The steiner minimal network for convex configurations

SupposeX is a convex configuration with radius of maximum curvaturer and at most one of the edges joining neighboring points has length strictly greater thanr. We use the variational approach to show the Steiner treeS coincides with the minimal spanning tree and consists of all these edges with a longest edge removed. This generalizes Graham's problem for points on a circle, which we had solved. In addition we describe the minimal spanning tree for certain convex configurations.     

Adsorption of polymers at the solution-solid interface. VII. Poly(acrylic acids), model compounds,and vapors on nylon fibers

Nylon fibers of various draw ratios have been used as the adsorbents in the adsorption of low molecular weight acids and of polyacrylic acids from solution. Polymer adsorption is lower than on a precipitated nylon and decreases with fiber orientation; however, propionic acid and glutaric acids do not show this dependence on nylon structure. The structural alterations consequent on fiber drawing also reduce swelling in hydroxylated liquids. The greater accessibility of the precipitated nylon is shown by vapor sorption and extends to nonhydroxylated vapors; however, the differences in behavior between polyamide powder and fiber cannot be fully attributed to the former possessing a porous structure, in the sense normally employed in surface chemistry.     

NMR studies of silver ion and 1,3,5-trinitrobenzene complexes

A study, using NMR techniques, of the propensity of complex formation of silver ions with hydrocarbons and heterocyclic compounds containing nitrogen, oxygen and sulphur is described. The results are compared with similar data using 1,3,5-trinitrobenzene as an acceptor. The differences in the two sets of data are interpreted in terms of localized and delocalized interactions in the formation of weak complexes.     

Giant resonances in f-p shell nuclei studied by inelastic scattering of 80 MeV 3He ions

Spectra of ⁴⁸Ti, ⁵⁶Fe, ⁵⁹Co and ⁶⁰Ni have been investigated for excitation energies of 10–20 MeV by inelastic scattering of 80 MeV ³He particles. Broad peaks with excitation energies of about $\text{63±1 A}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>3</mtext>}}\text{MeV}$ and FWHM of 6 MeV have been observed. Much narrower satellite peaks occur at excitation energies about $\text{51±1 A}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>3</mtext>}}\text{MeV}$. The angular distributions of the sum of these peaks were compared to extended optical model DWBA calculations which confirm the predominant L = 2 character of the GR. A tentative assignment of L = 2 or 4 was made for the 13.5 MeV satellite peak in ⁵⁶Fe. We see no definite evidence for the presence of an L = 1 GR component.     

High resolution proton magnetic resonance spectra of heterocyclics 1,4-benzodithiin and 1,4-benzodioxin

The proton magnetic resonance spectra of 1,4-benzodithiin and 1,4-benzodioxin have been fully analyzed. The effects of the substituent heteroatoms on the chemical shifts of the various protons in the two molecules have been discussed.     

A reduced gradient method for quadratic programs with quadratic constraints and lp-constrained lp-approximation problems

A dual algorithm is developed for solving a general class of nonlinear programs that properly contains all convex quadratic programs with quadratic constraints and l_p-constrained l_p-approximation problems. The general dual program to be solved has essentially linear constraints but the objective function is nondifferentiable when certain variables equal zero. Modifications to the reduced gradient method for linearly constrained problems are presented that overcome the numerical difficulties associated with the nondifferentiable objective function. These modifications permit ‘blocks’ of variables to move to and away from zero on certain iterations even though the objective function is nondifferentiable at points having a block of variables equal to zero.     

On causal dynamics without metrisation: Part II

An interpretation of an ordering relation as an operator is constructed by appeal to the notion of homotopy. A metric-free definition of a causal ordering is given, after carefully investigating, by means of the notion of ordered sets, what is crucial to the notion of causality. A metric-free condition is derived that a causal ordering operator must satisfy, and a concrete realisation is found to exist that embodies the paraphernalia of Minkowski space with Zeeman's fine topology. Moreover, it is seen that a coordinate space used to give a cartography to causally related events must have at least a hyperbolic metric.This work was undertaken and completed whilst the author was at the Post Office Research Station, Dollis Hill, London, N.W.2.     

Far infrared spectrum of acrylonitrile

Two bands in the far infrared spectrum of acrylonitrile have been studied under moderately high resolution. Accurate band centers are obtained and the rotational constants are analyzed in terms of Coriolis coupling between the in-plane and out-of-plane bending modes.