Relative locality of derivations

Let H and K be symmetric linear operators on a C¹-algebra $\text{U}$ with domains D(H) and D(K). H is defined to be strongly K-local if $\text{ω(K(A)}{}^1\text{K(A)) = 0}$ implies $\text{ω(H(A)}{}^1\text{H(A)) = 0}$ for A?D(H) ∩ D(K) and ω in the state space of $\text{U}$, and H is completely strongly K-local if $\text{Ω(K(A)}{}^1\text{K(A))=0}$ implies $\text{Ω(H(A)}{}^1\text{H(A))=0}$ for A ∈ D(H) ∩ D(K) and Ω in the state of $\text{U}$, and H is cpmpletely strongly K-local if $\text{H??}{}_{\mathrm{n}}$ is $\text{K??}{}_{\mathrm{n}}$-local on U?M_n for all n ? 1, where $\text{1}$_n is the identity on the n × n matrices M_n. If $\text{U}$ is abelian then strong locality and complete strong locality are equivalent. The main result states that if τ is a strongly continuous one-parameter group of ¹-automorphisms of $\text{U}$ with generator δ₀ and δ is a derivation which commutes with τ and is completely strongly δ₀-local then δ generates a group α of ¹-automorphisms of $\text{U}$. Various characterizations of α are given and the particular case of periodic τ is discussed.     

The graph isomorphism disease

The graph isomorphism problem—to devise a good algorithm for determining if two graphs are isomorphic—is of considerable practical importance, and is also of theoretical interest due to its relationship to the concept of NP-completeness. No efficient (i.e., polynomial-bound) algorithm for graph isomorphism is known, and it has been conjectured that no such algorithm can exist. Many papers on the subject have appeared, but progress has been slight; in fact, the intractable nature of the problem and the way that many graph theorists have been led to devote much time to it, recall those aspects of the four-color conjecture which prompted Harary to rechristen it the “four-color disease.” This paper surveys the present state of the art of isomorphism testing, discusses its relationship to NP-completeness, and indicates some of the difficulties inherent in this particularly elusive and challenging problem. A comprehensive bibliography of papers relating to the graph isomorphism problem is given.     

A versatile approach to analogues of the cannabinoid-like anti-emetic nonabine (BRL 4664)

Analogues of the cannabinoid-like anti-emetic nonabine (BRL 4664), in which the 4-pyridinyl and 1,2-dimethylheptyl groups have been replaced by other aryl and alky groups respectively, have been synthesised from the appropriate resorcinol by a novel procedure. The route has been shown to be of general applicability, although in the case of nonabine itself the procedure does not represent a viable alternative to that previously published.     

A serendipitous route to the benzidene nucleus. Structural elucidation by noeds.

The epimeric indenones (3) and (4) are readily synthesised and rapidly formed by treating bis(p-methoxyphenylcyclopropyl) ketone (2) with stannic chloride in nitromethane. The structure of (3) was determined by X-ray crystallography and that of (4) by NOEDS.     

A mild reduction of aliphatic nitro compounds to imines for further in situ reactions : A simple synthesis of pyrroles

Tributylphosphine-diphenyldisulphide reduces nitroalkanes to imines which can be trapped intramolecularly to give pyrroles.     

A non-factorial algorithm for canonical numbering of a graph

A graph certificate or canonical form is a short unique (up to isomorphism) representation of the graph. Thus two graphs are isomorphic iff their certificates are identical. In this paper an O(cⁿ) graph isomorphism algorithm which also yields a certificate of the graph is presented. The certificate produced by this algorithm is a canonical numbering of the vertices of the graph.     

On the existence of hydrodynamic fluctuation forces

The form of fluctuating hydrodynamic forces has been derived recently [R.B. Jones, Physica 105A (1981) 395] from the stochastic fluid equations of Landau and Lifshitz. We show by general physical arguments as well as by a direct calculation that such forces do not exist in the classical limit.     

The approximation of flows

Let U, V be two strongly continuous one-parameter groups of bounded operators on a Banach space $\text{X}$ with corresponding infinitesimal generators S, T. We prove the following: ∥U_t, ? V_t ∥ = O(t), t → 0, if and only if U = V; ∥U_t ? V_t∥ = O(t^α), t → 0; with 0 ? α ? 1, if and only if $\text{S = Ω(T + P)Ω}{}^{\mathrm{?1}}$, where Ω, P, are bounded operators on $\text{X}$ such that $\text{∥U}{}_{\mathrm{t}}\text{Ω ? ΩU}{}_{\mathrm{t}}\text{∥ = O(t}{}^{\mathrm{\alpha }}\text{), ∥U}{}_{\mathrm{t}}\text{P ? PU}{}_{\mathrm{t}}\text{∥ = ?O(t}{}^{\mathrm{\alpha }}\text{), t → 0; ∥U}{}_{\mathrm{t}}\text{? V}{}_{\mathrm{t}}\text{∥ = O(t)}$ if and only if $\text{S}{}^1\text{? T}{}^1$ has a bounded extension to $\text{X}$¹. Further results of this nature are inferred for semigroups, reflexive spaces, Hilbert spaces, and von Neumann algebras.     

Thermodynamic properties of a fluid of hard spheres with water-like dipole and quadrupole moments

The thermodynamic properties of a fluid of hard spheres with dipole and tetrahedral quadrupole moments are investigated using thermodynamic perturbation theory. The drpole-quadrupole coupling is found to be very important at values of the multipole moments that are close to those of water.