Flexible involutive meadows

We investigate a notion of inverse for neutrices inspired by Van den Berg and Koudjeti's decomposition of a neutrix as the product of a real number and an idempotent neutrix. We end up with an algebraic structure that can be characterized axiomatically and generalizes involutive meadows. The latter are algebraic structures where the inverse for multiplication is a total operation. As it turns out, the structures satisfying the axioms of flexible involutive meadows are of interest beyond nonstandard analysis

Axiomatics for the external numbers of nonstandard analysis

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

Assemblies as Semigroups

In this paper we give an algebraic characterization of assemblies in terms of bands of groups. We also consider substructures and homomorphisms of assemblies. We give many examples and counterexamples

A generalization of the Banach-Steinhaus theorem for finite part limits

It is well known, as follows from the Banach-Steinhaus theorem, that if a sequence $\left\{ y_{n}\right\} _{n=1}^{\infty}$ of linear continuous functionals in a Fr\'{e}chet space converges pointwise to a linear functional $Y,$ $Y\left( x\right) =\lim_{n\rightarrow\infty}\left\langle y_{n} ,x\right\rangle$ for all $x,$ then $Y$ is actually continuous. In this article we prove that in a Fr\'{e}chet space\ the continuity of $Y$ still holds if $Y$ is the \emph{finite part} of the limit of $\left\langle y_{n},x\right\rangle$ as $n\rightarrow\infty.$ We also show that the continuity of finite part limits holds for other classes of topological vector spaces, such as \textsl{LF}-spaces, \textsl{DFS}-spaces, and \textsl{DFS} $^{\ast}$-spaces,\ and give examples where it does not hold

Optimization with flexible objectives and constraints

A programação linear e a otimização não linear são estudadas do ponto de vista da análise não-standard, nos casos em que a função objetivo e/ou as restrições não são totalmente especificadas, permitindo de facto alguma imprecisão ou flexibilidade em termos de pequenas variações. A ordem de grandeza de tais variações será modelada por neutrizes, que são subgrupos convexos aditivos da reta real não-standard, e por números externos, que são a soma de um número real com uma neutrix. Esta abordagem preserva as características essenciais de imprecisão, mantendo regras de cálculo bastante fortes e eficazes. Funções, sequências e equações que envolvem números externos são designadas de flexíveis. Consideramse problemas de otimização com funções objetivo e/ou restrições flexíveis em que são dadas as condições necessárias e suficientes para a existência de soluções ótimas ou aproximadamente ótimas, tato para problemas de otimização linear como não linear. Para exemplificar a programação linear nesta configuração são estudados, sistemas flexíveis de equações lineares. As condições para a solubilidade de um sistema flexível por métodos usuais tais como a regra de Cramer e o mo todo de eliminação de Gauss-Jordan são estabelecidas. Além disso, é considerado um método de parâmetros para resolver sistemas flexíveis onde são apresentadas fórmulas de soluções dependendo dos parâmetros. O conjunto de soluções de um sistema flexível é expresso em termos de vetores externos e neutrizes. Para estudar a otimização não linear com objetivos e restrições flexíveis, são desenvolvidas ferramentas de análise para sucessões e funções flexíveis; Abstract: Optimization with flexible objectives and constraints Both linear programming and non-linear optimization are studied from the point of view of non-standard analysis, in cases where the objective function and/or the constraints are not fully specified, indeed allow for some imprecision or flexibility in terms of some limited shifts. The order of magnitude of such shifts will be modelled by neutrices, additive convex subgroups of the nonstandard real line and external numbers, sums of a neutrix and a non-standard real number. This approach captures essential features of imprecision, maintaining rather strong and effective rules of calculation. Functions, sequences and equations which involve external numbers are called flexible. We consider optimization problems with flexible objective functions and/or constraints. Necessary and sufficient conditions for the existence of optimal or approximate optimal solutions are given for both linear and non-linear optimization problems with flexible objective functions and constraints. To deal with linear programming in this setting, flexible systems of linear equations are studied. Conditions for the solvability of a flexible system by usual methods such as Cramer’s rule and Gauss-Jordan elimination are established. Also, a parameter method is considered to solve flexible systems. Formulas of solutions depending on parameters are presented. The set of solutions of a flexible system is expressed in terms of external vectors and neutrices. In order to investigate non-linear optimization with flexible objectives and constraints, we develop tools of analysis for both flexible sequences and functions