Analysis on the Levi-Civita field and computational applications

Keywords: Non-Archimedean analysis Levi-Civita fields Power series Measure theory and integration Optimization Computational applications This paper is dedicated to the loving memory of my brother Saïd Shamseddine (1968–2013). a b s t r a c t In this paper, we present an overview of some of our research on the Levi-Civita fields R and C. R (resp. C) is the smallest non-Archimedean field extension of the real (resp. complex) numbers that is Cauchy-complete and real closed (resp. algebraically closed); in fact, R is small enough to allow for the calculus on the field to be implemented on a computer and used in applications such as the fast and accurate computation of the derivatives of real functions as ''differential quotients'' up to very high orders. We summarize the convergence and analytical properties of power series, showing that they have the same smoothness behavior as real and complex power series; we present a Lebesgue-like measure and integration theory on the Levi-Civita field R; we discuss solutions to one-dimensional and multi-dimensional optimization problems based on continuity and differentiability concepts that are stronger than the topological ones; and we give a brief summary of the results of our ongoing work on developing a non-Archimedean operator theory on a Banach space over C. An overview of recent research on the Levi-Civita fields R and C will be presented. We recall that the elements of R and its complex counterpart C are functions from Q to R and C, respectively, with left-finite support (denoted by supp). That is, below every rational number q, there are only finitely many points where the given function does not vanish. For the further discussion, it is convenient to introduce the following terminology. Definition 1.1 (k; $; %). For x – 0 in R or C, we let kðxÞ ¼ minðsuppðxÞÞ, which exists because of the left-finiteness of supp(x); and we let kð0Þ ¼ þ1. Moreover, we denote the value of x at q 2 Q with brackets like x½qŠ. Given x; y – 0 in R or C, we say x $ y if kðxÞ ¼ kðyÞ; and we say x % y if kðxÞ ¼ kðyÞ and x½kðxފ ¼ y½kðyފ. At this point, these definitions may feel somewhat arbitrary; but after having introduced an order on R, we will see that k describes orders of magnitude, the relation % corresponds to agreement up to infinitely small relative error, while …