RECENT AND NEW PERSPECTIVES IN INTERVAL ANALYSIS

Introduced in the 1950s as a method to deal with the uncertainty of errors, Interval Analysis, a kind of generalization of real analysis in which real intervals replace real numbers, despite the progress made by research, still has some crucial open problems, including the need to standardize theory through a robust and consistent framework for both analysis and algebra. Therefore, this work aims to pursue a dual objective. First, it intends to offer an updated state of the art on the concepts, problems and techniques of Interval Analysis, with a specific focus on some theoretical aspects and on the calculus of interval-valued functions of a single real variable. Through an intensive use of the so-called midpoint-radius representation, more advantageous than conventional notations, the possible types of partial orders in the space of compact real intervals are studied; the use of the gH-difference and gH-differentiability is also extremely useful, above all for the concepts related to the study of functions: limits, derivatives, monotonicity, as well as the analysis of extreme points, concavity and convexity. The various topics, revisited and enriched with innovative notations (e.g. a new representation of complex numbers), acquire a more complete meaning and new application possibilities open up (e.g. at the q-calculus). The other goal to aspire to is to deepen the investigation from an algebraic point of view, also through unconventional approaches. In particular, thanks to the introduction of a new partial order with polarity characteristics with respect to already acquainted orders, it is possible to determine hitherto unexplored algebraic structures: some quite known, such as semirings and pre-semirings, others more unusual, like the so-called combined structures. Moreover, from a study on the complementation properties, interval Boolean structures are also configured and, finally, the construction of an interval quotient set leads us towards even more solid structures, such as a pseudoring. The graphic representations, which constitute a fundamental part of the work, accompany the entire discussion, providing interesting and explanatory examples that ensure greater clarity and expository completeness.