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.     

Classification of the centers and their isochronicity for a class of polynomial differential systems of arbitrary degree

In this paper we classify the centers localized at the origin of coordinates, and their isochronicity for the polynomial differential systems in R² of degree d that in complex notation z=x+iy can be written as where j is either 0 or 1. If j=0 then d?5 is an odd integer and n is an even integer satisfying 2?n?(d+1)/2. If j=1 then d?3 is an integer and n is an integer with converse parity with d and satisfying 0<n?[(d+1)/3] where [⋅] denotes the integer part function. Furthermore λ∈R and A,B,C,D∈C. Note that if d=3 and j=0, we are obtaining the generalization of the polynomial differential systems with cubic homogeneous nonlinearities studied in K.E. Malkin (1964) [17], N.I. Vulpe and K.S. Sibirskii (1988) [25], J. Llibre and C. Valls (2009) [15], and if d=2, j=1 and C=0, we are also obtaining as a particular case the quadratic polynomial differential systems studied in N.N. Bautin (1952) [2], H. Zoladek (1994) [26]. So the class of polynomial differential systems here studied is very general having arbitrary degree and containing the two more relevant subclasses in the history of the center problem for polynomial differential equations.     

Integrability and algebraic limit cycles for polynomial differential systems with homogeneous nonlinearities

We consider the class of polynomial differential equations , where P_n and Q_n are homogeneous polynomials of degree n. These systems have a focus at the origin if λ≠0, and have either a center or a focus if λ=0. Inside this class we identify a new subclass of Darbouxian integrable systems having either a focus or a center at the origin. Additionally, under generic conditions such Darbouxian integrable systems can have at most one limit cycle, and when it exists is algebraic. For the case n=2 and 3, we present new classes of Darbouxian integrable systems having a focus.     

The reversibility and the center problem

In this work we study the narrow relation between reversibility and the center problem and also between reversibility and the integrability problem. It is well known that an analytic system having either a non-degenerate or nilpotent center at the origin is analytically reversible or orbitally analytically reversible, respectively. In this paper we prove the existence of a smooth map that transforms an analytic system having a degenerate center at the origin with either an analytic first integral or a C^∞ inverse integrating factor into a reversible linear system (after rescaling the time). Moreover, if the degenerate center has an analytic or a C^∞ reversing symmetry, then the transformed system by the map also has a reversing symmetry. From the knowledge of a first integral near the center we give a procedure to detect reversing symmetries.     

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.     

Centers for the Kukles homogeneous systems with even degree

For the polynomial differential system $\dot{x}=-y$, $\dot{y}=x +Q_n(x,y)$, where $Q_n(x,y)$ is a homogeneous polynomial of degree $n$ there are the following two conjectures done in 1999. (1) Is it true that the previous system for $n \ge 2$ has a center at the origin if and only if its vector field is symmetric about one of the coordinate axes? (2) Is it true that the origin is an isochronous center of the previous system with the exception of the linear center only if the system has even degree? We give a step forward in the direction of proving both conjectures for all $n$ even. More precisely, we prove both conjectures in the case $n = 4$ and for $n\ge 6$ even under the assumption that if the system has a center or an isochronous center at the origin, then it is symmetric with respect to one of the coordinate axes, or it has a local analytic first integral which is continuous in the parameters of the system in a neighborhood of zero in the parameters space. The case of $n$ odd was studied in [8].     

Numerical procedures in electrochemical simulation

The use of both the Laplace transformation technique and the finite element method has been analyzed in an electrochemical system. The diffusion-controlled adsorption process of a species onto an electrode surface is studied in detail. © 1994 John Wiley & Sons, Inc.     

Dynamical Casimir Effect for Scalar Fields II (Energy Calculation)

The energy of the evolved vacuum state is calculated. From a frequency cut-off regularisation the divergent terms are separated and, in the 1 + 1 dimensional case they are removed with a mass renormalisation of the moving boundary. A renormalisation of the external force is also needed in 3 + 1 dimensions. PACS Subject Classifications: 42.50.Lc, 03.70.+k, 11.10.Ef.