Arithmetic sums of the form

where f is an arithmetic function and [ ] is the greatest integer function frequently occur in various situations in the theory of numbers and have much of interest in their own right. Two instances appear in the well-known results.

A Hausdorff-Young theorem is given for Lp-valued analytic functions on the open unit disc and estimates on such functions and their derivatives are deduced.

Un groupe abélien est dit super-décomposable si il ne possède pas de facteur direct indécomposable non nul. Les groupes super-décomposables doivent être nécessairement sans-torsion et un premier exemple en fut donné par A. L. S. Corner [1]. Un exemple différent apparait dans P. A. Krylov [4]. De plus P. Jambor et J. Bečvar, ont développé, dans [3] une généralisation de la notion de sous-groupe de base dans laquelle apparait de façon naturelle des groupes super-décomposables. Dans cette note, nous donnons d'abord un critère pour qu'un groupe soit super-décomposable et nous construisons ensuite une famille de groupes qui satisfont à ce critère, donnant ainsi de nouveaux exemples de groupe super-décomposable.

The duality between compact 0-dimensional semilattices and discrete semilattices studied by K. H. Hofmann et al. [2] is here extended to larger categories of topological semilattices.

We regard topological semilattices as objects in the category CvSl of convergence semilattices, believing CvSl to be the appropriate setting for this study. The referee objected to this at first, which prompted us to set forth more carefully our reasons for doing so, as follows.

Kakutani's Theorem states that every point convex and use multifunction ϕ defined on a compact and convex set in a Euclidean space has at least one fixed point. Some necessary conditions are given here which ϕ must satisfy if c is the unique fixed point of ϕ. It is e.g. shown that if the width of ϕ(c) is greater than zero, then ϕ cannot be lsc at c, and if in addition c lies on the boundary of ϕ(c), then there exists a sequence {xk} which converges to c and for which the width of the sets ϕ(xk) converges to zero. If the width of ϕ(c) is zero, then the width of ϕ(xk) converges to zero whenever the sequence {xk} converges to c, but in this case ϕ can be lsc at c.

Let S and T be locally compact topological semigroups and a semidirect product. Conditions are determined under which topological left amenability of S and T implies that of , and conversely. The results are used to show that for a large class of semigroups which are neither compact nor groups, various notions of topological left amenability coincide.

Le but de la présente note est de donner une preuve différente et beaucoup moins laborieuse que celle que l'on trouve dans [1] pour obtenir la représentation de étant des entiers positifs et f étant une fonction multiplicative.

A set of sentences T is called independent if for every . It is countably independent if every countable subset is independent. In flnitary first order logic, Lωω, the two notions coincide because of compactness. This is not the case for infinitary logic.

If is a von Neumann algebra that thas no nonzero finite discrete central projection, then there is no nontrivial compact derivation of into itself.

D. F. Dawson has considered several questions of the following nature. Suppose T is a regular matrix summability method. If A is a regular matrix and x is a sequence having a finite limit point, then there exists a subsequence y of x such that each finite limit point of x is a T-limit point of Ay. In the present paper, we show the regularity condition for A may be replaced by the requirement that A be a limit preserving bv to c map. This leads to summability characterizations for several classes of sequences.

Let β(n) = ∑p|nP and B(n) = ∑Pα||nαP denote the sum of distinct prime divisors of n and the sum of all prime divisors of n respectively. Both β(n) and B(n) are additive functions which are in a certain sense large (the average order of B(n) is π2n/(6 log n), [1]).

Let U be an n -dimensional vector space over a finite field of q elements. The number of elements of Λ2U of each irreducible length is found using the isomorphism of Λ2U with Hn, the space of n x n skew-symmetric matrices, and results due to Carlitz and MacWilliams on the number of skew-symmetric matrices of any given rank.

Let M be a differentiable manifold which admits the free action of a group Γ with compact quotient M’ = M/Γ. Suppose that the Γ action lifts to a Hermitian vector bundle E→M. If Γ leaves invariant a measure μ on M, then denote by L2(E) the completion of with respect to the inner product .

Let ∑ cj and ∑dj be, respectively, convergent and divergent series of positive terms and let ∑ aj be a third series of positive terms. It is well known, [1, pg. 275] that ∑ aj converges if lim sup(aj/cj)<+∞, but diverges if liminf(aj/dj)>0. In this note we prove a generalized version of this comparison test that relies not on term-by-term comparison of the series, but on the relative densities of the terms of the series.

The aim of this paper is to prove the following theorem:

Let R be a semiperfect ring. Let Q be a left R -module satisfying (a) Q is R-projective and (b) J(Q) is small in Q. Then Q is projective.

Suppose R is a commutative ring with identity. Let M be an R -module, and suppose f is a function from M to R. How do we characterize the property that f be a quadratic form?

Following L. Fuchs [1], we define a valuated vector space to be a vector space V with a valuation from V to a totally ordered set Γ in which every nonempty subset has a supremum. It is assumed that Γ has a maximum element . A standard model for Γ is a closed initial segment of ordinals with the symbol ∞ adjoined.

In the category of T1 -spaces and continuous maps, every space X has a unique maximal essential extension. X has an injective hull iff card X≤1.

Every locally convex space E of dimension at most c contains a hyperplane G with the following property: the linear hull of each bounded Banach disk in G is finite-dimensional.

It is shown in this paper that an immersion of a connected, closed n-manifold into another connected n-manifold is a diffeomorphism if and only if the induced homomorphism between the fundamental groups is surjective at some point. This is proved as a consequence of a more general assertion about topological spaces.