We consider the problem of language identification in the limit from positive data enumerated by functions restricted to the class of primitive recursive functions. Conditions are described under which a family of languages is identifiable. They depend on the behavior of inference machines when they are presented with enumeration of elements of unknown languages. How these conditions compare to cases when positive data are enumerated by recursive functions is also considered.

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An fg-coloring of a multigraph is a coloring of edges such that each color appears at each vertex v at most f(v) times and at each set of multiple edges joining vertices v and w at most g(vw) times. The minimum number of colors needed to fg-color a multigraph is called the fg-chromatic index of the multigraph. This paper proves a new upper bound on the fg-chromatic index. The proof immediately yields a polynomial time algorithm to fg-color a given multigraph using a number of colors not exceeding the upper bound. The worst-case ration of the algorithm is at most 3/2.

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The error of the Markoff's formula, which is a differentiation of the Newton forward- or backward-difference formula, is analysed. We firs derived the integral remainder for the Newton interpolation formula. This follows the Markoff's formula with integral remainder. The kernel functions (Peano kernels) of the integral remainder for the Markoff's formula, which are represented in terms of the truncated power functions, can be rewritten in terms of the normalized B-splines. We also derived the integeral remainders of modified and generalized Markoff's formulas. This principle can be applied to the Gregory's formula.

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This paper describes the techniques to vectorize four iterative methods at no expense of their inherent rate of convergence for solving systems of linear equations in fluid flow problems. The solvers included here are 1)Line Jacobi Method, 2)Line Gauss-Seidel Overrelaxation Method of Van Doormaal-Raithby, 3)Strongly Implicit Procedure of Stone, and 4)Modified Strongly Implicit Procedure of Schneider-Zedan. The enhanced performance of each solver is demonstrated on VP-400EX through the numerical simulation of two different flow fields.

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