Influence of catalyst depletion or deactivation on polymerization kinetics. III. Solution polymerization of acrylonitrile in N,N-dimethylformamide at −30°C.

The experimental results on homogeneous polymerization of acrylonitrile initiated with the sodium triethylthioisopropoxyaluminate, NaAlEt₃S(i-Pr), catalyst in DMF at ?30°C. are compared with the prediction of equations based on a postulated mechanism. The agreement between the calculated and observed number-average molecular weight combined with the kinetic data and the relationship between the conversion and the initial catalyst concentration provides a rigorous test concerning the validity of the equations and the mechanism of the polymerization. A plausible mechanism is postulated as follows: The initiation must be relatively fast in accordance with the rate equations and the growing polymer undergoes propagation, transfer (to monomer), and deactivation simultaneously. The infrared spectrum of the low molecular weight polymer prepared at a high catalyst concentration showed strong absorption at 2337, 2205, and 1620 cm.^?1 but no absorption at 900 cm.^?1, indicating that there are two nitriles in the polymer, one of which is conjugated. The possibility of having ? CH?CH₂ groups in the polymer is ruled out by the absence of the band at 900 cm.^?1. In view of these facts, it is concluded that the polymer has a ? CH?CHCN endgroup resulting from the transfer reaction.     

Instability criteria for steady flows of a perfect fluid

An instability criterion based on the positivity of a Lyapunov-type exponent is used to study the stability of the Euler equations governing the motion of an inviscid incompressible fluid. It is proved that any flow with exponential stretching of the fluid particles is unstable. In the case of an arbitrary axisymmetric steady integrable flow, a sufficient condition for instability is exhibited in terms of the curvature and the geodesic torsion of a stream line and the helicity of the flow.     

Generic eigenvalue multiplicities for two-parameter families of operators

A theorem about two-parameter families of Schrödinger operators in proved; the potential is parameter dependent.     

Volume Contents

Volume Contents

Volume Contents     

Techniques,computations, and conjectures for semi-topological <Emphasis Type="Italic">K</Emphasis>-theory

We establish the existence of an Atiyah-Hirzebruch-like spectral sequence relating the morphic cohomology groups of a smooth, quasi-projective complex variety to its semi-topological K-groups. This spectral sequence is compatible with (and, indeed, is built from) the motivic spectral sequence that relates the motivic cohomology and algebraic K-theory of varieties, and it is also compatible with the classical Atiyah-Hirzebruch spectral sequence in algebraic topology. In the second part of this paper, we use this spectral sequence in conjunction with another computational tool that we introduce — namely, a variation on the integral weight filtration of the Borel-Moore (singular) homology of complex varieties introduced by H. Gillet and C. Soulé – to compute the semi-topological K-theory of a large class of varieties. In particular, we prove that for curves, surfaces, toric varieties, projective rational three-folds, and related varieties, the semi-topological K-groups and topological K-groups are isomorphic in all degrees permitted by cohomological considerations. We also formulate integral conjectures relating semi-topological K-theory to topological K-theory analogous to more familiar conjectures (namely, the Quillen-Lichtenbaum and Beilinson-Lichtenbaum Conjectures) concerning mod-n algebraic K-theory and motivic cohomology. In particular, we prove a local vanishing result for morphic cohomology which enables us to formulate precisely a conjectural identification of morphic cohomology by A. Suslin. Our computations verify that these conjectures hold for the list of varieties above.Mathematics Subject Classification (2000): 19E20, 19E15, 14F43The first author was partially supported by the NSF and the NSAThe second author was supported by the Helen M. Galvin Fellowship of Northwestern UniversityThe third author was partially supported by the NSF and the NSA     

The unstable spectrum of the Navier–Stokes operator in the limit of vanishing viscosity

The Navier–Stokes equations for the motion of an incompressible fluid in three dimensions are considered. A partition of the evolution operator into high frequency and low frequency parts is derived. This decomposition is used to prove that the eigenvalues of the Navier–Stokes operator in the inviscid limit converge precisely to the eigenvalues of the Euler operator beyond the essential spectrum.