Neighborhood unions and the cycle cover number of a graph

For several years, the study of neighborhood unions of graphs has given rise to important structural consequences of graphs. In particular, neighborhood conditions that give rise to hamiltonian cycles have been considered in depth. In this paper we generalize these approaches to give a bound on the smallest number of cycles in G containing all the vertices of G. We show that if for all x, y ? V(G), |N(x) ∩ N(y)| ≧ 2n/5 + 1, then V(G) is coverable by at most two cycles. Several related results and extensions to t cycles are also given.     

Structure and reorientation kinetics of hydrogen passivated shallow impurities in silicon from vibrational spectroscopy

A survey is presented of what has been learned from vibrational spectroscopy and theory about the structure and reorientation kinetics of hydrogen-passivated shallow impurities in Si.     

Recent developments in the characterisation of microemulsions

Microemulsions are becoming increasingly complex systems by containing more sophisticated surfactants, polymers, biomolecules, inorganic nanoparticles, etc. The detailed understanding of such more complex systems requires increasingly more refined and comprehensive characterisation. This is typically done by the combination of complementary techniques and is aided by the fact that several experimental methods have been improved (such as electron microscopy) in recent times, new ones have become available (such as fluorescence correlation spectroscopy), and the theoretical understanding of structural data is advancing.     

Weak Attractive Ligand–Polymer and Related Interactions in Catalysis and Reactivity: Impact,Applications, and Modeling

The notion of weak attractive ligand–polymer interactions is introduced, and its potential application, importance, and conceptual links with “cooperative” ligand–substrate interactions are discussed. Synthetic models of weak attractive ligand–polymer interactions are described, in which intramolecular weak C? H???F? C interactions (the existence of which remains contentious) have been detected by NMR spectroscopy and neutron and X‐ray diffraction experiments. These C? H???F? C interactions carry important implications for the design of catalysts for olefin polymerization, because they provide support for the practical feasibility of ortho‐F???H_β ligand–polymer contacts proposed for living Group 4 fluorinated phenoxyimine catalysts. The notion of weak attractive noncovalent interactions between an “active” ligand and the growing polymer chain is a novel concept in polyolefin catalysis.     

Independence and graph homomorphisms graph homomorphisms

A graph with n vertices that contains no triangle and no 5-cycle and minimum degree exceeding n/4 contains an independent set with at least (3n)/7 vertices. This is best possible. The proof proceeds by producing a homomorphism to the 7-cycle and invoking the No Homomorphism Lemma. For k ≥ 4, a graph with n vertices, odd girth 2k+1, and minimum degree exceeding n/(k+1) contains an independent set with at least kn/(2k+1) vertices; however, we suspect this is not best possible. © 1993 John Wiley & Sons, Inc.     

Multiple crossover phenomena and scale hopping in two dimensions

We study the renormalization group for nearly marginal perturbations of a minimal conformal field theory M_p with p 1. To leading order in perturbation theory, we find a unique one-parameter family of “hopping trajectories” that is characterized by a staircase-like renormalization group flow of the C-function and the anomalous dimensions and that is related to a factorizable scattering theory recently solved by Al. B. Zamolodchikov. We argue that this system is described by interactions of the form . As a function of the relevant parameter t, it undergoes a phase transition with new critical exponents simultaneously governed by all fixed points M_p, M_p−1,…, M₃. Integrable lattice models represent different phases of the same integrable system that are distinguished by the sign of the irrelevant parameter .     

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.