In this article we investigate the behaviour of the omega function, which counts the number of prime factors of an integer with multiplicity, as one runs over those integers of the form $a + b$ where $a$ is from a set $A$ and $b$ is from a set $B$. We prove, for example, that if $A$ and $B$ are sufficiently dense subsets of the first $N$ positive integers and $k$ is a positive integer then the number of pairs $(a,b)$ for which the omega function of $a + b$ lies in a given residue class modulo $k$ is roughly the total number of pairs divided by $k$.

We consider the structure of certain intermediate domains betweena local Noetherian domain $R$ and an ideal-adic completion $R^{\ast}$ of $R$ that arise as the intersection of $R^{\ast}$ with a field containing $R$. In the case where the intersection domain $A$ can be expressed as a directed union of localized polynomial extension rings of $R$, the computation of $A$ is easier. We examine conditions for this to happen. We also present examples to motivate and illustrate the concepts considered.

Soit $k$ un corps de nombres. L'objectif de cet article est double. D'une part, on étudie l'ensemble des éléments du groupe des classes de $k$ qui sont réalisables par les classes de Steinitz des extensions galoisiennes de $k$ qui sont modérément ramifiées et dont le groupe de Galois est d'ordre 4 ou quaternionien d'ordre $8$. D'autre part, on décrit l'ensemble des classes de Steinitz des extensions quadratiques (resp. biquadratiques) de $k$ qui sont plongeables dans des extensions cycliques de degré 4 (resp. quaternioniennes de degré 8), sous l'hypothése de la ramification modérée.

We give sharp upper estimates for the difference circumradius minus inradius and for the angle between the radial vector (respect to the center of an inball) and the normal to the boundary of a compact $h$-convex domain in the hyperpolic space. We apply these estimates to get the limit at the infinity for the quotients Volume/Area and (Total $k$-mean curvature)/Area of a family of $h$-convex domains which expand over the whole space. The theorem for the first quotient gives an extension to arbitrary dimension of a result of Santaló and Yañez for the hyperbolic plane.

Conditions are found under which the norm of an operator on a Banach sequence space is determined by its action on decreasing, positive sequences. For the space $d(w,1)$, the norm and “lower bound” of such operators can be equated to the supremum and infimum of a certain sequence. These quantities are evaluated for the averaging, Copson and Hilbert operators, with the weighting sequence given either by $w = 1/n^{\alpha}$ or by the corresponding integral.

We establish a new class of monotonic trigonometric sums. Through a result of Andreev and Duren, our theorem provides information about the coefficients of certain Bloch functions.

We generalize to finitely connected planar domains the result of Joel Shapiro which gives a formula for the essential norm of a composition operator. In the process, we define and give some properties of a generalization of the Nevanlinna counting function and prove generalizations of the Littlewood inequality, the Littlewood-Paley identity, and change of variable formulas, as well.

The set of bounded point evaluations for $P^{t}(\mu)$ is determined for certain measures $\mu$ supported in the closed unit disk in the complex plane. Here $P^{t}(\mu)$ denotes the closure in L $L^{t}(\mu)$ of the polynomials in $z$.

We introduce a new norm (derived from Bombieri's norm for polynomials) on a class of functions on the complex plane. This norm is hilbertian, and can be viewed as a weighted $L_{2}$ norm (or a weighted $l_{2}$ norm). It allows us to give quantitative results of the following sort: If we solve $P(D)u = f$ (with boundary conditions), and if we modify $f$, how is the solution $u$ modified?

Controllable partitions of unity in $C(X)$ are partitions of unity whose supports fulfil a uniformity condition depending on the entropy numbers of the compact metric space $X$. We construct a chain of such partitions in $C([0,2]^{m})$ such that the span of any partition is a proper subspace of the span of the following one. This chain gives rise to approximation quantities for functions from $C([0,2]^{m})$ as well as for $C([0,2]^{m})$-valued operators and to corresponding Jackson type inequalities. Inverse inequalities are presented for Hölder continuous functions and operators.

Let $R$ be a commutative ring with identity. We consider conditions in order that there exists an embedding of $R$ in a local ring. This leads naturally to an examination of conditions in order that a quasilocal ring $(R,\mathbf{m})$ be dominated by a local ring. This, in turn, leads to a study of extensions of the residue field of a quasilocal ring.