A study of the synthesis and characterization of ethylene/dicyclopentadiene copolymers using a metallocene catalyst

Copolymers of ethylene/dicyclopentadiene were produced using a Me₂Si(Ind)₂ZrCl₂/methylaluminoxane catalyst system. The melting and crystallization points of the freshly prepared copolymers steadily decreased with increasing comonomer concentration. This was attributed to increased comonomer concentration in the polymer. When the comonomer incorporation, as measured by ¹³C NMR, is plotted against the comonomer concentration in the reactor, a plateau appears at concentrations higher than 0.12 mol/L. At concentrations greater than 0.12 mol/L time dependant crosslinking begins to be observed in the copolymers after exposure to air for several months. This crosslinking is also apparent in the thermosetting behavior of the copolymers when they are allowed sufficient time to crosslink. Copolymers with lower comonomer concentrations possess melting enthalpies even after several weeks, suggesting that there is a threshold concentration of 0.12 mol/L for the crosslinking process. Tensile tests of thermoplastic samples showed that incorporation of ca. 5 mol% of comonomer into the polyethylene main chain results in a semi-elastomeric material which possesses high strain recovery and whose strain hardening is similar to that observed for the homopolymer.     

On ac-number quantum τ-function

We first review the properties of the conventional -functions of the KP and Toda-lattice hierarchies. A straightforward generalization is then discussed. It corresponds to passing from differential to finite-difference equations; it does not involve, however, the concept of an operator-valued -function nor the one associated with non-Cartanian (levelk 1) algebras. The present study can be useful for better understandingq-free fields and their relation to ordinary free fields.E-mail: mironov@td.fian.free.net, mironov@grotte.teorfys.uu.se; morozov@vxitep.itep.msk.su, morozov@vxdesy.desy.de; vinet@ere.umontreal.ca, vinet@lps.umontreal.ca. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 1, pp. 119–131, July, 1994.     

Quantum spin chains with fractional revival

A systematic study of fractional revival at two sites in XX$XX$ quantum spin chains is presented. Analytic models with this phenomenon are obtained by combining two basic ways of realizing fractional revival in a spin chain. The first proceeds through isospectral deformations of spin chains with perfect state transfer. The second makes use of couplings provided by the recurrence coefficients of polynomials with a bi-lattice orthogonality grid. The latter method leads to analytic models previously identified that can exhibit perfect state transfer in addition to fractional revival.     

A Dirac–Dunkl Equation on S2 and the Bannai–Ito Algebra

The Dirac–Dunkl operator on the two-sphere associated to the \({{\mathbb{Z}_{2}^{3}}}\) reflection group is considered. Its symmetries are found and are shown to generate the Bannai–Ito algebra. Representations of the Bannai–Ito algebra are constructed using ladder operators. Eigenfunctions of the spherical Dirac–Dunkl operator are obtained using a Cauchy–Kovalevskaia extension theorem. These eigenfunctions, which correspond to Dunkl monogenics, are seen to support finite-dimensional irreducible representations of the Bannai–Ito algebra.     

Vitaminbestimmungen

Ohne Zusammenfassung     

Exact operator solution of the Calogero-Sutherland model

The wave functions of the Calogero-Sutherland model are known to be expressible in terms of Jack polynomials. A formula which allows to obtain the wave functions of the excited states by acting with a string of creation operators on the wave function of the ground state is presented and derived. The creation operators that enter in this formula of Rodrigues-type for the Jack polynomials involve Dunkl operators.     

Dynamical supersymmetry of the magnetic monopole and the 1/r 2-potential

We examine the recently discovered dynamical OSp(1, 1) supersymmetry of the Pauli Hamiltonian for a spin 1/2 particle with gyromagnetic ratio 2, in the presence of a Dirac magnetic monopole. Using this symmetry and algebraic methods only, we construct the spectrum and obtain the wave functions. At all but the lowest angular momenta, the states transform under a single irreducible representation of OSp(1, 1). On the lowest angular momentum states, it is impossible to define self-adjoint supercharges, and the states transform under an irreducible representation of SO(2, 1) only. The Hamiltonian is not self-adjoint in thes-wave sector, but admits a one parameter family of self-adjoint extensions. The full SO(2, 1) algebra can be realized only for two specific values of the parameter.The Pauli Hamiltonian is generalized to accommodate a ²/r ² potential. The new Hamiltonian exhibits a dynamical OSp(2, 1) supersymmetry. The spectrum and the wave functions are obtained. The states at all but the lowest angular momenta transform under the sum of two irreducible representations of OSp(2, 1). These two representations are distinguished by the chirality of their ground state. On the lowest angular momentum states, the OSp(2, 1) group is still realized, since the supercharges can all be rendered self-adjoint simultaneously, but the states only transform according to a single irreducible representation of OSp(2, 1). The chirality of the ground state for this representation is related to the signs of andeg. The Hamiltonian is not self-adjoint in thes-wave sector when ||<3/2. Only one of its self-adjoint extensions supports the OSp(2, 1) supersymmetry, and yields the wave functions obtained from the group theoretic approach. The supersymmetry is always spontaneously broken as there exists no normalizable zero energy states.The massless Dirac Hamiltonian in the presence of a magnetic monopole and a/r potential is related to a generator of an OSp(2, 1) superalgebra which also contains the Pauli Hamiltonian. This symmetry is used to generate the complete spectrum of the Dirac Hamiltonian.This work is supported in part through funds provided by the U.S. Department of Energy (DOE) under contract DE-AC02-76ER03069, and by the Natural Science and Engineering Research Council (NSERC) of Canada     

Simultaneous measurements of second-harmonic generation and two-photon photoelectric emission from Au

Received: 1 October 1998 / Accepted: 30 October 1998