Flat envelopes in commutative rings

We prove that ifR is a commutative ring such that each localization at a prime ideal has finite weak global dimension then everyR-module has a flat envelope if and only ifR is coherent and has weak global dimension less than or equal to two. The authors have been partially supported by the C.A.I.C.Y.T.     

Plasma polymerization of trifluoromethyl substituted aromatic compounds

Thin film coatings were prepared by polymerizing trifluoromethyl-substituted aromatic compounds in a glow discharge with low power levels and medium pressures. Smooth, continuous films which are adherent and insoluble in conventional solvents were produced. A significant amount of the fluorine content from the monomer was retained in the polymer. The trifluoromethyl group was not removed by the plasma conditions and appeared in the film as trifluoromethyl and difluoromethylene groups. The films were slightly less wettable than those of the unfluorinated films.     

Teachers' funds of knowledge and the teaching and learning of mathematics in multi-ethnic primary schools: Two teachers' views of linking home and school

In this paper we explore two teachers' views on the role(s) of parents, the local community and children's home lives in the learning of mathematics in primary school. We use Moll and Greenberg's concept of ‘funds of knowledge’ and apply it to the case studies of two teachers working in the UK context. Issues of teachers' professional experience, ethnicity, class and gender emerge as significant in examining similarities and differences in the teachers' beliefs, understandings and practices in the area of linking home and school. We end with a discussion of some implications for teacher education and professional development.     

N2 rotational energy distributions in cold,supersonic beams from electron excited fluorescence measurements of N+2

Ground-state rotational energy distributions of N₂ molecules produced in pure and He-seeded supersonic expansions have been determined by measurements of the N⁺₂ first negative band rotational line intensities produced by 800 eV electron impact on cooled pure and He-seeded N₂ supersonic beams. Sufficient spectral resolution was employed to resolve completely both P and R branches of the first negative bands. Rotational state distributions were obtained to much higher values of J than in previous investigations. The data show that at 800 eV, the electric dipole selection rule, |ΔJ| = 1, is consistent with the observed N⁺₂ emission bands and that the rotational energy distributions produced in the cooled, supersonic beam are non-Boltzmann with a large population in the first few rotational states followed by a long, high-energy fail to quite high J values.     

Dissipating the Langevin equation in the presence of an external stochastic potential

In the Langevin formalism, the delicate balance maintained between the fluctuations in the system and their corresponding dissipation may be upset by the presence of a secondary, space-dependent stochastic force, particularly in the low-friction regime. In prior work, the latter was dissipated self-consistently through an additional uniform (mean-field) friction [T. Shepherd and R. Hernandez, J. Chem. Phys. 115, 2430 (2001).] An alternative approach to ensure that equipartition is satisfied relies on the use of a space-dependent friction while ignoring nonlocal correlations. The approach is evaluated with respect to its ability to maintain constant temperature for two simple one-dimensional, stochastic potentials of mean force wherein the friction can be evaluated explicitly when there is no memory in the barriers. The use of a space-dependent friction is capable of providing qualitatively similar results to those obtained previously, but in extreme cases deviations from equipartition may be observed due to the neglect of the memory effects present in the stochastic potentials.     

Value Distribution of Interpolating Blaschke Products

A Blaschke product B with zero-sequence (a_n) is called almostinterpolating if the inequality lim inf_n(1 – |a_n|²)|B'(a_n)|