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In this note two theorems obtained by S. Zahid Ali Zenei in [11] are refined.

On construit dans [R] des representants explicites des fibres principaux de fibre S 3 au-dessus de S 7 ; on en deduit des actions libres et explicites de S 3 , dont les quotients sont des spheres exotiques de dimension 7. La description de ces actions est basee sur une formule de classification des fibres. Or cette formule est incorrecte. Le theoreme 1 corrige cette formule ainsi que la formule de classification ; le theoreme 2 exhibe les indices corrects des spheres exotiques de dimension 7 qui apparaissent comme quotient des actions libres de S 3 .

The Award Committee – Michael Gnewuch, University of Kaiserslautern, Germany and Friedrich Pillichshammer, Johannes Kepler University, Austria – determined that the following two papers exhibited exceptional merit and therefore awarded the prize to: ShuTezuka for his paper ‘‘On the discrepancy of generalizedNiederreiter sequences’’, which appeared in June–August, 2013. Vol. 29, pp. 240–247. Joos Heintz, Bart Kuijpers, and Andres Rojas Paredes for their paper ‘‘Software engineering and complexity in effective Algebraic Geometry’’, which appeared in February, 2013. Vol. 29, pp. 92–138. The $3000 prize will be divided between the winners. Each author will also receive a plaque.

Dans un article anterieur, nous avons etudie les langages m-disjonctifs et s-disjonctifs sur un alphabet X ainsi que les relations d'ordre partiel compatibles qui leur sont associees. Cependant, nous n'avions considere jusqu'a present que des alphabets d'au moins deux lettres. Dans cette note, nous traitons le cas d'un alphabet a une lettre et nous montrons que tout langage disjonctif sur un alphabet a une lettre est s-disjonctif

This is the first English translation of Bolzano's paper, Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetzes Resultat gewahren, wenigstens eine reelle Wurzel der Gleichung liege (Prague 1817). It has already appeared in French, Russian, and Czechoslovakian translations. The paper represents an important stage in the rigorous foundation of analysis and is one of the earliest occasions when the continuity of a function and the convergence of an infinite series are both defined and used correctly.

In this paper we will construct an example showing that the problem posed in [1] has a negative answer. Two more theorems on the subject treated in [1] will be included.Let Io = [0, 1], R the reals, and let, for A ⊂ R, Ao be the interior of A. Let {xn} be a sequence in [0, 1> such that 0 = x1 < x2 < … and lim xn = 1. For each n, let In be closed interval having x as its midpoint (except for n = 1 in which case x1 is the left endpoint of I1) such that In ∩ Im= ϕ, and the metric density relative to Io of at 1 is zero. Let Jn be a closed interval in In concentric with In (except for n = 1, where J1 has x1 as its left endpoint) whose length is half that of In

Suppose X is a simply-connected CW- complex whose homotopy groups are finite. For each n, let be the class of torsion groups whose p - primary components are zero for all primes p which do not figure in the homotopy groups πi(X) for i ≤ n.

(1985). Lars Valerian Ahlfors: Collected Papers, Volume I, 1929–1955, Volume II, 1954–1979. Edited by Gian-Carlo Rota. The American Mathematical Monthly: Vol. 92, No. 5, pp. 366-368.

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