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     

Learning to Make Sense of Division of Fractions: One K–8 Preservice Teacher's Perspective

Current reform documents in mathematics education recommend that teachers help students develop both conceptual and procedural understandings. However, teachers often do not possess the in-depth mathematical reasoning necessary to accomplish this goal. The purpose of this article is to provide one way in which preservice teachers can come to better understand the mathematics they will teach.     

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.     

The zeta function of a simplicial complex

Given a simplicial complex δ on vertices {1, …,n} and a fieldF we consider the subvariety of projective (n−1)-space overF consisting of points whose homogeneous coordinates have support in δ. We give a simple rational expression for the zeta function of this singular projective variety overF _q and show a close connection with the Betti numbers of the corresponding variety over ℂ. This connection is particularly simple in the case when Δ is Cohen-Macaulay.     

Reducing system of parameters and the Cohen-Macaulay property

Let R be a local ring and let (x ₁, …, x _r) be part of a system of parameters of a finitely generated R-module M, where r < dim_R M. We will show that if (y ₁, …, y _r) is part of a reducing system of parameters of M with (y ₁, …, y _r) M = (x ₁, …, x _r) M then (x ₁, …, x _r) is already reducing. Moreover, there is such a part of a reducing system of parameters of M iff for all primes P ε Supp M ∩ V _R(x ₁, …, x _r) with dim_R R/P = dim_R M − r the localization M _P of M at P is an r-dimensional Cohen-Macaulay module over R _P. Furthermore, we will show that M is a Cohen-Macaulay module iff y _d is a non zero divisor on M/(y ₁, …, y _d−1) M, where (y ₁, …, y _d) is a reducing system of parameters of M (d:= dim_R M).     

A 533-MHz BiCMOS superscalar RISC microprocessor

This 533-MHz BiCMOS very large scale integration (VLSI) implementation of the PowerPC architecture contains three pipelines and a large on-chip secondary cache to achieve a peak performance of 1600 MIPS. The 15 mm×10 mm die contains 2.7 M transistors (2M CMOS and 0.7 M bipolar) and dissipates less than 85 W. The die is fabricated in a six-level metal, 0.5-μm BiCMOS process and requires 3.6 and 2.1 V power supplies     

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.     

Quantum correlations in dual quantum measurements

 We analyze the quantum measurement properties of dual non-degenerate parametric amplifers in the twin-beam configuration, in the cascaded back-action-evasion configuration, and in Kerr-type photon-number quantum non-demolition measurements. It is found that Einstein-Podolsky-Rosen correlations can be obtained between the quadrature components of an idler mode and the sum of the readout of two signal modes. Furthermore, we discuss dual-mode quantum non-demolition measurements on the combination of two light modes, and the generation of number-state entanglement. Received: 12 April 1996/Revised version: 2 July 1996     

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’.