IEEE Transactions on Pattern Analysis and Machine Intelligence

What is modal logic

General Topology-I

This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

A first course in noncommutative rings, by T. Y. Lam. Pp. 385. £37 (pb), £62.50 (hb). 2001. ISBN 0 387 95325 6 (pb), 0 387 95183 0 (hb) (Springer-Verlag).

The notion of fuzziness as defined in this paper relates to situations in which the source of imprecision is not a random variable or a stochastic process, but rather a class or classes which do not possess sharply defined boundaries, e.g., the “class of bald men,” or the “class of numbers which are much greater than 10,” or the “class of adaptive systems,” etc. A basic concept which makes it possible to treat fuzziness in a quantitative manner is that of a fuzzy set, that is, a class in which there may be grades of membership intermediate between full membership and non-membership. Thus, a fuzzy set is characterized by a membership function which assigns to each object its grade of membership (a number lying between 0 and 1) in the fuzzy set. After a review of some of the relevant properties of fuzzy sets, the notions of a fuzzy system and a fuzzy class of systems are introduced and briefly analyzed. The paper closes with a section dealing with optimization under fuzzy constraints in which an approach to...

Fuzzy sets and systems

On interval fuzzy S-implications

On (G,N)-implications derived from grouping functions

Interval additive generators of interval t-norms and interval t-conorms

An interval is a continuum of real numbers, defined by its end-points. Interval analysis, proposed by R. Moore in the 50's, concerns the discovery of interval functions to produce bounds on the accuracy of numerical results that are guaranteed to be sharp and correct. The last criterion, correctness, is the main one since it establishes that the result of an interval computation must always contains the value of the related real function.Correctness rests on the ``Fundamental Theorem of Interval Arithmetic''. This theorem, induced us to define interval representations, which captures the fact that interval analysis works like a kind of language to express computations with real numbers, this is implicitly formulated at [HJE01] p. 1045, lemma 2.Until now the idea of intervals as representation of real numbers was not explicitly defined and its relation with some aspects of interval analysis was not explored. Here we show some of these relations in terms of the topological aspects of intervals (Scott and Moore topologies). The paper also defines what we call the canonical interval representation of a real function, which is the obvious best ``mathematical'' (not necessarily computable) interval representation of a real function **. The idea is to show some properties of correct interval algorithms giving a relationship with some other kind of functions, such as extensions and inclusion monotonic functions; and to show how this ideal object preserves the continuity of real functions into Scott and Moore topologies.

Regivan H. N. Santiago

Papers

On interval fuzzy S-implications

On (G,N)-implications derived from grouping functions

Interval additive generators of interval t-norms and interval t-conorms

Formal Aspects of Correctness and Optimality of Interval Computations

Lattice-valued overlap and quasi-overlap functions