully developed turbulent channel flow with heat transfer

Recently, with the advent of supercomputers, there has been considerable interest in the use of direct numerical simulation to obtain information about turbulent shear flow at low Reynolds number. This paper presents a pseudospectral technique to solve the full three-dimensional time-dependent Navier-Stokes and advection-diffusion equations without the use of subgrid-scale modelling. The technique has not been previously used for fully developed turbulent channel flow simulation and is based on methods applied in other contexts. The emphasis of this paper is to provide a reasonably detailed account of how the simulation is done rather than to present new calculations of turbulence. The details of an algorithm for turbulent channel flow simulation and the grid and time step sizes needed to integrate through transient behaviour to steady state turbulence have not been published before and are presented here. Results from a Cray-2 simulation of fully developed turbulent flow in a channel with heat transfer are presented along with a critical comparison between experiment and computation. The first- and second-order moments agree well with experimental measurements; the agreement is poor for higher-order moments such as the skewness and flatness near the walls of the channel. Detailed information given about the effects of spatial grid resolution on a computed results is important for estimating the size of the computation required to study various aspects of a turbulent flow.     

Fiber formation via solution spinning of the cellulose/ammonia/ammonium thiocyanate system

Fiber formation via the cellulose/ammonia/ammonium thiocyanate system by wet spinning has been investigated. This report presents a characterization of the structure and tensile properties of fibers spun under various coagulation conditions. Microscopic observations showed that the molecular size of coagulant was the dominant factor governing the crosssectional shape of the fibers. Density, birefringence, and crystallinity data indicated that a higher cellulose concentration and lower coagulation temperature favored development of a fiber with a denser and more oriented structure. Under optimum conditions, a welldefined fibrillar structure was obtained. Fiber tensile property measurements suggested the existence of a linear relationship between the fiber breaking tenacity and the product of the square of the Hermans' orientation factor and the infrared crystallinity index.     

Generalized Rotation numbers

A rooted graph is a pair (G,x), where G is a simple undirected graph and x ∈ V(G). If G is rooted at x, its kth rotation number h_k (G,x) is the minimum number of edges in a graph F of order |G| + k such that for every v ∈ V(F) we can find a copy of G in F with the root vertex x at v. When k = 0, this definition reduces to that of the rotation number h(G,x), which was introduced in [“On Rotation Numbers for Complete Bipartite Graphs,” University of Victoria, Department of Mathematics Report No. DM-186-IR (1979)] by E.J. Cockayne and P.J. Lorimer and subsequently calculated for complete multipartite graphs. In this paper, we estimate the kth rotation number for complete bipartite graphs G with root x in the larger vertex class, thereby generalizing results of B. Bollobás and E.J. Cockayne [“More Rotation Numbers for Complete Bipartite Graphs,” Journal of Graph Theory, Vol. 6 (1982), pp. 403–411], J. Haviland [“Cliques and Independent Sets,” Ph. D. thesis, University of Cambridge (1989)], and J. Haviland and A. Thomason [“Rotation Numbers for Complete Bipartite Graphs,” Journal of Graph Theory, Vol. 16 (1992), pp. 61–71]. © 1993 John Wiley & Sons, Inc.     

Determination of the Tolman cone angle from crystallographic parameters and a statistical analysis using the crystallographic data base

Summary A new method of evaluating the Tolman cone angle from X-ray structural data available from the Cambridge Crystallographic Data Base has been developed and a statistical analysis of the cone angles of the phosphines PPh₃, PMe₂Ph, PMePh₂, PMe₃, PEt₃ and PCy₃ (Cy = cyclohexyl) in transition metal complexes has been completed.     

On the Immersion Relation and an Embedding Problem for Infinite Graphs

Recent results on the immersion relation for graphs are applied to show that there exist (uncountable) graphs A and G such that G contains n disjoint copies of A for each positive integer n, but the graph N₀A consisting of infinitely many disjoint copies of A does not embed in G, where by ‘embedding’ we mean ‘immersion in the weak sense’.     

Particle production from signature change

We consider the (massless) scalar field on 2-dimensional manifolds whose metric changes signature and which admit a spacelike isometry. Choosing the wave equation so that there will be a conserved Klein-Gordon product implicitly determines the junction conditions one needs to impose in order to obtain global solutions. The resulting mix of positive and negative frequencies produced by the presence of Euclidean regions depends only on the total width of the regions, and not on the detailed form of the metric.This essay received honorable mention in the 1991 competition of the Gravity Research Foundation.     

Depth profiling of latex blends of fluorinated and fluorine‐free acrylates with laser‐induced secondary mass spectrometry

We explored phase separation and self‐assembly of perfluoroalkyl segments at the surface of polymer films obtained from latices of semifluorinated acrylate copolymers and the corresponding latex blends of nonfluorinated and semifluorinated polyacrylates. With laser‐induced secondary mass spectrometry the fluorine distribution was measured after annealing above the minimum film‐forming temperature of the polymers up to a depth of several micrometers. Depth profiles of a semifluorinated acrylate homopolymer and latex blends thereof with fluorine‐free alkylacrylates with 25, 50, and 75 mol % semifluorinated acrylate as well as a copolymer comprised of alkyl acrylate and semifluorinated acrylate (50/50 mol %) were investigated. In the case of latex blends containing both semifluorinated polyacrylates and fluorine‐free or low‐fluorine polymers, self‐assembly accounted for enrichment of the perfluoroalkyl segments at the surface. Coatings exhibiting low surface energy and having a substantially reduced total fluorine content were obtained. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 360–367, 2003