A canonical ordering of polybenzenes and polymantanes using a prime number factorization technique

The correlation described over twenty years ago by Matula between the prime factorization of integers and the class of alkanes is re-examined with a view to explaining the probable reason why there have, to date, been no major extensions of this idea. By considering the class of alkanes as a one-dimensional one-parameter system, a new perspective on the method is gained that is amenable to extension, but in a different direction than originally anticipated. With this new perspective, the classes of polybenzenes and polymantanes are seen to be the representatives of two- and three-dimensional one-parameter systems, respectively. A nomenclature, comparable to one that Matula used for alkanes, is created that gives a unique canonical name to all possible combinations of either polybenzene or polymantane modules. Such a nomenclature contains a built-in means of positioning the molecule in the field of interest in accordance with arbitrary pre-selected criteria, such as Patterson's rules, and also coding that indicates symmetries inherent in the structure of this molecule.     

History of the Development and Growth of Thermosetting Polymers

Fibrous proteins are cross-linked through the disulfide linkages of cystine; animal and vegetable proteins can be cross-linked by the reaction of tannic acid, chromic acid, or formaldehyde; and natural rubber can be vulcanized by heating with sulfur. However, most thermosetting polymers are synthetic polymers. These include saturated polyesters (Glyptals) synthesized by Smith in 1901, phenolic plastics produced by Baekeland in 1907, urea and melamine plastics produced by John and Henkel in 1918 and 1935, alkyds patented by Kienle in 1933, glycol maleates patented by Ellis in 1937, and epoxy resins patented by Schlack in 1933. Many of these developments were made prior to the advent of commercial synthetic thermoplastics. However, because their fabrication is more labor intensive than that of thermoplastics, thermosetting plastics now account for less than 20% of all plastics produced. Nevertheless, over 2.5 million tons of these thermosets are produced annually and the use of these cross-linked plastics continues to grow.     

Nonlinear two-point boundary value problems with multiple solutions

In the first part of this paper we study the convergence of finite difference methods to approximate the maximal solution of problems of the form:u+f (x, u)=0, with boundary conditions eitheru(0)=u(b)=0 oru(0)=u (b)=0, 0<b1. The functionf(x, u) satisfies several conditions that are explicitly given in § 1. This work extends earlier results of Parter (see references at the end).Since this problem has in general more than one solution we develop in the second part two algorithms to approximate solutions characterized by the number of their zeros in (0, 1). We include in the last Section numerical results and some additional comments on the implementation of the algorithms on a digital computer.The research of this author was supported by the Office of Naval Research under contract N^o N-0014-67-A-0128-004.Part of this research was carried on while this author was visiting the Mathematics Research Center, University of Wisconsin, during the summer of 1969.     

Nowhere-zero 6-flows

We prove that every graph with no isthmus has a nowhere-zero 6-flow, that is, a circulation in which the value of the flow through each edge is one of ±1, ±2,…, ±5. This improves Jaeger's 8-flow theorem, and approaches Tutte's 5-flow conjecture.     

A note on convergence of the multigrid V-cycle

Several recent papers have discussed the convergence of the multigrid V-cycle. In particular there are several results for the symmetric case: where the numbers of smoothings before the fine-to-coarse transfer and after the coarse-to-fine transfer are the same. In most instances, the smoother H=I?E^-1A has been limited to the case where E is positive definite and the eigenvalues h of H satisfy 0?h?1. In this note we extend these results to asymmetric V-cycles and the case where -b?h?1 with 0<b<1.     

Cycles in dense digraphs

Let G be a digraph (without parallel edges) such that every directed cycle has length at least four; let β(G) denote the size of the smallest subset X ? E(G) such that G?X has no directed cycles, and let γ(G) be the number of unordered pairs {u, v} of vertices such that u, v are nonadjacent in G. It is easy to see that if γ(G) = 0 then β(G) = 0; what can we say about β(G) if γ(G) is bounded?

We prove that in general β(G) ≤ γ(G). We conjecture that in fact β(G) ≤ ½γ(G) (this would be best possible if true), and prove this conjecture in two special cases:

• when V(G) is the union of two cliques
• when the vertices of G can be arranged in a circle such that if distinct u, v, w are in clockwise order and uw is a (directed) edge, then so are both uv, vw.

     

Electron super-rich radicals in the gas phase. A neutralization-reionization mass spectrometric and ab initio/RRKM study of diaminohydroxymethyl and triaminomethyl radicals

Diaminohydroxymethyl (1) and triaminomethyl (2) radicals were generated by femtosecond collisional electron transfer to their corresponding cations (1+ and 2+, respectively) and characterized by neutralization-reionization mass spectrometry and ab initio/RRKM calculations at correlated levels of theory up to CCSD(T)/aug-cc-pVTZ. Ion 1+ was generated by gas-phase protonation of urea which was predicted to occur preferentially at the carbonyl oxygen with the 298 K proton affinity that was calculated as PA = 875 kJ mol-1. Upon formation, radical 1 gains vibrational excitation through Franck-Condon effects and rapidly dissociates by loss of a hydrogen atom, so that no survivor ions are observed after reionization. Two conformers of 1, syn-1 and anti-1, were found computationally as local energy minima that interconverted rapidly by inversion at one of the amine groups with a <7 kJ mol-1 barrier. The lowest energy dissociation of radical 1 was loss of the hydroxyl hydrogen atom from anti-1 with ETS = 65 kJ mol-1. The other dissociation pathways of 1 were a hydroxyl hydrogen migration to an amine group followed by dissociation to H2N-C=O* and NH3. Ion 2+ was generated by protonation of gas-phase guanidine with a PA = 985 kJ mol-1. Electron transfer to 2+ was accompanied by large Franck-Condon effects that caused complete dissociation of radical 2 by loss of an H atom on the experimental time scale of 4 mus. Radicals 1 and 2 were calculated to have extremely low ionization energies, 4.75 and 4.29 eV, respectively, which belong to the lowest among organic molecules and bracket the ionization energy of atomic potassium (4.34 eV). The stabilities of amino group containing methyl radicals, *CH2NH2, *CH(NH2)2, and 2, were calculated from isodesmic hydrogen atom exchange with methane. The pi-donating NH2 groups were found to increase the stability of the substituted methyl radicals, but the stabilities did not correlate with the radical ionization energies.     

The calculation of resonance separatrices for the near-integrable case of the standard mapping

The standard mapping arises in many physical applications, including the analysis of nonlinear resonant acoustic oscillations in a closed tube. A perturbation expansion, in powers of the amplitude parameter, is given for the calculation of the fixed points of various orders and the associated separatrices. It is shown how exact homoclinic orbits can be calculated numerically. Explicit analytic expressions are given for the separatrices associated with the first four resonances when the perturbation parameter is small.