Anna Capietto

On the existence of two solutions with a prescribed number of zeros for a superlinear two-point boundary value problem

where the function f has superlinear growth at infinity and p grows at most linearly in u and u′. Our method is based on a continuation theorem for a coincidence equation of the form Lu = N(u, λ), with the parameter λ varying in the unit interval I. In [1] and the preceding papers [4], [9, §5.5] we introduced a continuous functional φ(u, λ) which takes integer values on large solutions and, under certain conditions, we were able to show that there are solutions of Lu = N(u, 1) with φ(u, 1) equal to some positive integer. In the applications to scalar ordinary differential equations φ was related to the number of zeros of the solutions and thus we could prove that the boundary value problems we considered have at least one solution with k zeros in [a, b] for each sufficiently large k.

Infinitely many radial solutions to a boundary value problem in a ball

In this paper we are concerned with the existence and multiplicity of radial solutions to the BVP whereB is an open ball in ℝK and u↦∇·(a(|∇u|)∇u) is a nonlinear differential operator (e.g. the plaplacian or the mean curvature operator). The function f is defined in a neighborhood of u=0 and satisfies a «sublinear»-type growth condition for u→0. We use a degree approach combined with a time-map technique. Multiplicity results are obtained also for nonlinearities of concave-convex type.

Multiplicity results for systems of asymptotically linear second order equations

Abstract We prove the existence and multiplicity of solutions, with prescribed nodal properties, for a BVP associated with a system of asymptotically linear second order equations. The applicability of an abstract continuation theorem is ensured by upper and lower bounds on the number of zeros of each component of a solution.

Stability and Bifurcation Theory for Non-Autonomous Differential Equations

Periodic differential operators have a rich mathematical theory as well as important physical applications. They have been the subject of intensive development for over a century and remain a fertile research area. This book lays out the theoretical foundations and then moves on to give a coherent account of more recent results, relating in particular to the eigenvalue and spectral theory of the Hill and Dirac equations. The book will be valuable to advanced students and academics both for general reference and as an introduction to active research topics.

An Automatized Method Based on LaTeX for the Realization of Accessible PDF Documents Containing Formulae

Assistive technologies for visually impaired people (screen readers and braille displays) perform satisfactorily with regard to digital documents containing alphabet characters, but they still have a long way to go as far as formulae and graphs are concerned. In general, the most spread digital documents are in PDF format. However, in the case of mathematical contents, they are not accessible at all, since formulae are usually unreadable by screen readers. Currently, a standard and fast method for inserting accessible formulae into a PDF document is still lacking despite it is a very important issue for spreading accessible digital scientific documents. In this paper, we propose a method for automatically generating a PDF document with mathematical contents accessible by assistive technologies for visually impaired people. Specifically, we have developed a LaTeX package that produces a final PDF document where the formulae are totally accessible by screen readers and braille displays.

On the accessibility of Moodle 2 by visually impaired users, with a focus on mathematical content

In this paper we study the accessibility by visually impaired people of the learning management system (LMS) Moodle 2. The study is conducted by testing four different visually impaired subjects, with different degrees of disability and performing different tasks connected to different roles in the LMS. A peculiar focus is given to the accessibility of content involving mathematics. At the end of the paper, some recommendations to improve the accessibility of Moodle 2 are given.

Axessibility: a LaTeX Package for Mathematical Formulae Accessibility in PDF Documents

Accessing mathematical formulae within digital documents is challenging for blind people. In particular, document formats designed for printing, such as PDF, structure math content for visual access only. While accessibility features exist to present PDF content non-visually, formulae support is limited to providing replacement text that can be read by a screen reader or displayed on a braille bar. However, the operation of inserting replacement text is left to document authors, who rarely provide such content. Furthermore, at best, description of the formulae are provided. Thus, conveying detailed understanding of complex formulae is nearly impossible. In this contribution we report our ongoing research on Axessibility, a LaTeX package framework that automates the process of making mathematical formulae accessible by providing the formulae LaTeX code as PDF replacement text. Axessibility is coupled with external scripts to automate its integration in existing documents, expand user shorthand macros to standard LaTeX representation, and custom screen reader dictionaries that improve formulae reading on screen readers.

Artificial Neural Networks and Fuzzy Logic for Recognizing Alphabet Characters and Mathematical Symbols

Optical Character Recognition software (OCR) are important tools for obtaining accessible texts. We propose the use of artificial neural networks (ANN) in order to develop pattern recognition algorithms capable of recognizing both normal texts and formulae. We present an original improvement of the backpropagation algorithm. Moreover, we describe a novel image segmentation algorithm that exploits fuzzy logic for separating touching characters.

The Maslov Index and Global Bifurcation for Nonlinear Boundary Value Problems

We first describe the notion of Maslov index and relate it to the concepts of moment of verticality and of phase-angles. We then illustrate the role of the Maslov index in the development of global bifurcation results for nonlinear first order differential systems in \({\mathbb{R}}^{2N}\) and for nonlinear planar Dirac-type systems.

On The Use of Time-Maps in Nonlinear Boundary Value Problems

Existence and multiplicity of solutions (with prescribed nodal properties) for the two-point boundary value problem associated to a second order ODE is proved. First, we consider an autonomous sublinear problem; secondly, we deal with a nonautonomous superlinear asymmetric one.