In a theorem and its corollary [1] Friedberg gave an enumeration of all the recursively enumerable sets without repetition and an enumeration of all the partial recursive functions without repetition. This note is to prove a similar theorem for the primitive recursive functions. The proof is only a classical one. We shall show that the theorem is intuitionistically unprovable in the sense of Kl...

We point out a generic correlation between the Friedberg-Lee symmetry of quark mass operators and the vanishing of quark masses. We make phenomenological explorations on two textures of quark mass matrices with the broken Friedberg-Lee symmetry. We present a new pattern of quark mass matrices in agreement with current experimental data. Both analytical and numerical results of our calculations ...

We consider hypothesis testing with smooth functional data by performing pointwise tests and applying a multiple comparisons procedure. Methods based on general inequalities (such as Bonferroni’s method) do not perform well because of the high correlation between observations at nearby points. We consider the multiple comparison procedure proposed by Westfall and Young (1993) and show that it a...

We prove various results on effective numberings and Friedberg numberings of families related to algorithmic randomness. The family of all Martin-Löf random left-computably enumerable reals has a Friedberg numbering, as does the family of all Π 1 classes of positive measure. On the other hand, the Π 1 classes contained in the Martin-Löf random reals do not even have an effective numbering, nor ...