3517036
Fixed Point Theory in Weak Second-Order Arithmetic
Abstract We develop a basic part of fixed point theory in the context of weak subsystems of second-order arithmetic. RCA0 is the system of recursive comprehension and Σ01 induction. WKL0 is RCA0 plus the weak Konig's lemma: every infinite tree of sequences of 0's and 1's has an infinite path. A topological space X is said to possess the fixed point property if every continuous function f:X→X has a point x ϵ X such that f(x) = x. Within WKL0 (indeed RCA0), we prove Brouwer's theorem asserting that every nonempty compact convex closed set C in R n has the fixed point property, provided that C is expressed as the completion of a countable subset of Q n. We then extend Brouwer's theorem to its infinite dimensional analogue (the Tychonoff-Schauder theorem for R N ) still within RCA0. As an application of this theorem, we prove the Cauchy-Peano theorem for ordinary differential equations within WKL0, which was first shown by Simpson without reference to the fixed point theorem. Within RCA0, we also prove the Markov-Kakutani theorem which asserts the existence of a common fixed point for certain families of affine mappings. Adapting Kakutani's ingenious proof for deducing the Hahn–Banach theorem from the Markov-Kakutani theorem, we also establish the Hahn-Banach theorem for seperable Banach spaces within WKL0, which was first shown by Brown and Simpson in a different way.
7016005
Non standard Analysis in WKL0
Within a weak subsystem of second-order arithmetic WKL0, we develop basic part of non-standard analysis up to the Peano existence theorem.
8895020
The strong soundness theorem for real closed fields and Hilbert’s Nullstellensatz in second order arithmetic
Abstract.By RCA0, we denote a subsystem of second order arithmetic based on Δ01 comprehension and Δ01 induction. We show within this system that the real number system R satisfies all the theorems (possibly with non-standard length) of the theory of real closed fields under an appropriate truth definition. This enables us to develop linear algebra and polynomial ring theory over real and complex numbers, so that we particularly obtain Hilbert’s Nullstellensatz in RCA0.
35660731
Some Properties of Fuzzy Sets of Type 2
The concept of fuzzy sets of type 2 has been defined by L. A. Zadeh as an extension of ordinary fuzzy sets. The fuzzy set of type 2 can be characterized by a fuzzy membership function the grade (or fuzzy grade) of which is a fuzzy set in the unit interval [0, 1] rather than a point in [0, 1]. This paper investigates the algebraic structures of fuzzy grades under the operations of join ⊔, meet ⊔, and negation ┐ which are defined by using the extension principle, and shows that convex fuzzy grades form a commutative semiring and normal convex fuzzy grades form a distributive lattice under ⊔ and ⊓. Moreover, the algebraic properties of fuzzy grades under the operations and which are slightly different from ⊔ and ⊓, respectively, are briefly discussed.
62238493
Two-Level Schemata and Generalized Links for Hypertext Database Models
5205647
Problems of Relational Database Design
In the logical design of a relational database, the theory of data dependencies has been playing an important role. Data depenencies are integrity constraints concerned with attribute relationships. The theory of functional dependencies (FDs) were successful to capture the important aspects of relational database design: semantics of data and logical scheme design (normalization). Several problems of FDs have been also pointed out, and many kinds of new dependencies have been introduced as the generalizations of FDs. Each of the researches on the theory of the new dependencies, semantics of data and the logical scheme design has made a great progress with few sufficient interactions. This leads to several problems when the data dependency theory is used in the actual relational database design.
1810071
Testing of Join Dependency Preserving by a Modified Chase Method
34835412
A Relational Data Language with Simplified Binary Relation Handling Capability
In order to develop a general purpose research information system, we have introduced a new relational data language RLB (Relational data Language with simplified Binary relation handling capability), which can handle bibliographic data as well as data handled by ordinary data base systems. RLB is based on relational calculus introduced by Codd, but allows queries to be written in relatively high level expressions by defining and using proper binary relations over a set of n-ary relations. The characteristics of this language are as follows: (1) Binary relations between attributes or compound attributes can be defined and used to formulate queries; (2) Familiar operations for binary relations such as a concatenation, a closure operation, Boolean operations and comparison operators are provided to simplify queries; (3) Binary relations are classified into several types by their corresponding graphs. This classification can be used for checking storage operations such as updating, insertion and deletion, as well as for selecting efficient physical storage structures; (4) Dependencies between attributes are also regarded as binary relations and thus, they are classified for the same objectives described in (3). In this classification, functional dependencies, multivalued dependencies and set - set dependencies are contained; (5) It can provide several facilities similar to mapping oriented languages (for example, SEQUEL) or network model based languages (for example, LSL). Furthermore, an output system for binary relations between domain compatible attributes is also described, which produces a tree expansion of the given graph.