Marco Abrate

On the synthesis of quantum Hall array resistance standards

Quantum Hall effect (QHE) is the basis of modern resistance metrology. In quantum Hall array resistance standards (QHARS), several individual QHE elements, each one with the same QHE resistance (typically half of the von Klitzing constant), are arranged in networks that realize resistance values close to decadic values (such as 1kor 100k� ), of direct interest for dissemination. The same decadic value can be approximated with different grades of precision, and even for the same approximation several networks of QHE elements can be conceived. This paper investigates the design of QHARS networks by giving methods to find a proper approximation of the resistance of interest, and to design the corresponding network with a small number of elements; results for several decadic case examples are given. The realization of these networks with multiterminal QHE elements requires a new multiple bridge connection, here described.

Groups and monoids of Pythagorean triples connected to conics

Abstract We define operations that give the set of all Pythagorean triples a structure of commutative monoid. In particular, we define these operations by using injections between integer triples and 3 × 3 matrices. Firstly, we completely characterize these injections that yield commutative monoids of integer triples. Secondly, we determine commutative monoids of Pythagorean triples characterizing some Pythagorean triple preserving matrices. Moreover, this study offers unexpectedly an original connection with groups over conics. Using this connection, we determine groups composed by Pythagorean triples with the studied operations.

Construction and Composition of Rooted Trees via Descent Functions

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.