We prove the compactness of certain Hankel operators on weighted Bergman spaces of harmonic functions on the unit ball in Rn.

This note is devoted to giving a conceptually simple proof of the Invertible Ideal Theorem [1, Theorem 4·6], namely that a prime ideal of a right Noetherian ring R minimal over an invertible ideal has rank at most one. In the commutative case this result may be easily deduced from the Principal Ideal Theorem by localizing and observing that an invertible ideal of a local ring is principal. Our proof is partially analogous in that it utilizes the Rees ring (denned below) in order to reduce the theorem to the case of a prime ideal minimal over an ideal generated by a single central element, which can be easily dealt with by adapting the commutative argument in [8]. The reader is also referred to the papers of Jategaonkar on the subject [5, 6, 7], particularly the last where another proof of the theorem appears which yields some additional information.

1. The problem of determining the state of stress in the vicinity of a penny-shaped crack which is opened by thermal means has been considered by Olesiak and Sneddon [1]. In that paper no simple closed expressions were given either for the stress-intensity factor at the tip of the crack or for the normal component of the surface displacement. The purpose of this note is to show how such expressions may be derived.

This paper is devoted to determine the connectedness of the branch loci of the moduli space of non-orientable unbordered Klein surfaces. We obtain a result similar to Nielsen's in order to determine topological conjugacy of automorphisms of prime order on such surfaces. Using this result we prove that the branch locus is connected for surfaces of topological genus 4 and 5.

In this paper, we study the bounded approximation property for the weighted space $\mathcal{HV}$(U) of holomorphic mappings defined on a balanced open subset U of a Banach space E and its predual $\mathcal{GV}$(U), where $\mathcal{V}$ is a countable family of weights. After obtaining an $\mathcal{S}$-absolute decomposition for the space $\mathcal{GV}$(U), we show that E has the bounded approximation property if and only if $\mathcal{GV}$(U) has. In case $\mathcal{V}$ consists of a single weight v, an analogous characterization for the metric approximation property for a Banach space E has been obtained in terms of the metric approximation property for the space $\mathcal{G}_v$(U).

Let $X$ be a Banach space and let $B(X)$ denote the space of bounded operators on $X$. Two elements $S,T\inB(X)$ are isometrically equivalent if there exists an invertible isometry $V$ such that $TV=VS$. If $X$ is a Hilbert space, then $V$ is a unitary operator and $S$ and $T$ are said to be unitarily equivalent.

In its original formulation Lang's theorem referred to a semilinear map on an n-dimensional vector space over the algebraic closure of GF(p): it fixes the vectors of a copy ofV(n, p^h) . In other words, every semilinear map defined over a finite field is equivalent by change of coordinates to a map induced by a field automorphism. We provide an elementary proof of the theorem independent of the theory of algebraic groups and, as a by-product of our investigation, obtain a convenient normal form for semilinear maps. We apply our theorem to classical groups and to projective geometry. In the latter application we uncover three simple yet surprising results.

We first study the regularised version of a modified two-component Camassa–Holm shallow water system and obtain the energy estimates of the corresponding approximate solutions. Then, we present a sufficient condition which guarantees that these approximate solutions converge to a low regularity weak solution of the modified two-component Camassa–Holm shallow water system.

A conjecture of reputable vintage states that c(G)≤b(G) + l for a finite p-group G of class c(G) and breadth b(G). This result has been proved in a medley of special cases and in particular whenever b(G)≤3. We now prove it for b(G) = 4.

The axiom of comonotonic independence for a preference ordering was introduced by Schmeidler [9]. It leads to the comonotonic additivity for the functional representing the preference ordering, which is necessarily a Choquet integral.

The aim of this paper is to illuminate the concepts of comonotonicity, comonotonic independence and comonotonic additivity. For example the seemingly weaker condition of weak comonotonic independence used by Chateauneuf in [2] is seen to be equivalent to comonotonic independence. Comonotonic additivity is characterized as additivity on chains of sets. From this the characterization of Choquet integrals in [4], [1], [8] follows easily.

From the classical differential equation of Jacobi fields, one naturally defines the Jacobi operator of a Riemannian manifold with respect to any tangent vector. A straightforward computation shows that any real, complex and quaternionic space forms satisfy that any two Jacobi operators commute. In this way, we classify the real hypersurfaces in quaternionic projective spaces all of whose tangent Jacobi operators commute.

Skew-gentle algebras are a generalisation of the well-known class of gentle algebras with which they share many common properties. In this work, using non-commutative Gröbner basis theory, we show that these algebras are strong Koszul and that the Koszul dual is again skew-gentle. We give a geometric model of their bounded derived categories in terms of polygonal dissections of surfaces with orbifold points, establishing a correspondence between curves in the orbifold and indecomposable objects. Moreover, we show that the orbifold dissections encode homological properties of skew-gentle algebras such as their singularity categories, their Gorenstein dimensions and derived invariants such as the determinant of their q-Cartan matrices.

Let μ ≠ 0 be an ultradistribution of Beurling type with compact support in the space . We investigate the range of the convolution operator Tμ on the space of non-quasianalytic functions of Beurling type associated with a weight w, in the case the operator is not surjective. It is proved that the range of TM always contains the space of real-analytic functions, and that it contains a smaller space of Beurling type for a weight σ ≥ ω if and only if the convolution operator is surjective on the smaller class.

C. Widland [14] has defined Weierstrass points on integral, projective Gorenstein curves. We show here that the Weierstrass points on a generic integral rational nodal curve have the minimal possible weights or, equivalently, that such a curve has the maximum possible number of distinct nonsingular Weierstrass points. Rational curves with g nodes arise in degeneration arguments involving smooth curves of genus g and they have also recently arisen in connection with g-soliton solutions to certain nonlinear partial differential equations [11], [13].

An oft-cited result of Peter Shiu bounds the mean value of a nonnegative multiplicative function over a coprime arithmetic progression. We prove a variant where the arithmetic progression is replaced by a sifted set. As an application, we show that the normalized square roots of −1 (mod m) are equidistributed (mod 1) as m runs through the shifted primes q − 1.

Let M(c) denote a 4n-dimensional quaternion space form of quaternion sectional curvature c, and let P(H) denote the 4n-dimensional quaternion projective space of constant quaternion sectional curvature 4. Let N be an n-dimensional Riemannian manifold isometrically immersed in M(c). We call N a totally real submanifold of M(c) if each tangent 2-plane of N is mapped into a totally real plane in M (c). B. Y. Chen and C. S. Houh proved in [1].

Let A be a commutative algebra, and let M be a bimodule over A. A derivation from A into M is a linear mapping D: A→M that satisfies

If M is only a left A-module, by a derivation from A into M we mean a linear mapping D: A→M such that

Each A-bimodule M is trivially a left module. However, unless it is commutative, i.e.

the two classes of linear operators from A into M characterized by (1) and (2), respectively, need not coincide.

The structure of an inverse monoid can be determined by the complete set of Schützenberger graphs of a presentation. Necessary and sufficient conditions for a collection of inverse X-graphs to be the complete set of Schützenberger graphs of some inverse monoid presentation are established and decidability results are obtained. Conditions for a single inverse X-graph to be a Schu¨tzenberger graph for some presentation are also obtained, and both problems are restricted to the case of Clifford monoids and E-unitary inverse monoids. Decidability and undecidability results are obtained for the case of finite graphs. It is also proved that the problem of embedding a finite inverse X-graph in the Cayley graph of a group is undecidable.

We prove that if X is a real Banach space, with dim X\ge 3, which contains subspace of codimension 1 which is 1-complemented in X and whose group of isometries is almost transitive then X is isometric to a Hilbert space. This partially answers the Banach-Mazur rotation problem and generalizes some recent related results.

Prime rings came into prominence when Posner characterized prime rings satisfying a polynomial identity [9]. The scarcity of invertible central elements made it difficult to generalize results from central simple and primitive algebras to prime rings. For example, we do not automatically have tensor products at our disposal. In [5], the first author introduced the Martindale ring of quotients Q(R) of a prime ring R in his theorem characterizing prime rings satisfying a generalized polynomial identity (GPI). Q(R) is a prime ring containing R whose center C is a field called the extended centroid of R. The central closure of R is the subring RC of Q(R) generated by R and C. RC is a closed prime ring since its extended centroid equals its center C. Hence we have a useful procedure for proving results about an arbitrary prime ring R. We first answer the question for closed prime rings and then apply to R the information obtained from RC. It should be noted that simple rings and free algebras of rank at least 2 are closed prime rings. For these reasons, closed prime rings are natural objects to study.