We review our algebraic framework for linear boundary problems (concentrating on ordinary differential equations). Its starting point is an appropriate algebraization of the domain of functions, which we have named integro-differential algebras. The algebraic treatment of boundary problems brings up two new algebraic structures whose symbolic representation and computational realization is based on canonical forms in certain commutative and noncommutative polynomial domains. The first of these, the ring of integro-differential operators, is used for both stating and solving linear boundary problems. The other structure, called integro-differential polynomials, is the key tool for describing extensions of integrodifferential algebras. We use the canonical simplifier for integro-differential polynomials for generating an automated proof establishing a canonical simplifier for integro-differential operators. Our approach is fully implemented in the TH∃OREM∀ system; some code fragments and sample computations are included.

Abstract. Archives are implemented as an extension of Theorema for representing mathematical repositories in a natural way. An archive can be conceived as one large formula in a language consisting of higher-order predicate logic together with a few constructs for structuring knowledge: attaching labels to subhierarchies, disambiguating symbols by the use of namespaces, importing symbols from other namespaces and specifying the domains of categories and functors as namespaces with variable operations. All these constructs are logic-internal in the sense that they have a natural translation to higher-order logic so that certain aspects of Mathematical Knowledge Management can be realized in the object logic itself. There are a variety of operations on archives, though in this paper we can only sketch a few of them: knowledge retrieval and theory exploration, merging and splitting, insertion and translation to predicate logic.

Archives are implemented as an extension of Theorema for representing large bodies of mathematics. They provide various constructs for organizing knowledge bases in a natural way: breaking formulae across cells, grouping them in a hierarchical structure, attaching labels to subhierarchies, disambiguating symbols by the use of namespaces, importing symbols from other namespaces, addressing the domains of categories and functors as namespaces with variable opera− tions. All constructs are logic–internal in the sense that they have a natural translation to higher–order logic so that �mathematical knowledge management � can be treated by the object logic itself. 1

Abstract not found