Fibonacci, Van der Corput and Riesz-Nágy

What do the three names in the title have in common? The purpose of this paper is to relate them in a new and, hopefully, interesting way. Starting with the Fibonacci numeration system — also known as Zeckendorff's system — we will pose ourselves the problem of extending it in a natural way to represent all real numbers in (0,1). We will see that this natural extension leads to what is known as the ?-system restricted to the unit interval. The resulting complete system of numeration replicates the uniqueness of the binary system which, in our opinion, is responsible for the possibility of defining the Van der Corput sequence in (0,1), a very special sequence which besides being uniformly distributed has one of the lowest discrepancy, a measure of the goodness of the uniformity.Lastly, combining the Fibonacci system and the binary in a very special way we will obtain a singular function, more specifically, the inverse of one of the family of Riesz-Nágy.     

Polynomial inverse integrating factors for quadratic differential systems

We characterize all the quadratic polynomial differential systems having a polynomial inverse integrating factor and provide explicit normal forms for such systems and for their associated first integrals. We also prove that these families of quadratic systems have no limit cycles.     

$${\mathbb{Z}_2\mathbb{Z}_4}$$-linear codes: rank and kernel

A code C{{\mathcal C}} is \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C{{\mathcal C}} by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary linear code). The corresponding binary codes of \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-additive codes under an extended Gray map are called \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes. In this paper, the invariants for \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the rank and dimension of the kernel, are studied. Specifically, given the algebraic parameters of \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the possible values of these two invariants, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with dimension of the kernel k is given. Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code for each possible pair (r, k) is given.     

Ion binding to polyelectrolytes: Monte Carlo simulations versus classical mean field theories

The influence of ion size and surface charge model in titrations of ionizable polyelectrolytes is studied by means of the Semi Grand Canonical Monte Carlo simulation method in the context of the primitive model. Three models describing a discrete distribution of charged functional groups on the polyelectrolyte and different values for the radius of the background electrolyte spanning from ionic to hydrated radii values were analyzed. The polyelectrolyte titrations were simulated by calculating the degree of ionization versus pH curves at two ionic strengths. The results allow us to quantify the impact of the sizes of the background salt ions and surface functional groups of the polyelectrolyte on the dissociation degree. This influence is explained in terms of the effectiveness of the screening of the charged surface sites. Finally, by comparison with the Non-Linear Poisson–Boltzmann model, the influence of ionic correlations and finite size of the solution ions is assessed.     

A Phenyl α-Nitronyl Nitroxide with a Forced Chiral Conformation

Summary.  A new phenyl α-nitronyl nitroxide bearing a hydroxyl group at position 2 of the aromatic ring and a chiral methyl lactate substituent at position 5 has been synthesized with the aim of combining the magnetic properties of this kind of radical with the optical properties endowed by the chiral group. The optically active compound forms intramolecular hydrogen bonds between the OH group and one of the NO groups and shows a large torsion angle between the two rings when compared with similar radicals with no substituent in this position. Therefore, the optical properties are distinct. The optical and magnetic properties of the new radical in both solution and solid state are presented. Received June 23, 2000. Accepted (revised) September 18, 2000     

Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields

We provide normal forms and the global phase portraits in the Poincaré disk for all the Hamiltonian linear type centers of linear plus cubic homogeneous planar polynomial vector fields.