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We generalize the results of a previous paper by one of the authors to show a relationship among a class of string solutions through O(d,d) transformations. The results are applied to a rotating black hole solution of three dimensional general relativity discussed recently. We extend the black hole solution to string theory and show its connection with the three dimensional black string with nonzero momentum through an O(d,d) transformation of the above type.

Abstract-This paper proposes a fuzzy set-based approach for handling relative orders of magnitude stated in terms of closeness and negligibility relations. At the semantic level, these relations are represented by means of fuzzy relations controlled by tolerance parameters. A set of sound inference rules, involving the tolerance parameters, is provided, in full accordance with the combination/projection principle underlying the approximate reasoning method of Zadeh. These rules ensure a local propagation of fuzzy closeness and negligibility relations. A numerical semantics is then attached to the symbolic computation process. Required properties of the tolerance parameter are investigated, in order to preserve the validity of the produced conclusions. The effect of the chaining of rules in the inference process can be controlled through the gradual deterioration of closeness and negligibility relations involved in the produced conclusions. Lastly, qualitative reasoning based on fuzzy closeness and negligibility relations is used for simplifying equations and solving them in an approximate way, as often done by engineers who reason about a mathematical model. The problem of handling qualitative probabilities in reasoning under uncertainty is also investigated in this perspective.Index Terms-Closeness, fuzzy numbers, fuzzy relations, negligibility, qualitative probabilities, relative orders of magnitude.

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It is shown that the previously known N = 3 and N = 4 superconformal algebras can be contracted consistently by singular scaling of some of the generators. For the later case, by a contraction which depends on the central term, we obtain a new N = 4 superconformal algebra which contains an SU (2) × U (1) 4 Kac-Moody subalgebra and has nonzero central extension. IP/BBSR/92-90hep-th/9301010