Abstract:
A subset X of a group G is a set of pairwise non-commuting ele- ments if ab 6= ba for any two distinct elements a and b in X. If jXj ? jY j for any other set of pairwise non-commuting elements Y in G, then X is said to be a maximal subset of pairwise non-commuting elements and the cardinality of such a subset is denoted by !(G). In this paper, among other thing, we prove that, for each positive integer n, there are only finitely many groups G, up to isoclinic, with !(G) = n (with exactly n centralizers).

Abstract:
We prove that infinite p-adically discrete sets have Diophantine definitions in large subrings of some number fields. First, if K is a totally real number field or a totally complex degree-2 extension of a totally real number field, then there exists a prime p of K and a set of K-primes S of density arbitrarily close to 1 such that there is an infinite p-adically discrete set that is Diophantine over the ring O_{K,S} of S-integers in K. Second, if K is a number field over which there exists an elliptic curve of rank 1, then there exists a set of K-primes S of density 1 and an infinite Diophantine subset of O_{K,S} that is v-adically discrete for every place v of K. Third, if K is a number field over which there exists an elliptic curve of rank 1, then there exists a set of K-primes S of density 1 such that there exists a Diophantine model of Z over O_{K,S}. This line of research is motivated by a question of Mazur concerning the distribution of rational points on varieties in a non-archimedean topology and questions concerning extensions of Hilbert's Tenth Problem to subrings of number fields.

Abstract:
In this paper, we propose the study of a conjecture whose positive solution would provide an example of a non-convex Chebyshev set in an infinite-dimensional real Hilbert space.

Abstract:
In this paper, we propose the study of a conjecture whose affirmative solution would provide an example of a non-convex Chebyshev set in an infinite-dimensional real Hilbert space.

Abstract:
On any closed Riemannian manifold of dimension greater than $7$, we construct examples of background physical coefficients for which the Einstein-Lichnerowicz equation possesses a non-compact set of positive solutions. This yields in particular the existence of an infinite number of positive solutions in such cases.

Abstract:
The motion of a spinning football brings forth the possible existence of a whole class of finite dynamical systems where there may be non-denumerably infinite number of fixed points. They defy the very traditional meaning of the fixed point that a point on the fixed point in the phase space should remain there forever, for, a fixed point can evolve as well! Under such considerations one can argue that a free-kicked football should be non-chaotic.

Abstract:
The concept of a uniform set is introduced for an ergodic, measure-preserving transformation on a non-atomic, infinite Lebesgue space. The uniform sets exist as much as they generate the underlying $\sigma$-algebra. This leads to the result that any ergodic, measure-preserving transformation on a non-atomic, infinite Lebesgue space is isomorphic to a minimal homeomorphism on a locally compact metric space which admits a unique, up to scaling, invariant Radon measure.

Abstract:
A clone on a set X is a set of finitary functions on X which contains the projections and which is closed under composition. The set of all clones on X forms a complete algebraic lattice Cl(X). We obtain several results on the structure of Cl(X) for infinite X. In the first chapter we prove the combinatorial result that if X is linearly ordered, then the median functions of different arity defined by that order all generate the same clone. The second chapter deals with clones containing the almost unary functions, that is, all functions whose value is determined by one of its variables up to a small set. We show that on X of regular cardinality, the set of such clones is always a countably infinite descending chain. The third chapter generalizes a result due to L. Heindorf from the countable to all uncountable X of regular cardinality, resulting in an explicit list of all clones containing the permutations but not all unary functions of X. Moreover, all maximal submonoids of the full transformation monoid which contain the permutations of X are determined, on all infinite X; this is an extension of a theorem by G. Gavrilov for countable base sets.

Abstract:
An example of an infinite dimensional and separable Banach space is given, that is not isomorphic to a subspace of l1 with no infinite equilateral sets.

Abstract:
This paper is a continuation of the study of topological properties of omega context free languages (omega-CFL). We proved before that the class of omega-CFL exhausts the hierarchy of Borel sets of finite rank, and that there exist some omega-CFL which are analytic but non Borel sets. We prove here that there exist some omega context free languages which are Borel sets of infinite (but not finite) rank, giving additional answer to questions of Lescow and Thomas [Logical Specifications of Infinite Computations, In:"A Decade of Concurrency", Springer LNCS 803 (1994), 583-621].