A groupoid generalisation of Leavitt path algebras

Let \(G\) be a locally compact, Hausdorff, étale groupoid whose unit space is totally disconnected. We show that the collection \(A(G)\) of locally-constant, compactly supported complex-valued functions on \(G\) is a dense \(*\) -subalgebra of \(C_c(G)\) and that it is universal for algebraic representations of the collection of compact open bisections of \(G\) . We also show that if \(G\) is the groupoid associated to a row-finite graph or \(k\) -graph with no sources, then \(A(G)\) is isomorphic to the associated Leavitt path algebra or Kumjian–Pask algebra. We prove versions of the Cuntz–Krieger and graded uniqueness theorems for \(A(G)\) .     

Co-universal C*-algebras associated to generalised graphs

We introduce P-graphs, which are generalisations of directed graphs in which paths have a degree in a semigroup P rather than a length in ?. We focus on semigroups P arising as part of a quasi-lattice ordered group (G, P) in the sense of Nica, and on P-graphs which are finitely aligned in the sense of Raeburn and Sims. We show that each finitely aligned P-graph admits a C*-algebra C*_min (Λ) which is co-universal for partialisometric representations of Λ which admit a coaction of G compatible with the P-valued length function. We also characterise when a homomorphism induced by the co-universal property is injective. Our results combined with those of Spielberg show that every Kirchberg algebra is Morita equivalent to C*_min (Λ) for some (?²* ?)-graph Λ.     

Combined CI-HY studies of atomic states. IV. The four lowest 1S and four lowest 1P states of He and the lowest 1S and 1P states of H−1

Combined CI -HY method calculations are reported for the ground and first three excited S states of He with an error on the order of 10^?7 a.u. within the same 120-term basis. For He ¹P, the four lowest states are obtained with an error ≤2 × 10^?6 a.u. within the same 102-term basis. H^?1 S and ¹P states are also treated by the same CI -HY technique. The utility of an spd Slater-type orbital, r, v = 0, 1 basis is investigated, with indications that it might be an excellent basis for states of first row atoms.     

Decoherence via the dynamical casimir effect

We derive a master equation for a mirror interacting with the vacuum field via radiation pressure. The dynamical Casimir effect leads to decoherence of a superposition state in a time scale that depends on the degree of "macroscopicity" of the state components, and which may be much shorter than the relaxation time scale. Coherent states are selected by the interaction as pointer states.     

Mean value theorems for a matrix valued derivative

The derivative of a function defined on a set of matrices is given utilizing the Moore-Penrose pseudoinverse, which yields a natural extension of differentiation in the univariate calculus. Several mean value theorems involving this derivative are established.     

Facilitating problem definition in teams

The problems that a team comes to define as needing to be worked on depend upon idiosyncratically defined subjective, qualitative and organizationally political factors affecting the individual members of the team. In this paper we shall argue that Operational Research tends to pay little attention to the process of problem definition and that when it does, the emphasis is towards determining an accurate or correct interpretation of the external situation that the team confronts and the organizational objectives which have been laid down for the team. We suggest, however, that Operational Researchers are particularly well placed to act in a way which pays attention to both these fields. This paper is addressed to those Operational Researchers who predominantly work with complex policy problems belonging to a team of decision-makers. We shall demonstrate that group techniques, new computer simulation technology, and attention to process can lead to a team negotiating a definition and redefinition of a problem so that solutions have a greater commitment from the team, and therefore a higher chance of being implemented.