Fragments of HA based on \Sigma_1-induction

In the first part of this paper we investigate the intuitionistic version of (in the language of ), using Kleene's recursive realizability techniques. Our treatment closely parallels the usual one for and establishes a number of nice properties for , e.g. existence of primitive recursive choice functions (this is established by different means also in [D94]). We then sharpen an unpublished theorem of Visser's to the effect that quantifier alternation alone is much less powerful intuitionistically than classically: together with induction over arbitrary prenex formulas is -conservative over . In the second part of the article we study the relation of to (in the usual arithmetical language). The situation here is markedly different from the classical case in that and are mutually incomparable, while is significantly stronger than as far as provably recursive functions are concerned: All primitive recursive functions can be proved total in whereas the provably recursive functions of are all majorized by polynomials over . 0 is unusual also in that it lacks closure under Markov's Rule . Received April 4, 1996     

The Erdös-Kac theorem for additive arithmetical semigroups

We show that the Erdös-Kac theorem for additive arithmetical semigroups can be proved under the condition that the counting function of elements has the asymptotics G(n) = q ⁿ (A + O(1/(lnn)^k) as n → ∞ with A > 0, q > 1, and arbitrary k ∈ ? and that P(n) = O(q ⁿ /n) for the number of prime elements of degree n. This improves a result of Zhang.     

Hydrodefluorination of pentafluoropyridine at rhodium using dihydrogen: detection of unusual rhodium hydrido complexes

The pentafluoropyridyl complex [Rh(4-C5NF4)(PEt3)3] (3) reacts with H2 to give initially the dihydrido complex cis-mer-[Rh(H)2(4-C5NF4)(PEt3)3] (6). Within a few hours 2,3,5,6-tetrafluoropyridine as well as two rhodium(III) complexes mer-[Rh(H)3(PEt3)3] (mer-) and fac-[Rh(H)3(PEt3)3] (fac-) are formed. A catalytic C-F activation process for the formation of 2,3,5,6-tetrafluoropyridine starting from pentafluoropyridine and dihydrogen using 3 as a catalyst has been developed. Reaction of [RhH(PEt3)3] (1) with hydrogen affords fac-[Rh(H)3(PEt3)3] (fac-7) and mer-[Rh(H)3(PEt3)3] (mer-7) in a ratio of 1 : 7.25 at 193 K. The latter complex represents the first mononuclear rhodium compound bearing trans-hydrides.     

The Turán-Kubilius inequality for additive arithmetical semigroups

We show that the Turán-Kubilius inequality holds for additive arithmetical semigroups satisfying the following conditions: G(n) = q ⁿ (A+O(1/ln n)) (where A > 0 and q > 1) for the number of elements of degree n and P(n) = O(q ⁿ /n) for the number of prime elements of degree n. This is an improvement of a result of Zhang. We also give some variants of the inequality under some stronger or weaker assumptions and applications for the prime divisor function ω and related functions. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 46, No. 3, pp. 457–471, July–September, 2006.