Rendiconti dell’Istituto di Matematica dell’Università di Trieste: an International Journal of Mathematics vol.51 (2019)

Rendiconti dell’Istituto di Matematica dell’Università di Trieste was founded in 1969 by Arno Predonzan, with the aim of publishing original research articles in all fields of mathematics and has been the first Italian mathematical journal to be published also on-line. The access to the electronic version of the journal is free. All published articles are available on-line. The journal can be obtained by subscription, or by reciprocity with other similar journals. Currently more than 100 exchange agreements with mathematics departments and institutes around the world have been entered in.

In the present paper, partly a survey, we discuss upper and lower bounds for finite group-actions on bounded surfaces, 3-dimensional handlebodies and closed handles, handlebodies in arbitrary dimensions and finite graphs (the common feature of these objects is that they have all free fundamental group).

This paper is a survey of recent existence results for solutions of first and second order nonlinear differential systems with nonlocal boundary conditions using methods based upon convexity, topological degree and maximum-principle like techniques.

For a polarized complex Abelian variety $A$ we study the function $N_A(t)$ counting the number of elliptic curves in $A$ with degree bounded by $t$. This extends our previous work in dimension two. We describe the collection of elliptic curves in the product $A = S \times F$ of an Abelian variety and an elliptic curve by means of an explicit parametrization, and in terms of the parametrization we express the degrees of elliptic curves relative to a split polarization. When this is applied to the self product $A = E^k$ of an elliptic curve, it turns out that an asymptotic estimate of the counting function $N_A(t)$ can be obtained from an asymptotic study of the degree form on the group of endomorphisms of the elliptic curve.

In this paper we study, for any subset\ $I$\ of $\mathbf{N}^{\ast}$ and for any strictly positive integer $k$, the Banach space $E_{I}$ of the bounded real sequences $\left\{ x_{n}\right\} _{n\in I}$, and a measure over $\left( \mathbf{R}^{I},\mathcal{B}^{(I)}\right) $ that generalizes the $k$-dimensional Lebesgue one. Moreover, we recall the main results about the differentiation theory over $E_{I}$. The main result of our paper is a change of variables' formula for the integration of the measurable real functions on $\left( \mathbf{R}^{I},\mathcal{B}^{(I)}\right) $. This change of variables is defined by some functions over an open subset of $E_{J}$, with values on $E_{I}$, called $\left( m,\sigma\right) $-general, with properties that generalize the analogous ones of the finite-dimensional diffeomorphisms.