The Dedekind-Mertens lemma and the contents of polynomials

Let be a commutative ring, let be an indeterminate, and let . There has been much recent work concerned with determining the Dedekind-Mertens number =min , especially on determining when = . In this note we introduce a universal Dedekind-Mertens number , which takes into account the fact that deg() + for any ring containing as a subring, and show that behaves more predictably than .

     

Operators with bounded conjugation orbits

For a bounded invertible operator on a complex Banach space let be the set of operators in for which Suppose that and is in A bound is given on in terms of the spectral radius of the commutator. Replacing the condition in by the weaker condition as for every 0$">, an extension of the Deddens-Stampfli-Williams results on the commutant of is given.

     

-semigroups generated by second order differential operators with general Wentzell boundary conditions

Let us consider the operator where is positive and continuous in and is equipped with the so-called generalized Wentzell boundary condition which is of the form at each boundary point, where This class of boundary conditions strictly includes Dirichlet, Neumann and Robin conditions.

Under suitable assumptions on , we prove that generates a positive -semigroup on and, hence, many previous (linear or nonlinear) results are extended substantially.

     

Purely infinite, simple -algebras arising from free product constructions. III

In the reduced free product of C-algebras, with respect to faithful states and , is purely infinite and simple if is a reduced crossed product for an infinite group, if is well behaved with respect to this crossed product decomposition, if and if is not a trace.     

Integral representation of linear functionals on spaces of unbounded functions

Let be a vector lattice of real functions on a set with , and let be a linear positive functional on . Conditions are given which imply the representation , , for some bounded charge . As an application, for any bounded charge on a field , the dual of is shown to be isometrically isomorphic to a suitable space of bounded charges on . In addition, it is proved that, under one more assumption on , is the integral with respect to a -additive bounded charge.

     

Are covering (enveloping) morphisms minimal?

We prove that for certain classes of modules such that direct sums of -covers ( -envelopes) are -covers ( -envelopes), -covering ( -enveloping) homomorphisms are always right (left) minimal. As a particular case we see that over noetherian rings, essential monomorphisms are left minimal. The same type of results are given when direct products of -covers are -covers. Finally we prove that over commutative noetherian rings, any direct product of flat covers of modules of finite length is a flat cover.     

Bivariate version of the Hahn-Sonine theorem

We consider orthogonal polynomials in two variables whose derivatives with respect to are orthogonal. We show that they satisfy a system of partial differential equations of the form where , , is a vector of polynomials in and for , and is an eigenvalue matrix of order for . Also we obtain several characterizations for these polynomials. Finally, we point out that our results are able to cover more examples than Bertran's.

     

Normalizers of the congruence subgroups of the Hecke group II

Let . Let be an ideal of and let be the maximal ideal of such that . Then . In particular, if is square free, then is self-normalized in .

     

Statistical extensions of some classical Tauberian theorems

Hardy's well-known Tauberian theorem for Cesàro means says that if the sequence satisfies and , then . In this paper it is shown that the hypothesis can be replaced by the weaker assumption of the statistical limit: st-lim , i.e., for every , . Similarly, the ``one-sided' Tauberian theorem of Landau and Schmidt's Tauberian theorem for the Abel method are extended by replacing and with st-lim and st-lim , respectively. The Hardy-Littlewood Tauberian theorem for Borel summability is also extended by replacing , where is a continuous parameter, with , and further replacing it by -st-lim , where is the Borel matrix method.

     

Maximum curves and isolated points of entire functions

Given , curves belonging to the set of points were defined by Hardy to be maximum curves. Clunie asked the question as to whether the set could also contain isolated points. This paper shows that maximum curves consist of analytic arcs and determines a necessary condition for such curves to intersect. Given two entire functions and , if the maximum curve of is the real axis, conditions are found so that the real axis is also a maximum curve for the product function . By means of these results an entire function of infinite order is constructed for which the set has an infinite number of isolated points. A polynomial is also constructed with an isolated point.

     

Principal values for the Cauchy integral and rectifiability

We give a geometric characterization of those positive finite measures on with the upper density finite at -almost every , such that the principal value of the Cauchy integral of ,

{\varepsilon}} \frac{1}{\xi-z}\, d\mu(\xi),\end{displaymath}">

exists for -almost all . This characterization is given in terms of the curvature of the measure . In particular, we get that for , -measurable (where is the Hausdorff -dimensional measure) with , if the principal value of the Cauchy integral of exists -almost everywhere in , then is rectifiable.

     

When products of selfadjoints are normal

Suppose that are two selfadjoint bounded operators on a Hilbert space . It is elementary to show that is selfadjoint precisely when . We answer the following question: Under what circumstances must be selfadjoint given that it is normal?     

Existence and uniqueness theorems for singular anisotropic quasilinear elliptic boundary value problems

On bounded domains we consider the anisotropic problems in with 1$"> and on and in with and on . Moreover, we generalize these boundary value problems to space-dimensions 2$">. Under geometric conditions on and monotonicity assumption on we prove existence and uniqueness of positive solutions.     

Convergence of paths for pseudo-contractive mappings in Banach spaces

Let be a real Banach space, let be a closed convex subset of , and let , from into , be a pseudo-contractive mapping (i.e. for all and 1)$">. Suppose the space has a uniformly Gâteaux differentiable norm, such that every closed bounded convex subset of enjoys the Fixed Point Property for nonexpansive self-mappings. Then the path , , defined by the equation is continuous and strongly converges to a fixed point of as , provided that satisfies the weakly inward condition.

     

A classification of prime segments in simple artinian rings

Let be a simple artinian ring. A valuation ring of is a Bézout order of so that is simple artinian, a Goldie prime is a prime ideal of so that is Goldie, and a prime segment of is a pair of neighbouring Goldie primes of A prime segment is archimedean if is equal to it is simple if and it is exceptional if In this last case, is a prime ideal of so that is not Goldie. Using the group of divisorial ideals, these results are applied to classify rank one valuation rings according to the structure of their ideal lattices. The exceptional case splits further into infinitely many cases depending on the minimal so that is not divisorial for

     

A new construction of semi-free actions on Menger manifolds

A new construction of semi-free actions on Menger manifolds is presented. As an application we prove a theorem about simultaneous coexistence of countably many semi-free actions of compact metric zero-dimensional groups with the prescribed fixed-point sets: Let be a compact metric zero-dimensional group, represented as the direct product of subgroups , a -manifold and (resp., ) its pseudo-interior (resp., pseudo-boundary). Then, given closed subsets of , there exists a -action on such that (1) and are invariant subsets of ; and (2) each is the fixed point set of any element .

     

Convex curves, Radon transforms and convolution operators defined by singular measures

Let be a convex curve in the plane and let be the arc-length measure of Let us rotate by an angle and let be the corresponding measure. Let . Then This is optimal for an arbitrary . Depending on the curvature of , this estimate can be improved by introducing mixed-norm estimates of the form where and are conjugate exponents.     

Composition operators on Dirichlet-type spaces

The Dirichlet-type space ) is the Banach space of functions analytic in the unit disc with derivatives belonging to the Bergman space . Let be an analytic self-map of the disc and define for . The operator is bounded (respectively, compact) if and only if a related measure is Carleson (respectively, compact Carleson). If is bounded (or compact) on , then the same behavior holds on ) and on the weighted Dirichlet space . Compactness on implies that is compact on the Hardy spaces and the angular derivative exists nowhere on the unit circle. Conditions are given which, together with the angular derivative condition, imply compactness on the space . Inner functions which induce bounded composition operators on are discussed briefly.

     

Betti numbers of modules of essentially monomial type

Let be a Noetherian local ring. In this paper we supply formulae for computing the ranks of syzygy and Betti numbers of -modules of essentially monomial type. These modules are defined with respect to various -regular sequences. For example, finite length modules of monomial type over regular local rings of dimension are modules of essentially monomial type with respect to -regular sequences of length . If a module is of essentially monomial type with respect to an -regular sequence of length , then the rank of its -th syzygy is at least and its -th Betti number is at least .

     

Embedding obstructions and -complexes

The vanishing of Van Kampen's obstruction is known to be necessary and sufficient for embeddability of a simplicial -complex into for , and it was recently shown to be incomplete for . We use algebraic-topological invariants of four-manifolds with boundary to introduce a sequence of higher embedding obstructions for a class of -complexes in .