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We assume that every element of a matrix has a small, individual error, and model it by an external number, which is the sum of a nonstandard real number and a neutrix, the latter being a convex (external) additive group. The algebraic properties of external numbers formalize common error analysis, with rules for calculation which are a sort of mellowed form of the axioms for real numbers.We model the propagation of errors in matrix calculus by the calculus of matrices with external numbers, and study its algebraic properties. Many classical properties continue to hold, sometimes stated in terms of inclusion instead of equality. There are notable exceptions, for which we give counterexamples and investigate suitable adaptations. In particular we study addition and multiplication of matrices, determinants, near inverses, and generalized notions of linear independence and rank.

Flexible systems are obtained from systems of linear equations by adding to the elements of the coefficient matrix and the right-hand side scalar neutrices, which are convex groups of (non-standard) real numbers. The neutrices model imprecisions, giving rise to calculation rules extending informal error calculus. Stability conditions for flexible systems are given in terms of relative imprecision and size of determinants. We then apply the explicit formula for the elements of the successive intermediate matrices of the Gauss-Jordan elimination procedure to find the solution of flexible systems, keeping track of the error terms at every stage. The solution respects the original imprecisions in the right-hand side and is the same as the one given by Cramer’s rule.

Systems of linear equations are studied with uncertainties in the coefficients, often representing imprecise data. Such imprecisions are seen as orders of magnitude. The orders of magnitude are not...

The elements of the successive intermediate matrices of the Gauss-Jordan elimination procedure have the form of quotients of minors. Instead of the proof using identities of determinants of \cite{Li}, a direct proof by induction is given., 8

The quotient class of a non-archimedean field is the set of cosets with respect to all of its additive convex subgroups. The algebraic operations on the quotient class are the Minkowski sum and product. We study the algebraic laws of these operations. Addition and multiplication have a common structure in terms of regular ordered semigroups. The two algebraic operations are related by an adapted distributivity law.

We give a characterization of the validity of the distributive law in a solid. There exists equivalence between the characterization and the modified axiom of distributivity valid in a solid.

In the setting of nonstandard analysis we introduce the notion of flexible sequence. The terms of flexible sequences are external numbers. These are a sort of analogue for the classical \emph{O$ (\cdot ) $} and \emph{o$ (\cdot ) $} notation for functions, and have algebraic properties similar to those of real numbers. The flexibility originates from the fact that external numbers are stable under some shifts, additions and multiplications. We introduce two forms of convergence, and study their relation. We show that the usual properties of convergence of sequences hold or can be adapted to these new notions of convergence and give some applications., 44 pages

Measures on the real line may be decomposed into a regular, singular and atomic part. The objective of the present article is to provide analogous decompositions for a class of nonstandard, discrete functions defined on an infinitesimal discretization of the real line.

We study discrete functions on equidistant and non-equidistant infinitesimal grids. We consider its difference quotients of higher order and give conditions for their near-equality to the corresponding derivatives. Important tools are the formula of Faa di Bruno for higher order derivatives and a discrete version of it. As an application of such transitions from the discrete to the continuous we extend the DeMoivre-Laplace Theorem to higher order: n-th order difference quotients of the binomial probability distribution tend to the corresponding n-th order partial differential quotients of the Gaussian distribution.

Neutrices are additive subgroups of a nonstandard model of the real numbers. An external number is the algebraic sum of a nonstandard real number and a neutrix. Due to the stability by some shifts, external numbers may be seen as mathematical models for orders of magnitude. The algebraic properties of external numbers gave rise to the so-called solids, which are extensions of ordered fields, having a restricted distributivity law. However, necessary and sufficient conditions can be given for distributivity to hold. In this article we develop an axiomatics for the external numbers. The axioms are similar to, but mostly somewhat weaker than the axioms for the real numbers and deal with algebraic rules, Dedekind completeness and the Archimedean property. A structure satisfying these axioms is called a complete arithmetical solid. We show that the external numbers form a complete arithmetical solid, implying the consistency of the axioms presented. We also show that the set of precise elements (elements with minimal magnitude) has a built-in nonstandard model of the rationals. Indeed the set of precise elements is situated between the nonstandard rationals and the nonstandard reals whereas the set of non-precise numbers is completely determined.

One method to determine the asymptotics of individual solutions of a difference equation is by solving an associated asymptotic functional equation. Here we study the behaviour of the solutions in an asymptotic neighbourhood of such individual solutions. We identify several types of attraction and repulsion, which range from almost orthogonality to almost parallelness. Necessary and sufficient conditions for these types of behaviour are given.

Neutrices are convex additive subgroups of the nonstandard space R k , most of them are external sets. Because of the convexity and the invariance under some translations and multiplications, external neutrices are models for orders of magnitude. One dimensional neutrices have been applied to asymptotics, singular perturbations, and statistics. This paper shows that in R k , with standard k , every neutrix is the direct sum of k neutrices of R . These components may be chosen to be orthogonal.

Foundations.- The strength of nonstandard analysis.- The virtue of simplicity.- Analysis of various practices of referring in classical or non standard mathematics.- Stratified analysis?.- ERNA at work.- The Sousa Pinto approach to nonstandard generalised functions.- Neutrices in more dimensions.- Number theory.- Nonstandard methods for additive and combinatorial number theory. A survey.- Nonstandard methods and the Erd?s-Turan conjecture.- Statistics, probability and measures.- Nonstandard likelihood ratio test in exponential families.- A finitary approach for the representation of the infinitesimal generator of a markovian semigroup.- On two recent applications of nonstandard analysis to the theory of financial markets.- Quantum Bernoulli experiments and quantum stochastic processes.- Applications of rich measure spaces formed from nonstandard models.- More on S-measures.- A Radon-Nikodym theorem for a vector-valued reference measure.- Differentiability of Loeb measures.- Differential systems and equations.- The power of Gateaux differentiability.- Nonstandard Palais-Smale conditions.- Averaging for ordinary differential equations and functional differential equations.- Path-space measure for stochastic differential equation with a coefficient of polynomial growth.- Optimal control for Navier-Stokes equations.- Local-in-time existence of strong solutions of the n-dimensional Burgers equation via discretizations.- Infinitesimals and education.- Calculus with infinitesimals.- Pre-University Analysis.

Neutrices are convex subgroups of the nonstandard real number system, most of them are external sets. They may also be viewed as modules over the external set of all limited numbers, as such non-noetherian. Because of the convexity and the invariance under some translations and multiplications, the external neutrices are appropriate models of orders of magnitude of numbers. Using their strong algebraic structure a calculus of external numbers has been developped. which includes solving of equations, and even an analysis, for the structure of external numbers has a property of completeness. This paper contains a further step, towards linear algebra and geometry. We show that in ℝ2 every neutrix is the direct sum of two neutrices of ℝ. The components may be chosen orthogonal.

In some branches of pure mathematics such as asymptotics, or mathematics applied to astronomy, physics or economics for example, we are often confronted with laborious calculations where we are asked to handle mathematical entities of different orders of magnitude. In many cases if we decide to push the calculations to the last detail, these same calculations become, if not impenetrable, at least unreadable. In such cases, the classical procedure is to replace the expressions which are too detailed by approximations and to give the “order of magnitude” of the error made. Physicians, astronomers, economists, computer scientists and others have some heuristic methods for the calculation of this order of magnitude. Asymptoticians, on their side, have developed formalisms which start by defining the notion of order of magnitude of a function and then give rules which allow some elementary calculations using this notion. The notation of Landau is perhaps a good example. (See [93].) We also mention the “Infinitarcalcul” of Du Bois-Reymond. (See [68].)

The preceding sections illustrated the use of 7 important tools of nonstandard asymptotic analysis: 1) External sets and relations 2) Permanence principles 3) The classification of convex external subsets of ℝ 4) The lemma of dominated approximation 5) The concentration lemma 6) The Laplace method with parameters 7) Shadow expansions In Chapters II and III we attempt to embed the examples of this chapter into a general theory of approximation. Chapters IV, V and VI are devoted to the tools.

We study the free boundary problem in a nonstandard setting of infinitesimal discretisations of the heat equation. In particular we derive regularity results of solutions and the free boundary, in terms of S-continuity and S-differentiability Nous étudions le problème de la frontière libre dans le contexte non-standard de discrétisations infinitésimales de l'équation de la chaleur. En particulier nous prouvons des résultats de régularité des solutions et de la frontière libre, en termes de S-continuité et de S-dérivabilité.

Systems of linear equations, called flexible systems, with coefficients having uncertainties of type o (.) or O (.) are studied. In some cases an exact solution may not exist but a general theorem that guarantees the existence of an admissible solution, in terms of inclusion, is resented. This admissible solution is produced by Cramer’s Rule; depending on the size of the uncertainties appearing in the matrix of coefficients and in the constant term vector some adaptations may be needed.