On the number of solutions to systems of Pell equations

We prove that, for positive integers a, b, c and d with c≠d, a>1, b>1, the number of simultaneous solutions in positive integers to ax²−cz²=1, by²−dz²=1 is at most two. This result is the best possible one. We prove a similar result for the system of equations x²−ay²=1, z²−bx²=1.     

Singular open book structures from real mappings

We define open book structures with singular bindings. Starting with an extension of Milnor’s results on local fibrations for germs with nonisolated singularity, we find classes of genuine real analytic mappings which yield such open book structures.     

Non‐effective Quantifier Elimination

Genera connections between quantifier elimination and decidability for first order theories are studied and exemplified.     

Organocuprates as initiators for the polymerization of methyl methacrylate in THF

Several types of lithium organocuprates (I) e.g. R₂CuLi, RR'CuLi, R₃CuLi₂, R₂CuLi₂X (X- CN, SCN), were prepared and used as polymerization initiators for methyl methacrylate in THF at temperatures ranging from −30 to +20 ^*C. Polymerization initiated by a mixed homocuprate, e.g. lithium n-butyl (thien-2-yl)cuprate was studied in more detail. Kinetic measurements indicated an overall large (−4 to −14 kcal) apparent negative activation energy of polymerization in the −30 + +20 ^*C temperature range. The reaction was shown to have a kinetic order of about 1.5 with respect to initiator concentration. Conductance measurements indicate that the equivalent conductance increases with initiator concentration in the 10⁻² + 10⁻¹ M range indicating the formation of multiple ions formed by intermolecular association. The data are consistent with a polymerization involving aggregated ion pairs of as yet undetermined composition. The negative activation energy is tentatively interpreted as due to the exothermic formation of aggregates existing as highly solvated ion pairs.     

Scaling coupling of reflecting Brownian motions and the hot spots problem

We introduce a new type of coupling of reflecting Brownian motions in smooth planar domains, called scaling coupling. We apply this to obtain monotonicity properties of antisymmetric second Neumann eigenfunctions of convex planar domains with one line of symmetry. In particular, this gives the proof of the hot spots conjecture for some known types of domains and some new ones.