A semiderivation of a ring R is an additive mapping f:R → R together with a function g:R → R such that f(xy) = f(x)g(y) + xf(y) = f(x)y + g(x)f(y) and f(g(x) ) = g(f(x)) for all x, y ∊ R. Motivating examples are derivations and mappings of the form x → x — g(x), g a ring endomorphism. A semiderivation f of R is centralizing on an ideal U if [f(u), u] is central for all u ∊ U. For R prime of char. ≠2, U a nonzero ideal of R, and 0 ≠ f a semiderivation of R we prove: (1) if f is centralizing on U then either R is commutative or f is essentially one of the motivating examples, (2) if [f(U), f(U) ] is central then R is commutative.

This note answers affirmatively a question of the author [4, p. 456], by producing an example of an integrally closed quasi-local non valuation domain of global dimension 3, each of whose overrings is a goingdown ring. Although [4, Proposition 4.5] shows that such an example cannot be constructed by means of restrained power series, an approach via the more general D+M construction succeeds.

In this paper, we define the η-invariant for a cusped hyperbolic 3-manifold and discuss some of its applications. Such an invariant detects the chirality of a hyperbolic knot or link and can be used to distinguish many links with homeomorphic complements.

The paper gives a good approximation of a Ck function on Rn and its derivatives by the restriction of an entire function on Cn and its derivatives respectively.

We give weighted norm inequalities for the maximal fractional operator ${{\mathcal{M}}_{q}},\beta $ of Hardy–Littlewood and the fractional integral ${{I}_{\gamma }}$. These inequalities are established between ${{\left( {{L}^{q}},\,{{L}^{p}} \right)}^{\alpha }}\left( X,\,d,\,\mu \right)$ spaces (which are superspaces of Lebesgue spaces ${{L}^{\alpha }}\left( X,\,d,\,\mu \right)$ and subspaces of amalgams $\left( {{L}^{q}},\,{{L}^{p}} \right)\left( X,d,\mu \right)$) and in the setting of space of homogeneous type $\left( X,d,\mu \right)$. The conditions on the weights are stated in terms of Orlicz norm.

Let l and k be positive integers. Then for each integer n ≥ 1, define d(n; l, k) to be the number of (positive) divisors of n which lie in the arithmetic progression I mod k. Note that d(n;1,1) = d(n), the ordinary divisor function.

Hinčin proved that any limit law, associated with a triangular array of infinitesimal random variables, is infinitely divisible. The analogous result for additive free convolution was proved earlier by Bercovici and Pata. In this paper we will prove corresponding results for the multiplicative free convolution of measures defined on the unit circle and on the positive half-line.

As a consequence of results due to Bourgain and Stegall, on a separable Banach space whose unit ball is not dentable, the set of norm attaining functionals has empty interior (in the norm topology). First we show that any Banach space can be renormed to fail this property. Then, our main positive result can be stated as follows: if a separable Banach space $X$ is very smooth or its bidual satisfies the ${{\mathcal{w}}^{*}}$-Mazur intersection property, then either $X$ is reflexive or the set of norm attaining functionals has empty interior, hence the same result holds if $X$ has the Mazur intersection property and so, if the norm of $X$ is Fréchet differentiable. However, we prove that smoothness is not a sufficient condition for the same conclusion.

We prove that given an operator space structure on a dual Banach space Y*, it is not necessarily the dual one of some operator space structure on Y. This allows us to show that Sakai's theorem providing the identification between C*-algebras having a predual and von Neumann algebras does not extend to the category of operator spaces. We also include a related result about completely bounded operators from B(ℓ2)* into the operator Hilbert space OH.

Dans ce travail, on étudie les propriétés de la conorme, c(a) d'un élément a d'une C*-algèbre A. Nous montrons que c(a) est aussi douée de la condition de B*- algèbre, c'est-à-dire qu'elle vérifie l'égalité: c(a)2 = c(a*a) = c(aa*) = c(a*)2. Nous remarquons que la conorme est invariante par restriction à toute sous-C*-algèbre de A, propriété connue pour le spectre.

Soit {an} une suite d'éléments réguliers telle que an → a régulier. Nous donnons des conditions nécessaires et suffisantes pour que où a† désigne l'inverse de Moore-Penrose de a.

In this paper diagonals of various orders in a (strict) convex polygon Pn are considered. The sums of lengths of diagonals of the same order are studied. A relationship between the number of consecutive diagonals which do not intersect a given maximal diagonal and lie on one side of it and the order of the smallest diagonal among them is established. Finally a new proof of a conjecture of P. Erdos, considered already in [1], is given.

J. Marica and J. Schönhein [4], using a theorem of M. Hall, Jr. [3], see below, proved that if any n − 1 arbitrarily chosen elements of the diagonal of an n × n array are prescribed, it is possible to complete the array to form an n × n latin square. This result answers affirmatively a special case of a conjecture of T. Evans [2], to the effect that an n × n incomplete latin square with at most n − 1 places occupied can be completed to an n × n latin square. When the complete diagonal is prescribed, it is easy to see that a counterexample is provided by the case that one letter appears n − 1 times on the diagonal and a second letter appears once. In the present paper, we prove that except in this case the completion to a full latin square is always possible. Completion to a symmetric latin square is also discussed.

Up to a duality, the known finite projective planes which are not translation planes all are equivalent to affine planes which contain the type of structure defined below to be a "derivable net". (Insofar as the known finite planes are concerned, this means that the intimate connection between projective geometry and linear algebra still holds for non-Desarguesian planes.)

Let $\{M_{n}\}_{n=1}^{\infty }$ be a sequence of expanding matrices with $M_{n}=\operatorname{diag}(p_{n},q_{n})$, and let $\{{\mathcal{D}}_{n}\}_{n=1}^{\infty }$ be a sequence of digit sets with ${\mathcal{D}}_{n}=\{(0,0)^{t},(a_{n},0)^{t},(0,b_{n})^{t},\pm (a_{n},b_{n})^{t}\}$, where $p_{n}$, $q_{n}$, $a_{n}$ and $b_{n}$ are positive integers for all $n\geqslant 1$. If $\sup _{n\geqslant 1}\{\frac{a_{n}}{p_{n}},\frac{b_{n}}{q_{n}}\}<\infty$, then the infinite convolution $\unicode[STIX]{x1D707}_{\{M_{n}\},\{{\mathcal{D}}_{n}\}}=\unicode[STIX]{x1D6FF}_{M_{1}^{-1}{\mathcal{D}}_{1}}\ast \unicode[STIX]{x1D6FF}_{(M_{1}M_{2})^{-1}{\mathcal{D}}_{2}}\ast \cdots \,$ is a Borel probability measure (Cantor–Dust–Moran measure). In this paper, we investigate whenever there exists a discrete set $\unicode[STIX]{x1D6EC}$ such that $\{e^{2\unicode[STIX]{x1D70B}i\langle \unicode[STIX]{x1D706},x\rangle }:\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}\}$ is an orthonormal basis for $L^{2}(\unicode[STIX]{x1D707}_{\{M_{n}\},\{{\mathcal{D}}_{n}\}})$.

This paper proves that if р < s, then 0 is the only function that multiplies a Bergman Lр space into a Bergman Ls space.

The origin and importance of multiparameter Sturm-Liouville problems in mathematical physics has recently been discussed by Atkinson [1, sects. 3 and 4], [2, introduction]. In spite of the importance of such problems, and in spite of the work done in this field by such early investigators as Klein and Hilbert, Atkinson points out that in recent years this field has been relatively neglected in contrast to the single parameter case. As an example, he states that as opposed to the one parameter case, the detailed behaviour of the eigenvalues and eigenfunctions in the multiparameter Sturm-Liouville case is still far from clear.

Let (X, d) be a metric space and Ti(i=l, 2) be self mappings of X. The purpose of this paper is to investigate the fixed and common fixed points of Ti, when the pair Ti(i=l, 2) satisfies a condition of the following type:

(1)

In this note we show the existence of a spread which is not a dual spread, thus answering a question in [1]. We also obtain some related results on spreads and partial spreads.

Let ∑ = PG(2t-l, F) be a projective space of odd dimension (2t-l, ≥2) over the field F. In accordance with [1] we make the following definitions. A partial spread S of ∑ is a collection of (t-l)-dimensional projective subspaces of ∑ which are pairwise disjoint (skew).

We study the asymptotic properties of positive solutions to the semilinear equation — Δu = f(x, u). Existence and asymptotic estimates are obtained for solutions in exterior domains, as well as entire solutions, for n ≧ 2. The study uses integral operator equations in Rn, and convergence theorems for solutions of Poisson's equation in bounded domains. A consequence of the method is that more precise estimates can be obtained for the growth of solutions at infinity, than have been obtained by other methods. As a special case the results are applied to the generalized Emden-Fowler equation — Δu = p(x)uγ, for γ > 0

Let t be a field and a finite dimensional t-algebra. Auslander-Reiten sequences [AR] play a fundamental rôle in the representation theory of ; in particular, they can be used to construct new indecomposable modules from known ones. For the latter reason I think it worthwile to point out certain torsion theories on the category of -modules, such that the category of -torsionfree modules has Auslander-Reiten sequences; thus giving another construction of indecomposable modules.