Limit Cycles of Piecewise Smooth Differential Equations on Two Dimensional Torus

In this paper we study the limit cycles of some classes of piecewise smooth vector fields defined in the two dimensional torus. The piecewise smooth vector fields that we consider are composed by linear, Ricatti with constant coefficients and perturbations of these one, which are given in (3). Considering these piecewise smooth vector fields we characterize the global dynamics, studying the upper bound of number of limit cycles, the existence of non-trivial recurrence and a continuum of periodic orbits. We also present a family of piecewise smooth vector fields that posses a finite number of fold points and, for this family we prove that for any 2k number of limit cycles there exists a piecewise smooth vector fields in this family that presents k number of limit cycles and prove that some classes of piecewise smooth vector fields presents a non-trivial recurrence or a continuum of periodic orbits.     

A three-dimensional lanthanide-organic radical open-framework

The reaction of a 3-connecting PTMTC radical with Tb(III) ions forms a three-dimensional open-framework with the formula [Tb(PTMTC)(DMF)(3)] (1), which associates a lattice complex T topology with large channels, guest-induced reversible crystal to amorphous transformations and ferromagnetic metal-radical interactions.     

Efficient preparation of N-phenylsulfenyl ketimines from oximes or nitro compounds without racemization of alpha-stereocenters

As N-sulfenyl imines (e.g., RR'C=N-SAr) can be readily transformed to their N-sulfinyl imines (RR'C=N-SOAr), N-sulfonyl imines (RR'C=N-SO2Ar), and N-sulfonyl oxaziridines, the very mild procedure developed to convert ketoximes and secondary nitro derivatives to N-arenesulfenyl ketimines constitutes a new and efficient route to all these series of compounds. The configuration of the alpha-stereocenters is retained.     

Quadratic systems with a rational first integral of degree 2: A complete classification in the coefficient space $$\mathbb{R}^{12} $$

A quadratic polynomial differential system can be identified with a single point of through the coefficients. Using the algebraic invariant theory we classify all the quadratic polynomial differential systems of having a rational first integral of degree 2. We show that there are only 24 topologically different phase portraits in the Poincaré disc associated to this family of quadratic systems up to a reversal of the sense of their orbits, and we provide a unique representative of every class modulo an affine change of variables and a rescalling of the time variable. Moreover, each one of these 24 representatives is determined by a set of invariant conditions and each respective first integral is given in invariant form directly in The authors are partially supported by a MEC/FEDER grant MTM2005-06098-C02-01, and a CONACIT grant number 2005SGR-00550. Partially supported by CRDF-MRDA CERIM-1006-06     

Transition metal bisdithiolene complexes based on extended ligands with fused tetrathiafulvalene and thiophene moieties: new single-component molecular metals

The gold and nickel bisdithiolene complexes based on new highly extended ligands incorporating fused tetrathiafulvalene and thiophene moieties (alpha-tdt=thiophenetetrathiafulvalenedithiolate and dtdt=dihydro- thiophenetetrathiafulvalenedithiolate), were prepared and characterised by using cyclic voltammetry, single crystal X-ray diffraction, EPR, magnetic susceptibility and electrical transport measurements. These complexes, initially obtained under anaerobic conditions as diamagnetic gold monoanic [nBu(4)N][Au(alpha-tdt)(2)] (4), [nBu(4)N][Au(dtdt)(2)] (3) and nickel dianionic species [(nBu(4)N)(2)][Ni(alpha-tdt)(2)] (8), [(nBu(4)N)(2)][Ni(dtdt)(2)] (7), can be easily oxidised to the stable neutral state just by air or iodine exposure. The monoanionic complexes crystallise in at least two polymorphs, all of which have good cation and anion segregation in alternated layers, the anion layers making a dense 2D network of short SS contacts. All of the neutral complexes, obtained as microcrystalline or quasi amorphous fine powder, present relatively large magnetic susceptibilities that correspond to effective magnetic moments in the range 1-3 mu(B) indicative of high spin states and very high electrical conductivity that in case of the Ni compound can reach sigma(RT) approximately 250 S cm(-1) with a clear metallic behaviour. These compounds are new examples of the still rare single-component molecular metals.