At first we investigate the similarity between the Kleinecke-Shirokov theorem for subnormal operators and the Fuglede-Putnam theorem and also we show an asymptotic version of this similarity. These results generalize results of Ackermans, van Eijndhoven and Martens. Also we show two theorems on degree of approximation on subnormal derivation ranges. These results generalize results of Stampfli on degree of approximation on normal derivation ranges. The purpose of this paper is to show that the Fuglede-Putnam-type theorem on normal operators can certainly be generalized to subnormal operators.

Every ideal of a Noetherian ring may be represented as a finite intersection of primary ideals. Each primary ideal may be decomposed as an irredundant intersection of irreducible ideals. It is shown that in the case that Q is an M-primary ideal of a local ring (R, M) satisfying the condition that Q: M = Q + Ms−1 where s is the index of Q, then all irreducible components of Q have index s. (Q is “index-unmixed”.) This condition is shown to hold in the case that Q is a power of the maximal ideal of a regular local ring, and also in other cases as illustrated by examples.

We show that for reflexive spaces X the density of numerical radius attaining operators in L(X) is equivalent to the density of numerical radius attaining operators in L(X*). As a consequence of this fact and of a result of Berg and Sims, we prove that for uniformly smooth spaces X the numerical radius attaining operators are dense in L(X).

In this article we give a new proof of the theorem that a positive even convex function on the real line, which vanishes at infinity, is the Fourier transform of an integrable function. Related results in several variables are also proved. As an application of our results we solve the factorization problem of Sobolev algebras.

It is well known that every local formation of finite soluble groups possesses three distinguished local definitions consisting of finite soluble groups: the minimal one, the full and integrated one, and the maximal one. As far as the first and the second of these are concerned, this statement remains true in the context of arbitrary finite groups. Doerk, Šemetkov, and Schmid have posed the problem of whether every local formation of finite groups has a distinguished (that is, unique) maximal local definition. In this paper a description of local formations with a unique maximal local definition is given, from which counter-examples emerge. Furthermore, a criterion for a formation function to be a local definition of a given local formation is obtained.

On a set X, let μ* be an outer measure and μ the measure induced by μ*. We show that if X is a finite set, then the measure μ is saturated. We give two examples of non-regular outer measures on an infinite set X which induce non-saturated and saturated measures, respectively. These answer a query posed by Wilansky.

A characterization of the completely reducible (zerosymmetric right) near-rings which are the direct sum of their socle of type 2 and of their 2-radical is given.

Let Ω be a bounded convex domain in R2 with a smooth boundary. Let 0 < γ < 1. Let be a solution, positive in Ω, of

Then the function uα is concave for α = (l–γ)/2.

This paper demonstrates a connection between two measures of discrepancy of sequences which arise in the theory of uniform distribution modulo one. The sequence formed from the non-negative integer multiples of an irrational number ξ is investigated and, by an application of the “Steinhaus Conjecture”, some values of the two discrepancies are obtained using continued fractions.

A result of Howson is that two finitely generated subgroups U and V of a free group have finitely generated intersection. Hanna Neumann showed further that, if m, n and N are the ranks of U, V and U ∩ V respectively, then N ≤ 2(m−1)(n−1) + 1, and Burns strengthened this, showing that N ≤ 2(m−1)(n−1) − m + 2 (if m ≤ n). This paper presents a new and simple proof of Burns' result. Further, the graph-theoretical ideas used provide still stronger bounds in certain special cases.

The paper considers a system of differential equations with impulse perturbations at fixed moments in time of the form

where x ∈ Rn, ε is a small parameter,

Sufficient conditions are found for the existence of the periodic solution of the given system in the critical and non-critical cases.

It would seem from results of Foltz, Lair, and Kelly that symmetric monoidal closed structures, and even monoidal biclosed ones, are quite rare on one-sorted algebraic or essentially-algebraic categories. They showed many such categories to admit no such structures at all, and others to admit only one or two; no such category is known to admit an infinite set of such structures.

Among concrete categories, topological ones are in some sense at the other extreme from essentially-algebraic ones; and one is led to ask whether a topological category may admit many such structures. On the category of topological spaces itself, only one such structure - in fact symmetric - is known; although Greve has shown it to admit a proper class of monoidal closed structures. One of our main results is a proof that none of these structures described by Greve, except the classical one, is biclosed.

Our other main result is that, nevertheless, there exist topological categories (of quasi-topological spaces) which admit a proper class of symmetric monoidal closed structures. Even if we insist (like most authors) that topological categories must be wellpowered, we can still exhibit ones with more such structures than any small cardinal.

This paper provides examples of indecomposable 2-groups with non-abelian automorphism groups in which every automorphism is central.

As is well known, the divergence of the set of constants known as the Lebesgue constants corresponding to a particular method of summability implies the existence of a continuous, periodic function whose Fourier series, summed by the method, diverges at a point, and of another such function the sums of whose Fourier series converge everywhere but not uniformly in the neighborhood of some point.

In 1961, Lorch and Newman established that if L(n; g) is the nth Lebesgue constant for the Hausdorff summability method corresponding to the weight function g(u), then

where

where the summation is taken over the jump discontinuities {εk} of g(u) and M{f(u)} denotes the mean value of the almost periodic function f(u).

In this paper, a partial extension of this result to the two dimensional analogue is obtained. This extension is summarized in Theorem 1.3.

We give a generalization of the notion of spectral maximal sub-spaces, for which some of the main results are still valid. We give an application of this theory on a class of operators, defined on some reflexive Banach space.

In an ordered Banach space which is orthogonally decomposable, we define the absolute value and its general properties are given. The results are used to study the properties of linear operators which satisfy Kato's inequality and the locality condition.

An example is given of a Banach space with normal structure which does not satisfy the geometrical conditions commonly expected to be related to normal structure.

Let KG be the group ring of the group G over the infinite field K, and let U(KG) be its group of units. If G is torsion, we obtain necessary and sufficient conditions for a finite subgroup H of G to be either normal or subnormal in U(KG). Actually, if H is subnormal in U(KG), we can handle not only the case H finite, but the precise assumptions depend on the characteristic of K.

We obtain an asymptotically sharp lower bound for the sum of the first k eigenvalues of the biharmonic operator.

In this note a theorem proved by Abu-Khuzam and Yaqub has been improved as follows: let R be a semi-simple ring such that for all x, y in R there exists a positive integer n = n(x, y) for which either (xy)n + (yx)n or (xy)n – (yx)n is central. Then R is commutative.