Multiplicities of interpoint distances in finite planar sets

What is the maximum number of unit distances between the vertices of a convex n-gon in the plane? We review known partial results for this and other open questions on multiple occurrences of the same interpoint distance in finite planar subsets. Some new results are proved for small n. Challenging conjectures, both old and new, are highlighted.     

On conservative approximation by linear polynomial operators an extension of the Bernstein's operator

In this work we study linear polynomial operators preserving some consecutive i-convexities and leaving invariant the polynomials up to a certain degree. First, we study the existence of an incom patibility between the conservation of certain i-convexities and the invariance of a space of polynomials. Interpolation properties are obtained and a theorem by Berens and DeVore about the Bernstein's operator is extended. Finally, from these results a generalized Bernstein's operator is obtained. This work was supported by Junta de Andalucia. Grupo de investigación: Matemática Aplicada. Código: 1107     

Community Health Care in Rochdale Family Health Services Authority

The National Health Service and the Social Services are engaged on a massive transfer of resources in which, wherever possible, care will be delivered to vulnerable patients in their own homes. This paper deals with two pilot medical practices in Rochdale. The needs of patients are considered on a subjective scale in five categories. It was confirmed that there was a consistency of subjective measurement between the nurses involved but there was no relation between these particular measures and the amount of nursing care being delivered. However, there was a strong relationship between age and the probability of being on the nursing care list, with a significant difference between the sexes. The model can lead to the identification of categories of patients who might have slipped through the net and also to estimates of the total home care nursing list in a practice.     

A Network Simulation of High-congestion Road-traffic Flows in Cities with Marine Container Terminals

Port cities where marine cargo terminals are located are generally near urban areas characterized by high-congestion road traffic. Changes in cargo traffic volumes into a marine terminal, or in the surrounding traffic arteries, which carry this traffic, can significantly affect the terminal's operations. Conversely, activity at the terminal can have an impact on the traffic levels and congestion for a considerable distance from the terminal. This paper demonstrates a methodology useful for studying the impact of road traffic flows on marine container terminals located in highly congested areas. This model was developed at the request of the Virginia Center for World Trade and was used to answer three planning questions in the port of Hampton Roads, Virginia, USA-what would be the impact of: opening a new section of interstate highway, a projected doubling of container traffic at one terminal, and a daily unit train in the vicinity of another terminal. The problem was made more challenging by limited data-collection funds. None the less, the model was deemed valid by a panel of traffic experts and officials from several major state and private agencies involved in marine traffic management. The model results were subsequently a factor in two major decisions related to terminal management.     

The Positivity of Intersection Multiplicities and Symbolic Powers of Prime Ideals

Serre's nonnegativity conjecture for intersection multiplicities has recently been proven by O. Gabber. In this paper we investigate Serre's positivity conjecture using the methods which he developed. We show in particular that the positivity conjecture has implications for properties of symbolic powers of prime ideals in regular local rings.     

The blow-up curve of solutions of mixed problems for semilinear wave equations with exponential nonlinearities in one space dimension, II

In this paper, we consider mixed problems with a timelike boundary derivative (or a Dirichlet) condition for semilinear wave equations with exponential nonlinearities in a quarter plane. The case when the boundary vector field is tangent to the characteristic which leaves the domain in the future is also considered. We show that solutions either are global or blow up on a C¹ curve which is spacelike except at the point where it meets the boundary; at that point, it is tangent to the characteristic which leaves the domain in the future.     

Steric Effects on Reaction Rates. Rate and equilibrium constants for oxidation of cyclanols

Equilibrium constants for oxidations of cyclanols by cyclohexanone in benzene have been determined in the presence of aluminium isopropoxide. The free energies of the equilibrium (ΔG_ox) are correlated with equilibrium constants for dissociation of cyanohydrins, rate constants for cyclanol oxidation with chromic acid, ketone reduction with sodium borohydride and trifluoroethanolysis of tosylates.     

Some reactions of tetrafluoroethylene oligomers

As part of a programme to prepare and evaluate a series of perfluoro- chemicals for use as inert fluids, the fluorinations of some tetrafluoroethylene oligomers over cobalt (III) fluoride have been studied.Fluorination of perfluoro-3,4-dimethylhex-3-ene (tetramer) and perfluoro-4-ethyl-3,4-dimethylhex-2-ene (pentamer) over CoF₃ at 230°C and l45°C respectively afforded the corresponding saturated fluorocarbons however, at 250°C, pentamer gave predominantly the saturated tetramer. The thermal behaviour of these saturated fluorocarbons alone and in the presence of bromine and toluene has been studied.Pyrolysis of pentamer over glass beads at 500°C gave perfluoro-1,2,3- trimethylcyclobutene and isomers of perfluoro-2,3-dimethylpenta-1,3- diene. Under similar conditions perfluoro-2-(1-ethyl-1-methylpropyl). 3-methylpent-1-ene (hexamer) gave perfluoro-1-methyl-2-(1-methyl- propyl)-cyclobut-1-ene and perfluoro-2-methyl-3-(1-methylpropyl)-buta- 1,3-diene.These reactions and the structural elucidation of the products will be discussed.     

Elektronische effekte in cyclopentadienyl-chrom-komplexen,die mehrere akzeptorliganden enthalten

The coordinative properties of the acceptor ligand CS are discussed on the basis of the new thiocarbonyl complexes CpCr(CO)(CS)(NO) (I), [CpCr(CS)(NO)₂]PF₆ (II), CpCr(CS)(L)(NO) (L = pyridine (III) or trimethylphosphane (IV) and related carbonyl complexes. The IR and NMR spectra (¹H, ¹³C, ³¹P) of the complexes indicate ligand—ligand interactions between CS and both the nitrosyl group and the cyclopentadienyl ring. An attempt is made to compare CS with other sulphur-containing acceptor ligands such as NS (in CpCr(CO)₂(NS)) and CS₂ (in CpCr(CS₂)(PMe₃)(NO)).