Giuseppina Anatriello

Tribonacci-like sequences and generalized Pascal's pyramids

A well-known result of Feinberg and Shannon states that the tribonacci sequence can be detected by the so-called Pascals pyramid. Here we will show that any tribonacci-like sequence can be obtained by the diagonals of the Feinbergs triangle associated to a suitable generalized Pascals pyramid. The results also extend similar properties of Fibonacci-like sequences.

Logarithmic spirals and continue triangles

In this article we will use some special triangles, to construct polygonal chains that describe the families of logarithmic spirals, among which are the celebrated Golden Spiral, Spira solaris and Pheidia Spiral.

Identification of Fully Measurable Grand Lebesgue Spaces

We consider the Banach function spaces, called fully measurable grand Lebesgue spaces, associated with the function norm where denotes the norm of the Lebesgue space of exponent , and and are measurable functions over a measure space , , and almost everywhere. We prove that every such space can be expressed equivalently replacing and with functions defined everywhere on the interval , decreasing and increasing, respectively (hence the full measurability assumption in the definition does not give an effective generalization with respect to the pointwise monotone assumption and the essential supremum can be replaced with the simple supremum). In particular, we show that, in the case of bounded , the class of fully measurable Lebesgue spaces coincides with the class of generalized grand Lebesgue spaces introduced by Capone, Formica, and Giova.

On an assumption of geometric foundation of numbers

In line with the latest positions of Gottlob Frege, this article puts forward the hypothesis that the cognitive bases of mathematics are geometric in nature. Starting from the geometry axioms of the Elements of Euclid, we introduce a geometric theory of proportions along the lines of the one introduced by Grassmann in Ausdehnungslehre in 1844. Assuming as axioms, the cognitive contents of the theorems of Pappus and Desargues, through their configurations, in an Euclidean plane a natural field structure can be identified that reveals the purely geometric nature of complex numbers. Reasoning based on figures is becoming a growing interdisciplinary field in logic, philosophy and cognitive sciences, and is also of considerable interest in the field of education, moreover, recently, it has been emphasized that the mutual assistance that geometry and complex numbers give is poorly pointed out in teaching and that a unitary vision of geometrical aspects and calculation can be clarifying.

Banach function norms via Cauchy polynomials and applications

Let X1,…,Xk be quasinormed spaces with quasinorms | ⋅ |j, j = 1,…,k, respectively. For any f = (f1,⋯,fk) ∈ X1 ×⋯× Xk let ρ(f) be the unique non-negative root of the Cauchy polynomial . We prove that ρ(⋅) (which in general cannot be expressed by radicals when k ≥ 5) is a quasinorm on X1 ×⋯× Xk, which we call root quasinorm, and we find a characterization of this quasinorm as limit of ratios of consecutive terms of a linear recurrence relation. If X1,…,Xk are normed, Banach or Banach function spaces, then the same construction gives respectively a normed, Banach or a Banach function space. Norms obtained as roots of polynomials are already known in the framework of the variable Lebesgue spaces, in the case of the exponent simple function with values 1,…,k. We investigate the properties of the root quasinorm and we establish a number of inequalities, which come from a rich literature of the past century.