Formalizing undefinedness arising in calculus

Abstract. Undefined terms are commonplace in mathematics, particularly in calculus. The traditional approach to undefinedness in mathematical practice is to treat undefined terms as legitimate, nondenoting terms that can be components of meaningful statements. The traditional approach enables statements about partial functions and undefined terms to be stated very concisely. Unfortunately, the traditional approach cannot be easily employed in a standard logic in which all functions are total and all terms are defined, but it can be directly formalized in a standard logic if the logic is modified slightly to admit undefined terms and statements about definedness. This paper demonstrates this by defining a version of simple type theory called Simple Type Theory with Undefinedness (sttwu) and then formalizing in sttwu examples of undefinedness arising in calculus. The examples are taken from M. Spivak’s well-known textbook Calculus.

Uma proposta de modelagem da lista de materiais Bill of materials' modeling: a proposal

Este trabalho propõe uma nova forma de estruturação da lista de materiais, para empresas fabricantes de bens de capital, apesar dos conceitos gerais serem passíveis de utilização em outros tipos de companhias de manufatura. A lista de materiais, como elemento fomentador da integração, é modelada com base em necessidades-chave de diversos usuários, desde o orçamento até a expedição. O objetivo é facilitar as atividades de administração de materiais e gestão da produção. A metodologia desenvolvida consiste em estruturar uma lista de materiais paralela, delineada em função do organograma fabril da empresa e da seqüência de fabricação do produto no tempo. Esses conceitos foram aplicados na formação de grupos de dispositivos dedicados, utilizados na fabricação de hidrogeradores, na SADE VIGESA S/A., fábrica de Araraquara.<br>This paper suggests a new idea to form a Bill of Materials (BOM), for heavy equipment manufacturing companies, with the general concepts applied in others kind of manufacturing companies. The bill of materials, as an essential element of integration, from marketing to delivery, aims to facilitate the activities of Production Planning and of the Control System. The methodology developed is based on a new approach to build parallel bill of materials, using concepts of pseudo levels and assemblies, mainly based on the industrial organizational structure of the company and on the sequence of product fabrication in time. The implementation of this system for dedicated fixtures resulted in expressive reduction of lead time, project hours, fabrication hours and costs