The author has previously associated to each commutative ring with unit $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\Bbbk $ and étale groupoid $\mathscr{G}$ with locally compact, Hausdorff, totally disconnected unit space a $\Bbbk $-algebra $\Bbbk \, \mathscr{G}$. The algebra $\Bbbk \, \mathscr{G}$ need not be unital, but it always has local units. The class of groupoid algebras includes group algebras, inverse semigroup algebras and Leavitt path algebras. In this paper we show that the category of unitary$\Bbbk \, \mathscr{G}$-modules is equivalent to the category of sheaves of $\Bbbk $-modules over $\mathscr{G}$. As a consequence, we obtain a new proof of a recent result that Morita equivalent groupoids have Morita equivalent algebras.

Shrinkande and Bhagwan Das (1970) showed how to extend a (4t – 1, 4t) row-orthogonal matrix with entries ± 1 to a Hadamard matrix of order 4t. Using a slightly different approach we consider extensions of (4t – k, 4t) row-orthogonal matrix to a Hadamard matrix of order 4t.

Let M=G/K be a generalized flag manifold, that is, an adjoint orbit of a compact, connected and semisimple Lie group G. We use a variational approach to find non-Kähler homogeneous Einstein metrics for flag manifolds with two isotropy summands. We also determine the nature of these Einstein metrics as critical points of the scalar curvature functional under fixed volume.

We show that a bounded linear operator on a uniformly convex space may be perturbed by a compact operator of arbitrarily small norm to yield an operator which attains its numerical radius.

In this paper we introduce and study the notion of dual continuous (d–continuous) modules. A decomposition theorem for a d–continuous module is proved; this generalizes all known decomposition theorems for quasi-projective modules. Besides we study the structure of d–continuous modules over some special types of rings.

We study the stochastic cubic nonlinear Schrödinger equation (SNLS) with an additive noise on the one-dimensional torus. In particular, we prove local well-posedness of the (renormalized) SNLS when the noise is almost space–time white noise. We also discuss a notion of criticality in this stochastic context, comparing the situation with the stochastic cubic heat equation (also known as the stochastic quantization equation).

We will prove the Theorem of Hartman-Grobman in a very general form. It states the topological equivalence of the flow of a nonlinear non-autonomous differential or difference equation with critical component to the flow of a partially linearized equation. The critical spectrum has not necessearily to be contained in the imaginary axis or the unit circle respectively. Further on we will employ the socalled calculus on measure chains within dynamical systems theory. Within this calculus the usual one dimensional time scales can be replaced by measure chains which are essentially closed subsets of R. The paper can be understood without knowledge of this calculus.

So our main theorem will be valid even for equations defined on very strange time scales such as sequences of closed intervals. This is especially interesting for applications within the theory of differential-difference equations or within numerical analysis of qualitative phenomena of dynamical systems.

It is shown that a unital lattice-ordered ring in which the square of every element is positive is embeddable in a product of totally ordered rings provided it is archimedean, semiperfect, or π-regular. Also, some canonical examples of unital l-domains with squares positive that are not totally ordered are discussed.

A system or family (Aγ: γ∈ N) of sets Aγ, indexed by the elements of a set N, is called an (a, b)-system if ¦N¦ = a and ¦Aγ¦ = b for γ ∈ N. Expressions such as “(a, <b)-system” are self-explanatory. The system (Aγ: γ∈N) is called a δ-system [1] if Aμ∩Aγ = Ap ∩ Aσ whenever μ, γ, ρ, σ ∈ N; μ≠ γ; ρ ≠ σ. If we want to indicate the cardinality ¦N¦ of the index set N then we speak of a δ(¦N¦) system. In [1] conditions on cardinals a, b, c were obtained which imply that every (a, b)-system contains a δ(c)-subsystem. In [2], for every choice of cardinals b, c such that the least cardinal a = fδ(b, c) was determined which has the property that every (a, < b)-system contains a δ(c)-subsystem.

The recent resurgence of interest in the groups introduced in Conway (1967), together with their analogues, is recorded in Johnson, Wamsley and Wright(to appear).

An inverse semigroup S is called E-unitary if the equations ea = e = e2 together imply a2 = a. In a previous paper the author showed that every inverse semigroup is an idempotent separating homomorphic image of an E-unitary inverse semigroup. The main question considered in this paper is the following. Given an inverse semigroup S give a method for constructing E-unitary inverse semigroups P and idempotent separating homomorphisms φ P → S in such a way that the structure of P as a P-semigroup is evident.

We determine which permutative varieties are saturated and classify all nontrivial permutation identities for the class of all globally idempotent semigroups.

For any bounded linear operator A in a Banach space, two generalized condition numbers, k(A) and k(A) are defined in this paper. These condition numbers may be applied to the perturbation analysis for the solution of ill-posed differential equations and bounded linear operator equations in infinite dimensional Banach spaces. Different expressions for the two generalized condition numbers are discussed in this paper and applied to the perturbation analysis of the operator equation.

Let (Хζ,λ) be a σ-finite measure space, and let ϕ be a non-singular measurable transformation from X into itself. Then a composition transformation Cϕ on L2(λ) is defined by Cϕf = f ∘ ϕ. If Cϕ is a bounded operator, then it is called a composition operator. The space L2(λ) is said to admit compact composition operators if there exists a ϕ such that Cϕ is compact. This note is a report on the spaces which admit or which do not admit compact composition operators.

The product of two subsets C, D of a group is defined as . The power Ce is defined inductively by C0 = {1}, Ce = CCe−1 = Ce−1C. It is known that in the alternating group An, n > 4, there is a conjugacy class C such that CC covers An. On the other hand, there is a conjugacy class D such that not only DD≠An, but even De≠An for e<[n/2]. It may be conjectured that as n ← ∞, almost all classes C satisfy C3 = An. In this article, it is shown that as n ← ∞, almost all classes C satisfy C4 = An.

Lef: G → G be a continuous map of a graph and let d(A) denote the derived set (or limit points) of A ⊂ G. We prove that d(Ω(f)) ⊂ λ (f) and the depth of f is at most three. We also prove that if f is piecewise monotone or has zero topological entropy, then the depth of f is at most two. Furthermore, we obtain some results on the topological structure of Ω(f).

In this note we study commutative Baer rings, uniting the abstract algebraic approach with the approach of [3] using minimal prime ideals. Some new characterisations of this class of rings are obtained, relations between the minimal prime ideals of a commutative Baer ring B and its algebra EB of idempotents are considered, and some results concerning the direct decomposition of commutative Baer rings are given. We then study Baer ideals, and finally state without proof a new construction of the Baer extension of a commutative semiprime ring.

We extend results of Andrews and Bressoud [‘Vanishing coefficients in infinite product expansions’, J. Aust. Math. Soc. Ser. A27(2) (1979), 199–202] on the vanishing of coefficients in the series expansions of certain infinite products. These results have the form that if

$$\begin{eqnarray}\frac{(q^{r-tk},q^{mk-(r-tk)};q^{mk})_{\infty }}{(\pm q^{r},\pm q^{mk-r};q^{mk})_{\infty }}=:\mathop{\sum }_{n=0}^{\infty }c_{n}q^{n}\end{eqnarray}$$

$k$

$m$

$s$

$t$

$r=sm+t$

$c_{kn-rs}$

Let 0 = λ0 < λ1 ≤ λ2 ≤ λ3 ≤ … denote the sequence of eigenvalues of the Laplacian of a compact minimal submanifold in a unit sphere. Yang and Yau obtained an upper bound on λn+1 in terms of λn and the sum λ1 + … + λn. In this note we shall prove an improved version of this upper bound by using the method of Hile and Protter.

We prove that a finitely generated semigroup whose word problem is a one-counter language has a linear growth function. This provides us with a very strong restriction on the structure of such a semigroup, which, in particular, yields an elementary proof of a result of Herbst, that a group with a one-counter word problem is virtually cyclic. We prove also that the word problem of a group is an intersection of finitely many one-counter languages if and only if the group is virtually abelian.