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Abstract: If a continuous function on a complete metric space has approximate roots and in a uniform manner at most one root, then it actually has a root. We validate this heuristic principle in Bishop-style constructive mathematics without countable choice, and thus can shed some more light on the role played by the completion when it comes to solving equations. [Copyright &y& Elsevier]

Abstract: Regular fuzzy logic functions are the functions that can be obtained by means of a finite number of compositions from a number of very simple starting functions, which are related to three-valued logic. In this work we prove that if a function f can be implicitly defined by a system of equations involving regular fuzzy logic functions, then f is itself a regular fuzzy logic function. The proof is based on results about regular Kleene algebras and Masao Mukaidono''s characterization of regular logic functions. [Copyright &y& Elsevier]

Abstract: Based upon Cartan''s geometric formulation of differential equations, Harrison and Estabrook proposed a geometric approach for the symmetries of differential equations. In this Letter, we extend Harrison and Estabrook''s approach to analyze the symmetries of differential-difference equations. The discrete exterior differential technique is applied in our approach. The Lie symmetry of -dimensional Toda equation is investigated by means of our approach. [Copyright &y& Elsevier]

Abstract: Let (z, w) ∈ ℂ × ℂ (zw) be a positive definite kernel and B a subset of ℂ. In this paper, we seek conditions in order that the restriction (z, w) ∈ B × B(zw) be strictly positive definite. Since this problem has been solved recently in the cases in which B is either ℂ or the unit circle in ℂ, our purpose here is twofold: to present some results we obtained when attempting to solve the problem for the above and other choices of B and to acquaint the audience with some other questions that remain. For two different classes of subsets, we completely characterize the strict positive definiteness of the kernel. We include a complete discussion of the case in which B is the unit circle of ℂ, making a comparison with the classical problem of strict positive definiteness on the real circle. [Copyright &y& Elsevier]