On Coxeter recurrences

An interesting family of recurrences of order n ≥ 2, which are globally (n+3)-periodic was introduced by Coxeter in 1971. We prove a surprising property of this family: ‘all’ the possible geometrical behaviours that linear real (n+3)-periodic recurrences can have are present inside the Coxeter recurrences.     

Upper bounds for the number of zeroes for some Abelian integrals

Consider the vector field x^′=−yG(x,y),y^′=xG(x,y)$x^{\prime }=-yG(x,y),y^{\prime }=xG(x,y)$, where the set of critical points {G(x,y)=0}$\{G(x,y)=0\}$ is formed by K$K$ straight lines, not passing through the origin and parallel to one or two orthogonal directions. We perturb it with a general polynomial perturbation of degree n$n$ and study the maximum number of limit cycles that can bifurcate from the period annulus of the origin in terms of K$K$ and n$n$. Our approach is based on the explicit computation of the Abelian integral that controls the bifurcation and on a new result for bounding the number of zeroes of a certain family of real functions. When we apply our results for K≤4$K\le 4$ we recover or improve some results obtained in several previous works.     

Neutron stars in Scalar-Tensor-Vector Gravity

Scalar-Tensor-Vector Gravity (STVG), also referred as Modified Gravity (MOG), is an alternative theory of the gravitational interaction. Its weak field approximation has been successfully used to describe Solar System observations, galaxy rotation curves, dynamics of clusters of galaxies, and cosmological data, without the imposition of dark components. The theory was formulated by John Moffat in 2006. In this work, we derive matter-sourced solutions of STVG and construct neutron star models. We aim at exploring STVG predictions about stellar structure in the strong gravity regime. Specifically, we represent spacetime with a static, spherically symmetric manifold, and model the stellar matter content with a perfect fluid energy-momentum tensor. We then derive the modified Tolman–Oppenheimer–Volkoff equation in STVG and integrate it for different equations of state. We find that STVG allows heavier neutron stars than General Relativity (GR). Maximum masses depend on a normalized parameter that quantifies the deviation from GR. The theory exhibits unusual predictions for extreme values of this parameter. We conclude that STVG admits suitable spherically symmetric solutions with matter sources, relevant for stellar structure. Since recent determinations of neutron stars masses violate some GR predictions, STVG appears as a viable candidate for a new gravity theory.     

UV solvatochromic shifts of 4-hydroxy-benzophenone and 2,4-dihydroxy-benzophenone in ethanol-acetonitrile mixtures

The anomalous UV spectroscopic behavior of 4-hydroxy-benzophenone and 2,4-dihydroxy-benzophenone in ethanol-acetonitrile mixtures has been investigated using theoretical and experimental methods. Band I of these compounds, associated to strong pi-->pi* transitions, suffers a blue shift when the polarity of the ethanol-acetonitrile solutions increases. The magnitude of the solvatochromic shifts suffered by the analyzed benzophenones (BPs) changes inversely with the planarity of their molecules. The experimental solvatochromic shifts of the compounds were correlated with the permittivity and the solvation parameters alpha and pi* that characterize the ethanol-acetonitrile mixtures used. In order to explain theoretically the observed solvatochromic shifts, it was proposed that in solution, the two compounds form an association complex of stoichiometry 1:1 with a molecule of ethanol. These complexes are formed by means of an intermolecular hydrogen bond between the hydrogen atom of 4-OH group of the solutes and the oxygen atom of ethanol. The calculations performed at the HRF/6-31G(d) level of theory using Onsager's and Tomasi's models showed that these solute-ethanol association complexes have an elevated thermodynamic stability. Good linear relations were obtained between the experimental absorption frequencies of BPs and the theoretical absorption frequencies of the association complexes. These frequencies were also very satisfactorily correlated with the properties of the above mentioned ethanol-acetonitrile mixtures. It was concluded that the magnitude of the analyzed solvatochromic shifts is determined by the degree of occurrence of solute-solvent interactions, which essentially depend on the polarity, polarizability and hydrogen bond donating ability of the ethanol-acetonitrile mixtures.     

First derivative of the period function with applications

Given a centre of a planar differential system, we extend the use of the Lie bracket to the determination of the monotonicity character of the period function. As far as we know, there are no general methods to study this function, and the use of commutators and Lie bracket was restricted to prove isochronicity. We give several examples and a special method which simplifies the computations when a first integral is known.     

Bifurcation of 2-periodic orbits from non-hyperbolic fixed points

We introduce the concept of 2-cyclicity for families of one-dimensional maps with a non-hyperbolic fixed point by analogy to the cyclicity for families of planar vector fields with a weak focus. This new concept is useful in order to study the number of 2-periodic orbits that can bifurcate from the fixed point. As an application we study the 2-cyclicity of some natural families of polynomial maps.     

Molecular Hydrogels from Bolaform Amino Acid Derivatives: A Structure–Properties Study Based on the Thermodynamics of Gel Solubilization

Insight is provided into the aggregation thermodynamics associated to hydrogel formation by molecular gelators derived from L ‐valine and L ‐isoleucine. Solubility data from NMR measurements are used to extract thermodynamic parameters for the aggregation in water. It is concluded that at room temperature and up to 55 °C, these systems form self‐assembled fibrillar networks in water with quite low or zero enthalpic component, whereas the entropy of the aggregation is favorable. These results are explained by considering that the hydrophobic effect is dominant in the self‐assembly. However, studies by NMR and IR spectroscopy reveal that intermolecular hydrogen bonding is also a key issue in the aggregation process of these molecules in water. The low enthalpy values measured for the self‐assembly process are ascribed to the result of a compensation of the favorable intermolecular hydrogen‐bond formation and the unfavorable enthalpy component of the hydrophobic effect. Additionally, it is shown that by using the hydrophobic character as a design parameter, enthalpy‐controlled hydrogel formation, as opposed to entropy‐controlled hydrogel formation, can be achieved in water if the gelator is polar enough. It is noteworthy that these two types of hydrogels, enthalpy‐versus entropy‐driven hydrogels, present quite different response to temperature changes in properties such as the minimum gelator concentration (mgc) or the rheological moduli. Finally, the presence of a polymorphic transition in a hydrogel upon heating above 70 °C is reported and ascribed to the weakening of the hydrophobic effect upon heating. The new soft polymorphic materials present dramatically different solubility and rheological properties. Altogether these results are aimed to contribute to the rational design of molecular hydrogelators, which could be used for the tailored preparation of this type of soft materials. The reported results could also provide ground for the rationale of different self‐assembly processes in aqueous media.     

A proof of Perkoʼs conjectures for the Bogdanov–Takens system

The Bogdanov–Takens system has at most one limit cycle and, in the parameter space, it exists between a Hopf and a saddle-loop bifurcation curves. The aim of this paper is to prove the Perko?s conjectures about some analytic properties of the saddle-loop bifurcation curve. Moreover, we provide sharp piecewise algebraic upper and lower bounds for this curve.     

Limit cycles for two families of cubic systems

In this paper we study the number of limit cycles of two families of cubic systems introduced in previous papers to model real phenomena. The first one is motivated by a model of star formation histories in giant spiral galaxies and the second one comes from a model of Volterra type. To prove our results we develop a new criterion on the non-existence of periodic orbits and we extend a well-known criterion on the uniqueness of limit cycles due to Kuang and Freedman. Both results allow to reduce the problem to the control of the sign of certain functions that are treated by algebraic tools. Moreover, in both cases, we prove that when the limit cycles exist they are non-algebraic.