Nous obtenons différentes identités de l'analyse combinatoire contenant des coefficients binomiaux de Gauss.

A general procedure is introduced for identifying categories of Cauchy spaces which have completions which possess a certain convergence property P, and for constructing completion functors on such categories. These results are applied to obtain a characterization of Cauchy spaces which allow T3 completions, and to construct T3 completions for categories of Cauchy groups and Cauchy lattices.

It is shown that for a finite group G to be isomorphic to a subgroup of SO(3) (or, equivalently, of PSL(2, C)) it is necessary and sufficient that G satisfies the property that the normalizer of every cyclic subgroup is either cyclic or dihedral.

The aim of this paper is two-fold: First we prove the Radotype inequality Here denote the weighted geometric means of

with where the pi are positive weights. Thereafter we investigate under which conditions the sequence

is convergent as n → ∞

We introduce a new constant in Banach spaces which implies, in certain cases, the weak- or weak*-normal structure.

Let R and S be rings, G and H abelian groups, and RG and SH the goup rings of G and H over R and S respectively. In this note we consider what relations must hold between G and H or between R and S if the group rings RG and SH are isomorphic. For example, it is shown that if R and S are integral domains of characteristic zero, G and H torsion abelian groups such that if G has an element of order p then p is not invertible in R, and RG and SH are isomorphic, then the rings R and 5 are isomorphic and the groups G and H are isomorphic.

Uncountable collections of continua of dimension m embeddable in En are investigated, where the difference between m and n is not restricted to one. Collections of isometric copies of continua equivalent to Menger universal continua and collections of continua analogous to G. S. Young's Tn-sets are the main considerations.

The Davison convolution of arithmetical functions f and g is defined by where K is a complex-valued function on the set of all ordered pairs (n, d) such that n is a positive integer and d is a positive divisor of n. In this paper we shall consider the arithmetical equations f(r) = g, f(r) = fg, f o g = h in f and the congruence (f o g)(n) = 0 (mod n), where f(r) is the iterate of f with respect to the Davison convolution.

Let P be an irreducible Markov matrix. Assume:

We study conditions for such a matrix to be nonrecurrent. If P is nonrecurrent we study the invariant vectors of P (invariant column vectors and invariant row vectors).

In this note, we develop the notion of a Zp-tower in a Demuskin group, and apply the results of Koch and Wingberg on the uniqueness of so-called Demuškin formations to give a classification of such towers in the case p ≠ 2.

We will be looking at quadratic polynomials having positive discriminant and having a long string of primes as initial values. We find conditions tantamount to this phenomenon involving another long string of primes for which the discriminant of the polynomial is a quadratic non-residue. Using the generalized Riemann hypothesis (GRH) we are able to determine all discriminants satisfying this connection.

Let U be a nonempty subset of a Banach space, S a left reversible semitopological semigroup, a continuous representation of S as lipschitzian mappings on U into itself, that is for each s ∊ S, there exists ks > 0 such that for x, y ∊ U. We first show that if there exists a closed subset C of U such that then S with lim sups has a common fixed point in a Hilbert space. Next, we prove that the theorem is valid in a Banach space E if lim sups

It is shown that under certain conditions the asymptotic relationship

between two Dirichlet series implies the same relationship with λn replaced by log λn.

In this paper we establish a relationship between the rational solutions (x(t), y(t)), over C(t), of the diophantine equation:

and the solutions which parametrize the elliptic curve E, y2 = 4x3 — g2x — g3 admitting complex multiplication by λ. We first characterize the form of all rational solutions of diophantine equation (1). The rational solutions are derivable from the subsititutions

in which μ = 0,ω1,ω2,ω1 + ω2 = ω3. Using techniques established in elliptic function theory, we prove that the complex multiplier λ, associated with a unique rational solution (x(t), y(t)), must be of a certain form. Next we construct all rational solutions of diophantine equation (1) by using the addition theorems valid for the Weierstrass function, Specific examples are finally worked out for the cases

A. Grothendieck has shown that if the space C(X) is a Banach dual then X is hyperstonean; moreover, the predual of C(X) is strongly unique. In this article we give a vector analogue of Grothendieck's result. We show that if E* is a reflexive Banach space and C(X, (E*, σ*)) denotes the space of continuous functions on X to E* when E* is provided with its weak* (= weak) topology then the full content of Grothendieck's theorem for C(X) can be established for C(X,(E*,σ*)). This improves a result previously obtained for the case in which E* is Hilbert space.

We construct a positive linear contraction T of all LP(X, μ)- spaces, 1 ≦ p ≦ ∞, μ(X) = 1 such that T1 = 1, T* 1 = 1 and also Tf > 0 a.e. for all f ≧ 0 a.e., f ≢ 0 but for which there is an f ∊ L∞ such that (Tnf — ∫ fdμ) does not converge in L1-norm. We also show that if T is a contraction of a Hilbert space H, there exists an isometry Q and a contraction R such that ∥Tnx - QnRx∥ —> 0 as n —» ∞ for all x in H

When a pair of adjacent C1 strata (X, Y) is Whitney (a)- regular at a point , there is, up to homeomorphism, just one germ at 0 of intersections S ∩ X where S is a Lipschitz transversal to Y at 0 (S is the graph of a Lipschitz map from The proof uses a blowing-up construction and a strong form of Verdier's (w)-regularity.

An R-module is a simple divisible module if it is a nonzero divisible module that has no proper non-zero divisible submodules. We study simple divisible modules and their endomorphism rings, give some examples and determine all simple divisible modules over some classes of rings.

Let G be a finite group. By a Moore G-space we mean a G-space X such that for each subgroup H of G the fixed point space XH is a simply connected Moore space of type (MH,n), where MH is an abelian group depending on H, and n is a fixed integer. By a co-Hopf G-space we mean a G-space with a G-equivariant comultiplication. In this note it is shown that, in contrast to the non-equivariant case, there exist Moore G-spaces which are not co-Hopf G-spaces.

Let d*(n) denote the number of unitary divisors of the positive integer n. For x > 1, let B(x) denote the number of integers n for which nd*(n) ≦ x. Balasubramanian and Ramachandra proved that there exists a positive constant β such that . In this note we give an explicit expression for β as an infinite product, namely where the product is over all primes p.