Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application

Summary The present paper, together with the previous one (Part 1: Theory, published in this journal) is intended to give an explicit method for computing all Lyapunov Characteristic Exponents of a dynamical system. After the general theory on such exponents developed in the first part, in the present paper the computational method is described (Chapter A) and some numerical examples for mappings on manifolds and for Hamiltonian systems are given (Chapter B).

---

Sommario Questo articolo, insieme con il precedente (Parte 1: Teoria, pubblicato in questa stessa rivista) è inteso a fornire un metodo esplicito per il calcolo di tutti gli esponenti caratteristici di Lvapunov per un sistema dinamico. Dopo la teoria generale su tali esponenti sviluppata nella prima parte, qui si illustra il metodo di calcolo (Capitolo A) e si danno esempi numerici per applicazioni di varietà in sè e per sistemi Hamiltoniani (Capitolo B).

     

Direct carbohydroximoylation of aromatics with primary nitroalkanes in triflic acid (TFSA).

Primary nitroalkanes react with aromatics in triflic acid to yield arylated oximes at 70–80°C.     

Mixed electric-magnetic second-order nonlinear optical response of helicenes

The mixed electric-magnetic second-order nonlinear optical responses of oriented films of helicenes have been computed ab initio using the random phase approximation method and compared to the pure electric-dipole counterpart. It turns out that the mixed electric-magnetic responses can be of the same order of magnitude as the pure electric-dipole counterpart when there is no donor/acceptor (D/A) substituent or these D/A pairs are weak, i.e., when the pure electric-dipole response is small. When adding strong D/A substituents, the pure electric-dipole response increases substantially and much more than its mixed electric-magnetic counterpart. Consequently, the ratio between the mixed electric-magnetic and pure electric responses decreases. Although there is no general rule, the mixed responses evolve as a function of substitution quasi similarly to the pure electric contribution. This study confirms therefore the possibility of tuning the mixed electric-magnetic response by employing appropriate chiral molecules.     

The configuration of β-halogenoenamines. The use of a new procedure for NOE measurements

The configurations of 16 different β-halogenoenamines have been determined using the proton NOE effect, and the same results were obtained as when the ²J(CH) coupling constants were used. A new procedure called the ‘repetitive shift frequency’ is presented for the NOE determinations which allows better reproducibility and improves the precision.     

Anion-receptor molecules: Synthesis of a chiral and functionalized binding subunit,a bicyclic guanidinium group derived from L- or D- asparagine

The optically active (4S,8S)-4, 8-bis(hydroxymethyl)-1,5,7-triazabicyclo[4.4.0]dec-5-ene ((S,S)- 1 ) has been synthesized in nine steps from L -asparagine with a total yield of 5.1%. Similarly, the enantiomer (R,R)- 1 has been prepared from D -asparagine. (S,S)- and (S,S)- 1 are representative examples of rigid and functionalized bicyclic guanidine systems and constitute useful intermediates in the construction of chiral selective anion-receptor molecules.