The Number of M-Sequences and f-Vectors

M -sequences (a.k.a. f-vectors for multicomplexes or O-sequences) in terms of the number of variables and a maximum degree. In particular, it is shown that the number of M-sequences for at most 2 variables are powers of two and for at most 3 variables are Bell numbers. We give an asymptotic estimate of the number of M-sequences when the number of variables is fixed. This leads to a new lower bound for the number of polytopes with few vertices. We also prove a similar recursive formula for the number of f-vectors for simplicial complexes. Keeping the maximum degree fixed we get the number of M-sequences and the number of f-vectors for simplicial complexes as polynomials in the number of variables and it is shown that these numbers are asymptotically equal. Received: February 28, 1996/Revised: February 26, 1998     

Synthesis and properties of sulfonated and crosslinked poly[(vinylidene fluoride)-graft-styrene] membranes

Crosslinked proton exchange membranes were prepared by electron beam grafting of styrene with divinylbenzene, DVB, or bis(vinyl phenyl)ethane, BVPE, on to poly(vinylidene fluoride), PVDF, followed by sulfonation. The area and thickness increase, crystallinity and mechanical properties of the membranes were investigated. All these properties were influenced by the crosslinker structure and its concentration. The crosslinkers increase the final degree of grafting. For noncrosslinked membranes it was about 80% and with 10% of BVPE crosslinker about 180% after 8 hr. The area increase with grafting is very similar for the noncrosslinked membranes and the differently crosslinked membranes, but the crosslinkers reduce swelling for sulfonated membranes swelled in boiling water. DVB reduces swelling more than BVPE. With grafting, the crystallinity of the PVDF phase decreases; with 10% BVPE the decrease is smallest and with 10% DVB largest. The mechanical properties of the sulfonated swelled membranes decrease with the degree of grafting, for a 10% DVB crosslinked membrane the strength decreases to zero at degrees of grafting higher than 75%. © 1998 John Wiley & Sons, Ltd.     

Voice Source Characteristics in Mongolian “Throat Singing” Studied with High-Speed Imaging Technique, Acoustic Spectra, and Inverse Filtering

Mongolian “throat singing” can be performed in different modes. In Mongolia, the bass-type is called Kargyraa. The voice source in bass-type throat singing was studied in one male singer. The subject alternated between modal voice and the throat singing mode. Vocal fold vibrations were observed with high-speed photography, using a computerized recording system. The spectral characteristics of the sound signal were analyzed. Kymographic image data were compared to the sound signal and flow inverse filtering data from the same singer were obtained on a separate occasion. It was found that the vocal folds vibrated at the same frequency throughout both modes of singing. During throat singing the ventricular folds vibrated with complete but short closures at half the frequency of the true vocal folds, covering every second vocal fold closure. Kymographic data confirmed the findings. The spectrum contained added subharmonics compared to modal voice. In the inverse filtered signal the amplitude of every second airflow pulse was considerably lowered. The ventricular folds appeared to modulate the sound by reducing the glottal flow of every other vocal fold vibratory cycle.     

Phase transitions for modified Erdős–Rényi processes

A fundamental and very well studied region of the Erdős–Rényi process is the phase transition at m∼n/2 edges in which a giant component suddenly appears. We examine the process beginning with an initial graph. We further examine the Bohman–Frieze process in which edges between isolated vertices are more likely. While the positions of the phase transitions vary, the three processes belong, roughly speaking, to the same universality class. In particular, the growth of the giant component in the barely supercritical region is linear in all cases.     

The protonation state of small carboxylic acids at the water surface from photoelectron spectroscopy

We report highly surface sensitive core-level photoelectron spectra of small carboxylic acids (formic, acetic and butyric acid) and their respective carboxylate conjugate base forms (formate, acetate and butyrate) in aqueous solution. The relative surface propensity of the carboxylic acids and carboxylates is obtained by monitoring their respective C1s signal intensities from a solution in which their bulk concentrations are equal. All the acids are found to be enriched at the surface relative to the corresponding carboxylates. By monitoring the PE signals of acetic acid and acetate as a function of total concentration, we find that the protonation of acetic acid is nearly complete in the interface layer. This is in agreement with literature surface tension data, from which it is inferred that the acids are enriched at the surface while (sodium) formate and acetate, but not butyrate, are depleted. For butyric acid, we conclude that the carboxylate form co-exists with the acid in the interface layer. The free energy cost of replacing an adsorbed butyric acid molecule with a butyrate ion at 1.0 M concentration is estimated to be >5 kJ mol(-1). By comparing concentration dependent surface excess data with the evolution of the corresponding photoemission signals it is furthermore possible to draw conclusions about how the distribution of molecules that contribute to the excess is altered with bulk concentration.     

Antimony Complexes for Electrocatalysis: Activity of a Main-Group Element in Proton Reduction

Main-group complexes are shown to be viable electrocatalysts for the H₂-evolution reaction (HER) from acid. A series of antimony porphyrins with varying axial ligands were synthesized for electrocatalysis applications. The proton-reduction catalytic properties of TPSb(OH)₂ (TP=5,10,15,20-tetra(p-tolyl)porphyrin) with two axial hydroxy ligands were studied in detail, demonstrating catalytic H₂ production. Experiments, in conjunction with quantum chemistry calculations, show that the catalytic cycle is driven via the redox activity of both the porphyrin ligand and the Sb center. This study brings insight into main group catalysis and the role of redox-active ligands during catalysis.     

The phase transition in inhomogeneous random graphs

The “classical” random graph models, in particular G(n,p), are “homogeneous,” in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, power‐law degree distributions. Thus there has been a lot of recent interest in defining and studying “inhomogeneous” random graph models. One of the most studied properties of these new models is their “robustness”, or, equivalently, the “phase transition” as an edge density parameter is varied. For G(n,p), p = c/n, the phase transition at c = 1 has been a central topic in the study of random graphs for well over 40 years. Many of the new inhomogeneous models are rather complicated; although there are exceptions, in most cases precise questions such as determining exactly the critical point of the phase transition are approachable only when there is independence between the edges. Fortunately, some models studied have this property already, and others can be approximated by models with independence. Here we introduce a very general model of an inhomogeneous random graph with (conditional) independence between the edges, which scales so that the number of edges is linear in the number of vertices. This scaling corresponds to the p = c/n scaling for G(n,p) used to study the phase transition; also, it seems to be a property of many large real‐world graphs. Our model includes as special cases many models previously studied. We show that, under one very weak assumption (that the expected number of edges is “what it should be”), many properties of the model can be determined, in particular the critical point of the phase transition, and the size of the giant component above the transition. We do this by relating our random graphs to branching processes, which are much easier to analyze. We also consider other properties of the model, showing, for example, that when there is a giant component, it is “stable”: for a typical random graph, no matter how we add or delete o(n) edges, the size of the giant component does not change by more than o(n). © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 31, 3–122, 2007