Some remarks on tensor differentiation

Summary The Jacobian of classical tensor analysis is generalized to a transformation operator containing second derivatives. This operator and the Jacobian may be used to formulate a second order tensor calculus. In this calculus, a simple contraction scheme includes as special cases Lie differentiation, the Lie bracket, covariant differentiation, and differentiation with respect to the tensor connections of Bompiani. To Enrico Bompiani on his scientific Jubilee     

The tensor product of function semimodules

Given two domains of functions with values in a field, the canonical map from the algebraic tensor product of the vector spaces of functions on the two domains to the vector space of functions on the product of the two domains is well known to be injective, but not generally surjective. By constructing explicit examples, we show that the corresponding map for semimodules of semiring-valued functions is in general not even injective. This impacts the formulation of topological quantum field theories over semirings. We also confirm the failure of surjectivity for functions with values in complete, additively idempotent semirings by describing a large family of functions that do not lie in the image.     

Direct sums of injective semimodules and direct products of projective semimodules over semirings

We prove that if the direct sum of a family of semimodules over a semiring S is an injective semimodule or if the direct product of a family of semimodules over S is a projective semimodule, then the cardinality of the subfamily consisting of all semimodules which are not modules is strictly less than the cardinality of S. As a consequence, we obtain semiring analogs of well-known characterizations of classical semisimple, quasi-Frobenius, and one-sided Noetherian rings.     

Some remarks on the Riemannian metrics on warped products

The purpose of the present paper is to study the following problem. Given a warped product M:=B×_f F, F=ø, give conditions that B and f must satisfy for the existence of the Riemannian simple double of M.This research was supported in part by Non-Directed Research Fund, Korea Research Foundation, 1989.     

Some remarks on a Telgarsky's conjecture concerning products of paracompact spaces

R. Telgarsky conjectured that if X is a paracompact space then the product X×Y is paracompact for every paracompact space Y if and only if the first player of the G(DC,X) game, introduced by R. Telgarsky, see [R. Telgarsky, Spaces defined by topological games, Fund. Math. 88 (1975) 193-223], has a winning strategy. The paper contains some results supporting this conjecture.     

Tensor products of idempotent semimodules. An algebraic approach

We study idempotent analogs of topological tensor products in the sense of A. Grothendieck. The basic concepts and results are expressed in purely algebraic terms. This is one of a series of papers on idempotent functional analysis. Dedicate to S. G. Krein on the occasion of his 80th birthday Translated fromMatermaticheskie Zametki, Vol. 65, No. 4, pp. 572–585, April, 1999.