On sums of powers of integers

K. Thanigasalam has shown that for any positive integer k the sequence of positive integers represented by $\text{x}{}_2{}^2\text{+ x}{}_3{}^3\text{+ x}{}_5{}^5\text{+ x}{}_1{}^{\mathrm{k}}$ has positive density. Here we prove that the asymptotic density of this sequence is 1, a similar result is proved for the sequence represented by $\text{x}{}_2{}^2\text{+ x}{}_3{}^3\text{+ x}{}_5{}^6\text{+ x}{}_1{}^{\mathrm{k}}$.     

Relaxations in the transition region of crosslinked polyesters. I. The β relaxation

The glass transition region of styrene-crosslinked poly(1,2-propylene phthalate fumarate) was studied using a torsion pendulum. The region was composed of two separate relaxations. The major relaxation was the glass transition of the whole network, while the minor relaxation, at a lower temperature, was ascribed to the polyester segments between the crosslinks. A comparison was made between these relaxations and these observed in related polyester networks.     

Nuclear and atomic models

A theorem concerning fermion interaction is postulated and applied to the problems of atomic (electronic) and nuclear physics. Model building basedsolely upon the postulate thatadjacent like fermions must be singlet paired accounts for the closed shells ofboth nuclear and atomic structure. The implied antiferromagnetic FCC lattice of protons and neutrons in alternating layers has been found previously to be the lowest-energy solid configuration of nuclear matter (N=P) (Canuto and Chitre, 1974). The buildup of the FCC lattice from a central tetrahedron reproduces all of the shells and subshells of the isotropic harmonic oscillator, which of course is the basis for the shell model. In atomic structure, the singlet pairing of adjacent electrons implies closed-shell structures uniquely at the six noble gases and the three noble metals, Ni, Pd, and Pt. The basis for the postulate concerning fermions is found in terms of classical electrodynamics; it is a microscopic corollary of Biot-Savart's law that parallel currents attract whereas antiparallel currents repel.     

Polymeric nucleophiles and cosolvents. The reactivity of poly(ethylene glycol) and its derivatives with benzyl bromide under solvolytic conditions

Pseudo first-order rate constants for the solvolytic displacement of benzyl bromide in 100% ethanol and in 80 and 60% aqueous ethanol (v/v) are reported. The effect of adding poly(ethylene glycol) (PEG) and PEG dithiols and diamines to the solvolytic solutions has been studied. The solvolytic rates are mildly retarded on adding PEG, MW 950, or PEG dimercaptoacetate, MW 548 or 748 (at a concentration of about 8% of the polymer) to reactions in absolute ethanol. In 80% ethanol, however, a rate acceleration by the polymers was observed, with the thiols exhibiting the largest effects. PEG di(3-mercaptopropionate) reacts slightly faster than the PEG dimercaptoacetates, but liberated acid affects the former's reaction. Three molecular weights of primary diamines structurally derived from propylene oxide-capped PEG were evaluated and found to increase solvolytic rates of benzyl bromide significantly. Kinetic evaluation and comparisons with reactions containing equivalent amine groups per gram in an appropriate model (ethanolamine) revealed that the polymer's amine groups are actively involved nucleophilically. Grunwald-Winstein plots and second-order rate constants revealed that the polymeric diamines caused the benzyl bromide to undergo polymer-assisted displacement in all solvent compositions. Since it was noted that the second-order rates increased as the PEG amine molecular weight increased, it appears that the PEG framework may be assisting the reaction.     

Probability distributions in fuzzy logic II

Probability distributions associated with several ‘iff’ ply operators are discussed. These exact distributions are compared with relevant normal approximants. Two possible applications are noted.