Radical Couplings as Key Steps for the Preparation of Derivatives of Nonactic Acid

Summary. Free radical couplings from furan, as cheap starting material, were studied in view of developing a rapid strategy en route to the synthesis of derivatives of nonactin. The chain containing the alcohol function was introduced in one or two steps in 86% yield. For the introduction of the second chain with the ester function two different coupling methods were tested. Starting from the advanced intermediates obtained nonactin derivatives can be prepared by catalytic hydrogenation of the furan ring.     

Symplectic Yang–Mills theory,Ricci tensor,and connections

A Yang–Mills theory in a purely symplectic framework is developed. The corresponding Euler–Lagrange equations are derived and first integrals are given. We relate the results to the work of Bourgeois and Cahen on preferred symplectic connections.     

Hospital Waiting List Management Models

This paper describes work carried out with hospital consultants in Lancaster District Health Authority to develop and use computerized waiting list management models. The ‘what if?’ models produced have been found to be a very useful tool in the constructive debate about problems faced by consultants and managers, and of the options available to them.     

Engel Values in Residually Finite Groups

The following theorem is proved. Let n be a positive integer and q a power of a prime p. There exists a number m = m(n, q) depending only on n and q such that if G is any residually finite group satisfying the identity ([x _1,n y ₁] ⋯ [x _m,n y _m ])^q ≡ 1, then the verbal subgroup of G corresponding to the nth Engel word is locally finite.     

Construct non-graded bi-Frobenius algebras via quivers

Using the quiver technique we construct a class of non-graded bi-Frobenius algebras. We also classify a class of graded bi-Frobenius algebras via certain equations of structure coefficients.     

Refraction of light in media

The roles of the magnetic field and electric field of the light are investigated when the light is refracted in the medium. The model of the electron cloud conductor is presented. Electron cloud in a molecule is treated as a conductor and the Faraday’s Law is applied to this conductor that is in the alternating magnetic field of the light. dB _M/dt of the light gives rise to an alternating induced current on the electron cloud conductor, and the light exchanges energy, i.e. the refractive energy, with the electron cloud conductor. Formulas of refractive index, which is the ratio of light speed in vacuum to that in the medium, are derived with this model. These formulas are tested with several mediums and Langevin’s diamagnetic susceptibility of helium gas, and the results are in good agreement with the measured data. The anisotropy and the nonlinearity of the refractive index are explained with the theory described in this work. Supported by Beijing Science and Technology New Star Program (Grant No. 952870400), the Beijing Municipal Commission of Education and the Excellent Young Teachers Program of Ministry of Education of China     

Systems of generalized quasivariational inclusions problems with applications to variational analysis and optimization problems

In this paper, we study an existence theorem of systems of generalized quasivariational inclusions problem. By this result, we establish the existence theorems of solutions of systems of generalized equations, systems of generalized vector quasiequilibrium problem, collective variational fixed point, systems of generalized quasiloose saddle point, systems of minimax theorem, mathematical program with systems of variational inclusions constraints, mathematical program with systems of equilibrium constraints and systems of bilevel problem and semi-infinite problem with systems of equilibrium problem constraints. This research was supported by the National Science Council of the Republic of China.