The Moore Bound for Irregular Graphs

 What is the largest number of edges in a graph of order n and girth g? For d-regular graphs, essentially the best known answer is provided by the Moore bound. This result is extended here to cover irregular graphs as well, yielding an affirmative answer to an old open problem ([4] p. 163, problem 10). Received: June 27, 2000 Final version received: July 3, 2001     

X-Ray Mapping and Interpretation of Scatter Diagrams

Electron beam induced quantitative X-ray mapping has become a very useful characterisation tool for determining the elemental distribution in materials, whether using energy dispersive spectroscopy or wavelength dispersive spectroscopy. The X-ray intensity distributions of the elements from an X-ray map allow us to generate two dimensional and ternary scatter diagrams thus converting spatial information into concentration dimensions, which is an important tool for displaying the spatial relationships of elements or correlated elements (phases) in materials. To best understand how to use this tool, we need to understand the production and features of the scatter diagram. The type of clustering observed in the scatter diagram, whether oval, linear or spherical, can give the major and trace element distributions within phases as well as qualitative and quantitative phase information. This paper demonstrates the generation of scatter diagrams, properties of scatter diagrams, interpretation of scatter diagrams and the advantages of scatter diagrams through the use of examples.     

Nuclear equation of state and compression modes

The nuclear matter (N = Z and no Coulomb interaction) incompressibility coefficient, K _nm , which is directly related to the curvature of the nuclear matter equation of state, is a very important physical quantity in the study of properties of nuclei, supernova collapse, neutron stars and heavy-ion collisions. We review the current status of K _nm and the experimental and theoretical methods used to determine the value of K _nm from the excitation crosssections σ(E) and the transition strength distributions S(E) of compression modes in nuclei. In particular, we will consider the isoscalar giant monopole resonance (ISGMR) and the isoscalar giant dipole resonance (ISGDR) and provide a simple explanation to the long standing problem of the conflicting results obtained for K _nm , deduced from experimental data on excitation cross sections for the ISGMR and data for the ISGDR.     

A new class of long-tailed pausing time densities for the CTRW

We present some asymptotic results for the family of pausing time densities having the asymptotic (t) property(t) [t ln¹⁺(t/T)]^–1. In particular, we show that for this class of pausing time densities the mean-squared displacement r ²(t) is asymptotically proportional to ln(t/T), and the asymptotic distribution of the displacement has a negative exponential form.     

Properties of noninteger moments in a first passage time problem

We calculate the moments t ^q , whereq is not necessarily an integer, of the first passage time to trapping for a simple diffusion problem in one dimension. If a characteristic length of the system isL and t _q ~L (q) asL, then we show that there is a phase transition atq=q _c such that whenq<q _c ,(g)=0, and forq>q _c , (q) is a linear function ofq. These analytical results can be used to explain results for large moments for diffusion on a hierarchic structure. We also show how to calculate noninteger moments in terms of characteristic functions.     

Synthesis and characteristics of microspheres of polystyrene derivatives

Two sytrene derivatives formylstyrene and styrene sulfonylcholoride, were synthesized. Polymeric microspheres in diameters ranging from 0.1 to 2 μm were synthesized by polymerization of chlormoethylstyrene, formylstyrene, and styrene sulfonylcholoride in organic solvents, in the presence of appropriate surfactants. The kinetics of microsphere formation were studied. The molecular weight distribution of the products was determined by gel permeation chromatography. Conditions for binding various amino ligands to the microspheres were also established.     

Shear-induced coalescence of emulsified oil drops

Crude oil droplets, when suspended in water, possess negative surface charges which give rise to double-layer repulsive forces between the drops. According to conventional DLVO theory, the magnitude of this repulsion (based on the measured zeta potential) is more than sufficient to prevent coalescence of the droplets. Indeed, when two such droplets were brought together on direct (i.e., "head-on") approach, coalescence was rarely observed. Upon oblique approach, however, the same droplets were seen to coalesce readily. An oblique encounter must necessarily give rise to lateral relative motion-or shearing-between the droplet surfaces. It is speculated that, if the charge distributions at the droplet surfaces were heterogeneous, lateral shearing would facilitate many encounters between surface patches of different zeta potentials across the intervening water film. If the repulsion across any local region were sufficiently weak to allow formation of an oil bridge across the water film, coalescence of the drops would follow inevitably. With the hypothesis of surface heterogeneity, it is not necessary to invoke any additional colloidal interactions (such as "hydrophobic forces") to account for the observed droplet-droplet coalescence. This finding may have important implications for the underlying mechanisms of emulsion stability in general and the commercial extraction of bitumen from oil sands in particular.     

Sigma-aromaticity and sigma-antiaromaticity in saturated inorganic rings

The magnetic properties of a series of inorganic saturated rings, (SiH2)n, (GeH2)n, (NH)n, (PH)n, (AsH)n, On, Sn, and Sen (n = 3-6), exhibit zigzag behavior with ring size resembling that of aromatic and antiaromatic Hückel pi-systems and (CH2)n rings. Computed GIAO-SCF nucleus-independent chemical shifts (NICS) and localized (LMO) NICS analysis indicate that the sigma-ring electrons are chiefly responsible for this zigzag behavior. This evidence for sigma-aromaticity is further supported by theoretical strain energy (TSE). The Hückel 4n + 2/4n aromaticity/antiaromaticity rule for pi-electron systems applies well to the smaller saturated rings.     

Bond dissociation energies and radical stabilization energies associated with model peptide-backbone radicals

Bond dissociation energies (BDEs) and radical stabilization energies (RSEs) have been calculated for a series of models that represent a glycine-containing peptide-backbone. High-level methods that have been used include W1, CBS-QB3, U-CBS-QB3, and G3X(MP2)-RAD. Simpler methods used include MP2, B3-LYP, BMK, and MPWB1K in association with the 6-311+G(3df,2p) basis set. We find that the high-level methods produce BDEs and RSEs that are in good agreement with one another. Of the simpler methods, RBMK and RMPWB1K achieve good accuracy for BDEs and RSEs for all the species that were examined. For monosubstituted carbon-centered radicals, we find that the stabilizing effect (as measured by RSEs) of carbonyl substituents (CX=O) ranges from 24.7 to 36.9 kJ mol(-1), with the largest stabilization occurring for the CH=O group. Amino groups (NHY) also stabilize a monosubstituted alpha-carbon radical, with the calculated RSEs ranging from 44.5 to 49.5 kJ mol(-1), the largest stabilization occurring for the NH2 group. In combination, NHY and CX=O substituents on a disubstituted carbon-centered radical produce a large stabilizing effect ranging from 82.0 to 125.8 kJ mol(-1). This translates to a captodative (synergistic) stabilization of 12.8 to 39.4 kJ mol(-1). For monosubstituted nitrogen-centered radicals, we find that the stabilizing effect of methyl and related (CH2Z) substituents ranges from 25.9 to 31.7 kJ mol(-1), the largest stabilization occurring for the CH3 group. Carbonyl substituents (CX=O) destabilize a nitrogen-centered radical relative to the corresponding closed-shell molecule, with the calculated RSEs ranging from -30.8 to -22.3 kJ mol(-1), the largest destabilization occurring for the CH=O group. In combination, CH2Z and CX=O substituents at a nitrogen radical center produce a destabilizing effect ranging from -19.0 to -0.2 kJ mol(-1). This translates to an additional destabilization associated with disubstitution of -18.6 to -7.8 kJ mol(-1).     

Doubly charged ion mass spectra: VI—Amines

Doubly charged ion mass spectra of 22 amines (2–10 carbon atoms) were determined using an Hitachi RMU-7L double focusing mass spectrometer. Molecular ions were not observed in the spectra of aliphatic amines. The most intense product ion peaks in the spectra of lower molecular weight amines resulted from hydrogen elimination from the molecular ion; however, as amine molecular weight increased the largest peaks resulted from both hydrogen and heavy atom elimination from the molecular ion. Dominant ions in the doubly charged ion spectra of lower molecular weight aliphatic amines were from reactions of [C_nH₃N]²⁺ (n:=2, 3, 4) type ions. The spectra of higher molecular weight aliphatic amines spanned a wide mass range. Appearance energies for some of the more prominent ions were measured in the range from 25 to 49 eV. A geometry optimized quantum mechanical self-consistent field molecular orbital treatment was used to compute the energies and structural parameters of prominent ions in the doubly charged ion mass spectra.