State sum models and simplicial cohomology

We study a class of subdivision invariant lattice models based on the gauge groupZ _p , with particular emphasis on the four dimensional example. This model is based upon the assignment of field variables to both the 1- and 2-dimensional simplices of the simplicial complex. The property of subdivision invariance is achieved when the coupling parameter is quantized and the field configurations are restricted to satisfy a type of mod-p flatness condition. By explicit computation of the partition function for the manifoldRP ³×S ¹, we establish that the theory has a quantum Hilbert space which differs from the classical one.Supported by Stichting voor Fundamenteel Onderzoek der Materie (FOM)     

The algebraic structure of cohomological field theory

The algebraic foundation of cohomological field theory is presented. It is shown that these theories are based upon realizations of an algebra which contains operators for both BRST and vector supersymmetry. Through a localization of this algebra, we construct a theory of cohomological gravity in arbitrary dimensions. The observables in the theory are polynomial, but generally non-local operators, and have a natural interpretation in terms of a universal bundle for gravity. As such, their correlation functions correspond to cohomology classes on moduli spaces of Riemannian connections. In this uniformization approach, different moduli spaces are obtained by introducing curvature singularities on codimension two submanifolds via a puncture operator. This puncture operator is constructed from a naturally occuring differential form of co-degree two in the theory, and we are led to speculate on connections between this continuum quantum field theory, and the discrete Regge calculus.     

Scheduling precedence graphs of bounded height

The existence of a schedule for a partially ordered set of unit length tasks on m identical processors is known to be NP-complete (J. D. Ullman, NP-complete scheduling problems, J. Comput. System Sci., 10 (1975), 384–393). The problem remains NP-complete even if we restrict the precedence graph to be of height bounded by a constant. (J. K. Lenkstra and A. H. G. Rinnooy Kan, Complexity of scheduling under precedence constraints, Operations Res., 26 (1978), 22–35; D. Dolev and M. K. Warmuth, “Scheduling Flat Graphs,” IBM Research Report RJ 3398, 1982). In these NP-completeness proofs the upper bound on the number of available processors varies with the problem instance. We present a polynomial algorithm for the case where the upper bound on the number of available processors and the height of the precedence graph are both constants.     

Oxoneptunium(v) as part of the framework of a polyoxometalate

(NH4)14Na4[(Np3W4O15)(H2O)3(BiW9O33)3].62H2O (1) and (NH4)14.5Na3.5[(Np3W4O15)(H2O)3(SbW9O33)3].40.5H2O (2) each contain three neptunyl(v) moieties encapsulated within heteropolyoxotungstate frameworks in which axial {NpO2}+ oxygens form one face of a WO6 octahedron.     

Avoided-crossing-based liquid-crystal photonic-bandgap notch filter

We demonstrate a highly tunable deep notch filter realized in a liquid-crystal photonic-bandgap (LCPBG) fiber. The filter is realized without inducing a long-period grating in the fiber but simply by filling a solid-core photonic-crystal fiber with a liquid crystal and exploiting avoided crossings within the bandgap of the LCPBG fiber. The filter is demonstrated experimentally and investigated using numerical simulations. A high degree of tuning of the spectral position of the deep notch is also demonstrated.     

Injective dimension of sheaves of rational vector spaces

The Cantor–Bendixson rank of a topological space X is a measure of the complexity of the topology of X. We will be interested primarily in the case that the space is profinite: Hausdorff, compact and totally disconnected. In this paper, we prove that the injective dimension of the abelian category of sheaves of $\mathbb{Q}$-modules over a profinite space X is determined by the Cantor–Bendixson rank of X.