The Arboreal Approach to Pairs of Involutions in Rank Two

We use ideas of J.-P. Serre to obtain a geometric classification of the integral p-adic rank two representations of the infinite dihedral group.     

Geometric configurations of singularities for quadratic differential systems with three distinct real simple finite singularities

In this work we classify, with respect to the geometric equivalence relation, the global configurations of singularities, finite and infinite, of quadratic differential systems possessing exactly three distinct finite simple singularities. This relation is finer than the topological equivalence relation which does not distinguish between a focus and a node or between a strong and a weak focus or between foci (or saddles) of different orders. Such distinctions are, however, important in the production of limit cycles close to the foci (or loops) in perturbations of the systems. The notion of geometric equivalence relation of configurations of singularities allows us to incorporate all these important geometric features which can be expressed in purely algebraic terms. The geometric classification of all configurations of singularities, finite and infinite, of quadratic systems was initiated in a work published in 2013 when the classification was done for systems with total multiplicity m _f of finite singularities less than or equal to one. That work was continued in an article which is due to appear in 2014 where the geometric classification of configurations of singularities was done for the case m _f = 2. In this article we go one step further and obtain the geometric classification of singularities, finite and infinite, for the subclass mentioned above. We obtain 147 geometrically distinct configurations of singularities for this family. We give here the global bifurcation diagram of configurations of singularities, both finite and infinite, with respect to the geometric equivalence relation, for this class of systems. The bifurcation set of this diagram is algebraic. The bifurcation diagram is done in the 12-dimensional space of parameters and it is expressed in terms of polynomial invariants, a fact which gives us an algorithm for determining the geometric configuration of singularities for any quadratic system in this particular class.     

Rank Two Integral Representations of the Infinite Dihedral Group

We give an effective classification of the representations of the infinite dihedral group in GL ₂(R) where R is either the valuation ring ?_(p) or the ring of p-adic integers.     

On the interaction of a beam of polar molecules with a static and a resonant RF field as a source of molecular interferences

A molecular beam interference model is presented based on a two-state interaction between a polar molecule and a resonant RF field as it occurs in the so-called C-field of a typical molecular beam electric resonant spectrometer. The treatment shows the onset of interferences in the beam transmission spectra as well as in its transverse profile. It is demonstrated how the molecular interferences are originated by the wavefunction phase shift introduced by the resonant RF field. Furthermore it is shown that for a given beam velocity and oscillating field frequency the fringes’ visibility depends on the strength of the RF field, i.e. the Rabi frequency, in the transmission spectra. Likewise the presence of a RF field gradient in the perpendicular beam direction gives rise to a peak structure in the transverse beam profile. The theoretical model was applied to simulate a variety of beam transmission spectra under resonant conditions as well as some experimental data already published by our group showing a satisfactory agreement between experimental and simulated data. Finally, the potentiality of this internal state molecular interferometer to carry out studies in matter-wave interferometry is remarked.     

Topological entropy of continuous self-maps on closed surfaces

The objective of this work is to present sufficient conditions for having positive topological entropy for continuous self-maps defined on a closed surface by using the action of this map on the homological groups of the closed surface.     

Periodic Solutions of the Duffing Differential Equation Revisited via the Averaging Theory

We use three different results of the averaging theory of first order for studying the existence of new periodic solutions in the two Duffing differential equations $\ddot y+ a \sin y= b \sin t$ and $\ddot y+a y-c y^3=b\sin t$, where $a$, $b$ and $c$ are real parameters.     

A density functional theory study of the structure and vibrational spectra of beta-carotene, capsanthin, and capsorubin

A theoretical study of the structure and the vibrational spectra of the beta-carotene molecule and its derivatives capsanthin and capsorubin is carried out. We first investigate systematically the theoretical method which provides the best results for beta-carotene by performing ab initio calculations at the HF/6-31G(d), SVWN/6-31G(d), PBE0/6-31G(d), BLYP/6-31G(d), B3LYP/6-31G(d), B3LYP/6-31G(d,p), B3LYP/6-311G(d), and B3LYP/6-311G(d,p) levels and by using previous theoretical results available in the literature obtained at the AM1 and BPW91/6-31G(d) levels. The influence of both the level of calculation and the size of the basis set used in the geometry optimization and in the determination of the IR and Raman spectra of this molecule is thus analyzed. It is confirmed that the hybrid functional B3LYP with the basis 6-31G(d) is the method that gives the best results as a whole. By use of this level of calculation, we next optimize the molecular geometries of related molecules of capsanthin and capsorubin, which to the best of our knowledge have only been studied at the semiempirical AM1 level. In addition we calculate the IR and Raman spectra of these molecules at the B3LYP/6-31G(d) level of theory. The results obtained for capsanthin show on the one hand that the double bond of the beta-ionone ring is outside the polyene chain plane, due to the repulsion between the hydrogen atoms of the ring methyl groups and the hydrogen atoms of the polyene chain, and on the other hand that the carbonyl double bond in the other headgroup is very close to planarity with the polyene chain, since in this case such a repulsion does not exist. For the molecule of capsorubin the two carbonyl groups also take the same coplanar orientation relative to the polyene chain. The IR and Raman spectra theoretically computed for these two molecules are finally compared with their experimental spectra and the vibrational normal modes of the main signals are interpreted.