Abstract:
Iron Overload Cardiomyopathy (IOC) is still the main cause of death in thalassemia major (TM) patients. Unfortunately, Conventional Echocardiography fails to predict early cardiac dysfunction. As Tissue Doppler Imaging (TDI) may demonstrate regional myocardial dysfunction, we wondered if exercise may reveal abnormalities at TDI which are not evident at rest. To try to evaluate left and right myocardial performances at rest and after maximal exercise by both conventional and TDI parameters, 46 beta-TM adult patients and 39 control subjects were enrolled. All patients had a liver iron quantification by Superconducting Quantum Interference Device (SQUID) and also a cardiac iron assessment by MRI (T2*): 38 TM patients had no evidence of cardiac iron overload. Whereas TM patients did not shown diastolic dysfunction and all of them presented a good global response to exercise, TDI detected a reduced increase of the S’ waves of left ventricle basal segment during exercise. This finding seems to have some weak but interesting relations with iron overload markers. In conclusion, in our study, exercise stress TDI-echocardiography was able to demonstrate subtle systolic abnormalities that were missed by Conventional Echocardiography. Further studies are required to determine the meaning and the clinical impact of these results.

Abstract:
This paper is essentially devoted to the study of some interesting relations among the well known operators $I^{(x)}$ (the interpolated Invert), $L^{(x)}$ (the interpolated Binomial) and Revert (that we call $\eta$). We prove that $I^{(x)}$ and $L^{(x)}$ are conjugated in the group $\Upsilon(R)$. Here $R$ is a commutative unitary ring. In the same group we see that $\eta$ transforms $I^{(x)}$ in $L^{(-x)}$ by conjugation. These facts are proved as corollaries of much more general results. Then we carefully analyze the action of these operators on the set $\mc{R}$ of second order linear recurrent sequences. While $I^{(x)}$ and $L^{(x)}$ transform $\mc{R}$ in itself, $\eta$ sends $\mc{R}$ in the set of moment sequences $\mu_n(h,k)$ of particular families of orthogonal polynomials, whose weight functions are explicitly computed. The moments come out to be generalized Motzkin numbers (if $R=\zz$, the Motzkin numbers are $\mu_n(-1,1)$). We give several interesting expressions of $\mu_n(h,k)$ in closed forms, and one recurrence relation. There is a fundamental sequence of moments, that generates all the other ones, $\mu_n(0,k)$. These moments are strongly related with Catalan numbers. This fact allows us to find, in the final part, a new identity on Catalan numbers by using orthogonality relations.

Abstract:
In this paper, we define a new product over $\mathbb{R}^{\infty}$, which allows us to obtain a group isomorphic to $\mathbb R^*$ with the usual product. This operation unexpectedly offers an interpretation of the R\'edei rational functions, making more clear some of their properties, and leads to another product, which generates a group structure over the Pell hyperbola. Finally, we join together these results, in order to evaluate solutions of Pell equation in an original way.

Abstract:
In this paper we study a general class of conics starting from a quotient field. We give a group structure over these conics generalizing the construction of a group over the Pell hyperbola. Furthermore, we generalize the definition of R\'edei rational functions in order to use them for evaluating powers of points over these conics. Finally, we study rational approximations of irrational numbers over conics, obtaining a new result for the approximation of quadratic irrationalities.

Abstract:
In this paper we study the action of a generalization of the Binomial interpolated operator on the set of linear recurrent sequences. We find how the zeros of characteristic polynomials are changed and we prove that a subset of these operators form a group, with respect to a well-defined composition law. Furthermore, we study a vast class of linear recurrent sequences fixed by these operators and many other interesting properties. Finally, we apply all the results to integer sequences, finding many relations and formulas involving Catalan numbers, Fibonacci numbers, Lucas numbers and triangular numbers.

Abstract:
In this paper, we present the problem of counting magic squares and we focus on the case of multiplicative magic squares of order 4. We give the exact number of normal multiplicative magic squares of order 4 with an original and complete proof, pointing out the role of the action of the symmetric group. Moreover, we provide a new representation for magic squares of order 4. Such representation allows the construction of magic squares in a very simple way, using essentially only five particular 4X4 matrices.

Abstract:
In this paper we study the action of the Binomial and Invert (interpolated) operators on the set of linear recurrent sequences. We prove that these operators preserve this set, and we determine how they change the characteristic polynomials. We show that these operators, with the aid of two other elementary operators (essentially the left and right shifts), can transform any impulse sequence (a linear recurrent sequence starting from $(0,...,0,1)$) into any other impulse sequence, by two processes that we call \emph{construction} and \emph{deconstruction}. Finally, we give some applications to polynomial sequences and pyramidal numbers. We also find a new identity on Fibonacci numbers, and we prove that $r$--bonacci numbers are a Bell polynomial transform of the $(r-1)$--bonacci numbers.

Abstract:
We propose a novel approach for studying rooted trees by using functions that we will call descent functions. We provide a construction method for rooted trees that allows to study their properties through the use of descent functions. Moreover, in this way, we are able to compose rooted trees with each other. Such a new composition of rooted trees is a very powerful tool applied in this paper in order to obtain important results as the creation of new rational and Pythagorean trees. 1. Introduction Trees represent one of the most important topics in the combinatorial theory. Generating trees whose nodes are elements of a structure may constitute an important step for studying the properties of the structure itself and for the structure visualization. In this paper, some structural properties of trees will be dealt in a novel way in order to make easy the tree generation for sets satisfying specific requirements. One of the main subjects of the present work is the so-called descent functions. Identifying functions of this type allows to reveal a simple and direct technique to the generation of trees on sets equipped with a weight function over a partially ordered set. Furthermore, the introduced technique yields to a substantial simplification of the study of tree levels and easily and directly finds the degree of any node. Moreover, we introduce a composition of trees, which represents a completely new and very powerful tool for the generation of trees. This operation gives the feasibility of generating a new tree starting from a given pair of trees associated with a generating system once we assign a partition of the system itself. We present some examples to show the power of this tool. In particular, the composition of trees will be used to construct new examples of trees of rational numbers (rational trees) and of trees of primitive Pythagorean triples (Pythagorean trees). These applications are of special interest because nowadays only three examples of the first type [1–4] and two of the latter are known [5–7]. On the other hand, one can observe that the used techniques allow to generate an infinite number of trees of either type, and that the shown applications are merely representative. The techniques of explicit construction of the trees described here show some features that arise from properties of the algebraic elements involved, making themselves natural and rigorous. 2. Tree Construction via Descent Functions In this section, we present a new and useful approach to rooted trees by using particular functions that we will call descent

Abstract:
In this paper we study how to accelerate the convergence of the ratios (x_n) of generalized Fibonacci sequences. In particular, we provide recurrent formulas in order to generate subsequences (x_{g_n}) for every linear recurrent sequence (g_n) of order 2. Using these formulas we prove that some approximation methods, as secant, Newton, Halley and Householder methods, can generate subsequences of (x_n). Moreover, interesting properties on Fibonacci numbers arise as an application. Finally, we apply all the results to the convergents of a particular continued fraction which represents quadratic irrationalities.

Abstract:
This paper is devoted to the study of eigen-sequences for some important operators acting on sequences. Using functional equations involving generating functions, we completely solve the problem of characterizing the fixed sequences for the Generalized Binomial operator. We give some applications to integer sequences. In particular we show how we can generate fixed sequences for Generalized Binomial and their relation with the Worpitzky transform. We illustrate this fact with some interesting examples and identities, related to Fibonacci, Catalan, Motzkin and Euler numbers. Finally we find the eigen-sequences for the mutual compositions of the operators Interpolated Invert, Generalized Binomial and Revert.