One usually defines the notion of a computable real number by using recursive functions. However, there is a simple way due to A. Mostowski to characterize the computable real numbers by using only primitive recursive functions. We prove Mostowski’s result differently and apply it to get other simple characterizations of this kind. For instance, a real number is shown to be computable if and on...

We define the following recursive function T (n): T (0) = 0 T (1) = 0 T (2) = 1 T (n) = T n 2 + T n 2 + 1 + T n 2 − 1 + T n 2 + 1 (1.1) (n ≥ 3).

In this article we look into characterizing primitive groups in the following way. Given a primitive group we single out a subset of its generators such that these generators alone (the so-called primitive generators) imply the group is primitive. The remaining generators ensure transitivity or comply with specific features of the group. We show that, other than the symmetric and alternating gr...

The excedance number for Sn is known to have an Eulerian distribution. Nevertheless, the classical proof uses descents rather than excedances. We present a direct recursive proof which seems to be folklore and extend it to the colored permutation groups Gr,n. The generalized recursion yields some interesting connection to Stirling numbers of the second kind. We also show some logconcavity resul...

We give a precise characterization of parameter free n and n induction schemata I n and I n in terms of re ection principles This allows us to show that I n is conservative over I n w r t boolean combinations of n sentences for n In particular we give a positive answer to a question by R Kaye whether the provably recursive functions of I are exactly the primitive recursive ones We also obtain s...

We establish an asymptotic expansion for a class of partial theta functions generalizing a result found in Ramanujan’s second notebook. Properties of the coefficients in this more general asymptotic expansion are studied, with connections made to combinatorics and a certain Dirichlet series.

We study for s ∈ N the functions ξk(s) = 1 Γ(s) R ∞ 0 t et−1 Lik(1−e )dt, and more generally ξk1,...,kr (s) = 1 Γ(s) R ∞ 0 t et−1 Lik1,...,kr (1 − e )dt, introduced by Arakawa and Kaneko [2] and relate them with (finite) multiple zeta functions, partially answering a question of [2]. In particular, we give an alternative proof of a result of Ohno [8].

The classical propositional logic is known to be sound and complete with respect to the set semantics that interprets connectives as set operations. The paper extends propositional language by a new binary modality that corresponds to partial recursive function type constructor under the above interpretation. The cases of deterministic and non-deterministic functions are considered and for both...

The classical propositional logic is known to be sound and complete with respect to the set semantics that interprets connectives as set operations. The paper extends propositional language by a new binary modality that corresponds to partial recursive function type constructor under the above interpretation. The cases of deterministic and non-deterministic functions are considered and for both...