Hend Dawood

Interval arithmetic is a fundamental and reliable mathematical machinery for scientific computing and for addressing uncertainty in general. In order to apply interval mathematics to real life uncertainty problems, one needs a computerized (machine) version thereof, and so, this article is devoted to some mathematical notions concerning the algebraic system of machine interval arithmetic. After formalizing some purely mathematical ingredients of particular importance for the purpose at hand, we give formal characterizations of the algebras of real intervals and machine intervals along with describing the need for interval computations to cope with uncertainty problems. Thereupon, we prove some algebraic and order-theoretic results concerning the structure of machine intervals.

keywords: Interval mathematics, Machine interval arithmetic, Outward rounding, Floating-point arithmetic, Machine monotonicity, Dense orders, Orderability of intervals, Symmetricity, Singletonicity, Subdistributive semiring, S-semiring.

In this report, we set up an axiomatic system of algorithmic differentiation and deduce its fundamental algebraic properties.

In this report, we study some of the order-theoretic and algebraic aspects of interval mathematics.

The aim of this paper is to provide a metamathematical investigation of the theory of intervals with the requisite predicate calculi and axiomatic set theory.