Polyfluorocycloalkenes. Part XV [1]. Reactions of decafluorocyclohexene with secondary amines

Pyrrolidine, 3,3,4,4-tetrafluoropyrrolidine, and dimethylamine, react with decafluorocyclohexene to give the non-basic enamines, 1-pyrrolidino-, 1-(3′,3′,4′,4′-tetrafluoropyrrolidino)-, and 1-dimethylamino-nonafluorocyclohex- 1-ene, respectively.     

Experimental study of pionic Ca and Ti and examination of the pion-nuclear potential for low-Z nuclei

We report the measurement of the strong interaction shifts and widths of the 3d → 2p transitions in pionic ^{40,42,43,44,48}Ca and ^46,48,50Ti. Using these new data in combination with earlier low-Z pionic data (6 ≦ Z ≦ 16), we have studied, in a purely phenomenological way, the pion-nuclear optical potential. Employing nuclear-structure information from Hartree-Fock calculations and measured charge densities, we have fitted the pion-nuclear potential parameters to the pionic atom data. The effect of adding an isovector dependence to the s- and p-wave two-nucleon terms has been investigated. We have also explored the sensitivity of these data to the value of the Lorentz-Lorenz parameter ξ. A small but definite sensitivity to the latter parameter has been observed with the best fit value of ξ being significantly greater than 1. We have used the phenomenological potentials to determine the neutron radii of the Ca and Ti isotopes and, in fact, find reasonable agreement with Hartree-Fock predictions and with the results of other hadronic probes.     

Characterization of the Darboux point for particular classes of problems

A minimal sufficient condition for global optimality involving the Darboux point, analogous to the minimal sufficient condition of local optimality involving the conjugate point, is presented. The Darboux point is then characterized for optimal control problems with linear dynamics, cost functionals with a general terminal state term and an integrand quadratic in the state and control, and general terminal conditions. The Darboux point is shown to be the supremum of a sequence of conjugate points. If the terminal state term is quadratic, along with a scalar quadratic boundary condition, then the Darboux point is also the time at which the Riccati matrix becomes unbounded, giving a characterization of the unboundedness of the Riccati matrix at points which are not in general conjugate points.This research was supported by the National Science Foundation under Grant No. GK-30115.This is Definition 2.1 of Ref. 1.     

Existence of ground states and KMS states for approximately inner dynamics

A strongly continuous one parameter group of *-automorphisms of aC*-algebra with unit is said to be approximately inner if it can be approximated strongly by inner one parameter groups of *-automorphisms. It is shown that an approximately inner one parameter group of *-automorphisms has a ground state and, if there exists a trace state, a KMS state for all inverse temperatures. It follows that quantum lattice systems have ground states and KMS states. Conditions that a strongly continuous one parameter group of *-automorphisms of a UHF algebra be approximately inner are given in terms of the unbounded derivation which generates the automorphism group.     

A precision determination of the radial charge parameters and the quadrupole moment of 181Ta using muonic X-rays

We report the measurement of the energies of the 4f → 3d, 3d → 2p and 2p → 1s atomic transitions in muonic ¹⁸¹Ta. Using transitions to and from the 1s and 2p levels, as well as the hyperfine splittings of the 2p and 3d states, we obtained the radial charge parameters assuming deformed Fermi distributions. We found the intrinsic static quadrupole moment Q₀ = 6.82 ± 0.06 e · b, in excellent agreement with Coulomb excitation results. We verified that the dynamic (corresponding to transitions to the first excited state) and the static (ground state) E2 moments were equal to within 1.1 %. The intrinsic hexadecapole moment was determined to be Π₀ = ?0.12 ± 0.40 e · b².     

Heisenberg model and a random walk on the permutation group

It is shown that the isotropic Heisenberg model can be analysed in terms of a random walk on the permutation group. This approach makes it intuitively clear why the Heisenberg model exhibits long range order or ferrogmagnetic behavior in three dimensions and not in two and one dimensions. This approach to the Heisenberg model lends itself to computer analysis.Work supported in part by the National Science Foundation     

The Darboux point

A theory of global optimality based upon the Darboux-point concept is developed. A definition is proposed for the Darboux point, and the Darboux point is shown to exist on nonglobally optimal trajectories under relatively general conditions. A mutually exclusive classification of Darboux points is noted, and several properties are proved for one of these classes (the Type-1 Darboux point). Numerous examples are included to illustrate the Darboux-point definition and properties.     

Resistance inequalities for KMS states of the isotropic Heisenberg model

Inequalities which show that the spin correlations between spins at two lattice sites is bounded by a constant times the inverse square root of the electrical resistance between the lattice sites is proved for KMS states of the isotropic Heisenberg model. The resistance is calculated using the inverse of the coefficients occuring in the Heisenberg Hamiltionian as the resistances between neighboring lattice sites.Work supported in part by a National Science Foundation GrantOn leave from the University of Pennsylvania, Philadelphia, Pa. 19174, USA